Fractional Integrals and Derivatives




FRACTIONAL INTEGRALS
AND DERIVATIVES
Theory and Applications
Stefan G. Samko
Rostov State University, Russia
Anatoly A. Kilbas
Belorussian State University, Minsk, Belarus
Oleg I. Marichev
Belorussian State University, Minsk, Belarus
Gordon and Breach Science Publishers
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Copyright @ 1993 by OPA (Amsterdam) B.V. All rights reserved. Published
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Originally published in Russian by Nauka i Tekhnika, Minsk in 1987 as
HHTerpanY H UpOH3BoHe6Horo uopRKa H HeKoTope
HX upanozeHHR
@ 1987 Nauka i Tekhnika, Minsk
Library of Congress Cataloging-in-Publication Data
Samko, S.G. (Stefan Grigor'evich)
[lntegraly i proizvodnye drobnogo poriadka i nekotorye ikh
prilozheniia. English]
Fractional integrals and derivatives: theory and applications /
Stefan G. Samko, Anatoly A. Kilbas. Oleg I. Marichev.
p. em.
Includes bibliographical references and index.
ISBN 2-8-8124-864-0
1. Integral equations. 2. Fractional integrals I. Kilbas, A.
A. (Anatolii Aleksandrovich) II. Marichev, 0.1. (Oleg Igorevich)
Ill. Title.
QA431.S2413 1993 93-26071
515'.43 -- dc20 CIP
No part of this book may be reproduced or utilized in any fonn or by any
means, electronic or mechanical, including photocopying and recording,
or by any infonnation storage or retrieval system, without pennission in
writing from the publisher. Printed in Singapore.

CONTENTS
Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Preface to the English edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xxiii
Notation of the main forms of fractional integrals and derivatives xxv
Brief historical outline................................................. xxvii
Chapter 1 - Fractional Integrals and Derivatives on an
Interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
ti 1. Preliminaries............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. The spaces H). and H).(p).. . .. .. . .. .. .. . .. .. .. . .. .. .. .. .. .. 1
1.2. The spaces Lp and Lp(p) .. .. .. . .. .  . .. .. .. .. .. . .. .. .. .. .. .. 7
1.3. Some special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4. Integral transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . 23
ti2. Riemann-Liouville Fractional Integrals and Derivatives. . . . 28
2.1. The Abel integral equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2. On the solvability of the Abel equation in the space of
integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3. Definition of fractional integrals and derivatives and their
simplest properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4. Fractional integrals and derivatives of complex order. . . . . . . . 38
2.5. Fractional integrals of some elementary functions..... .. .. .. 40
2.6. Fractional integration and differentiation as reciprocal
operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7. Composition formulae. Connection with semigroups of
operators. . . . . . . . .. . . . . . . .. . . . . . .. . . . .. . . . . . . . . . . . . . . . . .. .. 46

vi
CONTENTS
ti3. The Fractional Integrals of Holder and Summable Functions 53
3.1. Mapping properties in the space H). . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2. Mapping properties in the space H$(p) " ... .. .. .. .. ... . . .. . 57
3.3. Mapping properties in the space L,.. .. ..... .. ., .. ... .. .. .. . 66
304. Mapping properties in the space L,(p). .. .. .. .., .. .. .. ... .. . 70
ti4. Bibliographical Remarks and Additional Information to
Chapter 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1. Historical notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2. Survey of other results (relating to U-3) . . . . . . . . . . . . . . . . . . . 84
Chapter 2 - Fractional Integrals and Derivatives on the
Real Axis and Half-Axis....................................... 93
ti5. The Main Properties of Fractional Integrals and Derivatives 93
5.1. Definitions and elementary properties....... ....... .. ... .... 93
5.2. Fractional integrals of Holderian functions. . . . . . .. . . . . . . . . . . 98
5.3. Fractional integrals of sumrnable functions. . . . . . . . . . . . . . . . . . 102
504. The Marchaud fractional derivative. . . . . . . . . . . . . . . . . . . . . . . . . 109
5.5. The finite part of integrals due to Hadamard. . . . . . . . . . . . . . . 112
5.6. Properties of finite differences and Marchaud fractional
derivatives of order Q > 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.7. Connection with fractional power of operators. ., .. ... .. .. .. 120
ti6. Representation of Function by Fractional Integrals of
L,-F\1nctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.1. The space [Ot(L,) .. .............. .. .. .................. .. .. 122
6.2. Inversion of fractional integrals of L,-functions .,. j ... .. .. .. 123
6.3. Characterization of the space [Ot(L,) .... .............. .. .. . 127
6 A. Sufficiency conditions for the represent ability of functions by
_ fractional integrals. .. .. . .. .. .. .. .. . .. .. .. . .. .. .. .. . .. .. .. .. 131
6.5. On the integral modulus of continuity of [Ot(Lp)-functions... 136
ti7. Integral Transforms of Fractional Integrals and Derivatives 137
7.1. The Fourier transform.. .. .. .. ..... .. ....... .. .. .. .. ... .. ... 137
7.2. The Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3. The Mellin transform.. .... ......... ..... .. .. ..... .. .. .. .. .. 142
ti8. Fractional Integrals and Derivatives of Generalized
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.1. Preliminary ideas... ..... .. .. ..... .. ....... .. .. ..... .... ... 145
8.2. The case of the axis R l . Lizorkin's space of test functions.. . 146

CONTENTS
vii
8.3. Schwartz's approach........................................ 154
8.4. The case of the half-axis. The approach via the adjoint
operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.5. McBride's spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.6. The, case of an interval. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
ti9. Bibliographical Remarks and Additional Information to
Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . .-. . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.1. Historical notes. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.2. Survey of other results (relating to SS 5-8) . . . . . . . . . . . . . . . . . . 163
9.3. Tables of fractioal integrals and derivatives. . . . . . . . . . . . . . . . 172
Chapter 3 - Further Properties of Fractional Integrals
and Derivatives................................................... 175
ti10. Compositions of Fractional Integrals and Derivatives with
Weights. . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
10.1. Compositions of two one-sided integrals with power weights. 176
10.2. Compositions of two-sided integrals with power weights.. .. . 189
10.3. Compositions of several integrals with power weights. .. ... .. 191
10.4. Compositions with exponential and power-exponential
weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
till. Connection between Fractional Integrals and the Singular
Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .,' . . . . . . . . . . . . . . . . . . . . . . . . 199
11.1. The singular operator S . .. . .. .. . . . .. . .. . .. .. .. .. .. . .. .. .. .. 199
11.2. The case of the whole line.. .. .. .. .. . .. .. .. .. .. . .. .. .. .. . .. . 202
11.3. The case of an interval and a half-axis. . . . . .. .. . . . .. .. . . . . . . 204
11.4. Some other composition relations............ .. ........... " 210
ti12. Fractional Integrals of the Potential Type................ ... 213
12.1. The real axis. The Riesz and Feller potentials............... 214
12.2. On the "truncation" of the Riesz potential to the half-axis. . 218
12.3. The case of the half-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
12.4. The case of a finite interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
ti13. Functions Representable by Fractional Integrals on an
Interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
13.1. The Marchaud fractional derivative oil an interval.. .. .. ... . 224
13.2. Characterization of fractional integrals of functions in L, . . . 229
13.3. Continuation, restriction and "sewing" of fractional integrals 234
13.4. Characterization of fractional integrals of H8lderian functions 238

viii
CONTENTS
13.5. Fractional integration in the union of weighted Holder spaces 246
13.6. Fractional. integrals and derivatives of functions with a
prescribed continuity modulus.. ... .. .. .. .. . .. .. . . .. . . . .. . . . 249
fi14. Miscellaneous Results for Fractional Integro-differentiation
of Functions of a Real Variable.. .. . .. .. .. .. .. . .. .. .. .. . .. .. .. 254
14.1. Lipschitz spaces H; and if;......... .................. ..... 254
14.2. Mapping properties of fractional integration in H;. ..... .. .. 256
14.3. Fractional integrals and derivatives of functions which are
given on the whole line and belong to H; on every finite
interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
14.4. Fractional derivatives of absolutely continuous functions. . . . 267
14.5. The Riesz mean value theorem and inequalities for fractional
integrals and derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
14.6. Fractional integration and the summation of series and
integrals. . . . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
fi15. The Generalized Leibniz Rule.. .. .. .. .. .. ... .... ..... .. . .. .. . 277
15.1. Fractional integro-differentiation of analytic functions on the
real axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
15.2. The generalized Leibniz rule.. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. . 280
fi16. Asymptotic Expansions of Fractional Integrals.............. 285
16.1. Definitions and properties of asymptotic expansions......... 285
16.2. The case of a power asymptotic expansion. . . . . . . . . . . . . . . . . . 287
16.3. The case of a power-logarithmic asymptotic expansion...... 294
16.4. The case of a power-exponential asymptotic expansion. . . . . . 297
16.5. The asymptotic solution of Abel's equation.......... .... ... . 299
fi17. Bibliographical Remarks and Additional Information to
Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
17.1. Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
17.2. Survey of other results (relating to U 10-16)................ 305
Chapter 4 - Other Forms of Fractional Integrals and
Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
fi18. Direct Modifications and Generalizations of Riemann-
Liouville Fractional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
18.1. Erdelyi-Kober-type operators. .. . . . . .. . . . . . . . . . . . .. . . . . . . . . . 322
18.2. Fractional integrals of a function by another function. . . . . . . 325
18.3. Hadamard fractional integro-differentiation ............ ..... 329

CONTENTS
ix
18.4. One-dimensional modification of Bessel fractional integra-
differentiation and the spaces H',' = L; ... .. ... ...... .. .... 333
18.5. The Chen fractional integral.. .. .. .. .. . .. .. .. . .. .. .. .. . .. .. . 338
18.6. Dzherbashyan's generalized fractional integral. .. .. .. ... .. .. - 344
fi19. Weyl Fractional Integrals and Derivatives of Periodic
functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
19.1. Definitions. Connections with Fourier series. .. .. .. ... .. .. .. 347
19.2. Elementary properties of Weyl fractional integrals. . . . . . . . . . 352
19.3. Other forms of fractional integration of periodic functions. ., 354
19.4. The coincidence of Weyl and Marchaud fractional derivatives 356
19.5. The represent ability of periodic functions by the Weyl
fractional integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
19.6. Weyl fractional integration and differentiation in the space of
Holderian functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
19.7. Weyl fractional integrals and derivatives of periodic functions
in H: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
19.8. The Bernstein inequality for fractional integrals of trigona-
metric polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
fi20. An Approach to Fractional Integro-differentiation via
Fractional Differences (The Griinwald-Letnikov Approach) 371
20.1. Differences of a fractional order and their properties.. .... .. 371
20.2. Coincidence of the Griinwald-Letnikov derivative with the
Marchaud derivative. The periodic case. . . . . . . . . . . . . . . . . . . . . 376
20.3. Coincidence of the Griinwald-Letnikov derivative with the
Marchaud derivative. The non-periodic case.. .. .. ..... .. .. . 382
20.4. Griinwald-Letnikov fractional differentiation on a finite
interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
fi21. Operators with Power-Logarithmic Kernels. . . . . . . . . . . . . . . . . 388
21.1. Mapping properties in the space H). . .. . .. .. .. .. . .. .. .. .. . .. 389
21.2. Mapping properties in the space H$(p)..................... 396
21.3. Mapping properties in the space L,.. .. .. .. . .. .. .. .. . .. .. .. . 401
21.4. Mapping properties in the space L,(p). .. .. .. ... .. .. .. ... .. . 404
21.5. Asymptotic expansions...... ....... .. .. ..... .. ..... .. " .. .. 411
fi22. Fractional Integrals and Derivatives in the Complex Plane 414
22.1. Definitions and the main properties of fractional integra-
differentiation in the complex plane. .. .. .. ... .. .. .... ... .. .. 416
22.2. Fractional integra-differentiation of analytic functions. . . . . . . 420
22.3. Generalization of fractional integra-differentiation of analytic
functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

x
CONTENTS
fi23. Bibliographical Remarks and Additional Information to
Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
23.1. Historical notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
23.2. Survey of other results (relating to SS 18-22)............... . 436
23.3. Answers to some questions put at the Conference on
Fractional Calculus (New Haven, 1974) .... .. .. .. ... .... .. " 455
Chapter 5 - Fractional Integro-differentiation of
Functions of Many Variables.. .. ....... .. .................. .. 457
fi24. Partial and Mixed Integrals and Derivatives of Fractional
Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
24.1. The multidimensional Abel integral equation. . . . . . . . . . . . . . . 458
24.2. Partial and mixed fractional integrals and derivatives. .. .. " 459
24.3. The case of two variables. Tensor product of operators.. .. " 463
24.4. Mapping properties of fractional integration operators in the
spaces Lp(Rn) (with mixed norm) .. .. ..... ................. 464
24.5. Connection with a singular integral. . . . . . . . . . . . . . . . . . . . . . . . . 466
24.6. Partial and mixed fractional derivatives in the Marchaud
form. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 468
24.7. Characterization of fractional integrals of functions in Lp( R2) 471
24.8. Integral transform of fractional integrals and derivatives. .. . 473
24.9. Lizorkin function space invariant relative to fractional integra-
differentiation. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
24.10. Fractional derivatives and integrals of periodic functions of
many variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
24.11. Griinwald-Letnikov fractional differentiation. ......... .. .. .. 479
24.12. Operators of the polypotential type......................... 480
fi25. Riesz Fractional Integro-differentiation...................... 483
25.1. Preliminaries............................................... 484
25.2. The Riesz potential and its Fourier transform. Invariant
Lizorkin space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
25.3. Mapping properties of the operator la. in the spaces L,(R")
and L,(Rn;p).............................................. 494
25.4. Riesz differentiation (hypersingular integrals) . . . . . . . . . . . . . . . 498
25.5. Unilateral Riesz potentials.. .. .. .. .. . .. .. .. .. . .. .. . .. .. .. .. . 502
fi26. Hypersingular Integrals and the Space of Riesz Potentials 505
26.1. Investigation of the normalizing constants dn,'(o) as functions
of the parameter 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

CONTENTS
xi
26.2. Convergence of the hypersingular integral for smooth func-
tions and diminution of order I to I > 2[cr/2] in the case of a
non-centered difference.... .. .. ..... .. " .. ....... .. .. ..... .. 510
26.3. The hypersingular integral as an inverse of a Riesz potential 512
26.4. Hypersingular integrals with homogeneous characteristics. .. 518
26.5. Hypersingular integral with a homogeneous characteristic as
a convolution with the distribution. . . . . . . . . . . . . . . . . . . . . . . . . 525
26.6. Representation of differential operators in partial derivatives
by hypersingular integrals .. . .. .. .. .. . .. .. .. .. .. . .. .. .. .. . .. 527
26.7. The space [01 (L,) of Riesz potentials and its characterization
in terms of hypersingular integrals. The space L:,r (R n ) . . . . . 532
fi27. Bessel Fractional Integro-differentiation . . . . . . . . . . . . . . . . . . . . . 538
27.1. The Bessel kernel and its properties. . . . . . . . . . . . . . . . . . . . . . . . 538
27.2. Connections with Poisson, Gauss-Weierstrass and
metaharmonic continuation semigroups.................... 541
27.3. The space of Bessel potentials .. .. . .. .. .. .. .. . .. .. .. .. . .. .. . 543
27.4. The realization of (E_)0I/2, cr > 0, in terms of hyper singular
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
fi28. Other Forms of Multidimensional Fractional Integro-
differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
28.1. Riesz potential with Lorentz distance (hyperbolic Riesz
potentials) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
28.2. Parabolic potentials.. .. .. .. .. .. .. . .. .. .. .. . .. .. .. .. . .. .. .. . 562
28.3. The realization of the fractional powers (-z + :t ) 01/2 and
(E - z + :t )0I/2, cr > 0, in terms of a hypersingular integral 565
28.4. Pyramidal analogues of mixed fractional integrals and
derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
fi29. Bibliographical Remarks and Additional Information to
Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
29.1. Historical notes. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. . .. .. .. .. . 580
29.2. Survey of other results (relating to SS 24-28) . . . . . . . . . . . . . . . . 584
Chapter 6 - Applications to Integral Equations of the
First Kind with Power and Power-Logarithmic Kernels 605
fi30. The Generalized Abel Integral Equatioll . . . . . . . . . . . . . . . . . . . . 606
30.1. The dominant singular integral equation. . . . . . . . . . . . . . . . . . . . 606
30.2. The generalized Abel equation on the whole axis.. .. .. ... .. . 610
30.3. The generalized Abel equation on an interval... ... .. .. . . .. . 616
30.4. The case of constant coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

xii
CONTENTS
fi3t. The Noether Nature of the Equation of the First Kind with
Power-Type Kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
31.1. Preliminaries on Noether operators..... .... ..... .. .. .. ... .. 630
31.2. The equation on the axis .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. 634
31.3. Equations on a finite interval.. .. . .. .. .. .. . .. .. .. .. . .. .. . . . . 646
31.4. On the stability of solutions.. .. .. . .. .. .. . .. . .. .. . .. .. .. .. . . 657
fi32. Equations with Power-Logarithmic Kernels... . . . . . . . . . . . . . . 659
32.1. Special Volterra functions and some of their properties.. .. ., 661
32.2. The solution of equations with integer non-negative powers
of logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664
32.3. The solution of equations with real powers of logarithms. . . . 667
fi33. The Noether Nature of Equations of the First Kind with
Power-Logarithmic Kernels... .. .. ....... ....... .... .... ..... . 672
33.1. Imbedding theorems for the ranges of the operators 1:/ and
I:! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673
33.2. Connection between the operators with power-logarithmic
kernels and singular operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
33.3. The Noether nature of equation (33.1)........ ., .. ....... .. . 681
fi34. Bibliographical Remarks and Additional Information to
Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
34.1. Historical notes.. .. .. .. .. .. .. .. .. .. . .. . .. .. .. .. .. . .. .. .. .. . 684
34.2. Survey of other results (relating to U 30-33).. . . . . . . . . . . . . . . 687
Chapter 7 - Integral Equations of the First Kind with
Special Functions as Kernels.................................. 695
fi35. Some Equations with Homogeneous Kernels Involving
Gauss and Legendre Functions. ......... .. ............ ...... . 696
35.1. Equations with the Gauss function. . . . . . . . . . . . . . . . . . . . . . . . . 696
35.2. Equations with the Legendre function.. ........... .. ....... 699
fi36. Fractional Integrals and Derivatives as Integral Transforms 703
36.1. Definition of the G-transform. The spaces !Dt;'(L) and Lc,..,)
and their characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704
36.2. Existence, mapping properties and representations of the
G-transform: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709
36.3. Factorization of the G-transform ....... ................ .. .. 713
36.4. Inversion of the G-transform .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. 716
36.5. The mapping properties, factorization and inversion of
fractional integrals in the spaces !Dt;, (L) and L c,..,) ..... . . . 720

CONTENTS xiii
36.6. Other examples of factorization.. .. .. ... " ., .. ... .. ., .. ... " 722
36.7. Mapping properties of the G-transform on fractional integrals
and derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726
36.8. Index laws for fractional integrals and derivatives. . . . . . . . . . . 727
fi37. Equations with Non-Homogeneous Kernels................. 730
37.1. Equations with difference kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . 731
37.2. Generalized operators of Hankel and Erdelyi-Kober
transforms. ......... ..... .... ......... .... ..... .... " .., .. . 737
37.3. Non-convolution operators with Bessel functions in kernels. . 741
37.4. Equation of compositional type. .... ..... .. .. '" .. .. .. . .. . . . 746
37.5. The W-transform and its inversion.. .... .. ... " .. .. ..... .. . 752
37.6. Application of fractional integrals to the inversion of the
W -transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
fi38. Applications of Fractional Integra-differentiation to the
Investigation of Dual Integral Equations . . . . . . . . . . . . . . . . . . . . 761
38.1. Dual Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
38.2. Triple equations. .. .. .. . .. .. .. . .. .. .. .. .. .. . .. .. .. .. . .. .. .. . 768
fi39. Bibliographical Remarks and Additional Information to
Chapter 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
39.1. Historical notes. .. .. .. . .. .. .. .. . .. .. .. .. . .. .. .. .. . .. .. .. .. . 772
39.2. Survey of other results (relating to SS 35-38) . . . . . . . . . . . . . . . . 775
Chapter 8 - Applications to Differential Equations....... 795
fi40. Integral Representations for Solution of Partial Differential
Equations of the Second Order via Analytic Functions and
Their Applications to Boundary Value Problems. . . . . . . . . . . 795
40.1. Preliminaries............................................... 796
40.2. The representation of solutions of generalized Helmholtz
two-axially symmetric equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800
40.3. Boundary value problems for the generalized Helmholtz two-
axially symmetric equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809
fi41. Euler-Poisson-Darboux Equation............................. 812
41.1. Representations for solutions of the Euler-Poisson-Darboux
equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813
41.2. Classical and generalized solutions of the Cauchy problem.. 819
41.3. The half-homogeneous Cauchy problem in multidimensional
half-space. . '. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823
41.4. The weighted Dirichlet and Neumann problems in a half-plane 826

xiv
CONTENTS
fi42. Ordinary Differential Equations of Fractional Order . . . . . . . 829
42.1. The Cauchy-type problem for differential equations and
systems of differential equations of fractional order of general
form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830
42.2. The CauchY'7type problem for linear differential equation of
fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837
42.3. The Dirichlet-type problem for linear differential equation of
fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843
42.4. Solution of the linear differential equation of fractional order
with constant coefficients in the space of generalized functions 846
42.5. The application of fractional differentiation to differential
equations of integer order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849
fi43. Bibliographical Remarks and Additional Information to
Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856
43.1. Historical notes. .. .. . .. .. .. .. . .. .. .. .. .. . .. . .. .. .. .. .. .. .. . 856
43.2. Survey of other results (relating to SS 40-42).. .. .. .... '" .. . 858
Bibliography. . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873
Author Index........................................................... 953
Subject Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965
Index of Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973

FOREWORD
The concepts of fractional differentiation and integration are usually associated
with the name of Liouville. However, the creators of differential and integral
calculus had already considered derivatives not only of integer order, but of
fractional order too. On reading this book we learn that fractional derivatives were
the subjects of Leibniz's study. Euler also took an interest in fractional derivatjves.
Liouville, Abel, Riemann, Letnikov, Weyl, Hadamard and many other well-known
mathematicians of the past and present influenced the development of fractional
integro-differentiation, which has now become a significant topic in mathematical
analysis.
Integrals and derivatives of integer order are the normal integrals and
derivatives of analysis. But in the case of fractional order these ideas manifest
their own peculiar features. Because various modifications arise naturally in
different situations, the interconnections between these modifications need to be
investigated.
Fractional derivatives and integrals have many uses and they themselves have
arisen from certain requirements in applications.
Although there are many individual research papers on fractional derivatives
and integrals, a unifying monograph on the topic has never been published. The
present book fills this gap. It is written by a prominent specialist in mathematical
analysis, Prof. Dr. S.G. Samko (Rostov State University), together with Dr.
A.A. Kilbas and Dr. 0.1. Marichev (Belorussian State University).
This monograph is an extensive, yet compact, exposition on the present state
of mathematical research on fractional integro-differentiation.
The authors themselves have made valuable contributions to the theory of
fractional integration and differentiation and so it is natural that their own results
take up a rather conspicuous amount of space in this book.
The book is constructed in the following way. The main sections in each
chapter acquaint the reader with the fundamental questions. The general theorems
are proved in full as a rule, although the reader is sometimes referred to original
sources. All chapters include a. historical resume with results and sources. Many
statements which supplement the fundamental text are given here without proof.

xvi
FOREWORD
Part of the text is devoted to the case of one variable, and the rest to several
variables. The multidimensional case is especially interesting. Only in special
cases does it reduce to combining the known one-dimensional results. In the case
of many variables such subjects as the fractional integra-differentiation, theory of
Riesz, hypersingular integrals, Bessel fractional integra-differentiation, fractional
power of hyperbolic, parabolic differential operators and others are considered.
This book also contains a chapter devoted to integral equations with power or
logarithmic-power kernels. Here the integra-differential operators already discussed
are applied to solve rather general integral equations. In the process, the necessity
of using the classical results of Muskhelishvili and Gahov become apparent. The
authors of this monograph, who come from the Gahov school, are masters in these
methods and have had much to do with their development.
As well as this the book also contains a great deal of theory on integral
equations of the first kind with special functions in the kernel, the solutions of
which are obtained by means of fractional integra-differentiation. In the final
chapter applications to some problems in differential equations are given.
The presentation of the monograph uses simple everyday language based on
the knowledge of differential and integral calculus, usually well within the limits of
courses taught in physics, mathematics and engineering faculties. This makes the
book easily understandable to a wide circle of readers. The book will be of interest
to anyone interested in mathematical analysis. It may serve as an introduction
to questions connected with the idea of fractional integration and differentiation.
There is no doubt that the book will be useful to specialists both as a reference
book with its large bibliography and as a subject of study.
I believe that the monograph will be a success, and I wish it good luck.
Academician S.M. Nikol'skii
Steklov Institute of Mathematics,
Moscow

PREFACE
TO THE ENGLISH EDITION
This book contains not only a logical presentation of the main principles of
fractional calculus, but also surveys the numerous special investigations involving
fractional integro.-differentiation. Thus, while translating the book into English
we were unable to avoid adding to these surveys papers which appeared after
the Russian edition of the book was published in 1987. This was not an easy
task: fractional calculus has not ceased in its development. We may refer, for
example, to the 3rd Conference on Fractional Calculus, held in Tokyo, 1989 (see its
proceedings in "Fractional Calculus and Applications", Ed. K. Nishimoto, Nihon
Univ., Japan, 1990) and to the fact that about four hundred new references appear
in the Bibliography of the English translation. Thus the English edition is rather
expanded mainly due to the reviewing of U 4, 9, 17, 23, 29, 34, 39 and 43. Most
additions were made to SS 23 andc29, and certain parts of S 43 were rewritten.
All the surveys, including the additions, are theoretical in character. We are not
concerned in this book with the applied aspects of fractional analysis, such as in
engineering, modelling, mechanics and so on. We draw your attention, however, to
the papers by Simak [1] (1987), Bagley [1] (1990) and the book by Gorenflo and
Vessela [3] (1991) which appeared after the Russian edition was published. Large
bibliographies of applied investigation may by found in these publications.
Our method of citing references is to emphasize the year of publication in the
historical commentaries US 4.1, 9.1, 17.1,23.1,29.1,34.1,39.1,43.1). In the main
body of the text and in the reviews (SS 4.2, 9.2, 17.2, 23.2, 29.2, 34.2, 39.2, 43.2)
the year of publication is not indicated, except perhaps in the few cases when it
may be important.
There are also some slight alterations in the main text of the book. They were
made either in connection with new information or to improve the presentation. .
The Russian text was rewritten in English by S.G. Samko (Preface, Brief
Historical Outline, Foreword, U 2,4-9, 12-14, 17-20, 22-31) and A.A. Kilbas (the
remaining sections) and we hope that the reader will appreciate the extent of this
great achievement.

xviii
PREFACE TO THE ENGLISH EDITION
We would like to emphasize that our interests are mainly in the field of real
analysis. That is why fractional calculus in the complex plane, although considered
in S 22, plays second fiddle in this book in comparison with fractional analysis
of one and many real variables. Nevertheless, we present a fairly comprehensive
survey of investigations within the framework of complex analysis in S 23.
We would like to express our thanks to Prof. R. Gorenflo for helpful
information on some recent publications.
Last but not least, our deep thanks are owed to Galina Smirnova, Yulia
Zhdanova, Tatyana Bessonova and Igor Tarasyuk for the patient and careful typing
of this manuscript.

PREFACE
The field of mathematical analysis entitled fractional calculus, which deals with
the investigation and applications of derivatives and integrals of arbitrary (real
or complex) order has a long history described in the Brief Historical Outline.
It is a complex topic having interconnections with various problems of function
theory, integral and differential equations, and other branches of analysis. It has
been continually developed, stimulated by ideas and results in various fields of
mathematical analysis. Fractional calculus of functions of one and many variables
continues to be developed intensively. This is demonstrated both by the many
publications - hundreds of papers in the past years - and by the international
conferences devoted to the problems of fractional calculus. The first such conference
was held in 1974 (New Haven, USA; Proceedings in "Fractional calculus and
its applications", Ed. B. Ross, Lect. Notes Math., 1975, v. 457), the second
- in 1984 (Glasgow, Great Britain; Proceedings in "Fractional calculus", Eds.
A.C. McBride and G.F. Roach, Res. Notes Math., 1985, v. 138). Considering
the long history of the development of fractional analysis it is a surprising fact
that few if any monographs devoted to this topic have appeared. Indeed world
mathematical literature can not point to any book which would thoroughly and
comprehensively reflect the achievements of this theory. The only book specially
devoted to fractional calculus by Oldham and Spanier [1] (1974) was written by
specialists in the applied problems of chemistry, and contained only a presentation
of some classical points of the theory. The main attention in the book was focused
on the evaluation of fractional integrals and derivatives of concrete functions, and
to applications to diffusion problems. Books which contain a chapter or a section
concerning certain questions in the field of fractional calculus are mentioned, for
example, Zygmund [6], Dzherbashyan [2], Sneddon [3], [6], Butzer and Nessel [1],
Butzer and Trebels [2], Davis [3], [4], Okikiolu [7], Samko [31], Fenyo and Stolle
[1]. The publication of the little known thesis of Marke [1] (1942), in Danish, by
Copenhagen University, is also of certain interest for specialists. Lastly we have
singled out papers which contain historical outlines of the development of fractional
calculus. The first such outline appeared in the paper by Letnikov [2] (1868).
There are also historical outlines in the papers by Davis [3], [4], Mikolas [6], Ross

xx
PREFACE
[1]-[3], Tremblay [1, p. 12-19]. In the main they are devoted to the classical period
of the development of fractional calculus.
Perhaps the reasons for the absence of a unifying monograph on fractional
calculus was because of the very rapid development of the theory of fractional
integro-differentiation in the last decades, and also its multifarious branching
especially in the case of many variables.
The absence of such a monograph in a way became a hindrance to the
development of fractional calculus. Some results, amongst which were those of
a fundamental and essential nature, were published in original papers, several of
which were difficult to find and were little known. This inevitably created a
situation where investigators wasted much effort obtaining results already known
or readily derived from known ones. Also some papers contained mistakes caused
by the incorrect interpretation of the basic ideas of the theory.
Indeed the history of fractional calculus is replete with many papers where
results already known were rediscovered, sometimes by the same methods that
predecessors had used, and sometimes by quite other means. This situation was
aggravated by the existence of many different approaches to fractional integro-
differentiation, and consequently by many different fields in fractional calculus.
Comparison of these approaches was seldom carried out and was comparatively
little known. A researcher starting in the field often encountered inconvenience
caused by the necessity to orientate himself or herself in the many diverse definitions
of fractional integro-differentiation and in the enormous flow of publications.
The authors of this book have their interests in the theory of integral operators,
function theory, integral and differential equations and special functions, and they
have used the apparatus of fractional integro-differentiation in their investigations
since 1967. In the authors' investigations the necessity of obtaining results in
the theory of fractional integro-differentiation arose frequently, and gradually the
interests of at least the first of the authors shifted to the fractional calculus - firstly
to functions of a single variable, and since 1974 to functions of many variables.
In their work the authors gradually arrived at the idea of writing a book which
would reflect the modern state of fractional calculus, and present its applications to
the theory of integral operators and integral and differential equations. The wide
bibliographical search undertaken by the authors and the analysis of the enormous
number of papers strengthened the authors' idea. An essential role was also
played by the fact that since 1968 the first author had given lectures on fractional
integra-differentiation of functions of one and many variables to undergraduates
and postgraduates of Rostov State University.
The temptation to present all the important results of the theory known up
to now with complete proofs was great. However, such an approach would require
a multi-volumed edition. Thus, the authors found it more expedient to single
out from the main text the distinctive historical-surveying sections which complete
every chapter. These sections US 4, 9, 17, 23, 29, 34, 39, 43) provide historical
commentaries to the content of the preceding chapter, and contain the discussion
and formulation of results which are close to the subject matter of the chapter, but
were not included in the main text. These commentaries and results are divided

PREFACE
xxi
into paragraphs, whose enumeration are relevant to the corresponding section. For
example, S 4.1 contains historical information relevant to Chapter 1 and consists
of paragraphs giving information on each of the U 2.1-3.2 of the chapter. The
second subsection, S 4.2, presents a survey of results on the subject of Chapter 1
and consists of S S 2.1-2.7 and 3.1.-3.4. The authors were faced with the
difficult problem of selecting material for the main text, and the authors' tastes
naturally influenced this choice. The results presented in the main body of the
chapters are, as a rule, given with the complete proof.
The first five chapters of the book contain the presentation of the theory of
fractional integro-differentiation itself. Chapters 1-4 deal with functions of one
variable and Chapter 5 with functions of many variables. Chapters 6-8 contain
applications to integral and differential equations. The application of fractional
integro-differentiation to multidimensional integral equations is not discussed in
this book. Such applications may be found in the book by Samko [31] and in the
review paper by Samko and Umarkhadzhiev [3].
We call the reader's attention to S 23.3, where some questions which were
posed at the 1st Conference on the Fractional Calculus (New Haven, 1974) were
answered and S 9.3, which presents tables of fractional integrals and derivatives of
some elementary and special functions.
There are a large number of references given in this book covering a great
number of publications describing both theory and applications. The reader will
find references to many papers which may prove to be new both to specialists and
historians in fractional calculus.
The authors do not concern themselves with the theory of fractional power of
operators, as this would lead the text too far astray, although this topic is touched
upon episodically as, for example, in S 5.7. The book also deals with the
symbolical calculus of Hoole and Heavyside, and does not use the ideas of G- and
H -functions of many variables, as the theory of these functions is only in the first
stage of development.
The significance of fractional integrals and derivatives or of Abel-type equations
in application must especially be emphasized. This mathematical apparatus is
used in various sciences such as physics, mechanics, chemistry and others. After
the known Abel problem on the tautochrone (Abel [1] (1823» the first applications
were made by Liouville [1] (1832) to problems of geometry, physics and mechanics.
Amongst them we may find the Laplace problem concerning the influence of an
infinite rectilinear conductor on a magnet, the Ampere problem of the interaction
of two such conductors, problems connected with attraction of bodies, the problem
of the heat distribution in a ball, the Gauss problem of approximate quadratures
and others. The survey of the applied problems considered by Liouville in the
paper by Letnikov [4] (1874), 21-44, is worth seeing in this connection.
There are many papers of purely applied character which use the methods
of fractional calculus, but this book does not deal with applications in other
fields except mathematics. The applications of fractional calculus to integral and
differential equations presented in Chapters 6-8 are of theoretical mathematical
character themselves. The reader interested in purely applied aspects of fractional

xxii
PREFACE
calculus should refer to the following publications: Oldham and Spanier [1],
mentioned above, which contains the chapter "Application to diffusion problems";
the paper [2] by the same authors which contains a large list of papers with
applications to chemical physics, hydrology, random processes, viscoelasticity,
gravitation theory and so on; the book "Abel inversion and its generalization"
(Novosibirsk.. 1978) and, in particular, the introductory paper by Preobrazhenskii
[1] in this book; and the Proceedings of the 1st Conference on the Fractional
Calculus, mentioned above. Some other publications are of relevance as well: the
books by Tseitlin [1] (1984) in particular, pp. 275-276, Yu.I. Babenko [1] (1986) and
the papers by Brenke [1] (1922), ROthe [1] (1931), Rabotnov [1] (1948), Bykov and
Botashev [1] (1965), Shermergor (1966), Fedosov [1] (1978), Gomes and Pestana [1]
(1978), Zaganescu [I), [2] (1982), Bagley and Torvik [1] (1986), Koeller [1] (1986),
Gorenflo and Vessel a [2] (1986).
Finally note that the term ''fractional'' integra-differentiation is used
throughout this book. Sometimes this word arouses objections since the order
of "fractional" integra-differentiation is an arbitrary number, not necessarily a
fractional one. However, the authors consider it inexpedient to change this
historically established term.
The authors hope that they have succeeded in presenting the various
approaches to fractional integra-differentiation, and in acquainting the reader
with the interconnections between them and in clarifying the question about the
complete coincidence of some of these approaches, including the coincidence of the
domains of definition.
Sections  2,4-6, 8,9, 12-14, 17-20 (except 18.1) 22-31, and the Brief Historical
Outline were written by S.G. Samko.
Sections n 15, 16, 21, 28.4, 32 and 33 were written by A.A. Kilbas.
Sections SS 7 (except S 7.1), 10, 35-38, 40-42 were written by 0.1. Marichev.
Sections is 3, 11, 34 were cowritten by Samko and Kilbas, n 39, 43 and 18.1
by Kilbas and Marichev and S 1 by Samko, Kilbas and Marichev.
The authors note that B.S. Rubin read a considerable part of the manuscript
and made a number of valuable suggestions. This was also done by Vu Kim Than
with respect to some sections. Some materials prepared by N .A. Virchenko were
used in S 38, by Vu Kim Than - in S 36, by S.B. Yakubovich - in S 37.5 and S 37.6
and by V.S. Adamchik and A.V. Didenko - in S 42. Qualified assistance in the
preparation of the manuscript was also given by V.A. Nogin and B.G. Vakulov.
Useful information and assistance in finding a number of papers were rendered by
R.G. Buschman, I.H. Dimovski, B. Fisher, H.-J. Glaeske, R. Johnson, S.L. Kalla,
K.S. Kolbig, E.R. Love, A.C. McBride, M. Mikolas, B. Muckenhoupt, K. Nishimoto,
S. Owa, B. Ross, M. Saigo, R. Wheeden. The authors express their gratitude to all
of them.

INTRODUCTION
The subject of this book is differentiation DOt and integration [Ot of arbitrary
order and some of their applications to integral and differential equations. Most
of the theory of such operations is concerned with functions of one variable, but
Chapter 5 is concerned with different forms of fractional integra-differentiation of
functions of many variables. The discussion is presented mainly for functions of a
real variable, but S 22 is devoted to functions of a complex variable.
In order to make the book suitable for as wide a readership as possible we
start the discussion from simpler properties and statements and pass from special
cases to general ones. For this reason we deliberately investigate "model" cases
before considering the most general problem. In many cases, especially towards
the beginning of the book, we prefer proposition with simpler forms and proofs.
More complicated cases are dealt with in later chapters or in final sections of the
appropriate chapters.
The characteristic feature of the book is an exposition of practically all known
forms of fractional integra-differentiation and their comparison with each other. In
many cases not only the identical nature of different forms to each other in certain
spaces of functions is proved, but the coincidence of their domains of definition is
also shown.
Another distinguishing feature of the book is that we stress the problem
of the represent ability of a function j(z) by the fractional integral j = [Otcp, a > 0,
of a function cp belonging to one or another given space X. This question is
investigated in all situations considered in the book - for functions of one and many
variables, in periodic and non-periodic cases, on the whole real axis or in the whole
space and in a finite interval - for all forms of fractional integra-differentiation. As
a rule X is L,-space or Holder space H). or a similar weighted space.
As a matter of fact the represent ability of a function j(z) by the fractional
integral j = [Ot cp of order a studied in the book is a more important fact than the
existence of fractional derivative of order a of a function j(z). We reveal general
situations when the existence, in one or other sense, of a fractional derivative DOt j
of a function j(z) is equivalent to the representability of the latter by a fractional
integral. Then it is easy, in particular, to answer the question: why does the

xxiv
INTRODUCTION
existence of one or other form DOl I of a fractional derivative lead to the existence
of a derivative DOl I, (3 < Q, of the same form?
Finally, one more distinctive feature of the book is our endeavour to unify the
notation of various forms of fractional integra-differentiation. It is impossible to do
without such a unification in a book containing many versions of fractional integrals
and derivatives. The reader should immediately note the sign f: in the notation
of fractional integrals and derivatives of functions of one variable. These signs
mean the choice of the left-sided and right-sided fractional integra-differentiation
connected with the left-sided and right-sided translations I( Z =F t), respectively.
A consideration of both of these two forms is caused not only by the desire to
achieve a common form of presentation, but mainly by the existence of interesting
connections between the above forms of fractional integra-differentiation and the
applications discussed in the book.

NOTATION OF THE MAIN FORMS
OF FRACTIONAL INTEGRALS
AND DERIVATIVES
I+cp
Icp
1:+ cp
1&_ cp
I:+;gcp, I:+;z"cp
1:+;u ,'1 cp
I&_ou nCP
, ,.,
It,aCP. Kq,aCP
1'1,aCP, K'1,aCP
Iacp
1 (01)
% cP
a-%cp
J:+cP
Ie; cP
1:0 cP, Its cP
Gacp
G%cp
Vf
V:+f, V&_f
- Liouville left-sided fractional integral (S 5.2)
- Liouville right-sided fractional integral (S 5.3)
- Riemann-Liouville left-sided fractional integral (S 2.17)
- Riemann-Liouville right-sided fractional integral (S 2.18)
- fractional integrals of one function by another (SS 18.24,
18.38-18.41)
- Erdelyi-Kober-type left-sided operator (SS 18.1, 18.2)
- Erdelyi-Kober-type right-sided operator US 18.3, 18.4)
- Kober operators (U 18.5, 18.6)
- Erdelyi-Kober operators (S 18.8)
- Riesz potential (SS 12.1, 25.1)
- Weyl fractional integrals of periodic functions (U 19.5,
19.7)
- Hadamard fractional integration (UI8.42-18.44)
- Griinwald-Letnikov fractional integral (S 20.46)
- Chen fractional integral (S 18.80)
- Riemann-Liouville fractional integrals in the complex plane
(SS 22.8, 22.17-22.20)
- Bessel fractional integration (U 18.61, 27.8)
- modifications of Bessel fractional integration (S 18.63)
- Liouville fractional derivatives US 5.6, 5.7)
- Riemann-Liouville fractional derivatives (U 2.22, 2.23,
2.32, 2.33)

xxvi MAIN FORMS OF FRACTIONAL INTEGRALS AND DERIVATIVES
'D:+;gf
Df
DOt f
D+f, Db_f
'Df
'DOt) f
D Ot) f
f)
'Dof, 'Dtsf
'DC; f
(E:I::'D)Ot, (E:l::D)Ot
- fractional derivatives of one function by another U 18.29)
- Marchaud fractional derivatives (SS 5.57, 5.58, &.80)
- Riesz fractional derivative (S 25.59)
- analogues of Marchaud fractional derivative in an interval
(SS 13.2, 13.5)
- Hadamard fractional derivatives (U 18.56, 18.57)
- Weyl fractional derivatives of periodic functions (S 19.17)
- Weyl-Marchaud fractional derivatives of periodic functions
U 19.18)
- Griinwald-Letnikov fractional derivatives U 20.7)
- Riemann-Liouville fractional derivatives in the complex
plane (U 22.3, 22.21)
- Chen fractional derivatives U 18.87)
- modifications of Bessel fractional differentiation (SS 18.71,
18.72 )

BRIEF HISTORICAL OUTLINE
The idea of generalizing the notion of differentiation d:£:) to noninteger orders of p appeared at
the birth of the differential calculus itself. The first attempt to discuss such an idea recorded in
history was contained in the correspondence of Leibniz. In one of his letters to Leibniz conceming
the theorem on the differentiation of a product of fWlctions, Bemoulli asked about the meaning of
this theorem in the case of noninteger order of differentiation. Leibniz in his letters to L'Hopital
(1695) and to Wallis (1697), see Leibniz [1, pp.301-302], [2, p.25], made some remarks on the
possibility of considering differentials and derivatives of order 1/2. The interested reader can
find other details of Leibniz's correspondence conceming these ideas on fractional differentiation
described by Ross [2].
It was Euler [1, p.56] (1738) who took the first step. He observed that the result of
the evaluation of the derivative dd:P. of the power fWlction :cO has a meaning for non-integer
p. Laplace [1, pp.85, 156] (1812) proposed the idea of differentiation of non-integer order for
functions representable by an integral f T(t)t- Z dt. In the treatise of Lacroix [1, ppA09-410]
(1820) the idea of Euler was repeated and the exact formula for the evaluation of the derivative
d 1 / 2 Z. .
dZ 1 / 2 was already given.
The next step was taken by Fourier [1] (1822), who suggested the idea of using the equality
00 00
dP Lx) = 2 J >.' d>' J J(t) cos(>.x - t>. + p1I" /2)dt
-00 -00
(1)
in order to define the derivative for non-integer order. This was the first definition for the
derivative of arbitrary positive order suitable for any sufficiently "good" fWlction, not necessarily
a power function.
The examples mentioned above may be regarded as a prehistory of fractional integra-
differentiation. The proper history of fractional calculus began with the papers by Abel and
Liouville. In the papers by Abel [1] (1823), [2] (1826) the integral equation
Z
J Ip(t)dt - - J( - ),
... x > a, 0 < I' < 1,
(x - t)#J
o
(2)
was solved in connection with the tautochrone problem. The solution in both papers was given
for arbitrary I' e (0,1), although the tautochrone problem itself leads to the case I' = 1/2. We
emphasize this because of the widely spread delusion that Abel himself solved equation (2) for
I' = 1/2 only. Although Abel's investigations were not penormed in the spirit of the idea of
how to generalize differentiation, they played an enormous role in the development of these ideas.

xxviii
BRIEF HISTORICAL OUTLINE
The reason was that the left-hand side of Abel's equation represents, as it will become obvious
later, the fractional integral operation of order 1 - 1', while the inversion of this equation leads to
fractional differentiation. However, the notions of fractional integra-differentiation in such a form
were shaped somewhat later.
In 1832-1837 a series of papers by Liouville [1]-[8] appeared which made him by right the
real creator of the substantial theory of fractional integra-differentiation. It did not yet reach
its completed form as in the further developments of later investigators, but in these papers
far-reaching and important ideas were proposed. The initial definition suggested by Liouville
[1] (1832) was based on the formula for differentiating an exponential function and is relevant
00
to functions f(x), which may be expanded as the series f(x) = L: cke°Jez. For such functions
k=O
Liouville's definition is
00
D' f(x) = L ckae°Jez
k=O
for any complex p. The restrictiveness of such a definition is evidently connected with the
convergence of the series. Starting from his definition (3), Liouville [I, p.7) obtained the formula
for the differentiation of a power. function. Moreover in the same paper on p.8 he derived, though
not quite rigorously from the modem point of view, the formula
(3)
00
D-' f(x) = (-I)r(P) f !p(x + t)t,-ldt, -00 < x < 00, Rep > O.
o
(4)
This is now called the Liouville form of fractional integration with the factor ( -1)P being omitted.
On pp.11-69 of [1] many applications to problems of geometry, physics, mechanics etc. were
considered. A list of these problems has already been gi ven in the preface.
In further papers, Liouville [2]-[8] developed and applied the ideas he introduced. Among
the results obtained an idea in the first paper [I, p.l06] (1832) that is, to define the fractional
derivative as a limit of a difference quotient  t f / hJ', is especially worth mentioning (where t f
is a difference of fractional order). However, Liouville gave no essential development of this idea,
except for example in his paper [6, p.224] (1835), where he obtained Fourier's formula (1) for
non-integer p, based on this idea. He also evaluated fractional derivatives of some elementary
functions via this approach. This idea was more deeply considered in papers by Griinwald [1]
(1867) and Letnikov [1] (1868).
The papers by Liouville [3] (1832) and [8] (1837) were the first ones that contained an
application of fractional calculus to the solution of some types of linear ordinary differential
equations. In another paper [7) (1835) Liouville considered the effect of a change of variable in
fractional derivatives and integrals. Here the idea of fractional integra-differentiation of a function
by another one was contained in embryo. This idea was more distinctly formulated 30 years later
in a paper by Holmgren [1] (1865-1866). (The interested reader can find Liouville's biography and
a general analysis of his contribution to the development of mathematics in a recently published
book by Liitzen [1].)
Next in significance to the works by Liouville was the paper by Riemann [1]. Paper [1]
written by Riemann in 1847 when just a student, was only published in 1876, ten years after his
death. Riemann had arrived at the expression
Z
1 f !p(t)dt
r(a) (x _ t)l-a' x> 0,
o
(5)
for fractional integration, and since that time this has become one of the main formulae of
fractional integration together with Liouville's construction (4). It is necessary to note here that

BRIEF HISTORICAL OUTLINE
xxix
both Liouville and Riemann dealt with the scalled "complementary" functions which arise when
one attempts to treat fractional differentiation of order a as fractional integration of order -a -
see also the Historical Notes in 4.1 and 9.1 below in this connection.
In 1861 an Abel-type integral equation with the kernel (x 2 - t 2 )-1/2 was solved by
Joachimsthal [1]. A more general equation of such a kind with the kernel [T(X) - T(S)]a-l, a > 0,
was considered 74 years later by Sato [1] (1935) although its solution was in fact already known
to Hohngren [1] (1865-1866).
This latter paper by Holmgren should be especially noted. He used result (5) as a definition
of fractional integration and gave a detailed investigation together with applications to the
solution of ordinary differential equations. The merit of Holmgren's work is in the fact that he
was the first who gave up the "complementary" functions and consciously suggested that one
consider fractional differentiation as an operation inverse to fractional integration. Some years
later Letnikov [1]-[4] (1868-1874), who was not aware of Holmgren's paper, expounded the theory
of fractional integro-differentiation from the same point of view. Holmgren's paper remained
little known both to contemporaries and to later generations of mathematicians and therefore it
was undeservedly little cited. In spite of which Holmgren was the first, after some rather fonnal
arguments of Liouville, who gave a rigorous proof of Leibniz's role for the fractional derivative
Da (uv) of a product of two functions. He also gave such a formula with a remainder in the
integral fonn. Further, he was also the first to introduce the notion of fractional integration of
one functon by another and gave a detailed investigation of the compositions of the fonn
D;ll(Z)fI (x)D;;(z) ... D;:(z/n(x)u(x),
(6)
where Ij (x) denotes an operator of multiplication by a function Ij (x). Moreover he considered
firstly partial and mixed derivatives of functions of two variables. Holmgren [2] (1867) also pushed
well ahead in the application of fractional integrals to ordinary differential equations which was
begun by Liouville [3] (1832).
Grunwald [1] (1867) and Letnikov [1] (1868) developed an approach to fractional
differentiation based on the definition
D a I(x) = lim (hI)(x) .
h-O h a
(7)
While the arguments of the first author were rather fonnal, the latter gave a rigorous and
thorough construction of the theory of fractional integra-differentiation on the basis of such a
definition. Letnikov had in particular shown that thus defined D-a 1 coincides with Liouville's
expression (4) for Rea > 0 and with Riem&IUl's definition under the appropriate interpretation
of the fractional difference (h J)(x). He proved the semi group property within the framework of
definition (7).
A paper by Letnikov [2] (1868) was the first that contained a comprehensive historical survey
of the development of fractional calculus. In a long paper [4] (1874) by Letnikov a complete
theory of fractional integra-differentiation was constructed on the basis of definitions (4) and (5).
The detailed and comprehensive application of this theory to the solution of differential equations
was also given. The reader may find in 42 the solutions of some ordinary differential equations
which go back to this paper by Letnikov.
We note that there was an interesting pQ.blication by Sludskii [1] (1889) on the biography of
Letnikov and his works. See also Nekrasov and Pokrovskii [1] (1889).
Many papers were published other than those of Liouville, Riemann, Holmgren, Grunwald
and Letnikov. We may mention, for example, the papers by Peacock [1] (1833), Greatheed
[I], [2] (1839), Kelland [1]-[3] (1840-1851), Center [1]-[4] (1848-1849), Tardy [1] (1858) and
M.E. VashchenkZakharchenko [1] (1861). Some of them contained polemics with predecessors.
These were connected either with the idea of complementary functions or with the seeming
contradiction between Liouville's and Riem&IUl's definitions, though this contradiction seems

xxx
BRIEF HISTORICAL OUTLINE
far-fetched from the present-day point of view. Others developed or specified small points of the
subject and do not contain fWld&mental ideas.
A further period in the history of fractional calculus is connected with the Cauchy fonnula
1<')(z) = L 1 f(t)dt
211"i (t - z),+1
I:.
(8)
for analytic fWlctions in a complex plane. The direct extension of this fonnula to non-integer
val.ues of p leads to difficulties arising from the multivaluedness of the function (t - z)-p-l and
therefore it depends on the location of the curve £, sUIToWlding the point z and on a cut C
defining a branch of the fWlction (t - z)-,-I. Such an extension was first made by Sonine [1]
(1870), [2] (1872), who showed in the case of analytic functions that this new approach coincided
(with that of Riemann) for Rep < 0 (d. (5»:
%
1<p)(z) = ----2- 1 f(t)dt ,
r( -p) (z - t)I+,
%0
(9)
the integration path being al.ong the interval. [zo, z] in the complex plane, where Zo is the
intersection of the curve £, with the brunch cut C.
It should be emphasized that fractional calculus had developed from its origin in the complex
plane, see for example the papers by Liouville [1]-[8] (1832-1837). The fonnula
1
(Iof)(z) = (z -( zo)a 1 (1 - t)a f[(l - t)zo + tz]dt
r a)
o
in the complex plane, evidently being a modification of (9), had appeared in the paper by
Hohngren [I, p.1] (1865-1866).
Letnikov [3] (1872) made the important remark that £, being a circle, the Cauchy-Sonine
fonnula (8) transfonned into the fonn
211'
1<')(z) = r(1 + p) 1 e-i,8 fez + re i8 )dO, r = Iz - zol.
211"rP
o
(10)
We note that Sonine [2] (1872) continued the investigation of the Leibniz fonnula for Da(uv)
which was begun by Liouville and Holmgren.
We emphasize the priority of Sonine in the extension of the Cauchy fonnula (8) to noninteger
p.
Besides the development of the fractional. calculus itself Sonine [4], [5] (1884) began the
investigation of entities more general than fractional. integrals, together with their applications to
integral equations with special. functions in the kernel. He obtained a solution of the Abel type
equation
:&
1 k(x - t)<p(t)dt = f(x), x> a,
a
with an arbitrary kernel k(x) satisfying certain assumption [4], [5] (1884). fu particular he
fOWld the solution of such an integral equation with the Bessel fWlction in the kernel, the latter

BRIEF HISTORICAL OUTLINE
xxxi
frequently occurring in applications. We note that many years later this result was rediscovered
by other investigators. Details concerning Sonine's ideas may be seen in Chapter I, 4.2 (note
2.4), chapter 7, 39.1 (notes refering to 37.1 and 37.2) and Chapter 7, 39.2 (note refering to
.37.3) below.
In 1888-1891 Nekrasov (1)-[4] gave applications of fractional integro-differentiation in the
form (8) to the integration of high order differential equations. He was also the first to give
a procedure for reducing some multidimensional integrals to double integrals via the evaluation
of compositions of the form (6). The latter represents the solution of differential equations
considered by him.
On the eve of the 20th century a comprehensive paper by Hadamaro [1] (1892) appeared.
The idea of fractional differentiation of an analytic fWlction via differentiation of its Taylor series
00
DOl f( ) _ '" r(k + 1) (_ )1:-01
Z - L.J r(k + 1 _ 01) CI: z zo ,
1:=0
f(l:) (zo)
CI: = k!
(11)
although known before Hadamaro's paper, was used here as an effective working mathematical
tool, Wlderstood as the fractional differentiation of an analytic fWlction in a disk by a radial
variable. Since then any method using (11) is usually named after Hadamard. We note that
Hadamard dealt with fractional integration in the form of
1
]01 f(x) = r:) f (1 - )0I-1 f(z)d,
o
(12)
which led him further to consider generalized fractional integrals of the fonn
1
f v()f(z)d(.
o
(13)
However Hadamard did not develop this idea, although he had considered the case v() =
(_ln)0I-1. Many years later a substantial theory of generalized integration (13) was
created by Dzherbashyan [4] (1967), [5] (1968).
In 1915 Hardy and Riesz [1] used fractional integration for the summation of divergent
series. "Nonnal means" by Riesz, well known since then in mathematical analysis, represent
fractional integrals of a partial sum of a series (see 14.8).
The development of mathematical analysis and of function theory led to the appearance
of new fonns of fractional integra-differentiation. Weyl [1] (1917) defined fractional integration
suitable for periodic fWlctions:
00
]0I)1p '" L (ik)-OIlpl:eib,
1:= -00
211'
Ipl: = 2 f e-il:tlp(t)dt,
o
IpO = 0,
(14)
It is realized as a convolution
211'
]OI) Ip = 2- f \)%(x - t)lp(t)dt
211"
o
(15)

xxxii
BRIEF HISTORICAL OUTLINE
with a certain special fWlction \II%(x). He had shown that the fractional integrals (14)-(15) may
be written in the form
:&
01 1 f
1+1p= reOl)
Ip(t)dt
(x - t)1-0I '
00
[0I1p =  f Ip(t)dt ,
- reOl) (t - X)1-0I
:&
o < 01 < I,
(16)
-00
provided that the integrals in (16) of a periodic function Ip were Wlderstood to be conventionally
convergent ones over an infinite interval, the assumption IpO = 0 guaranteeing this convergence.
For this reason fractional integrals over infinite intervals, especially the integral 1Ip, were named
in many later papers as Weyl integrals, even in those cases when they were considered as
absolutely convergent. This fact is a historical misunderstanding. Fractional integration over
an infinite interval had first appeared in the papers of Liouville (see (4», whereas Weyl arrived
at the integrals in (16) from (14){15), with specific interpretation of these integrals for periodic
fWlctions only satisfying the condition IpO = 0
Thus fractional integrals over infinite intervals considered for arbitrary (non-periodic)
functions have no direct relation to Weyl's ideas. Therefore, though paying homage to his
profoWld ideas and to their influence on the later development of fractional calculus, we still
consider it expedient to name constlUctions such as (16) as Liouville fractional integrals, the.first
being left-hand sided and the second right-hand sided.
Weyl was also the first to prove that a fWlction lex) has a continuous derivative of order 01
if it is Lipshitzian of order .x > 01. A certain analogue of this assertion for non periodic fWlctions
was soon obtained by Montel [1] (1918). More exact theorems of fractional integra-differentiation
of Lipshitzian functions were later proved by Hardy and Littlewood in 1928 - see below.
The above mentioned paper [1] (1918) by Montel contained a number of important features.
Here an analogue of Bernstein's inequality for fractional derivatives of algebraic polynomials on a
finite interval was first given. Such an inequality in the complex domain was later established by
Sewell [1] (1935), [2] (1937). Montel obtained also a generalization of Bernstein's theorem on the
rate of approximation of differentiable fWlctions by algebraic polynomials in the case of fractional
differentiability. The beginnings of analogous assertions for fWlctions of two variables were also
contained in his paper.
In 1922-1923 Riesz [1] proved the mean value theorem for fractional integrals and an
important corollary stating that if
:&
IIp(x)1 :5 vex), f Ip(t)(x - t)OI-ldt :5 w(x), x > a,
(I
vex), w(x) being increasing fWlctions, then
:&
f Ip(t)(x - t)I1- 1 dt :5 c[v(x)]l-.6/ OI [w(x)].6/ OI , 0 < {3 < 01.
(I
(17)
This is equivalent to a statement of the property of the logaritlunic convexity of a monotonic
majorant of a fractional integral.
In 1924 an original paper by Zeilon [1] appeared in which the connection between left-hand
sided and right-hand sided fractional integration via the singular integral operator with Cauchy
kernel (x - y) -1 was obtained for the first time using a certain formalism -see pp. 9 and 7 of
the cited paper. We note that although it was written rather formally and with some errors in
formulae, this paper contained very interesting and original ideas which were not noticed either
by contemporal'ies, or by later investigators. The first of the authors of this book discovered

BRIEF HISTORICAL OUTLINE
xxxiii
the before mentioned paper by a lucky chance in 1985. The constructive connection between
left-hand sided and right-hand sided integration mentioned above was obtained in the final form
and in various situations by different mathematicians and much later in 1965-1967.
Special tribute should be paid to the paper by Marchaud [1] (1927), where a new form of
fractional differentiation
00
(D a J}(,,) = c J (!2") dt, a> 0,
o
(18)
was introduced, (U)(x) being a finite difference of order I, > a, 1 = 1,2,..., while c is a
normalizing constant. This form coincides with that of Liouville:
:&
1 d n J
(D a l)(x) = na _ n) dx n
l(t)dt
(x - t)a-n+1 '
n = [a] + 1,
(19)
-00
for sufficiently "good" functions lex). It is very important to note, however, that the construction
given by (18) has a significant advantage in comparison with (19): namely it is applicable to
functions with essentially more "bad" behaviour at infinity. In particular lex) in (18) may have
a growth at infinity of order less than a which is impossible in (19). Expressions of type (18)
are now named as Marchaud fractional derivatives. We note also that the difference form (18)
in the case 1 = I, 0 < a < 1 had appeared earlier by accident in the paper by Weyl [1] (1917)
on periodic functions, but no development of this idea itself was suggested. Marchaud gave a
thorough investigation of the constructions in (18). In particular he realized that the integral
in (18) was to be understood in general as a limit of the corresponding truncated integral over
(E, 00), the integral not necessary being absolutely convergent.
Moreover, Marchaud suggested that a more general form with generalized difference
I
L: Cil(x - kit) be considered instead of (J)(x) in (18). Here the ki are arbitrary increasing
i=O
positive numbers, the coefficients Ci being uniquely defined (up to a multiple constant) by the
I .
conditions L: cikf = 0, j = 0,1,... ,1- 1. He showed that if lex) is represented by a fractional
i=O
Liouville integral 1 = 1+ <p, <p being "good", then the fractional derivative (18) with such a
generalized difference coincides with the Liouville fractional derivative, and therefore does not
depend on the choice of ki. Marchaud also introduced partial and mixed fractional derivatives of
the type (18) for functions of twc;» variables.
It is difficult to overvalue the result of Hardy and Littlewood [3] (1928) on the mapping
property of the fractional integration operator from Lp into Lq, q-I = 1'-1 - a. This result is
known as the Hardy-Littlewood theorem with limiting exponent. It was extended by the same
authors [5] (1932) to the case of spaces Hp of functions analytic in the circle, and by Sobolev [1]
(1938) to the multidimensional case for Riesz potentials, - see (20) below - with q-I = 1'-1 - aln,
1 < l' < nla. Such theorems with limiting exponent profO\mdly influenced not only fractional
calculus, but function theory and functional analysis in general.
Hardy and Littlewood [3] (1928) also proved theorems on mapping properties of fractional
integra-differentiation in Lipshitzian spaces. Let
H = {l(x): lex) e Lip'x, a  x  b, lea) = O}.
They showed that 1:+ maps H into H+a, ,X + a < I, while 1)+ maps H, ,X > a, into H-a
The rapid development of Lebesgue integral theory naturally influenced fractional calculus.
In 1928 Tonelli [1] noted the role of absolute coptinuity in finding the Lebesgue integrable solution

xxxiv
BRIEF HISTORICAL OUTLINE
of Abel's integral equation (2). In the final form this role was revealed by Tamarkin [1] (1930)
who obtained necessary and sufficient conditions for the solvability of the Abel equation in Ll'
In 1930 the Griinwald-Letnikov approach mentioned above, was generalized by Post [2],
who suggested the constlUction a(V)f, where a(z) is analytic in Iz - 11 < 1, the case a(z) = za
corresponding to the Griinwald-Letnikov approach itself. He proved the Leibniz JUle for such a
generalized differentiation. The generalized differentiation Va log V, I,O(V) log V was later studied
by Davis [3, pp.78-85] (1936), on the basis of another approach.
In 1931 Watanabe [1] gave a proof of the following form
V::+(Jg) = t (k: fJ) (V:.;:p-. J}(V'g)
k=-oo
for Leibniz' JUle for analytic fWlctions.
In 1934 ZygmWld [1] proved the analogue of the Hardy-Littlewood mapping theorem for
p = 1 for the case of Weyl fractional integrals of periodic fWlctions, showing their mapping
property from Ll (log+ Ld l - a into Ll/(l-a)'
Two years later Riesz [2], [3] (1936) (see also [4] (1938), [6] (1949» introduced fractional
integration of fWlctions of many variables as potential type operators, commonly known since
then as Riesz potentials. One of these potentials is formally a negative fractional power (_)a/2
of the Laplace operator  = .f;.. + . . . + ..f!.r. and is realized as
oZi OZ..
Ialp = c f l,O(y)dy ,
Ix - yln-a
Rft.
(20)
where R n is an n-dimensional Euclidean space, 0 < a < n, c is a normalizing constant. Another
potential is same the power of the hyperbolic differential operator in partial derivatives and it is
introduced as
f l,O(x-y) d
c [r(y)]n-a y, a > n - 2,
K+
(21)
with Lorentz distance r(y) = v y - y - .., - ya. Here K+ = {y: y; > y + ... + y}. The
latter potential proved to be an effective method of solving the Cauchy problem for hyperbolic
differential equations in partial derivatives as was demonstrated by Riesz [6] (1949), [8] (1967
(1939» himself.
In the cited papers of Riesz (see also the paper [7]) the idea of introducing fractional
integra-differentiation by means 'of analytic continuation with respect to the parameter a fOWld
its constructive realization. fu particular he showed that the potential (21) defined for a > n - 2
or Rea > n - 2 may be analytically continued into the whole complex plane except for a finite
number of poles.
In 1938 a detailed investigation of conventionally convergent fractional integrals
N
It. 1,0 = _ ( 1 ) lim f l,O(x - t)t a - l dt
r a N-oo
o
was Wldertaken by Love [1] for fWlctions which did not necessary vanish at infinity. He introduced

BRIEF HISTORICAL OUTLINE
xxxv
the class I>., 0 < a  ,X of functions for which
z+T
f <p(t)dt  weT),
z
00
f t>'-ldw(t) < 00
1
and showed that I+, <p exists as a limit which is unifonn in x if <p E I>.. The case of almost periodic
functions was specially treated in this paper.
A fonnula for fractional integration by parts
b b
f (V+f)(X)9(X)dx = f f(X)(V b _9)(X)dx,
a a
which proved to be very useful in fractional analysis was given by Love and Young [1] (1938).
Nagy [1] (1938) proved in the periodic case a Bernstein-type theorem on the rate of
approximation of fractional integrals by trigonometric polynomials. For trigonometric sums
f(x) = L: h: eib he obtained the Favard type inequality
Iklm
IIIa) fllLoo  -;'lIfIlLoo'
m
We note that he also gave some interesting properties of the kernel \)+(x) [see (15)] which
are of importance in the theory of approximation by trigonometric polynomials.
In the paper by Bang [1] (1941), little known even to specialists, a similar inequality
IIIa) filL  c ( min I'xk I ) -a IIfllL
;;I; 00 lkN 00
N
was proved for trigonometric sums I(x) = L: akei>'kZ, 'xk =F O. Bang also proved the
k=l
Bernstein-type inequality
( ) 2 1 - a ( ) a
IIV%a fllLoo  r(2 _ a) 19;N I'xkl IIfllL oo '
0< a < 1,
for such sums. The latter inequality was independently obtained by Civin [1] (1940), [2] (1941)
in the more general situation but with a worse constant.
As is seen from the display of these important results, fractional calculus had long ago
ceased to be a subject on its own and became an inalienable part of function theory.
In 1940 the papers by Erdelyi [4] and by Erdelyi and Kober [1] appeared, where the following
modification
Z
2x- 2 (a+'1) f
r(a) (x 2 - t 2 )a- 1 t 2 '1+1<p(t)dt,
o
00
2x 2 '1 f
_ (t 2 _ x2)a-l t l-2a-2'1<p(t)dt
r(a)
Z

xxxvi
BRIEF HISTORICAL OUTLINE
of fractional integrals were introduced. These proved to be very useful in applications to
integral operators, integral and differential equations. ThesE: entities and their generalizations are
now known as ErdeIyi-Kober fractional integrals. Note however that the- ideas connected with
fractional differentiation "by the function x 2 " or "by the function ...;x" were contained long ago
in the paper by Liouville [1, p.lO] (1832).
In the papers by ErdeIyi and Kober cited above and in the paper by Kober [1] (1940)
the Mellin transfonn of fractional integrals was investigated, thus starting the approach to
fractional calculus based on the correspondence between the operator (5) and multiplication by
r(1 - a - 8)/r(1- 8). The idea of using such a correspondence with respect to Fourier transforms
goes back to Fourier [1] (1822), see (1). It was properly developed much later by Kober [3]
(1941), who was the first to put the correspondence between fractional integration I+, and the
multiplication by (ix)-a in Fourier transfo images on a rigorous basis.
Kober [2] (1941) was also the first to introduce integration of purely imaginary order, and
proved its mapping property in the space L2'
The same year C06sar [1] (1941) introduced a useful modification of Liouville fractional
differentiation in the fonn
N
- (1 ) lim .!!... J (t - x)-a f(t)dt.
r 1 - a N -00 dx
:&
This is applicable, like the Marchaud fractional derivative, to functions with "bad" behaviour at
infinity.
Having surveyed fractional calculus up to WW II we end our historical appraisal. It would
require too much space even to itemise just the important results obtained during the years since
1941. We pay tribute to investigators of recent decades by citing the names of mathematicians
who have made a valuable scientific contribution to fractional calculus development from 1941
until the present. These are M.A. Al-B&ssaln, L.S. B06anquet, P.L. Butzer, M.M. Dzherbashyan,
A. Erdelyi, T.M. Flett, Ch. Fox, S.G. Gindikin, S.L. Kalla, I.A. Kipriyanov, H. Kober, P.I. Lizorkin,
E.R. Love, A.C. McBride, M. Mikolu, S.M. Nikol'skii, K. Nishimoto, 1.1. Ogievetskii, R.O. O'Neil,
T.J. Osler, S. Owa, B. Ross, M. Saigo, tN. Sneddon, H.M. Srivastava, A.F. Timan, U. Westphal,
A. Zygmund and others.
The reader may find infonnation on investigations in fractional calculus till 1990 in the
surveying ( 4, 9, 17, 23, 29, 34, 39, 43) of the book.
The authors would like to pay tribute to the role of Prof. B.Ross: his activity of many years
and his energy have greatly promoted both the wide popularization of the fractional calculus and
the realization ofthe 1st International Conference on this theme (New Haven, 1974). The authors
also should like to mark out the fruitful role of Prof. A.C. McBride and Prof. G.F. Roach,
the organizers of the 2nd Int Conference (Glasgow, 1984) and the successful activity of Prof.
K. Nishimoto, the organizer of the splendid 3rd Conference (Tokyo, 1989), the authors being
happy to have been participants of the latter conference.

Chapter 1. Fractional Integrals
and Derivatives on an Interval
The present chapter is of fundamental importance. Proceeding from the solution of
the Abel integral equation we introduce fractional integrals and derivatives. They
were first considered by Riemann and Liouville and hence are named Riemann-
Liouville fractional integrals and derivatives. Such constructions generalize the
ideas of ordinary integration and differentiation and are one of the main forms of
one-dimensional fractional integro-differentiation in this book.
The simplest properties of Riemann-Liouville fractional integrals and
derivatives will be proved. The most important of these are (a) the mutual
inversion of the operators of fractional integration and differentiation, and (b)
the sernigroup property applicable to them, which will be proved for functions in
certain spaces. As we shall see later on these properties are valid for other forms
of one- and multidimensional fractional integra-differentiation operators. The
mapping properties of fractional integration operators are investigated in Holder
spaces H>' and in Lp-spaces and in similar weighted spaces H(p) and L,(p).
Various notions and statements known in mathematical analysis which are
repeatedly used throughout this book are discussed in the first section of the
chapter.
 1. Preliminaries
We exhibit here certain ideas and propositions in mathematical analysis necessary
for our purposes. Topics such as Holder weighted spaces H(p), the weighted
spaces L,(p) of summable functions, special functions and integral transforms are
discussed.
1.1. The spaces H>' and H>'(p)
Let n = [a, b], -00 $ a < b $ 00, so n may be a finite interval, a half-line
or the whole line. We use the standard notations such as Rl = (-00,00), and

2 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
R = [0,00) for the whole real line and half-line respectively and we denote by R 1
a real line supplemented by one infinite point. We introduce here, for use in the
future, the spaces of Holderian functions on a finite interval, a half-line and the
whole line although in this chapter they will only be required in the case of a finite
interval.
First let 0 be a finite interval. The function /(x) given on 0, is said to satisfy
the Holder condition of order A on 0 if
I/(xd - /(x2)1  Al x 1 - x21).
(1.1)
for any Xl, X2 E 0, where A is a constant and A is the Holder exponent. It is clear
that if the function /(x) satisfies the Holder condition it is continuous on O.
Definition 1.1. Let 0 be a finite interval. We denote by H). = H).(O) the space
of all functions which in general are complex valued, and satisfying the Holder
condition of a fixed order A on O.
It is simple to see that under such a definition only the case 0 < A  1 is
of interest, since if A > 1 then the space H). contains only constant functions
/(x) == const. It follows from (1.1) that /'(x) == 0 if A > 1. In this connection (see
Definition 1.6 below) we will set HO(O) = C(O).
We shall also need the space
h). = h).(O)
( 1.2)
of functions /(x) satisfying a condition stronger than (1.1), namely
/(X2) - /(X1)  0
I X 2 - xII
as X2  Xl
(1.3)
for all Xl E O. It is clear that h). c H)...
The space Hl(O) is often called a Lipschitz space. We further give the
definition of the space AC(O) of absolutely continuous functions. This space is
wider than H 1 (O).
Definition 1.2. A function /(x) is called absolutely continuous on an interval
0, if for any  > 0 there exists a 6 > 0 such that for any finite set of pairwise
n
noniTltersecting intervals [aI:, bl:] C 0, k = 1,2,.. ., n, such that L: (bl: - al:) < 6,
1:=1
n
the inequality L:.I/(bl:) - /(al:)1 <  holds. The space of these functions is denoted
k=l
by AC(O).

 1. PRELIMINARIES
3
It is known (see the books of Kolmogorov and Fomin [1, p.338] or Nikol'skii
[8, p.368-369]) that the space AC(O) coincides with the space of primitives of
Lebesgue summable functions:
:&
f(z) E AC(O) <=> f(z) = c + f cp(t)dt,
a
b
f Icp(t)ldt < 00.
a
(1.4)
Therefore absolutely continuous functions have a summable derivative f'(z) almost
everywhere. The converse of this statement, however, is not true., i.e. absolute
continuity does not follow from the existence of a summable derivative almost
everywhere; this fact will have influence in the theory of fractional integra-
differentiationj see S 2.6.
It is apparent that H1(O) C AC(O)j the inverse imbedding is not true. E.g.,
f(z) = (z - a)a E AC(O), but (z - a)a  H 1 (O) if 0 < Q < 1, since the condition
(1.1) with A = 1 does not hold at the point z = a.
Definition 1.3. Let us denote by AC"(O), where n = 1,2,... and 0 is an
interval, the space of functions f(z) which have continuous derivatives up to order
n - 1 on 0 with f(n-1)(z) E AC(O).
It is clear that AC1(O) = AC(O) and similarly to (1.4), the space AC"(O)
consists of functions representable by an n-multiple Lebesgue integral, with a
variable upper limit of a summable function, the constant in (1.4) being replaced
by a polynomial of order n - 1 (see Lemma 2.4 in S 2).
Later on a modification of the space AC(O) for the case when 0 is a line will
be indicated in S 6.3.
Now let 0 be the whole line or a half-line. In this case the definition of the
space H).(O) needs an additional specification of "Holder" behaviour at infinity.
Namely, a function f( z) is said to satisfy the Holder condition in the neighbourhood
of infinity if
1 1 1 I ).
If(zt) - f(Z2)1 ::;; A - - -
Z1 Z2
for all Z1 and Z2 with sufficiently large absolute values.
(1.5)
Definition 1.1'. Let {} be a line or a half-line. We denote by H). = H).({}) the
space of functions satisfying the Holder condition (1.1) for any finite interval of 0
and the condition (1.5) in the neighbourhood of infinity.
We note that the two conditions (1.1) and (1.5) defining the space H).(O)
for an infinite interval {} are equivalent to a single condition or a "global" Holder
condition
IZl - z21).
If(Zl) - f(Z2)1 ::;; A (1 + I Z ll».(1 + I Z 21».
which can be checked directly.
(1.6)

4 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
The following lemma states that combining two Holderian functions gives a
Holderian function again.
Lemma 1.1. Let fh = [a, c], fh = [c, b), -00  a < c < b  00, and 0 = [a, b). If
/(z) E H>'(OI) and /(z) E H>'(02) and /(c - 0) = /(c + 0), then /(z) E H>'(O).
One may find a proof of Lemma 1 in the book of Muskhelishvili [1, p.21], for
example.
We shall also need the following Holder weighted spaces.
Definition 1.4. Let p(z) be a nonnegative function. The space of functions /(z)
such that p(z)/(z) E H>'(O) is denoted by H>'(p) = H>'(Oj p).
In what follows a weight function p(z) will be a power function coupled to a
finite number of points
n
p(z) = II Iz - zkl#Jk,
k=1
where Ilk are real numbers and Zk E O. The case
(1.7)
p(z) = (z - at(b - z)f3, a  z  b,
(1.8)
will be the most prevalent. If 0 contains the point at infinity then it is expedient
to take the weight (1.7) in the form
n
p(z) = (1 + z2)#J/2 II Iz - zkl#Jk,
k=1
(1.9)
i.e. to couple it to the point z = 00. While considering the weight (1.9) we shall
use the notation
n
Ilo = -Il - L Ilk
k=1
(1.10)
the exponent of the weight at infinity. By the definition of the space H>'(p),
functions in this space are represented by the form
I(z) = ;(; , lo(z) E H'.
(1.11)
We shall need also the following subspace in H>'(p).
Definition 1.5. Let p(z) be given by (1.7) or (1.9). We denote by H(p) =
H(Oj p) a set of those functions in H>'(p), for which /O(Zk) = 0 and /0(00) = 0,

 1. PRELIMIN ARIES
5
the latter in the case when 0 contains the point at infinity, in the representation
(1.11). We denote by H a space of functions from H)., which vanish at x = a and
x = b.
We note that we shall often write H>., H).(p), H(p) instead of H).(O),
H).(p, 0), H(p, 0) in those cases when there is no chance of misunderstanding.
We shall also use the weighted spaces
h(p) = {f(x) : p(x)f(x) E h)., p(x)f(x)lz=a = p(x)f(x)lz=b = O},
(1.12)
where h). is the space (1.2), (1.3) and p(x) is the weight (1.8).
The spaces we have introduced are linear spaces easily equipped with norms.
Thus, when 0 is an interval we put
IIfIlH" = max If(x)1 + sup If(xd - f(2)1 .
zEn zl,zEn IXI - x21
ZlZ
(1.13)
The second term in (1.13) is an infimum of all possible values of a constant A in
(1.1). The space H). is complete with respect to the norm (1.13), i.e. it is a Banach
space, the proof being given for example in the book of Muskhelishvili [I, p.173].
When 0 is a whole line of a half-line the norm is introduced on the bases of (1.6)
by the relation
IIfIlH' = maxlf("')1 + sup (1 + 1"'11)'(1 + 1"'21)' 1f(I"'J) - f2)1 .
zEn zl,zEn Xl - X2
ZlZ
(1.14)
It is possible to make sure of the completeness of H).(O) with respect to the
norm (1.14), for example, by mapping the whole line (the half-line) onto the
circle (half-circle) by means of a fractional-linear transformation and using the
completeness of H).-space for any bounded curve - Muskhelishvili [1, p.173]. The
space (1.2) is also known to be a closed subspace in H). - Krein, Petunin and
Semenov [1, p.269]).
In the weighted case a norm is introduced on the basis of (1.11) by
IIfIlH"(p) = IIfoIlH'"
(1.15)
the completeness of H).(p) with respect to this norm being obvious owing to the
isometry (1.15) between the spaces H).(p) and H)..

6 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
The following useful property holds: if f(x) E H>'([a, b]) and 0 < a < A, then
g(z) = f\Z) - c) E H>-a([a,b]), a  c  b;
x-c
IIgIlH"-o  KllfIlH'"
(1.16)
where K does not depend on f(x), see, for example, Muskhelishvili [1, p.22].
Let L 1 (0) be a space of Lebesgue integrable functions on O. If 0 is a finite
interval and p(x) is the weight (1.7) the following imbeddings
C 1 (0) C H;(O;p) C L 1 (0)
( 1.17)
are valid together with the norm inequalities
KillfliL l  IIfIlH;(p)  K211flle l
( 1.18)
provided that A  !lk < A + 1, k = 1,2,..., n (see the definition of IIfliL l in
(1.26». Here and below Cm(O) denotes a space of functions which are m times
continuously differentiable on 0 with norm
m
IIflle m = max L If(k)(x)l, m = 0,1,2,. .. ,
zEO k=O
(1.19)
IIflle o == IIflle.
We denote by C(f' = C(f' (R 1 ) a space of infinitely differentiable finite functions
in R 1 .
The space introduced below is the extension of the space H>'(O) to the values
A> 1.
Definition 1.6. Let A = m + 0", where m = 0,1,2,... and 0 < 0" < 1. We say that
f(x) E H>'(O), if f(x) E cm(o) and f(m)(x) E HU(O); and
IIfIlH" = IIflle m + 1I/(m)IIHcr.
(1.20)
When A is an integer, one has often to deal with a somewhat wider space of
functions with the Holder (Lipschitz) condition containing a logarithmic multiplier
as in Theorem 3.1 and 3.2 in S 3. In this connection we give the following definition.

 1. PRELIMINARIES
7
Definition 1.7. Let A = m + u, where m = 0,1,2,..., and 0 < u < 1. We say
that f{x) E H).,k = H).,k{O), k E Ri, if f{x) E cm{O) and
If(m)(z + h) - f(m)(z)1 :S Alhl u (In II )' ,
1
Ihl < 2'
(1.21)
and
IIfllH",k = IIfllc m + sup
z,z+hEO
Ihl$1/2
If(m){x + h) - f(m){x)1
Ihl u (In rtr) k
( 1.22)
By analogy with (1.2) we introduce the space h).,k = h).,k{O) of functions
which satisfy a condition stronger than (1.21), namely
. f(m){x + h) - f(m){x)
hm k = 0,
h-O Ihlu (lnrkr)
1
x,x + hE 0, 0 < Ihl < 2'
(1.23)
Similarly we have the weighted spaces of functions H;,k(p), H).,k(p) and hJk{p)
analogous to (1.5) and (1.11) and (1.12).
Remark 1.1. Although it is very convenient to use Holder spaces in various
problems, (and they are widely applied in this book), they have one essential
shortcoming: they are not separable. There exist no "good" dense subsets in H).
and If).(p) and it is impossible to approximate the function f{x) E H). by "better"
functions in the norm of the space H)., since a closure of "good" functions in the
norm H). gives h)., but not H>., (see, for example, Krein, Petunin and Semenov
[1, p.269]). This "negative" property of the space H). will nowhere be displayed
in our considerations. However we should bear it in mind if we try to construct
approximation methods for the solution of equations studied in Chapters 6 and 7
in those cases when they are considered in H). or H).{p).
The integral Holder spaces H; will be considered in S 14. The proximity of
the values of J(xd and f(X2) will be estimated not in a uniform metric as in (1.1),
but in an integral one.
1.2. The spaces L, and L,(p)
We assume that the reader is familiar with Lebesgue measurability of functions
and the Lebesgue integral. Again let 0 = [a, b), -00 ::;; a < b ::;; 00. We denote by
L, = L,(O) the set of all Lebesgue measurable functions f(x), complex valued in
general for which f IJ(x)l"dx < 00, where 1 ::;; p < 00. We set
o

8 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
II/IIL.(O) = {[ 1/(z)l"dz } 11.
If p = 00 the space L,(O) is defined as the set of all measurable functions with
a finite norm
( 1. 24 )
II/IILoo(O) = esssup I/(x)l,
zEO
( 1. 25)
where esssup I/(x)1 is an essential maximum of the function I/(x)1 - see details in
Nikol'skii [6, p.12-13].
Everywhere below we assume that 1  p  00.
As usual two equivalent functions, i.e. differing on a set of zero measure, are
considered to be equal to one element of the space L,(O). That is, they are not
distinguished as elements of this space.
For norms (1.24) and (1.25) we shall also use the notations
11/11" = II/lIL p = 1I/IILp(o).
(1.26)
Let us give some properties of the spaces L,:
a) The Minkowsky inequality
III + gIlLp(o)  II/IILp(o) + IIgIlLp(o),
(1.27)
so that L,,(O) is a normed space. It is also known that L,(O) is a complete space;
b) The Holder inequality
f I/(x)g(x)ldx  II/IILp(O) IIgIILpl(o) I pi = p/(p - 1), (1.28)
o
where I(x) E L,(O), g(x) E L,,(O). Index pi, which is connected with p by the
relation
1 1
- + - = 1, (1.29)
p pi
is called conjugate to p. We note that (1.28) is true if 1  p  00 (pi = 00, if p = 1,
and pi = 1, ifp = (0).
From (1.28) the generalized Holder inequality
f 1/1 (x) .. . Im(x )Idx  11/1 IIL pl (0) .. . 111m IIL p ", (0), (1.30)
o

 1. PRELIMINARIES
9
m
follows where h:(z) E L,,, (0), k = 1,2,.. ., m, L: l/Pk = 1.
k=I
We see that the imbedding
L'1 (0) C L'2 (0), IIJIIL p2 (0) ::; cllJIIL pl (0), PI > P2  1,
(1.31 )
is derived from Holder inequality (1.28), 0 being a finite interval;
c) Fubini's theorem which allows us to interchange the order of integration in
repeated integrals:
Theorem 1.1. Let 0 1 = [a, b], O 2 = [c, d], -00 ::; a < b ::; 00, "';00 ::; c < d ::; 00,
and let J(z, y) be a measurable function defined on 0 1 x O 2 . If at least one of the
integrals
1 dz 1 J(z, y)dy,
0 1 O 2
1 dy 1 J(z, y)dx,
O 2 0 1
11 J(x,y)dxdy,
0 1 X02
is absolutely convergent then they coincide.
The following particular case of Fubini's theorem holds, namely
b z b b
1 dz 1 J(z, y)dy = 1 dy 1 J(z, y)dz
a a a y
(1.32)
assuming that one of these integrals is absolutely convergent. The latter relation
is called the Dirichlet formula.
The generalized Minkowsky inequality
{/ dz 1 f(z,y)dy P}"P ::;1 dy {/ If(Z,Y)IPdZ}"P , (1.33)
adjoining the Fubini theorem is also true;
d) the property of mean continuity for functions in L,.
Lemma 1.2. Let J(z) E L,,(O), 1 ::; P < 00. Then
1 IJ(z + h) - J(z)I'dz  0
o
(1.34 )

10 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
as h --+ 0, the function J(z) is continued by zero for z + h ft 0;
e) let C (0) be the space of all infinitely differentiable functions finite on O.
Finiteness on 0 means that J(z) == 0 in the neighbourhood of the end-points z = a
and z = b of the set 0 = [a, b], -00  a < b  00. The space C (0) is dense in
L,(O), 1  p < 00. In the case when 0 is a finite integral, a set of all polynomials
is dense in L,,(O), 1  p < 00;
f) the so called the Lebesgue dominated convergence theorem on passage to a
limit under the integral sign:
Theorem 1.2. Let the function J(z, h) have a summable majorant: If(z, h)1 
F(z), where F(z) does not depend on the parameter hand F(z) E L 1 (0). If
lim J(z, h) exists for almost all z, then
h-O
lim f f(z, h)dz = f lim J(z, h)dz.
h-O h-O
(1 n
(1.35)
The proof of the above properties can be found in the books by Kolmogorov and
Fomin [1], Nikol'skii [8] and N atanson [1].
We shall also need the following statement.
00
Theorem 1.3. Let K(t) E L 1 (R1) and J K(t)dt = 1. Then the averaging
-00
00 00
f K:(t)f(x - d)dt =  f KG) f(x - t)dt
-00 -00
(1.36)
of the function f(z) E L,,(R 1 ), 1  p < 00, converges to J(x) as c --+ 0 in L,(R1)_
norm. Moreover, if IK(t)1  A(ltl), wher A(r) E L1(R), {4nd monotonically
decreases then the averaging (1.36) converges to f(x) almost everywhere.
The proof of convergence in L,,-norm in this theorem is well-known and simple.
See the proof of almost everywhere convergence in the book by Stein [2, p.77].
The periodic analogue of Theorem 1.3, which we shall need in future is also
worth mentioning.
Theorem 1.3'. Let the function kE(t) satisfy conditions:
2 2
1) J kE(t)dt = 21r; 2) J IkE(t)ldt  M, where M does not depend on c;
o 0

 1. PRELIMINARIES
11
211"
3) lim J Ik£(t)ldt = 0 for any fJ > O. Then
£-0 0
211'
_ 2 1 f k£(t)cp(z - t)dt -----+ cp(z)
1r £-0
o
provided (that cp(z) E L,(0,21r), 1  p < 00 (or cp(z) E C([0,21r») in L, (or C)
nonn.
The assertion of Theorem 1.3' is well-known in the theory of Fourier series
and it is easily provided by the Minkowsky inequality
211' 211' 6
2 f k£(t)[cp(z - t) - cp(z)]dt  211cpll f Ik£(t)ldt + f Ik£(t)llIcp(z - t) - cp(z)lIdt
o 6 0
 (211cpll + M)7J -+ 0
by the choice of () in the second term and € in the first one.
Sometimes Theorems 1.3 and 1.3' are called the identity approximation
theorems.
Definition 1.8. Let p(z) be a nonnegative function. We denote by L,(p) =
L,(Ojp) the space of functions f(z), measurable on 0 for which
{ } l
II/IIL.(,) = [P(Z)I/(Z)IPdZ < 00.
We shall deal only with weights of the form (1.7) and (1.9).
The space L,(p) is a Banach space in view of the isometry
IIfIlLp(p) = IIpl/' fIlLp(O).
( 1.37)
Owing to (1.37), an analogue of the Holder inequality for weighted spaces
f If(z)g(z)ldz  IIfIlL p (p)lIgIIL p l(p 1 - p l), 1 < p < 00. (1.38)
o
follows from (1.28).

12 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
In the sequel we shall have to deal with convolution integral operators
00
h * cp = (h * cp)(x) = f h(x - t)cp(t)dt
-00
( 1. 39)
where it is clear that h * cp = cp * h. The boundedness theorem in L,-spaces known
as Young's theorem is true for them.
Theorem 1.4. If h(t) E L1(R I ), cp(t) E L,(R 1 ), then (h * cp)(x) E L,(RI),
1  p  00, and the inequality
IIh * cpll,  IIhlh IIcpll,
( 1.40)
is valid.
Let us give another theorem on the boundedness of operators with a
homogeneous kernel in L,-spaces. We recall that the function k(x, t), x > 0, t > 0,
is said to be a homogeneous function of degree a, if the equality
k(AX, At) = Aak(x, t), A > 0,
(1.41)
holds. In particular, setting A = t- I , we come to the conclusion that any
homogeneous function of degree a may be represented in the form
k(x, t) = taka (T) .
(1.42)
Theorem 1.5. Let k(x, t) be a homogeneous function of degree -1. If
00 00
K = f Ik(x, 1)lx- 1 /" dx = f Ik(l, t)lt-I/'dt < 00,
o a
( 1. 43)
then the integral operator
00
Kcp = (Kcp)(x) = f k(x, t)cp(t)dt
a
(1.44)
is bounded in L,(O,oo), 1  p < 00, and IIKCPIiL p  KllcpllL p ' where K is given by
(1.43).

 1. PRELIMINARIES
13
Proof. First of all we observe that the integrals (1.43) coincide. One may
check this by the change of variable t = l/x and taking the homogeneous
property k(l, x-l) = xk(x, 1) into account. Then representing (1.44) in the form
00
Krp = J k( 1, t) rp( tx )dt and applying the generalized Minkowsky inequality (1.33)
o
we obtain
00 { 00 } llP
IIKCPIiL. :5 ! Ik(l,t)ldt ! Icp(tz)I'dz = KllcpIlL..
.
The latter theorem is due to Hardy, Littlewood and Polya [1].
We shall also formulate the theorem on the boundedness of the convolution
operators (1.39) in L,(Rl). This theorem is known as the Fourier-multiplier
theorem - Mikhlin [2, p.239] and is proved in Nikol'skii [6, p.59] and Stein
[2, p.128]). It is formulated in terms of the Fourier transform h(x) of the kernel
h(x). As for the Fourier transform, see 1.4.
Theorem 1.6. Let the function m(x) = h(x) satisfy conditions
Im(x)1  c, Ixm'(x)1  c, x E R 1 .
(1.41')
Then the operator given by :F-1m:Frp (on a dense set) is bounded in the space
Lp(Rl), 1 < p < 00.
Corollary. Let
00
Arp = f a(x - t)rp(t)dt,
-00
00
Brp = f b(x - t)rp(t)dt.
-00
If the functions a(x)/b(x) and b(x)/a(x) satisfy the conditions (1.41'), then
A(L,) = B(Lp), 1 < p < 00.
Lastly we give the following Banach theorem.
Theorem 1. 7. Let A and B be linear bounded operators in a Banach space X. If
Arp == Brp for rp in the set which is dense in X, then Arp == Brp for all rp E X.

14 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
1.3. Some special functions
Here we give the definitions and the simplest properties of some special symbols
and functions, more detailed information may be found in the book by Erdelyi,
Magnus, Oberhettinger and Tricomi [1] and Prudnikov, Brychkov and Marichev
[1-3].
A. The Pochhammer symbol (z)n with integer n is defined by
(z)n=z(z+I)...(z+n-l), n=I,2,..., (z)o=1. (1.45)
It is clear that
(Z)n = (-It(1 - n - z)n, (l)n = n!,
( 1.46)
(Z)n = f(z + n)/f(z),
( 1.47)
where f(z) is given by (1.54). Equation (1.47) can be used for introducing the
symbol (z)n with complex n.
B. Binomial coefficients are defined by the formula
( a ) _ (-I)n(-a)n _ (_I)n-l a f(n-a)
n - n! - f(1 - a)f(n + 1) .
( 1.48)
In particular when a = m, m = 1,2,..., we have
(:) - n!(mm n)! if m  n
and (1.49)
(:) = 0 if 0  m < n.
In the case of arbitrary (complex) /3 and a, a 1= -1, -2,.. ., we set
( a ) _ f(a + 1) _ sin(/3 - a)1I" f(a + l)f(/3 - a) (1.50)
/3 - f(/3+1)f(a-/3+1) - 11" f(/3+1)'
It follows from (1.66) and (1.50) that
1(;)1  pl:a
(1.51)

 1. PRELIMINARIES
15
for any fixed Q (f; -1, -2,...) and real /3 -+ +00.
We also note the relations
_ . ( Q ) _ ( j - Q - 1 )
( 1)1 . - . ,
) )
(1.52)
t (;)(kj) = (n;p).
(1.53)
see Prudnikov, Brychkov and Marichev [1; 4.2.5.13]).
C. The gamma-function f(z). The Euler integral of the second kind
00
f(z) = f zZ- l e- z dz, Rez > 0,
o
(1.54)
is called the gamma-function. It is obviously convergent for all z E C, for which
Rez > O. Here zz-1 = e(z-l)ln&. The gamma-function is extended to the
half-plane Rez  0, z f; 0, -1, -2,.. ., by analytic continuation of this integral.
Thus, the reduction formula
f(z + 1) = zf(z), Rez > 0,
( 1. 55)
obtained from (1.54) by integration by parts, yields the equality
f(z) = f(z+n) = f(z+n) ,
(z )n z (z + 1) . .. (z + n - 1)
(1.56)
Rez>-n, n=I,2,..., zf;0,-I,-2,...,
which allows us to carry out such an analytic continuation into the half-plane
Rez > -n with any n. There exist some other methods of analytic continuation
on the basis of Cauchy-Saalschiitz, Euler-Gauss and of other formulae.
It follows from (1.56) that the function f(z) is analytic everywhere in
the complex plane except z = 0, -1, -2,...,. where it has simple poles and is
represented by the formula
f(z) = [(_1)1: /k!(z + k)][1 + O(z + k»), z -+ -k, k = 0,1,2,...
(1.57)
The latter is obtained from (1.56) as z -+ 1 - n after replacing n - 1 by k.
More exact representations for O(z + k) may be found in Marichev [10, p.42]).

16 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Here and everywhere in the book the equality j(z) = O(g(z», z -+ a, means
Ij(z)/g(z)1 < M < 00 as Iz - al < c. The relation j(z) = o(g(z», z -+ a, means
that lim i.ip = 0, and equivalence, j(z) "V g(z), Z -+ a, means that lim M = 1.
z-a z-a
The coefficient of (z + k)-I in the neighbourhood of the pole z = -k from
(1.57) is called a residue of the gamma-function and the relation between this
function and its residue is denoted by
res r(z) = (_1)1: /k!, k = 0,1,2,...
z=-I:
( 1.58)
We observe the obvious property r(n) = (n - I)!, n = 1,2,..., r(l) = 1, and
formulate some other ones:
a) the general difference equation
r(z + n) = (z)nf(z),
f(z _ n) = f(z)
(z -n)n
(_I)n
= (1 - z)n f(z);
(1.59)
b) the functional equation
f(z)f(l- z) = 1r/sinz1r, f(I/2) = Vi;
(1.60)
c) the duplication formula (Legendre formula)
2 2z -I ( 1 )
f(2z) = Vi f(z)f z + 2
(1.61 )
and more the general Gauss-Legendre multiplication theorem
mmz-I/2 m-l ( k )
f(mz) = (21r)(m-l)/2 g f z + m '
m= 2,3,...;
(1.62)
d) the asymptotic Stirling formula
r(z) = y'2;zz-1/2e- Z [1 +O(I/z)], largzl < 1r, Z -+ 00, (1.63)

 1. PRELIMINARIES
17
and its corollaries
n! = v'21rn (n/e)n[1 + O(I/n»), n -+ 00,
(1.64)
Ir(z + iy)1 = y'2;lyIZ-l/2 e -,..I,I/2[1 + O(I/y»), y -+ 00; (1.65)
e) the ratio expansion of two gamma-functions at infinity
N
r(z + a) = a-b  CI: + a-b o( -N-l )
r ( b ) z  Ii: Z Z ,
z + 1:=0 Z
Co = 1, 1 arg(z + a)1 < 1r, Izl -+ 00,
( 1.66)
where the coefficients CI: are expressed in terms of generalized Bernoulli polynomials
by the formula CI: = (-I)k-a)k B:-b+l(a), see Luke [1, p.20). Generalized Bernoulli
polynomials may be found in Gel'fond [1, Chapter 4]).
In some cases we shall also use the logarithmic derivative of the gamma-
function, called by the Euler psi-function
d r'(z)
1jJ(z) = dz In r(z) = r(z) '
( 1.67)
D. The beta-function B(z, w). The Euler integral of the first kind
1
B(z, w) = f zZ-I(1 - z)W-1dz, Rez > 0, Rew> 0,
o
(1.68)
is called the beta-function. It is connected with the gamma-function by the relation
B( ) _ r(z)r(w)
z,w - r(z+w)'
(1.69)
The integral (1.68) makes sense when Rez = 0 or Rew = 0 (z f; 0, w f; 0). In this
case it is understood to be conditionally convergent. In particular, there exists the
limit
1-£
B(z, i9) = lim f zZ-I(I_ z)i8- 1 dz, Rez > 0, 9 f; 0,
c-o
o
which coincides with the analytic continuation of B(z, w) with respect to w onto
the values Rew = 0, w f; O.
(1.70)

18 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
The integral
00
f (t - Z)a-l(t - y)/J-1dt = (Z - y)a+/J-1B(a, 1 - a - (3),
:&
(1. 71)
z > y, 0 < Rea < 1 - Re{3,
which is reduced to (1.68) by the substitution t = Y + (z - y)-l will be useful
later.
E. The Gauss hypergeometric function is defined in the unit disk as the sum
of the hypergeometric series
 (a)l:(bh zlc
2 F l(a,b;e;z) = L.J (eh k! '
1:=0
(1. 72)
Its parameters a, band c and the variable z may be complex (c f; 0, -1, -2,...)
and (a) I: is the Pochhammer symbol (1.45). The series is convergent for Izl < 1
and for Izi = 1, Re(c - a - b) > O. For other values of z the Gauss hypergeometric
function is defined as an analytic continuation of the series. One of the methods of
such a continuation is the Euler integral representation
1
2 F 1(a,b;c; z) = r(b)? _ b) f t 6 - 1 (1_ t)"-6-1(1_ zt)-"dt,
o
( 1. 73)
o < Reb < Ree, 1 arg(1 - z)1 < 11',
in which the right-hand side is defined under the indicated conditions insuring
the convergence of the integral. The condition 1 arg(1 - z)1 < 11' means that the
function is considered in the complex plane z with the cut (1, (0), joining the
singular points z = 1 and z = 00 of the Gauss hypergeometric function. It should
be noted that in (1.73) we choose the principal branch (1 - tz)-O = e-01n(1-b),
where In(1 - tz) is real for z E [0,1].
One may find a most extensive list of particular cases and properties of the
Gauss hypergeometric function in Erdelyi, Magnus, Oberhettinger and Tricomi [1]
and in Prudnikov, Brychkov and Marichev [1-3]. Here we put down only some of

 1. PRELIMINARIES
19
the simplest properties of this function:
2Fl(a,b;c;z) = 2 F l(b,a;c;z),
2Fl(a,b;b;z) = (1- z)-O,
(1.74)
(1.75)
(1.76)
2Fl(a,b;c;0) = 2Fl(0,b;c;z) = 1,
r(c)r(c - a - b)
2 F l(a,b;c; 1) = r(c _ a)r(c _ b)' Re(c - a - b) > 0, (1.77)
d k (a)k(b)t
dz k 2F1 (a,b;c;z)= (C)k 2Fl(a+k,b+k;c+k;z) (1.78)
and note that many important special functions are defined via the Gauss
hypergeometric function. Thus, the associated Legendre function P!:(z) is
represented as
1 ( Z+I ) I-'/2 ( l-Z )
P:(z) = r(l _ J.l) z _ 1 2 F l -v,1 + v; 1- J.l;  '
( 1. 79)
z  [-1,1],
1 ( l+z ) I-'/2 ( l-Z )
P:(z) = r(1 _ IJ) 1- Z 2 F l -v,1 + v; 1 - IJ;"2 '
( 1. 80)
- 1 < z < 1,
and the confluent hypergeometric function and Bessel functions defined in F and
G below are obtained from the Gauss hypergeometric function by a passage to a
limi t.
F. The confluent h,lpergeometric function or Kummer function is defined by
the formula
1 Fl (a; c; z) = f: « : » :  = 6 2Fl (G, b; c; i), Izi < 00. (1.81)
k=O
G. Bessel functions JII(z), I I1 (z), KII(z) are defined on the basis of the function
00 zk . ( z )
OF1(C; z) = L: ( c ) kk! = } lFl a;c; a '
1:=0
Izl < 00,
(1.82)

20 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
by the following formulae:
J.(z) = r(v 1) Gr of! (v+ 1;- 2 )
00 (_I)I:(z/2)21:+v
=  r(v+ k + l)k!
( 1.83)
is Bessel function of the first kind,
00 ( /2) 21:+V
I ( ) -  z _ - 7riv/2 J ( . )
v Z - L..., r ( k l)k ' - e v zz
1:=0 v + + .
(1.84)
is modified Bessel function, and
7r
Kv(z) = . [I-v(z) - Iv(z)], v f; 0, :i:l, :i:2,. ..,
2 sm V7r
( 1. 85 )
Kn(z) = lim Kv(z), n = 0, :i:l, :i:2,...
v-n
are known as Mcdonald functions. It is clear that K_v(z) = Kv(z).
Here we also note the formula
00
f pV+1Jv(pa) (a/2)1J-1
(p2 + 1)1J dp = r(Jj) K v - IJ + 1 (a),
o
( 1. 86)
a > 0, -1 < Rev < 2ReJj - 1/2,
which we will need in the future - Prudnikov, Brychkov and Marichev [2, 2.12.4.28]).
H. The generalized Riemann zeta-function (Riemann-Hurwitz function) is
defined by the series
00
(s,a) = L(a+ m)-8, Res> 1.
m=O
(1.87)
The analytic continuation of this series to other values of s is realized via the
Hurwitz result
( ) _ 2f(l- s) [ . S7r E oo cos27rma S7r E oo Sin27rma ]
s, a - (2 )1 sm 2 1 + cos 2 1 .
7r -8 m -8 m -8
m=l m=l
( 1. 88)

 1. PRELIMINARIES
21
See Whittaker and Watson [1, p.63] where the properties of this function may be
found in more detail. When s = 0, -1, -2,... the function coincides with the
Bernoulli polynomials Bn(z) up to a constant multiplier:
(-n,a) =
Bn+1(a)
n + 1 '
n = 0,1,2, . . .
(1.89)
I. The Mittag-LeJJler function is an entire function defined by the series
00 1c
Ea(z) = E r(Q: + 1) ' Q > O.
1c=O
(1. 90)
A sum of a more general series
00 1c
E a ,.8(z) = [; r(a: + ,8)' a> 0, p > 0,
(1.91)
is also called the Mittag-Leffler function. Thus, Ea(z) = Ea,1 (z). The following
result
00
1 e-ttfJ-1 E (taz)dt = 
a ,fJ 1 - z'
o
Izl < 1,
(1.92)
is known (see Erdelyi, Magnus, Oberhettinger and Tricomi [1, 18.1(26») and
this may be used to give the formula for the Laplace transform of the function
%(J-l Ea.fJ(za): namely
1 00 a-(J
e-,((J-1 Ea,fJ(a) = ; _ l ' Rep> 1,
o
(1.93)
the Laplace transform is discussed in S 1.4. In particular, if {3 = 1 we have
00
1 e-'( Ea(a) = 1 , Rep> 1.
p _ p1-a
o
(1.94)
For more details on the Mittag-Leffler function we refer to Erdelyi, Magnus,

22 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Oberhettinger and Tricomi [1, Chapter 18] and M.M. Dzhebashyan [2, Chapters 3,4].
Meijer's G-function of the order (m,n,p,q), where 0  m  q and 0  n  p,
is a function defined by the Mellin-Barnes integral
Gr;:t ( z I (a p ) ) = G;:t ( z I a1, . . . ,a p )
(b q ) b 1 ,...,b q
= 2i f
L
m n
n f( b j + s) n f( 1 - aj - s)
j=l j=l -3 d
, q Z s,
n f(aj + s) n f(1 - b j - s)
;=n+1 j=m+1
( 1. 95 )
where an infinite contour L separates all left poles s = -bj - k, j = 1,2, ..., m,
k = 0,1,2,... , of the numerator from the right ones s = 1- aj + k, j = 1,2,. .., n,
k = 0,1,2,.... Under suitable conditions it may be one of the three types,
L = L-oo, L+ oo or Lioo. In particular, it may even by a straight line
L = (/ - ioo, / + ioo). A description of the contours L;j;oo and L ioo and a more
detailed list of properties and particular cases of the G-function may be found
in Prudnikov, Brychkov and Marichev [3], see also Luke [1]. Here we give a few
relations
G mn ( z I (a,» ) = G mn ( ! 1 1 - (b q ) ) ,
,q ( b q ) qp Z 1 - (a p )
zOtG mn ( z I (a p » ) = G mn ( z I (a,) + a )
,q (b q ) ,q (b q ) + a '
(1. 96)
( 1.97)
G (z I) = e- z ,
11 ( I 0, 1/2 ) 1
G 22 X 0, 1/2 = 1("(1 - x)'
(1.98)
G10 ( I a ) = (1 - x)+-1
11 X 0 r( a) ,
Gal ( I a ) = (x - 1)+-1
11 x 0 f(a)'
( 1. 99)
where Yt. is a symbol of a truncated power function
y+ = yOt, Y > 0; y+ = 0, y < o.
( 1.100)
We also note that the functions (1.72), (1.79)-(1.85) and also (1.90) and (1.91)
with rational a and many other important special functions are particular cases of
Meijer's G-function.

 1. PRELIMINARIES
23
1.4. Integral transforms
As is known classical integral transforms are of the form
00
(K<p)(x) = f k(x, t)<p(t)dt = g(X),
-00
(1.101)
where k(x, t) is some given function (the kernel of the transform), <p(t) is the
original function given in a certain space, and g(x) is the transform of the function
<p(t). Among the most important integral transforms are the Fourier transform
with k(x,t) = e izt and the Mellin transform with k(x,t) = t z - 1 in (1.44). They
are connected with each other by changes of variables and functions (see Marichev
[10, p.31]) and have wide applications.
All other classical integral transforms may be divided into two classes: (a)
convolution transforms with homogeneous kernels of the type (1.44) and (1.42) and
(b) transforms with respect to indices or parameters of special functions included
in the kernels.
In the class of convolution-type transforms the following ones of the form
(1.44) are mostly known and we list them below. (For the definitions of the special
functions mentioned see Prudnikov, Brychkov and Marichev [3]).
The Laplace transform (k(x, t) = e- zt );
the sine and cosine Fourier transform (k(x,t) = sinxt and cosxt);
the Hankel transform (k(x,t) = VZtJII(xt»;
the Stieltjes transform (k(x, t) = r(p)(x + t)-');
the Hilbert transform which is a singular integral (k(x, t) = 1I'-t(t-x)-1 in (1.101»;
the Meijer transform (k(x,t) = VZtKII(xt»;
the transform with the Neumann function in the kernel (k(x, t) = yZtYII(xt»;
the transform with Struve function in the kernel (k(x, t) = VZtHII(xt»;
the generalized Laplace transform with the parabolic cylinder function in the kernel
(k(z, t) = 2-1I/2e- zt / 2 D II ( v'2Xt»;
the 1F1-transform (k(z,t) = IFt(a;c;-xt»;
the generalized Meijer transform (k(x, t) = (xt)"-1/2e- d W M ,,,(xt»;
the Gauss hypergeometric transform (k(x,t) = rlb) 2Ft (a,b;c;-));
the Love transform (k(", t) = (S;;r 2Fl (a, b; c; 1 - ) );
the Buschman transform (k(x, t) = (x 2 - t 2 )+>'/2 p; ());
the Narain G-transform (k(z, t) = Gn (xt I :: )); etc, etc. The main concerns
of this book such as the Riemann-Liouville integrals of fractional order, with
left-hand sided kernel k(x,t) = (x - t)+-1/r(Q) and right-hand sided kernel
k(x, t) = (t - x)+-1/r(Q) belong to the first class.
The class of transforms with respect to indices includes
the Kontorovich-Lebedev transform (k(x, t) = Kiz(t));

24 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
the Mehler-Fock transform (k(x, t) = Pi-1/2(t), t > 1; k(x, t) = 0, t < 1);
th W . t  (k( t) rtffl,n+2 (t 1 1-II+iZ,I-II-iZ,(QP) )) t t
e Imp ranslorm x, = 1.7,+2,f (b p ) ; e c, e c.
One may find the theory of integral transforms in the books by Ditkin and
Prudnikov [1], M.M. Dzherbashyan [2], Lebedev [3] and Titchmarsh [1]. Later we
shall need only the simplest information from this theory, given below.
A. The Fourier transform of a function rp(x) of a real variable -00 < x < 00,
is defined as follows
(.1'rp)(x) = .1'{rp(t);x} = cp(x)
00
= f eiztrp(t)dt.
-00
( 1.1 02)
Sometimes it is useful to write it in the form
00 .
d f elzt - 1
(.1'rp)(x) = - d . rp(t)dt.
x at
-00
( 1.103)
The inverse Fourier transform is given by the formula
(.1'-lg)(X) = g(x) = 2 g( -x)
00
=  f e-iztg(t)dt,
21r
-00
( 1.104)
which has slight differences from (1..t02). The integrals in (1.102) and (1.104)
converge absolutely for functions rp,g E L 1 (R 1 ) and in the norm of the space
L 2 (Rl) for rp, 9 E L2(RI). L 1 -theory and L 2 -theory of the Fourier integral are well
known, see Kolmogorov and Fomin [1], Nikol'skii [8] and Stein and Weiss [3].
The Fourier transform of the function rp(x) E L 1 (R 1 ) is a bounded continuous
function which tends to zero as Ixl--+ 00 by the Riemann-Lebesgue theorem. The
rate of decrease of (Frp)(x) at infinity is connected with the smoothness of the
function rp(x). This connection is given by the simple relations
.1'{D n rp(t); x} = (-ixt(.1'rp)(x),
D n (.1' rp )(x) = .1' {( itt rp(t); x},
(1.105)
( 1.106)
where D n = d"'" n = 1, 2, . . .. These equations are valid for sufficiently good

 1. PRELIMINARIES
25
functions, for example, functions, which are continuously differentiable up to order
n and such that cp(I:}(x) E L1(R 1 ), k = O,I,2,...,n.
The application of the Fourier transform to the convolution operator (1.39) is
especially to be noted. If h(x),cp(x) E L1(Rl), then (h * cp)(x) E L1(Rl) and the
result
F {(h * cp)(t); x} = (.1'h)(x)(Fcp )(x), (1.107)
called the Fourier convolution theorem is true. It holds if h(x) E L1(Rl),
cp(x) E L 2 (Rl) (then h * cp E L2(RI», or if h(x),cp(x) E L 2 (Rl) (then (h * cp)(x) is
continuous, bounded and vanishes at infinity).
B. The sine- and cosine Fourier transform of a function cp(x», x > 0, are
defined as follows
00
Fccp = (Fccp)(x) = / cp(t) cosxtdt,
o
( 1.108)
00
.1'/lCP = (.1'/lCP)(x) = / cp(t) sin xtdt,
o
(1.109)
and the inverse ones are of the form
00
- 2 /
(Fe Ig)(X) =;: g(t)cosxtdt,
o
( 1.110)
00
(F;lg)(x) =  / g(t)sinxtdt,
o
(1.111)
respectively.
C. The Mellin transform of a function cp(x), x > 0, is defined as follows
00
CP*(8) = VJt{cp(t); s} = / t/l-1cp(t)dt,
o
(1.112)
and its inverse is given by the formula
'1 +i 00
cp(x) = VJt-I{cp*(s);x} = 2i / cp*(S)X-'d8, 'Y = Res.
'Y- ioo
(1.113)

26 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
These relations are derived from (1.102) and (1.104) if we replace <p(t) by <p(e t )
and ix by 8. By replacing h( t) and x by h( e t ) and In x respectively, we obtain the
Mellin convolution relation
00
(h 0 <p)(x) = f h (T) <p(t) t .
o
(1.114)
from the Fourier convolution (1.39). The convolution theorem (1.107) with respect
to (1.114) takes the form
(h 0 <Pt(s) = h-(s)<p-(s).
( 1.115)
Substituting the expression (1.115) instead of <p-(8) in (1.113) and taking (1.114)
into account we obtain the Parseval relation
00 'Y+ioo
f h (T) <p(t) t = 2i f h-(s)<p-(s)x- 8 ds.
o 'Y-ioo
(1.116)
If we denote by +-+ the correspondence between a function and its Mellin
transform then relations of the general type
<p(ax) +-+ a- 8 <p-(s); x£w<p(x) +-+ <p-(8 + a);
rp(x') +-+ Ipl-l<p_(s/p), p 1= 0; <p(x- 1 ) +-+ <p-(-8);
(xn<p(x»(n) +-+ (1 - s)n<P-(s), x1+ t <p(k)(xd = 0,
(1.117)
k=0,1,...,n-1, Xl =0,00,
are easily proved together with the Mellin transform formulae of some important
functions
e- Z +-+ f(s), Res> 0;
(1 - x)+-l f(s)
f(a) +-+ f(s + a) ' Rea, Res> 0;
(1 - x)+-l
+-+
f(a)
f(l- a - s)
f(1 - s) ,
o < Rea < 1- Res;

 1. PRELIMINARIES 27
r(p)(1 + x)-P +-+ r(s)r(p - S), 0 < Res < Repj
1 r(s)r(l-s)
11'(1 - x) +-+ r(s + 1/2)r(I/2 _ s) = ctgS1l', 0 < Res < Ij
r(s + 11/2)
J II (2Vi) +-+ r(II/2 + 1 _ s) ' -ReIl/2 < Res < 3/4j
r(a) r(s)r(a - s)
r(c) lFl(aj Cj -x) +-+ r(c _ s) , 0 < Res < Reaj
r(a)r(b) F ( b" ) +-+ r(s)r(a-s)r(b-s)
r(C) 2 1 a, ,c,-x r(c-s)'
(1.118)
o < Res < Rea, Rebj
(1 - x)+-l r(s)r(s + c - a - b)
r(c) 2 F l(a, bj c; 1 - x) +-+ r(s + c _ a)r(s + c _ b) '
Re c > 0, Re s > 0, Re (s + c - a - b) > O.
Comparing (1.95), (1.113) and (1.118) with each other it is not difficult to observe
that if we can take a vertical line L = (1'-;00,1'+;00) as L in (1.95) without losing
convergence of the integral, then the ratio of the products of gamma-function of
general type being under the integral sign (1.95) is the Mellin transform of the
G-function. Particular cases of such a ratio are given on the right-hand sides of
(1.118), the left-hand sides being particular cases of the G-function (see (1.98) and
(1.99».
One can obtain more detailed information about properties of the Mellin
transform and tables of results in Marichev [10] and Prudnikov, Brychkov and
Marichev [3].
D. The Laplace transform of a function cp(x), 0 < x < 00, is defined as follows
00
Lcp = (Lcp)(p) = L{cp(t),p} = f e-ptcp(t)dt,
o
(1.119)
and its inverse is given by the formula
'Y+ioo
(L-1g)(x) = L-1{g(p)jx} =  f ePZg(p)dp, l' = Rep > Po.
211'1
'Y- ioo
( 1.120)
By using the Mellin transform (1.112) one may obtain another form for the inverse

28 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Laplace transform
'Y+ioo
(L- 1 g)(z) = 2i J r(: s) g'(l- s)ds, Res = ')' < 1.
'Y- ioo
(1.121)
see Marichev [10, formula (8.29)].
The integral
:&
[h * cp] = [h * cp]( x) = J h(X - t)cp(t)dt.
o
( 1.122)
is the convolution relation for the Laplace transform. The convolution theorem
(1.107) with respect to (1.122) yields the form
L[h * cp](p) = (Lh)(p)(Lcp)(p).
(1.123)
One may obtain the Laplace transform from the Fourier transform (1.102)
by restricting functions with the condition cp(t) = 0 for t < 0, and replacing the
variable ix by a complex variable p. Its properties are considered in detail, for
example, in Ditkin and Prudnikov [1].
We also point out the following formula for the Laplace transform
n-l
(Lcp(n»(p) = pn(Lcp)(p) - L pn-I:-Icp<n)(O),
1:=0
( 1.124)
which is easily proved by integration by parts provided that the corresponding
integrals exist.
 2. Riemann-Liouville Fractional
Integrals and Derivatives
The idea of fractional integration is closely connected with Abel's integral equation.
Thus it is convenient to start from the solution of this equation.
First we give its formal solution. Afterwards we prove the exact theorem on
solvability in the class of absolutely integrable functions. Having the solution of
Abel equation at our disposal we may realize constructively fractional differentiation
as an operation inverse to fractional integration. Proceeding from this idea we shall
give the corresponding definitions. This section concerns the first simple properties
of fradional integra-differentiation.

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 29
2.1. The Abel integral equation
The integral equation
:&
1 f cp( t)dt
r(a) (x - t)l-a = f(x),
o
x> 0,
(2.1)
where 0 < a < 1, is called Abel's equation. Let a > -00 and suppose that equation
i$ considered on a finite interval [a, b]. The factor l/r(a) is chosen for convenience
by reasons which will become clear below.
Equation (2.1) may be solved in the following way. Changing x to t and t
to s respectively in (2.1), multiplying both sides of the equation by (x - t)-a and
integrating we have
:& t :&
f dt f cp(s)ds r f f(t)dt
(x-t)a (t-s)l-a = (a) (x-t)a '
a a a
(2.2)
Interchanging the order of integration in the left-hand side by Dirichlet formula
(1.32) we arrive at
:&:& :&
f d f dt r f f( t)dt
cp(s) s (x _ t)a(1 _ s)l-a = (a) (x _ t)a '
a a a
The inner integral is easily evaluated after the change of variable t = s + r( x - s)
and application of the formulae (1.68), (1.69):
:& 1
f (x - t)-a(t - st-1dt = f ra-l(1- r)-adr
, 0
= B(a, 1- a) = r(a)r(1- a).
Therefore
:& :&
f 1 f f(t)dt
cp(s)ds = r(1 _ a) (x - t)a '
a a
(2.3)

30 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Hence after differentiation we have:
:J:
1 d f J(t)dt
cp(x) = f(l- a) dx (x - t)a '
a
(2.4)
So if (2.1) has a solution, this solution is necessarily given by (2.4) and
therefore it is unique.
For simplicity we have considered the case 0 < a < 1 in (2.1). The case a = 1
is evident, while the case a > 1 is reduced to the case 0 < a < 1, by differentiating
both parts in (2.1). The solution of Abel equation with a > 1 is in fact contained
in Theorem 2.4.
Quite analogously the Abel equation of the form
b
1 f cp(t)dt
f(a) (t-x)l-a =J(x),
:J:
x  b,
(2.5)
is considered and instead of (2.4) one obtains for 0 < a < 1 the following inversion
formula
b
1 d f J(t)dt
cp(x) = - f(1 _ a) dx (t - x)a '
:J:
(2.6)
2.2. On the solvability of the Abel equation in the space of
integrable functions
Let us clarify under what conditions on J(x) the Abel equation is solvable. In
order to formulate the main result of this subsection (Theorem 2.1) we introduce
notation
:J:
1 f J(t)dt
!I-a(x) = f(1 - a) (x - t)a '
a
(2.7)
It is evident that
b b
f Ift-a(z)ldz:<; r(2  <» f 1/(t)l(6 - t)l- a dt,
a a
(2.8)
so J(x) E LI(a, b) implies that ft-a(x) E L 1 (a, b) as well.

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 31
Theorem 2.1. Abel equation (2.1) with 0 < a < 1 is solvable in L 1 (a, b) il and
only if
/1-OI(X) E AC([a, b» and h-OI(a) = o.
(2.9)
These conditions being satisfied the equation has a unique solution given by (2.4).
roo/. Necessity. Let (2.1) be solvable in L 1 (a,6). Then all considerations of the
.ove subsection are true, the possibility of changing the order of integration in
(2.2) being proved with the aid of Fubini Theorem 1.1. Thus (2.3) is valid. Hence
we obtain (2.9) in view of (4.1).
Sufficiency. Since /1-OI(X) E AC([a,b]), we have If-OI(x) = d /1-OI(x) E
L1(a, b). So the function given by (2.4) exists almost everywhere and belongs to
L 1 (a, b). Let us show that it is indeed a solution of (2.1). For this purpose we
substitute it into the left-hand side of (2.1) and denote the result by g(x), i.e.
:&
1 f 1_OI(t) ( )
r(a) (x _ tP_OI dt = 9 x .
a
(2.10)
We shall show that g(x) = I(x), which proves the theorem. (2.10) is an equation
of type (2.1) with respect to If-OI(x). It is certainly solvable since it is merely a
notation. So by (2.4) we have
:&
, 1 d f g(t)dt
l1-0I(x) = r(1 - a) dx (x - t)OI .
a
Le. 1_OI(x) = g_OI(x). Functions /1-OI(X) and g1-0I(X) are absolutely continuous,
the first by assumption, the second by virtue of (2.3) with g(x) in the right-hand
side. Hence h-OI(x) - g1-0I(X) = c; note that the condition of absolute continuity
is essential in this reasoning: it cannot be weakened to continuity, since it is known
that there are continuous but not absolutely continuous functions different from
constant and having the derivative equal to zero almost everywhere. - Natanson
[1, p.201]). We have fI-OI(a) = 0 by conjecture, while g1-0I(a) = 0 because (2.10) is
:&
8 solvable equation. Hence c = 0, so f J):tt) dt = O. The latter is an equation of
a
the form (2.1). The uniqueness of its solution leads to the relation I(t) - g(t) = 0,
which completes the proof. .
The criterion of solvability for Abel's equation is given in Theorem 2.1 in
terms of the auxiliary function l1-0I(X). The following,lelIllIJ.a and corollary give a
simple sufficient solvability condition in terms of the function I(x) itself.

32 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Lemma 2.1. If f(x) E AC([a, b]), then !1-a(x) E AC([a, b)) and
I,-a(z) = r(2  n) [/(a)(z - a)'-a + i I'(t)(z - t)i-a dt ] . (2.11)
t
Proof. In view of (1.4) we may substitute the function f(t) = f(a) + J f'(s)ds
a
into (2.7) so that
:& t
f(a) 1-01 1 j dt j ,
l1-a(x) = r(2 _ a) (x - a) + r(2 _ a) (x _ t)a f (s)ds.
a a
(2.12)
The first term here is an absolutely continuous function because (x - a)l-a =
:&
(1 - a) J(t - a)-adt. Since
a
j :& dt j t f'(s)ds = j :& (j t f'(S)dS ) dt
(x - t)a (t - s)a
a a a a
(2.13)
which may be verified by direct interchange of order of integration in both parts of
the equation, the second term in (2.12) is also a primitive of a summable function
and hence it is absolutely continuous. The representation (2.11) follows from (2.12)
after the interchange of the order of integration. This completes the proof. .
Corollary. If f(x) E AC([a,b]), then Abel's equation (2.1) with 0 < a < 1 IS
solvable in L 1 ( a, b) and its solution (2.4) may herein be represented in the form
[ :& ]
1 f(a) f'(s)ds
cp(z) = r(l- n) (z - a)a + ! (z - s)a .
(2.14)
Indeed the solvability conditions (2.9) are satisfied owing to Lemma 2.1 and
(2.12) and (2.13). Since I;'(x) = d f1-a(X) we observe that (2.14) may be obtained
by differentiating (2.11), the differentiation itself under the sign of an integral being
easily proved with the aid of (2.13).
We should also like to emphasize that we have simultaneously obtained a new
form, (2.14), of Abel's integral equation inversion, which is applicable to absolutely
continuous right-hand sides f(x).

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 33
Similarly to Theorem 2.1 one may show that (2.5) is solvable in L 1 (a,b) if and
only if it-OI(X) E AC([a, b]) and it-OI(b) = 0, where
b
- 1 f f(t)dt
I1-OI(x) = r(1 _ Q) (t _ X)OI '
:&
O<Q<1.
The solution (2.6) of (2.5) with f(x) E AC([a, b]) may be written down similarly to
(2.14) as follows
[ b ]
1 f(b) f'(s)ds
I"(t) = r(1-Q) (b-tja - ! (o-t)O .
(2.15)
Let us note that in S 14 we shall give one more form of Abel equation inversion,
see (14.29) and (14.30).
2.3. Definition of fractional integrals and derivatives and their
simplest properties
For an n-fold integral there is a well known formula
:&:&:& :&
f dz: f dz:... f I"(z:)dz = (n  I)! f (z - tr'l"(t)dt (2.16)
(I a (I a
easily proved by induction. Since (n - I)! = r(n) we observe that the right-hand
side of (2.16) may have a meaning for non-integer values of n. So it is natural to
define the integration of a non-integer order as follows.
Definition 2.1. Let <p(x) E Ll(a, b). The integrals
:&
01 del 1 f <p(t)
(1(1+ <p)(x ) = r(Q) (x _ t)1-OI dt,
a
x> a,
(2.17)
b
Q del 1 f <p(t)
(lb_<p)(x) = r(Q) (t _ x)l_OI dt ,
:&
x< a,
(2.18)
where Q > 0, are called fractional integrals of the order Q. They are sometimes
called left-sided and right-sided fractional integrals respectively. The accepted names
for the integrals (2.17) and (2.18) are the Riemann-Liouville fractional integrals.

34 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Thus a fractional integral is a construction already known to us by consideration
of Abel's equation.
We shall more often use left-sided fractional integration resorting sometimes
to the designation /a(x) = (1+/)(x) as in (2.7).
Fractional integrals (2.17), (2.18) are defined for functions rp(x) E Ll(a, b),
existing almost everywhere. In Theorem 2.6 and in S 3 below we shall consider in
detail mapping properties of operators 1+, I b _ in the spaces Lp(a, b) of summable
functions as well as in the spaces of Holderian functions.
Note a simple relation
QP;+ = I b _Q, Q1b_ = I:+Q,
(2.19)
between operators 1:+ and I b _, Q being the "reflection operator': (Qrp)(x) =
rp(a + b - x).
The relation
b b
f rp(x)(I:+1/J)(x)dx = f 1/J(x)(Ib_)(x)dx
o 0
(2.20)
is valid. It is usually called the formula for fractional integration by parts - see also
(2.64). One can prove (2.20) directly by interchanging the order of integration by
Dirichlet formula (1.32) in the left-hand side for example. Equation (2.20) is true if
\rp(x) E Lp, 1jJ(x) E Lq, lip + l/q  1 + a, p  1, q  1,
with p f; 1, q f; 1 in the case lip + l/q = 1 + a. We shall prove (2.20) under these
assumptions in S 3.3.
Fractional integration has the property
I a 1 /J - l a +/J
0+ o+rp - 0+ rp,
[ a 1 /J - l a +/J
b- b-cp - b- '1',
a > 0, /3 > o.
(2.21)
Equations (2.21) are satisfied in any point for <p(t) E C([a, b» and in almost every
point for <p(t) E Ll(a,b). They are true in any point even for cp(t) E L 1 (a,b) if
Q + (3  1. The proof of (2.21) is direct
:& t
a (J - 1 f dt f '1'( r )dr
1 0 +1 0 +<p - r(a)r(,B) (x - t)l-a (t - r)l-/J '
o 0

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 35
and by interchanging the order of integration by Fubini's theorem and setting
t = r + s( x - r), we have
:&
01 fj - B(a, p) f cp( r)dr
Io+Io+cp - r(a)r(p) (x - r)l-OI-fj '
o
which gives (2.21).
The result in (2.21) is called a semigroup property of fractional integration.
We shall also consider this property for fractional differentiation in S 2.7.
As for fractional differentiation, it is natural to introduce it as an operation
inverse to fractional integration. In view of the inversion of Abel's equation (2.1)
or (2.5) which was obtained above we arrive at the following definition.
Definition 2.2. For functions f(x) given in the interval [a, b), each of the
expressIons
:&
01 1 d f f(t)dt
('Do+f)(x) = r(l- a) dx (x - t)OI '
o
(2.22)
('Db_f)(x) =
b
1 d f f(t)dt
r(l- a) dx (t - X)OI'
:&
(2.23 )
is called a fractional derivative of order a, 0 < Q < 1, left-handed and right-handed
respectively.
Fractional derivatives (2.22) and (2.23) are usually named Riemann-Liouville
derivatives.
Note that we have defined fractional integrals for any a > 0, while fractional
derivatives are for the moment introduced only for orders 0 < a <: 1. Before
passing to the case a  1 we give a simple and sufficient condition for the existence
of fractional derivatives.
Lemma 2.2. Let /(x) E AC([a,b]), then 'D:+f and 'Db_f exist almost everywhere
for 0 < a < 1. Moreover 'D:+/, 'D6_/ E L,(a, 6), 1  r < l/a, and
[ :& ]
01 I /(a) /'(t)dt
'0.+/ = r(l- 0) (0: - a)" + ! (0: - t)" '
(2.24)

36 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
[ b ]
va f = 1 f(b) _ f'(t)dt .
6- r(l- n) (b - Z)a ! (t _ z)a
(2.25)
This lemma's assertion follows from the Corollary of Lemma 2.1, the condition
V<;+f, Vb_f E Lr(a, b), 1  r < I/Ot, being derived directly from (2.24) and (2.25).
Later in S 14 we shall give results like (2.24) and (2.25) for fractional derivatives
in the case when f'(t) is not necessarily integrable in the point t = a or t = b
respectively, - (14.29) and (14.30).
Note also that in SS 13.2 and 13.3 we shall prove other sufficient conditions
for the existence of fractional derivatives which are more useful in applications,
allowing a function f(x) to have integrable singularities in particular. Let us
indicate in this connection an example of the function f(x) = (x -a)-IJ, 0 < fl < 1,
for which (V:+f)(x) is defined. The direct evaluation with the aid of properties of
beta and gamma functions leads to the Euler formula
( Va f )( ) _ r(1 - fl) 1
a+ X - r(1 _ fl - Ot) (x - a)IJ+a '
(2.26 )
in particular
('D::+f)(z) == 0 if I(z) = (z _ )l-a
(2.27)
The fractional derivative (2.26) is an integrable function if fl + Ot < 1. This
situation will be typical US 13.2, 13.3) in the sense that a function f(x) with an
integrable singularity has an integrable fractional derivative V<;+f, if the order of
the singularity of f(x) is less than 1 - Ot.
Remark 2.1. Assertion (2.27) means that the function (x - a)a-l plays the same
role for the fractional derivative 1)<;+f as a constant does in usual differentiation.
We pass now to fractional derivatives of large orders Ot  1. We shall use
standard notations: [Ot) meaning the integral part of a number Ot and {Ot} meaning
its "fractional" part, 0  {Ot} < 1, so that
Ot = [Ot) + {a}.
(2.28 )
If Ot is an integer, the derivative of order Ot is understood in the sense of usual
differentiation:
V::+ = ( d: r. Vr_ = (-  r, n = 1,2,3....
(2.29)

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 37
If a is not an integer, it is natural to introduce V<;+I, V b _ I by the relations
( d ) [a] ( d ) [a]+l
va I  _ V{a} 1 = _ 1 1 -{a} 1
a+ dx a+ dx a+'
(2.30)
( d ) [a] ( d ) [a]+l
va 1 ':2 __ V{a} 1 = __ 1 1 -{a} 1
b- dx b- dx b-'
(2.31 )
So
:&
-na _ 1 ( d ) n f j(t)dt )
va+1 - r(n _ a) dx (x _ t)a-n+1 ' n = [a + 1,
a
(2.32)
b
-na _ (_I)n (  ) n f I(t)dt )
vb_1 - r(n _ a) dx (t _ x)a-n+1 ' n = [a + 1.
:&
(2.33 )
We shall also use the notations
V:+I = I;; I = (1:+)-1 I, a  0,
(2.34)
assuming that each of them implies the derivative (2.22), (2.32), the definitions
V b _ I = Ib"_a I being interpreted similarly. We shall sometimes use the notation
( d )a I(x) = (vg+/)(x) instead of vg+, see S 42.
A sufficient condition for the existence of derivatives (2.22), (2.23) is
:&
f I(t)dt E AC[a]([a, b».
(x - t){a}
a
where Ac[a]([a, b» is a class introduced by Definition 1.3. This condition holds if
I(x) E Ac[a]([a, b».
It is not difficult to verify that (2.26) is true for any a > 0 and similarly for
(2.27) we have
('D:+/)(x) == 0 if I(x) = (x - a)a-A:, k = 1,2,...,1 + [a). (2.35)

38 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
2.4. Fractional integrals and derivatives of complex order
The operations of fractional integration 1:+, lb_ and of fractional differentiation
'D:+, 'Dr_ introduced in S 2.3 for real a > 0 may be easily made meaningful for
any complex value of a with Rea > 0, the case Rea = 0 being discussed below in
particular. For this purpose it is sufficient to elucidate a choice of the branch of
the multivalued power function T a - 1 , a E C. Let us assume that
T a = TaO[cos(8InT) + isin(8 In T)], a = ao + i8, T> O.
(2.36)
everywhere below. Then the statement of Lemma 2.1, and the results in (2.24),
(2.25) (with 0 < Rea < 1), (2.26), (2.27), (2.34) remain true if we replace [a] by
[Rea] in the definitions (2.30)-(2.33).
It is clear that integrals (or derivatives) of a complex order a (Rea f; 0)
represent an analytic continuation - in the parameter a - of fractional integrals (or
derivatives respectively) originally defined for Ima = O.
In the case of purely imaginary order the fractional derivatives defined similarly
to (2.22) by the formula
:e
i8 1 d f i8
'Da+1 = r(l- i8) dx (x - t) I(t)dt,
a
(2.37)
make sense. One cannot however use (2.17) for the definition of fractional integrals
of purely imaginary order because of the divergence of the integral for a = i8. So
it has become accepted to define fractional integrals of purely imaginary order as
li8 1 - .!!. 11+i8 1 Therefore
a+ - d:e a+ .
:e
'8 d.;! 1 d f i8
I:. + 1 - r(1 + i8) dx (x - t) I(t)dt,
a
(2.38)
b
'8 de! 1 d f '8
Ib_1 = r(1 + i8) dx (t - x)' I(t)dt,
:e
(2.39 )
In order to complete the definition of fractional integro-differentiation for all
a E C, it remains to introduce the identity operator .
ode! 0
'Da+cP= la+CP = cP
(2.40 )
for Q = 0 which is in evident agreement with (2.38).

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 39
As was to be expected, there is no essential difference between integrals and
derivatives of purely imaginary order unlike the case Rea f; 0, cf. (2.37) and
(2.38). Operators I and 'D8+ by their nature are closer to singular operators, but
not to integration or differentiation, so that the name "operations of integration
and differentiation" is merely a conditional one for them.
Lemma 2.3. If f(x) E AC([a, b», then 'D'+f exists for all x and it may be
represented in the form (2.24) with a = i8.
The prool of Lemma 2.3 is quite similar to those of Lemma 2.1 and of its
Corollary.
The condition f(x) E AC([a, b]) is redundant for existence of integrals
(derivatives) of purely imaginary order, see S 4.2 (note 2.10). Below, in Lemma
8.2, we shall see that 'D'+f may be well defined for functions f E Lp, P > 1, and
that the operator 'D,+ is bounded in the space Lp, P > 1.
In Theorem 2.2 below we give sufficient conditions for the existence of
fractional derivatives of an arbitrary complex order a, Rea  0, the simpler cases
o < a < 1 and Rea = 0 being considered apart in Lemmas 2.2 and 2.3 below.
Since the theorem will be stated in terms of the class Acn (see Definition 1.3), we
give first the characterization of this class.
Lemma 2.4. The space Acn([a, b» consists of those and only those functions
f(x), which are represented in the form
1 j :& n-l
f(x) = (n _ I)! (x - t)n-lcp(t)dt + L CI:(x - a)l:.
a 1:=0
(2.41 )
where cp(t) E L 1 (a, b), Ck being arbitrary constants.
The proof of lemma follows immediately from the definition of the space
ACn([a, b» and from (1.4) and (2.16).
Note that in (2.41) we have
cp(t) = I(n)(t), Ck = I(k)(a)/k!.
(2.42 )
Theorem 2.2. Let Rea  0 and f(x) E ACn([a, b)), n = [Rea] + 1. Then'D:+f
exists almost everywhere and may be represented in the form
n-l f(II) ( a ) L_ 1 j :& I(n) ( t ) dt
'D a - x - a iii a
0+1 -  r(1 + k - a) ( ) + r(n - a) (x - t)a-n+l .
11_0 0
(2.43)

40 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Proof. Since J(x) E AC", we have the representation (2.41). Substituting it into
(2.32) and taking (2.42) into account we obtain (2.43) after simple transformations.
It is not difficult to see that (2.20), (2.21) remain valid for complex values of
0:,{3 if Reo: > 0, Rep > 0, (and l/p+l/q < I+Reo: for (2.20». The same concerns
Theorem 2.1, Lemma 2.1 and the Corollary of the latter with 0 < Reo: < 1.
Note at last the validity of the following lemma.
Lemma 2.5. Let rp(t) E L 1 (a,b). The homogeneous Abel integral equation
1<;+rp = 0 has only the trivial solution rp(x) == 0 (almost everywhere) Jor any 0: with
Reo: > O.
Proof. Let us denote m = [Reo:]. First let Reo: f; 1,2,.... Differentiating m
times the equality 1:+ = 0 we have p;+mrp = O. Here 0 < Re(o: - m) < 1 so
then rp == 0 in view of Theorem 2.1 the validity of which for complex exponent has
already been noted. If 0: = m - i8, we differentiate m - 1 times the result 1:+-rp = 0
:&
and arrive at J(x - t)-i8rp(t)dt = O. The case '8 = 0 is clear. If 8 1= 0, then,
a :&-£ t
operating similarly to (2.2), we have J (:&_:;1+ 0 ' J(t - s)-i8rp(s)ds = 0, c > O.
a a
Interchanging the order of integration by Fubini's theorem with the subsequent
:&-E 1- r.
change t = s+(x-s) of variable we obtain J rp(s)ds J i8(l_)-i8-1d = O.
a 0
Since rp( s) ELl, the passage to the limit is possible under the first integral sign
if the inner integral converges as c --+ O. The latter is true by virtue of (1.70).
:&
So letting c --+ 0, we have that B(1 - i8, i8) J rp(s)ds = O. Hence rp(s) == 0 almost
everywhere which completes the proof. . a
2.5. Fractional integrals of some elementary functions
In the following formulae we suppose that a E C and 1:+rp = V;;rp in the case
Reo: < O.
1. For power functions rp(x) = (x - a),6-1, rp(x) = (b - x).8- 1 , Re{3 > 0, we
have respectively
l a r({3) ( "" _ a) a+,6-1 E C
a+rp = r(o: + {3) .., , 0: ,
l a In = r({3) ( b _ X ) a+,6-1 C
b- T r( 0: + {3) , 0: E ,
\
(2.44)
(2.45 )
These formulae adjoin (2.26) and may be proved by direct evaluation.
2. In the more general case rp(x) = (x - a),6-1(b- x)'1-1 Gauss hypergeometric

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 41
function (1.72) appears:
1 01 - (b ) "(-1 r({3) ( ) 01+.6-1 F (1 IJ. IJ. x - a )
o+<p- -a r(Q+{3) x-a 2 1 -"Y,,.,,Q+,.,, b-a '
a < x < b,
(2.46 )
where Re{3 > 0 and "Y is arbitrary. One may obtain (2.46) by simple transformations
of the Euler representation (1.73). A similar formula may be also written down for
a function <p(x) = (x - a)p-l(x - Cp-l, where c < a:
1:+'1' = (a - C)'-I r(() fJ) (z - a)o+II- 1 2 F 1 (1 - 'Y. fJ; <> + fJ; - : =: ) .
c < a < x,
(2.47)
with Re{3 > 0 and arbitrary "Y. Note also useful particular cases of (2.46), (2.47),
obtained with the aid of (1.75):
01 [ (x - a)P-l ] _ 1 r({3) (x - a)0I+.6- 1
10+ (b-X)OI+.6 - (b-a) OI r(Q+{3) (b-x).6 '
a < x < b,
(2.48 )
[ (x-a).6- 1 ] 1 r({3) (x_a)0I+.6- 1
[01 _
0+ (X-C)OI+.6 - (a-c)OIr(Q+{3) (x-c).6 '
c < a < x,
(2.49 )
3. Let <p(x) = (x - a).6- 1 I n (x - a), Re{3 > O. Then
1:+[(z-a)II-lln(z-a») = r(() fJ) (z-at+II-I[t/>(fJ)-t/>(fJ+<»+ln(z-a»). (2.50)
where ,p(z) is the Euler psi-function (1.67). Indeed after the change t = a + s(x - a)
of variable in the integral
:&
1 01 =  f (t - a)P- 1 In(t - a) dt
o+<p r(Q) (x - t)I-OI
o
in the case ReQ > 0 we obtain I:+<p = [r(Q»)-I(x - a)0I+.6- 1 [Cl + c2ln(x - a»),

42 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
where
1
f sP-lln s
Cl = (1- s)I-OI ds ,
o
1
C2 = f (1- s)OI-l sP- 1 ds.
o
Here C2 = B(a, p) while Cl is evaluated by differentiating the equality C2 = B(a, p)
with respect to p.
4. For the function <p(x) = COB ";X - a/";x - a we have
1 01 l/'J - 201-1/2 /:i ( x _ a ) (201-1)/4J ( . 
a+r - Y"1t 01-1/2 V iI.i - uj,
Rea  0,
(2.51 )
where JII(z) is the Bessel function (1.83). One may obtain (2.51) by expanding
COB x in a Taylor series. The relation (2.51) is the Poisson formula, in fact, which
is known in the theory of Bessel functions and has the form
1
J (z) = (Z/2)" f eizt(1 - t 2 )"-1/2dt
II r(v + 1/2)..fi .
-1
(2.52)
The equations (2.51), (2.52) may be obtained from each other by simple changes
of variables.
5. By similar expansion in the series one may derive the results
01 01-1 r.= ( x_a ) 0I-I/2 A(x-a) ( A(x-a) )
la+[(x-a) cOBA(x-a») = y1f -y cos 2 J OI - 1 / 2 2 '
Rea > O.
(2.53)
1:+[(x-a)IJ/2J IJ (v;=a)) = 2 01 (x-a)(IJ+0I)/2J IJ + 0I ( ";x - a) , a E C, ReJJ>-1.
(2.54)
In the case JJ = -1/2 (2.54) leads to (2.51).
We do not dwell here on the fractional integra-differentiation of the exponential
function e'Yz and of the trigonometric functions. This is connected with the fact
that the Riemann form of fractional integration used for a f; -00 (or b f; +(0) does
not give the natural result of the type 100(e'YZ) = "(-OIe'Y x . We may obtain such a
formula if instead of 1:+ we use Liouville form of fractional integra-differentiation
which corresponds to the case a = -00 (or b = +(0). So we shall consider fractional
integration of exponential and trigonometric functions in S 5.1 below. See also
S 9.3, where there are brief tables of fractional integrals of various elementary and
special functions with information about other such tables and about the methods
of evaluation of such integrals.

fi 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 43
2.6. Fractional integration and differentiation as reciprocal
operations
It is well known that ordinary differentiation and integration are reciprocal
:&
operations if the latter is applied first, i.e. (d/dx) J cp(t)dt = cp(x). However
a
:&
f V"(t)dt f; V'(x) in general case because of the appearance of the constant -V'(a).
G
In the same way (d/dx)n 1:+ V' == V', but 1::+V'(n) f; V', differing from V' by a
polynomial of the order n - 1. Similarly we shall always have V:+I:+V' == V', but
If+V:+V' does not necessarily coincide with cp(x) because the functions (x - a)a-I:,
i, = 1,2,..., [Re a] + 1, may arise, the linear combinations of which play the role
of polynomials for fractional differentiation, see (2.35). Theorem 2.4 which is
proved below elucidates the matter. It is convenient to consider the following as a
preliminary.
Definition 2.3. Let 1:+(L,), Rea> 0, denote the space of functions f(x),
represented by the left-sided fractional integral of order a of a summable function:
1= 1:+V', V' E L,(a, b), 1 :5 p < 00.
The characterization of the space 1:+(L 1 ) is given by the following theorem
generalizing Theorem 2.1.
Theorem 2.3. In order that f(x) E 1:+(L 1 ), Rea > 0, it as necessary and
sufficient that
fn-a(x) ':21::; a f E ACn([a, b]),
where n = [Re a] + 1 and that
(2.55 )
(I:) ( )
fn-a a = 0,
k = 0, 1,2,...,n- 1.
(2.56)
Proof. Necessity. Let f = 1:+cp, V' E L 1 (a, b). In view of the semigroup property
(2.21) we have r;:;a f = 1:+V', V' E L 1 (a, b), and the conditions (2.55), (2.56) follow
then from Lemma 2.4.
Sufficiency. Conditions (2.55), (2.56) being satisfied we can represent fn-a(x),
according to Lemma 2.4 as fn-a(x) = 1:+cp, where V' E L 1 (a, b). Consequently
I:+ a / = 1:+V' = I::; a 1:+ V' owing to the semi group property (2.21). Hence
1:+ a (f - 1:+V') = O. Since Re(n - a) > 0 we have f - 1:+V' = 0 by lemma 2.5
which completes the proof. .
Let us emphasize in connection with definition 2.3 that the representability
of a function f(x) by a fractional integral of order a and the existence of a
fractional derivative of this order for f( x) are different things. Thus the function

44 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
I(x) = (x - a)a-1, 0 < Rea < 1, already familiar to us has a fractional derivative
which is equal to zero - (2.35). However the function (x - a)a-1 may not be
represented by a fractional integral of order a for any summable functions. The
reason for this is that h-a(a) f; 0, in the case of this function, so the conditions
(2.56) are not satisfied. The reader acquainted with distribution theory will
understand that the function (x - a)a-1 may be the fractional integral of order a
only of a distribution, namely the Dirac delta-function 6(x - a) - see S 8.1.
Let us focus on the idea itself of the existence of a fractional derivative. For
simplicity let 0 < Rea < 1. If we say that V<;+I = (djdx)I+a 1 exists almost
everywhere, then we must take into account the following. It is known that the
existence of a summable derivative g'(x) of a function g(x) does not yet guarantee
:&
the restoration of g(x) by the primitive, i.e. J g'(t)dt 1= g(x) + c in general - see
a
for example Natanson's book [1, p.199]. Moreover, there exists (ibid., p.201) such
a monotone continuous function g(x) t const that g'(x) = 0 almost everywhere.
This has already been mentioned in the proof of Theorem 2.1. These "exotic"
effects are removed if we deal with absolutely continuous functions. We remind the
reader that integration by parts in the Lebesgue integral is in general possible only
for absolutely continuous functions, the latter being already employed in the proof
of Theorem 2.3.
For these reasons it is clear that the supposition " the fractional derivative
V<;+I exists almost everywhere and is summable" is insufficient for developing a
satisfactory theory, i.e. it is insufficient for a function I(x) to be represented by a
fractional integral of order a. So it is necessary to make this supposition stronger.
For this purpose we give the following
Definition 2.4. Let Rea> O. A function I(x) E L1(a, b) "is said to have a
summable fractional derivative V+I, if 1:+ a 1 E ACn([a, b]), n = [Rea] + 1.
In other words this definition uses an idea employing only the first of two
conditions (2.55), (2.56) characterizing the space I<;+(Ld.
Remark 2.2. If V+I = (djdx)n 1:+ a 1 exists in the usual sense, i.e. 1:+ a 1 is
differentiable n times at every point, then, evidently, I(x) has a derivative in the
sense of Definition 2.4.
We have found necessary to dwell in detail on the above reasons and to give,
in particular, Definition 2.4 because confusing the two ideas - existence of the
fractional derivative and representability of a function by the fractional integral -
and sometimes the careless interpretation of the first of these notions, has caused
errors in the papers of many authors.
The following theorem which is the main one in this subsection deals with the
question mentioned in its heading.
Theorem 2.4. Let Re a > O. Then the equality
V+ 1<;+ cp = cp( x )
(2.57)

S 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 45
U valid for any summable function <p(x) while
1:+V:+f = f(x)
(2.58)
u satisfied for
f(x) E 1:+(L 1 ).
(2.59)
If we assume that instead of (2.59) a function f(x) E Ll(a,b) has a summable
'rltnvative V:+f (in the sense of Definition e.4), then (2.58) is not true in general
and is to be replaced by the result
n-l ( ) a-t-l
l a -na f - f( ) "" x - a -1(n-k-l) ( )
a+va+ - x - L.J r(a _ k) In-a a ,
k=O
(2.60)
where n = [Rea] + 1 and fn-a(x) = I::; a f. In particular we have
l a -na f = f( ) _ h-a(a) ( _ ) a-l
a+va+ x r(a) x a .
(2.61)
for 0 < Re a < 1.
Proof. We have
z t
va la - 1 tf1 f dt f <p(s)ds
0+ a+<P - r(a)r(n _ a) dx n (x - t)a-n+l (t _ s)l-a'
a a
Interchanging the order of integration and evaluating the inner integral we
obtain that
z
V::+I::+'P = f(ln) dn f 'P(,)(z - .t-1d..
a
(2.62)
Then (2.57) follows from (2.62) by (2.16).
Furthermore (2.58) with the assumption (2.59) immediately follows from
(2.57). We also note that (2.58) was in fact obtained in the sufficiency part of the
proof of Theorem 2.3. It remains for us to prove (2.60). For this purpose we repeat
arguments that were used in the sufficiency part of the proof of Theorem 2.3, the
difference being only in the fact that the terms outside the integral do not now
disappear and give the additional sum in (2.60). .

46 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Corollary 1. The following analogue of Taylor's theorem
n-l (1)Ot+j f)( )
f(x) = L a+. a (x - a)Ot+j + Rn(x), Rea > 0,
j=-n r(a + J + 1)
(2.63)
is valid, where Rn(x) = (1:: n 1)::n f)(x) and f(x) is assumed to have a summable
derivative 1):t n f in the sense of Definition e.4.
Indeed, (2.63) is an evident reformulation of the property (2.60).
Let us also note that some generalization of the formula (2.60) is given in S 4.2
(note 2.8).
Corollary 2. The formula
& &
f f(x)(1):+g)(x)dx = f g(x)(1)b_f)(x)dx, ° < Rea < 1, (2.64)
a a
is valid under the assumption that f(x) E If_(Lp), g(x) E 1:+(L q ), p-l + q-l $
1 +a.
Indeed (2.64) follows from (2.20) if we denote 1J b _f = I;'(x), 1):+g = 1jJ(x) and
take (2.58) into account.
A simple sufficiency condition for functions f(x), g(x) to satisfy (2.64) is that
f(x),g(x) E C([a,b]) and that (1J:+g)(x), (1J b _f)(x) exist at every point x E [a,b]
and are continuous. Below, in S 14 we shall give less restrictive sufficient conditions
- see the corollary of Theorem 14.4.
2.7. Composition formulae. Connection with semigroups of
operators
In the following theorem we find it convenient to use the unified notation of (2.34)
both for fractional integrals and fractional derivatives assuming that 1:+ = 1J;;':
for Rea < 0.
Theorem 2.5. The relation
l Ot I /J - I Ot+/J
a+ a+1;' - a+ I;'
(2.65)
is valid in any of the following cases:
1) Re{3 > 0, Re(a + (3) > 0, I;'(x) E L 1 (a,b);
2) Re{3 < 0, Rea > 0, I;'(x) E 1;:(L 1 );
3) Rea < 0, Re(a + (3) < 0, I;'(x) E 1;;-/J(Ld,
the cases a = 0, {3 = ° and a + {3 = ° being also admiSsible for real a and {3.

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 47
Proof. 1) In the case Rea > 0, Re,8 > 0 the semigroup property (2.65) is already
established in (2.21). Let us consider the case Rea = 0, Re,8 > 0, letting a = iO.
Then
x x
i9 f3 - 1 d f f i9 f3-1
Ia+la+cp- r(,8)r(1 +iO) dx cp(s)ds (x-t) (t-s) dt
a 8
x
_ B(I + iO,,8) d f ( )( _ ) i9+f3d _ li9+f3+1
- r(,8)r(1 + iO) dx cp s X S S - dx a+ cpo
a
(2.66)
Since Re(iO + ,8 + 1) = 1 + Re,8 > 1 and (2.65) is already proved in the case
x
Rea > 0, Re,8 > 0, we have 1+f3+1cp = 1!+1+f3 cp = J(I:+f3 cp)(t)dt, so (2.65)
a
with a = iO follows from (2.66).
It remains in case 1) to consider the situation when Rea < O. We have
1 01 I f3 -n -Oi l -OI+f3+OI -n -Oi l -OI l Ol+f3
a+ a+ cp = v a + a+ cp = V a + a+ a+ cp.
(2.67)
This last is valid in view of (2.65) because Re( -a) > 0 and Re(a +,8) > O.
Applying then (2.57), we obtain (2.65) from (2.67) in the case Rea < 0 as well.
2) In the case Re,8 < 0, Rea > 0 we have by the assumption that cp = r;':1/J,
where ,p ELI, so l::t f3 cp = 1:: f3 1;':,p. Since Re(a +,8+ (-,8» > 0, we obtain
from here according to case 1) that 1::f3 cp = 1:+ 1/J = 1:+ V;: cp = l:+l:+cp.
3) In the remaining case we have by assumption that cp = 1;;-f31/J, 1/J E L1,
and hence l:+I:+cp = l:+I:+l;;-f31/J = l:+r;;;1/J according to case 1). So
I:+I:+cp = V;': 1;;;1/J which leads to (2.65) by (2.57)
. It remains to note that the cases a = 0, ,8 = 0 are trivial, while the case
a + ,8 = 0 coincides with (2.57) or (2.58), which completes the proof. .
Remark 2.3. We have not covered the following cases in Theorem 2.5: 1) Re,8 = 0,
Rea > 0; 2) Re(a +,8) = 0, Re,8 > 0 but the case a +,8 = 0, Re,8 > 0 is contained
in Theorem 2.4; 3) Rea = 0, Re,8 < O. It may be shown that (2.65) is true in these
cases if we construct the class of admissible functions by the following conditions
espectively: 1) there exists a summable derivative V;: cp of purely imaginary
order; 2) there exists a summable derivative V;;-f3 cp of purely imaginary order;
3) there exist a summable derivatives V;: cp and V;;-f3 cp (in the sense of Definition
2.4).
Remark 2.4. In the case 2) of Theorem 2.5 (2.65) is not valid unless the condition
cp E 1;;: (LI) is saisfied. If instead of this condition we require only that a function
cp(x) has a summable fractional derivative (in the sense of Definition 2.4), then

48 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
(2.65) is replaced by the relation
n-l (n-k-l) ( )
101 I fJ _ 1 00 + fJ _  'Pn+fJ a ( _ ) OI-k-l
a+ a+'P - a+ 'P L..J r ( Ol _ k ) x a .
k=O
(2.68)
where n = [-Re,8] + 1 and 'Pn+fJ(x) = 1::t fJ 'P' Relation (2.68) is derived from
(2.60) with the aid of (2.65).
The property (2.65) of fractional integrals and derivatives is called a semigroup
property. This is connected with a notion of a semigroup of operators. We give the
corresponding definition, but for simplicity we let the parameter Ol be real.
Definition 2.5. A one-parameter family of linear bounded operators TOI' Ol  0, in
a Banach space X is said to be a semigroup if
TOITfJ = TOI+fJ, Ol  0, {3  0,
(2.69)
TOI;? = I;? , I;? E X.
(2.70)
A semigroup is called strongly continuous if
lim IITOI'P - TOIo'Plix = 0, 0  Olo < 00,
01-010
(2.71 )
for any 'P EX. It is called continuous in uniform (operator) topology if the limit
(2.71) ezists in the operator topology, i. e. lim IITOI - T 01 0 II = 0 when Ol -+ Olo.
In view of (2.69) it is easy to see that a semigroup strongly continuous for
Ol = 0 is inevitably strongly continuous for all Ol  O.
Theorem 2.6. Operators of fractional integration form a semigroup in L,(a, b),
p  1, which is continuous in uniform topology for all Ol > 0 and strongly continuous
for all Ol  O.
Proof. First let us note that fractional integration operators are bounded 10
L,(a, b), Le. the following estimates
(b - a)ReOI
III:+I;?IILp(a,b)  ReOllr(Ol) I III;?IILp(a,b) , ReOl > 0,
(b - a)ReOI
IIl b -'PIIL p (a,b)  ReOllr(Ol), III;?IILp(a,b)' ReOl > 0,
(2.72)
(2.73 )

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 49
hold. They may be verified by simple operations using the generalized Minkowski's
inequality. The property (2.69), that is the relation (2.65) is proved in Theorem
2.5. It remains to clear up the character of continuity of the semigroup. Let
ao > O. We have
:&
1010 01 _ [ 1 1 ] f ep(t)dt
a+ep - la+ep - r(ao) - r(a) (x - t)l-Olo
a
:&
+  f [(x - t)OIO-l - (x - t)OI-l]ep(t)dt
r(a)
a
=Aep + Bep.
Let us estimate IIAepIlL p , IIBepIlL p ' In view of (2.72)
I r(ao) I (b - a)OIo
IIAepIlLp 1- r(a) r(l+ao) lIepIlL p '
(2.74)
Letting ep(x) to be zero outside [a, b), we obtain
6-a
1 f 11 - t Ol - Oio I
IBepl  r(a) t l - Olo lep(x - t)ldt,
o
Consequently, owing to Minkowski's inequality (1.33)
6-a ( 6 ) 1/,
1 1 t Ol - OiO
II BcpIiL.  r(,,) I I l-a. I dt ! Icp( z - t) I" dz
(2.75)
6-a
1 f 11 - t Ol - OiO I
 r(a) tl-Olo dtllepllL p '
o
Gathering estimates (2.74), (2.75) we obtain
6-a
11(1:+ - I:.t)epll < 1 1 _ r(ao) I (b - a)OIo +  f 11 - t Ol - OiO I dt.
lIepll - r(a) r(1 + ao) r(a) t l - Olo
o
We note that we may pass to the limit as a -+ ao in the right hand side integral.

50 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Taking also into account that r( a) is continuous for a > 0 and f( a) 1= 0 we arrive
at the result lim 111:+ - I:+, II = O.
a-ao
Let ao = O. Let us show that
lim III:+cp - CPIiL p = O.
a-O
(2.76)
We have
x-a
1';+", - '" = r(I",) J t a - 1 ",(", - t)dt - ",(",)
o
x-a
= a J cp( x - t) - cp( x) dt (x) [ (x - a) a _I ]
f(l + a) t 1 - a + cp f(l + a)
o
=Ucp+Vcp,
so IIl:+cp - CPIiL p  IIUCPIiL p + IIVCPIlL p ' It is clear that
b
IIV ",lit::; J 1",(",)1" I I-+a - 11" d",.
a
The passage to the limit under the integral sign here is possible by the Lebesgue
dominated convergence Theorem 1.2. So lim IIV CPIiL p = O. In order to estimate
a-O+
U cp further, we approximate a function cp( x) by a polynomial P( x) in the L,-space
norm (see the property f) of the space Lp in S 1.2). Then
IIUcpllL p  IIU(cp - P)IIL p + IIU PIILp'
(2.77)
We apply the generalized Minkowski's inequality to the first term, cp(x) being
continued as usual by zero outside [a, b], and obtain
2(b - a)a
IIU(cp':'- P)IIL p  f(l + a) IIcp - PIiLp < constc.
For the second term in (2.77) we have

 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES 51
b-a
IU PI  r(l: a) j to max IP'(t)lclt -;;-:0' 0,
o
which completes the estimates and the proof. .
The continuity of the semigroup 1:+ at the point a = 0 may be formulated
not only in terms of L,-space convergence, but in terms of almost everywhere
convergence as well. Consider the following
Definition 2.6. A point Xo is said to be a Lebesgue point of a function
<p(x) E Ll(a, b) if
t
lim ! j [<p(xo - s) - <p(xo)]ds = O.
t-O t
o
(2.78)
It is known that almost all points Xo E [a, b) are Lebesgue points of a function
<p(x) E Ll(a,b) - see for example Zygmund's book [2, p.ll1).
Theorem 2.7. Let <p(x) E Ll(a, b). Then
lim(I:+<p)(x) = <p(x),
0'-0
(2.79 )
for any Lebesgue point of a function <p(x) and consequently almost everywhere on
[a, b).
Prool. Let Xo be a Lebesgue point of <p(x). Let us denote
Zo t
(t) = j <p(s)ds = j <p(xo - s)ds.
zo-t 0
(2.80)
We have
t
(t) 1 j
- - <p(xo) = - [<p(xo - s) - <p(xo»)ds  0
t t
o
by (2.78). Therefore (t) = t[<p(xo) + b(t»), where b(t) is a bounded function such
that Ib(t)1 < € as soon as 0 < t < 6 = 6(€). We have

52 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
o-a zo-a
I:+'P = r(la) ] to-1'P(ZO - t)dt = r(l a ) ] to-l
o 0
o-a
(ZO - a) 1 (t) I 1 - a ] (t)dt
- r(a)(zo - a)l-a - r(a) ti-a t=O + r(a) t 2 - a
o
zo-a
1 - a ] a-i (ZO - a)
= r(a) t b(t)dt + r(a)(zo _ a)l-a
o
zo-a 6
1 - a ( ) ] a-l d 1 - a ( ) ] a-i b( )d
+ r( a) <p Zo t t + r( a) <p Zo t t t.
o 0
Hence
a (zo - a) [ 1 - a a ]
(Ia+<P)(zO) - <p(zo) = r(a)(zo _ a)l-a + <p(zo) ar(a) (zo - a) - 1
6 zo-a
I-a ] 1 I-a ] 1
+ r(a) t a - b(t)dt + r(a) t a - b(t)dt.
o 0
One may pass to the limit under the last integral sign. So
lim I(I:+<p)(zo) - <p(zo)1 1<p(zo)1 lim [ r;l- a ) (ZO - a)O -1 ]
a-O+ a-O+ + a
6
+ lim l r - ( a ) ] ta-lb(t)dt
a-O+ a
o
I . 1 - a a
:::; 1m r(1 ) u € = €.
a-O+ + a
Since € is arbitrary we obtain (2.79) from here. Theorem 2.7 is proved. .
Remark 2.5. The convergence of the operator 1:+ to the identity as a -+ +0 may
be characterized more precisely:
(l:+'P)(Z) = 'P(z) + a [-r'(I)'P(Z) + :z i In(z - t)'P(t)dt] + 0(01),

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 53
where 0(0:)/0: -+ 0 for almost all x. This result is easily obtained if we compare
the quotient [1:+1;' - 1;']/0: with its limit as 0: -+ +0, the latter being evaluated by
L'Hospital rule. See Hille and Phillips' book [1, p.677] about such a limit which is
known as an infinitesimal generator in the theory of operator semigroups.
fi 3. The Fractional Integrals of Holder and
Summable Functions
We consider the mapping properties of fractional integration operators in Holder
spaces and in the spaces of summable functions. The theorems of this section show
that fractional integration not only keeps invariant, but also essentially improves
the properties of functions. The results are formulated and proved for the fractional
integrals 1:+1;'. One may reformulate them for the fractional integrals 11:_ I;' by
virtue of (2.19).
It must be emphasized that the results of this section given in the SS 3.1-3.2
on the mapping properties of fractional integration operators from H into H;+a,
or from H(p) into H;+a(p) in the weighted case, will be extended in S 13.6 to the
Holder spaces H, when the Holder condition is of the form
II;'(x + h) - l;'(x)1  cw(h)
where w(h) is a given continuous nondecreasing function, w(O) = O.
Here and everywhere below c, Cl, C2,'" will denote absolute constants which
do not depend on the variables x, h, etc. Such different constants may be denoted
by the same let ter.
Everywhere in this section [a, b) is a finite interval except in S 3.4 where the
case of the half-axis -00 < a < b = 00 is admitted in .Theorem 3.7 and Lemmas
3.2 and 3.3.
3.1. Mapping properties in the space H>'
Theorems 3.1-3.4 below show that generally speaking fractional integration of order
0: improves the order A of the Holder property by 0:. The case when A + 0: is an
integer is a special one. It leads to the space H>.,l - see Definition 1.7. We begin
with the main case 0  A  1, 0 < 0: < 1.
Theorem 3.1. Let I;'(x) E H>'([a, b», 0  A  1, 0 < 0: < 1. Then the fractional
integral 1:+1;' has the form
I+'P = r() 0) (z - a)a + .p(z),
(3.1)

54 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
where 1/J(x) E H>'+Ot if A + a 1= 1 and 1/J(x) E H>'+Ot,l if A + a.= 1; moreover the
estimate
11/J(x) I  A(x - a)>'+Ot
(3.2)
holds.
Proof. Representing Pi+cp as
x x
lOt cp = cp(a) f (x - t)Ot-idt +  f cp(t) - cp(a) dt
a+ r(a) r(a) (x-t)1-Ot
a a
we obtain (3.1) where
x
1/J(x) =  f cp(t) - cp(a) dt.
r(a) (x - t)1-Ot
a
It is clear that the inequality
x
11/J(x)1  IIcpIlH"r-i(a) f(t - a)>'(x - t)Ot-idt
a
is true. Changing the variable t = a + s(x - a) we deduce (3.2).
We prove that 1/J(x) E H>'+Ot or 1/J(x) E H>'+Ot,l. The cas A + a  1 is to be
considered first. For brevity we set g(x) = <p(x) - cp(a), so that
Ig(x)1  A(x - a)>'.
(3.3)
Let h > 0; X,x + hE [a,b]. We have
x-a x-a
1/J ( x h ) -1/J ( x) = ( f g(x - t)dt - f g(x - t) dt )
+ r(a) (t + h)1-Ot t i - Ot
-h 0
o
_ g(x) [(x-a+ht-(x-a)Ot]+ f g(x-t)-g(x) dt
r(1 + a) r(a) (t + h)1-Ot
-h
x-a
+ r(l",) f [(t + h)o-l - to-1]fg(z - t) - g(z)]dt
o
=h + J 2 + J 3 .
(3.4)

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 55
If h  z - a, then according to (3.3) we find
A
IJ11  r(1 + a) (z - a)>'I(z - a + hYJt - (z - a)al  ch>.+a.
If 0 < h < z - a, then by (3.3) and the inequality (1 + t)a - 1  at, t > 0, we have
the estimate
IJd r(I",) (.,-a)>+aI(I+ .,a r -11
 ch(z - a)>.+a-l  ch>.+a.
Then
o >.
I I A f it I >.+a
J2  r(a) (t + h)l-a  ch .
-h
Finally we estimate J 3 :
z-a
IJ I <  f t>'[ta-l - (t + h)a-l]dt
3 - r(a)
o
r-.
-r-
Ah>.+a f
= - t>'[t a - l - (t + It- l ]dt.
r(a) .
o
(3.5)
Hence the estimate IJ 3 1  ch>.+a, A + a  1, holds if z - a  h. If z - a > hand
A + a < 1 we have IJ 3 1  ch>.+a also, in view of the convergence of the integral
(3.5) at infinity, since
Ila-' - (I + W-'I = ,a-I [1- (1 +  r- ' ]  ct a - 2
if t > 1. If A + a = 1 then (3.5) yields the estimate
IJ.I  Ah+a(c+ j,+a-2dl)
1
z-a 1
 Cl h + C2 h In   ch In h

56 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
provided that 0 < h < 1/2. Collecting the estimates for J 1 , J 2 , J 3 we complete the
proof of the theorem in the case  + a  1.
We find it more convenient to give the proof of the remaining case  + a > 1 in
another place after introducing and investigating the so-called Marchaud fractional
derivatives. So we refer the reader for the proof of the case  + a > 1 to S 13.4 -
see the text following Lemma 13.1. .
The case of an arbitrary a > 0 and  > 0 will be considered in Theorem 3.2
below. As a preliminary we give some corollaries from Theorem 3.1.
Corollary 1. The operator
:e
 f cp(t) - cp(a) dt 0 < a < 1,
r(a) (x - t)l-a '
a
is bounded from H)., 0    1, into H).+a if  + a 1= 1 and into H).+a,l if
+a=1.
Corollary 2. The operator 1:+ is bounded from C = H O into H a .
It is easy to see, indeed, that 1:+ is bounded from Loo into H a :
:e
r(a)lf(x + h) - f(x)1  sup { f [(X - t)a-l - (x + h - t)a-l]dt
a<:e<b
- - a
:e+t
+ f (z + h - tt-ll1cpIlL } :5 challcpIIL,
:e
where f(x) = I:+cp and h > O.
Theorem 3.2. Let cp(x) E H).,   O. Then the fractional integral I:+cp, a > 0,
has the following form
a _  cp(l:) (a) a+k
Ia+cp - L.J r(a + k + 1) (x - a) + 1/J(x),
k=O
(3.6)
where m is a maximal integer such that m < , and

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 57
H).+OI
.p(z) E { H+o.l
if A + Q is not an integer or if A
and Q are integers
if A + Q is an integer but A and
Q are not integers.
Theorem 3.2 is deduced from Theorem 3.1 if we take into account that the
function
.p(z) = f(lOt) j [1"(1) - t I"('!(a) (I - a)'] (z - I)o-ldt
a k_O
has a derivative of order m + [Q):
Z
,p(m+[OI])(z) - 1 j [<p(m)(t) _ <p(m)(a)](t _ a){OI}-ldt
- r( { Q } ) .
a
3.2. Mapping properties in the space H(p)
We recall that <p(z) E H6(p) implies p(z)<p(z) E H). and p(z)<p(z)lz=zlr = 0 in
all points Zk to which the weight p(z) given by (1.7) is "fixed". We begin the
investigation with the main cases p(z) = (z - a)#J or p(z) = (b - Z)II.
Remark 3.1. Everywhere below when considering fractional integration 1:+ in
the space H6(p) we assume that p(z)<p(z)lz=a = 0 is independent of the fact
whether the weight p(z) is fixed to the point z = a or not.
Theorem 3.3. Let 0 < A < 1, A + Q < 1. The operator 1:+ is bounded from
H6(p) into H+OI(p) if p(z) = (z - a)#J, p. < A + 1, or p(z) = (b - Z)II, v> A + Q.
Proof. We emphasize that p. may be negative and v have no upper bound.
1. The case p(z) = (z - a)#J, JJ < A + 1. Let <p(z) E H6(p), so that
<p(z) = (z - a)-#Jg(z), where g(z) E H>., g(a) = O. We are to prove that
G( z) = j ( :: ) #J (Z::CII E H+OI and IIGIIHA+CII  cllgllHA. We represent G( z) as
a
Z Z
j g(t)dt j (z - a)#J - (t - a)#J
G(z) = (z _ t)1-OI + (t _ a)#J(z _ t)1-OI g(t)dt
a a
= G1(z) + G2(Z).

58 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Here G 1 (x) E H;+OI by Theorem 3.1. For G 2 (x) we have
z+h
f (x + h - a)IJ - (t - a)IJ
G 2 (x + h) - G 2 (x) = ( ) ( h )1- g(t)dt
t-aIJx+ -t 01
z
z
+ [(x + h - a)IJ - (x - a)IJ) f ( ) r<')d )1-
t-a IJ x+ -t 01
a
z
f ( X - a ) IJ - ( t - a ) IJ
+ [(x + h - t)OI-l - (x - t)OI-l]g(t)dt
(t - a)IJ
a
= J 1 + J2 + J 3 .
(3.7)
Further we shall need the inequalities
Ix IJ - yIJI $ c(x - y)x IJ - 1 , X  Y > 0, fl  0,
Ix IJ - yIJ I :5 Ifll(x - y)yIJ- 1 , X  Y > 0, fl :5 1,
(3.8)
(3.9)
the constant c being independent on x and y. The inequalities (3.8) and (3.9) are
proved, for example, by means of the mean value theore.
We estimate J 1 . Using the inequality Ig(t)1 :5l1gIlH,,(t - a». and (3.9) we find
z+h
f (x + h - t)OI dt
IJ 1 1 :5 cllgllH" (t - a)1-).
z
x+h
f (x + h - t)OIdt ).+01
:5 cllgllH" (t _ x)1-). = ctllgllH" h
z
when fl :5 1. As for the case fl  1 we use (3.8) to estimate J 1 :
z+h
I J I ( h - ) IJ-l f (x + h - t)OIdt
1 <cx+ a ( ) 
- t - a IJ-"
z
:S (z +:':: a)'-P T (I - ,>P- '
z

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 59
Hence according to (3.8) we derive the estimate
IJ 1 1  cha(x + h - a)IJ-1[(x + h - a».+l- IJ - (x - a».+l- IJ ]
 ch a + 1 (x + h - a».-l  ch).+a.
(3.10)
In the case when x - a  h we have
z). z+h).
I J I hIJ f (t - a) -IJdt h IJ f (t - a) -IJdt
2  c (x + h _ t)l-a  C (x-+ h - t)l-a
a a
 chIJ(x + h - a».+a- IJ  ch).+a
if JJ > 0 and
z
f (t - a».-IJdt
I J 21  c(x - a)IJ (x + h _ t)l-a
a
z
f ( t - a » .-IJdt
< C ( x - a ) IJ < C ( x - a » .+a < ch).+a
- (x - t)l-a - -
a
if p < O. As for the case x - a > h we have
z
I J I ch f (t - a».-IJdt ch h ). + a
< - < c
2 - (x - a)l-IJ (x - t)1-a - (x - a)l-).-a -
a
if JJ  1 and
z+h
I J I ch f (t - a».-IJdt = ch < ch).+a
2  (h + x - a)l-IJ (x - t)l-a (h + x - a)l-).-a -
a
if JJ > 1 according to (3.9) and (3.8) respectively.
Finally, changing the variable t = a + s(x - a) we obtain
1
IJ.I  IIgIlHA(Z - a)+a f I'-" - 'II (1 - . + Z  a r-' - (1- B)a-'Id',
o

60 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
If x - a $ h, then IJ 3 1 $ cllgIlH"h>.+a and if x - a > h then
1
f I >.-# >'1
IJ 3 1 $ cllgllH"(x - a)>.+a-lh (1- s)2a ds $ cllgIlH"h>.+a.
o
Collecting the estimates for J 1, J 2 , J 3 we have
IG 2 (x + h) - G2(x)1 $ cllgIlH"h>.+a.
Using this estimate and the inequality IGdx)1 $ cllgllH"(x - a)>.+a we deduce that
G(x) E H;+a.
2. The case p(x) = (b - X)", II > A + a. Now I;'(x) = (b - X)-II g(x) where
g(x) E H>' and g(a) = g(b) = 0 in accordance with Remark 3.1. We are to prove
that
:e
G ( x ) = f ( b - x ) II g(t)dt H>.+a
b - t (x - t)l-a E ,
IIGIIH"+O $ cllgllH"
a
:e
and that G(a) = G(b) = O. Since IG(x)1 $ c f(t - a)>'(x - t)a-1dt as x --+ a, then
a
the condition G( a) = 0 becomes obvious. If x --+ b, then
b-a
IG(x)1 $ (b - X)" f t>'-II{t + x - b)a-1dt.
b-:e
Hence after the change of variable t = (b - x) we have
b-.
1>=r
IG(x)1 $ (b - x)>.+a f t>'-II{t - 1)a- 1 dt
1
00
( b ) >.+a f dt
$ - x t"->'(t _ 1)l-a
1
and so G(b) = O. To prove the Holder property of the function G(x) we represent

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 61
it in the form
:z: :z:
f g(t)dt f (b - z)" - (b - t)"
G(z) = (z _ t)l-a + (b _ t),,(z _ t)l-a g(t)dt
a a
= Gl(z) + G2(Z).
Here Gl(z) E H>.+a and IIGIIIHA+cr  cllgllHA by Theorem 3.1. Assuming that
z + h E (a, b) for G 2 (z) we have:
G 2 (z + h) - G2(Z) = J l + J 2 + J 3 ,
where
:z:+h
f (b-z-h)"-(b-t)"
J l = (b _ t),,(z + h _ 1)l-a g(t)dt,
:z:
:z:
J - [( b - h - Z ) " - ( b - Z ) " ] f g(t)dt
2- (b-t)"(z+h-t)l-a'
a
:z:
J = f (b - z)" - (b - t)" [(z + h _ t)a-l _ ( z _ t ) a-l ] g ( t ) dt.
3 (b _ t)"
a
Using the estimate Ig(t)1  IIgIlH" (b - t)>' and (3.8) we obtain
:z:+h
IJll  c f (z + h - t)a(b - t)>'-ldt
:z:
h
= c f a(b-z- h+)>'-l
o
from whence after the change of variable  = (b - z - h)s it follows that
h/(b-:z:-h)
IJII  c(b - z - h)>.+a f sa(1 + s)>'-lds.
o

62 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
If b - x - h  h, then
h/(b-z-h)
IJ11c(b-"-h)+a[1+ J s+a-lds] ch+a.
1
But if b - x - h  h, then
h/(b-z-h)
IJd  c(b - x - h)>'+Ol J sOlds ::; ch>'+Ol.
o
For J2, applying (3.8) again we have
z
J (b-t)>'-lldt
I J 21  ch(b - x)"-1 (x + h _ t)l-Ol
a
z
h(b - ) 1I-1 J (b-t)>'-lldt
<c x ()l '
- X - t -Ol
a
00
Now we change the variable x-t = (b-x). Then IJ21 ::; ch(b-x)>'+Ol-l J Ol-l(l+
o
)>'-II  since 1/ > A + Q. Hence, I J 21 ::; ch>'+Ol.
It remains to estimate J 3 . By substituting t = b - s(b - x) we have
,-.
t=r 1
IJ.I  c J 118_I (b - ")*1 (s - 1 + b ., r- - (s - W-lldsliuIlH"
1
Applying (3.9) we obtain
b-.
t=r
1 I ( >'+Ol h J 11 - s"lds >.+Ol
IJ 3  cllg IH" b - x) b _ x S"->'(S _ 1)2-Ol  ch
1
since b - x  h and A + Q < 1. The theorem is thus proved. .
A similar statement for the weight p( x) = (x - a)#J (b - X)" is easily derived
from Theorem 3.3. Namely, the following theorem holds.

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 63
Theorem 3.3'. Let 0 < A < 1 and A + a < 1. The operator 1:+ is bounded from
H(p) into H;+a(p) with p(x) = (x - a)#J(b - xt, p. < A + 1, II > A + a.
Proof. Let rp(x) E H(p). We choose any point c E (a, b) and introduce the
functions
( ) _ { rp( x ),
rp(J x -
rp( c) ,
x :5 c,
x  c,
{ 0,
rpb(X) =
rp( x) - rp( c) ,
x:5 c,
x  c,
so that rp(x) = rp(J(x) + rpb(X). Here rp(J(x) E H(p(J) and rpb(X) E H(Pb) where for
brevity we have denoted
p(J(X) = (x - a)#J, Pb(X) = (b - xt.
Let us check, for example, that rp(J(x) E H(p(J)' We have
( ) ( ) _ { g(X)/Pb(X),
P(J X rp(J X -
p(J(x)rp(c),
x:5 c,
x  c,
where g(x) = p(x)rp(x) E H>'([a, c]), g(a) = O. Since the functions [Pb(X)]-l
and p(J(x)rp(c) are infinitely differentiable on [a,c] and [c,b], respectively, then
g(x)/ Pb(X) E H>'([a, c]) and p(J(x) E H>'([c, b)). Then p(J(x )rp«x) as a continuous
"sewing" of functions from H>'([a, c)) and H>'([c, b)) also belongs to H>'([a, b]).
It is also clear from the above arguments that IIrp(JIIH"(PG) :5 cllrpIlH"(p) and
IIrpbIlH"(Pb) :5 cllrpIlH"(p)' By Theorem 3.3 we have
III:+rp(JIIH"+O(p.) :5 cllrp(JIIH"(PG) :5 cllrpIIH"(p)
and
III:+rpbIlH"+O(PG) :5 CllrpbIlH"(pl» :5 cllrpIlH"(p)'
Noting that (I:+rpb)(x) == 0 if x :5 c we find that II 1:+ rpb IIH"+'" (p) :5
cIIII:+rpbIlH"+"'('b)' Taking also into account that Pb(X) E H>.+a and II> A + a we
obtain III:+rp(JIIH"+o(p) :5 c2I1 I :+rp(JIIH"+o(p.)' As a result we have
III:+rpIlH"+O(p) :5 cIlI:+rp(JIIH"+o(PG) + cIlI\rpbIIH"+O'(p/» :5 cllrpIIH"(p)
which was required. This completes the proof of the theorem. .
Lastly we extend Theorem 3.3 and Theorem 3.3' to the case of the general
weight (1.7). For this purpose we have first to prove the lemma providing the

64 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
assertion analogous to Theorem 3.3 for integrals with the kernel (x + t)a-l, x > 0,
t > O. This lemma will naturally hold under weaker conditions.
Lemma 3.1. Let <p(x), 0 < x ::; I, admit the estimate 1<p(x)1 ::; kx-'Y, Q < 'Y < 1.
Then
I
/( x ) = 1 <p(t)dt E Ha+/J ([ O n. x'Y+/J )
(t+x)1-a ,IJ,
o
(3.11)
for any {3  0 such that Q + (3 ::; 1, and 1I/IIH"'+JJ(z'Y+JJ)  ck, c being independent
on <p( x ).
Proof. We have to prove that
I
(x) = x'Y+/J 1 <p(t)(t + xt-1dt E Ha+/J([O, ij).
o
Assuming that I = 1 and h > 0 we have
1
I(x + h) - (x)1 ::;kl(x + h)'Y+/J - x'Y+.81 l(x + h + tt-1t-'Ydt
o
1
+ kx'Y+/J 1 t-'Y[(t + x)a-l - (t + x + h)a-l]dt
o
=k(l + 2)'
The inequality (3.8) and the change of variable t = (x + h)s give the estimate
l/(z+h)
l  c1h(x + h)a+/J- 1 1 s-'Y(1 + s)a-1ds
o
 ch(x + h)a+/J- 1  cha+/J.
Using (3.9) we have

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 65
1
2 $ X-r+/J h f t--r(t + x)a- 2 dt
o
l/z
= CX a +,8-1 h f (t + l)a- 2 t--r dt
o
and we obtain the estimate 2 $ cha+/J if x 2: h. If x $ h, then
1
2 $ 2x-r+,8 f t--r(t + x)a- 1 dt
o
l/z
= 2x a +,8 f s--r(s + l)a- 1 ds $ cha+/J.
o
Collecting the estimates for 1 and 2 we complete the proof of the lemma. .
Finally, the following theorem concerns the case of a general weight of the
form
n
p(x) = II Ix - XI:I#Jk, a = Xl < X2 < ... < X n $ b.
k=l
(3.12)
Theorem 3.4. Let p(x) be the weight (3.12), A + a < 1 and the following
conditions
1) PI < A + 1;
!) A + a < Pk < A + 1, Ie = 2, .. . ,n - 1;
9) A + a < Pn < A + 1 if X n < b and A + a < Pn if X n = b hold. Then the
operator 1:+ is bounded from H6(p) into H;+a(p).
The proof of this theorem is in fact a consequence of Theorem 3.3, Theorem 3.3'
. { n - 1 if X n = h,
and Lemma 3.1. We find it convement to denote n = Let
n if X n < b.
( ) { ep( x ),
epic x =
0,
Xk $ x $ Xk+1,
X  [Xk, XI:+1],
Ie = 1,2, . . . ,n,
n
80 that ep( x) = E epk (x). It is clear that
1:=1
1) (I:+epk)(x) == 0 if x < Xk (Ie  2),

66 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
:&
2) (I:+cpl:)(x) = rla) J cp(t)(x - t)a- 1 dt if XI: < X < XI:+1 (1  k  ii),
:&Ic
:&Ic+l
3) (I:+cpl:)(x) = rla) J cp(t)(x _t)a- 1 dt if XI:+1 < X < XI:+2 (1  k  ii -1),
:&Ic
4) (I:+cpl:)(x) is an infinitely differentiable function if X  XI:+2'
The factor Ix - XI: IIJIc being infinitely differentiable beyond the points XI:, we
easily obtain that I:+cpl: E H+a(p) on [a, b]. Here we use Theorem 3.3' with
k = 1,..., n - 1 and Theorem 3.3 with k = nand X n < bin 2), and Lemma 3.1 -
after the change of variable t = XI:+1 -  in 3) - in 3).
Remark 3.2. The condition A + a < Ill:, k = 2,..., n, in Theorem 3.4 may not be
weakened. If A + a  Ill:, Theorem 3.4 is wrong. Let, for example, p(x) = (b - x)IJ,
Il  A + a. It is easy to check for the function cp(x) = (x - a)>'(b - x)>,-IJ E H$(p)
that
:&
IJ ( a ) _ (b - x)IJ f (t - a)>'dt ( >.+a )
(b - X) Ia+cp (X) - r(a) (b _ t)IJ->'(x _ t)l-a 1= 0 (b - x)
a
as X --+ b so that I:+cp  H;+a(p).
3.3. Mapping properties in the space L1'
Fractional integrals are known to keep, at least, the space L,(a,b) invariant, see
(2.72) and (2.73). The statements revealing how much the fractional integral I:+cp
is "better" than the function cp(x) E L1' are more important. The situation here is
such that the fractional integral belongs to Lq with q > p if 0 < a < lip, and it
even proves to be a continuous (Holderian) function which belongs to Ha-l/, or
H a - 1 /"I/ 1" , see Definitions 1.6, 1.7, if a > l/p. The following theorem is known
as the Hardy-Littlewood theorem with limiting exponent.
Theorem 3.5. If 0 < a < 1, 1 < p < l/a, then the fractional integration operator
1:+ is bounded from L, into Lq with q = p/(1 - ap).
The proof of this theorem requires more refined methods than those we have
used until now. The presentation of the mathematical technique necessary for
the proof would take us far aside. Thus we shall give only the proof of a
simpler assertion, namely 1:+ is bounded from L1" 1  P < 1/ a, into Lr' where
1  r < q = p/(1 - ap). We refer the reader interested in the proof of the case
r = q, and also in consideration of the cases p = 1 and p = l/a not contained in
Theorem 3.5 to the bibliographical notes in SS 4 and 9. See also S 4.2 (notes 3.2,
3.3). In view of (1.31) it is sufficient to take r > p. We set c = (l/r - l/q)/2. Then

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 67
we have
:&
r(a) 1 I.:'+cp 1  J (lcp(t)I(z - t)'- ) Icp(t)I'-(Z - t)'- dt.
a
Using the generalized Holder inequality (1.30) with n = 3, Pl = r, P2 = rpl(r - p)
and P3 = pi we obtain
:& 
r(a)II.:'+cpl  (J Icp(t)I'(z - t)r<-ldt) ·
a
:& -:& 
X (J Icp(t)I'dt) ' · (J(Z - W'-ldt) '
a a
:& 
cllcpll: (J Icp(t)I'(z - t)'<-ldt) '.
a
Hence
b b 1
III.:'+cpIiL.  cllcpll: (J Icp(t)I'dt J Iz - W-1dZ) ..
a a
 cllcpll  IICPIII p = cllcpll,.
.
Corollary. Relation (2.20) concerning fractional integration by parts holds if
cp(x) E L" 1/J(x) E Lq, lip + 1/q  1 + a, but p 1= 1, q 1= 1 in the case
lip + 1/q = 1 + a.
In fact, we consider the case lip + l/q = 1 + a taking the imbedding (1.31)
into account. By Theorem 3.5, integrals on the left- and right-hand sides of (2.20)
are absolutely convergent - apply the Holder inequality. So the interchange of the
order of integration leading to (2.20) is justified by Fubini's theorem.
Theorem 3.6. If a > 0, P > l/a, then the fractional integration operator
1:+ is bounded from L,(a, b) into HOt-l!P(a, b) if a - lip 1= 1,2,..., and into
HOt-l!p,I!,' (a, b) if a - lip = 1,2,..., and
(I.:'+cp)(z) = o(Z - at-; ) as z --+ a.
(3.13)

68 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Proof. We obtain (3.13) using the Holder inequality
z: .1 z: 
II:+cpl  r(} (J Icp(t}I'dt) · (J (z - t}(O-I),' dt) ·
a a
z: .1
 c(z - ar (J Icp(t}I'dt) '.
a
Further, we consider the case a - lip  1 at first. For x, x + h E [a, b] we have
1 z: J +h _
(1:+cp)(x + h) - (I:+cp)(x) = f(a) (x + h - tt lcp(t)dt
z:
z:
+ r(1",) J [(z + h - W- I - (z - W-I]cp(t}dt = 1 1 +1 2 . (3.14)
a
Using the Holder inequality (1.28) we find
z:+h .1 z:+h 
lId ::; r() ( J Icp(t}1" dt ) · ( J (z + h - t}(O-I),' dt) ·
z: z:
 ch a -  IIcpllL p '
z: 
11 2 1  Ii (J I(z + h - W- I - (z - t}O-II" dt) ·
a
r-.
-r- 1
 Ir(",WlhO-lIcpIlL.( J 100-1_(0+ W-II"dt) ".
o
If x - a =:; h, then an estimate for 12 is clear. If x - a > h, then we use (3.9) and
(A + B)I/" =:; A 1/,' + Bl/,' to obtain

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 69
(x-a)lh 1/1"
11 2 1  hO-I/'IIIOIlL. [0 1 + 02 J '<0-2),' dB]
1
[ ( h ) l -a+ll'P ]
 ha-ll'PIII;'IIL p C3 + C4 (X _ a)
C4 being additionally multiplied by (In xha ) 1/,' in the case a -  = 1. Hence we
derive the estimate
{ cha-l/'III;'IILp
11 2 1  ch Iln till" IIl;'lIL p
if a - lip < 1,
if a - lip = 1.
Collecting the estimates for 1 1 and 12 we complete the proof of the theorem when
a - lip  1.
Let now a - lip> 1. Then k < a - lip  k + 1, k = 1,2,... , and this case is
reduced to the previous one by direct differentiation:
dl: I a l a-I:
dxl: a+1;' = a+ 1;', 0 < a - k  1,
using the definitions of the spaces H>' and H>',I: in the case A > 1. .
Corollary. Theorem 3.6 holds in a stronger form
1:+ : L,(a, b) -+ ha-l/'([a, b», 0 < lip < a < 1 + lip,
where h>' is the space (1.2).
Proof. For I;' E L,(a, b) and any c > 0 the equality I;' = Pe + I;'e holds where Pe
is a polynomial and IIl;'eliL p < c by the property d) of the space L,. -. see S 1.2.
Hence Theorem 3.6 shows that
I(I:+I;')(x + t) - (I:+I;')(x)1 1(I:+Pe)(x + t) - (I:+Pe)(x)1
+ I(I:+l;'e)(x + t) - (I:+l;'e)(x)1
clltla + C2 Itla-l/'lIl;'e II,
=o(ltl a - l/ ,).

70 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Remark 3.3. The statement on the boundedness of the operator 1:+ from Loo
into H OI mentioned above (the Corollary of Theorem 3.1) corresponds to the case
p = 00 in Theorem 3.6.
3.4. Mapping properties in the space L,(p)
N ow we consider the mapping properties of fractional integration operators in the
weighted spaces L,(p) with weight (3.12) where a = Xl < X2 < ... < X n = b.
It proves to be that even in the case of the simplest weight p( x) = Ix - dl"
concentrated in one point d of the interval [a, b), that the restrictions on the
exponent turn out to be essentially different depending on the fact as to whether
the point d of this interval coincide with the end-points of the interval [a, b) or is an
inner point of it. We find it convenient to admit here into consideration a half-axis
as well: -00 < a < b  00.
We begin our consideration with the weight p( x) = (x - a)". The following
lemma about the commutation of fractional integrals with power functions allows us
to pass directly from the "nonweighted" theorem on the boundedness of fractional
integration operators to their weighted analogues. The notation p"(x) = (x - a)"
will be used in Lemmas 3.2, 3.3 and in Theorem 3.7, 3.8 unlike that used above.
Lemma 3.2. Let 0 < a < 1, cp E L,(O,/), 0 < I  00, 1 < p < 00, and
J.l > -1 + IIp. Then the equalities
Ig+p"cp = p" Ig+(cp + A 1 CP),
p" Ig+cp = Ig+p"(cp + A 1 CP),
(3.15)
(3.16)
hold, Ai being bounded in L,(O, I) operators in the form
:&
Ai = 1['-lJ.lsina1r j Ai(X,t)cp(t)dt, i = 1,2,
o
1 j :& ( Y - t ) 01 ( t ) " dy
A 1 (x,t) = - - --,
t-x x-y Y Y
t
1 j :& ( Y - t ) 01 ( Y ) " dy
A 2 (x,t) = - - --.
x-t x-y x y
t
The case p = 1, J.l > 0 as in Lemma 10.1 below, shows that if cp E L 1 (0, I) for (3.15)
and (I In xl + l)cp(x) E L 1 (0, I) for (3.16), then the above equalities remain true

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 71
and the operators (3.15) and (3.16) are bounded and (Ilnxl + 1)(<p(x) + Al<P(X» E
Ll(O,/),-<p(x) + A2<p(X) E L1(O,I), 0< 1 < 00.
prool. We show the boundedness of the operators Ai. The kernel Al (x, t) is
homogeneous of degree -1 and
1
f a(1 - )-a
IA 1 (I,t)1 = t IJ [t +(1- t»)l+IJ  ct\
o
A = { fl,
a-g,
fl < a,
p  a,
where ° < g < a. In view of this estimate, the operator Al is bounded in L,(O, I)
by Theorem 1.5. The boundedness of the operator A 2 is proved similarly. Now we
verify (3.15). We have
x IJ 1 0 +(<p + Al<P) =x IJ 1 0 +<p
:& :&
- psin a1l' IJ f IJ ( )dt f (y - t)ady
1I'r( a) x t <P t yl+1J
o t
:&
f (x - r)a-l(r - y)-a
X dr.
r-t
11
Using (11.4) from S 11 to evaluate inner integrals on the right-hand side we obtain
(3.15). Relation (3.16) is proved similarly. .
The following statement generalizes Lemma 3.2 to the case of weighted spaces
and an arbitrary a > O.
Lemma 3.3. Let a > 0, a f; 1,2,3,..., <P E L,([O,ij,), 0 < I  00,1 < p < 00,
m = [a], (3 < p - 1 + min(pp, 0). Then (3.15) and (3.16) are true where all Ai are
bounded operators in L,([O, ij,) and are defined by the equalities
m :&
Al<P = AIJ,a)<p = ?: Cj f Alj(X, r)<p(r)dr,
J=O 0
A 2 <P = p-IJ A -IJ,a) pIJ<p,
1 .
f a+m- J
_ IJ m-j  j-IJ-m-l
Alj(X, r) - r (x - r) (1 _ ){a} [r + (x - r)] ,
o
( m + 1 ) m!(j - m - P)m-j+l
Cj = j (m - j)!r(a)r(l- {a})'

72 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Equality (3.15) is verified by direct application of the fractional differentiation
operator vg+ to p-/J I+plJcp. The result (3.16) follows from (3.15). The
boundedness of the operators Ai, i = 1,2 follows from Theorem 1.5.
We now go on to consider the fractional integrals themselves.
Theorem 3.7. Let a > 0, -00 < a < b $ 00, p  1, p. < p - 1, 0 < a < m + IIp,
q = p/[l- (a - m)p]; and 0 $ m  a if p f; 1 and 0 < m  a if p = 1. Then the
operator 1:+ is bounded from L,([a,b],p/J) into Lq ([a,b],p(-m)q).
Proof. Let, at first, 1 < p < 00. We set cP = p-'i CPo, CPo E L,. If m < a, then, by
(3.15) we have (we may suppose that a = 0):
I e.-m I OI I e. / OI-m _e. I OI-m/.
p p 0+ cP  cp p 0+ P p CPo = 0+ 0/,
111/JIlL p  cllcpllL p '
and then all that remains is to use Theorem 3.5 - see also Theorem 5.3. If
a = m, then p'i- OI IC+cp-'i is the operator with a homogeneous kernel of degree
-1. Therefore the theorem follows from Theorem 1.5. In the case p = 1 the Holder
inequality (1.28) is to be used. In fact, we have
II:
II+I :5 r(l",) 1 {(z - y)a-l y -71(y)l; } {Y"I(!I)I}'" dy
o
II: 1. II: 
:5 r() (1 (z - y)<a-l)f!l-"(f-l)I(Y)ld!l)' (1 yI' 1(y)ldY)' ,
o 0
from whence
· I II: I
IIp"-m I+IIL. :5 r/",) IIp"lIt (1 z("-m)fdz 1 (z - y)(a-l)fy-"(f-l)I(Y)ldY) i
o 0
I 
:5 r/",) IIp"lIt (1 yl'k(!I)I(Y)ldY)' :5 cllpl'IIL, ,
o
since
00
k(y) = y-/Jq 1 x(/J-m)q(x - y)(OI-l)qdx
11
00
= 1 (/J-m)q( - 1)(OI-l)qd < 00.
1

i 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 73
The theorem is thus proved. .
The important particular case J.l = 0 and q = p in Theorem 3.7 is to be noted:
b 1.
{J( - a)-a'I(I+cp)()I'd}'  cllcpIlL"
a
(3.17)
1 < p < 00, -00 < a < b  00.
The inequality
b 1
{ J (b - )-a'I(I+cp)()I'd}'  cllcpIlL"
a
(3.18)
1 < p < l/a, -00 < a < b < 00,
also holds. One may prove it directly, but it is much easier to refer to (5.45) proved
below in S 5.3 from which it follows immediately.
The inequalities (3.17) and (3.18) do not hold in the case p = 1. For example,
the former is replaced by
b b
J J b-a
(x - a)-al(I:+cp)(x)ldx  c Icp(x)lln -dx,
x-a
a a
(3.17')
which is proved by simple estimates.
A particular case of Theorem 3.7 when p = 1, a = 0 (a = m, J.l = -c) is also
to be emphasized:
b b
J x-a-£I(I<f+cp)(x)ldx  c J x-£lcp(x)ldx, b> 0, a > 0, c> 0,
o 0
where c = c(c). This relation does not hold if c = 0 (lim c(c) = 00). In the case
£-0
c = 0 the inequalities
b b
J >.b+l J >. Ib+l
x-a In --;:-1(I<f+cp)(x)ldx  c In + --;:-lcp(x)ldx,
o 0
(3.17")

74 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
00 &
1 >.b+l 1 >. b+l
z-Iln -,;-zl(Izl-a<p)(z)ldz  c In +1 -,;-zl<p(z)ldz,
& 0
( 3.17"')
where Q > 0,   0, b > 0 hold. The former is proved by simple estimates after
changing the order of integration on the left-hand side. The latter is derived from
the former if we replace z by z-l, b by b- 1 and <p(z) by z-2<p(z-I).
The following theorem specifies Theorem 3.7 for the finite interval [a, b] when
p> l/Q.
Theorem 3.8. Let -00 < a < b < 00. If 1 < p < 00, 0 < Q - IIp < 1, then the
operator 1:+ is bounded from L,(pI") into H-l/'(pI"/'), JJ < p - 1; and
(I:+<p)(z) = 0 (z - a)a-) as z --+ a.
(3.19)
Prool. When applying the Holder inequality to I:+<p where <p(z) = (z -
a)-IJ/P<po(z), <po(z) E L,([a, b]), we obtain the estimate (3.19):
( X ) 11, ( X ) IIp'
1 1 :+ <pI  I I <po(t)I'dt l(t - a)-IJ/(P-1)(z - t)(a-1)" dt
a a
x 11
 C(x - a)a-(H"J/. (1 lo(t)l.dt) '.
a
To prove the boundedness of the operator 1:+ form L,(PP) into H-l/'(pp/')
we use the commutation relation (3.15). We have pIJI'I:+p-IJI,<po = I:+1/J,
1I"IIL p  cll<poliL p which yields the wanted result by Theorem 3.6. .
Corollary. Theorem 3.8 holds also in a stronger form
1:+ : L,(pIJ)  h-; (pi-) ,
1 1
o < - < Q < - + 1,
p p
where h(r) is the weighted space (1.12).
Proof. Any function in L, can be approximated by infinitely differentiable
functions which are finite on (a, b) as seen in property e) of functions in L, in
S 1.2. Hence, for <p E L,(pI") we have the representation <p = at: + <Pt: where
at: is an infinitely differentiable function on [a, b] and 1I<pt: IILp(pP) < c. Denoting

i .3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 75
4/ = /(X + t) - I(x), 1 = I:+cp, we have
IA (p I) 1  IA (p I:+a e ) 1 + IA (p I:+CPe) I. (3.20)
Theorem 3.8 gives the estimate IA(P#'/' 1:+ CPe) I  cllCPe IILp(P" )h a - l /, . Since
tie E L,(pIJ) for any q > p then by Theorem 3.8 again we find
IA (p I:+a e ) 1  cha-lIaeIlLq(p") = 0 (ha-;)
as h --+ 0+. Substituting the obtained estimates into (3.20) we obtain IA (p f) 1 
ha-;(ce: + 0(1». This proves the corollary. .
Now we are to consider the case of weight p(x) = Ix - dl IJ , a < d  b < 00.
Theorem 3.9. Let 1 < p < 1/0: and fl < P - 1, the latter in the case d < b only.
Then the operator 1:+ is bounded from L,(p), p(x) = Ix - dl IJ , into Lq(r) where
q = p/(1 - o:p), r(x) = Ix - dill and
v> -1
if fl  o:p - 1,
v = flq/P if fl> o:p-l,
(3.21 )
Proof. Let cp(t) = It - dl-IJ/',p(t), ,p(t) E L,(a, b). The functions cP and 1/J may be
considered to be nonnegative. Then for x E (a, d) we have
r(o:)(d - x)i(I:+cp)(x) = Al + A 2 ,
:& :&
where Al = (d - x)lI/ q - IJ/ , I(x - t)a- I 1/J(t)dt and A 2 = (d - X)II/q I[(d - t)-IJlp-
a a
(d - x)-IJ/,](x - t)a-l,p(t)dt. From (3.21) it follows that v/q - fl/P  O. Therefore,
by Theorem 3.5 we obtain
IIA1IILq(a,d)  cll,pIlLp(a,d)  cIlCPIlLp([a,b];p)'
(3.22)

76 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
A similar estimate has to be proved for A 2 . If P.  0, then
A = ( d - X ) -  f :& [( d - x ) 1-'11' _ 1 ] 1/J(t)dt < 2A
2 d-t (x-t)l-a - 1,
a
and we can use (3.22). If p. < 0, we have
:&
A 2 = (d - x)"lq f[(d - t)¥ - (d - x)¥](x - t)a- 1 1/J(t)dt.
a
If p. $ ap - 1, then according to (3.8) we obtain
:&
,,-c f L+.Iel 1
A2 $ c(d - x)--q (d - t)" p - (x - t)a1/J(t)dt
a
d
$ c(d - x) ,,;c f(d - t)i+¥+a-11/J(t)dt
a
$ c( d - x) ,,;c II 1/J1I Lp( a,d) E Lq (a, d),
where 0 < c < II + 1. If p. > ap - 1, we represent A 2 as
:& 
A 2 = lci (d-x) f (d-)¥-l f 1/J(t)dt
p (x - t)l-a
a a
:& I!.
< C f ( d - x ) p g()d
- d- d-'
a
(3.23)
where 9 = I:+1/J E Lq(a, d), IIgIIL,,(a,d)  cll1/JIIL p (a,d)' We note that the integral
operator on the right-hand side of (3.23) is bounded in L,(a, d) by Theorem 1.5.
We estimate (x - d)" lq (I:+cp)(x) for x E (d, b). We have
d:& 
 a (f f) (x-d)"1/J(t)dt ,
r(a)(x - d)" Ia+cp = + I!. = B 1 + B 2 .
It - dip (x - t)l-a
a d
(3.24 )

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 77
If fl  o:p - 1, 0 < c < min(1 - 0:, V + 1), for B1 we derive
d e.
 j (d - t)- p 1/J(t)dt
B 1 < ( X - d ) "
- (x - t)l-a-e
/I
 c(x - d) "; (1::e+1/J) (d)
 c(x - d) "; 111/JIIL p (/I,d) E Lq(d, b).
If fl > o:p - 1, then we replace the function (x - t)a-1 in B1 by the relation
:&
( ) a-1 sinO:1r (d ) a j (x-r)a-1(r-d)-a d
x-t = - -t r
1r r-t
d
- see (11.4). Interchanging the order of integration we obtain
B sin 0:11' ( d) e. j :& (r - d)-  g( r)dr
1- - x- p
- 11' (x - r)l-a '
d
d
g(r) = j ( d - t ) a- 1/J(t)dt .
r-d r-t
/I
By Theorem 1.5 we have IIgIILp(d,b)  clitPIILp(/I,b)' Hence, by Theorem 3.7, we find
IIB1IIL,,(d,b)  cllgIlLp(/I,b)  clitPIILp(/I,b)'
The required estimate for the second term in (3.24) is obtained by application
of Theorem 3.7, which completes the proof. .
Remark 3.4. If flIp  0: - IIp in Theorem 3.9 we can not take v = flqlp. Indeed,
taking I;'(x) = Id - xl- IJ1 , for x E (, d) we obtain

II II j  (x - t)a-1
(d - x)i(I:+I;')(x)  c(d - x)i e. dt
(d - t) p
/I
 c(d - x)i ft Lq.

78 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
The analogue of Theorem 3.9 holds for general power weights as well:
n n
p(x) = II Ix - xl:l IJk , r(x) = II Ix - xl:l llk ,
1:=1 1:=1
(3.25)
a = Xl < .. . < X n = b.
We state without proof the corresponding proposition for the operator 1:+.
Theorem 3.10. Let 1 < p < 00, fll: < p - 1 for k = 1,2,..., n - 1; 0 $ m $ a,
0< a < m+ *, q = l-(:-m), ' III = (- m)q, III: = (- m)q, if fll: > ap-l
and IIIe > (a - * - m )q, if fll: $ ap - 1, k = 2,..., n. Then the operator 1:+ is
bounded from L,(p) into Lq(r).
We investigate the problem whether the fractional integrals I:+tp, tp E L,(p),
belong to weighted Holder spaces if a > lip. Firstly we consider the weight
p(x) = Ix - dl IJ , a < d $ b.
Theorem 3.11. Let 1 < p < 00, lip < a < 1 + lip, and 0 < fl < p - 1 if
a < d < band 0 < fl < 00 if d = b. Then the fractional integration operator 1:+ is
bounded from L,(p), p(x) = Ix - dl IJ into Hr;in(IJ/"a-l/')(pl/,) if p f; ap - 1 and
into H-l/"l/" (pl/p) if fl = ap - 1.
Proof. Let II = flip, tp(x) = Ix - dl- lI tpo(x), tpo E L,. We are to prove that
:& { min(lI,a-;)
X - d II tpo(t)dt Ho
fix) = ! I t - d I (x - t)'-a E H;-.;r
if II -J. a - 1
..,... "
if II = a - 1,
,
> { IIfIlHmiD("'-;)
clitpoliL p - IIfll _l..
H p p
if II -J. a - 1
r p'
if II = a - 1.
,
(3.26)
We fix the point al E (a, d). Then, by Theorem 3.8, we have f(x) E
H a - 1 /'([a, ad), and f(a) = 0 and IIfIlH-l/p([o,ol])  clitpollL p ' We demonstrate
that f(x) vanishes if x = d and satisfies (3.26) on [al,d]. Let x E (al,d). Then we
have
01 :&
fix) = (J + J) I ; : I" (:(:a = u(x) + v(x).
o 01

i 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 79
We estimate u(x):
01 1
II (/ (d-t)-IIP'dt ) 7
lu(x)1  c(d - x) IIcpoliL p (a1 _ t)(l-a)p'
o
 c(d - x)"l1cpoIlLp'
For v( x) we have
:t: , l
( / (x - t)(a-1)p dt ) pr
Iv(x)1  (d - x)"IICPoliL p (d _ t)" P ' .
01
(3.27)
If II < Q - IIp, then
:t: 1/ '
Iv(z)1  (d - z)" lI'PoIlL. (/ (z - t)(a-1-_»>' dt) P
01
 c( d - x)" IICPoIlL p '
H II> Q - IIp, then, after the change of variable t = x - (d - x), we get
( "'4-_:1 ) l/p'
Iv(x)1  (d-x)a-tllcpoIiL p / (a-1)p'(1+)-IIP'
01
 c(d-xt-;lIcpoIiLp.
In the case when II = Q - IIp we similarly obtain
"'4-- 1
1 _'"
Iv(z)1  (d - z)"II'PoIlL. (/ (a-1)p'  + / C1)
o 1
 c(d - x)"lIcpoIlLp(l + Iln(d - X)I).
since we take x - a1 > d - x.
It follows from the above estimates that f(d - 0) = O. It is similarly proved
that f( d + 0) = 0 if d < b.
To estimate the Holder norm of the functiol\ J() when x E [aI, d), we

80 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
represent /( x) as
:& :&
1 cpo(t)dt 1 (d - x)" - (d - t)" d
/(x) = (x _ t)l-a + (d _ t)/(x _ t)l-a cpo(t) t
o 0
= h(x) + h(x).
By Theorem 3.8 we have IIhIlH"'-l/ P ([o,b]) :5 cllcpoIlLp(o,b)'
For /2 we get h(x + h) - 12 (x) = it + J2 + J3, al :5 x < x + h :5 d, it, J 2 and
J 3 being the same as in (3.7) but with (x - a)# replaced by (d - x)/. We estimate
J 1 considering for simplicity the case II :5 1 only. We have
:&+h 1
IJ 1 1 ::; IICPoIlL. ( 1 (d - W",' (0: + h - t)(a-1),'I(d - t)" - (d - 0: - h)" I" dt) "
x
:&+h 
::; cllCPo ilL. ( 1 (d - t)-",' (0: + h - ti a+"-l),' dt) ·
:&
:&+h 1
::; cllCPoilL. ( 1 (0: + h - t)(a-1)" dt)" ::; cha-;IICPoIlL.'
:&
Estimating J2 we find
:& I l
(I (d - t)-IIP dt ) ;r
IJ21 :5 ch/IlCPoliL p (x + h _ t)(a-l)pl
o
01 1:& I 1
(I (d - t)-IIP dt 1 (d - t)-IIP dt ) pr
:5 ch/IlCPoliL p (al _ t)(1-a)p l + (x _ t)(1-a)p l
o 01
:& I.
[ ( 1 (d - t)-IIP dt ) 7 ]
:5 ch/IICPoilL p 1 + (x _ t)(1-a)p l .
01
The integral in parenthesis coincides with the integral in (3.27) which has already
been estimated. Therefore, we have IJ 2 1 :5 ch/IlCPoIlLp(o,b) if II < a - lip. If II 
a-lip, then the expression in square brackets is estimated by l+c(d-x)a-lI-l/p :5
r- G 1
h
1 + ch a - lI - l / p if II > a - lip and by 1 + J (l-"')Ptl+)"P' :5 c (1 + In t) if
o
II = a - 1, which yields the required result for J 1 .
P

 3. THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS 81
We go further by estimating J2. We fix any point 6 E (a,aI) and represent J 3

6 (11 :&
J. = (/ + / + /) (d - zl; t - t)" [(z+ h - w- 1 - (z - W- 1 ]<{Jo(t)dt
(I 6 (11
= J 31 + J 3 2 + J 33 .
It is obvious that
6
IJ 31 1  ch / ICPo(t)dt  chllcpoliLp
o
tor the first term and that
(11 
IJ'21 $ cll<{JoIlL. (/ I(z - t)a-l - (z + h - t)a-lJ" dt)'
6
 cha-lIcpoIiLp
for the second one. Lastly we have
:& vp' 
IJ..I $ cll<{JOIlL. (J ( = =: ) I(z - tt- 1 - (z +h - W- 1 1" dt ). ,
(11
whence I J 33 I  ch a - 1 / p llcpoIIL p '
The inequality (3.26) on the interval [ai, d] follows from the above estimates.
For [d, b), if d < b, the arguments are similar. The theorem is thus proved. .
Remark 3.5. The dependence of the order of the Holder property of the fractional
integral / = I:+cp on the correlation between pIp and Q - IIp is caused by the
behavior of the integral / = 1:+ cp at the point x = d. This behaviour can be
described by varying the exponent of a weight function for /(x). In this case,
the Holder order is not changed and is equal to Q - IIp. We state without proof
the theorem on the boundedness of the fractional integration operator in such an
interpretation for a general power weight (3.25). One may find another variant of
a similar theorem in S 4.2 (note 3.1).
Theorem 3.12. Let 1 < p < 00, IIp < Q < 1 + IIp and Pk < P - 1 for
Ie = 1,2,..., n - 1. Then the operator 1:+ i8 bounded from Lp(p) into H;-l/ P (r),

82 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
and even into h-1/P(r), if r(x) = (x - a)/Jdp n Ix - x1:1 6 k, A = {k : k E
l:eA
{2, 3, . . . , n}, Ill: > O} and 61: = Ill:/P if Ill: > a:p - 1 and 61: = a: + £1: - l/p, £1: > 0,
if Ill:  a:p - 1.
The question naturally arises concerning mapping properties of fractional
integration in the weighted spaces Lp(p) with an arbitrary, not necessary power,
weight. In this case, weight functions satisfying the so-called Muckenhoupt-type
condition are considered. We shall not consider this question here, but observe
that one may find the some information in S 9.2 (note 5.8). See also Theorem 25.4
and S 29.2 (notes 25.8 and 26.11) in the multi-dimensional case.
 4. Bibliographical Remarks and
Additional Information to Chapter 1
4.1. Historical notes
The survey of papers concerning the origin and development of the main ideas of fractional
calculus was given in the brief historical outline at the beginning of the book. Here our historical
comments refer to the contents of this chapter only.
Notes to  2.1 and 2.2. Equation (2.1) has associated with the name of Abel, who was
the first person, who considered and solved this equation for 0 < cv < 1 in cOImection with the
tautochrone problem - Abel [1] (1823), [2] (1826). The false notion that Abel dealt only with the
case cv = 1/2 has already been noted in the "Brief historical outline". The Abel type equation
:&
J(x2 - s2)-1/2<p(s)ds = l(x) was solved by Joahimstal [1] (1861). A more general equation of
o
such a type:
:&
/[T(X) - T(S)]cv-1<p(s)ds = l(x), cv > 0,
o
with a monotone increasing function T(X) was in fact solved by Sato [1] (1935), although its
solution was already known to Holmgren [1] (1865-1866). Equation (4.1) occurs frequently in
many applications. We note for example its employment in the theory of generalized analytic
functions, Polozhii [1] (1964), [2, p.236] (1965), [3, p.186] (1973).
The solution of (2.1) developed in  2.1 is well known. The criteria of its solvability in
L1 (a, b) given in Theorems 2.1 and 2.3 is less known. They were given by Tamarkin [I, Theorem 4]
(1930) for cv > O. The proof of Tamarkin's theorem was expounded in Dzherbashyan's book
[2]. The role of absolute continuity in the theory of fractional integrals of sununable functions
was thoroughly cleared up by Tamarkin [1] (1930), although Tonelli [1] (1928) had earlier used
the idea of absolute continuity for solving the Abel equation and had in particular obtained the
statement of Corollary of Lemma 2.1 and (2.14) with 1 e AC([a, b]). The criteria of solvability of
the Abel equation in the space of continuous functions was in fact obtained in the book of Bacher
[1, p.8-9] (1909) the conditions in (2.9) being then replaced by the conditions h-cv(x) e C 1 ([a, b]),
h_cv(a) = O.
Notes to  2.3. Definition 2.1 goes back to Riemann [1] (1847) (published in 1876).
Definition 2.2 of fractional differentiation is also contained in this paper. As well as
Liouville, Riemann dealt with the so-called "complementary" ftmctions when he defined fractional
differentiation. These functions are in fact power functions with arbitrary constant coefficients
which were introduced by Liouville and Riemann in order that the relation I+ D+ 1 = 1 be
(4.1)

 4. ADDITIONAL INFORMATION TO CHAPTER 1.
83
valid for all admissible 1, d. (2.60). We mention the polemics in the paper by Cayley [1] (1880)
connected with the "complementary" functions. The first people who deliberately refused to
use "complementary" functions were Hohngren [1] (1865-1866), and independently, Letnikov [4]
(1874). Avoiding these functions Holmgren and Letnikov introduced fractional differentiation as
an operation which is the left inverse to fractional integration. This approach is widely used in the
modem analysis. They constructed the foundations of fractional calculus via such an approach.
Relation (2.20) was established by Love, Young [1] (1938).
Equation (2.26) was known to Euler [1, p.56] (1738) in the sense that he introduced the
fractional derivative of a power function by (2.26) as a definition.
Notes to  2.4. The order a of fractional integro-differentiation was already taken as
complex in the papers by Liouville, Riemann, GrUnwald, Letnikov, Sonine and others. Fractional
integrals of purely imaginary order a = i8 were introduced by Kober [2] (1941), by means of
Mellin transfonn, which enabled one to consider such a fractional integration as an operation
continuous in L2(O, 00). Forms (2.37) and (2.38) of fractional integration of purely imaginary
order arose in Kalisch [1] (1967), where it was shown that they generate operators bounded in
Lp(O, I), 1 < p < 00. See such statement in Lemma 8.2 below for the case of the whole real
axis. The paper by Fisher [2] (1971) also concerned the investigation of purely imaginary order
fractional integration in the space Lp. The detailed investigation of composition fonnulae, the
so-ca.lled index laws, for purely imaginary order fractional integrals was undertaken by Love [4],
[5] (1971-1972).
Theorem 2.2 is in fact contained in Tamarkin [1] (1930). The result (2.43) itself was given
for sufficiently "good" functions by Holmgren [I, p.7], (1865-1866), and Letnikov [1, p.26], (1868).
Notes to  2.5. Relations (2.44)-(2.54) for the evaluation of fractional integrals of
elementary functions were known long ago. In particular, (2.44) and (2.45) are in fact due to
Euler [1] (1738), who introduced them by definition, while (2.48) and (2.49) may be found in
Hohngren [I, p.27), (1865-1866). The relation in (2.50) with (3 = 1 for logarithmic functions was
given by Letnikov [I, p.37], (1868). The result in (2.54) is a modification of a Sonine's fonnula,
known as a first Sonine integral - Sonine [3] (1880) or his book [6, p.206].
In connection with the derivation of (2.50) let us note that the idea of obtaining results
containing power-logarithmic functions from the corresponding expressions for purely power
functions by differentiating with respect to the power exponent is due to Volterra [2] (1916).
Using this method one may obtain results for the fractional integration of functions <p(x) =
(x - a).6- l ln m (x - a) in particular - see Table 9.1 in  9.3 below.
Notes to  2.6. The space [:+[Ll (a, b)] as an object in itself arose originally in
Dzherbashyan and Nersesyan [4] (1960), [5] (1961), while the space [a[Lp(a, b)] appeared in
Samko (7) (1969) for the case (a,b) = Rl, and in Rubin [1] (1972) for the case of a finite interval.
Theorem 2.3 was given by Tamarkin [1] (1930). The reasons for the insufficiency of the almost
everywhere existence of summable derivative 1):+1 for repreenting the function l(x) by the
fractional integral of order a which were discussed after Theorem 2.3 are essential. The error of
the sufficiency of this condition is contained in a number of investigations on fractional calculus.
Definition 2.4 used in this book enabled us to give a strict proof of (2.58) which has been "proved"
by different authors under the wrong assumption that 1):+1 exists almost everywhere and is
sununable only.
Notes to  2.7. The semi group property (2.65) of fractional integration operators was
proved in the papers by Riemann, Hohngren, and Letnikov. Among the mathematicians of the
19th century the latter gave the most exact and complete exposition of this property, see Letnikov
[4, ch.II) (1874) and [5] (1874). It is relevant to refer to the characterization of the properties of
fractional integration from the point of view of semigroup theory which was given in the book of
Hille and Phillips [I, Ch.23, p.674-690]. Theorem 2.5 was proved by various authors at different
times under diverse assumptions. In the case of integrals understood in the Denjoy sense this
theorem was proved by Bosanquet [1] (1931) for a > 0, {3 > OJ the case of Stieltjes integrals,
summable in the Cesaro sense, was treated in Isaacs [2] (1960). A consideration of (2.65), more
complete than in Theorem 2.5, was given by Love [5] (1972) which included the cases Rea = 0
and Re{3 = O. Theorem 2.6 was firstly fonnulated and proved to all appearance by Hille - Hille

84 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
and Phillips [I, p.675]. The proof of Theorem 2.7 is adopted from the book of Dzherbashyan
[2, p.568]. For the analogous multidimensional assertion for Riesz fractional integration we refer
to  29.2 (note 25.17).
Notes to  3.1 and 3.2. The first result concerning fractional differentiation of Holderian
functions is due to Weyl [1] (1917). He showed - in the periodic case which will be considered
in  19 - that functions satisfying the Holder condition of order .x had continuous fractional
derivatives of order a < .x. A similar statement for fractional Riemann-Liouville derivatives in the
non-periodic case considered in  3 was obtained by Montel [1] (1918). For this purpose he used
Bernstein-type theorems on the rate of approximation of a function by algebraic polynomials. The
exact result on the mapping properties of fractional integra-differentiation within the framework
of H>'-spaces are due to Hardy and Littlewood [3] (1928). They obtained Theorem 3.1 on mapping
properties of 1:+ from H to H;+a, .x + a < 1 (as well as Lemma 3.1 on such properties for
V:+ from H6, .x > a, to H-a which is proved in  13 below). Theorems 3.3, 3.3' and 3.4 on
mapping properties of 1:+ in the spaces H6(p) with a power weight were proved by Rubin [7]
(1974) under the restriction 0 < JJl < .x + 1 on the power exponent relating to the point x = a.
These theorems are proved here under the assumption JJl < ,\ + 1 as in the paper by Rubin [22]
(1986). We followed these Rubin's papers, except Theorem 3.4, which is proved in a different way.
We note the paper by Chen [1] (1959) who considered the mapping properties in weighted
Holder spaces for certain integral operators, generalizing the case of fractional integration. The
result in this paper includes Theorem 3.3 with 0  .x < 1.
Notes to  3.3 and 3.4. Theorems 3.5-3.7 were established by Hardy and Littlewood
[1] (1925), [3] (1928), see also  9.1 (notes to  5.3) in this cOIUlection. A simple assertion that
1:+ maps Lp into Lr with r < pl(1 - ap) was first noted by Hardy [1] (1917). The fonnulated
specification for the case a - lip = 1,2,... was given by Kilbas. Theorem 3.8 was obtained by
Karapetyants and Rubin [2] (1984). We gave a shorter proof given by Rubin [22] (1986) on the
basis of Lemma 3.2 on continuation. The proof of Theorem 3.7 is also based on Lemma 3.2.
It was given in Rubin [22] (1986), see also [17, p.529] (1983), and by Karapetyants and Rubin
[2] (1984). The proof of Theorem 3.7 in the case p = 1 is due to Flett [3] (1958). Lemma 3.2
was proved by Karapetyants and Rubin [1] (1982), [3] (1986). A more general Lemma 3.3 was
obtained by Rubin [17, p.529] (1983). The proof of corollaries of Theorems 3.6 and 3.8 were noted
in Karapetyants and Rubin [2] (1984).
Theorem 3.10 was proved by Karapetyants and Rubin [2] (1984) in the case m = 0, and
Rubin [22] (1986) in the case m e [0, a]. We gave here a simpler proof of this theorem in the
particular case only. We refer to Theorem 3.9, when the weight is related to a single point x = d.
This .proof was obtained by Kilbas and Rubin independently and it has not been published earlier.
Theorem 3.11 was proved by Karapetyants and Rubin [2] (1984) in the case II '# a - lip. The
case II = a - lip was considered by Kilbas, but not published earlier. Theorem 3.12 was proved
by Rubin [22] (1986).
4.2. Survey of other results (relating to fifi 1-3)
2.1. The solution of Abel's equation on an arc in the complex plane and of the more general
equation
t
f P(t - T)
( ) <p(T)dT = f(t), 0 < a < 1.
t-T a
a
was given by Sakalyuk [3] in the case when P(t) is a polynomial and t,T e £, £ is a smooth arc
with the end-points a and b.
2.2. Bosanquet [1] considered the Abel equation when its solution is not necessarily
Lebesgue integrable, the integrals being understood in the Denjoy sense.
2.3. There are l1l.aDY papel'8 concerning approximate solution of Abel's equation (2.1).
We note some of them. The paper by Whittaker [1] appears to be the first one. The method

 4. ADDITIONAL INFORMATION TO CHAPTER 1.
85
of approximation of the solution by means of Jacobi polynomials was suggested by Fettis [1].
Orthogonal polynomials were used for the same purpose by Minerbo and Levy [1]. A numerical
x
method of solving the more general equation Jk(x, t)(x- t)-a<p(t)dt = I(x), x> 0, was suggested
o
by Weiss [1] and Weiss and Anderssen [1]. Edels, Hearne and Young [1] discussed a numerical
method in the particular case a = 1/2 in cOIUlection with problems arising in the theory of
electrical discharge in a gas. This case was also considered in the paper by Gorenflo and Kovetz
[1], which was motivated by applications in spectroscopy, the suggested method taking into
account the specific application. The solution of the Abel equation with a perturbation was given
by Gorenflo [1] by means of the method of quadrature optimization. The development and the
survey of various approximation methods were given in a series of papers by Gorenflo [2], [3],
[5], [6]. We note that Gorenflo [3] suggested a method which was convenient in applications,
which gives the approximation by a sum of a "good" function and the finite combination of
step-functions with unknown jumps. The method of splines was applied by Doctorskii and Osipov
[1], Voskoboinikov [1] and Medvedev [1]. Various approximation methods can also be found in
the papers Atkinson [2], Balasubramanian, Norie and de Vries [1], Bronner [1], Chan and Lu [1].
Eggermont [I], [2], Frie [I], Gerlach and Wolfersdorf [I], Zheludev [2], Kosarev [I], Lubich [1],
Malinovski and Smarzewski [1], Sirola and Andersen [1], Ugniewski [I], [2] and Voskoboinikov [2].
One can find a brief exposition of the idea of the method of an asymptotic solution for Abel
equation in  16.5 below.
2.4. Sonine [4], [5], [6, p.148] generalized the Abel integral equation by considering the
equation
:&
f k(x - t)<p(t)dt = I(x), x> a.
a
(4.2)
He gave its exact solution
:&
<p(x) =  f 1(:& - t)/(t)dt
dx
a
(4.2')
under the assumption that for the kernel k(x) there exists the function I(x) such that
:&
f k(t)/(x - t)dt = 1 for x > o.
o
(4.2")
The existence of such a function I(x) is guaranteed for example in the case of the kernels of the
00  00
form k(x) = x a - 1 g(x), g(x) = E all x" , ao '# o. In this case I(x) = x-ah(x), h(x) = E b"x",
11=0 11=0
o < a < 1 (Wick [1]). The wider class of kernels k(x) satisfying Sonine's condition was pointed
out by Rubin [14, p.62-63].
We note that Volterra [1] used the method of solving the Abel equation (2.1) in order to
reduce the more general equation
:&
f ( k(x, t » <p(t)dt = I(x), 0 < a < I,
x-ta
a
with a continuous function k(x, t), to an integral equation of the second kind with a kernel without
a singularity. Volterra called this approach the kernel transformation method. Such reductions
may be seen in  31.3 below in greater detail.

86 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
2.5. The solution of Abel type integral equation of the second kind
:&
,X f <p(t)dt
<p(x) - rea) (x _ t)l-a = lex),
o
x > 0, a > 0,
(4.3)
was given by Hille and Tamarkin [1] in tenns of the Mittag-Lemer special functions (1.90):
:&
<p(x) = d: f Ea['x(x - t)a]l(t)dt.
o
(4.4)
One may verify (4.4) by applying the Laplace transform with (1.93) for the transfonn of the
Mittag-Lemer function and (7.14) below, taken into account.
Equation (4.3) with a = 1/2 was used by Gorenflo [4] for solving some applied problems
connected with the Newtonian heating of a homogeneous haIf-space by radiation from outside
across the boundary.
Equation (4.4) defines (up to the change of ,X by 1/'x) the resolvent (,XE - I+)-l of the
operator I+ of fractional integration. In this connection it is necessary to mention the papers
by Hille [1, 2], concerning the investigation of this resolvent, as well as considerating the special
properties of fractional integration operators in the general context of linear operator spectral
theory and ergodic theory - Hille and Phillips [1]. The resolvent (E - 'xM)-l of a more general
integral operator
:&
M", = f [ (x t;-1 + Ml (x, I)] ",(I)dl,
o
was investigated by Hromov [1], Matsnev [1], [2] and Matsnev and Hromov [1] with the aid of
the properties of fractional differentiation 'DO t and of asymptotics of the Mittag-Lemer function,
Mdx, t) being a function sufficiently smooth eyond t = x and having a behaviour "better" than
(x - t)a-l as t _ x. In this connection see also Kabanov [1, 2].
We note that Hille and Tamarkin [2] obtained bounds of eigenvalues of operators of fractional
integration type - see  23.2 (note 19.10). The extension of this result to the so-called singular
values was given by Faber and Wing [1].
The equation
00
<p(x) - rta) f (t - x)a-l<p(t)dt = 0, x e R l , 0 < a < 1,
:&
with ,X > 0 has the non-trivial solution <p(x) = Ce- o :&, a = ,Xl/a, which is its general solution
(Hardy and Titchmarsh [1], see also Titchmarsh [1]).
Brakhage, Nickel and Rieder [1] gave a closed fonn solution of (4.3) in terms of elementary
functions in the case when a is rational: a = mln < 1. The case a = 1/2 for the equation of
the type (4.3) considered on the whole real line was known to Liouville [5, p.285], (1834). The
equation
n-l :&
'" 'xl: f <p(t)dt
<p(x) - L.J real:) (x _ t)l-a k = lex),
1:=0 0
more general than (4.3) but with rational ale = kIn was solved by K06titzin [1] and Rieder [1].
The latter paper also contains an investigation of a system of equations in the fonn of (4.5).
x> 0,
(4.5)

 4. ADDITIONAL INFORMATION TO CHAPTER 1.
87
The equation
:&
,X f <p(t)dt
<p(x) - x Oi (x _ t)I-OI = l(x),
o
0< x < I,
with fixed aingularity at x = 0 is solvable in closed form. The corresponding homogeneous equation
L__ . ._1 . . al I . I 1 . L [0 1] 1 if ' r (1 +OI-l /)
PAD m generAl. a nontrivi 80 utlon, .lor examp e, m P" < P < 00, ,,> r( 01 )r( 1-1 p) ' see
Mihailov [1, p.29].
Let us also note that in Davis [I, p.l05] (see also [2]) there is given a procedure for solving
the equation
:&
( x ) - f P(t)<p(t)dt = le x ) x> a
<p (x - t)I-OI '
a
(4.6)
in the case when pet) is a polynomial and 01 is rational.
Equ&tiOlUl (4.3), (4.5) and (4.6) a4joint to differential (or integro-differential) equations of
fractional order, which are in general considered in  42.
In the applied problems nonlinear Abel equations arise. We cite, for example, Schneider [I],
which contains the simplest case [<p(x)]I+OIIP - 'x(1g+ <p)(x) = 0 with a special value of ,x, and
remark that there is a great deal of investigations of non-linear Abel equations of the second kind
with a non-linearity of a general type under the integral sign. We note Dinghas [1] as one of the
first such papers and the comprehensive works by Gorenflo and Vessela [I, 2], where a detailed
survey of the investigations with applications is given. See al80 Lubich [2] for a numerical method
of solving a non-linear Abel-type equations of the second kind.
We note that if we choose lex) == 1 in (4.3) and (4.4) and substitute then <p(x) = EOI(,XxOl)
into (4.3), we obtain the following relation
:&
2- f EOI('xtOl) dt = EOI(,XxOl) - I, 01 > 0,
r(o) (x - t)I-OI
o
(4.7)
for the fractional integration of a Mittag-Lemer function.
We alao note the related expression
:&
f tp-l E ( t 201 )
....!- 201,p dt = xp-1[E (x Ol ) _ E ( x 201 )]
rea) (x _ t)I-OI OI,P 201,P ,
o
01 > 0, (3 > 0,
(4.8)
which may be checked by directly applying the Laplace transfonn with the aid of (1.93). Equations
(4.7) and (4.8) were obtained in the papers by Humbert and Agarwal [1] and Agarwal [I], where
one may find other relevant relations, see also the book of Dzhebashyan [2, Ch.III, 1].
2.6. Equation (2.20) of fractional integration by parts allows one to construct new
biorthogonal function system using a given system <Pn(X), V1m(X) of such a kind:
b
f <Pn(x)V1m(x)dx = cSn,m, n,m = 1,2,...,
a
(4.9)

88 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Namely, let
+n(") = .(,,)1.\'+(.....), "m(") = .() V+ ( ":: ).
u(x), vex) being arbitrary functions, u(x)  0, vex)  o. Then it follows from (2.20), even if only
formaJIy, that n(X), Wn(X) alao satisfy (4.9). This idea is due to Erdelyi [3], who constructed
by this method new biorthogonal system in terms of hypergeometric functions 2f'1 and #'2 based
on Jacobi polynomials
""n(X) = 2f'1 (-n, J3 + ni "'Yi x), m(X) = cx'Y-I(1 - x).8-'Y 2f'1 (-m, J3 + mi "'Yi x),
2f'1 being the Gauss hypergeometric function (1.72). He gave other such systems as well. We
refer to  9.2 (note 5.4) and also  23.2 (note 18.7) in connection with this method in other cases.
The investigations of Pacchiarotti and Zanelli [1] and Zanelli [3], concerning fractional derivatives
of Legendre and Jacobi polynomials adjoin in a sense to Erdelyi's idea.
Equation (2.20) had earlier been used by Erdelyi [I, 2] in order to obtain some representations
for the Gauss hypergeometric functions, - see also similar representations for Appel functions FI,
F2, F4 in Manocha [I], [2].
C.M. Joshi [1] used fractional integrals (2.17) and (2.18) in order to obtain integral
representations for three Lauricella functions of three variables - Erdelyi, Magnus, Oberhettinger
and Tricomi [1, 5.14].
2.7. The following relation
n-l
"" (x - xo)a+k a+k
f(x) = L.J rea + k + 1) (V a + f)(xo) + Rn,m,
k=-m
with m < a, x > Xo  a and
Zo
Rn = ( la+nV a + n f )( x ) + 1 f( X - t ) a-m-I ( V a - m - 1 f )( t ) dt
, m Zo a+.. rea _ m) a+
a
due to Y. Watanabe [1, p.31], is the generalization of Taylor's expansion (2.63). From
here we obtain (2.63) if m = -n and Xo - a. However, the generalized Taylor series
00
f(x + h) = L: r(:;l) (V::/ r f)(x) with fractional derivatives was already formally
m=-oo
written down by Riemann [1]. The proof of the validity of such an expansion for certain classes
of functions was undertaken by Hardy [3] both for finite and infinite a.
2.8. Another variant of generalizing Taylor's series was suggested by Dzherbaahyan and
Nersesyan [1], [2]. Namely, let ao = 0, ai,"', am be an increasing sequence of real numbers such
that 0 < ak - at_l  I, k = 1,2,..., m. Let x > O. We introduce the notation
v(Ok) f _ll-(ak-ak-dVI+ak-l f
- 0+ 0+
( 4.10)
and remark that V(ak) f  V+f in general. The "fractional derivative" v(ak) f differs from the
Riemann-Liouville fractional derivative V+f by a finite sum of power functions as seen from

 4. ADDITIONAL INFORMATION TO CHAPTER 1.
89
(2.68). This fact allows us to anive at the generalized Taylor expansion
m-l (a) / :1:
I(x) = ,, (V Ic 1)(0) xalc + 1 (x _ t)a",-I(1'(a...) I)(t)dt
L.J r(1 + al:) r(1 + am)
1:=0 0
(4.11 )
(Dzherbashyan and Nersesyan [I, p.88j 2]) for functions I(x) having all continuous used derivatives.
In the cited papers the authors demonstrate the usefulness of introducing the derivatives
00
(4.10) in the problem of evaluating coefficients of the general power series I(x) = L: al:x alc ,
1:=0
( 1)(0.) 1)( 0 ) ll . b " th . . fi h d . . f f . .
Clk = r(l+alc) , as we as In 0 t8J.IUng e crIterIa or t e ecomposltlon 0 unctIOns In
the Dirichlet series. This approach to investigation of functions decomposable in generalized
power series or Dirichlet series was further developed by Dzherbashyan and Saakyan [1]. They
considered a generalization of Bernstein's theorem for absolutely monotone functions to the case
of the caUed < p > - absolutely monotone functions. The definition of this idea is based
on the fractional integro-differentiation of the form (4.10). Further generalizations of absolutely
monotone functions were given in this way by Saakyan [1], [2] and Dzherbashyan and Saakyan [2].
We note also that in the latter papers a generalization of the Taylor expansion was
suggested, which was associated with the Mittag-Lemer function and with generalized fractional
n-l
differentiation of the type n (1'': + >'jE)/, p > I, which was extnded later on to the case
j=O
o < p < 1 by Saakyan [3].
We remark also that Osler [3] dealt with the Taylor expansion of the form I(z) =
00
L: (1'a+aI: I)(ZO)(z - zo)a+d 1r(1 + a + ak) in the complex plane, the particular case of
1:=-00
such an expansion being earlier considered by Fabian [3]. We refer also to Osler [6] where a
certain integral analogue of the Taylor expansion in the complex plane is given. In  7.3 below
we di8CU88 a certain such integral analogue, different from that considered by Osler (see Remark
7.3). In connection with the generalization of the Taylor expansion (2.63), Badalyan [1] obtained
a relation of the type (4.11) for constructions more complicated than fractional derivatives.
2.9. The idea of differentiation with respect to the power exponent widely employed by
VolteITa [2] for evaluating integrals of power-logarithmic functions was developed by Rubin [10],
[14]. One may follow Rubin [10] and apply fractional integration with respect to {3 in order
to evaluate fractional integrals of the functions (x - a).6 -1 In II (1 I (x - a» (for x - a < 1). The
paper by Rubin [14] contains the generalization of this approach based on the application of
the convolution operation with respect to the variable p. One may obtain in this way relations
for evaluating fractional integrals of functions of the type (x - a).6- 1 fi (Inl: :I:a ) >'Ic , where
1:=0
lnk: = !nln... :, -00 < >'1: < 00, (3 > o.
....
k
2.10. Love [4] gave sufficient conditions for the existence of the fractional integral (2.38)
of purely imaginary order. He showed, for example, that if a function I is integrable over
6
[0,00] and the condition !t-1Wl(J,t)dt < 00,6> 0, is satisfied, where Wl(J,t) is the integral
o
modulus of continuity of I (see (13.24) below), then the fractional integral 1 exists for any
real 8. Corq,pare this with Theorem 13.5, which contains sufficient conditions for the existence of
the fractional derivative 1':+1. It was also shown by Love [4j 5, p.388r that 11 exists for a
function I e Ll(a,b) if and only if 1+i' I e AC([a,b])j then 11 e Ll(a,b) and the relation
l;i' J = I holds almost everywhere on (a, b).

90 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
2.11. The estimate (2.72) is a particular case of the inequalities
lIe"g Icf+""lIp  rc:: 1) lIe"g""lIp,
lIe. g Icf+""lIp  ) ( : ) aIm lIe"g""lI" ,
where Ot > 0, II > 0 and II lIP is the norm in Lp(O, b), get) e Cm([O,b]), m  I, g(I:)(t)  0,
k = 1,...,m -I, g(m)(t)  -a  0 (a > 0 in the second inequality). These inequalities were
proved by Bukhgeim [I], [2, p.46] and used by him in [2] in the investigation of the inverse
problems of reconstructing differential equations by given traces of their solutions.
2.12. Let X = Lp(O, I), 1  p < 00, or X = C([O, 1]) and let {la}a>O be a family of linear
operators in X. Is it true that the conditions -

(It ",,)(x) = I ",,(t)dt, lalfJ = la+fJ' a,p > 0, (1a"")(X)  0
o
for ",,(x)  0 define the family 101 uniquely so that
:&
(1a"")(x) = (Ig+",,)(x) = rta) I ",,(t)(x - t)a- 1 dt
o
(4.12)
x> a,
for all"" e X? The question was put by J. Lew at the Conference on Fractional Calculus in 1974,
see Osler [9, p.397]. The paper by Cartwright and McMullen [1] contains in fact a positive answer
to, this question under additional assumption that the mapping a - 101 is continuous from R
into L(X - X) in any Hausdorff topology.
2.13. We note the paper by Spain [1] who discussed the idea of interpolation of
integra-differentiation 1 01 by the expression
00 00 :&
1 01 sin'lfa L (_I)I:",,(I:)(x) L (-1)1: 1 I( ) 1:-1 ( )d
""=- + -- x-t ""t t
'If a-k a+k(k-l)! '
1:=0 1:=1 a
00
based on the interpolation relation F( 01) = sin", a", L: ( -12( 1:) . However, this approach
1:=-00
was not further developed because of the evident difficulties connected with consideration of the
composition [a IfJ "".
2.14. Based on the fractional integration operator 1(a)(x) = (1:+1)(x) Zanelli [I, 2]
introduced the notion of fractional order variation of a function as follows: v(a)(J; [a, b)) =
b b
-O f I h l- 1 11(l_a)(x+ h) - 1(1_a)(x)ldx which coincides formally with f 1(1'\1)(x)ldx. He
a a
investigated the connection of fractional differentiation 1':+1 and of fractional variation v(a)
with Stieltjes approximating polynomials and certain weighted means.
2.15. Based on (2.53) and (2.54), Penell [1] and Thielman [1] obtained expansions of the
integral ([2",,)(x) into series of Fourier-Bessel type by the known expansion of a function ",,(x)
into the series of trigonometric or Bessel functions, respectively.

 4. ADDITIONAL INFORMATION TO CHAPTER 1.
91
2.16. Let
pF',[CVl'''''CVP;X] = f: (cvdl:...(cvph: xl:
{31t.. ., {3, ({3dt .. . ({3,)1: k!
1:=0
be the generalized hypergeometric function (Erdelyi, Magnus, Oberhettinger and Tricomi [1,4.1]).
Misra [1] proved the following Rodrigues type formula
F. [ -n,cvl""'CVP;X ]
p+l , {3
(31, . . ., ,
r({3d ... r({3,) I-fJ ""cvp-fJ"
x "vO
r(cvd... r(cvp) +
X xCVp-fJ,,-l V CVp - 1 -fJ"-l XCVp-l -fJ,,-
0+
x xCV3-fJVCV-fJXCV-fJIVCV1-fJl [ XCV1-l ( l_ X ) n ]
0+ 0+ .
Here Rodrigues type formulae for classical polynomials are contained as particular cases. We alao
note that Koschmieder [2] used fractional derivatives to obtain some properties of the function
,Rf'
3.1. The variant of Theorem 3.12 when the weight is not changed, but the Holderian
uponent is variable (see Remark 3.5) is as follows. Let 1-'1 < p - I, 0 < J.l.1: < p - I,
k = 2,3,..., n -I, J.l.n > O. H l/p < cv < 1 + l/p, the operator 1+ is bounded from Lp([a, b), p)
Into H6([CI, b); p), where
{ min(cv - l/p, J.l./p) ,
,X=
-e + J.l./p,
if J.l. '# cvp - I,
if J.l. = cvp - I,
with J.l. = min J.l.I:, e > 0 (Karapetyants and Rubin [2]).
t2
3.2. Hardy and Littlewood's Theorem 3.5 has the following analogue for p = 1:
{ } II,
1'(I+<p)(")'fdr $ C + C i 1<p(")I(In+ 1<p(")I)'/fd,,,
1
q--
- l-cv'
with C depending only on cv and (a,b) (Flett [3]). A similar inequality had earlier been proved
by Zygmund [1] in the periodic case for Weyl fractional integrals - see  23.2 (note 19.7) below.
3.3. Let us consider in more detail the limiting case p = I/cv in Hardy and Littlewood's
theorem 3.5. Hardy and Littlewood [1] already noted that
1(Lp) C n Lr(a,b),
rl I
lip
but 10+ (L p ) rt. Loo(a, b).
( 4.13)
The latter is confinned by the example 1 = 1 <Pe where
<p.(") = I" - ;I- t (In I"  I f1¥ E L,(O, 1), 0 <. <, -1. (4.14)
l-
It is easily shown for this function that lex)  c (In 2/:-x ) p as x - 2/e - 0 (Karapetyants
and Rubin [4]).

92 CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Relations (4.13) lead to the natural idea. of constructing an intennediate space X, Loo C
X C U Lr, containing the range 1(Lp) and "close", if possible, to 1(Lp). We mention
rl
here two ways of constructing such a space. The first one is based on local properties of functions
I from lf (Lp), while the second one deals with the asymptotic behaviour of Lr-norms of
functions I when r - 00.
A. Let BMO(a, b) be the space of functions lex) E Ll(a,b) such that
11111- = sup mIl < 00,
Ic(a,b)
(4.15)
where
ml 1= II f I/(x) - iJldx,
I
iJ = II f I(x)dx.
J
(4.16)
Estimating the nonn (4.15) we may verify that
1(Lp) C BMO(a, b),
III <p1I-  GII<pllp.
Such an assertion was apparently first obtained by Peetre [1] in the multidimensional case for
Riesz potentials, although a similar result was earlier noted by Stein and Zygmund [3]. We refer
to paper by Reimann and Rychener [1] in this connection. As for BMO-space8 (the spaces of
bounded mean oscillation) in the case of a finite interval [a, b), one may refer for example to the
books by Kashin and Saakyan [I, Ch.5] and Garnett [I, Ch.5].
B. Karapetyants and Rubin [1] introduced the Banach space X -y (a, b) of functions I (x)
( En Lr(a,b» ) with the finite nonn sup(r--Yll/llr) and showed that the operator I,
rl rl
1 < p < 00, is bounded from Lp(a, b) into X-y(a, b) if "y  lip' and unbounded if"Y < lip'. The
proof is based on the fact that IIIIILp-Lr = O(r 1 / p' ) as r - 00.
We note that BMO(a, b) rt. X-y(a, b) and X-y(a,b) rt. BMO(a, b) for 0 < "y < 1. The first is
attested by the example lex) = lnx giving the relation II/lIr = (1 + 0(1» as r - 00 in the case
a = 0, b = 1. The example cOITesponding to the second assertion is given by
lex) = { O ( 1 ) "
In z-1/2
for 0 < x < 1/2,
for 1/2 < x < 1.
These examples were suggested by Karapetyants and Rubin [3].
One may find the development of the ideas discussed in  17.2 (note 13.1) below.
3.4. Let lex) E LlCa,b). The simple fact that (I:+/)(x) exists almost everywhere admits
the following sharpening: any point x, where (I:+/)(x) exists is its (left) Lebesgue point, see
(2.78) (Love (10». In this connection ass also  17.2 (note 12.6).

Chapter 2. Fractional Integrals and
Derivatives on the Real Axis
and Half-axis
The present chapter contains investigations about fractional integrals and
derivatives on an infinite interval. The functions under consideration are to
be chosen so that the corresponding integrals converge at infinity. We shall deal
with functions "vanishing" at infinity in an appropriate manner, e.g. with functions
in Lp(Rl) such that 1 < p < 1/0: or in Lp(R 1 ; p), when the condition on p may be
weakened owing to the weight p(x), or with weighted Holderian functions vanishing
at infinity. Fractional integrals will absolutely converge for such functions. One
may treat fractional integrals more widely by considering them as conditionally
convergent:
:&
1 f rp(t)dt =
r(o:) (x-t)l-a
-00
:&
1 lim f rp(t)dt
reo:) N-oo (x - t)l-a'
:&-N
Then one may admit locally summable functions rp( t), not necessarily vanishing at
:&+T
infinity if the behaviour of their means J rp(t)dt is subjected to certain conditions.
:&
Such a treatment will not concern here but we shall deal with it both in the
periodic (S 19, see (19.20) for example) and in the non-periodic cases U 14.3). See
also additional information in S 9.2 (notes 5.1-5.3, 5.10).
 5. The Main Properties of Fractional
Integrals and Derivatives
5.1. Definitions and elementary properties
The fractional integrals given by (2.17) and (2.18) are easily extended from the
case of a finite interval [a, b) to the case of a half-axis or axis. Properly speaking

94 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
definitions (2.17) or (2.18) by themselves may be used on the half-axis (a,oo) or
( -00, b) respectively due to the variable limit of integration. We shall use the
notations of (2.17) and (2.18) for the corresponding half-axis and write
:&
a 1 f cp(t)dt
(Io+cp)(x) = r(a) (x _ tp-OI ' 0 < X < 00.
o
(5.1)
We denote fractional integrals on the whole real axis by
:&
a 1 f cp(t)dt
(I+cp)(x) = r(a) (x - t)1-OI '
-00
-00 < X < 00,
(5.2)
00
a 1 f cp(t)dt
(I_cp)(x) = r(a) (t - x)1-OI '
:&
-00 < X < 00.
(5.3)
One may write (5.2) and (5.3) as a convolution:
00
(I)(.,) = rtn) f t%-l(., - t)dt
-00
(5.4)
00
= r(l n ) f ta-l(.,:i: t)dt,
o
where
{ tOl-l
t+- 1 = '
0,
t>O
t < 0'
(5.5)
{ 0
t Ol - 1 - '
- - I t I 0l - 1 ,
t>O
t<O
Fractional integrals l are defined for functions cp E Lp ( -00,00) if 0 < a < 1
and 1  p < l/a. Indeed
1 00
(I+) = r(1 n ) f ta-l(., - t)dt + r() f ta-l(., - t)dt.
o 1
The existence of the first term may be proved for almost all x for example using

 5. THE MAIN PROPERTIES OF FRACTIONAL INTEGRALS
95
inequality (1.33), while the second one exists for all x by the Holder inequality
(1.28) when 1  p < I/Ot.
In a similar fashion to (2.22) and (2.23) Liouville fractional derivatives
:&
01 1 d f f(t)dt
(1)+f)(x) = r(I-Ot)dx (X-t)OI '
-00
(5.6)
00
1 d f f(t)dt
(1)f)(x) =
r(l- Ot) dx (t - X)OI'
:&
-00 < x < 00,
are introduced in the case 0 < Ot < 1. If Ot  1, we set
00
( -n0i f)( x ) = (:l:I)n iF f tn-0I-1 f( 'P:t: t)dt, [ ] + 1
V::I: r( n _ Ot) dx n .... T n = Ot .
o
(5.7)
analogously to (2.30). In the case of the half-line (0,00) we consider
:&
1)01 x_I cr f f(t)dt
( o+f)( ) - r(n _ Ot) dx n (x _ t)OI-n+l '
o
(5.8)
00
1)01 (_I)n cr f f(t)dt
( _f)(x) = r(n _ Ot) dx n (t _ x)OI-n+l .
:&
The connection between Ii-tp and Itp analogous to (2.19) has the form
QItp = I;Qtp, (Qtp)(x) = tp(-x), -00 < x < 00. (5.9)
The operators I satisfy simple rules of commutation with the operators of
translation and dilatation. Let us introduce the notation
(Thtp)(x) = tp(x - h), x, h E Rl,
(IT 6 tp)(x) = tp(6x), x E R 1 , 6 > O.
(5.10)
(5.11)

96
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
It is easily verified that
Th If:. tp = If:. Th tp
(5.12)
(5.13)
II6If:.tp = 6 01 If:.II6tp.
Property (5.13) is also valid on the half-axis for the fractional integral Ig+tp i.e.
II6I+tp = 6 01 I+II6tp.
(5.14)
The semigroup property
I OI I fJ II') - 1 01 +/311')
+ +T - + T,
IItp = Ic:.+/3 tp.
(5.15)
is true as in the case of a finite interval. If the function tp is "sufficiently good",
then (5.15) holds for all a > 0, (3 > O. It may be verified in a similar fashion to
(2.21). Within the framework of the space Lp(Rl) (5.15) is valid for such a > 0,
{3 > 0 that Q + (3 < l/p - see S 5.2 below in connection with mapping properties of
I in Lp(Rl).
The fractional integration by parts formulae
00 00
J <p(x)(It.1/J)(x)dx = J 1/J(x)(Itp)(x)dx,
(5.16)
-00
-00
00 00
J f(x)(V:g)(x)dx = J g(x)(Vf)(x)dx.
-00 -00
(5.17)
are valid. Equation (5.16) may be obtained similarly to the analogous formula
(2.20) by directly interchanging the order of integration. Formula (5.17) follows
from (5.16) by rewriting I+1/J = g, Itp = f. In this fashion these formulae may
be proved for "sufficiently good" functions. It is not difficult to prove (5.16) for
tp(x) E Lp, 1/J(x) E Lr, p> l,r > 1,I/p+ l/r = 1 +a. It may be achieved with the
aid of Theorem 5.3 analogously to the proof of formula (2.20). The proof of (5.17)
in the case of summable fractional derivatives is given in Corollary 2 of Theorem
6.2 below.
Equations (5.16) and (5.17) hold on the half-axis as well. So we have
00 00
J tp(x)(Ig+1/J)(x)dx = J 1/J(x)(Itp)(x)dx.
o 0
(5.16')

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS
97
in the notation of (5.1) and (5.3).
Integrals of purely imaginary order a = i8 are defined similarly to (2.38):
00
'8 _ 1 d f i8
(r+ <p)(x) - r(1 + i8) dx t <p(x - t)dt
o
(5.18)
for "sufficiently good" functions.
We indicate now some elementary functions <p(x), for which If:<p may also be
evaluated in terms of elementary functions.
1. For <p(t) = e%t we have
(I<p)(x) = e%Z, Rea > 0,
(5.19)
where the signs chosen in the right-hand and left-hand sides must be the same.
00
Indeed, I+.(e Z ) = rlOl) J ez-ttOl-1dt = e Z . More generally we have
o
I(e%az) = a-OIe:!:az, Rea > 0, Rea > 0,
(5.20)
which is proved similarly to (5.19) in view of (7.5), obtained in S 7 below. Compare
also (5.20) with equations (22.26) and (22.27).
2. Equations
:!:a
If:(e:!:a sin bx) = (a 2 : b 2 )01/2 sin(bx =F a<p),
(5.21 )
%az
I(e%az cosbx) = (a 2 : b 2 )01/2 cos(bx =F a<p),
(5.22)
follow from (5.20) under the assumption that a  0, b  0, a 2 + b 2 > 0,
<p = arg(a + bi) E [0,71'/2] and Rea > 0, excluding the case a = 0, when
o < Rea < 1. In the latter case
I (sin bx) = b- OI sin (bx T a 2 7r ) ,
I(cosbx) = 6- 01 Cot (bx T a 2 7r ) .
(5.23)

98
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
{ (x - a),6-1
3. For <p(x) = '
0,
x> a,
Rep> 0, we have by (2.44)
x $ a,
{  ( ) a+,6-1
([+<p)(x) = rra+7f) x - a ,
0,
x> a,
x $ a.
4. If 0 < Rea < ReJJ the equations
[a [ r(JJ) ] _ %2f! r(JJ - a)
+ (l:i:ix)# -e (l:i:ix)#-a'
(5.24 )
[a [ r(JJ) ] _ r(JJ - a)
- (x:i: i)# - (x:i: i)#-a '
(5.25 )
are valid with the power functions (1:i: ix)#, (x:i: i)# being understood in the
normal way, i.e. as corresponding values of the main branch of the analytic function
z# in the complex plane with the cut along the positive half-axis
# - I 1 # i# arg % I . 1 - 0
z - z e , 1m arg z %=t+i£,t>O - .
£_+0
(5.26)
By choosing (5.26) we may write that
(:i:ix + 1)# = (1 + x 2 )lj e%i#arctgz, x E R 1 ,
(x :i: i)# = (1 + x2) e%i#arcctgz, x E R 1 .
(5.27)
Equation (5.24) may be rewritten as
[a [ r(p) ] = e%a1ri r(p - a) .
+ (x:i: i)# (x:i: i)#-a
(5.28)
We find it convenient to prove equations (5.24) and (5.25) later, at the end of S 7.1.
For the moment we remark only that these formulae are deducible one from each
other in view of the connection given in (5.9), provided that we take into account
that (1:i: ix)#lz=- = eTiWIr/2(:i: i)# by (5.27). Thus it will be sufficient to prove
only one of the equations, either (5.24) or (5.25).
5.2. Fractional integrals of Holderian functions
Results of this subsection are similar to those of S 3, the difference being in the
specialness due to the presence of infinity on the axis Rl or half-axis. We start

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS
99
with the case of weighted Holderian functions on the half-axis R = [0,00]. We
consider fractional integrals
:&
la -  f cp(t)dt
o+cp - r(Q) (x - t)l-a'
o
(5.29)
00
I a -  f cp(t)dt
-cp - r(Q) (t _ x)l-a ' X > o.
:&
The statement about their Holderian property on the half-axis will be obtained by
reducing it to the case of a finite interval via the following lemma.
Lemma 5.1. The transformation y = 1/(x + 1) maps the space H>'(Rjp),
p = p(x), x > 0, onto the space H>'([O, 1]; r), where
r = r(y) = p[(1 - y)/y), 0 < y < 1.
(5.30)
The lemma's proof may be obtained by direct verification. Note that the
change of variable y = 1/(x + 1) transforms the Holderian condition (1.6) into
(1.1).
Theorem 5.1. Let cp(x) E H&CR; p), where
n
p(x) = (1 + x)#J II Ix - XI:I#Jk, 0 = Xl < X2 < ... < X n < 00
1:=1
(5.31 )
Let also  + Q < 1 and  + Q < Ill: <  + 1, k = 2,3,..., n. If
n
PI <  + 1, P + LPk < 1-,
1:=1
(5.32)
then IC+cp E H;+a(11.; p.). If +Q < PI < + 1 (or III = 0) or Q- < p+ t PI:,
k+l
then Icp E H;+aCRjp.). Here p.(x) = (1 + x)-2a p (x) in both cases.
Prool. Theorem 5.1 is reduced to Theorem 3.4 by changing variables. Indeed the

100 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
substitutions y = 1/(x + 1), r = 1/(t + 1) lead to
z 1 ( 1 )
f cp(t)dt - I-a f cp -T dr
(x - t)l-a - y r 1 + a ( r - y)l-a'
o y
(5.33)
00 y ( 1 )
f cp(t)dt I-a f cp 7' dr
(t - X )l-a = y r 1 + a (y - r)1-a'
z 0
(5.34)
By Lemma 5.1 the change y = 1/(x + 1) maps the space H>'(R;p) onto the space
H>'([O, 1]; r(y» with weight
n
r(y) = p[(1 - y)/y] = c II Iy - YI:I"k,
1:=0
(5.35 )
n
where Yo = 0, P.o = -p. - L: Pic, and YI: = (1 + XI:)-I, k = 1,2,..., n. Applying
1:=1
Theorem 3.4 to the integrals (5.33) and (5.34) we obtain the proof of the theorem
after simple transformations. .
Corollary 1. In the case p(x) = xll(1 +x)" the operator Ig+ maps H6(Rjp) into
H;+a(Rj p.), A + Q < 1, p.(x) = x ll (1 + x),,-2a if
II < A + 1, p. + II < 1 - A.
(5.36)
The same is true for the operator I if the conditions (5.36) are replaced by
A + Q < II < A + 1, Q - A < p. + II.
(5.37)
We also note a particular case of this corollary, namely if cp(x) E H>'([O, 00]),
cp(O) = cp(oo) = 0, then for Q < 0, A + Q < 1 we have
z
f cp(t)dt (x)
(x - t)1-a - (1 + x)2a'
o
(5.38)
where (x) E H>.+a([o, 00]), (O) = (oo) = 0.
Statements of the type (5.38) are of interest for functions cp(x) E H>'(R)
without the additional assumptions cp(O) = cp( 00) = 0. For this purpose we

 5. THE MAIN PROPERTIES OF FRACTIONAL INTEGRALS 101
denote u(x) = <p(oo) + «10\.) , so that u(O) = <p(0), u(oo) = <p(oo). Since
I+[(1 + x)-l-a] = r(l+:;(l+Z) by (2.49), we have
<p(0) x a <p(oo) x a + l
la u = + .
0+ r(I+Q)I+x r(I+Q)I+x
Applying then (5.38) to the function <p(x) - u(x), we obtain the following
Corollary 2. If <p(x) E H>'([O, (0», then
Z
f <p(t)dt _ <p(0)  + <p(oo) x a + l + (x)
(x - t)I-a - Q 1 + x Q 1 + x (1 + x)2a'
o
for Q > 0,  + Q < 1 with (x) E H>.+a([o, (0)), (O) = (oo) = O.
Now we give a theorem for fractional integrals on the whole real line which
is similar to Theorem 5.1 but essentially different due to other conditions on the
parameters.
Theorem 5.2.. Let <p(x) E H6Ck 1 ;p), where
n
p(x) = (1 + x2) II Ix - XI:I#Jk, -00 < Xl < ... < X n < 00. (5.39)
1:=1
n
If +Q < 1, +Q < Pk < + 1, k = 1,...,n, and Q- < J.l+ L: Pk < 1-,
1:=1
then I+<p,I<p E H;+a(Rl;p.j, where p.(x) = (1 + x 2 )-a p (x).
Prool. In view of (5.9) it is sufficient to consider the fractional integral
f(x) = I<p only. Since we may transfer the origin to the point Xl by
translation, it is clear (due to (5.12)) that we have to prove the theorem for
the case 0 = Xl < X2 < ... < X n < 00. It is not difficult to see then by
Lemma 1.1 that f(x) E H;+a(Rlj p.) if and only if f(x) E H;+a(R; p.) and
. n
f(-x) E H;+a(R;PIP2)' where PI = X#Jl, P2 = (1 + x 2 )-P, (3 = -2Q + L: Pk.
k=2
We have taken into account here that 0 < CI  Pl1:();:()  C2 < 00 for X E R.i.
The assertion that f(x) E H;+a(R; p*) is contained in Theorem 5.1. Further for
X > 0 we have
Z 00
1 f <p( -t)dt 1 f <p(t)dt
f( -x) = r(a) (x - t)l-a + r(a) (t + x)l-a '
o 0

102
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
The first term here belongs to H;+OI(Rj P1fJ2) by Theorem 5.1 as well. Let us
denote the second term by G(x). It is an infinitely differentiable function for
o < x < 00. In order to specify its behaviour when x  0 and x  00, we set
2 00
f cp(t)dt f cp(t)dt
r(a)G(x) = (t + x)l-OI + (t + x)l-OI = G 1 (x) + G2(X).
o 2
For 0 =:; x =:; 1 we see that G(x) E H;+OI([O, l]jP1) by Lemma 3.1 for G 1 (x) and
by the infinite differentiability of G2(X). For x  1 it is easy to show that
Gdx) E H+0I([1,oo]jP1P2) taking into account the infinite differentiability of
G 1 (x) for x  1 and the obvious behaviour of G 1 (1/y) in a neighbourhood of the
point y = O. As for G2(X), x  1, we have after the substitution t = 1'-1 + 1,
x = y-1 - 1 that
00 1
f cp(t)dt 1-01 f 1/J( T)dT
(t + x)l-OI = Y (1' + y)l-OI'
2 0
(5040 )
where 1/J(T) = T- 1 - OI cp () E H([O, 1];P3), by Lemma 5.1 and P3(T) =
T1+OI p eT ) = T 1 + 0I +#J°p4(T) with weight P4(T) not already being "attached" to
n
the point l' = 0, and Ilo = -Il - L: Ille (see (5.35». Since T 1 + 0I +#J01/J( 1') E H>'([O, 1])
1e=1
and T 1 + 0I +#J0,p( T)IT=O = 0, we see that 11/J( 1')1  CT--Y, 'Y = 1 + a + Ilo - A, in the
neighbourhood of the point l' = O. Applying Lemma 3.1 with {3 = A (which is
possible under the condition a - A < -Ilo < 1- A) we see that the right-hand side
of (5040) belongs to H+OI([O, l]jy 201 +#J° p4 (y» if we take into account its infinite
differentiability for y> O. Then the left hand side belongs to H;+OI(R;p.(x» by
Lemma 5.1. .
Remark 5.1. Some results on mapping properties of fractional integrals in the
spaces of smooth functions on the axis or half-axis may be found in S 8.2, 804.
5.3. Fractional integrals of summable functions
We consider here fractional integrals of functions cp E Lp, given on the axis or
half-axis. The difference from the case of a finite interval is the following. In the
case of a finite interval the fractional integration operators were defined (see S 3.3)
on any space Lp, 1 $ p  00, and mapped Lp, 1 < p < 00, into Lq with any q such
that 1  q $ p/(1 - ap) when ap < 1 and 1  q < 00 when ap  1. But for the
case of the axis or half-axis these operators are well defined for 1  p < l/a and
may map Lp into Lq for 1 < p < l/a and q = p(1 - ap) only. More exactly, the
Hardy-Littlewood Theorem 3.5 in the case of the whole axis holds in the following
form.

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS 103
Theorem 5.3. Let 1 $ p $ 00, 1 $ q $ 00, Q > O. Operators If: are bounded from
L p (R 1 ) to Lq(R1) if and only if 0 < Q < 1,1 < p < I/Q and q = p/(I- QP).
We omit the proof of this theorem as well as that of Theorem 3.5 (see
references in S 9) and demonstrate here only the simple proof of the necessity of
conditions Q E (0,1), p E (1, I/Q), q = p/(1 - QP). Let III.tCPllq $ cllcpllp. Then
IIl.+II.scpllq $ cllII.scplip as well, II.s being the operator (5.11). By (5.13) and the
equality IIII.scplip = 6- 1/p IIcplip we obtain IIl.+cpllq $ c6a+-;- IIcplip. Letting 6 to tend
to 0 and to 00, we note that this inequality may be valid only for l/q = l/p - Q.
Since q > 0 we have p < 1/ Q. It remains to exclude the case p = 1. The function
{ Iln -'1 !
cp( x) = z z '
0,
0< x < !,
x  (0, 1/2),
'Y> 1,
(5.41 )
is an example of a function in L 1 (R 1 ) such that (l.+cp)(x)  L l /(1_a)(R 1 ) if
1 < 'Y < 2 - Q. Indeed for 0 < x < 1/2 we have
z
f dt x a - 1 In 1 -'Y 1
r(Q)(Icp)(x) > x a - 1 t In'Y l/t = 'Y _ 1 z ,
o
(5.42 )
so l.+cp E L 1 /(1_a)(R 1 ) only when ('Y - 1)/(2 - Q) > 1, i.e. 'Y> 2 - Q.
It is evident that Theorem 5.3 is valid also for fractional integrals (5.29) on
the half-axis (0,00).
Some information for the case p = 1 may be found in Theorem 5.6.
We give the following weighted variant of Theorem 5.3 for the case of the
half-axis.
Theorem 5.4. Let
1  p < 00,
1
0< Q < m +-,
P
(5.43 )
o $ m $ Q,
p
q=
1 - (Q - m)p
with m f; 0 for p = 1. Then operators 1, I+ are bounded from Lp(R;xl') into
Lf(R; XII), v = (/l/p - m)q:
{ CO } 11q { CO } 1/P
! z"lecp)(z)!fdz  K ! z" Icp(z) I"dz
(5.44)

104 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
(similarly for IC+tp), where p < p - 1 for the operator IC+ and p > Otp - 1 for the
operator I .
The statement of Theorem 5.4 for operator IC+ was earlier proved in Theorem
3.7 and it may be obtained for I from its validity for IC+ by substituting
l/x = y, y-1-atp(x) = tp1(Y),
Pl = -P + p + Otp - 2, V1 = -11- 2 + (1 - Ot)q
in view of the equality (IC+tp)(x) = yl-a(Itpd(y).
Let us note the important particular cases of Theorem 5.4. The case P = 0,
m = 0 leads to Theorem 5.3, while the case p = 0, m = Ot gives the inequalities
00 00
f xapl(Itp)(x)IPdx  K P f Itp(x)IPdx,
o 0
(5.45 )
1  p < I/Ot, 0 < Ot < 1;
00 00
f xapl(Io+tp)(x)IPdx  K P f Itp(x)IPdx,
o 0
(5.46 )
1 < p < 00, Ot > 0,
known as Hardy inequaliti.s. One may also obtain these inequalities independently
of Theorem 5.4 by using Theorem 1.5 by limiting oneself to the condition
1 < p < l/a in (5.45). Theorem 1.5 gives herein the value of the constant K:
K=rG-o)/rG),
K = r (  ) / r ( 0 +  )
for the inequalities (5.45) and (5.46) respectively. It is not difficult to show that
these constants are sharp.
We find it convenient to single out the case m = Ot (i.e. the case
q = p) in Theorem 5.4 by rephrasing it (after the notational change p/p = -1',
tp(x) = x 7 f(x» in the following way.
The operators x fJ Ix7 and x fJ IC+x 7 , Ot > 0, are bounded from Lp(Ri) into

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS
Lp(R; X- p (a+,6+'Y», p  1, if (Q + ...,.)p < 1 and (...,. + l)p > 1 respectively:
105
00 00
f x-(a+,6+'Y)Plx,6 Ix'Y f(x)IPdx  KP f If(x)IPdx,
o 0
( 5.45')
1  p < 00, (Q + ...,.)p < 1, Q > 0;
00 00
f x-(a+.8+'Y)Plx,6 Io+x'Y f(x)IPdx  K P f If(x)IPdx,
o 0
(5.46')
lp<oo, (...,.+I)p> 1, Q>O.
In particular, the operators x,6 Ix'Y and x,6 I+x'Y are bounded in Lp(R) in the
case Q + fJ +...,. = 0 under the above conditions.
The inequalities (5.45') and (5.46') with x- pRe (a+,6+'Y) instead of x-(a+,6+'Y)p
are valid for complex values of Q, p, ...,. also if pRe(Q +...,.) < 1 and p(Re...,. + 1) > 1
respectively, in both the following cases ReQ > 0, 1  p < 00 or ReQ = 0, Q f; 0,
1 < p < 00. We observe that the purely imaginary case Q = i8 may be considered
by representing x-'Y Ix'Y in the form I + (x-'Y Ix'Y - I) where the first term
may be treated by Lemma 8.2 and the second one by Theorem 1.5.
In the bounding cases of restrictions to inequalities (5.45') and (5.46) when
p = 1 and ...,. = 1 - Q or ...,. = 0 respectively, the integrals on the left-hand side of
these inequalities may diverge. In these cases we arrive at the inequalities (3.17"')
and (3.17") instead of (5.45') and (5.46'). The inequalities (3.17"') and (3.17")
with A = 0 show in particular that the operator x-I Ixl-a is bounded from
L 1 (b, 00); In k.:p-x) into L 1 (b, (0), and x-a 10+ is bounded from Ll (0, b); In )
into L 1 (0, b), 0 < b < 00.
We give now without proof the generalization of Theorem 5.4 for the case of a
general power weight (5.39). We consider x E n where n may be the half-axis R
or axis Rl. In the case n = R we set
o = Xl < .. . < X n < 00.
(5.47)
Let us denote
{ (-m)q
VI: = ( Q _ * _ m ) q + C I:
for Ill: < QP - 1,
for Ill:  QP - l,cl: > O.

106 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Let Po = -P - PI - ... - Pn. We also let
V) = _ P1Q _ t VI: _ { POq/p for Po > 1- p,
P 1:=2 € - q/P' for Po  1 - p, € > 0,
( 2 )  { Iloq/p
V oo = -mq - L.JVI: - ,
1:=1 € - q/p
for Po > 1 - p,
for Ilo  1 - p, € > 0,
V) = - ( IlO + m ) q - tVI:'
P 1:=1
in order to define weighted functions of the type (5.39):
! (1 + X)II) x(#Jl/p-m)q fI Ix - XI: 11110 for n = R,
1:-2
r+(x) = n-
(1 + Ixl)lI) n Ix - XI: 11110 for n = R1,
1:=1
{ (l+x)IIi:) fI Ix-xl:lllk for n=R,
r_(x) = 1:=1
r+(x) for n = R1.
Theorem 5.5. Let 1 < p < 00, 0 :$ m :$ a, 0 < a < m + lip, and let p(x) be the
weight (5.39) satisfying condition (5.47) in the case of the half-line. Let
Il/c < P - 1, k = 2,3,... ,n.
( 5.48)
If in addition to (5.48) we also have PI < p - 1, then the operator Ig+ is bounded
from Lp(R,p) into Lq(R,r+). If in addition to (5.48) we have that Il < 1- ap,
then I is bounded from Lp(R,p) into Lq(R,r_). Finally the operators l
are bounded from Lp(R1,p) into L q (R 1 ,r:i;) if in addition to (5.48) we have that
III < p - 1, Po < 1 - ap.

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS 107
We note the particularly useful case of Theorem 5.5:
 
tll"I"I(I+)(")I'd" } · :S c tll"I"I(")I"d" }' (5.49)
l<p<oo, Qp-l<JJ<p-l,
1 1 1
--Q<-<-
p - q - p'
l+v l+JJ
- = - - Q.
q P
An inequality similar to (5.45) is valid in the case of the whole axis Rl as well,
namely
{ 00 } l
_£ I., l-a"I(I )(., ) I" d., :S KII II",
1 < p < I/Q.
(5.50)
This is contained in Theorem 5.5, but may be obtained also with aid of Theorem
1.5 by passing to the half-axis.
We prove now a simple theorem, which applies to the case p = 1 and is useful
in applications.
Theorem 5.6. Let J(x) = I+cp or J(x) = Icp. If cp(x) E Ll(R;p),
p(x) = (1 + x)#J, Q - 1 < fl  0, then

{i<1 + .,)"1/(")1' d., }' :S K i(1 + ")"I(")ld." (5.51)
where 1  r < 1/(1 - Q) and v = r(1 - Q + fl) - 1 except the case J(x) = I+cp,
JJ = 0, when v < r(1 - Q) - 1.
Proof. Let J(x) = Icp and let A be the left-hand side of (5.51). We have
00 ( t ) l/r
A :S r(1 Q ) ! 1(t)1 ! (1+ .,)"(t - .,)(a-I)r d., dt
by the generalized Minkowsky inequality (1.33). The substitution x = t - (1 + t)

108 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
gives
t e/(H1)
f (1 + x t (t - x )(a-1)r dx = (1 + t)/J r f (1 - )" (a-1)r   c(1 + t)/J r
o 0
in view of the inequalities v > -1 and (Q - l)r > -1. Thus the above estimate for
A leads to (5.51). The case f(x) = Io+<p is treated similarly, the only difference
being that we deal with the integral
00 00
f (1 + x t(x - t)(a-1)r dx = (1 + tt+ 1 +(a-1)r f (a-1)r( + 1)". .
t 0
Remark 5.2. Theorem 5.6 is valid on the axis also in the case of the weight
(1 + Ixl)/J, where Q - 1 < fl  0, v = r(1 - Q + fl) - 1 if fl f; 0 and v < r(l- Q) - 1
if JJ = O. The proof is similar.
We conclude this subsection by constructing a space of functions summable on
the axis R1 which is invariant with respect to fractional integration. The spaces
Lp or Lp(p) with a power weight do not possess this property. We introduce the
space Lp,w with exponential weight by defining the norm
{ 00 } l/p
1I<pIIL.,. = _! e-w'I<p(tjl'dt ,
1  p < 00.
(5.52)
We denote also by C w = C w (R 1 ) the space of functions <p(t) such that e-wt<p(t) E
C(R1), 1I<pllc... = max e-wtl<p(t)1 and for brevity set Lp,w = C w in the case p = 00.
Theorem 5.7. The operators l, Q > 0, are bounded in Lp,w, 1  p  00, and
II Ia l1 < { (p/lwl)a, 1  p < 00,
:i: Lp,...-Lp,... - I I -a
w , p = 00,
(5.53 )
and
IIlIIL1,...-Ll,... = IIIlIc..._c... = Iwl- a .
(5.54)
if:i:w > 0, respectively.

 5. THE MAIN PROPERTIES OF FRACTIONAL INTEGRALS
109
Proof. For w > 0 we have
00 { 00 } l/P
r(n)II I .t<PIIL..w  / to-1dt 1 .-"'"I<P(Z - t)lPdz
(5.55)
00
= f e-1fta-ldtllcpIlLp.... = r(a) (5) a IIcpIlL p ,....
o
In order to obtain (5.54) it remains for us to observe that IIIIIL1.'" = Iwl-allcpIlL1,...
for non-negative functions cp(x), which may be perceived from the operations in
(5.55), and that IIIcpllc... = Iwl-allcpllc... for the function cp(x) = e WZ . .
5.4. The Marchaud fractional derivative
Liouville's fractional derivatives (5.6) on the axis R 1 may be reduced in general to
a more convenient form than (5.6). Let us suppose temporarily that a function
f(x) is sufficiently "good", for example j(x) is continuously differentiable and with
its derivative, j'(x), vanishes at infinity as Ixla-l-e, c > O. We suppose that
o < a < 1. We have
00
(1)';-1)(z) = r(l  n) :z f ,-0 f(z - t)dt
o
00
= r(l  n) f ,-0 f'(z - t)dt
o
(5.56)
00 00
= r(1 c:. a) f f'(x - t)dt f a
o t
00
_ a f j(x)-j(X-) 
- r(l-a) l+a .
o
Let us denote
00
(D a f)(x) = a f j(x) - j(x - t) dt
+ r(1 - a) t 1 + a
o
Z
- a f j(x) - j(t) dt
- r(1 - a) (x - t)l+a '
-00
(5.57)

110
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
so D./ == 'D+.f for sufficiently "good" functions f(x).
Similar transformations lead to the expression
00
( D a f )( x ) = a f f(x) - f(x + t) -00 < x < 00,
- r(1 - a) t 1 + a '
o
(5.58)
which replaces 'D+.f. Constructions (5.57) and (5.58) will be called Marchaud
fractional derivatives.
It is clear that integrals (5.57) and (5.58) exist under more general assumptions
for the function f(x): the above restrictions were needed in order to realize the
simple transformation (5.56) from 'D+.f to D+.f. It is evident that integrals (5.57)
and (5.58) exist for example for bounded functions satisfying the local Holder
condition of order A > a. This may be weakened to A = a if one takes functions
f(x) belonging locally to the space Ha,-a, a > 1 and bounded at infinity.
It is natural to ask the question whether 'D+.f == D+.f not only for "sufficiently
good" functions, but also for all those functions f(x) for which 'D+.f and D+.f exist
(almost everywhere for example). Does D+.f exist if'D./ exists and vice versa?
The second question may be answered in the negative at once: D+f exists
for the function f(x) == const and Df == 0, while 'Df does not exist for
f(x) = const. In general, let f(x) be locally Holderian of order A > a and
nonvanishing at infinity, for example, tending to a constant or even growing (!) as
Ixl a - e . Then D+.f exists. This is not true for 'D+f which requires better behaviour
of f( x) at infinity.
The answer to the first question is more difficult at least because the domains
of definition of the operators 'D+.f and D+f prove to be different from each other.
This difference is closely connected with the problem of inversion of fractional
integrals. Which version
DJ1f> == If> or VI1f> == If>
is more natural? The second version has already been used in the case of a finite
interval (see S 2.6). In the case of the axis R 1 , the situation is as follows: if
If> E L", then the first version will work for all admissible values of p, 1  p < l/a,
while the second version fits the case p = 1 only (see S 6.2 below). So Marchaud
fractional derivatives D+. f are more convenient on the R 1 than Liouville fractional
derivatives V+.f as they allow more freedom for f(x) at infinity.
It goes without saying that the differences between V+f and D+f discussed
above and connected with their behaviour at infinity will not be present in the case
of a finite interval.
In future Marchaud fractional derivatives for "not very good" functions f(x)

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS
111
will be understood to be conditionally convergent. Namely, let
00
( Da f )( x ) = a j f(x) - f(x =F t) dt.
%,e r(1 - a) t 1 + a
e
(5.59)
Then by definition
Df = lim D ef,
e-O ·
where the character of the convergence will be defined by the problems under
consideration. Thus, we shall treat the passage to a limit in (5.60) in the norm of
the space Lp while studying the inversion DII;' = 1;', I;' E Lp, in S 6.2.
The expressions in (5.59) will be termed truncated Marchaud fractional
derivatives.
Let us note the properties of Marchaud fractional derivatives similar to (5.9),
(5.12) and (5.13):
(5.60 )
II6Df = 6- a D% TI 6f.
(5.61 )
(5.62)
QD%f = D%Qf, ThDf = D%Thf,
Remark 5.3. One may obtain Marchaud fractional derivatives on the half-axis
(0, (0) analogously to the equalities (5.56) - (5.58). The derivative (Vf)(x)
is transformed to (Df)(x), x > 0, without changes, but instead of (vg+f)(x),
x > 0, we obtain, following (5.56):
:&
a 1 f(O) 1 j f'(x - t)
(Vo+f)(x) = r(1 _ a) x a + r(1 _ a) t a dt
o
= f(O) 1 + 1 j :& f'(x _ t) ( a j :& -l-a +  ) dt
r(1 - a) x a r(1 - a) x a
o t
:&
f(x) a j f(X)-f(x-t) d
= + t,
r(1- a)x a r(1 - a) t 1 + a
o
so
:&
D a d.:f f(x) a j f(X)-f(t)dt
o+f - r(1 - a)xa + r(1 - a) (x - t)l+a '
o
(5.63)
x > 0, ° < a < 1,

112
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
which plays the role of the "left-hand sided" Marchaud derivative on the half-axis
(0,00).
Constructions similar to (5.63) will be used on the finite interval too - see
S 13.1.
As for Marchaud fractional derivatives of order a > 1 we shall consider them
in S 5.6 below.
5.5. The finite part of integrals due to Hadamard
00
Comparing the Marchaud fractional derivative D1 = rcla) J I(Z::/(Z) dt with
o
fractional integrals If, we see that Df is formally obtained from If if we
replace a by -a. Subtraction of f(x) here provides the convergence of the integral.
Thus D f are closely connected with ideas concerning divergent integrals. We
elaborate on some of these ideas.
Definition 5.1. Let a function (t) be integrable on an interval c < t < A for any
A > 0 and 0 < c < A. The function (t) is said to possess the Hadamard property
at the point t = 0 if there exist constants ale, b and Ale > 0 such that
A N
f (t)dt = L alec->'k + b In  + Jo(c),
£ le= 1
(5.64 )
where lim Jo(c) exists and is finite. By definition
£-0
A
p.f. f (t)dt = lim Jo(c).
£-0
o
(5.65)
The limit (5.65) is called a finite part (partie finie) of the divergent integral
A
J (t)dt in the Hadamard sense or simply an integral in the Hadamard sense. The
o
constructive realization of the function Jo(c) is sometimes called a regularization
A
of the integral J (t)dt.
o
It is not difficult to see that constants ale, b, Ale in (5.64) do not depend on A.
If (t) is integrable at infinity, by definition we put
00 A 00
p.f. f (t)dt = p.r. f (t)dt + f (t)dt
o 0 A
(5.66)

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS
113
and it easy to see that this definition does not depend on the choice of A.
00
Now we return to D.! and consider the divergent integral J J(:ldt . The
o
next lemma holds.
ma 5.2. Let 0 < a < 1 and let f(x) be locally Holderian of order  > a.
Then the function (t) = f(x - t)t- 1 - a possesses the Hadamard property at the
point t = 0 for each x and if If(t)1  cltl a - e , € > 0, as t -+ -00, then
00 00
f j f(x-t) d = j f(x-t)-f(X) d
p. . t 1 + a t t 1 + a t.
o 0
The proof of this lemma may be obtained by direct verification of condition (5.64)
and definitions (5.65) and (5.66).
Lemma 5.2 states that
D%f = p.f.Ia f, 0 < a < 1.
(5.67)
One may also say that (D/)(x) represents for any x the analytic continuation
of the function (Ia f)(x) from the half-plane Rea < O. Also this continuation
is extended to the half-plane Re a <  for the functions f( x) mentioned in
Lemma 5.2. This follows from the analyticity of the functions l(a) = If
and 2(a) = 1)a f in the half-planes Rea > 0, Rea < 0 respectively (for
sufficiently "good" functions f) and from the coincidence of their boundary values:
lim l(a) = lim 2(a) = rima f.
Rea-O+ Rea-O-
The conclusion D+f = p.r. I;: f, 0 < a < 1, similar to (5.67) is valid also for
the Marchaud fractional derivative (5.63).
The interpretation of Df in (5.67) indicates the way how one may make
Marchaud derivatives meaningful for a  1. For this purpose we give the
regularization of the divergent integral Ii-a f, a > 0, in the next lemma.
Lemma 5.3. Let locally f(x) E em and let f(m)(x) satisfy locally the Holder
condition of order, 0   < 1. Then the function (t) = f(x - t)t- 1 - a possesses
the Hadamard property at the point t = 0 for any x if Rea < m +. If also

114 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
If(t)1 =:; cltl a -£ for t -+ -00, then
m Ie
00 1 f(x -t) - E(-I)I:Lrf(l:)(x)
1 / f(x-t) 1 / 1:=0 Ie.
r( -a) p.r. t1+a dt = r( -a) t 1 + a dt
o 0
(5.68 )
00
1 / f(x - t) m (_1)1: f(Ie)(x)
+- dt+ ,
r(-a) t 1 + a  k! r(-a)(k - a)
o 1:_0
where Rea < m + A, a f; 0,1,2, . . .
The prool of this lemma is obtained by direct verification.
We note that after the choice m = [a] the equality (5.68) may be rewritten as
follows
[a] Ie
00 00 f(x - t) - L:  f(I:)(x)
1 / f(x - t) 1 / 1:=0 1:.
r(_a) p.f. t1+a dt = r(-a) t 1 + a dt,
o 0
a 1= 0, 1, 2, . . . ,
(5.68')
where the integral on the right-hand side converges absolutely for the functions
mentioned in Lemma 5.3. Result (5.68') has an advantage over (5.68) in being
more compact.
In view of (5.67) and of the analyticity of the right-hand side in (5.68) with
respect to a, it is natural to use (5.68) for defining the fractional derivative of order
a, Rea > O. Let us show that such a definition agrees well with the definition (5.7)
in the case of sufficiently "good" functions f.
Theorem 5.8. Let f(x) satisfy the assumptions of Lemma 5.3 with m  [a] + 1.
Then the Liouville fractional derivative 'Dt./ coincides with (5.68) for any a such
that Rea > 0, a f; 1,2,. . .
Proof. Let {3 = a - n + 1, n = [a] + 1 (0 < (3 < 1) in correspondence with (5.7).
For the "truncated" Liouville derivative we have
:&-£ 00 n-l I:
 / f(t)dt = (-I)n ( {3) / f(x - t)dt + '" (-1) ({3)1: f(n-l-I:) ( x _ )
dxn (x _ t)f3 n tn+f3 L.J cl:+f3 c ,
-00 _ £ 1:=0
(5.69 )

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS 115
which may be proved by direct differentiation of the left-hand side integral.
Regularizing the right-hand side integral as in (5.68), we obtain
/ 00 /(x - t)dt = / 1 [ /(X _ t) _  (_1)1: (t _ g)1: /(11:) ( x _ g) ] 
tn+fJ L.J k! tn+fJ
£ £ 1:=0
(5.70)
+ / 00 /(x - t) dt +  (_I)n-I:-1 al:(g) /(n-t-1)(x _ €)
tn+fJ L.J (n - k - I)! gl:+fJ
1 1:=0
with the designation
1 I;,"
al:(g) = gf3+1: /(t - €t- 1 -l: t -n- f3 dt = / n-1-1:( + 1)-n-f3d.
£ 0
Substituting (5.70) into (5.69) we arrive at the equality
00 1 n-l I:
:;n / f(; I) dl = (-1t(,B)n{ / [f(" - t) -1; (-k? (I - £» f(»(., - e)] ,::
£ £ 1:-0
00
+ / /(x-t) dt+  (-1)1: /(n-I:-1)(x-€) [ ({3)1: _ ({3)n aA:(g) ]} .
t n + fJ  gA:+fJ (n - k - I)! (5 1)
1 1:-0 .7
00 00
We have at(€) = f n-1-A:(1 +)-n-f3 - f n-I:-1(1 +)-n-f3d. Here the first
o .!.=.!.
'"
integral is easily reduced to the beta function. Changing the variable  + 1 = l/gt
in the second integral, we obtain
1
a (g) - r(n - k)r(k + (3) _ €1:+f3 / (1 _ €tt-1-l:tfJ+I:-1dt
I: - r( n + (3) .
o
So the second square bracket in (5.71) is equal to
1
({3)n gl:+f3 / (1 _ d)n-1-l:tf3+1:-1dt ,.", ({3)1: €1:+f3
(n - k - I)! £-0 (k + (3)(n - k - I)! .
o

116 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Then passage to the limit in (5.71) as e -+ 0 is easily realized. So we have that the
Liouville fractional derivative
00
v'+l = r(n  a) :n J I(z -t)rPdI
o
indeed coincides with the right-hand side of (5.68) taking into account that
(-I)n(P)n = r(n - n)/r( -n) by (1.46) and (1.47). .
5.6. Properties of finite differences and Marchaud fractional
derivatives of order a > 1
Equations (5.57) and (5.58), defining Marchaud fractional derivatives, may be
extended to the case n > 1. One of the ways which comes to mind is to procede in
a similar fashion to (5.7) by putting n = n + {n}, n = [n] and introduce
00
D a f = lf1 D{a} f = {n} J f(n)(x) - f(n){x - t) dt.
+ dx n + r(I-{n} t 1 +{a}
o
It is possible however to choose another way by introducing differences of higher
order, that is, I > 1 in (5.57) and (5.58) instead of the first order difference. We
shall elaborate on this latter way. It will be preferable in some aspects because it
shows directly the analytic dependence of D.J on the parameter n. Firstly we
consider some simple properties of finite differences.
In terms of the translation Th we introduce
1
(dLJ)(z) = (E - TIJ 1= (;(-l)' G)f(Z - kh), (5.72)
which is said to be a finite difference of order I of a function f(x) with a step h
and with center at the point x.
We shall need the following function of the parameter n:
A,(a) = t(-l).-IG ) k a , n > O.
k=O
(5.73 )
It arises as 8 finite difference of the power function: (Aif)(O) = -Al(n) for

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS 117
f(x) = Ixl£w. The following property of this function:
A,{a) = 0 for a = 1,2,...,1- 1,
(5.74)
will be important for us. It follows from the obvious equality
A,(m) = - (. :z r (1- 41.=1.
(5.74')
As for non-integer values of a I it may be shown that A,( a) f; 0 for a E Rl,
a f; 1,2,...,1- 1, see (5.81) below and Lemma 26.1 in Chapter 5.
Lemma 5.4. Let f(x) E Cm(R 1 ) and let I  m. Then
1 ,
(d/)(z) = (m h : I)! /(1 - u)m-1 (_1)m-. k m G) fCm)(z - khu)du. (5.75)
o k_O
Proof. By Taylor's expansion (with the remainder in the integral form) we have
f(x - kh) =  (_h)i fCi)(x) + {-kh)m / 1 {I _ u)m-l f(m){x _ khu)du.
 I! (fin - I)!
1=0 0
So for the differences (5.72) we obtain the equality
(ALf)(x) = - 'E (_.)i f(i)(x)A,{i)
i=O I.
1 ,
+ h m / (1 _ u)m-I "'( _1)m-k k m ( I ) f(m){x - khu)du,
(fin - I)! L.J k
o k=O
which yields (5.75) in view of (5.74). .
Corollary 1. If f(x) E Cm(R 1 ) and f(m)(x) is bounded, then
I(ALf)(x)1  clhl m sup If(m)(x)l, I  fin,

(5.76)

116 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Then passage to the limit in (5.71) as € -+ 0 is easily realized. So we have that the
Liouville fractional derivative
00
01 1 lf1 f f( ) - (j
'D + f = r( n _ Q) dx n x - t t dt
o
indeed coincides with the right-hand side of (5.68) taking into account that
(-I)n(P)n = r(n - Q)/r(-Q) by (1.46) and (1.47). .
5.6. Properties of finite differences and Marchaud fractional
derivatives of order a > 1
Equations (5.57) and (5.58), defining Marchaud fractional derivatives, may be
extended to the case Q > 1. One of the ways which comes to mind is to procede in
a similar fashion to (5.7) by putting Q = n + {Q}, n = [Q] and introduce
00
DOl f = IF D{OI} f = {Q} f f(n)(x) - f(n)(x - t) dt.
+ dx n + r(l- {Q} t 1 +{0I}
o
It is possible however to choose another way by inroducing differences of higher
order, that is, I > 1 in (5.57) and (5.58) instead of the first order difference. We
shall elaborate on this latter way. It will be preferable in some aspects because it
shows directly the analytic dependence of D./ on the parameter Q. Firstly we
consider some simple properties of finite differences.
In terms of the translation Th we introduce
I
(di/)(z) = (E - 1',.)' / = (-1)' G)/(z - kh), (5.72)
which is said to be a finite difference of order I of a function f(x) with a step h
and with center at the point x.
We shall need the following function of the parameter Q:
,
A,(Q) = E(_I)A:M'l (  ) kOl, Q > O.
A:=O
(5.73)
It arises as a finite difference of the power function: (Aif)(O) = -A,(Q) for

 5. THE MAIN PROPERTIES OF FRACTIONAL INTEGRALS 117
f(x) = Ixl£w. The following property of this function:
A,(a) = 0 for a = 1,2,...,1- 1,
(5.74)
will be important for us. It follows from the obvious equality
( d ) m ,
A,(m) = - x dx (1 - x)'IJ:=l'
(5.74')
As for non-integer values of a, it may be shown that A,(o:) f; 0 for a E Rl,
a f; 1,2,...,1- J, see (5.81) below and Lemma 26.1 in Chapter 5.
Lemma 5.4. Let f(x) E Cm(Rl) and let I  m. Then
1 ,
(df)(z) = (m h : I)! /(1- u)m-l (-lr'km G).t<m)(z - khu)du. (5.75)
Proof. By Taylor's expansion (with the remainder in the integral form) we have
. 1
m-l ( kh)1 ( kh)m f
f(x - kh) = ?: -T- f(i)(x) + (: _ I)! (1- u)m-l f(m)(x - khu)du.
1=0 0
So for the differences (5.72) we obtain the equality
m-l .
(Af)(x) = - E (-.)I f(i)(x)A,(i)
i=O I.
1 ,
+ (m h : I)! /(1- u)m-l (_l)m-'km G).t<m)(z - khu)du,
which yields (5.75) in view of (5.74). .
Corollary 1. If f(x) E Cm(R 1 ) and /(m)(x) is bounded, then
I(Af)(x)1  clhl m sup Ilm)(x)l, I  m,
J:
(5.76)

120
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
5.7. Connection with fractional powers of operators
The operator 1)+ of fractional differentiation may be considered as a fractional
power of the differentiation operator:
1)+ = (d/dxY\ It. = (d/dx)-OI
(5.82)
under the appropriate interpretation of fractional power of an operator. In fact this
was the main model in mind in the development of the abstract theory of fractional
powers in Banach spaces. We refer the reader who wishes to become more familiar
with this theory to the books by Krasnosel'skii, Zabreiko et al. [1] and by Yosida
[1]. We mention only very briefly the simplest definitions in this theory and show
that they include the case of fractional integro-differentiation in a suitable setting.
Let X be a Banach space and let {Tt}, t  0, be a strongly continuous
semigroup in X (see Definition 2.5). The operator
A = lim !C1t - E)
t-O+ t
(X)
(5.83 )
is said to be a generator (or infinitesimal operator) of the semigroup Tt. It is
known (see e.g. Dunford and Schwartz's book [1], p.660) that the domain D(A)
of the operator A is dense in X and that A is a closed operator. The equality
Tt = etA is valid, at the least formally, the exact meaning being 1t = lim etA",
h-O
Ah = t(T h - E».
We shall consider fractional powers (_A)OI for operators A, which are
generators of strongly continuous semigroups. A positive power of an operator A
is defined by the formula
00
(_A)acp = r( Q) J ,-a-'(T,cp - cp)dt,
o
(5.84)
o < a < 1, cp E D( A),
compared with the Marchaud formula (5.57). The integral of a function of scalar
argument t with values in a Banach space is understood here as a Bochner integral
- see for example the book by Hille, Phillips [I, Ch.III, Section 1] about the latter
notion. Equation (5.84) is usually referred to as Balakrishnan's formula.
When Q  1 we may define the fractional power (_A)OI, following (5.80), by

 5. THE MAIN PROPEFUIES OF FRACTIONAL INTEGRALS
121
bile equality
00
(-A)'\? =  1 t-I-Ot(E - Tt)'cpdt,
x( Q , I)
o
where E is the identity operator, I > Q and X(Q, I) is the constant (5.81).
A negative power of the operator -A may be defined for 0 < Q < 1 by the
equality
(5.85)
00
(_A)acp = r(l n ) 1 t a - 1 T,cpdt, cp E x,
o
but the difference with (5.84) is here in the fact that the integral (5.86) may prove
to be divergent at infinity, unless additional assumptions on the semigroup 1t are
made. A simple condition providing convergence of this integral for all Q > 0 is
(5.86)
111tllx  M e- Et , c > o.
(5.87)
It is clear that in order to realize (5.82) we must represent the operator
A = d/ dx as the generator of a semigroup 1', which in view of (5.83) is the
semigroup of the translation operators:
(1tf)(x) = f(x - t).
(5.88 )
The problem, however, is how to choose the space X so that the semigroup Tt
is strongly continuous and (5.87) is satisfied. The spaces Lp(Rl), C(Rl) do not
match this intent since IITt II = 1 for them. We shall use the spaces Lp,w, C w for
this purpose (see Theorem 5.7 above).
Lemma 5.5. The semigroup (5.88) is strongly continuous in the space Lp,w(Rl),
1  p  00, and
IITtIlLp....
- e -t
- ,
1  p < 00;
IITtllc... = e- wt .
The proof of the Lemma may be obtained by direct verification.
Lemma 5.5 allows us to state that the integral (5.86) converges in the norm of
the space Lp,w with w > 0 and from (5.86) we have
( d ) -Ot 1 1 00
dx cp = r(Q) tOt-1cp(x - t)dt = I+cp,
o

122 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
where cp E Lp,w 1  p  00, w > 0, and the above mentioned L",w - convergence
(at infinity) of the integral is implied. As for (5.84), we obtain
00
( .!!... ) a =  f cp(x - t) - cp(x) dt = na
dx cp r( -a) t 1 + a +cp,
o
where cp E D(A) = {cp(t) : cp'(t) E Lp,w}, W > 0, and 0 < a < 1. We may
analogously treat the case a  1 based on (5.85).
fi 6. Representation of Functions by Fractional
Integrals of Lp-Functions
In S 5.3 we have considered fractional integrals [cp of functions cp E Lp.
Now we shall discuss such fractional integrals in greater detail and give their
characterization.
6.1. The space [a(L,,)
Let us denote the images of the fractional integration operators by [( L,,):
[(Lp) = {f: f(x) = [cp, cp E Lp(R 1 )},
0< a < 1, 1  p < l/a.
In reality they coincide with each other if 1 < p < l/a and we denote
[a(L p )t!2[:(L,,) = I(Lp), 1 < p < l/a,
(6.1)
but we find it convenient to put off the proof of this coincidence until S 11.2.
By Theorem 5.3
[a(L p ) C Lq(Rl), q = p/(I- ap),
(6.2)
and by Hardy's inequality (5.45)
[a(L,,) C Lp(Rl; lxi-a,,).
(6.3)

 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF Lp-FUNCTIONS 123
Let us note that
Lp(R I j IXI- ap ) (/:. Lq(Rl), Lq(RI) (/:. Lp(RI; Ixl- ap ). (6.4)
The first is obvious, while the second is illustrated by the example of the function
which is equal to j(x) = Ixl-I/q In-lIp Ixl for Ixl > 2 and to f(x) = 0 for Ixl < 2.
The imbeddings (6.2), (6.3) together with (6.4) mean that
[a(L p ) f; Lq(R I ),
[a(L p ) 1= Lp(Rlj Ixl-ap).
(6.5)
So, in view of Theorem 5.3, the space [a(L p ) does not coincide with any space
Lr (R 1 ), 1 $ r $ 00. It does not coincide with any space Lr (R I ; p), either.
Therefore the space [a(L p ) needs to be characterized. Subsection S 6.3 is devoted
to this purpose. Firstly we consider the inversion of fractional integrals [+cp,
cp E Lp, by means of Marchaud derivatives in S 6.2, which will provide the necessity
part of the characterization of the space [a(L p )'
In S 18 below we shall consider the following modification of the fractional
integration operator [, the so-called Bessel fractional integration, see S 18.4:
00
G a cp =  f ta-Ie-tcp(x =F t)dt
% r(a)
o
which differs from [ by the presence of the decreasing exponential factor. In
contrast with [ this integral is defined on functions cp(t) E Lp(RI) for all
1 :5 p  00. Besides G(Lp) C Lp by Young's Theorem 1.4, while I(Lp)  Lp. It
is important to note that
Lp n [a(L p ) = G+(L p ) = G(Lp), 1 < p < l/a,
the proof of which will be given in S 18.4.
6.2. Inversion of fractional integrals of Lp-functions
Liouville differentiation, 'D, inverts fractional integrals Icp in the frames of
:&
Lp-spaces: 'D[cp == cp, cp E Lp(RI), only for p = 1, since 'Dt.[+Cp == d J cp(t)dt,
-00
which assumes summability of cp( x) at infinity. As was already noted in S 5.4, in
the case p > 1 we shall use Marchaud derivatives, instead of 'Dj, treating them
as convergent in the norm of Lp(RI) - see (5.60).
The next lemma gives a useful representation of the truncated Marchaud
fractional derivatives.

124 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Lemma 6.1. For a function f(x) = It.tp, represented by fractional integral with a
density tp(x) E Lp(Rl), 1  p < I/Ot, the truncated fractional derivative D+,ef has
the following representation
00
(D+,ef)(x) = f (t)tp(x - €t)dt,
o
(6.6)
where the kernel
(t) = sinOt1l't+ - (t -1)+ E L1(R1)
11' t
(6.7)
has the properties:
00
f (t)dt = 1 and (t)  0,
o
(6.8)
Proof. For t > 0 we have
I(z) - I(z - t) = r:) {i <p(z - t{){a-ld{ -1 <p(z - I{)({ - l)a-ld{ } ,
so
00
f(x) - f(x - t) = t Oi f k()tp(x - t)d,
o
(6.9)
where
1 { OI-l 0 <  < 1,
k() = r(Ot) OI-l'_ ( _ 1)01-1,  > 1.
(6.10)
We note that k() E L1(R) and that
00
f k()d = O.
o
(6.11)

 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF Lp-FUNCTIONS 125
In view of (6.9) we obtain the relations
a a 1 00 dt 1 00 (  )
(D+,ef)(x) = r(1 _ a) t2 k t cp(x - )
e 0
00 Ue
a 1 cp( x -  1
= r(l- a)   k(s)ds
o 0
00 t
a 1 cp( x - ct) 1
= f(l- a) t dt k(s)ds.
o 0
Here
t
1 k(s)ds = a- i tf(1 - a)K(t) (6.12)
o
which may be proved by direct evaluation of the left-hand side. Then (6.8) becomes
evident. .
We note that in a generalization of (6.6) the representation
00
(D+,ef)(x) = 1 K"a(t)cp(x - ct)dt
o
(6.6')
may be similarly obtained for arbitrary a > 0, where the truncated Marchaud
derivative (5.80') is used in the left-hand side and
1
L: (-I)I:()(t - k)+
Kl,a(t) = 1:=:(a,l)r(1 + a)t
(6.7')
It is not difficult to show that
00
Kl,a(t) E Li(R i ) and 1 K"a(t)dt = 1.
o
(6.8')
Theorem 6.1. Let f(x) = lcp, cp E Lp(R i ), 1 :5 p < l/a. Then
cp(x) = (D%f)(x),
(6.13)

126 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
where DI is to be understood as
(D%/)(x) = lim(D%,el)(x),
e-O
(L p )
(6.14)
The limit in (6.14) also exists almost everywhere.
Lemma 6.1 has paved the way for the proof of this theorem. Indeed, by (6.6)
and (6.8) we have
00
(D+,e/)(x) - cp(x) = f K(t)[cp(x - ct) - cp(x)]dt.
o
Applying the generalized Minkowsky inequality, we obtain:
00
f e-O
IID+,el - cpll"  K(t)lIcp(x - ct) - cp(x )1I"dt ---+ 0
o
in view of Lebesgue dominated convergence Theorem 1.2 and property (1.34).
In agreement with the definition (6.14), (6.13) is proved. The existence almost
everywhere of the limit lim D+ el, IE [OI(Lp), follows from Theorem 1.3.
e-O '
Note that Lemma 6.1 and Theorem 6.1 yield the inequality
IID+.e/ll"  IID+/llp, IE I+.(L,,), 1  p < I/Ot. (6.15)
Indeed, in view of (6.8) and (6.13) we have from (6.6):
IID+,elll"  ilK 111 IIcplip = IIcplip = IID+/II".
The inequality (6.15) implies the equality
lim IID+ e/ll" = sup IID+ e/ll"
e-O ' e>O '
(6.16)
for I E [+.(L,,). In fact, the inequality obtained from (6.16) after replacing = by
, is obvious. The inverse inequality follows from (6.15) in correspondence with
(6.14).
It follows from Theorem 6.1 that I cp == 0, cp E L", only in the case cp( t) == O.

 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF Lp-FUNCTIONS 127
Thus we may introduce the norm in 10/ (L p ) by the relation
IIfllrO(L p ) = IIcpllL p ' f = I+cp.
(6.17)
The space IO/(L p ) with norm (6.17) is a Banach space as an isometric to Lp.
6.3. Characterization of the space [O/(L p )
The next theorem gives the characterization of the space 10/ (L p ) in terms of
truncated Marchaud fractional derivatives (cf. the characterization of this space in
Theorem 20.5 and 20.4 in terms of Lp-behaviour of finite differences of fractional
order) .
Theorem 6.2. The necessary and sufficient conditions for f(x) E IO/(L p ),
1 < p < 1/0:, are
1) one of two following conditions is valid:
lim D+ £f E Lp,
£-0 '
(L p )
(6.18)
sup IID+,£fll p < 00,
£>0 '
(6.19)
e) f(x) E Lr(Rl), where r = q = p/(1 - o:p) in the necessity pan and r is
arbitrary (1  r < 00) in the sufficiency part.
Proof. The necessity in this theorem is a simple fact, being a corollary of
Hardy-Littlewood's Theorem 5.3, of Theorem 6.1 and of (6.15). The sufficiency
part is more complicated.
Let f E Lr and suppose that one of the conditions (6.18), (6.19) is valid. We
are to show that there exists a function cp E Lp such that
f = 1+ cp
(6.20)
(then f E IO/(L p ». Instead of (6.20) we shall prove the result
f(x) - f(x - h) = (I+cp)(x) - (I+cp)(x - h)
(6.21 )

128
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
for any h > O. Let us denote
00
(Ah<P)(z) = f ah(z - t)rp(t)dt,
-00
(6.22)
1
ah(t) = r(Q) [t+- 1 - (t - h)+-l].
So the desired result (6.21) is I(z) - I(z - h) = (Ahrp)(z). We note that Ah is a
convolution operator with the summable kernel ah(t) E L1(R 1 ) and therefore the
composition AhD+.,£ is, for a fixed c > 0, a bounded operator in Lr(Rl) for all
r  1. Functions I(z) being sufficiently good, e.g. in e, we have
AhD+,£1 = D+,£Ahl = (D+,£Ii./)(z) - (D+,£I+/)(z - h).
Hence in virtue of the representation (6.6)
00
AhD+.,£1 = f K(t)[/(z - ct) - I(z - h - ct»)dt.
o
(6.23)
Since err is dense in Lr, (6.23) holds for all 1 E Lr in view of the boundedness of
operators on the left and right sides.
The required result (6.21) will be obtained from (6.23) by letting c --+ O. In
view of (6.8) the right-hand side in (6.23) converges to I(z) - I(z - h) in Lr-norm.
Consequently, there exists the limit of the left-hand side and so
lim AhD + a £1 = I(z) - I(z - h).
£-0 '
(6.24 )
Let (6.18) be valid. Since the operator Ah is bounded in Lp, the limit
lim AhD+ £f = Ah(lim D+ £/) = Ahrp.
£-0 ' £-0'
(L p ) (L p )
exists, where rp = D+.I E L". Since AhD+,cf converges both in Lr- and in
L,,-norm, the limit functions must coincide almost everywhere and we obtain from
(6.24) the result Ahrp = I(z) - I(z - h), which coincides with (6.21).
If the condition (6.19) is valid, we may choose a sequence c1c --+ 0 such that
D+,£..I weakly converges in L" (a bounded set in L", p > 1, is weakly compact,
see for example Dunford and Schwartz [1], p.314). Since any bounded operator is

 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF Lp-FUNCTIONS 129
both strongly and weakly continuous, we again obtain (6.21) from (6.24) by similar
arguments.
Equation (6.21) is thus proved. It remains to observe that if differences
of functions coincide identically, then the functions itself may differ only by a
constant. Therefore (6.21) yields (6.20), in view of the fact that I, [t.tp belong to
Lr, L, respectively. .
Corollary 1. The norm (6.17) in the space [Ot(L p ) is equivalent to the norms
lilli, + IIlim D+ ,:flip,
£-0 I
(L p )
(6.25)
lilli, + sup IID+,£/lIp, q = p/(1 - ap).
£>0
(6.26)
Corollary 2. The relation lor fractional integration by parts
00 00
f l(x)(D+g)(x)dx = f g(x)(D/)(x)dx
-00 -00
(6.27)
(with Marchaud fractional derivatives; cf. (5.17)) is valid under assumptions that
Df E Lp, D+g E Lr, I E L" gEL" where p > 1, r > 1, * +  = 1 + a and
1=1_", 1=1_",
. <A" r <A.
fndeed, these assumptions being satisfied, then I E [Ot(L p ), 9 E [Ot(Lr) and
therefore (6.27) follows from (5.16).
In order to formulate another corollary let us introduce the space of functions
in Lr(Rl), which have fractional (Marchaud) derivative in Lp(RI):
L p Ot r(Rl) = {/(x) : IE Lr, lim D+ £1 E L p }.
, £-0 I
(L p )
( 6.28)
Corollary 3. Let 0 < a < 1, 1 < p < l/a, 1 ::; r < 00. Then
L;,r(R 1 ) = Lr n [Ot(Lp).
(6.29 )
Remark 6.1. In view of (5.49) the following weighted variant of Theorem
6.2 is valid: a function f(x) is representable by a fractional integral I+tp with

130 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
<p E L p (R 1 ; IxIP), np - 1 < JJ < P - 1, p> 1, if and only if
f(x) E Lr(R1j Ixlll),
1 1 1
--n<-<-
- - ,
P r P
1+11 I+JJ
-=--n
r P
and
sup IID+,efIlL p (Rl j l z l") < 00.
e>O
We add to the characterization of the space [Ot(L p ) the following idea
concerning the behaviour of functions f(x) E P:X(L p ) at infinity: fractional integrals
f(x) = (I<p)(x) of nonnegative (or nonpositive) functions <p(x) have a "bad"
behaviour as x -+ :i:oo respectively. Namely, they vanish at most as clxl 0t - 1
however rapid the vanishing of the function <pC x) was - see the estimate (7.15)
below. So if a function f(x) is real valued and f(x) E LrnIOt(L p ), 1 :$ r :$ 1/(I-n),
then (D1)(x) necessarily changes sign on the axis. For the case r = 1 more exact
information may be given as follows:
00
If f(x) E L1 n [Ot(L p ), 1 :$ P < I/n, then J (Dtef)(x)dx = 0 for any c > O.
-00
00
If p = 1, then J (Df)(x)dx = 0 too.
-00
Indeed, the above relation for Dte f may be obtained by direct integration of
(5.59). As for Df in the case p = 1, it is sufficient to integrate (6.6) over the axis.
In the conclusion of this section we consider the characterization of the spaces
I(Ld, 0 < n < 1. They may be characterized similarly to the case of the finite
integral (see Theorem 2.1) in terms of the absolute continuity of the functions
00
It_a(z) = r(I  Cl) J I(z 'f t)t-adt.
o
Definition 6.1. We say that f(x) E AC(R 1 ), if f(x) is absolutely continuous
on any finite interval and has a bounded variation on the closed real line R1
(completed by two infinite points).
The statement that the function f(x) belongs to the class AC(R 1 ) is equivalent
z
to its represent ability in the form f(x) = J <p(t)dt + c, where <pet) E L 1 (R1).
-00
One might define the class AC(R 1 ) with the aid of mapping onto the
finite interval. Namely, let x = x(y) be a continuously differentiable one-
to-one mapping of the interval [0,1] onto the closed axis [-00,00] and let
f(y) = f[x(y)]. It may be shown that the definition of AC(R1) by the relation
AC(R 1 ) = {f(x) : f(y) E AC[O, In is equivalent to the definition given above.

 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF Lp-FUNCTIONS 131
Theorem 6.3. In order that I(x) E I(Ll), it is necessary and sufficient that
If_Ot(x) E AC(RI) and If_Ot(=foo) = 0 under the corresponding choice 01 signs.
The proof of the theorem is similar to the case of finite interval (see Theorem
2.1).
6.4. Sufficiency conditions for the representability of functions
by fractional integrals
Noting that
00
II Ot III a f w,(/, t)
D+,£ ,  r(1 _ a) t 1 + Ot dt,
£
( 6.30)
where
wp(/, t) = sup II/(x + T) - l(x)II"
O<T<t
we see that Theorem 6.2 immediately yields the following theorem.
(6.31)
00
Theorem 6.4. If I E Lq(Rl), q = p/(1 - ap) and J t-1-Otw,(/, t)dt < 00, then
°
I E IOt(L,), 1 < p < I/Q.
Let us give simple sufficiency conditions for a Holderian function I( x) to
belong to the space IOt(L p ). First we prove the following auxiliary estimates which
will be repeatedly used in the book.
For the integral
00
A ( x ) - f dt
a,b,c - Itla(1 + It1)6(1 + Ix - tl)c'
-00
with a < 1 and a + b + c > 1, the estimate
{ (I + Ixl)- min(a+b,c,a+6+c-l), if max(c, a + b) f; 1,
Aa,b,c(X)  K (1 + Ixl)l-a-b-c In(2 + lxI), if max(c, a + b) = 1,
(6.32)
is valid where K does not depend on x.
Proof. Since the function Aa,b,c(X) is bounded, it is sufficient to estimate it for
Ixl-+ 00. We represent it in the form
00
A ( ) I I l-a-b-c f dT
a,b,c X = X b c '
-00 ITla (ITI +) (IT -11 +)

132 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Hence for Ixl -+ 00 we have
( 1/2 3/ )
Aa,b,c(X) 5 KlxI 1 - a - b - c + IxI 1 - a - b - c f + f .
-1/2 1/2
It is evident that
1/2 1/2
J 1 = f 5 K f Irl-a(lrl + 1/lxl)-bdr
-1/2 -1/2
Izl/2
= 2Klxl a + b - 1 f r-a(r + 1)-bdr
o
So for Ixl -+ 00 we have it 5 K(lxl a + b - 1 + 1), if a+b f; 1, and J1 5 K In(2+ Ix!),
if a + b = 1. Similarly,
3/2 1/2
f  K f (II + II ) -e d  K{ ::-: z;, ::.
1/2 -1/2
Gathering estimates we obtain (6.32). .
Theorem 6.5. If f(x) E H>'(R1), A > a, then
I(D+,ef)(x)1  e(1 + Ixl}->'-o, a < A < 1, (6.33)
I(D+,ef)(x)1 5 e(1 + Ix!)->'-o In(2 + Ix!), A = 1, (6.34)
where e does not depend on x and €. If herein A> max(Q, -Q+l/p) and f(oo) = 0,
then f(x) E ]O(L,).
Proo/. In view of the Holderian condition (1.6) on R1 we obtain the inequality
00
I(D't,./)(z)1  (1+1IxD f t'+a-(l  Ix - tD '
o
Applying here (6.32), we obtain (6.33) and (6.34). If f(oo) = 0 and A > -Q + l/p,

 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF Lp-FUNCTIONS 133
then I E L(/I l/q = -a + l/p. Furthermore it follows from (6.33), (6.34) that
sup IID+..e/ll, < 00. Then IE [Ot(L,) by Theorem 6.2. .
e>O
The next theorem gives sufficiency conditions in the weighted terms.
Theorem 6.6. [11(z) = IZI,.(IZD'" where g(z) E H>'CR 1 ), then
I(D+,e/)(z)1  clzl- IJ - Ot (1 + Izl)- min(II.1- IJ ),
(6.35)
where  > a, -a < p.  1, II > a and c doesn't depend on z and c. In the case
II + JJ = lone more factor In(2 + Izl) is needed in (6.35). If besides
1 1
- - II < JJ < -,
q q
1 1
- = - - a,
q p
( 6.36)
then I(z) E [Ot(L,).
Proof. Denoting p(z) = IzIIJ(l + Izl)lI, we have
00
( D + Ot , ef)(z) 1 f g(z - t) - g(z) dt
r(-a)p(z) t 1 + Ot
e
00
1 f [ 1 1 ] g(z - t) d
+ r( -a) p(z - t) - p(z) t 1 + Ot t
e
=Ae(z) + Be(z).
The estimate for Ae(z) follows from (6.32):
IAe(z)1  clzl- IJ ( 1 + Izl)-II-Ot->'
(6.37)
(with appearance of the factor In(2 + Izl) in the case  = 1). For estimating Be(z)
we represent it as
00
B z - 1 1 f [ 1 _  ] g(z - t) dt
e( ) - r( -a) (1 + IZI)II Iz - tl IJ IzlIJ t 1 + Ot
e
00
1 f [ 1 1 ] g(z - t)dt
+ r( -a) (1 + Iz - tI)ll - (1 + Izl)" t 1 + Ot lz - tl IJ
e
=B:(z) + B:(z).

134 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
The estimate for B;(x) is obtained without difficulties
00
IB:(x)1  c(1 + Ix/)-II J t- 1 - a [lxl- IJ -Ix - ll- lJ ]dt
o
00
 clxl- IJ - a (1 + Ix/)-II J Itl- 1 - a ll - 11 - tl-IJldt
-00
= cdxl- IJ - a (1 + Ix/)-II.
Further,
00
2 C J I 1 1 I t-l-adt
IB£ (x)1  Ixla+1J (1 + IX/)II - (1 + Ixlll - tsignxl)lI 11 - tsignxlIJ
o
00
 clxl ll - a - IJ (1 + Ixl)-II J t ll - a - 1 11 - tl- IJ (1 + Ixlll - tl)-II dt
o
clxlll-a-IJ J clxlll-a-IJ
= (1 + Ixl)lI ... + (1 + Ixl)lI
It-ll>1/2
J
It-ll<1/2
"
= U£(x) + (x).
In the first of these integral we have It - 11 > (t + 1)/5, so
00
u clxl- a - IJ J dt Cl
£(x)  (1 + Ixl)lI t 1 + a - lI (1 + t)II+ IJ = Ixl a + IJ (1 + Ix 1)11 .
o
Further,
1
(x)  clxl ll - IJ - a (1 + Ixl)-II J -IJ(1 + Ixl)-II d
o
Ixl
= clxl ll - IJ - a (1 + Ixl)-II J t- IJ (1 + t)-II dt
o
(6.38 )
 clxl-IJ-a(1 + Ixl)min(II,l-lI)
(with the additional factor In(2+ Ixl) in the case 11+ p. = 1). Gathering inequalities,

 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF L,-FUNCTIONS 135
we see that Be(x) is estimated as in (6.38). Then the truncated derivative D+.ef
has the same estimate, in view of (6.37). Thereby (6.35) is proved. This being
obtained, the remaining statement of the theorem follows from Theorem 6.2. .
The next theorem, similar to Theorem 6.6, may be proved correspondingly.
Theorem 6.6'. ]f f(x) E H>'CR 1 ), A > a, -a + IIp < p. < IIp, then
J(z)-J(O) E ]OI ( L )
I z l  , .
N ow we shall prove the following theorem.
Theorem 6.7. The space ]OI(L,) is invariant relative to the operator of
multiplication by a function a(x) E H>'CR 1 ), A > a, so
lIafllrO(L p )  KliallH" IIfllrO(L p )'
where the constant K does not depend on a and f.
Proof. We verify the conditions of Theorem 6.2. The requirement af E L, is
obvious and lIafll,  lIallH" II/II,. Further,
D+.e(af) = a(x)D+,e/ + Ae/,
where
00
a f a( x) - a( x - t)
Aef = r(1 _ a) t 1 +01 f(x - t)dt.
e
It is clear that lIaD+,efll,  lIaIlH"IID+,e/II,. As for Ae/ we have
00 { 00 } lh
a dt I/(x - t)I'dx
IIAdli.  r( 1 - "') lIallH' ! I' +a- > _£ (1 + 1:1:1»' (1 + 1:1: - I I»'
or, after application of the Holder inequality with the exponents qlp and (qlp),
00 { 00 } OI
Q dt dx
IIAdli.  r( 1 - "') lIallH' 11/11. ! I '+a- _£ (1 + 1:l:1)1 a( 1 + 1:1: - II)I a
In view of (6.32) the repeated integral here converges, so we have finally
IID+,e(af)lI,  cliallH" IIfllrO(L p ), which completes the proof. .

136 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Remark 6.2. Equation (11.36), being proved in S 11 below, states that
multiplication by the step function 8(x) = (1 + signx)/2 leaves the space [Ot(L p )
invariant, 1 < p < 1/0:. Basing on this fact we shall give in subsection 11.4 another
condition sufficient for the inclusion f(x) E [Ot(L p ), which admits discontinuous
functions (see Corollary 2 of Theorem 11.6).
6.5. On the integral modulus of continuity of [Ot(L,)-functions
We conclude this section by certain simple properties of the continuity modulus
(6.31). Although in general f(x) E Lp(Rl) for f(x) E [Ot(L,), we have however
f(x) - f(x - h) E Lp(R 1 ) for any h. This follows from (6.9). Moreover, the
following statements are valid for f(x) E [Ot(L p ), 1 < p < 1/0::
1) wp(f, t)  ctOtIlD+.fll"
(6.39)
(6.40 )
2) wp(f, t) = o(t Ot ) when t --+ O.
Indeed (6.39) follows from (6.9) by application of the generalized Minkowsky
inequality (1.30). In order to obtain (6.39) we rewrite (6.9) as follows
00
f(x) - f(x - t) = tOt j K()[cp(X - t) - cp(x)], cp = D+.f,
o
by taking (6.11) into account. Hence, applying the generalized Minkowsky
inequality again, we obtain
00
wp(f, t)  tOt j IK()lwp( cp, t)d = o(t Ot ).
o
(6.41)
It is easy to derive the estimate
00
wp(f. t) :S cltaWp(. t) + C2 t j Wf.) df..
t
(6.42)
from (6.41), Cl and C2 not depending on t. By simple steps one may also obtain
the estimate
A A 00
j W,(f,t) dt< 2 j w'(CP,t) dt+ 3AOt j w'(CP,t) dt. (6.43)
t 1 + Ot - r(l+a) t r(l+o:) t 1 + Ot
o 0 A

 7. INTEGRAL TRANSFORMS OF FRACTIONAL INTEGRALS 137
f 7. Integral Transforms of Fractional Integrals
and Derivatives
We exhibit here the results of applying Fourier, Laplace and Mellin transforms
to fractional integrals and derivatives. Preliminaries concerning these transforms,
have been given in S 1.4.
7.1. The Fourier transform
The main assertion of this section is the following equation for the Fourier transform
of the fractional integral
:F(Icp) = (x)/(=Fix)a, 0 < Rea < 1.
(7.1)
The function (=Fix)a is to be understood in analogy with (5.26) as
(=Fix)a = ealnlzl=Fapsignz.
(7.2)
If a is real, we also write
(=Fix)a = Ixlae=F "';i signx.
(7.3)
In the case of fractional differentiation we shall have the similar equation
:F('D%cp) = (=Fix)a(x), Rea  O.
(7.4)
Equations (7.1) and (7.4) evidently link to (1.105), generalizing the latter to the
case of non-integer order.
We provide the auxiliary equation
00
f ta-1e-ztdt = f(a)/za, z 1= 0,
o
(7.5)
before proving (7.1). Here Rea> 0 when Rez > 0, 0 < Rea < 1 when Rez = 0
and the principal value of the function za, analytic in the right half-plane, is
chosen so that za is positive for z = x > 0 in the case of real a. Let us prove
this equation. We know it is valid for z = x > 0 by virtue of (1.54). Thus it is
true when Re z > 0, because the left-hand and right-hand sides are analytic in the

138
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
half-plane Rez > O. It remains to consider the boundary case Rez = 0, z 1= 0, i.e.
00
1 ta-le-iztdt = f(a)(ix)-a, 0 < Rea < 1, x 1= 0, (7.6)
o
the condition Re a < 1 providing convergence of the left-hand side integral at
infinity. To prove (7.6) we carry out the substitution itx = z:
00
1 ta-le-iztdt = (ix)-a 1 za-le-zdz,
o £
(7.7)
where £, is the imaginary half-axis (0, ioo) for x > 0 and the half-axis (-ioo, 0) for
x < O. Since le- z I exponentially vanishes in the right half-plane as Izi --+ 00, we
00
have J za-le-zdz = J xa-le-zdx = f(a) by the Cauchy integral theorem. Hence
£ 0
(7.6) follows.
The justification of (7.1) is given by the following theorem.
Theorem 7.1. Equation (7.1) is valid for 0 < Rea < 1 and rp(x) E L1(R 1 ), the
Fourier transform :1' in (7.1) being understood as in (1.103).
Proof. By (1.103) we have
00 t
:1' ( l a ) -  1 eizt - 1 dt 1 rp(s)ds
+rp - f(a) dx it (t - sp-a
-00 -00
1 d 1 00 1 00 eizt - 1
= -- s ds dt.
f( a) dx rp() it(t - S )l-a
-00 ,
The interchange of order of integration is made possible here by Fubini's Theorem
1.1. Further, after differentiating we obtain
00 00.
1 d 1 1 e'zt dt
:1'(/+rp) = f(a) dx rp(s)ds (t - s)l-a
-00 ,
00 00 00
1 1 . 1 eiXT dr rp(x) 1 eixtdt
-- se'X'ds --- -
- f(a) rp( ) r 1 - a - f(a) t 1 - a '
-00 0 0
Hence we arrive at (7.1) in view of (7.6). .

 7. INTEGRAL TRANSFORMS OF FRACTIONAL INTEGRALS
139
Remark 7.1. Equation (7.1) is not extended to values Rea  1 in its direct form,
since the left-hand side in (7.1) may not exist even for very smooth functions,
x
e.g. cp E C8'\ if Rea  1. Indeed, if a = 1 we have (lcp)(x) = J cp(t)dt, so
-00
(lcp)(x)  const as x --+ +00 and therefore the Fourier transform .1'1cp does
not exist in the usual sense. If a > 1 let us take the function cp E Crr, to be
nonnegative and positive on some interval [a, b]. Then for x > b
b
(1+",)(:>:)  r(I",) J (:>: - W-'",(t)dt
a
(7.8)
. (x - b)a - (x - a)'w
> mm cp( t )
- atb f(a + 1) ,
so that (I.+cp)(x)  Cx a - 1 as x --+ +00 and the Fourier transform .1'1.+cp does not
exist in the usual sense again. In S 8.2 below we shall extend (7.1) to all values of
a, Re a > 0, in the case of the specially constructed space of functions cp( x).
As for (7.4), it is valid for Rea > 0 for all sufficiently smooth functions, for
example, those which are differentiable up to the order n = [Rea] + 1, and vanish
sufficiently rapidly at infinity together with their derivatives. One may verify this
by rewriting V+I as Vi-I = I+- a I(n) and applying then (7.1) and (1.105).
We wish to illustrate the applicability of (7.1) to the evaluation of certain
integrals. Namely, we shall prove (5.24) with the aid of (7.1). First let
Rep > 1 in (5.24). Then (1:i: ix)-/J E L1(R 1 ) and so (see Theorem 7.1)
.1'It. [ (lr;),. ] = (o .1' [ (H:X)" ] by (7.1). Equation (7.5) yields
00
J t/J-le=Fteixtdt = r(J.l) ,ReJ.l > O.
% (1 =F ix)/J
-00
(7.9)
In view of the inversion formula (1.104) for Fourier transforms this means
[ 1 ] _ 211' /J-l =F X
.1' (1:i: ix )/J - r(p) x% e ,
(7.10)
which implies .1' 1'+ [ (1r;),. ] = 211' (::;o e=F x . Applying the inverse Fourier
transform to the latter equation, we arrive at the result (5.24) by taking
(7.9) into account with J.l replaced by J.l- a and x by -x. The above arguments
are true for Re(J.l - a) > 1, but (5.24) itself is valid for Re(p - a) > 0 due to the
analyticity with respect to J.l.

140
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
In conclusion we give here formulae for the evaluation of the cosine- and
sine-Fourier transforms of the fractional integrals Ig+cp and Ir.p on the half-axis
under the assumption that 0 < a < 1:
(1 01 ) -01 ( a1r . a1r )
:Fe o+Cp = X cos T:FeCP - sm T:F,cp , X > 0,
(7.11 )
(1 01 ) 01 ( . a1r a1r )
:F, o+Cp = x- sm T:FeCP + cos 2:F,CP , X > 0
(7.12)
In the case of I cP one should replace the sign before sin( a1r /2) by its opposite
value on the right-hand sides of these equations. These relations are easily obtained
from (7.1) by separating real and imaginary parts.
7.2. The Laplace transform
It follows from (1.122) that the fractional order integral (Igcp)(x), Rea> 0, is the
Laplace convolution of the form
(Ig+'P}(Z) = ['P(Z). ;t:; ] , Rea> o.
(7.13)
Therefore, using the convolution theorem (1.123) for the Laplace transform of the
fractional integral Ig+ cP, it is not difficult to obtain the result
(LI+cp)(p) = p-OI(Lcp)(p),
(7.14)
which is also true for sufficiently good functions cp if Rea < O. We do not consider
here the operator I<.!. since the composition LI<.!. cp leads to a complicated operator
containing the Kummer function IF1(a; c; z) in the kernel (see  36).
To prove (7.14) we need the following lemma.
Lemma 7.1. Let cp(x) E Ll(a,b) for any b> a and let the estimate
/cp(x)/  Ae'ox if X > b, A,po - const, Po  0,
(7.15)
holds and Rea> O. Then, for x > b + 1, the inequalities
1(I:+cp)(x)/  Be'ox, if Po > p,
I(I:+cp)(x) I  Bxmax(O,Reo-l), if Po = 0, B - const.
(7.16)

 7. INTEGRAL TRANSFORMS OF FRACTIONAL INTEGRALS 141
are valid.
Proof. Let x > b + 1. For simplicity we assume a to be real. Then we have
b x
1 f A f epot
1(1:+cp)(x) I  r(a) (x - t)a- 1 Icp(t)ldt + r(a) (x _ tF-a dt
a b
x-b
Ae'OX f e-PoT
< c 1 x max (O,a-1) + - -dr
- f(a) r 1 - a
o
< C x max (O,a-1) + C e POX
_ 1 2 ,
This completes the proof. .
Theorem 7.2. Let Rea > O. Then (7.14) with Rep > Po holds for functions cp(x),
satisfying the conditions of Lemma 7.1 with a = O.
Proof. The applicability of the Laplace transform in the case Re p > Po follows
from Lemma 7.1 and the fact that if cp E L 1 (0, b), then Ig+cp E L 1 (0, b) (see U 2
and 3). Equation (7.14) itself is verified by direct calculation, changing the order
of integration by the Fubini theorem and using (7.5). .
Theorem 7.3. Let -n < Rea  1 - n, n = 1,2,.... If cp(x) E ACn([O, b]),
<p(j)(0) = 0, j = 0,1,2,.. ., n - 1, for any b > 0 and the estimate (7.15) holds, then
(7.14) is true for Rep > Po.
Proof. Since (Io+cp)(x) = (d/dx)n(Iolncp)(x), Rea + n > 0, according to (2.32),
we, first, apply (1.124) taking into consideration that (d/ dx)i (Iln cp)( x) = 0
with x = 0 and j = 0,1,2,..., n - 1 follow from the conditions cp(j)(O) = 0,
j = 0,1,2,..., n - 1. Then we use Theorem 7.2 with respect to the integral I:ncp.
The theorem is thus proved. .
We note that (7.14) is used when solving various integral and differential
equations in Chapters 7 and 8.
Remark 7.2. Equation (7.14) and the inversion of the Laplace transform (1.120)
yield the following representation for the operator 10+ via the Laplace operators L
and L -1, namely
(I+cp)(x) = L -lx- a Lcp(x).
Another result analogous to (7.17) is also valid. This is
(7.17)
(Icp)(x) = Lx-a L-1cp(x),
(7.18)

142
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
which after substituting cp = L1/J follows from
(1L1/J)(x) = Lx-a1/J(x).
(7.19)
This is checked by direct evaluation for sufficiently good functions 1/J(x), if
o < Re a < 1.
7.3. The Mellin transform
Equations (1.105), (7.1), (7.4) and (7.14) show us that integro-differentiation of an
arbitrary order a is reduced in Fourier and Laplace transforms to multiplication
by the power functions (=t=ix)-a and p-a respectively. The results of (1.117) show
that when we apply the Mellin transform to a derivative of integer order n, its
transform is multiplied by the product (1- s)n = r(1 + n - s)/r(l- s). The latter
circumstance with respect to fractional integrals and derivatives lead us to the
relations
( a J( » . ( ) r(1 - a - s) J . ( )
10+ x s = r(l-s) s+a,
Re(s + a) < 1,
(7.20)
(If(")n8) = r() a/ '(8 + a),
Res> 0,
(7.21)
which after the substitution J(x) = x-acp(x), according to (1.117) take the form
( l a - a ( » . ( ) r (1 - a - s) . ( )
o+x cp x s = r(l- s) cp s,
Re(a+ s) < 1,
(7.22)
(l.,-acp(.,)n8) = r()a) CP'(8),
Re s < 1.
(7.23 )
The conditions for when these formulae are valid are contained in the following
theorems.
Theorem 7.4. Let Rea> 0 and J(t)t 8 + a - 1 E L 1 (0,00). Then (7.20) holds if
Res < 1- Rea while (7.21) holds if Res > O.
The proof is carried out in the same way as the proof of Theorem 7.2. Given
conditions on a and s ensure the existence of the inner integrals.
Theorem 7.5. Let -n < Rea  1 - n, n = 1,2,..., J(t) E cn([o, b]), b
being any positive number, and let J(t)ta+'-l E L 1 (0,00). Then (7.20) holds if

 7. INTEGRAL TRANSFORMS OF FRACTIONAL INTEGRALS 143
Res < 1 - Rea and conditions
x 8 - 1c (Ig: 1c J)(X) = 0 for x = 0, x = 00, k = 1,2, .. ., n,
(7.24 )
are satisfied, while (7.21) holds if Res > 0 and the conditions
x8-1c(I+1c f)(x) = 0 for x = 0, x = 00, k = 1,2,... , n,
(7.25 )
are satisfied.
Proof. By the condition f(t) E Cn([O, b]) the fractional derivative (I+J)(x) exists.
We write it down in the form (I+J)(x) = d"" (Ig:n J)(x) (see (2.32».
Then we apply the Mellin transform to it and integrate by parts n times, thus
00
(I+f(z})*(s) = f z'-l <:::l (I:nf)(z)
o
n-l
= L(1- S)1c X8 -1c-l IgI 1c + 1 J(x)I=o
1c=O
00
+ (1 - s)n f x,-n-l IgIn J(x)dx.
o
By (7.24) the integrated terms are equal to zero. Applying Theorem 7.4 and having
replaced a by a + n and s by s - n, respectively, we obtain the result
(I+f(z»*(s) = (1- s)n g =:   r(s + n).
Hence, .we deduce (7.20). The case of the integral I is considered in a similar way.
The theorem is thus proved. .
Together with (7.22) and (7.23) the following theorem, characterizing the
result of the application of fractional integrals and derivatives to the inverse Mellin
transform, is used below as well.
Theorem 7.6. Let J-(s) E L2(1/2 - ioo, 1/2 + ioo), 7J  min(O, Re(a - b». Then

144
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
the equations
1/2+ioo
b l lI-b -II f
x 0+ X
1/2-ioo
s" f. (S)X- 8 ds =
1/2+ioo
f " r( 1 - a - s) . ( ) 8
S r(1 _ b _ S) f S X- ds,
1/2-ioo
Rea < 1/2, Reb < 1/2;
(7.26)
1/2+ioo
x b I-bx-II f s" f.(s)x- 8 ds =
1/2-ioo
1/2+ioo
f " r(b+s) f . ( ) -8 d
s r(a+s) s x S,
1/2-ioo
Rea > -1/2, Reb> -1/2.
(7.27)
hold.
The proof follows from the existence of the integrals in (7.26) and (7.27) under
the formulated conditions on the parameters and function, from their absolute
convergence almost everywhere and, as a result of this, from the possibility of
interchanging the order of integration in the left-hand sides of (7.26) and (7.27).
After doing this the evaluation of the inner integral is carried out by (1.68).
We also note that  10, 18 and 36 contain various composition formulae
reflecting the result of the application of some other integral transforms to the
operators Ig+ and I, and their generalizations and modifications (see also   9,
23 and 39).
00
Remark 7.3. The fractional integrals r( a)(If)(x) = J t a - 1 f(x =r: t)dt, Rea > 0,
o
are the Mellin transforms of the functions rp::t:(t) = f(x=r:t), x being fixed. Applying
the inverse Mellin transform (1.113) we obtain the following representations of the
function f(x) via its fractional integrals, namely
al +ioo
f(x =r: r) = 2i f r(a)(If)(x)r-ada, a1 = Rea > 0,
al -ioo
(7.28)
with r > 0 and under the respective choice of signs. The formula may evidently be
interpreted as an integral analogue of the Taylor series expansion. Further, taking
in particular x = 0, we have
al +ioo
f(=r:r) =  f r(a)(I*f)(O)r-ada, r > O.
21r1
at -ioo
(7.29)

 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS 145
This means that f{x) may be restored by the values of its fractional integrals
(Itf)(O) only at one point if the latter are known for all a on some line
Rea = a} > O.
-fi 8. Fractional Integrals and Derivatives of
Generalized Functions
We assume that the reader has some minimal knowledge concerning generalized
functions. A generalized function is treated as a continuous functional on one or
another space of test functions. Various such spaces are used depending on the
problem in hand, in order to take into account the particular characteristics of the
problem. This will become obvious in the context of the present section.
.1. Preliminary ideas
We shall consider generalized functions over 0, where 0 is the real axis or half-axis.
Only S 8.5 will contain brief indications in the case of 0 being a finite interval. We
cho,ose test functions on 0 to be infinitely differentiable at the interior points of
o with prescribed behaviour at the endpoints of O. The value of the generalized
function f as a functional on the test function cp will be denoted by (f, cp), The
generalized function is called regular if there exists a locally integrable function
f(x) such that J f(x)cp(x)dx exists for each test function cp{x) and
{}
(f, cp) = f J{x)cp{x)dx
{1
(8.1)
It is assumed that the bilinear form (f, cp) is chosen in such a way that it coincides
with (8.1) in the case of a regular generalized function.
The space X = X(O) of test functions is assumed to be a topological vector
space. We denote by X' = X'(O) the topological dual space of X, i.e. the space of
continuous linear functionals on X.
Let us recall the notion of a generalized function concentrated at a point. A
generalized function f E X' is said to be zero on an open set G, if (f, cp) = 0 for
each test function which is zero beyond G. The union OJ of all open sets where
f = 0 is called a null set of the function f. The complement of the null set with
respect to 0 is said to be the support of the generalized function and is denoted by
supp f = 0 \ OJ. We say that the generalized function is concentrated at the point
Xo, if supp f is this point Xo.
The well-known Dirac function 6(x - xo), Xo E 0, and its derivatives defined
by
(6(1:){x - xo),cp) = (-l)A:cp(A:){xo);

146
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
provide examples of generalized functions, concentrated at a point.
The inverse statement is also true namely, any functional I, concentrated
N
at the point xo, is of the form I = L: cI:6(1:)(x - xo), - Vladimirov [2, p.52] or
1:=0
Gel'fand and Shilov [1, p.149].
There are two main ways to define fractional integrals and derivatives of
generalized functions. The first goes back to Schwartz [1] and is based on the
definition of a fractional integral as a convolution
1 a-l I
r(a) x:i: *
(8.2)
of the function r() x%-l with generalized function I - see S 8.3. This way is well
suited to the case of the half-line. The second way, which is more common, is
based on using the adjoint operator. Namely, starting with (2.20) and (5.16) for
fractional integration by parts one may introduce
(1:+1, cp) = (I, I_cp)
(8.3)
by definition, 11:_, I and fractional derivatives being defined similarly. The
approach via (8.3) will be correct if 11:_ continuously maps the space of test
functions X into itself. Sometimes a more general treatment is admitted when
I and I+I are considered as generalized functions on different spaces of test
functions X and Y so that I E X', I+I E Y ' and then 11:_ must map continuously
Y into X.
We shall outline (8.2) very briefly in S 8.3. The main attention is paid to (8.3).
This is considered in S 8.2 in the case of the axis R l while S 8.4 deals with the case
of the half-axis.
8.2. The case of the axis R l . Lizorkin's space of test functions
The well-known space S of Schwartz test functions (which are infinitely
differentiable and rapidly vanish at infinity together with all derivatives) as
well as the class C C S of finite infinitely differentiable functions is poorly
adapted for fractional integrals and derivatives. It is obvious that functions Icp,
1)cp, where cp E S, are infinitely differentiable, but whose behaviour at infinity is
not in general, sufficiently good - see Remark 7.1 and the inequality in (7.8) -
which show that the fractional integral It. cp even of a function cp E C may have
a "bad" behaviour as x --+ +00. Taking this into account we may choose such
infinitely differentiable test functions which are "good" as x --+ -00 and "bad" as
x --+ +00. We may introduce for example the space 6+ of infinitely differentiable
functions which have for x --+ -00 the same behaviour as functions in S and have
a slow growth for x --+ +00. The latter means that for each k = 0, 1, 2, . " there

 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS 147
exists m such that sup(1 + Ixl)-mlrp(I:)(x)1 < 00. The space 6+ and the similar
space 6_ with the "bad" behaviour of functions for x --+ -00 are invariant relative
to fractional integration It. and I respectively. Indeed, for rp E 6+ and any p we
have
00
1 (1 + Ixl)m dl: Iarp l < (1 + Ixl)m f Irp(I:)(x - t)1 dt
dx lc + - f(a) t 1 - a
o
00
f ta-ldt
 c(1 + Ixl)m (1 + Ixl + t)p
o
as x --+ -00. Taking p to be sufficiently large (p > a + 1) we obtain
00
1(1 + IzDm d' [.t'P1 :S c(l + IzDm+a+'-. f ta-I(l + Wa-1dt
o
(8.4)
 const
with p > m + a + 1 now. It is easily shown that It. preserves a slow growth as
x --+ +00.
The topology in 6:1: is easily defined by means of a countable set of norms
which embrace the "power" vanishing at one infinity and the "power" growth at
another.
The spaces 6:1: are, in a sense, adjacent to the spaces L"w (see (5.52» of
summable functions, which are also invariant with respect to fractional integration.
The inconvenience of the spaces 6:1: is in the fact that we have to consider
fractional integration It. and I on different spaces of test functions 6+ and 6-.
A more significant shortcoming is that generalized functions on 6:1: are to vanish
as x --+ :1:00, so, for example, power functions do not belong to 6.
Following Lizorkin, let us introduce the subspace  C S, invariant with respect
to fractional integration and differentiation. The idea of introducing such a space
will be clear in Fourier transforms. Indeed, by (7.1) the action of fractional
integration is reduced to dividing the Fourier transform by (=Fix)a:
:F(Irp)(x) = (=Fix )-a(x).
(8.5)
The required invariance will be achieved if the function (x) does not become
worse after division by (=Fix)a. So we introduce the space W of functions 1/J(x) E S,
which equal zero at the point x = 0 together with all their derivatives:
W = {1/J :,p E S, 1/J(1:)(0) = 0, k = 0,1,2,...}.

148
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
An example of a function 1/J E W is 1/J(x) = e-z-z-.
Definition 8.1. The space of Fourier transforms of functions in W is said to be a
Lizorkin space and is defined as
 = {cp : cp E S, IjJ E W}.
00
Since 0 = 1jJ(I:)(O) = J eio.tcp(t)tl:dt, the space  may be characterized as
-00
the space of Schwartzian test functions cp(x) E S, which are orthogonal to all
polynomials:
00
I tl:cp(t)dt = 0, k = 0,1,2,. ..
-00
(8.6)
We recall that (8.5) was proved for 0 < Rea < 1 (see Remark 7.1). Let us
show that in the case of functions cp(x) E  it is valid for all a.
Lemma 8.1. Equation (8.5) with cp E  is valid if Rea  O.
Proof. Let first 1  Rea < 2. (The case 0 < Rea < 1 is contained in Theorem
7.1.) Then
N t
:FI+cp = r-l(a) lim I eiztdt I (t - s)a-lcp(s)ds
N-oo
-00 -00
(8.7)
N N-8
= r-l(a) lim I cp(s)eiZ8ds I a-leiz{d.
N-oo
-00 0
If Re a f; 1 integration by parts yields the formula
1 [ . I N ( S ) a-l
:FIcp = r(a)ix Joo Na-lelzN 1- N cp(s)ds
-00
N N -8 ]
- (a - 1) I cp(s)e iZ8 ds I a-2eiz{d .
-00 0
The first term here tends to zero by (8.6) (apply L'Hopital's rule), while
in the second term it is possible to pass to the limit directly due to the
assumption Re a < 2. Taking into account the value of integral (7.6) we obtain
:Fl a (a-l)r(a-l)  ( ) h . h . d
+cp = ( -iz ) "'r(a ) cp X , W lC was reqUIre .

 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS 149
The case a = 1 is simple:
N
f eizN - e i . z 1
.1' licp = lim cp(s) . ds = - -:-cp(x)
N-oo IX IX
-00
in view of (8.6). We consider now the singular case Rea = 1, a f; 1, so that
a = 1 + i8, 8 f; O. It is easily seen that
N N-.
f cp( s )e iz ' ds f iS eiz  --+ 0 as N --+ 00
N-I 0
for cp(x) E . Therefore from (8.7)
N-I [ I N-' ]
.1' 1 01 cp =  lim f cp( s )e iz . ds f iS d( eiz - 1) + f iS deiz .
+ Ixf(a) N-oo
-00 0 I
Integrating by parts and carrying out simple transformations, we have
N-I N-I. S
.1'I OI cp = - lim { f cp(s)eiz'ds - NiSeizN f ( 1 -  ) ' cp(s)ds
+ Ixf(a) N-oo N
-00 -00
N-I N-,
+ i8 f.;.. 'P( s )ds f .;.:({) Ii{ }.
-00 0
{ 1,  < 1 . A
where X() = . The first term here obvIously tends to <p(x), the second
0,  > 1
term tends to zero due to the property (8.6), while in the third term the passage
to the limit is possible under the integral sign. So
01 1 [ cp( x) 8 A ]
.1'1+cp = - f(a) -y;- + ;A(x)cp(x) ,
(8.8)
where the notation
00.
A ( x ) = f e'z - () 
I-IS
o

150
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
is used. By passing to the limit it is not difficult to derive that A( x) =
r(iB)(-ix)-i8 - (iB)-l from (7.6) which turns (8.8) into the equality F1t.tp =
(-ix )-1-i8 <p(x). Finally, let Rea > 2. This may be reduced subsequently to
already considered cases due to the semigroup property I+.tp = IiI+-1cp, tp E .
Similarly the case Rea = 0 follows from the case Rea = 1 due to the semigroup
property Itp = d I+i8tp, tp E , and (7.4). .
We now apply Lemma 8.1 to fractional integration of purely imaginary order.
Lemma 8.2. Operators I, B E R 1 , defined on , are extendible to the bounded
operator in Lp(Rl), 1 < p < 00.
Proof. By lemma 8.1. the action of the operator I!t IS reduced 10 Fourier
transforms to multiplication by the bounded function
( -ix) -i8 = e( 1r/2)8sign x [cos( BIn Ixl) - i sin( BIn Ixl)]
at least on the set , which is dense in Lp( Rl) - see  9.2 (note 8.1). It is not
difficult to show that this function satisfies conditions (1.41'). Hence, by Theorem
1.6, operator I!t is bounded in Lp(Rl). .
It is easily deduced from Lemma 8.2. that the operator I is bounded in
L,(a, b), 1 < p < 00.
From Definition 8.1 and Lemma 8.1 it follows immediately that space  is
indeed invariant relative to fractional integration and differentiation of any order.
Space  may be considered as a topological vector space with the topology
of the space S. We recall that the latter is generated by the countable family of
seminorms sup(1 + x 2 )m/2Itp(l:)(x) I. Space  is closed in S. In fact it is known
x
(Gel'fand and Shilov [1, p.155]) that the topology, defined in S by the S-topology
in the space :F(S) = 8 of Fourier images, coincides with the initial convergence in
S. So it is sufficient to show that W is closed in S, which is evident.
One may define a topology in \Ii which embraces the behaviour of functions
1/J(x) not only at infinity, but for x -- 0 as well, namely by means of a countable
number of seminorms sup[(l + x 2 )m/2Ixl-'I1/J(t)(x)l]. This topology coincides with
x
that of S for functions 1/J E W. This may be checked by separating the cases Ixl < 1
and Ixl > 1 and applying the Taylor expansion with the remainder in integral form
in the first case.
Remark 8.1. Space  does not contain real-valued functions everywhere different
from zero. This follows from (8.6) with k = O.
The space of linear continuous functionals on <I> will be denoted by ' as is
usual. Let us compare ' and 8'. We begin by comparing Wi with 8'. Since W is
closed in 8, we may identify W' with the quotient-space of the Schwartzian space

 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS 151
5' modulo the subspace w of functionals in S' having W as a null space, i.e.
w' = s' /w
(8.9)
where w = {I : ! E S', (I, 1/J) = 0,1/J E w}. We have used the known general
fact: namely if M is a closed subspace in a linear topological space E, then
M' = E' / M 1. , where M 1. is the space of all functionals in E', which are orthogonal
to M. It follows from the definition of space W, that wb consists of functions
concentrated at the point x = O. Then - see S 8.1 - wb consists of linear
combinations of the delta-function and its derivatives. It is well known that 6(1:)(x)
is the Fourier transform of the power function, that is .1'{(-it)l:;x} = 211'6(1:)(x),
the Fourier transform being understood in the sense of generalized functions:
(j,cp) = (/,1jJ),
(8.10)
where cp E S or cp E . Consequently, in the representation
' = s' /,
(8.11)
similar to (8.9), we have the space
 = {I: I E S',(!,cp) = O,cp E },
(8.12)
consisting of polynomials. Therefore' may be considered as a quotient space
5' / P modulo the subspace P of all polynomials. In other words, , may be
obtained from 5' by "sifting out" polynomials, i.e. two functionals in S' differing
by a polynomial are indistinguishable as elements of ' .
Following (8.3) let us define fractional integration 1:l: of a generalized function
I E , by the equality
(I:l:/, cp) = (I, lcp), cp E . (8.13)
Fractional differentiation corresponds to the case Rea < O. Definition (8.13) is
proper for all a. In effect, we have
(I, Icp) = (j, ifY,) = (j, (:i:ix)-a1jJ(x»
(8.14)
in accordance with (8.10). Since 1jJ(x) E W here and multiplication by (:i:ix )-a is a
continuous operation in W, we have that (I, 1 cp) is a continuous functional on .
Remark 8.2. Equation (8.13) may serve as a definition of the fractional integral
for a function I(x) E L,(RI) with p  l/a when integrals 11/ do not exist in the

152 CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
usual sense, being divergent at infinity. In this case [I is a generalized function.
But this generalized function is such that its difference
dhl = I(z + h) - I(z),
defined in the proper way: (dhl,cp) = (/,d_hCP), is a normal function for all
h E Rl and dhl E L,(R1), if a < 1. This assertion is easily derived using the
representation (6.9). In the case a  1 a similar assertion may be obtained by
bringing in differences of higher order - see the exact formulation in a similar
situation for the multidimensional case in Lemma 26.4.
Let us consider some examples. Take equations (5.24) and (5.25). The
function I(z) = (1:i: iz)-IJ may be considered as an element in ' for any J.l
since .:1"[(1 :i: iz)-IJ] E W' by (7.10), the Fourier transform being understood in
accordance with (8.10). The situation j(z) == 0, corresponds to the case J.l = -m,
m = 0,1,2,..., which is in agreement with the fact that , does not contain
polynomials. Based on the usual rule (8.10) we have that the equations
P;w[(1 :i: iz)-IJ] - r(J.l - a) e%ia1r/2(1 :i: iz)a-IJ
+ - r(J.l) ,
(8.15)
I[(z:!: i)-"] = f(;(:)",) (z:!: it-" (8.16)
are valid for all complex J.l except the case J.l = a - m, m = 0,1,2,... For this case
we have equations
(_I)m-1
[a[(1 :i: iz)m-a] = (1 :i: iz)m In(1 :i: iz), (8.17)
+ m!r( a - J.l)
( _1)m-1
[[(z:i: i)m-a] = (z:i: i)m In(z:i: i), (8.18)
m!r(a - J.l)
where In(1 :i: iz) = In  :i: iarctgz, In(z:i: i) = In  :i: iarcctgz. Let us
prove for example (8.17). For I(z) = (1 + iz)-IJ we obtain the relation
(I./, cp) = (.:1"-1(1 + iz)-IJ, (iz)-a(z»
by (8.13), (8.14) and (8.5). Taking (7.9) into account we have
00
1 f IJ-1 x
(I./,cp) = f(J.l) z(iz): (z)dz.
-00

 8 FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS 153
Since <,3(1:)(0) = 0, k = 0,1,2, . . . , we observe that (1+/, cp) is a functional analytic
with respect to p. on the whole complex plane after the determination (I+/, cp) = 0
at the points p. = -m, m = 0,1,2,.... Then it follows from (8.15) that
00
(l a [(1 + ix)m-a],cp) = lim r(p. - 0:) f (1 + ix)a-IJcp(x)dx.
+ IJ-a-m r(p.)
-00
00
Since f (x + i)mcp(x)dx = 0 by (8.6), we may apply L'Hopital's rule, which yields
-00
(8.17).
Now we evaluate the fractional integral of the delta-function and of its
derivatives. We have
(I6(1:),cp) = (6(1:),Icp) = (-1)1: ::I: (Icp)I:J:=o = (-l)I:I(cp(I:»I:J:=o,
So 16(1:) is the functional which acts in accordance with formula
I: 00
(I a 6(1:) cp) = (-1) f ta-1cp(I:)(:i:t)dt
%' r(a)
o
or
00
( I a 6(1:) cp) = (-1)1: f ta-I:-l cp( t ) dt
:i:, r( a - k) %
-00
(8.19)
in the case Rea > k. Equation (8.19) means that
la6(1:) - (:i:l)1: ta-I:-l Rea > k
% - r( a - k) %' .
(8.20)
It is easily shown that (8.20) may be extended to the values Rea  k. The
generalized function t%-I:-l is to be treated then in the sense of regularization - see
(5.68). The points a = k - m, m = 0, 1,2, . . . , are herein singular, corresponding
to the equality 1+6(1:) = 6(m).
We conclude our consideration, of the space  by the following remark.
Besides its evident advantages - simplicity of definition, clarity of operations with
Fourier-images -  has an essential shortcoming: it is lacking in multipliers.
Namely, if m(x)cp(x) E  for all cp(x) E , then m(x) may be nothing else but a
00
polynomial. In reality, in this case we have f m(x)cp(x)dx = 0 for all cp(x) E ,
-00

154
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
but it was shown above that the space defined by (8.12) and orthogonal to ,
consists of polynomials only.
8.3. Schwartz's approach
We shall consider generalized functiQns on the test function space K 1 = COO(Rd.
Following Schwartz [1] we define a convolution of two generalized functions in
terms of a direct product.
Let f(x) and g(y) be generalized functions of variables x and y respectively.
A functional f x g, given on test functions cp(x, y) E K 2 = C{f(R 2 ) by the formula
(f X g, cp) = (f(x), (g(y), cp(x, y»))
is said to be the direct product of f(x) and g(y). Since
00 00
(f * g, cp) = f f f(x)g(y)cp(x + y)dxdy
-00 -00
for regular functions, it is natural to introduce
(f * g, cp) = (f(x) X g(y), cp(x + y))
(8.21)
for generalized functions f(x) and g(y), i.e. the convolution f * 9 is defined as the
value of the functional f(x) x g(y) of two variables on test functions cp(x,y) of the
form cp(x + y). The function cp(x + y) however is not finite in R 2 , and so it is not
a test function of two variables. It is easily seen that (8.21) will make sense for
generalized functions f(x), g(y), which have for example support on the positive
half-axis. Indeed, let the support of the test function cp(x) be in the interval [-a,a].
Since f(x) = 0, x < 0, g(y) = 0, y < 0, the functional (8.21) will not change if we
replace the function cp(x + y) by a finite function 1/J(x, y), of two variables which
coincides with cp(x + y) in the triangle 0  x  a, 0  y  a, x + y  a.
Based on the above notion of convolution we introduce the fractional integral
of a generalized function f E [( as
( 0'-1 )
(Ig+f,cp) = (It-f,cp) = ;(a) * f,cp
(8.22)
in the case when f is supported on the half-axis x > O. For such functions we have
that It-I is also supported on this half-axis. Equation (8.22) is applicable for all a
with the usual interpretation of the generalized function x-l in the case Rea < 0

 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS 155
- see (5.68), for example - except for the cases Q = 0, -1, -2,.. ., when (8.22) is
replaces by the rule of direct differentiation of a generalized function.
We shall sum up the consideration of correctness of Schwartz's approach by
the following theorem with K+ denoting the space of generalized functions f E K
supported on the half-axis R.
Theorem 8.1. If f E K+, then IC+f E K+ also for any Q E C. Besides
Ig+Ig+f = I:{j f, Q,P E C,
and for each f E K+ there exists a unique generalized function 9 E K+ such that
/ = IC+g, Q E C.
The proof is obtained directly from the definition of IC+f for f E K+.
8.4. The case of the half-axis. The approach via the adjoint
operator
The simplest way to define the test function spaces, which are invariant relative to
fractional integration I or I+ is as follows. Let S(R) be the restriction of the
space S = S(Rl) on the half-axis, so that the functions f(x) E S(R) are infinitely
differentiable on [0,00) and rapidly vanish together with all derivatives as x --+ 00.
It is easily shown that the operator I preserves the space S(R). Further, let
e = 6(R) be the space of infinitely differentiable functions <p(x) on [0,00),
which slowly grow as x --+ 00 and such that <p(A:)(0) = 0, k = 0,1,2,.... It may
be easily verified that this space is invariant relative to the operator IC+. The
spaces S(R) and 6(R) have an essential shortcoming, however. The spaces of
generalized functions on S(R) and on 6(R) do not contain functions which
grow as x --+ O. The second of these spaces does not even contain functions which
are bounded or slowly vanish at infinity. It is thus not suitable for applications.
As in the case of the whole axis it will be the test function space  of the Lizorkin
type, modified for the case of the half-axis, which is more convenient.
We introduce such a space. Let
S+ = S+(R) = {w : w E COO(R+);
lim x ' w(m)(x) = 0, I, m, = 0,1,2,... }.
z-O,OO
(8.23)

156
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
We define
+ = (14) = {w : wE S+;
00
f xtw(x)dx = 0, k = 0,1,2,...}.
o
(8.24)
Thus functions of the space + have the property that their Mellin transform is
equal to zero at the integer points s = 1,2,3,. . .. Let us introduce a topology in
the spaces S+ and + by the norms
sup sup(1 + x)tlw(m)(x)l, k = 0,1,2,...
mk z>o
It is easily shown that the spaces S+ and  + are complete in this topology.
The function
x( "') = exp ( - In  ", ) sin G In'" ) (8.25)
gives an example of a function in +. This may be checked directly from the fact
that the Mellin transform of this function is
00 00
f xZ-lx(x)dx = 2 f e-t/4sh(zt)sin tdt
o 0
(8.26 )
= 2.;;ieZ-7r /4 sin 1rZ
- see the tables of Prudnikov, Brychkov and Marichev [1,2.5.57.1].
We note that the space + is rich in elements. One may show that the Mellin
convolution (see (1.114» of functions wand 1/J belongs to + for any ,p E S+ and
w E+.
Owing to the invariance of the space  = (Rl) relative to the operator I
we may easily prove the following lemma.
Lemma 8.3. The operator I+, Rea > 0, maps the space + isomorphically onto
itself
Indeed, let co : + --+  be the operator of continuation by zero from the half-
axis to R 1 and let P+ be the restriction operator onto R, so that I+ = P+I+.co,
Since co(+) C  = (Rl), we obtain Ig+(+)  + by the invariance of (Rl)
relative to the operator I+.. The inverse imbedding is obtained by a similar
consideration of fractional differentiation vg+.

 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS 157
Based on Lemma 8.3 we may define the operator I, Re a > 0, in the space
+ dual to  + by the equation
(I/,w) = (I, Ig+w), w E +.
(8.27)
Lemma 8.3 immediately yields the following.
Corollary. The operator I, Rea> 0, maps the space + isomorphically onto
itselt
Equation (8.27) may serve as a definition of the fractional integral II for
I E L,(R) in the case p  l/a, though this integral diverges in general in this
case if understood in usual sense - see also Remark 8.2. This occurrence is an
vantage of the test functions space +. The spaces F",.. discussed briefly below
have no such advantage.
To formulate the analogue of Lemma 8.3 for the operator I we need to define
initially the following spaces of test functions
+ = {w : w E S+, i w(z)zo->dz = 0, Ii: = 1,2,... } ,
equipped with the topology of the space S+.
Lemma 8.4. The operator I, Rea> 0, maps + isomorphically onto +a.
The proof of this lemma is easily derived from Lemma 8.3 if we observe that
(Irp)(x) = x 2a (W I+ Wrp)(x) , (Wrp)(x) = x- 1 - a w(l/x).
Corollary. The operator Ig+, Rea > 0, defined by the equality (Ig+l,w) =
(I, Iw), w E +, maps the space (+a)' isomorphically onto (+)'.
Remark 8.3. Powers of x, x>' are elements of the space + if A f; 0,1,2,... and
of the space (+)' if a - A f; 1,2,... (for excluded values of A powers x>' are not
distinguishable from zero as elements of the corresponding space).
8.5. McBride's spaces
We discuss now very briefly another approach to fractional integra-differentiation
of generalized functions which was developed by McBride. Let J" be the space of
functions infinitely differentiable on (0,00) with a support in [0, ij and such that
sup Ix-,+l:rp(I:)(x)1 < 00
z>o
(8.28)

158
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
for each k = 0,1,2,... (cr. (8.23», p being an arbitrary real number. Norms (8.28)
generate a topology in .1, = U 1.1,1.
First of all we observe that power functions x>' belong to the dual space .1;
if A > p - 1. Multiplication by x>' is a continuous operation from .11' onto .1 q if
A = q - p. More interesting will be the statement about the action of the operator
I in .1,. To formulate this we introduce one more space, namely the space .10 of
finite infinitely differentiable functions which is defined (analogously to the space
.1,) by the (semi)norms:
sup Ix-l:(1 + Iln xl}-lrp(I:)(x)l, k = 0,1,2,...
z>o
Lemma 8.5. The operator I maps.1, continuously into .1 q with q $ p + Rea, il
p+ Rea < 0, and with q = 0, ifp+ Rea> O. In the case p+ Rea = 0 the operator
I maps .1, continuously into the space .10 .
The proof is obtained by simple estimates and is thus omitted.
Lemma 8.5 allows us to define the fractional integral I+f for generalized
functions f E .1;, q $ 0, by the equation
(I+f, rp) = (I, Irp), rp E .1"
(8.29)
where p = q - a if q < 0, and p > -a if q = O. One may define I+I for f E (.1 0 )'
by (8.29) as well, taking rp E .1-a in this case. It is easily shown that I+ maps .1:
into .1:- a if q < 0, and (.10.)' into .1:- a if q = O.
Some other ways are known of how to construct spaces of infinitely differentiable
functions which are similarly relevant to fractional integration. We mention for
example space F',IJ' defined as follows. By F'P' 1  P  00, we denote the space
defined by a countable set of (semi)norms
,t( rp) = IItkrp(k)(t)IILp(R)'
It is then said that rp E F',IJ' if t- lJ rp(t) E F'P' The topology in F'P,IJ is naturally
generated by the seminorms
,t,lJ(rp) = ,(t-lJrp).
It is known that the operators I+ and I map F',IJ continuously into F,.IJ+ a
for all values of ReJJ except a countable set of values ReJJ. We do not consider
these and other spaces here in detail, but make the following remark only - see
references in S 9.

 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS 159
Remark 8.4. The equation
(I/, cp) = (I, Io+cp), cp E Fq,IJ'
does not define the fractional integral I/, where I E Lp(R), for all p  1/0: no
matter what p > -1, q  1 may be.
Indeed, let cp E Fq,1J' We take I(x) = x l - 2 /1'(1 + x)-2/ p' E Lp(R),
cp(x) = x lJ (1 + x)-(l+£)/q E Fq,IJ' C > O. Then
00
f xlJ+adx
(1/, cp)  c I(x) (1 + x)(1+£)/q
o
(8.30)
00
f x1+IJ+a-2/Pdx
= c (1 + x )(1+£)/q+2/pl
o
in view of the inequality
:z:
1 f ylJ(x - y)a-1 cx lJ + a
la -- d >
o+cp - r(o:) (1 + y)(1+£)/q y - (1 + x)(1+£)/q'
o
The integral in (8.30) diverges in the case c = 1, 0: = 3 for all p > -1 and
q E [1,00).
8.6. The case of an interval
The approach via the adjoint operator, i.e. the equation
(I:+I, cp) = (I, Ib_cp)
(8.31 )
(see (2.20» may be used now. In view of (8.31) we may consider the fractional
integral IC:+1 of generalized functions I, if the latter are defined on the test
functions cp, which form the space X, as invariant relative to fractional integration
If_. It is easily seen that the space
Cb([a, b)) = {cp: cp(x) E COO ([a, b]), cp(l:) (b) = 0, k = 0,1,2,...} (8.32)
is suitable for this aim. It is invariant relative to fractional differentiation 'D b _ too.

160
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
A similar space
C([a, b]) = {cp : cp(x) E COO ([a, b]), cp(A:)(a) = 0, k = 0,1,2,...}
is invariant relative to fractional integration I+ and fractional differentiation V:+.
By simple arguments the following statement may be proved: Abel's equation
I+cp = I with I E X', X = Ch([a, b]), has a unique solution cp = V:+/ in the
space X' of generalized functions, this solution being understood in the sense that
('D+I,w) = (I, 'Db_ W ) , W E X.
 9. Bibliographical Remarks and
Additional Information to Chapter 2
9.1. Historical notes
Notes to  5.1. Fractional integro-differentiation of the fonn I <p (but differing indeed from I <p
by the factor (_l)a) appeared in the paper by Liouville [1, p.8] (1832). Liouville arrived at this
expression by fonnally transfonning his original definition of fractional integration of functions,
represented by series of exponential functions, or by an integral of such a function, we refer also
to his papers [2] (1832), [5] (1834). The fractional integral It. <p in the fonn (5.4) may be fOWid
in Letnikov [1, p.28] (1868). Liouville had to deal with the so called complementary functions,
which gave him much trouble - Liouville [2, p.94-105] (1832), [4] (1834). The representation of
fractional calculus free of this idea was given by Letnikov [1] (1868). The solution of the Abel
equation I<p = f on the whole line was first obtained by Liouville [5, p.277] (1834) in the fonn
<p = _I!..-a f'. As regards integrals of purely imaginary order (5.18), see the references in  4.1
(notes to  2.4). Equations (5.21)-(5.23) were fOWid by Liouville [2, p.121-123] (1832), a rather
strict proof being later given by Letnikov [1, p.38-44] (1868). Relations of the type (5.25) are
contained in the hand-book by Erdelyi, Magnus, Oberhettinger and Tricomi [2, 13.2 (7)]. The
result (5.16) first appeared in Kober [1] (1940) in the case of the half-axis.
Notes to  5.2. Theorems 5.1, 5.2 were proved by Rubin [22] (1986). We also note that
transfonnations of the type (5.34) were known long ago, as for example, Isaacs [1], p.175 (1953).
Notes to  5.3. Theorem 5.3 was proved by Hardy and Littlewood [3] (1928) by means
of re&lTangement of fWictions - Hardy, Littlewood and Polya [2, p.348]. There is known a proof
of this theorem which is elementary in the sense that it uses no other means except a long
chain of Holder and Minkowsky inequalities - Solonnikov [1] (1962). There are proofs based on
interpolation methods - see the references in  29.1 (notes to  25.3) concerning similar theorem
for multidimensional (Riesz) fractional integration. In the proof of the necessity conditions of
Theorem 5.3 we follow Stein [2, p.139]. The counterexample (5.41) was given by Hardy and
Littlewood [3] (1928). In the same paper Theorem 5.4 was also proved for the operator Ig+
but Wider more strict assumptions concerning the involved parameters. The case #J = ap - 1
was considered in the paper [6, p.363] (1936) of the same authors. Theorem 5.4, Wider the
assumptions stated in this theorem, was proved for the operator Ig+ by Flett [3] (1958). The
case p = q = 1, #J < -1 was dealt with in B06anquet [2, p.13] (1934). The proof for the operator
I in the case m = a, suitable for p = 1, can be also found in Miller [1] (1959). Theorem 5.4
for the operator I was proved by Okikiolu [3a] (1966), but with m ::/= o. In this paper he also
proved inequalities of the type (5.49) for the operator I - 1+ under assumptions given in (5.49)
b . h 1 1
ut WIt ;; - a < q'
The statement of Theorem 5.4 for the operator Ig+ was in fact already made in Section 3 -
see Theorem 3.7 and  4.1 (notes to  3.3, 3.4).

 9. ADDITIONAL INFORMATION TO CHAPTER 2
161
Theorem 5.5 which generalizes Theorem 5.4 in the case p ::/= 1, was proved by Rubin [22]
(1986).
Theorem 5.7 with p = 1 is given in Hille and Phillips [1, p.681].
Notes to  5.4-5.6. The integral (5.57) was already present in Weyl [1, p.302] (1917). Its
appearance here had, however, an episodic character. As an independent object of investigation,
the fractional derivative in the form (5.57) and in more general form (5.80), arose in Marchaud
[1] (1927), where it was comprehensively studied, we refer also to  9.2 (note 5.11) below. So it is
now accepted that (5.57) - (5.58) is referred to as the Marchaud fractional derivative. Definition
5.1 goes back to Hadamard, who introduced the idea of the finite part of the integral - Hadamard
[2, ch. III].
The value of the normalizing factor in (5.80) in the form (5.81) was already known to
Marchaud [1] (1927). He also used the "truncated" expressions (5.59), (5.80'). The Laplace
transforms of these truncated derivatives were evaluated by Berens and Westphal. [2].
Notes to  5.7. The representation of the semigroup theory and of fractional powers of
operators may be found in many books and papers. We refer, for example, to the books by
Dunford and Schwartz [1], Y06ida [2], Hille and Phillips [1], Butzer and Berens [1], Krasn06el'skii,
Zabreiko, Pustyl'nik and Sobolevskii [1]. The definition (5.84) of a fractional power of an
operator (the abstract analogue of the Marchaud derivative) is due to Balakrishnan [3] (1960).
We note Lions and Peetre [1, p.54] (1964) where the truncated construction (5.85) was used to
characterize the domain of definition of the fractional power (_A)a in the case of integer a.
This was generalized to the case of arbitrary a > 0 by Berens, Butzer and Westphal [1] (1968).
The realization of fractional integra-differentiation in the form (5.82), (5.84) in the spaces C w is
contained in Babev and Tarasov [1] (1978). The paper by Hughes [2] (1977) is also relevant.
It develops the theory of semigroups of unbounded operators which includes the case of the
Riemann-Liouville operators [+ in Lp(O,oo). See also Lanford and Robinson [1] (1989). In
connection with the theory of fractional powers of operators many papers may be mentioned.
We indicate, for example, Krasnosel'skii and Sobolevskii [1], Balakrishnan [1]-[3], Yosida [1],
J. Watanabe [1], Komatsu [1]-[6], Kato [2]-[4], Butzer and Berens [1], Westphal. [1], [2], Howel and
Westphal. [1], Y06hinaga [2], Yoshikawa [1], Hirsch [1], Fattorini [1].
Notes to  6.1-6.3. The space [a(L p ), 1 < p < l/a, of fractional integrals of Lp-functions
as an independent object for investigation appeared in Samko [7] (1969), [9] (1970), [11] (1971),
[14] (1973). The representation in Section 6 is based on [14] and follows in part the book by
Samko [31,  1].
Theorem 6.1 and Lenuna 6.1 were proved in Samko [14] (1973), see also Samko [31, pp.9-14].
The statement of Theorem 6.1 was known earlier in the case p = 2 for functions <p e L2(Rl)
which have compact support - see Stein and Zygmund [2, p.253] (1965), where the modification
(12.1) of fractional integration was considered - and in the case of a half-axis - see Berens and
Westphal. [2] (1968) - under the additional assumption that the function J(x) itself belongs to
Lp. The characterization of the space [a(L p ) presented in Theorem 6.2, was given in Samko [14]
in terms of (6.18), but in a weaker form. As regards the version (6.19), it was proved by Samko
[23] (1977) in the multidimensional case for Riesz fractional integrals. The characterization of the
space [a(L p ) n Lp in terms of (6.19) with the strong restriction 1 < p < 1(2a) in the sufficiency
part was given by Herson and Heywood [1] (1974). In the complete form the proof of Theorem
6.2 was presented in Samko [31]. The space (6.28) first appeared in Samko [17], [18] (1976),
[20] (1977) in the multidimensional case. Its characterization in the form (6.29) was obtained in
Samko [17] (1976) for p  r  p/(1 - ap) in the multidimensional case as well. The property,
stated in Remark 6.1, was proved in Samko [27], [31] (1978). The statement on the change of sign
of real-valued functions in Lr n [a(L p ), 1  r < 1/1 - a, was noted in Samko [14]. Theorem 6.3
was given in Samko [13] (1971).
Notes to  6.4 and 6.5. The sufficiency test of Theorem 6.4 was noted in Samko [14]
(1973). Theorem 6.5 in a more general form and in the multidimensional case was obtained by
Samko [26]. Theorem 6.7 was proved in Samko [14]. The assertion (6.40) is due to Hardy and
Littlewood [3] (1928), who considered the periodic case, the non-periodic example being noted in
Samko [8] (1969), [14] (1973). The estimate (6.42) is close to (13.62), see below. The estimate
(6.43) was given in Samko [27,  1] (1978).

162
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Notes to  7.1. Relation (7.1) was first given by Kober [3, Lemma 3] (1941) under the
asswnptions 0 < a  1, rP(t), Itl-arP(t) E LlCR1) (c!. Theorem 7.1).
Notes to  7.2. Relation (7.14) was apparently first noted by Doetsch [1, p.301] (1937)
with a > 0 or -n - 1 < a  n and 'P(O) = 'P'(O) = .., = 'P(n-l)(O) = O. It was, however, well
known earlier in the case of integer a. Relation (7.14) was proved in Widder [2] (1946) (the first
ed. in 1941) in tenns of the inverse Laplace transform - see Subsection 9.2 (note 7.3) below.
Lemma 7.1 and Theorems 7.2 and 7.3 have not been published elsewhere.
Notes to  7.3. Expressions (7.20) and (7.21) were first published by Kober [1, Theorems
5(a), 5(b)], 1940, in the space L,,(O,oo), a> 0 and 1  p < 00 for (7.17) and 1  p < l/a for
(7.18). Theorem 7.6 was established by Vu Kim Tuan [2] (1985). Relation (7.28) with the formal
interpretation of fractional integro-differentiation was already written down by Lambe [1] (1939).
Its strict formulation for fractional derivatives treated in the complex plane as a generalization of
the Cauchy relation (see (22.33") or (22.21» can be found in Mikolas [3a] (1962).
Notes to  8.1 and 8.2. The idea of introducing the space <) was suggested by
Semyanistyi [1] (1960). It was developed by Lizorkin [1] (1963), who introduced this concept into
function theory practice and widely used it in the theory of Liouville differentiation of one and
many variables. We refer also to the paper by Lizorkin [5] (1969). We consider it in Subsection
8.2 while discussing the space <). We note also Yoshinaga [1] (1964), where some properties of <)
were investigated.
The space V of test functions, close to the space 6-, was introduced by Veber [2] (1974).
This space differs from 6_ by the right-sided finiteness of its functions. It is also invariant relative
to fractional integra-differentiation I, V. Veber [4] (1976) investigated also convolutions and
Fourier transforms in V' in cOIUlection with the applications to differential equations of fractional
order.
The statement of the type of Lemma 8.2 on the boundedness of fractional integration
operators I in L,(O,I) was first noted in Kalisch [1] (1967).
Relations (8.17)-(8.20) have apparently not been noted elsewhere. The case k = 0 in
(8.20) was indicated in Bredimas [1, p.23] (1973), within the framework of Schwartz's approach
and under a different interpretation of fractional integro-differentiation - via fractional order
differences, see  9.2 (note 8.5) below. We deal with such an interpretation in the case of ordinary
functions in  20.
Notes to  8.3. The approach to fractional integro-differentiation of generalized functions
presented here is due to Schwartz [1, v.2, p.30] (1950). It is represented in Gel'fand and Shilov
[2, p.149].
Notes, to  8.4. This approach via an adjoint operator was developed on the half-axis in
the papers by ErdeIyi and McBride [1] (1970), Erdelyi [15] (1972), [17] (1975), McBride [2] (1975),
[4] (1977). In the brief representation of this approach we followed the papers by McBride [2],
[4], where the spaces J", J, and F,IJ were in particular defined and investigated. The further
development of results obtained there and other similar is discussed in the book by McBride [6]
(1979).
The space <)+, <)+. of Lizorkin's type in the case of the half-axis were introduced in Rubin
[27] (1987), where fractional integration in the spaces <)+, (<)+.)' was studied.
Notes to  8.5. The invariance of the spaces cgo ([a, b]), Cr([a,b]) under fractional
integration was observed in the paper by Veber and Urdoletova [1] (1974), where this was the
starting point of considering fractional integro-differentiation of generalized functions on a finite
interval [a, b]. Later on such spaces were used in the paper by Estrada and Kanwal [1] (1985),
devoted to solving - in generalized functions - various classes of integral equations including
singular ones (with Cauchy kernel), the Abel integral equation and others. The solution in
generalized functions of the equation
x
f 'P(-r)d-r
[hex) _ h(-r)]l-a = lex), x> a
a

 9. ADDITIONAL INFORMATION TO CHAPTER 2
163
was in particular given there, we refer to fractional integration operators connected with such as
equation in  18.2 and in  23.1 (notes to  18.2).
9.2. Survey of other results (relating to  5-8)
5.1. Liouville fractional differentiation (Vf)(x), 0 < a < 1, may be considered in the fonn
N
(Va J)(x) = - 1 lim..!!.- f J(t)dt
- r(1 - a) N-oo dx (t - x)a
:e
(9.1)
instead of (5.6) (Cossar [1]). This definition is convenient for its applicability to functions with
worse behaviour at infinity in comparison with (5.6). From this point of view the approach (9.1)
has something in common with the Marchaud definition (5.58) of the fractional derivative. For
(9.1) Cossar [1] proved the relation
00
( Va J )( ) ( Va J)( ) a f J(t)dt b
- x = b- x - r(1 _ a) (t _ x)l+a' x < .
b
The construction (9.1) was used in Bosanquet [8] for studying the conditions of the existence of
locally summable solutions of Abel's equation over an infinite interval. In this paper the following
statements were in particular proved: i) if J(x) is representable by a conventionally convergent
fractional integral J(x) = (l<p)(x), 0 < a < 1, then necessarily <pete) = (VJ)(x) where (VJ)(x)
is the Cossar derivative (9.1). The extension of this inversion statement to the case of the integral
I<p, interpreted as (C,p)-summable at infinity was later on given by Isaacs [3]. ii) in order that
J(x) be represented by the conventionally convergent integral J(x) = (1<p)(x), 0 < a < I, of a
locally summable function <p(x), it is necessary and sufficient that I;: a J e AC([a, b]) for all a
and b and
b 00
f (t - x)a-1dt f (s - t)-a-l J(s)ds - 0
:e b
as b - 00 for almost all x. This is an extension of Tamarkin's Theorem 2.1 to the case of the
whole real line and of locally summable solutions.
The above statements were extended by Choudhary [1] to the equations
00
f k(t - x)<p(t)dt = J(x), x > O.
:e
of Sonine type (see  4.2 (note 2.3)) under the assumption that there exists a kemell(t) such that
(4.2") holds.
The COS8ar derivative was used by Trebels [1], [2] in his investigations of fractional
differentiation in the space BVa(R) in connection with Fourier-multiplier problems, this space
being a natural generalization of the bounded variation function space and well-suited to problems
of fractional calculus. We refer also to Gasper and Trebels [1]-[4] where the Cossar derivative
found an application in consideration of more complicated function spaces W BVq,'Y aimed at
Hankel multiplier problems. We refer to Carbery, Gasper and Trebels [I], where these spaces were
shown to coincide with the space RL(q,) of the localized Riemann-Liouville fractional integrals.

164
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Gasper and Trebels also considered in [1]-[4] Fourier-Jacobi multiplier problems in tenus of spaces
wbvq,., connected with discrete fractional differences introduced by Bosanquet.
5.2. The "truncated" fractional integral
:&
1 f Ip(t)dt
(It.,NIp)(x) = na) (x _ t)l-a '
:&-N
N>O,
may be considered. IT Ip(t) e L,(R 1 ), 1 < 'P < l/a, then It.,NIp -+ It.1p almost everywhere and in
the norm of Lq(Rl), q = p/(I- ap). The almost everywhere convergence is obtained by means of
the Holder inequality. The convergence in Lq may be obtained by the Banach-Steinhaus theorem.
One may also use the representation
:&-N
(la NIp)(x) = I(x) _ sina1l" f ( N ) a I(t)dt ,
+, 11" x - N - t x - t
I = It. Ip
-00
- Samko [14, Theorem 2 and Lemma 2].
5.3. A thorough investigation of conventionally convergent fractional integrals It. Ip =
N
l/na) lim f Ip(x - t)t a - 1 dt of functions Ip(t) which do not necessarily vanish at infinity was
N-oo o
caJTied out by Love [1]. He defined the space I>., 0 < a $ .x < 1, of functions Ip( t) for which
there exists a function weT), T > 0 (depending on Ip(t» such that I :&jT Ip(t)dtl $ weT) and
00 :&
f t>'-ldw(t) < 00. It was shown that if Ip e I>., then I+1p exists as a uniform limit for x if .x> a,
1
and as a limit for almost all x if .x = a. The case of ahnost periodic functions (unifonnly almost
periodic functions and the almost periodic functions of Stepanov) was especially considered in
this paper. Some results concerning fractional integrals of functions non vanishing at infinity may
also be found in Geisberg [3] and Bosanquet [8].
5.4. One may use the results (5.16), (5.17), (5.16') offractional integration by parts similarly
to the case of a finite interval as in  4.2 (note 2.6), to construct biorthogonal systems of functions.
ErdeIyi [3], who suggested this idea, constructed biorthogonal systems of functions on the half-axis
(0,00) expressed in tenns of confluent hypergeometrical functions, starting from (5.16'). A given
initial biorthogonal system {lpn,1/1m} (see  4.2 (note 2.6» was chosen as Ipn(X) = La)(2x),
n
1/1n(x) = e-2:&xaLa>C2x) where La)(x) = ;hx-aex £;'(e-xx n + a ) = ?: (j) (-,)J are the
Laguerre polynomials. J = 0
&.5 Laguerre polynomials La)(x) (see above) admit a representation by the fraCtional
integral
La)(x) = (n + a)e:& l:n-a(e-:&xn)
n
00
= (n + a) 2- f e-tt-a-n-l(x + t)ndt
n na)
o
under the assumption that Rea < -no This was used by Srivastava [4] for the evaluation of
fractional integrals of functions which have the form e- 8t pet), pet) being a polynomial.

 9. ADDITIONAL INFORMATION TO CHAPTER 2
165
5.6 The fractional integration operator taken in the form
x-'I -01 j :&
IJ:OI'P = r(OI) (x - t)OI-It'l'P(t)dt
o
(notation due to Erdelyi [4]) is bounded in the space L,(R) if fl > -lIp'. This follows from
Theorem 1.5. In Erdelyi [4] a modification of the operator (9.2) was suggested in the form
(9.2)
-'1- 01 { j :&
1:'0I'P = x r(0I) (x - t)OI-I t'l 'P(t)dt
o
(9.3)
m-I 00
- :E<-I)'("  1)""_'_1 j t'+'V'<t)dt}
t=O 0
which differs from (9.2) by a finite-dimensional operator. Such as operator is bounded in
L,(R), 1 < l' < 00, under the choice m = 0 if fl > -lIp' and m = [-fl-1/p'] if fl < -lIp',
fl - 1/1''# -1, -2,.... The expression
r.t r(fl + IIp' - i)
(rot 'I,OI'P)(S) = r(fl + 01 + IIp' _ i) (rot'P)(s),
s = i + IIp,
for the Mellin transform (1.112) of the fractional integral (9.3) was also obtained in this paper in
the case 'P e Lp(R I ), 1  p  2 (c!. (7.22». Similar results were also obtained for the right-hand
sided integral of the type (9.3) with a variable lower limit of integration.
The boundedness of the operator (9.3) in the weighted space L,(R,xIJ) was considered by
Rooney [4].
5.T. By extending Theorem 5.5 we give here a statement similar to Theorem 3.12 in the case
of an infinite interval. Namely, we consider the boundedness of fractional integration operaton
from the space L,(O,p) into the Holderian space H-I/'(O;r), IIp < 01 < 1 + IIp, 0 being the
axis or half-axis. Let p(x) be the weight function (5.39). Let us introduce the notations
{ J1.l:/p
61: =
OI+et -lIp
if IJI: > OIp - I,
if 0 < J1.1: :5 OIp - 1, el: > 0,
Al = {k : k e {2,... ,n}, J1.1: > O},
A2={k:ke{1,...,n}, IJI:>O};
{ -01 + (2 - IJo)/p if J1.o > 1 - p,
6) = -01 - J1. p l - L 61: + 1 - 01 - eo + IIp, eO > 0, if 2 - OIp - P < J1.0  P - I,
teAl 1 if J1.o:5 2 - Otp - 1',
J1.0 = - J1. - J1.I - . . . - J1.n;
6) = -2 (01 - !. ) - L 61: _ { J1.olp ,
p l:eA 2 e - IIp
if J1.0 > 1 - p,
if J1.0 :5 1 - p, e > 0,

166
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
6) = -2a + (2 - IJo)/p - L 6lei
le eA
(1) f.1. 6
{ (I + x)600 x P n Ix - xlel le ,
keA 1
r+(x) = ()
(1 + Ixl)600 n Ix - X/c 16.,
leeA
6(3) 6
{ (I + x) 00 n Ix - xle lie,
r_(x) = leeA
r+(x),
. 1
for 0 = R+,
for 0 = h,I,
. 1
for O=R+,
for 0 = Rl.
The following statement is valid (Rubin [22]). Let 1 < p < 00, IIp < a < IIp + 1, p(x) =
n
(1 + x 2 )1-'/2 n Ix - xle II-'Ie, let the assumption (5.47) be satisfied in the case Xle e 0 = R
le=1
and let I-'le satisfy (5.48) with J.l.l < p - 1 also. The operator 10+ is bounded from Lp(R; p)
into 'H-I/P(R.,r+). The prime here appears in the case J.l.O $ 2 - ap - p and implies the
absence of the null effect at infinity, i.e. in general lim r+(x)(Io+<p)(x) '# 0, for <p e Lp(R; p).
:r:-oo
H both J.l.l < p - 1 and J.l.Q < 1 - ap, then the operators 1 are bounded from Lp(O; p) into
H a - 1 / p ( r"I. )
o u,r% .
5.8. There are generalizations of a weighted Theorem 5.5 to the case of one- and two-
weighted estimates with arbitrary weights. Andersen and Heinig [1] obtained sufficient conditions
for the boundedness of the fractional integral operators (5.1) and (5.3), and of more general
Erdelyi-Kober operators (18.1) and (18.3), from Lp(R;p) into L,(R;r). They proved, for
example, the following result for the fractional integral (5.1): let 1  p  q  00, 0 < a < 1 and
let p(x) and rex) be nonnegative weight functions on R; if there exists {3, 0  {3  I, such that
{  ) 1/ 9 ( A ) 1/ P
Fa-I (11. A) = V 1(1 - A)( a-I)p r( 1)If dl ! I(A - t)( a-I )(I-P)P' p( 1)1-" dl :S c < +00
for all A > 0, then 10+ is bounded from Lp(R; p) into L,(R; r).
Andersen and Sawyer [1] found necessary and sufficient conditions for the fractional integrals
(5.1) and (5.3) on the half-axis to admit a one-weighted (i.e. r = p) estimate. They proved, for
example, that if 0 < a < 1, 1 < p < -k, t =  - a, then the operator 1+ is bounded from
Lp(R, p) into L,(R, p) if and only if the inequality
[ a+h ] ; [ a ] ;r
sup !.. f pet)' dt !.. f p(t)_p' dt  c < 00
O<h<a<oo h h
a a-h
holds. They also found necessary and sufficient conditions for the weight p (or r) to have the
property that there exists the weight r (respectively p) such that 10+ or 1 is bounded from
Lp(R;p) into L,(R;r).
Stepanov [1]-[5] completely solved the problem of two-weighted estimates for the fractional
integral (5.1) in the case a  1. He proved, for example, that for arbitrary nonnegative functions
p(x) and rex) on R the operator 1+, a  1, is bounded from Lp(R,p) into L,(R;r),

 9. ADDITIONAL INFORMATION TO CHAPTER 2
167
1 < p  q < 00 if and only if
max sup Fa-dP, A) < 00
,6=0,1 A>O
(see Fa -1 (P, A) above). He also obtained similar results for the cases p = 1 and q = I, and
in terms of the function Fa-1(P' A) fOWld conditions for the operator I+ to be compact from
Lp (R ' p) into Lq (R ' r). The results analogous to listed ones were obtained by Stepanov also
in the case 1  q < p < 00. In the case 1 < p = q < 00 other conditions were obtained by
Martin-Reyes and Sawyer [1].
Stromberg and Wheeden [1] gave sufficient conditions for multidimensional fractional
integrals (see  29 below) and in particular for the fractional integrals (5.2) and (5.3) on the
real axis to admit polynomial weight estimates. We fonnulate one of their results for It:. Let
1 < p < 00, 0 < IIp - l/q  a, P = a - IIp + l/q. Let also
n
Q(x) = II Ix - xl:l#Jk,
1:=1
n
-00 < xl < x2 < .. . < Xn < +00, L J.l.1:  P,
1:=1
p(x) = IQ(x)IPw(x),
[ n ( ) -fJk ] q
rex) = IQ(x)I(1 + Ixl)-,6 II Ix - xI:I w(x)qlp
1 + Ix - xI: I
1:=1
where PI: = min(J.l.I:' P) and let a nonnegative measurable fWlction w(x) satisfy the so-called
Muckenhoupt condition
( ) p-1
II_lIp'
jIj ! w(")d,, jIj! we,,) do:  c < +00.
1 < p < 00,
(known also as Ap-condition) for every interval I on R 1 . Then If, is bOWlded from L p (Rl j p)
into Lq(R1j r).
5.9. Hardy and Littlewood's Theorem 5.3 on the mapping property of fractional integration
from Lp into Lq, q = pi (1 - ap), was extended to Orlicz spaces by O'Neil [2]. Let L At (R 1 )
be an Orlicz space generated by an N -fWlction. We use here the terminology of the book by
Krasnosel'skii and Rutitskii [1]. O'Neil [2] showed that if
uM' ( u)
1) M(u)  p > 1,
1
f M-1(u)
2) u 1 + a du < 00,
o
then the fractional integration operator I, 0 < a < 1, maps LAt(R1) continuously into Le(R1)
:&
where a-I (x) = f M-1(u)u- 1 - a du. A somewhat weaker assertion was given by Sharpley [1]
o 
using a different method. We note that O(x) is an. N-function Wlder the assumption 1). We must
add that O'Neil dealt with the multidimensional case.
See also Gel'man and Yasakov [1] concerning the mapping properties of potential type
operators in the spaces LAt(O), mesO < 00.
5.10. Geisberg [3] considered Marchaud fractional derivatives (Dt:J)(x), 0 < a < 1, x e RI,
of functions lex) which are locally Holderian of order more than a, bOWlded and such that there

168
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
o
exists lim N-l f I(x)dx = c. He in particular showed that I+Di.f == I(x) - c for such
N-oo -N
functions.
5.11. Marchaud [1] also defined and investigated a construction more general than what
we refer to as the Marchaud fractional derivative - (5.80). That is Marchand considered (see
pp.348-351 of his paper) constructions with generalized finite differences:
1
f oo L: ci/(x - kit)
OI ( 1 i=O d
I x) = "'Y(OI) t l +OI t,
o
I = [01] + 1,
where ki are arbitrary positive increasing numbers, and the coefficients Ci are bounded by
1 .
conditions L: Cikf = 0, j = 0, 1,... ,I - I, the nonnalizing constant being equal to "'Y(OI) =
i=O
1
r( -01) L: Ciki- In the case 01 = 1,2,... one must pass to a limit as 01 - m = 1,2,.... He
i=O
showed that if I(x) = 1+1p then the Liouville fractional derivative V+I coincides with IOI(X) and
thus doesn't depend on the choice of ki' Under the choice ki = i, Ci = (_I)i (D the expression
for 1 01 (x) coincided with (5.80).
5.12. The so-called divided differences [Xl, ..., xn+1; J] = ([X2, ..., xn+l i I] - [Xl, ...,xn i 1])/
(Xn+l - Xn), [Xi I] = I(x), in connection with fractional integra-differentiation were considered
by Popoviciu [1]. He showed in particular (see p.39 of his paper) that if such differences are
bounded, then I(x) has continuous fractional derivatives of order 01 < n.
5.13. In the paper by Gearhart [I], where fractional integration was considered from the
point of view of semigroup theory, subspaces in £2(0,00) were there defined which were invariant
relative to the left translation (Ttf)(x) = I(x + t), t > 0, and such that the operator 1 was
bounded in these subspaces. For these purposes the approximation operators
00
WOI I = 2- f tOl-1e-etT I dt e > 0
e r(0I) t ,
o
(9.4)
were introduced - compare this with the modification (18.64) of fractional integration - which
were the unitary equivalent to Toeplitz operators on Hardy subspaces H2(R). Based on
some results of Beurling and Lax for Toeplitz operators conceming spaces invariant relative to
translation, the author constructed a series of spaces Me L2(O, 00) which admit a strong limit
for W:. He also considered the question as to whether this limit coincides with the direct fonn of
the operator 1. The limiting case 01 = ifl of operators of purely imaginary order is also treated
in the paper.
5.14. The possibility for 1+ Ip or 11p being a constant identically: (1) (I+ Ip)(x) == C,
X e R, or (2) (1Ip)(x) == c, X e R, may be easily cleared up. Let Ip(x) be locally integrable,
then (1) holds if and only if I(x) = (c/r(l- OI»X-OI for 0 < 01 < 1 and c = 0, Ip(x) == 0 for 01  1.
00
In the case when f (1 + t)- OI IIp(t)ldt < 00, (2) holds if and only if c = 0 and Ip(x) == 0 (Roberts
o
[1]).
6.1. There is an extension of the characterization of the space 1 00 (£p) to the case of the
space 1 00 [L p (p)] with a general weight satisfying the so called Muckenhoupt-Wheeden condition.
It is given in Andersen [I], and is fonnulated in  29.2 (note 26.11).
6.2. Theorem 6.2 on the characterization of fractional integrals of functions in Lp was
extended to Orlicz space using O'Neil's results (see note 5.9 above) by Samko and Chuvenkov [1].

 9. ADDITIONAL INFORMATION TO CHAPTER 2
169
Nlunely, let conditions 1) and 2) from note 5.9 be satisfied and let fWlctions M(x) alid C(x) from
Dote 5.9 satisfy the 2-condition (Krasnosel'skii and Rutitskii [1]). In order that J(x) E la(L M ),
 < a < 1, it is necessary and sufficient that J(x) E Le(R1) and that (D+,£J)(x) converges in
th norm of L M (R1) as e - 0.
A Note that the function C(x) satisfies the 2-condition if and only if there exists a constant
a > ° such that
:&
f M-1(u) du < /I M- 1 (u) x  xo.
u a u - u a '
o
6.3. Let a * <p be a convolution operator. It maps continuously
i) Lr(Rl), 1  r < p, into la(L p ) if a(x) E la(L q ), l/q = 1 - l/r + lip;
ii) L p (R1) into la(L p ) if a(x) E 1+(L 1 ) or /I(x) E 1(Ld;
iii) la(L r ) into la(L p ) if 1 < r  p < l/a and a(x) E Lq(R1), l/q = 1 - l/r + lip (Samko
[14], [27, p.8]). This statement may be deduced from the relation
a * <p = 1+.(D+a * <p).
The validity of the latter is easily obtained Wlder the above assumptions by taking into acCOWlt
the obviousness of the equality for "good" fWlctions and bOWldedness of the operators in the left-
and right-hand sides in the corresponding spaces.
6.4. Let Wr(J, t) be the integral continuity modulus (6.31) of a function J(x) on R. The
following generalization:
wr(Io+<p, t)  cll<pllpta+l/r-1/p, p  r  pl(1 - ap),
(9.5)
of the estimate (6.39) is valid, 1 < p < l/a. It is a particular case of the general estimate
wr(K<p, t)  w,,(K, t)II<pllp, lip + lis = 1 + l/r
:&
for the operator K<p = J K(x - t)<p(t)dt (Karapetyants).
o
7.1. Equation (7.1) for the Fourier transform of a fractional integral and proved for
<p E L1(R 1 ) is extended to functions <p E L p (R 1 ), 1 < p  2,1 < p < l/a, if the convergence of
the Fourier transforms F<p and F(I<p) in (9.1) is treated in the norm of Lp' and Lq" respectively,
q = p(1 - ap)-1 - Okikiolu [2] - where the Riesz potential over Rl was considered. We note that
the proof in [2] may be simplified if one uses generalized functions on the space () studied in  8.2.
Added in proof. The estimate liFe 10+ <pllp  cll<pllp, 0< a < I, p = 2/(1 + a), is worth of
mentioning (Titchmarsh [I], s. 4.12).
7.2. In the book by Widder [2, pp.73 and 74] the following statement similar to Theorem
7.2 was proved.
00
Theorem 9.1. Let <p(x) E LICO, b) Jor every b > ° and let the integral J e-Potl<p(t)ldt be
o
convergent. Then the relation
'Y+ioo
 f (L<p)(p) e P :& dp = { (10+ <p)(x) <
211"1 pa 0,
'Y- ioo
x  0,
x < 0,

170
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
i8 valid for a  1, 'Y > PO, 'Y > 0 or for 0 < a  1 'Provided that the function <p(u) h48 a bounded
variation in a neighbourhood of the 'Point u = x, x  o.
00
7.3. Let Spr.p = rl'P) J (; be the Stieltje8 transform. The expression I-lSpr.p = SIr.p
o
is valid. In the paper by Widder [1] the inversion formula is obtained for the generalized Stieltjes
transform
00
f da(t)
(x + t)'P = f(x), l' > 0,
o
by applying the fractional integral (5.3), the complex-valued fWlction aCt) having a bounded
variation on every interval [0, R], R> 0, of the real axis.
7.4. The question, whether functions f(x) E L'P(R) have fractional derivatives Dt. f E
L'P(R) was answered in tenns of the Laplace transform by Berens and Westphal [1].
7.5. Let Wk,m(X) be the Whittaker function. Varma [1] defined integral transform
00
(Wr.p)(x) = x f e- xt / 2 (xt)m-l/2W k ,m(xt)f(t)dt,
o
(9.6)
which generalizes the Laplace integral transform (1.119), the latter being obtained from (9.6) in
the case k + m = 1/2. This transform was called later on the Varma tran8form. Let Itf+ <p and
Ir.p be fractional integrals (5.1) and (5.3). The Laplace transform of the functions x->'(Itf+r.p)(x)
and x->'-1 (Ir.p)(I/x) coincides with the Varma transform (Kalla [1]). Similarly the Laplace
transform of the functions (I+<p)(x2) and x2>'-2(Ir.p)(x-2) is the Meijer transfonn, defined in
 1.4 (Bora and Saxena [1] and Martie [1], [2]). The integral transfonns of Vanna and Stiltjes (see
note 7.3 above) as well as integral transfonns of the general kind (1.101) of fractional integrals
were considered by Kalla [1], [5], [8].
It was shown in K.J. Srivastava [1], that the generalized Meijer transform
00
Wk,mr.p(x) = x f (xt)-k-l/2 e -xt/2W k + 1 / 2 ,m(xt)r.p(t)dt
o
(9.7)
= x-m-k (W k + l / 2 ,m Ck-mr.p(t))(x)
of functions xa/2+k-m(Itf+tm-k<p(t))(x) and xa(I_k+a/2,ar.p)(x) where I;t;a is an operator
(9.2), leads to a transform of the type (9.7).
Bhise [1] defined an integral transform
00
( G )( x ) = X f Gm+1'0 ( xt l 'lI+a.....,'7m+am ) ( t ) dt
r.p m,m+l '71, ..,'7m,P r.p
o
(9.8)
with the Meijer G-function (1.95) in the kemel, which was called the Meijer-Laplace transform.
It is reduced to the Vanna transform (9.6) in the case al = ... = a m -l = 0, am = -1/2 - m - k,
11m = 2m, p = 0 and to the generalized Meijer transfonn (9.7) in the case al = ... = a m -l = 0,
am = p = -m - k, 11m = m - k. In the papers by Mathur [1], [2] the integral transfonns (9.8) of
functions X-P-II (Itf+ t" r.p(t))(x) and (I;t:ar.p)(x) were found.

 9. ADDITIONAL INFORMATION TO CHAPTER 2
171
7.6. A series of papers is concemed with the evaluation of special type of integral transfom1S
of fractional integrals. Mathur [2] found Meijer integral transfonns defined in  1.4, and the
00
transfonn x- 1 J 1 (.X, J.'i IIi -t/x)cp(t)dt with the Gauss function (1.72) in the kemelfor functions
o
x ll (lg+cp(v'i»(x 2 ) and xll(Ig+CP)(x), respectively. He gave an application to finding fractional
integrals of Fox's H -function. In Singh [2] integral transfonn with the Struve function HII (x) in
the kemel (see  1.4) of the function xll+1/2(IC+ cp)(x 2 ) was evaluated. In Rakesh [1] the integral
transfonn with Fox's H-function, defined in Gupta and Mittal [I], of fractional integrals (5.2) and
(5.3) was found.
7.7. Let F (a1 , .. . , an) = Ln f be the n-dimensional Laplace transfonn and let the
designation G(a) = AnF(a1'"'' an) denote that G(a) is the one-dimensional Laplace transform
of a function f(t, t,..., t). In connection with problems of non-linear system theory Conlan and
Koh [1] showed that if
1
F(ab" .,an) = ----tT F1 (ab a2,"', a m -1,a m +1,'" ,an),

then functions G(a) = AnF(at....,an) and G1(a) = An-1F1(a1,...,am-1,am+b...,Sn) are
related to each other by the operation of fractional differentiation: G(a) = (:[; 1'+ G1 (a), see
also Koh and Conlan [1].
7.8. In the paper by Smith [1] fractional integrals (5.3) and fractional derivatives (5.6)
00
were applied to obtain connections between Laplace-Stieltjes transfonns J e-xttPda(t) with
o
p > 0 and p = 0, as well as inversion formulae for these transfonns. Here a(t) is a real valued
function of a bounded variation on every interval (O,R), R > 0, satisfying conditions a(O+) = 0,
a(t) = 2-1[a(t + 0) + a(t - 0)].
7.9. A formula of the type (7.28) - the integral analogue of Taylor's series expansion -
was applied in Lambe [1] to derive integral representations for some special functions namely the
Gauss and Legendre confluent hypergeometric functions. This was only done formally.
8.1. The space () considered in subsection 8.2 is dense in Lp(R1), 1 < p < 00: for any
function f e Lp and each e > 0 there exists a function cp(x) e () such that IIf - cpllp < e. This
was proved by Lizorkin [5] by means of averages which were called by him completely balanced.
In Samko [29] another proof of this denseness was given.
8.2. In the papers by Lamb [2], [3] a certain approach to the consideration of fractional
integration I of generalized functions on the whole real line was developed. The investigation
was based on the theory of fractional powers of operators in Frechet spaces, developed by the
author - Lamb [1]. Operators I were shown to be fractional powers of operators 11 treated in
the spaces
'D,., = {<p : <p E Coo (R 1 ), :. [e -,,, <p(,,)] E L,(R 1 ), k = 0, 1,2, . .. } ,
and were shown to realize a homeomorphism of the space 1''1','" onto itself, provided that J.' > 0
in the case of positive sign, and J.' < 0 in the case of negative sign. See also Lamb [4], where these
statements were proved via Fourier multipliers teclmique.
x
8.3. In Sk6mik [I], [2] the fractional integration J ex:l-t:l {xr(-l cp(x - t)dt and
o
the corresponding fractional differentiation were considered in spaces of generalized functions.
The main attention was paid to defining the spaces of generalized functions where fractional
differentiation is uniquely inverted. In order to "sift out" generalized functions which infringe this
uniqueness, spaces of generalized functions were defined which were equal to zero at the point
x = 0 in a certain sense given by Lojasewich and others.

172
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
8.4. Let F p . 1J be the space of test functions defined by McBride [2], [3] -  8.4. Ahuja [1]
considered mapping properties of the Erdelyi-type operators (9.3) and of more general ones in the
spaces F p . IJ .
8.5. An interesting development of Schwartz's approach, connected with differences of
fractional order, was suggested in a series of papers by Bredimas [1]-[5]. The fractional
differentiation operator was defined in these papers as
00
lim ""'(-1)1: ( CV k )f(x - kh)/h a
h_OL...J
1:=0
(9.9)
(compare such an approach in  20 in the case of ordinary functions). It was shown that under the
appropriate interpretation of this limit, the restriction of the operator (9.9) onto the subspace of
the Schwartzian generalized function with a support on the half-axis R coincides with fractional
integration (of order -cv) of generalized functions given by Schwartz and represented in  8.3.
8.6. McBride [7] defined fractional powers of ordinary differential operators
L = xal'Dxo'Dx03.. .x o "'Dx o "+ 1 1, 'D = d/dx,
(9.10)
and considered their properties in the space Fp,lJ of the generalized functions discussed in  8.4.
I n+l I
Here the number m = n - ?: ai plays an essential role. The case m ':# 0 was treated in
1=1
McBride [7], [8] by means of ErdeJ.yi-Kober operators (see (18.1)-(18.2». The case m = 0 was
investigated by Lamb and McBride [1] via the spectral approach used in the theory of fractional
powers of operators; the results for m = 0 were shown to be obtainable as limiting cases of the
results for m > 0 in a sense.
Index laws la 1f3 = la+f3 and la x- a -f3 If3 = x- f3 r a +f3 x-a for fractional integro-
differentiation (see  10 for the second of these laws) were extended in McBride [8] to the
operators in (9.10) and operators L' = (_I)nxo..l+l'Dxo.....xo'Dxoll, 'D = d/dx, and in
McBride [9] to the operators Ta defined via Mellin transforms by the relation
(rotTalp)(x) = h(:-Y) (rotlp)(x + cv-y),
where h is a given fixed function, -y being a fixed number.
Observe also the fractional calculus of operators Ta in the surveying paper by McBride [II],
where one may find other references as well.
8.7. Vladimirov [3] extended the Schwartz approach (8.22) to the case of distributions over
the field of p-adic nwnbers. He studied some properties of the corresponding fractional integration
and, in particular, proved the semigroup property.
9.3. Tables of fractional integrals and derivatives
We do not provide large tables of fractional integrals and derivatives here, but confine ourselves
to the small Tables 9.1-9.3 below for fractional integro-differentiation 1:+ of Riemann-Liouville
and for that of Liouville l and refer to other tables in literature. Good tables of a rather large
nwnber of fractional integrals of various elementary and special functions are contained in the
handbook of Erdelyi, Magnus, Oberhettinger and Tricomi [3, ch.13]. A great deal of fractional
integrals of various functions may be found in the handbooks of Prudnikov, Brychkov and
Marichev [1]-[3] in the broken representation, but not being singled out especially as fractional
integration fonnulae. Comparatively small but interesting tables are contained also in the papers
by Higgins [5], Oldham and Spanier [1], Tremblay [1, pp.91-92, 426-433], Lavoie, Osler and
Tremblay [1], Lavoie, Tremblay and Osler [I], Nishimoto [6], Osler [I], [3].

 9. ADDITIONAL INFORMATION TO CHAPTER 2
Table 9.1
173
",,(x), x>a (1:+ ",,)(x) , x>a, aeC
1 (x-a),6-1 rr:) (x_a)a+,6-1, Re{3>O
2 (x:%c)'Y-l ( a%c ) ..,-1 ( )a ( a-z)
r( a+l ) x-a -#'1 1,1-"'Yi a+li 'iIC '
a:%c>O, "'YeC
3 (x-a),6-1 (b-x)'Y-l  ( z_a ) cr+-1 ( z-a)
r a+fJ ( b-a ) I-'" -#'1 {3, 1-"'Yi a +{3i o::a '
Re{3>O, "'YeC, a<x<b
4 ( z_a ) -1  ( z_a ) cr+-1
(b-z)cr+ ) r a+,6 ( b-a ) cr ( b-z ) ' Re{3>O, a<x<b
5 (x- a),6-1 (x:%c)7-1  ( z_a ) cr+-1 ( a-z)
r a+fJ ( a%c )l -'" -#'1 {3, 1-"'Yi a+{3i OR '
Re{3>O, "'YeC, a:%c>O
6  z a r - 1  ( z_a ) cr+-1
Z%C cr + r a+fJ ( a%c ) cr ( z%c ) ' Re{3>O, a:%c>O
( z_a ) cr ( ) 2a
7  1 z-a Re 1
(z*c)cr+l /  r ( a+l / 2 )   + ."tHC , a>-,
a:%c>O
e>'z A >.
8 r()>.cr "'Y(a, ,xx- 'xa)=e a(x_a)a El,a+l(.\x- 'xa)
9 (x- a),6 -1 e>'z ; (x-a)a+ fJ -I 1 FtC{3i a+{3i 'xx-'xa), Re{3>O
10 (x_a)a- 1 e 2i >'z ..;:i(2,X)1/2-a (x - a )a-l/2e i >.(z+a) J a_l/2('xx- 'xa),
Re a > °
11 {sin >.(z-a)} i-(I*I)/ ( )a
cos>.(z-a) 2r ( a+l ) x-a
xhFI (Ii a+ Ii a(x-a))=flFI (Ii a+li -i'x(x-a)))
12 ein >. vz -a } ..;:i (>.) 1/2-a (x_a)(2a+l)/4 {Jcr+l/(>. vz-a) }
sh >. ..p=;i ,. Icr+l/(>' vz-a)
13 (x_a)fJ-1 ein >.(z-a)} i-(I*I)/ ( _ )a+,6-1
cos>.(z-a) , 2 r a+fJ x a
Re{3> -(1 :%1)/2 xhFI ({3i a+{3i a(x-a))=flFl ({3i a+{3i -a(x-a)))
14 (x_a)a-l {sin 2>.(z-a)} ..;:i(z-a)a-l/2 {sin(>.z->.a)}J (.\x-'xa)
cos 2>.(z-a) , 2A C08(>.z->.a) a-l/2
Rea>-(I:%I)/2
15 ( _ ) -1/2 {COB >...p=;i} ..;:i (>.) 1/2-a (x_a)(2a-l)/4 {Jcr_l/(>' vz-a) }
x a ch >' v z - a 'J lcr_l/(>' vz-a)
In(x-a) 
16 r ;+1 (1n(x-a)+tb(I)-tb(a+l)]
17 (x-a)fJ- 1 ln(x-a) rr:) (x - a )a+,6 -1 [tb({3) - tb( a + {3) + In(x - a)),
Re{3>O
18 (x-a)fJ- 1 ln m (x-a) (x_a)a+,6-1 E (r;) b ( rr:» ) lnm-I:(x-a),
1:=0
Re{3>O, m=I,2,,,.
19 (X-Cl)II/2JII ('x v'x- a ) (2/ ,X)a(x-a)(a+II)/2Ja+II('x v'x-a ), ReIl>-1
20 (X_Cl)(2a-3)/4 ..;:i (T )a-l/2 J a _ 1 / 2 ('x v'x-a )Ya_l/2('x v'x-a ),
XY a _ 1 / 2 (2'x v'x-a ) Rea>O
21 (x_a)(2a-3)/4 ..;:i(z-a)a-l/2 ( vx=a> ( vx=a>
1f -X- Ia_l/2'x x-a Ka_l/2'x x-a,
XKa_l/2(2'x v'x-a) Rea>O
22 (x_a)fJ- 1 rr:) (x- a)a+,6-11 (I',lIi a+{3i.\x- 'xa), Re{3>O
X-#'1 (I', IIi {3i 'x(x-a))
23 (x-a)fJ -1 E p ,,6«x-a)lJ) (x_a)a+,6-1 E IJ ,a+,6«x-a)IJ), Rel'>O, Re{3>O

174
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Table 9.2
Ip(x), xeR I (I+Ip)(x), xeR 1 , aeC
1 (b- ax)'Y-l r  l-a-'Y ) (b )a+'Y- 1 0 b R ( )
r 1-'1 ) 00 - ax , a> , ax<, e a+"Y < 1
2 1 rw-) e%a1l'i/2.1 Re(J.I.-a»O J.I.::FO, -1, -2,...
(l%ix),. '" ( 1%lx ) " 0' ,
3 ( ),6-1 ( )a+,6-1
x-a + r a+,6 x- 0 + ' Re#>O
4 e>'x A-ae>.x, Re'x>O
5 {sin >.x } ,X -a {sin( >'x-a1l' /2) } A>O, Rea< 1
cos >.x c08(>.x-a1l'/2) ,
6 >.x {sin 'YX} eAr {sin(-yx-alp)} -arct /,X) Re,X>O, "Y>O
e cos'Y x (>'+'Y)o/ cos(-yx-alp) , Ip- g(-y ,
Table 9.3
Ip(x) , xeRI (1Ip)(x), xeRl, aeC
1 x'Y- 1 , x>O r (l -a-'Y ) a+'Y- 1 Re( + )<1 >0
r(1-'Y) x , a "Y ,x
2 (ax+b)'Y-l g = 'Yj (ax+b)a+'Y-l, Re(a+"Y)<I, larg(a/b)I<1I"
3 1  [( x+O )( X  l/ Rea>-1
[ (x+o)(x+b ) ] O+l /  r( a+l / 2 ) ( v'x+o+ X+b)20'
4 e->'x ,X-ae->.x, Re,X>O
5 e->'P 2a+l/211"-1/2a 1/2-a x (2a+l)/4 K a+l/2(AVX), Re 'x>0
6 { sin >.x } ,X_a{sin(>.x+a1l'/2)} 'x>0 Rea<1
cos >.x cos(>.x+a1l'/2)' ,
7 {sin >'P} Vi (2) a-l/2 x(2a+l)/4{ Y- O - 1 / 2 (>'P)} , 'x>0, Rea< 1/2
cos >. P X J -O-1/2(>'P)
8 ->.x {sin 'YX} e- >or {sin(-yx+alp) } ( / )
e cos'yx (>'+'Y2)O /  cos('Yx+alp) , "Y>O, Re'x>'O, Ip=arctg "Y ,X
9 x-v /2JII (Ax) (2/ A)a x( a-II )/2 JII-a('x.../X), A>O, Re(2a - II) <3/2
10 x-II/2 {y,,(>.x)} (2)a (a-II)/2{Y"-o(>'P)} {>.>o, Re(2a-II)<3/2}
K,,(>.x) X x K,,-o(>'P) , Re>.>O
We note also that many fractional integrals and derivatives of elementary and non-elementary
functions were found long ago by Letnikov [6]-[8], [10], [11], (1882-1888).
Tables 9.1-9.3 above contain both fractional integrals and derivatives. The relations in these
tables may be obtained provided that we take Rea>O first, and extended then to the case ReaO
by analytic continuation with respect to the parameter a. We note also that the condition
Re#>O connected with the convergence of an integral at the point x=a may also be omitted if
the integral 1+ is understood in the sense of integration along the Pochhanuner loop described
in  22.2. Some relations are given in the two-level representation as accepted in the handbooks
of Prudnikov, Brychkov and Marichev [1]-[3], see also [4]. Many relations from the mentioned
tables and many new relations can also be obtained by the method suggested by Marichev [10],
[11] - consider a general expression discussed in  36.9. Adamchik and Marichev [1] implemented
this algorithm for calculating integrals of hypergeometric functions, and in particular of fractional
integrals of the Meijer G-function defined in (36.3) in the computer algebra REDUCE system.

Chapter 3. Further Properties of
Fractional Integrals and Derivatives
In this chapter we shall continue our investigation of the properties of Riemann-
Liouville fractional integrals and derivatives on finite and infinite intervals of
the real line. Firstly problems will be considered which will be important in
Chapters 6 and 7 when studying integral equations of the first kind. Topics
such as the compositions of fractional integrals and derivatives with power and
exponential weights, connections between fractional and singular integrals, and
linear combinations of left-sided and right-sided fractional integrals with each other
are discussed. Then the characterization of fractional integrals of Lp-functions and
of functions in weighted Holder spaces Ht(p) on an interval will be given, and
various aspects of fractional integra-differentiation in the theory of functions of real
variable considered. These aspects such as the mapping properties of fractional
integra-differentiation operators in Lipschitz spaces H; and iI;, the fractional
differentiation of absolutely continuous functions, inequalities for fractional integrals
and derivatives, connections of fractional calculus with problems of summability
of series and integrals, etc. will be studied. In conclusion generalizations of the
classical Leibniz rule concerning the derivative of the product of two functions will
be discussed, and asymptotic expansions of fractional integrals near the end points
of an interval will be derived.
 10. Compositions of :Fractional Integrals and
Derivatives with Weights
In this section we consider the compositions of fractional integra-differentiation
operators with power, exponential and power-exponential weights in Lp-space,
1  p < 00. We also investigate the problem of the commutability for such
operators. Our main attention is paid to the simple compositions of the type
x.., Ig+x 6 . It should be noted that properties of these operators depend on their
domains of definition and on their parameters. Integrals of purely imaginary order

176
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
cannot be defined in the whole space L 1 and their operators are not bounded in
this space. Therefore, as a rule such integrals will not be considered here.
We shall use the following specific notation. Let a1,"', an, a n +1 be
complex numbers such that there exists only a single number a j such that
Reaj = min(Rea1,"', Rea n , Rea n +1)' Then we put a n +1 = ° and introduce the
function
! 0,
-a.
m(a1,...,a n )= J'
Reai > 0, i = 1,2,. . . ,n,
if there exists aj such that
Reaj < min(O, Real,. .., Reaj-1,
Reaj+1' . . . , Rea n ).
(10.1 )
If there exist aj and aA: such that aj f; aA: but
Reaj = ReaA: = min(O, Real, ..., Rea n )
then the function m( a1, . . . ,an) is not defined. We also introduce special function
spaces
) { Lp(a, b) if p > 1, 0  a < b  00,
L .(a,b =
p, L 1 « a, b); pn x I + 1) if p = 1, 0  a < b  00,
(10.2)
m(a). _ { Lp(O, b) if Rea> 0,
10+ (Lp(O,b» - l o + a (L p ,.(O,b» if Rea < 0;
m(a). _ { Lp,.(O, b) if Rea > 0,
10+ (Lp,.(O, b» - l o + a (L p (O, b» if Rea < OJ
Im(a). ( L « a 00 ) . x9 » = { Lp«a, 00); x 9 ) if Rea > 0,
- p,' l:a(Lp,.«a,00);x 9 » ifRea<O;
Im(a). ( L « a 00 ) . x9 » = { Lp,.«a, 00); x 9 ) if Rea > 0,
- p,." I: a (L p «a,00);x 9 » ifRea<O.
We note that L 1 ,.(a, b) C L 1 (a, b) and L 1 ,.(a, b) = L 1 (a, b) if 0 < a < b < 00
and for example x-l}n- 2 x E L 1 (0, b) but x-l}n- 2 x rt L 1 ,.(0, b).
(10.3)
10.1. Compositions of two one-sided integrals with power
weights
As it was noted in S 2 that a semigroup property given in (2.65) is fulfilled for the
fractional integrals and derivatives defined in (2.17)-(2.18) and (2.32)-(2.34)

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 177
1:+I:+f(x) = 1:+I:+f(x),
If_ 1 :_f(x) = 1:_lf_f(x),
1:+1:+f(x) = 1::/J f(x),
1:_lf_f(x) = 1_+/J f(x),
(lOA)
(10.5)
According to Theorem 2.5 and Remark 2.3 we obtain the following results.
Theorem 10.1. Let a and (3 be complex numbers such that the functions
m(Q,{3, a + (3) and m(a, a + (3) are defined. Then the first set of relations in (lOA)
and (10.5) are valid when f E (:Ot,/J.Ot+/J)(Lp(a, b» and f E 1(Ot./J.Ot+I')(Lp(a, b»
and the second set of results in (10.4) and (10.5) are valid when f E
1:;Ot.Ot+/J)(Lp(a,b» and f E 1(Ot,Ot+/J)(Lp(a,b» if -00 < a < b < +00, 1  p < 00
respectively. The results in (10.5) are also valid when b = +00 and instead of
the space Lp(a,b) we shall take the weighted space Lp«a,00);x p (M+l)-2) when
M = m(a,-a, {3, -{3, a + (3,-a - (3).
Remark 10.1. If the functions m(a,{3, a + (3) and m(a, a + (3) are not defined
then the statements of Theorem 10.1 are still valid under appropriate assumptions
on the spaces of functions f(x). These spaces should be contracted by conditions
of the existence of summable derivatives 1)-"'; f where 'Yj are two or four
points of 0, a, {3, a + {3 or two or three points of 0, a, a + {3 lying on the line
{"I : Re'Y = min(O, Rea, Re{3, Re(a + (3»} or {"I : Re'Y = min(O, Rea, Re(a + (3»},
respectively.
Proof. Theorem 10.1 and Remark 10.1 are direct corollaries of Theorems 2.4,
2.5 and Remark 2.3, if we take into account the imbedding Lp(a,b) C L 1 (a,b)
for p > 1 on a finite interval (a, b), the notation in (10.1), the property
1+(Lp(a, b» = Lp(a, b) and the substitution x = a + b - y while passing from 1:+
to If_ when b < 00. If b = +00, the theorem follows from Theorem 10.6 and from
the relation in (10.6) if we replace x by x-I and f(x) by x- Ot -I'-lf(x- 1 ) and take
(10.1) into account. .
In view of the second result in (10.5) with b = 00 and the second one in
(10.4) with a = 0 after the above replacements we have the following relations with
weighted factors
x Ot Ig+x- Ot -/J 1+x/J f(x) = 1+lg+/(x) = 1;1/J f(x),
x Ot lx-Ot-/J lx/J I(x) = 11/(x) = 1+/J I(x).
(10.6)
(10.7)
It is therefore natural to consider first the question about properties of the operators
x" 1+x6. Here and below the factors x'" and x 6 imply operators of multiplication
by the functions x'" and x 6 in the indicated order. The exact description of the
action of these operators is given by the following assertion.

178
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Lemma 10.1. Let 1/J(x) E L,,(O, b), 0 < b < 00, 1  p < 00, p(1 + ReJ.l) > 1 and
Rea 1= O. Then the representation 1/J(x) = 10+ xl' J(x), J(x) E 1;;:(1). (L,,(O, b»,
takes place if and only if 1/J(x) = x IJ 1 0 +g(x) where g(x) E 1;;:Ot).(L p ,.(0,b» or
1/J(x) = x lJ -£ 1 0 +X£gl(X) where gl(X) E 1;;:(1). (L" (0, b», p(1 + Reg) > 1. The
operators 10+ xl' , x 1J 1 0 + and x lJ -£ 1 0 +x£ where Rea> 0, are bounded from these
spaces to the space Lp«O, b); x-,,(Ot+IJ».
Proof. The case Rea> O. Part I. Necessity. Let 1/J(x) be represented in the
form 1/J(x) = 10+ xl' J(x), J(x) E Lp(O, b), 1  p < 00. We prove that there exists a
function g(x) E Lp,.(O, b» such that the relation
19+x lJ J(x) = x IJ 1g+g(x)
(IQ.8)
is valid. We obtain this function by applying the operator 1 0 + Ot x- 1J to (10.8) and
write it in the form
d
g(x) = dx (xX(x»
where X(x) = (X J)(x) = (x- 1 1J.;Ot x Ot)(x- IJ - 0t 1 0 +x lJ )J(x). Applying now the
inequality in (5.46') twice to the operators x- 1 1J.;Ot x Ot and x- IJ - Ot 10+ xl' we
obtain that X is bounded from L,,(O, b) to L,,(O, b) provided that the conditions
o < Rea < 1, p(l + ReJ.l) > 1, 1  p < 00, are fulfilled. Hence if J(x) E Lp(O, b)
then X(x) = x-116+9(x) E Lp(O, b). Therefore g(x) E Lp(O, b) when p > 1 - see
also Lemma 3.2 and (3.15). If p= 1 then the conditions X-116+9(X) E L 1 (0,b)
and (IIn xl + l)g(x) E L} (0, b) are equivalent. This fact follows, for example, from
(3.17") with a = 1 and A = 0 or from the relations
b x b x
f X-I f Ig(t)ldtdx = f f Ig(t)ldtdln x
o 0 0 0
x b
= In x f I g(t)ldt l x=b - f In xlg(x)ldx.
x=+o
o 0
(10.8')
The above equivalence follows from these relations, for example, when b < 1. So
we have proved by (10.2) that g(x) E Lp,.(O,b) provided that J(x) E Lp(O,b).
Sufficiency. Let 1/J(x) be such that 1/J(x) = x IJ 1 0 +g(x), g(x) E Lp,.(O, b),
1  p < 00. We prove that there exists a function J(x) E L,,(O, b) such that (10.8)
is valid. According to (10.8) and (3.16) we can take cP + A2CP as J and cP as g. By
Lemma 3.2 J = cP + A 2 CP E L,,(O, b) provided that 9 = cP E L,,(O, b) when p > 1.
But in the case p = 1 the space L 1 (0, b) should be contracted by the condition
(I In xl + l)g(x) E L 1 (0, b). This fact follows from the representation of the operator

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 179
(A 2 1;')(x) as X --+ O. This representation can be obtained from the relations for A 2
in Lemma 3.2 which are still valid when p = 1. Then we have
-1 j :& ( Y ) a ( y ) # dy
A 2 (x,t),..,., X - --
t-O X - Y X Y
o
= x- 1 B(a + Il, 1 - a)
== A2(X, 0), (A 2 1;')(x)
:&
,..,., 1r- 1Il sina1r j A 2(x,O)I;'(t)dt
t-O
o
= [B(a,ll)t 1x - 11 J+I;'(x).
The last representation and (10.8') with replacing 9 by I;' lead to the equivalence of
the conditions A 2 1;' E L 1 (O,b) and (IInxl + 1)1;' E L 1 (O,b). So if 9 = I;' E L",.(O,b)
then f E L,(O, b).
The case Rea> O. Part II. Necessity. Let 1jJ(x) be such that 1jJ(x) =
Ig+x# f(x), f(x) E L,(O, b), 1  p < 00, and p(1 + Reg) > 1. Then according to
the necessity proved above there exists a function 9 E L".(O, b) such that (10.8) is
valid. By the sufficiency proved above for such a function 9( x) there exists another
function 91 E L,(O, b) such that the relation x£ Ig+9(X) = Ig+x£ g1(X) similar to
(10.8) is valid. Excluding Ig+9(X) from these relations we obtain the analogue of
(10.8)
Ig+x# f(x) = x#-£ Ig+x£91 (x)
(10.9)
which yields the representation for 1jJ in the form 1jJ = x#-£ Ig+ x£ 91 (x), 91 (x) E
L,,(O, b).
The sufficiency in this case is proved similarly or on the basis of the necessity
by interchanging g and Il, 91 and f with each other.
The case Rea < O. Part I. Necessity. Let 1jJ(x) be such that 1jJ = Ig+x# f(x),
f(x) E 1 0 ': (L". (0, b». This means f(x) = 1 0 ': fi(x) and 1jJ(x) = Ig+x# 1 0 ': fi(x),
where fi(x) E L",.(O,b). Since p(1 + Rell) > 1 and fi(x) E L",.(O,b) but
- Re a > 0 then by using the proof of sufficiency, proved above in Part I and the
property Ig+I o + a = E we arrive at the relations
1jJ = Ig+(x# 1 0 + a f;) = Ig+(l o + a x# 12)
= x# 12 = x# Ig+(l o ': 12) = x# 1+9,
12 E L,,(O, b),

180
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
where 9 = 10+0112 E 1 0 + 0I (L,(0, b».
The sufficiency in this case is proved by inverse arguments with respect to the
last relations.
The case Rea < O. Part II. This proof is quite similar to that in the case
Rea> 0, Part II.
The last statement of the lemma follows from the above arguments and from
(5.46'). .
Using now the Dirichlet formula given in (1.32) we evaluate the composition
(X/1 I+ -{j1 x -011 )(x{j 1+-{jx-OI)f(x)
z t
J ( t ) 0I1-{j1-1 J ( t ) 0I-{j-1
=X{j1 x - t{j-0I1 dt - r r- 0I2 f( r)dr
r(a1-/h) r(a2-{32)
o 0
z
J (x - r)0I1+0I-{j1-{j-1
=X{j1 r{j-0I1-0I f( r)dr
r(a1 - (3})r(a2 - (32)
o
1
X J uOl-{j2-1(1 - U) 0I 1-{j1- 1 (l_ u(l - x/r»{j-0I1du,
o
Re(a1 - /31) > 0, Re(a2 - (32) > O.
Hence by (1.73) we obtain the following representation
(X{j1 1+ -(j1 X-Oil )( x{j I+-{j2 x - 01 2 )f( x)
z
J (x - r) 0I 1 +0I-{j1-{j-1
=X{j1
r(al + a2 - /31 - /32)
o
X 2Fl (a1 - {32, a2 - /32; a1 + a2 - /31 - (32; 1 - x/r)
X r{j-0I1-0I f(r)dr.
(10.10)
If we interchange the operators in the parentheses on the left-hand side of
(10.10) with each other, then the indices 1 and 2 will be interchanged with each
other too. This leads to the operator obtained from the right-hand side of (10.10)
by using the self-transformation formula:
2F1(a, b; c; z) = (1 - zt-a-b2F1(C - a, c - b; c; z)
(10.11)
Erdelyi, Magnus, Oberhettinger and Tricomi [1, 2.1.4(23)]. Thus the operators on

 to. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 181
the left-hand side of (10.10) commute
(X{jl Ig.;. -{jl x- al )(X{j2 Ig+-{j2 x -a 2 )!(x)
= (X{j2 Ig+.-{j2 x - a2 )(x{j1 Ig.;.-{j} x-a} )!(x)
(10.12)
under appropriate assumptions on the parameters.
We denote by T12' F and T12, T 21 the operators on both sides of (10.10) and
(10.12), respectively. Domains of definition of these operators and the ranges onto
which these domains are boundedly mapped essentially depend on the parameters
of the operators and firstly on the signs of Re(a1 - /3d and Re(a2 - /32)' These
domains may be taken to be the space L,,(O, b), 0 < b < 00, 1  p < 00, or some
subspaces of L,(O, b) defined by the corresponding conditions of representability. To
describe these spaces more conveniently in terms of representability via x--y lci+ 1/J(x)
and 1/J E L".(O, b) the parameters in the condition p(1 + ReJJ) in Lemma 10.1 must
satisfy additional assumptions. It should be noted that Lemma 10.1 itself gives the
main technique for such a description. In the cases when the above conditions and
the domains of definition of the operators intersect with each other the values of
the corresponding operators coincide on this intersection, this leading to (10.10) or
(10.12).
All the connections between T12' T 21 and F are given in Theorem 10.2 and
Table 10.1 where a = a1 + a2 - /31 - /32'
Theorem 10.2. Let some of the conditions in Aj from Table 10.1 be fulfilled.
Then the operators Bj are defined on the spaces C j C L,,(O, b), 1  p < 00, and are
bounded from C j onto Dj C L,(O, b) and the relations Ej are fulfilled.
Proof. Without loss of generality we can consider all parameters a1, a2, /31, /32 as
real numbers. We first suppose that the conditions A 1 are fulfilled. Then it follows
from (5.46') that the operator x{j2Ig+.-{j2 x -a 2 is bounded in L,,(O, b) provided
that p  1, p(1 - a2) > 1. Therefore T 12 is bounded in L,(O, b) if additionally
p(1 - ad > 1. By using Fubini's theorem the relation in (10.10) is easily proved by
direct evaluation on the set of sufficiently "good" functions !(x) dense in L,(O, b).
Therefore the operator F on the right-hand side of (10.10) is also bounded on this
dense set, and hence it is bounded on the whole space L,(O, b). So by Theorem
1.7 (10.10) is fulfilled in L,(O, b) under the assumptions AI. Using the relation
in (10.11) and the symmetry of the conditions Al with the indices 1 and 2 being
interchanged with each other we obtain (10.12) from (10.10).
Similar arguments on the basis of the inequality in (3.17"), instead of that in
(5.46"), which applicable twice when A = 1 and A = 0 yield the statement of our
theorem in the last case j = 11.
lt should be emphasized that the ranges onto which the operators T I2
and T 21 map from L,(O, b) do not coincide, because the operator T 12 maps
onto {X{jl Ig.+-{jl x{j2- a l- a 2Ig+.-{j21/J(x), 1/J E L".(O,b)}, when p(1 - a2) > 1 and

182
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Table 10.1
f A. B. C. D. E.
J J J J J
1 Re(a1-{3d>0, Re(a2-{32»0, T12, F, Lp(O, b) into Lp(O,b) ( 10.10),
p(l-Read>l, p(I- Rea 2»1 T21 (10.12)
2 Re(a1-{3d>0, Re(a2-{32»0, T12, F Lp(O, b) {x- o Ig+ 1/1, (10.10)
1'[1 + Re({32 - a1 - a2)] > 1, 1/1EL p ,.(O, b)}
p(I-Rea2» 1
3 Re(a1-{3d>0, Re(a2-{32»0, T12' T 21 Lp(O, b) {x- o Ig+ 1/1, (10.12)
p[I+Re({31-a1-a2)]>I, 1/1EL".(O, b)}
1'[1 + Re({32 -a1-a2)]> 1
4 Re(a1 -{3d<o, Re(a2 -{32»0, T12' F L,(O, b) into L,(O, b) (10.10)
p(l-Read>l, p(I-Rea2»I,
p(I-Re{3d>l, Rea>O
5 Re( a1 - {3d < 0, Re( a2 - {32) >0, T12' F L,(O, b) {x-o Ig+ 1/1, (10.10)
p(I-Rea2»I, p(I-Re{3d>l, 1/1EL".(O, b)}
p[I+Re({32- a 1- a 2)]>I,
Rea>O
6 Re(a1-{3d<0, Re(a2-{32»0, T 1 2 {x- o 10+ 1/1, Lp(O, b) -
p[I+Re(a1 -{31-{32)]>1, 1/1EL p ,.(O, b)}
p(l-Re{3d> 1, Rea<O
7 Re(a1-{3d>0, Re(a2-{32)<0, T12, F {xo2-1321132-021/1 {x131 -01 1 01 -131 X (10.10)
p(I-Read>1, p(I-Re{32»1, T21 0+ ' 0+ ' (10.12)
Rea>O 1/1EL p ,.(O, b)} xEL".(O, b)}
8 Re(a1-{3d>0, Re(a2-{32)<0, T12, T21 {xo2-1321132-021/1 {xI31-01101 -131 X (10.12)
0+ ' 0+ '
pel-Read> 1, p(l-Re {32»1, 1/1EL p ,.(O, b)} xELp,.(O, b)}
Rea<O
9 Re(a1-{3d<0, Re(a2-{32)<0, T12 {X O 1 0 ':1/1, L,(O, b) -
p[I+Re(a1-{31-{32)]>I, 1/1EL p ,.(O, b)}
p(I-Re{3d>1
10 Re(a1 -{3d<o, Re(a2 -{32) <0, T12, T21 {X O 1 0 ':1/1, Lp(O, b) (10.12)
p[I+Re(a1-{31-{32)]>I, 1/1EL".(O, b)}
p[1+Re(a2-{31-{32)]>1
11 1'=1, a1 =a2=0, T12, F, LiC(O, b)j L1 (0, b) (10.10),
Re{31 <0, Re{32<0 T21 ln 2 x+1) ( 10.12)
the operator T21 maps onto {xI32Ig+.-132xI31-01-02Ig+.-131.,p(x), .,p E Lp,.(O,b)} if
p(l - a1) > 1. The above ranges coincide with each other and with the space
{x- O Ig+rp(x), rp E Lp,.(O, b), a = a1 + a2 - /31 - /32} provided that the above
conditions are contracted to p(1 + /31 - a1 - a2) > 1 and p(1 + /32 - a1 - a2) > 1
and hence Lemma 10.1 can be applied to the operators x 131 Ig -131 and x 132 Ig+.-13 2 .
This yields the statements of the theorem in the cases j = 3 and j = 2.
Let now the conditions in A4 be given. Then in view of Theorem 1.5 the
operator F is bounded in Lp(O, b). Indeed we set
k(x, r) = (1 - r/x)-12F1(a1 - /32, a2 - /32; a; 1- x/r)(x/r)0+131-1r-1

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 183
and consider the second integral in (1.43):
1
1(1- r)a-1 2 Fl(al - /h,a2 - /32; a; 1- r-l)r-a-131-1/Pdr, Rea> O.
o
According to the relation
2Fl(a, b; c; z) = O(z-O) + O(z-b), a - b 1= 0, :1:1, :1:2,..., z --+ 00
( 10.13)
(Erdelyi, Magnus, Oberhettinger and Tricomi [1, 2.10 (2)]), the last integral is
1 1
convergent together with the integrals J r- a2 - 1 / P dr and J r- al - 1 / P dr which are
o 0
also convergent when p(l - a2) > 1 and p(l - at) > 1 respectively. So in this
case Theorem 1.5 can be applied to the operator F and this operator is bounded
in Lp(O,b). We apply the operator zalIg+-alz-131, which is bounded in Lp(O,b)
if p  1 and p(1 - (3t) > 1 - see (5.46'), to F and obtain the composition
zalIg+-alz-131F bounded in Lp(O,b). By direct evaluation it is not difficult
to prove the relation zalIg+-alz-131F = z132I+-132z-a2 /(z). Applying now the
inverse operator Z131 I+ -131 Z-al to both sides of this result we finally arrive at
(10.10) under the conditions in A4. Under stricter conditions, As, we can prove by
using Lemma 10.1 as was done in the case j = 2, that the ranges onto which the
operators T 12 and F act can be represented in the form {z-aI+"p, "p E Lp,_(O,b)}.
Let now the conditions in A6 be given and /(z) = zalo+a"p-(z), "p- E Lp,_(O, b).
We apply Lemma 10.1 to the operators zal-131-132Io;"p-(z), Ig+.- a lz a l-131-132 when
a > 0 and to the operator z131 I+ -131 Z-131 when a = a1 - /31 < 0, p. = -/31. As a
result we obtain
T 12 /(Z) = z1311";" -131 z-a 1 +132 I+-132 zal-131-132 Ig+ -al +132- a 2"p- (z)
= z131I..;..-131z132-alIg+-alzal-131-132"p1(Z)
- z 131Ial-131z-1311131-al/.- ( z )
- 0+ 0+ 0/1
_ 131-131 I al-131 1 131-al/. ( ) _ /. ( )
- z 0+ 0+ 0/2 z - 0/2 Z
where "p1,,,p2 E Lp(O, b) and "pi E Lp,_(O, b). This means that the operator T 12 is
bounded from C 6 onto Lp(O, b) under the conditions in A6.
If the conditions in A7 or As are valid except the condition on a, then by
Lemma 10.1 and the assumption p(1- /32) > 1 the function /(z) from C 7 = C s can
be represented in the form I(z) = Z a 2 Igt.-a2z-132cp where cp E Lp(O, b). Hence the
fractional derivative cp(z) = z132I+-132z-a2 /(z) exists and by p(l - ad > 1 the

184
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
operator T 12 is defined and bounded on C7. Moreover according to Lemma 10.1
D7 = Ds, and again Tl2f can be represented in the form
Tl2f(x) = x{J1I+-f3lx-0t1cp(X) = xf3l-OtlI+-f31X, X E Lp,.(O, b).
Applying the bounded operator xOt'J Ig+-Ot'J x- f3 'J to both sides of this relation
we obtain the composition on the right-hand side in which the order can be
interchanged in view of the corresponding assumptions concerning the type AI:
Ot'J I f3'J-0t2 -f3'J T f( ) - f3l / Otl -f3 l -011 01 2 I f32- Ot 'J -f32 ( )
x 0+ X 12 X - X 0+ x x 0+ x cp x
= x f3l I+-f3lx-Otl f(x).
Applying the inverse operator x f3 'J I+-f3'Jx-Ot'J to both sides of this relation we
arrive at (10.12): Tl2f = T 21 f. If in addition Rea> 0, then T 1 2 and T 21 remain
bounded operators the values of which can be written as the right-hand side of
(10.10), i.e. Ff, after direct evaluations.
In the case of the conditions in A g the proof is similar to that in the case in
A6.
Finally we prove (10.12) under the conditions in AlO, which are stricter than
in A g . We set T 12 f = g, where 9 E L,,(O, b). It follows from AlO that p(1 - {31) > 1
and p(1 - (32) > 1. Therefore according to the analogue of Al the relations
xOtllf3l-0t1x-f31xOt'J1f3'J-0t'Jx-f3'J g - xOt21f3'J-0t'Jx-f3'JxOtllfJl-0t1x-f3l g - f
0+ 0+ - 0+ 0+ -
are valid. Applying the operators T 1 2 and T 21 to these relations we obtain 9 = Tl2f
and 9 = T21f which yield (10.12). .
Remark 10.2. The conditions in Aj in Table 1 are sufficient. In some cases for
p > 1 they can be extended when Re(aj - (3j) = O. This follows from remarks
made after (5.46') about the boundedness of the corresponding fractional integrals
of purely imaginary order.
N ow we consider the compositions of two right-sided fractional integro-
differentiation operators with b = +00. If we replace x by l/x, a by 1 - {3,
(3 by 1- a and f(x) by f(x- 1 ) then the operator xf3I+f3x-Otf(x) becomes
x f3 I-f3 x- Ot f( x). This fact follows from the relation
x f3 I;f3 x- Ot f(x) = yl-Ot I-f3 yf3- 1 cp (y), xy = 1, f(x) = cp(y).
(10.14)
Using the above substitutions and (10.11) from (10.10) and (10.12) we have the

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 185
relations
(x,61 Ic:.. I -,61 X-Oil )(x,6 Ic:..-,6 X-OI )f( X)
00
f (r - X)OII+OI-,6I-,6-1
=X,61 +,6-0I1
r(crl + cr2 - PI - (2)
:&
X 2FI(crl - PI, crl - P2; crl + cr2 - PI - P2; 1 - rlx)
X r-OI f( r)dr,
(x,61 Ic:.. I -,61 X-Oil )(x,6 Ic:..-,6 X-OI )f( X)
= (x,6 Ic:..-,6 X-OI )(X,61 Ic:.. I -,61 X-Oil )f(x).
(10.15)
( 10.16)
Similarly after replacing x by l/x, tjJ(x) by x Ol - 1 1/J(1Ix), f(x) by X- 0I - 1 f(l/x)
and g(x) by x-0I-1g(l/x) we obtain the following result from Lemma 10.1.
Lemma 10.2. Let tjJ(x) E L,«a,oo);x'-0I,-2), 0 < a < 00, 1  p < 00,
p(1 + Rep) > 1, and Recr f; O. Then 1/J(x) = Ix-IJ f(x), where f(x) E
I(0I).(L,«(a,oo);x,+a'-2», a = signRecr. cr, if and only if tjJ(x) = x-IJIg(x)
where g(x) E I(0I).(L'J.«a,oo);x,+a'-2» or if 1/J(x) = x£-IJIx-£g1(x) where
g1(X) E I(0I).(L'J.«a,oo);x,+a'-2», p(1 + Rec) > 1.
To state an analogue of Theorem 10.1, in Table 1 we replace cri by 1 - Pi,
Pi by l-cri, X by 1/x, L,(O,b) by L,((a,oo);x- 2 ), 0 < a < 00, and in column
Ej (10.10) by (10.15) and (10.12) by (10.16). We denote by T12' F and T21 the
corresponding operators in (10.15) and (10.16). We call this rearranged Table by
Table 10.1' but for brevity, it has not been printed. Then from Theorem 10.2 we
obtain the corresponding result.
Theorem 10.3. Let some of the conditions in Aj from Table 10.1' be fulfilled.
Then the operators Bj are defined on the spaces Cj C L,(a, 00); x- 2 ), 1  p < 00,
and are bounded from Cj onto Dj C L,((a,oo);x- 2 ) and the relations in Ej from
Table 10.1' are fulfilled.
The right-hand sides of (10.10) and (10.15) can be transformed by using the
relation
2 F ,(a, b; c; z) = (1 - Z)-02 F , (a, c - b;c; z  1 )
(Erdelyi, Magnus, Oberhettinger and Tricomi [1,2.1.4(22)]). This leads to
two results of such a kind. We set c = crl + cr2 - PI - P2 and make the
substitutions a = cr1 - P2, b = cr2 - P2, rp(r) = r,6-OII-OI f(r); a = cr1 - P2, b =
cr1 - P1, rp(r) = r-OI f(r); a = crl - P2, b = cr2 - P2, rp(r) = r{J-OII-OI f(r); and
( 10.17)

186
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
a = al - {32, b = al - (31J rp( r) = r-a'J f( r) in (10.10), the above two results and
(10.15), respectively. We also introduce the notation:
j z (x _ rY- 1 ( X )
1Io+(a, b)rp(x) == r(c) 2 F l a, b; c; 1 -;: tp(r)dr,
o
( 10.18)
j z (x _ r)c-l ( r )
21o+(a,b)rp(x):: r(c) 2 F l a,b;c; 1- -; tp(r)dr,
o
(10.19)
C _ j oo (r _ x )C-I ( .. x )
3 1 _(a,b)rp(x) = r(c) 2 F l a,b,c, 1-;: rp(r)dr,
Z
(10.20)
j oo (r _ x)c-l ( r )
4 1 :(a,b)rp(x):: r(c) 2 F l a,b;c;I-; rp(r)dr.
Z
(10.21 )
Carrying out the above transformations and taking the commutation of (10.12)
and (10.16) into account we obtain the following compositional expansions for the
operators in (10.18)-(10.21)
l/O+(a, b)rp(x) = 10;b x -a 18+xarp(x),
110+(a, b)rp(x) = x c - a - b 18+xa-c 1 0 ;6 x brp(x);
3 1 : (a, b)rp(x) = x c - a - b lxa-c 1:- 6 x b rp(x),
3 1 :(a, b)rp(x) = 1:- b x- a lxarp(x);
4 1 :(a, b)rp(x) = x a lx-a 1:- b rp(x),
4 1 :(a, b)rp(x) = x b r:..-bx a - c lxc-a-6rp(x).
( 10.22)
( 10.23)
(10.24)
( 10.25)
( 10.26)
( 10.27)
( 10.28)
(10.29)
I c ( b) ( ) a [ b -a I c-b c-b ( )
2 0+ a, rp x = x o+x 0+ x rp x ,
I c ( b) ( ) b / c-b a-c i b c-a-b ( )
2 0+ a, rp x = x 0+ x o+x rp x ;
From (10.11), (10.17) and other simple properties of the Gauss function we
are able to write the relations connecting the operators (10.18)-(10.21) with each
other
jl C (a,b) = jlC(b,a) = x<-l)J(a+b-C)jr(c - a,c _ b)x<-l)J(c-a-b)
( 10.30)
j = 1,2,3,4;

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 187
jIC(a,b) = X-lJj+1IC(a,c - b)x lJ , j = 1,3;
(10.31 )
jJC(a,b)cp(x) = y 1 - C 4_jI C (a,b)CP1(Y), xy = 1,
CPI(Y) = xc+1cp(x), j = 1,2,3,4;
( 10.32)
the indices 0+ and - being omitted for brevity, and the results for special cases of
the above operators
jI+(a,O) = I+,
. I C ( ) _ (-I)jIJ I c (_1)j-11J
J 0+ a, c - x o+x ,
j = 1,2,
( 10.33)
(10.34)
j I.. ( a, 0) = I,
. I c ( ) _ ( -I)j IJ I c ( -I)j -11J
J _ a, c - x _x ,
j = 3,4.
After the above substitutions and transformations with the operators in (10.18)-
(10.21) from Theorems 10.2 and 10.3 we obtain the following result.
Theorem 10.4. Let Rec > 0, 1  p < 00, 0 < e < d < 00 and some of the
conditions in Aj from Table 10.2 be fulfilled. Then the corresponding operator Bj
is defined on the space Cj and is bounded from C j onto Dj and the relations in Ej
from Table 10.2 are fulfilled.
Proof. We consider the case j = 1 in Table 10.2. We represent the right-hand side
of (10.22) in the form XC(x- C I+bxb)(x-IJ-b I+xlJ)cp(x) and construct the operator
of T 12 -type from (10.10). For such an operator we write the conditions in the cases
j = 2,5 of Theorem 10.2. In these cases the condition p[1 + Re(,82 - al - (2)] > 1
when p = 1 is violated but it is not essential. All the above conditions lead to the
conditions of the case j = 1 of Theorem 10.4. Similar conditions in the case j = 7
of Theorem 10.2 yield the conditions in the case j = 2 of Theorem 10.4.
While considering the relations in (10.23)-(10.25) we use similar arguments
and prove that the above correspondence of conditions is totally repeated. We also
note that some inequalities which naturally follow from the other inequalities are
excluded.
While studying the cases j = 9,...,16 in Table 10.2 we use (10.32) and
afterwards the cases j = 5,6,7,8,3,4,1,2 of this Table but relative to the function
cpdx) = x-c-Icp(l/x). After that we obtain the conditions on functions and
parameters indicated in the cases j = 9,... ,16. .
Remark 10.3. We denote by lI+(a,b), 2I+(a,b), 3Id_(a,b), 4Id_(a,b) the
operators of the form (10.18)-(10.21) with the limits of integration (e, x) and
(x, d), 0 < e < d < 00, instead of (0, x) and (x, 00), respectively. Theorem 10.4 is
also valid for these operators but the conditions in Aj including p must be omitted.
In other conditions instead of the intervals (O,d) and (e,oo) we shall take (e,d)
and instead of indices 0+ and - in I a we shall take e+ and d-, respectively. We
shall also take the relation Lp,.(e, d) = L,(e, d) into account.

188
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Table 10.2
f A. B. C. D. E.
J J J J J
1 Reb>O, p(l+Rea»l 1 1 0+(a,b) Lp(O, d) /0+ (Lp,.(O, d)) (10.22)
2 Reb<O, p(l+Reb»l lo+(a, b) {x b /Of/J, {xb[c-b X (10.22)
0+ '
p[l+Re(a+b)]>l f/JELp,.(O, d)} xELp,.(O, d)}
3 Re(c-b»O, p(l+Reb»l, 110+(a, b) Lp(O, d) /O+(Lp,.(O, d)) (10.23)
p[l+Re(a+b-c)]> 1
4 Re(b-c»O, p(l+Rea»l 1/0+(a, b) { c-b lb-c f/J {x c - b [b X (10.23)
x 0+' 0+ '
f/JELp,.(O, d)} XELp,.(O, d)}
5 Re(c-b»O, p(l-Rea»l, 210+ (a, b) Lp (0, d) 1 0 +(L p ,.(O, d)) (10.24)
6 Re(b-c»O, 210+ (a, b) { c-b I b - c f/J {x c - b I b X (10.24)
x 0+' 0+ '
p[l+Re(c- a-b)]> I, f/JELp,.(O, d)} XELp,.(O, d)}
p[l + Re( c - b)] > 1
7 Reb>O, p(l-Reb»l, 210+(1J, b) Lp,.(O, d) 1 0 +(L p ,.(0, d)) (10.25)
p[l+Re(c-a- b)]> 1
8 Reb<O, p[l+Re(c-a)]>l 2 1 0+(a,b) {x b lo1/J, {x b l C - b x (10.25)
0+ '
f/JELp,.(O, d)} xELp,.(O,d)}
9 Re(c-b»O, p(l-Rea»l 31(a, b) Lp,.«e, 00): 1(Lp,.«e, 00): (10.26)
x pc + p - 2 ) xP c + p - 2 ))
10 Re(b-c»O, 31(a, b) {x- C 1-cf/J, /(Lp,.«e, 00): (10.26)
p[l+Re(c-b)]>l, wE Lp,.«e, OO)i xP+ pb - 2 ))
p[l+Re(c- a-b)]> 1 x p (1+b-c)-2)}
11 Reb>O, p(l-Reb»l, 31(a, b) Lp,.«e,oo)i 1(Lp).«e, 00): (10.27)
p[l+Re(c- a-b)]> 1 x pc + p - 2 ) xP+ pc - 2 ))
12 Reb<O, p[l+Re(c-a)]>l 31(a, b) {x- C 1:6 f/J, 1:.- b (L p ,.( (e, 00): (10.27)
wELp,.«e, OO)i x p (1+c-b)-2))
x p - pb - 2 )}
13 Re(c-b»O, p(l+Reb»l, i(a, b) Lp,.«e, OO)i 1(Lp,.« e, 00): (10.28)
p[l+Re(a+b-c)]> 1 x pc + p - 2 ) x pc + p - 2 ))
14 Re(b-c»O, p(l+Rea»l 41(a, b) {x- C 1-cf/J, 1(Lp,.«e, 00): (10.28)
WELp,.«e,oo); x Pb + p - 2 ))
x p (1+b-c)-2)}
15 Reb>O, p(l+Rea»l i(a,b) Lp,.«e, OO)i 1(Lp,.«e, 00): (10.29)
x pc + p - 2 ) x pc + p - 2 ))
16 Reb<O, p(l+Reb»l i(a, b) {x- C l:b f/J, 1:.- b (L p ,.«e, 00): (10.29)
p[l+Re(b+a)]>l wELp,.«e, oo)i x p (1+c-b)-2))
x p - pb - 2 )}
Remark 10.3 follows from the possibility of defining the integrand by zero
from (e, d) to (0, 00), and from the boundedness in Lp ( e, d) of the operator of
multiplication by the power function x'Y with arbitrary /. On the basis of this, the
conditions in Aj including p and connected with the influence of the weight x.., at
zero and infinity can be excluded.
In conclusion of this subsection we note that Remark 10.2 can be transferred
to Theorems 10.3 and lOA.

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 189
10.2. Compositions of two-sided integrals with power weights
We now find the conditions of commutability for two operators x Pl I+ -Pl x- al and
zfJ I-fJ x-a with power weights, and obtain some important representations
of such compositions. For this we use a slightly different approach to the above
operators.
If we apply the inverse Mellin transform given in (1.113) to (7.17) and (7.21)
and then replace a by a - (3, f(x) by x-a f(x) and s by s + (3 we obtain the
representations
"Y+ioo
8 I a - P -a f( ) =  f r(1 - a - s) f - ( ) -8 d
:r 0+ x x 21ri r(I-,8-s) sx s,
"Y- ioo
( 10.35)
Re(a+s)<I,
"Y+ ioo
P l a - fJ -a f( ) =  f r({3 + s) f - ( ) -8 d
x _ x x 2 . r( ) s x s,
1r2 a + s
"Y- ioo
( 10.36)
Re({3 + s) > 0,
where f-(s) is the Mellin transform of f(x) as in (1.112). These relations are also
valid for any a - (3 for a sufficiently good function f(x). Details of this may be
seen in the beginning of S 36.
If we construct the composition of the left-hand sides of (10.35) and (10.36)
with different indices and use the Parseval relation for Mellin transform given in
(10.116) we arrive at the relation
(zP l I+ -Pl x- al )(xP I-P'J x-a)f(x)
"Y+ ioo
=  f r(1 - al - S)r({32 + s) f-(s)x- 8 ds,
21ri r(1 - ,81 - S)r(a2 + s)
"Y- ioo
( 10.37)
-Re{32 < Res < 1 - Real.
Since the gamma-multipliers of the integrand in (10.37) can be interchanged with
each other then the operators in the left-hand sides of (10.37) commute for the
corresponding functions. Sufficient conditions for such commutability and for the
boundedness of the above operators are given by the following statement.
Theorem 10.5. Let Re(al - (3!) > 0, Re(a2 - (32) > 0, p(1 - Real) > 1,

190
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
pRef32 > -1 and J(x) E Lp(O, 00), 1  p < 00. Then the relations
(X{Jl I+ -{Jl X-OIl )(X{J2 I2-{J2 X-0I2)J(X)
= (X{J2 I2-{J2x-0I2)( X{Jl I+ -(Jl X-OIl )J( x)
00
= j ko () J(t) t ,
o
ko(y)
r(1 + f32 - crdy{J2
r( cr 2 - (32)r(1 + f32 - /3d
x 2Fl(1 + f32 - crl, 1 + f32 - cr2; 1 + f32 - /3l;Y)' Y < 1,
ko(Y) = r(1 + f32 - crdyOll-l
r(crl - f3dr(1 + cr2 - crd
( 10.38)
x 2Fl(1 + f3l - crl,1 + f32 - crl; 1 + cr2 - crl;y-l), y> 1,
are valid and all operators in (10.38) are bounded in L,,(O, 00).
Proof. The boundedness of the above operators in L,,(O,oo) follows from the
relations in (5.45') and (5.46') applied by one or another order in succession under
the assumptions of theorem. The coincidence of the compositions in (10.38) with
each other follows from (10.37) and their representation via the right-hand side of
(10.38) is proved by the direct evaluation of these compositions, or of the integral
in (10.37). This is carried out by using the theory of residues or by Slater's
theorem. (Marichev [10, Theorem 17].) .
We now indicate the important cases of such compositions.
A. Let in (10.37) cr} = 1 - 2a, Q2 = C - 2a, /31 = 1 - c, f32 = O. By using
12.24(1) in Marichev [10, Section 10] and the Parseval relation in (1.116) we obtain
the following compositional expansion via commutative operators
J OO t20-l ( 1 4xt )
(x + t)20 2 F l a, a + 2"; c; (x + t)2 J(t)dt
o
= B(c,c - 2a)(xl-clo+20x20-l)(I:-20x20-c)J(x),
( 10.39)
Re(c - 2a) > 0, 2pRea> 1, J(x) E Lp(O, 00), 1  p < 00.
B. Let in (10.37) crl = cr, cr2 = /31 = cr/2, /32 = O. Then similarly by using

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 191
1.5(1) in Marichev [10, Section 10] we have
00
J I., I-a f(t)dt = 2f(",) cos "'2" (.,a/2 2.,-a)(I/2.,-a/2)f("), (10040)
o
ReQ> 0, p(l- ReQ) > 1, f(x) E L,,(O, (0), 1  p < 00.
The details concerning these operators may be found in S 12.3, in particular,
compare (10.40) with (12.39).
C. Let in (10.37) Ql = /32 = 0, Q2 = /31 = 1/2. Then similarly by using 2.4(1)
in Marichev [10, Section 10] we find the expansions:
X 1 / 2 1;;/2 1/2x-l/2 f(x) = (Sf)(x),
11}2 1;:/2 f(x) = (Sf)(x),
(10.41)
where (Sf)(x) is the singular integral given in (11.1) with a = 0 and b = 00.
D. Let in (10.37) Ql = /32 = -Q, Q2 = /31 = 0 or Ql = /32 = 0, Q2 = /31 = Q.
Then the gamma-functions of the integrand in (10.37) are transformed to
the functions 8ins(:)1I" = COSQ1r + sinQ1I'ctg(s - Q)1r and 8in8\::)1I' = COBQ1r +
sin Q1I' ctgS1l', respectively. According to (10.41) the operators COB Q1I' E +
sin Q1I' x-aSx a and COB Q1I' E +sin Q1I'S, where E is the identity operator, correspond
to the above functions. Making the changes f = cp and 1 0 ': f = cp we obtain from
(10.37) the relations in (11.27) and (11.29). These will be proved by another
method in S 11.
10.3. Compositions of several integrals with power weights
We construct the composition of the type (10.37) of three operators x{3;I-{3;x-a;
or x{3; 1;-{3; x-a;, j = 1,2,3, and choose the parameters Qj, /3j such that all
gamma-functions of the integrand in (10.37) will be cancelled. Then on the basis
of (1.113) we obtain the following relations
I6+x a I+x" I+xfJ f(x) = f(x),
f!..xa Ix" Ix{3 f(x) = f(x),
( 10.42)
( 10.43)
where Q + /3 + 'Y = O. It is obvious that if we exclude 'Y in these relations then
we shall arrive at (10.6) and (10.7) equivalent to them. Our results are given by
Theorem 10.6 and Table 10.3.

192
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Theorem 10.6. Let a and /3 be complex numbers such that the function
m(a, a + (3) is defined, see (10.1). Then (10.6) and (10.42) are fulfilled when
J(x) = XU l;;:a,a+ (3 )x tJ .,p(x), where .,p(x) E L,,(O, b), 0 < b < 00, 1  p < 00,
and (10.7) and (10.48) are fulfilled when J(x) = XU 1(a,a+f3)xtJ.,p(x), where
.,p(x) E L,,(a, 00), 0 < a < 00, 1 < p < [max(lReal, IRe/3L IRe(a + (3)I]-I. The
parameters u and v are given in Table 10.3. If the function m( a, a + /3) is not
defined then the above statements are also valid under the additional assumptions
on J(x) obtained from Remark 10.1 after replacing x by x-I and J(x) by
x-a-f3-1 /(x- 1 ).
Table 10.3
Condition Value
1 Rea > 0, Re{3 > 0 u=v=o
2 Re{3 < 0, Re(a + {3) > 0 u = -{3, v = 0
3 Rea < 0, Re(a + {3) > 0 'U = -{3, V = Ot + {3
4 Re{3 > 0, Re(a + {3) < 0 u = -{3, v = 0
5 Rea > 0, Re(a +{3) < 0 'U = 0, V = a
6 Rea < 0, Re{3 < 0 u=v=o
Prool. Let Rea> 0, Re{3 > 0 and J E L,,(O, b). Then the condition p(1 + Re/3) >
p  1 and the inequality in (5.46') yield I;'(x) = x a - f3 1+xf3 J(x) E Lp(O, b). Hence
the composition on the left-hand side of (10.6) exists in L,,(O, b). The right-hand
side of (10.6) also exists in L,,(O, b) under the conditions of the theorem. We verify
(10.6) on the set of sufficiently "good" functions dense in Lp(O, b) and then can
extend it to the whole space L,,(O, b) by using Theorem 1.7.
In the case of right-sided operator the condition pRe (a + (3) < 1 follows from
(5.46') instead of the condition p(1 + Re/3) > I, which is fulfilled automatically.
The other five cases are reduced to the first case by using the statement for
the first case but relative to the corresponding analogue of (10.6):
x a + f3 1;':x- a I:f3 x- f3 (x f3 J(x» = 1+(xf3 J(x»,
x-a Ir: f3x - f3 l o + a x a + f3 .,p(x) = 1t+.,p(x),
f3 I - a -f3 a l f3 -a-f3 /. ( ) - l -a/. ( )
x 0+ x 0+ X 0/ X - 0+ 0/ X ,
( 10.44)
-a-f3 l a f3 I -a-f3 a/. ( ) 1 -f3/. ( )
X o+x 0+ X 0/ X = 0+ 0/ x ,
-P I -a a+f3 1 -f3 -a/. ( ) _ I -Ot-f3 /. ( )
X 0+ X 0+ X 0/ X - 0+ 0/ X .
Here the functions .,p(x) are connected with J(x) by means of the relations indicated
in the conditions of the theorem and in Table 10.3. In reality for example, the
two first relations in (10.44) can be obtained from (10.6) by applying the operator

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 193
Za+p Io':z-a to (10.6) and by the substitution ,p(z) = z-a-p IC+zfJ f(z) and these
relations are altered from (10.6) by replacing {3 by -{3, Q by Q + {3 and f by
zfJ f and {3 by Q + {3, Q by -Q and f by ,p, respectively. After we have written
the corresponding condition 1 in Table 10.3 for the above relations and made the
inverse passage to (10.6) we shall arrive at conditions 2 and 3 in Table 10.3. In
the case of the right-sided operators similar statements obtained from (10.44) by
replacing the index 0+ by - are valid also. The six conditions obtained above can
be united into the conditions given in Theorem 10.6, by using (10.1).
The last statement of the theorem follows from Theorem 10.1, Remark 10.1
and the above substitutions connecting (10.4), (10.5) with (10.7) and (10.6). In
the case Re(Q + (3 + 1') > 0 the compositions on the left-hand sides of (10.42) and
(10.43) can be written via integral operators involving the Gorn function
"  (a)k(b)k(a')/(b'), k 1
F 3 (a,a,b,bjcjz,y)=L.J () kill xy,O<x,y<1.
k,I=O C k+1 ..
( 10.45)
(Erdelyi, Magnus, Oberhettinger and Tricomi [1, 5.7.1.(8»), in the kernel. Indeed,
the composition of two fractional integrals, for example, of the type (10.22) leads
to the operator given in (10.18). Applying one ,more fractional integral with
power weight to the above operator we obtain the following representation for the
composition of three fractional integrals of the form (10.42):
I z (x - t)a-l I t (t - r)c-l ( t )
Io+xfJ1Io+(a,b)cp(x) = r(Q) tPdt r(c) 2 F l a,bjc;I-; cp(r)dr
o 0
z z
= I cp(r)dr I tp ( x - t ) a-l ( t - r ) C-l F ( a b'c'l - ! ) dt
r(Q)r(c) 2 1 '" r
o T
Z 00 Z
= I cp(r)dr ,, (a)k(b)k(-r)-k I tp(x - t)a-l(t - r)c+k-1dt
r(Q)r(c)  (chk!
o k-O T
Z
= 1 cp(r)dr  (a)k(b)k( -r)-kr(Q)r(c + kHx - r)a+c+k-l x P (-{3)/(Q), ( 1 _  ) '
r(Q)r(c) L (c)A;k!r(Q + C + k)(Q + C + k)ll! x
o /1;,1-0
Z
I (x - r)a+c-l
=x P ( ) F3(a,Q,b,-{3;Q+c;l-x/r,l-r/x)cp(r)dr,
r Q+c
o
Re(Q + c) > O.
(10.46)

194
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
During the above evaluations we assumed Rea> 0, ReP > 0 and applied Fubini's
theorem, the substitution t = x - TJ(x - r) and the expansion of the inner integral
in hypergeometric series by using (1.72). All these evaluations are correct and
lead to the right-hand side of (10.46) under the conditions of convergence of the
corresponding series. After that these conditions can be weakened or even removed
by using the principal of analytic continuations of the relations if we denote by
F3 not only the double series in (10.45) but its analytic continuation beyond the
domain 0 < x, y < 1. By the same method the condition Rea> 0 can be weakened
till the condition Re(a + c) > o. .
If we replace a by a', P by -b' and a + c by c in (10.46) and take (10.22) and
(10.35) into account we obtain the following analogue of (10.37):
j z (x _ r)c-l ( X r )
f(c) F3 a, a', b, b ' ; c; 1 - ;,1 -;; cp(r)dr
o
'Y+ ioo
1 j f(1 + a - c - s)f(l + b - c - s)f(1 - a' - b ' - s) _
- - cp.(s + c)x 8ds
211"i f(1 + a + b - c - s )f( 1 - a' - s )f(1 - b ' - s) ,
'Y- ioo
Res < 1 + Re(a - c), 1 + Re(b - c), 1 - Re(a' + b').
( 10.47)
According to (10.35) each pair of the gamma-functions, one function in the
numerator and one in the denominator, corresponds to a single fractional integral
with power weights. Hence, six gamma-functions in (10.47) correspond to six
variants of compositions of three such fractional integrals, and integrals in each
composition can be arranged in different order using six ways. If we also take into
account that each such an arrangement corresponds to six variants of conditions
connected with different variants of signs of the orders of these integrals, then it
is obvious that writing all variants of compositional expansions for the operator
given in (10.47) and the conditions such as in Table 10.2 is not convenient.
By analogy with (10.37) the compositions of any numbers of the fractional
operators in (10.35) and (10.36) lead us to the relation
m n
II (x fJj I; -fJj x -OIj) II (x 6 ,. I" -6,. X -'1,. )f( x)
j=1 k=1
'1 + ioo
=  j IT f(1 - aj - s) IT f(6 k + s) f.(s)x- 8 ds
211"i . f(I-,Bj-s) f("Yk+S)
'Y- ioo 3=1 k=l
00
- j Jm (  1 (OI)m,(-y).. ) t dt
- m+n,m+n t (6)..,(fJ)m f() t '
o
( 10.48)

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 195
involving the Meijer G-function given in (1.95).
The necessary condition for the second relation in (10.48) to be fulfilled is
the condition Re C,< "'i - /1i) + f. (r. - 6.») > 0 - see Theorem 36.3 and the
inequality in (36.21).
Sufficient conditions for the operators in the parentheses on the left-hand side
of (10.48) to be commutative is given by the following statement.
Theorem 10.7. Let f(x) E Lp(O, (0), p(l- Reaj) > 1, j = 1,2,...,m,
pRe61: > -1, k = 1,2,...,n, and p  1, Re(aj - (3j) > 0, Re("Y1: - 61:) > 0 or
p> I, Re(aj - (3j)  0, Re("Y1: - 61:)  O. Then the operators in the left-hand side
of (10.48) commute. If additionally Re C. ("'i - /1i) + .t,<r. - 6.») > 0, then
the left and the right-hand sides of (10.48), without the middle, are bounded from
Lp(O, 00) to Lp(O, 00) and coincide with each other.
The proof follows from the boundedness of the left-hand side of (10.48) in
Lp(O,oo) which is valid under the conditions of the theorem. Here the arguments
are the same as those in the case j = 1 when proving Theorem 10.2, but with
taking into account the remark after the relation in (5.46'). .
As in the cases of Theorems 10.1 and 10.2 the conditions in Theorem 10.7 may
be weakened and extended to negative Re(aj - (3j) or Re("Y1: - 61:), and the space
Lp(O,oo) can be contracted to the corresponding space of functions represented by
fractional integrals of other functions in Lp(O, (0).
In conclusion of this subsection we note that commutability is valid for the
products of more general operators of the form xl:fj l;;;.. x-I:Ot and x mfj 1"1!- x-mOt
under the appropriate conditions (see S 18.2).
10.4. Compositions with exponential and power-exponential
weights
We first consider compositions of two left-sided integrals with exponential weights.
The following statement is true.
Lemma 10.3. Let 1/J(x) E Lp(O, b), 0 < b < 00,1 :5 p < 00, and Rea f; O. Then
the representation 1/J(x) = e>'z Ig+e->'z f(x), f(x) E IOt)(Lp(O, b», takes place if
and only if 1/J(x) E -Ot)(Lp(O, b».
Proof. In the case Rea > 0 the lemma directly follows from Lemma 31.4 since
according to this lemma the operator e>'z Ig+e->'z, Rea> 0, is bounded from
Lp(O, b), 0 < b < 00, 1 :5 p < 00, onto Ig+(Lp(O, b». The case Rea < 0 is
considered in the same way as it was carried out in Lemma 10.1. .
According to Lemma 10.3 we are able to obtain the following result which is
an analogue of Theorem 10.1.

196
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Theorem 10.8. Let a and /3 be complex numbers such that the function
m(a, (3, a + (3) is defined. Then the operators 1+ and e>'z Ig+e->'z commute in the
space IOt,I3,Ot+(3)(Lp(O, b», 0 < b < 00, 1  p < 00, i.e. the relation
lOt e>'x 1 13 e->.z f( x ) - e>'z 1 13 e->'z lOt f( x )
0+ 0+ - 0+ 0+
( 10.49)
is valid. If Re(a + /3) > 0, Re/3i= 0, and f(x) E 1(3)(L,,(O, b» then
z
f ( )Ot+I3-1
Ig+e>'z Ig+e->'z f(x) = . x ( r ) 1 F l (/3; a + /3; A(X - r»f( r)dr.
fa+/3
o
( 10.50)
IfRe(a + (3) > 0, Rea 1= 0, and f(x) E 1(Ot)(Lp(O,b» then
z
f ( )Ot+I3-1
e>'z Ig+e->'z Ig+f(x) = x; + /3) 1F1({3; a + {3;A(X - r»f(r)dr.
o
( 10.51)
When Re(a + (3) > 0 the operators in (10.50) and (10.51) are bounded from the
above spaces onto 1:I3(Lp(O, b».
Proof. It is sufficient to prove (10.49)-(10.51) in the space L 1 (0, b). By Fubini's
theorem we make some evaluations and use the integral representation 6.5(1) in
the hand-book by Erdelyi, Magnus, Oberhettinger and Tricomi [1]. As a result we
have
z t
f ( t ) Ot-l f ( t ) 13-1
lOt e>'z 1 13 e->.x f( x ) = x - e>.tdt - r e->'T f( r ) dr
0+ 0+ f(a) f(/3)
o 0
x 1
f ( X - r ) Ot+I3-1 f
= f(r)dr ul3- 1 (1_ ut-1e>,u(z-T)du
f(a)f(/3)
o 0
x
f (x - r)Ot+I3-1
= f(a+/3) lF 1 (/3;a+/3;A(x-r»f(r)dr,
o
Rea> 0, Re/3 > O.
( 10.52)
Now if we apply 6.3(7) from the above handbook and use the relation in (10.52)
but in inverse order then we arrive at (10.51). The relation in (10.49) follows from
(10.50) and (10.51) in the case Rea > 0 and Re{3 > O. The last conditions can be

 10. COMPOSITIONS OF FRACTIONAL INTEGRALS AND DERIVATIVES 197
weakened by replacing them with the conditions in Theorem 10.8 which guarantee
the convergence of all integrals in (10.49)-(10.51). .
We note that (10.49) enables us to obtain the composition of any numbers of
the operators 1;+e:J:>'x with alternative signs for AX via the composition of only
two such operators. For example, the relations
fa e->'x 1/J e>'x fl e->,x f( x ) - I a e->'X ( e>'x I/J e->'x fl f( x »
0+ 0+ 0+ - 0+ 0+ 0+
- Ia+/Je->,xl f( x )
- 0+ 0+'
(10.53)
I a e->'x 1/J e>'x fl e->'x 1 6 e>,x f( x ) - I a e->'X ( I/J+ e>'x 1 6 f( x »
0+ 0+ 0+ 0+ - 0+ 0+ 0+
= 1:/J+ e->'x Ig+e>'x f(x),
( 10.54)
By analogy with (10.18)-(10.21) we introduce the operators
x
f (x r)c-l
(1'>'f)(x)= (c) lFl(a;C;A(x-r»f(r)dr,
o
(10.55)
00
f (r x)c-l
(la,>. f)(x) = (c) lFl(a; c; A(r - x»f(r)dr,
x
( 10.56)
Rec> 0, ReA> O.
It is obvious that
I C,O,>. - 1 c ,a,0 - I c 1 c,c'>. - e >,x I c e ->.x
0+ - 0+ - 0+ ' 0+ - 0+ ,
10,>. = la,o = 1:, lc.>' = e>'x l:e->'x.
( 10.57)
From (10.50) and (10.51) after the substitutions /3 = a and a + /3 = c we
obtain the relations
(I'>' f)(x) = 1+ae>'x 18+e->'x f(x),
(I'>' f)(x) = e>'x 18+e->'x 1+a f(x).
( 10.58)
(10.59)
The relations
(la.>. f)(x) = 1:-a e >.x le->'x f(x),
(la,>. f)(x) = e>'x le->'x 1:- a f(x)
( 10.60)
(10.61)

198
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
are proved similarly. To investigate the operators in (10.58)-(10.61) we need the
following result which follows from Theorems 18.2 and 18.3.
Lemma 10.4. Let ReA> 0, Rea > 0, 1  p < (Rea)-l. Then the operator
e>'z Ie->'z f(x) is bounded from Lp(a, 00), 0 < a < 00, onto I(Lp(a,oo».
(10.58)-(10.61) are valid under the conditions given in the following statement
which can be easily verified.
Theorem 10.9. Let Rec > 0 and let some of the conditions in Aj from
Table 10.4 be fulfilled. Then the operators Bj are defined on the spaces
Cj C Lp(O, b), 0 < b < 00, 1  p < 00, and are bounded from Cj onto Dj and the
relations in Ej are fulfilled.
Table 10.4
j Ai Bj Cj Dj Ej
1 Rea> 0 [c,a,). Lp(O,b) 10+ (L p (0, b» (10.58)
0+
2 Rea < 0 Ic,a,>' 1 0 ': (L p (0, b» Ig:;a (Lp (0, b» (10.58)
0+
3 Re(c-a»O [c,a, >. Lp (0, b) 18+(L p (0,b» (10.59)
0+
4 Re(c-a)<O [c,a,>. Ig.;c (L p (0, b» Ig+(L p (O,b» (10.59)
0+
5 Re a > 0, Re A > 0, 1:': a ,>. Lp(a,oo) 1(Lp(a, 00» (10.60)
pmax(Rea,Rec) < 1
6 Rea < 0, ReA> 0, 1:': a ,>. {e>'z l: a e->'z 1/1, 1:-a(Lp(a,oo» (10.60)
pRe (c - a) < 1 1/1 E Lp(a, oo)}
7 Re(c - a) > 0, ReA > 0, I:,:a,>. Lp(a,oo) I(Lp (a, 00» (10.61 )
pmax(Rec,Re(c - a» < 1
8 Re(c - a) < 0, ReA> 0, l:,:a,>. 1-C(Lp(a, 00» {e>'z I'!.. e->'z 1/1, (10.61)
pRe a < 1 1/1 E Lp(a,oo)}
Compositions of many-sided fractional integrals with exponential weights leads
us, in general, to integral operators of highly cumbersome form except in some
special cases. One of these cases is considered here. Using 2.3.6.10 in the handbook
by Prudnikov, Brychkov and Marichev [1] we evaluate the following composition:
z 00
r!.e->< IIf+f(z) = / ;l:; /(1 - z)a-l(t - T)a-le-'<dt
o z
00 00
+ / f(r)dr / (t _ x)a-l(t _ rt-1e->.tdt
r2(a)
z T
Al/2-a / 00 ( A )
- y'1rr(a) Ix - rl a - 1 / 2 e->,(z+T)/2I<a_l/2 "2lx - rl f(r)dr,
o
Re a > 0,
( 10.62)

 11. FRACTIONAL INTEGRALS AND THE SINGULAR OPERATOR 199
where KII(z) is the Macdonald function given in (1.85). For the operator in (10.62)
the corresponding analogue of Theorem 10.9 can be obviously proved but the
property of (10.43)-type is wrong.
The composition of two one-sided fractional integrals with power-exponential
weights leads us to the relation
[a x- 6 e>'z [13 x 6 e->.z f( x )
0+ 0+
J z (x - t)a+ 13 -1 ( X )
= r(a + {3) l {3, 6; a + {3; 1 - t' A(X - t) f(t)dt,
o (10.63)
where l is one of the Humbert functions:
 ( IJ 6' "Y' X y) =  ({3)j+k(6)j  y 1c. I x l < 1
1 P, , '" L...J ( ) . 'Ik l '
j,1c=O / J+1c]. .
( 10.64)
Erdelyi, Magnus, Oberhettinger and Tricomi [1, 5.7.1(20)] with a correction of the
misprint in the index in the numerator, ( ({3)m instead of ({3)n). The relation in
(10.63) is proved by the same way as (10.50).
 11. Connection between Fractional Integrals
and the Singular Operator
In this section we discuss the connection between the fractional integration
operators and the singular integral operator. It is shown how a left-hand sided
fractional integration is expressed via a right-hand sided one (and vice versa) in
terms of the singular operator. We first give the necessary preliminary information
concerning the singular operator.
11.1. The singular operator S
As before let n = [a, b], -00  a < b  00. We consider the singular integral
b
(Scp)(x) =  J cp(t)dt , x E (a, b),
7r t-x
(11.1)
a
the convergence being understood in the principal value sense. One may become
acquainted with the theory of such integrals and with the proofs of Theorems
11.1-11.3 given below in the books of Gahov [1], Muskhelishvili [1] and Gohberg
and Krupnik [4]. Here we give the properties of the operator S, which we shall

200
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
repeatedly use in the future. An important property of the operator S is its
boundedness in the spaces H(p) and Lp(p). Let p(x) be the weight (1.7) or (1.9),
and Xl = a, X n = b in the case when a and b are the finite points.
Theorem 11.1. If mesO < 00, 0 < A < I, then the operator S is bounded
n
in the space H and in the space H6(p), p(x) = n Ix-xkIIJk, Xlc EO, if
k=l
A < Ilk < A + 1 (k = 1, .. . ,n).
Theorem 11.2. If mesO = 00, 0 < A < 1, then the operator S is bounded in the
n
space H6(p), p(x) = (1 + x 2 )IJ/2 n Ix - xkl IJk , Xk E 0, if A < Ilk < A + 1 (k =
k=l
n
1,... ,n) and -A < P + L: PI: < 1 - A.
k=l
Theorem 11.3. The operator S is bounded in the space Lp(p), 1 < p < 00,
n
p(x) = (1 + x 2 )IJ/2 n Ix - xkl IJk , X/c EO, if -1 < Pk < P - 1 (k = 1,..., n) and if
k=l
n
(in the case mesO = 00) -1 < Il + L: Pk < P - 1.
k=l
In the case 0 = R 1 the explicit value of the norm of the operator S in the
space Lp(R1) is known namely
IISII = tg [-n/2 min(p, pi)].
( 11.1 ')
see S.K. Pichorides [1]. The equation S-1 = -S is also known to hold when
o = R 1 , and therefore
S2cp = _cp,
(11.2)
and the Fourier transform of the singular integral Scp is given by the equation
(Scp){ x) = isign x( x),
(11.3)
In the case of a finite interval [a, b] the following equalities
.!. j b (t _ a)IJ-1(b _ t)-IJ dt = { sin\J7r I =: IIJ-1 bx ' if x < a or x > b,
1r a t - x -ctgJl1r «;l:l , if a < x < b (11.4)
(0 < ReJl < 1);

 11. FRACTIONAL INTEGRALS AND-'fHE SINGULAR OPERATOR 201
.!. 1 6 (  ) I'  = { Bin11J1( [1 _I = II'] ,
11' b-t t-x 1 [ 1 ( x_a ) lJ ]
a  - COB p1l' 6-x '
if X < a or x > b,
if a < x < b
(11.5 )
(-1 < Rep < 1);
6
1 (t - a)II-1(b - t)IJ- 1
dt=
t-x
a
( 6-a ) ,,+11-1 ( .. 6-a )
6-x B(p,lIhF 1 1, p, p + II, 6-x '
if x < a or x > b;
(x - a)II-1(b - x)IJ- 1 1rctgp1l' - (b - a)IJ+ II - 2
B( 1 ) F (2 1 . 2 . 6-X ) (11.6)
x p - , II 2 1 - P - II, , - p, 6-a '
if a < x < b (Re p > 0, Re II > 0)
are valid - see Prudnikov, Brychkov and Marichev [1; 2.2.6.5-8] and ErdtHyi,
Magnus, Oberhettinger and Tricomi [2, 15.2(33)]. We note that if x f/. (a, b), the
integrals (11.4)-(11.6) are evaluated by a simple change of variable. Thus, for
example, if x < a, the change of variable s = :::=: transforms (11.4) to the
beta-function and the same change of variable yields (11.6) if x > b according to
(1.73). If a < x < b, the required equalities can be obtained by means of the
Sokhotskii formula, known in the theory of singular integrals - see Gahov [1, p.38].
The following results
6
1 dt
(t - a)(b - t)  It - yp-a(t - x)
a
1rctg Tsign (y - x)
- (x - a)a/2(b - x)a/2Ix _ yI1-a'
(11. 7)
6
1 sign(t - y)dt
!.:t:2. !:J:2.
(t-a)  (b-t) 2 It-yI1-a(t-x)
a
1rtg Q2
_ 2
- !.:t:2. ( !:J:2. 1 1 '
(x - a)  b - x)  Ix - y -a
( 11.8)
where a < x < b, a < y < b, 0 < a < 1, are also valid - see Gahov [1, p.530-531].
Finally, we note the formula for interchangng singular integrals
6 6 b b
1  1 cp(y, t)dt = -1I'2cp(x, x) + 1 dt 1 cp(y, t)dy ,
Y - x t - y (y - x)(t - y)
a a a a
(11.9)
named the Poincare-Bertrand formula - see Gahov [1, p.63].

202
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
11.2. The case of the whole line
The relations connecting the fractional integrals I+' cp and I cp with the singular
operator S have the simplest form in the case of the complete line.
Theorem 11.4. Let a < a < I, and cp(x) E Lp(R 1 ). Then the fractional integrals
I+' and I and the singular operator S are connected with each other by the
equalities
Icp = cosa7rI+cp+sina1rSIcp, 1  p < l/a,
(11.10)
Icp = cosa1rIcp - sina7rSIcp, 1  p < l/a,
(11.11)
SII:cp = 11:Scp, 1 < p < l/a.
(11.12)
Proof. First let cp E ego. We consider the integral
00 t
SI a -! f  f cp(r)dr
+cp- 7r t-xf(a) (t-r)l-a'
-00 -00
Here we can interchange the order of integration. Such an operation is known
to be valid in the case when one of the integrals is singular - see Gahov [1] and
Muskhelishvili [1]. Therefore, we have
00 00
SI<p = ..r,,) f <p(r)dr f (I - r)la(t - z) '
-00 T
After substituting t = r + s/(1 - s), the inner integral becomes
00 1
f dt f sa-l(1- s)-ads
(t - r)l-a(t - x) - r - x + s(1 + x - r)
T 0
{ -2:- ( r _ X ) a-l
sin a'll" '
- -7rctga1r(x - r)a-l,
x < r,
X> r,

 11. FRACTIONAL INTEGRALS AND THE SINGULAR OPERATOR 203
according to (11.4). Therefore, we find
[ 00 Z ]
SI a _ 1 cp(r)dr _ cp(r)dr
+\0 - sina..r(a) ! (T - z)l-a _! (z - T)l-a '
which yields (11.10) for cp E C{f. In view of the density of C{f in Lp(Rl) this
equality is extended to Lp(R 1 ) by the Banach theorem 1.7 owing to the boundedness
of the operators I from Lp(Rl) into Lq(Rn), q = p/(1 - ap) (Theorem 5.3) and
the boundedness of the operator S in Lq(Rl) (Theorem 11.3). In the case p = 1 it
is necessary to use Theorem 5.6 instead of Theorem 5.3 and Remark 5.2.
Equality (11.11) follows from (11.10) by virtue of (5.9) and the relation
QS = -SQ where Qcp = cp(-x).
The commutation relation (11.12) is obvious for good functions owing to the
fact that it is possible to pass to Fourier transforms for such functions - see (7.11)
and (11.3). As for the functions in Lp, it remains to refer to the boundedness of
the operators I and S. .
Corollary 1. The coincidence of the images
I+(L p ) = I(Lp>,:Yla(Lp), 1 < p < l/a,
is valid (see (6.1»). The space Ia(L p ) is invariant relative to the operator S.
Indeed, the equation Icp = I+(cosa7f'cp =F sina7f'Scp) holds, and therefore a
coincidence of images follows from the boundedness of the operator S in Lp.
Corollary 2. II a > 0 and (3 > 0, then the relation
P+I = [cos({3 - a)7f'E + sin({3 - a)1f'S]II,
is valid, E being the identity operator.
Corollary 3. The Marchaud derivatives D.! and DI are related by the equation
DI = cosa1l'D1 - sin a1f'SD/, IE Ia(L p )'
(11.11')
To obtain (11.11') it is sufficient to designate Icp = I and, in accordance
with Theorem 6.1, to apply the operator D+ to both sides of (11.11) and to take
(11.12) into account.

204
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Corollary 4. For any 0 < a < 1 and 1 < p < l/a the inequality
IIDfllp  AIID+fll p , f E Ia(L p ),
(11.11")
is true where A = Icosa7r1 +sina7rtg[7r/2min(p,p')].
Indeed, this estimate follows from (11.11') and (11.1').
Remark 11.1. Relations (11.10)-(11.12) are true for I;'(x) E Lp(p), p(x)
n
(1 + x 2 )1J/2 n Ix - xl:l lJk , XI: E R 1 , if -1 < Ill: < p - 1 (k = 1,..., n), ap - 1 <
1:=1
n
Il + L: Ill: < P - 1, p> 1. This follows from the Hardy-Littlewood-type Theorem
1:=1
5.5 and from Theorem 11.3 according to the Banach theorem 1.7.
Remark 11.2. In the theory of singular integrals (Gahov [1, p.43]) a simple
formula for differentiation of the singular integral, (Sf)(n) = S(f(n», is known. It
is also valid for the case of fractional differentiation, thus
DI(Sf) = S(DIf), f E Ia(L p ), 1 < p < l/a,
(11.13)
which is a paraphrase of the commutation property (11.12).
11.3. The case of an interval and a half-axis
Now let us find out how the fractional integrals 1<;+1;' and If_I;' are connected
with each other in the case of a finite interval [a, b]. These relations are more
complicated than (11.10)-(11.12) because of the influence of the end-points of the
interval. The following question arises: what is the result of the composition
V:+lf_ (or Vf_I<;+)? It is natural to expect that the operators V:+ and If_
annihilate each other in a sense, and their composition is an operator bounded, for
example, in the usual spaces H>' and Lp. We shall see that it contains the weighted
singular operator.
Assuming for the present that I;' E C, let us consider the composition
x b
a a sin a7r d f dt f( ) a 1 ( )d
V(J+Ib-I;'=-- d ( ) T-t - I;'T T
1r X x-ta
(J t
b
sin a7r d f ( )d
= -- I;' T T
7r dx
min(x,T)
f (x - t) - a ( T - t) a-l dt.
(J
(J

 11. FRACTIONAL INTEGRALS AND THE SINGULAR OPERATOR 205
We introduce the function
sin a1l' d [ :& / -£ / T 1
J£(x) =----;- dx cp(r)dr (x - t)-a(r - t)a- dt
a a
b :&
+ / I"(T)dT / (z - waiT - t)a-1dt]
:&+£ a
and set
T
K:l(X, r) ':1 / (x - t)-a( r - t)a-ldt
a

:11:-"
= / sa-l(1 + s)-ads, r < x,
o
:&
K:2(X, r)':1 / (x - t)-a(r - t)a-ldt
a
£:.!.
"-:II:
= / s-a(1 + s)a-lds, r> x,
o
Then we have
sin a1l' [
J£(x) =- K:l(X, X - c)cp(x - c) - K: 2 (x, x + c)cp(x + c)
11'
+ sin a1l' ( :& / -£ + / b ) ( r _ a ) a cp(r)dr ].
11' x-a r-x
(I :&+£
Hence we obtain
b
V:+I&_cp = lim J£(x) = sina1r { / ( r - a ) a cp(r)dr
£-0 1r X - a r - X
a
+ lim cp(x)[K:l(x, x - c) - K:2(X, x + c)]}.
£-0

206
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Evaluating the limit we find
lim[Kl(x, x - c) - K2(X, x + c)]
£-0
 z-.
= lim { j  [sa-I(1 + s)-a - s-a(1 + s)a-l]ds - j  s-a(1 + s)a-l]ds}
£-0
o
a:-:-c
j OO [ sa-l (I+S)a-l ]
= - ds
(1 + s)a sa
o
= 1I'ctg Q1I'
by 2.2.12.3 from Prudnikov, Brychkov and Marichev [1]. Therefore we have
b
-na l a ( ) sinQ1I' j ( r - a ) a cp(r)dr
V a + b-CP = COSQ1I'cp x + - - -
7r x-a r-x
a
(11.14)
and the relation sought for is obtained. Now knowing the explicit form of the
relation, it is easy to give its rigorous justification. It will be given for the functions
cP E L". We shall obtain other relations by the way.
Theorem 11.5. Let 0 < Q < 1 and
ra(x)=x-a, rb(x)=b-x.
(11.15)
Then the fractional integrals I:+cp, II:_cp and the singular operator S are connected
with each other by the relations
11:_ cp = COS Q1I'1:+cp + sin Q1r 1:+ r;; ° Sr:cp, cp E Lp, p> 1, (11.16)
1:+cp = COSQ1I'II:_cp - sinQ7rll:_r"baSrbCP, cP E Lp, p> 1, ( 11.17)
Ib_cp = cos Q1I'1:+cp + sin Q1rrb Sr"b a 1:+cp, cp E Lp, p  1, (11.18)
I:+cp = cos Q1rl b _cp - sin Q7rr: Sr;;a II:_cp, cp E Lp, p  1, (11.19)
which hold almost everywhere ifpQ  1 and for each x E (a,b), ifpQ > 1.
Proof. First let cp E ego. (11.16) follows from (11.14) since 1:+VC:+f = f.
This can also be checked by direct interchanging the order of integration in the

 11. FRACTIONAL INTEGRALS AND THE SINGULAR OPERATOR 207
right-hand side of (11.16). As for (11.18), we prove it by direct verification, thus
b b
a S  l a - (b - X)a f ( )d f (t - r)a-l(b - t)-a d
rb a a+ cP - cP r r t.
r b 7r t - X
a T
As mentioned previously the interchange of the integration order of singular
and absolutely convergent integrals is possible. Hence, according to (11.4), the
right-hand side is equal to
b x
1 f cp( r )dr ctg a7r f cp( r )dr
r(a)sina7r (r-x)l-a - r(a) (x-r)l-a'
x a
which yields (11.18). Relations (11.17) and (11.19) follow from (11.16) and (11.18),
respectively, if we use (2.19) and take into account that QS = -SQ.
Thus (11.16)-(11.19) have been proved for cp E ego. The operator r;aSr is
bounded in L", 1 < p < l/a, by Theorem 11.3. Then in view of the boundedness
of the operators P;+ and lb_ from L" into Lq, q = p/(I- ap), we can deduce that
the validity of (11.16) for the dense set ego in L", in accordance with Theorem 1.7,
implies its validity for all functions cp E L", 1 < p < l/a.
Relations (11.17)-(11.19) for cp E L", 1 < p < l/a, are proved in the similar
way. By the imbedding L p1 (a,b) C L"2(a, b), Pl > P2, these relations are also true
in the case p  l/a.
To clarify the validity of the relations in each point x E (a, b) in the case
P > l/a we write an easily checked result
1 S a 1 a-l C
- raCP = a_l Sra cp+ ( ) '
r ra X - a a
b
- ! f cp(t)dt
C- ,
7r (t - ap-a
a
( 11.20)
assuming that cp E L", p > l/a. Since the operator r-aSr-l is bounded, by
Theorem 11.3, in Lp if p > l/a, then the right-hand side in (11.16) is a Holder
function in accordance with Theorem 3.6. Therefore, (11.16) holds at each point.
The other relations are justified similarly.
It remains to consider the case p = 1 in (11.18) and (11.19). Similarly to
(11.20) we obtain
a s 1 ! a-I s I-a ! Cl
r a r = r a r a + (x _ a) 1- a '
b
C = .!.. f !(t)dt
7r (t - a)a
a

208
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
provided that the integrals on the right-hand side exist. Therefore, for f = f b _ cP
by simple calculations we find
b
1 1 ( b - X ) a-l 1
raS-f a cP = r a - 1 S_I a cP - f ( l- a ) cp( t ) dt.
a r a b- a a-I b- 7r
a ra
a
(11.21 )
Hence, the boundedness of the operator r:Sr;a I b _ from L 1 into Lr, 1 < r < 1/1-a
is seen. Indeed, this is obvious for the second term on the right-hand side and
follows from Theorems 5.6 and 11.3 for the first one. The case p = 1 in (11.18) is
considered similarly. .
Corollary 1. The spaces of functions representable by the fractional integrals
P;+cP and Ib_cp with the density cp E Lp coincide with each other if 1 < p < 1/a.
Indeed, according to (11.16) and (11.17) we have
I a f a ( . - a s a )
b-CP = a+ COS a7rcp + sm a7rr a ra cP ,
f a f a ( . -a s a )
a+CP = b- cos a7rcp - sm a1rr b rb cP ,
and therefore corollary follows from the boundedne5s of the operators r;a Sr: and
r b a Srb in L" if p < l/a.
Corollary 2. The following commutation relations
f:+r;a Sr:cp = rb Sr;a I:+cp,
(11.22)
Ib_r;as rbCP = r:Sr;a Ib_cp,
are valid.
Similarly to (11.13) we may interpret (11.22) as the formulae of fractional
differentiation of the singular integral
D+ (1 6 ( b - x ) a f(t)dt ) = 1 6 (  ) a (D+f)(t) dt,
b-t t-x t-a t-x
a a
D b _ (l b (  ) a f(t)dt ) = l b (  ) a (Db_f)(t) dt,
t-a i-x x-a t-x
a a
( 11.22')
for f E fa(L p ), 1 < p < l/a.

 11. FRACTIONAL INTEGRALS AND THE SINGULAR OPERATOR 209
Corollary 3. Together with (11.16)-(11.19) the following relations
b
I a a' a 1 a-I 1 f cp(t)dt
b-CP = cosa1rla+cp+sma1rla+----=rSra cp+ r( ) ( )l '
r a a t-a -a (11.23)
b
I a I a' a 1 S a-I 1 f cp(t)dt
a+CP = cos a1r b- cP - sm a1r I b _ r b a-I rb cP + r ( a ) (b _ t)l-a '
a ( 11. 24 )
b
I a I a' a-I s 1 a (x - a)a-I f ( )d
b-CP = cos a1r a+CP + sm a1rr b a-I Ia+CP + r() cP t t,
r b a a (11.25)
b
I a I a' a-I S 1 a (b - x)a-I f ( )d
a+cp=coSa1rb_cp-sma1rra a_I 1b-CP+ r() cpt t,
ra a a (11.26)
are also valid where 0 < a < 2, cp(x) E Lp(a, b) and p > max(l, l/a) in (11.23)
and (11.24) and 1  p < 00 in (11.25) and (11.26).
The assertions of Corollary 3 follow directly from (11.16)-(11.19) on the basis
of relations similar to (11.20) and (11.21). The proof of their validity for cP E Lp
with the mentioned values of p is carried out as in Theorem 11.5 for (11.16)-(11.19).
Remark 11.3. By Theorem 3.12 (11.16)-(11.19) already proved are also valid
n
for cp(x) E Lp([a,b),p),p(x) = n Ix - xl:l#Jk, X/c E [a, b), if -1 < Ill: < p-l (k =
1:=1
1,...,n).
In the case of n = R = [0, 00) the connection between the fractional
integration operators Ig+ and I and the similar operator S can be obtained from
(11.16)-(11.19) by means of the passage to a limit as a --+ 0 and b --+ 00. These
relations are given by the following equalities,
Icp = cos a1rIg+cp + sin a1rIg+x- a Sxacp,
( 11.27)
( 11.28)
(11.29)
(11.30)
Ig+cp = cosa1rIcp - sina1rIScp,
Icp = cos a1rIg+cp + sin a1rSIg_cp,
Ig+cp = cos a1r Icp - sin a1rx a Sx- a Icp,
where cp(x) E Lp(R), 1 < p < l/a, in (11.27), (11.28) and (11.30) and 1  p < l/a
in (11.29). We also note that (11.27) and (11.30) follow from (11.16) and (11.19),
while (11.28) follows from (11.11) in view of (11.12), and (11.29) follows from

210
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
(11.10). Moreover, (11.27)-(11.30) lead to commutation relations
Ig+x- a Sx a = SIg+, x a Sx- a r. = I<:'S
(11.31)
of the type (11.20).
Similar ly, (11.23)-( 11. 26) can be transferred to the half-axis R.
We conclude this subsection with the following assertion.
Lemma 11.1. Let 0 < a < 1. The operators
00
A ( ) sin a1r 1 cp(t)dt
cp = cosa1rcp x - - -,
1r t-x
o
x> 0,
( 11.32)
sin a1r 1 00 ( t ) a cp(t)dt
Bcp = cos a1rcp(x) + - - -,
1r X t-x
o
x> 0,
(11.33)
are mutually inverse: ABcp == BAcp == cp for cp(x) E Lp(R), 1 < p < l/a.
Proof. This lemma follows from comparison of (11.27) and (11.28). Really,
they have the form I<:'cp = Ig+Bcp and Ig+cp = I<:'Acp. Hence it is obvious
that ABcp == BAcp == cp for sufficiently good functions cp(x), and then it is
true for all cp( x) E Lp (R) by the boundedness of the operators A and B in
Lp(R), 1 < p < l/a, which follows from Theorem 11.3.
We note that the identities ABcp == BAcp == cp may be checked by the direct
evaluation of the compositions AB and BA by the Poincare-Bertrand formula
(11.9). .
11.4. Some other composition relations
We show that if J(x) E Ia(L p ), then "a truncation" of the function f(x) also
belongs to I a (L p ). Namely, let -00  a < b  00 and
PabCP = { cp(x),
0,
x E [a,b],
x f/. [a, b],
(11.34)
but in the case of the half-axes [a, b] = R or [a, b] = R: we write
P _ { cp(x),
+cp -
0,
x> 0,
x < 0,
{ 0,
P - cp = cp( x) ,
x> 0,
x < O.
( 11. 35 )

 11. FRACTIONAL INTEGRALS AND THE SINGULAR OPERATOR 211
We shall also prove that P(JbI+. == I+.Ncp, cp E Lp, 1 < p < 1/0:, N being a bounded
operator in Lp.
Theorem 11.6. Let cp(x) E L,(Rl), 1 < p < 1/0:. Then
P+I+.cp = I+.1/J, 1/J E L,(R 1 ),
( 11.36)
where
{ cp( x ) + sin mr J oo ( .!. ) a tp( -t) dt
1/J( x) = 11' 0 x x+t '
0,
x> 0,
( 11.37)
X < O.
Proof. First we note that 1/J(x) E Lp by Theorem 1.5. Then it is clear that
(I+.1/J)(x) = 0 if X < O. It remains to prove (I.+1/J)(x) = (I.+cp)(x) if X > O. We have
x 0 x ) 0' t 0'
(I a/. )( ) 1 f tp( t  dt sinall' J ( ) a ( )d f (x-t - t- dt A d . t
+ 'f' x = r(a) (x-t 1-0 + lI'r(a) -T cp T T t-T . ccor 109 0
o -00 0
(11.4) we obtain
o
(I'i-,p)(z) = (1;:+'I')(z) + r(lr» J (z'l'tl)a = (1'i-'I')(z),
-00
which completes the proof. .
We emphasize that Theorem 11.6 means that if the function f( x) is
representable by the fractional integral of a function in Lp(Rl), 1 < p < 1/0:, then
the function f+tx) = f(x) if x > 0, and f+(x) = 0 if x < 0 has the same property.
Corollary 1. Let cp(x) E L,(R 1 ), 1 < p < 1/0:. Then
PabI+.cp = I+.1/J, 1/J E Lp(R 1 ),
(11.38)
where
1/J(x) =
0,
sin all' 00 ( t ) a tp ( (J-t  dt
cp(x) + -;r- J z=a (x+t-(J 1-0 '
o
sin all' [ 1 ( -L ) a tp«(J-tdt
11' 0 X-(J (x+t-(J 1-0
00 ( t ) a tp( b-t  dt ]
- £ i=a (x+t-b 1-0 ,
x < a,
a < x < b,
x> b.

212
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Corollary 2. Let a(x) be a piecewise constant function with a finite number
of point discontinuous. Then a(x)f(x) E [a(L p ) for f(x) E [a(L p ) and
lIafIlI"'(L p )  cllfIlI"'(L p )'
Theorem 11.7. Let f(x) be a piecewise Holder function on the axis with a finite
number of points of discontinuities and with Holder exponent A > max( a, -a + l/p)
and let f(oo) = O. Then f(x) E [a(L p )'
Proof. For simplicity we assume f(x) to have only one discontinuity at x = O. We
introduce the functions
f1(x) = { /(x),
wt(x),
x  0,
x < 0,
f2(x) = { W2(X), x > 0,
/(x), x  0,
having chosen the functions Wl(X),W2(X) E Cgo such that Wl(O) = /(+0), W2(0) =
/(-0). Then /;(x) E H>'(Rl) and /;(00) = 0, i = 1,2. Therefore, /;(x) E [a(L p )
by Theorem 6.5. It is obvious that /(x) = 8(x)f1(x) + 8( -x)f2(x), where
8(x) = (1 + signx)/2, so that /(x) E [a(L p ) by Theorem 11.6. .
It is clear that we can obtain the analogous relation P _ [cp = Ig from
(11.36) with the function g(x), written similarly to (11.37). The following theorem
represents P_Icp as 1<+9' We denote
00
SaCP = ! f 1  l a cp(t)dt .
7r x t-x
-00
( 11.39)
Theorem 11.8. For cp(x) E Lp(Rl), 1 < p < l/a, the relation
P_[cp = J<+.Ncp,
( 11.40)
is valid where the operator N given by
N cp = cos a7r P _cp + sin a7rScp - sin a7r P+SaP +cp
(11.41)
00 00
! sin a", J (t)dt _ sin mr J ( .t ) a !£ffi.dt X > 0 ,
'" ,;-t '" ,; ,;-t '
-00 0
- 00
. .,,(t\dt
cos a7rcp(x) + sm",a", J , x < 0,
-00
(11.42)
is bounded in Lp(R 1 ).

 12. FRACTIONAL INTEGRALS OF THE POTENTIAL TYPE
213
Proof. Applying (11.10) we have
P_Icp = (E - P+)I-t(cOS a1rcp + sin a1rScp).
(11.43)
We rewrite (11.36) as
P+I-tcp = I-t P +(cp-sina7rS Ot P_cp).
(11.44)
Substituting (11.44) into (11.43) we obtain
P_Icp =I-t(cos a7rP_cp + sin a1r P_Scp
+ sin a1r cos a7rP+S Ot P_cp + sin 2 a7rP+S Ot P_Scp).
(11.45)
Interchanging the order of integration in the last term we find according to (11.9)
that
1 1 00 1 00 tOtdt
P+SOtP_Scp = - 1r 2 Z Ot cp(r)dr (t + z)(t + r)
-00 0
if z > O. Here, the inner integral is equal to P
00
1 tOtdt 1r IrIOtIl( r) - zOt
(t + z )(t + r) = sin a1r r _ z ,z > 0,
o
( 11.46)
where 1I( r) = 1 if r > 0, and 1I( r) = cos a1r if r < O. (see Prudnikov, Brychkov,
and Marichev [1; 2.2.4.25, 26]).
Therefore,
1
P+SOtP_Scp = (P+S - P+SOtP+ - cosa1rP+SOtP_)cp,
SID a1r
(11.47)
After that (11.45) turns into the desired relation (11.40). The boundedness of the
operator (11.41) in L,(Rl) in the case 1 < p < l/a follows from that of S and SOt
in L,(Rl) (see Theorem 11.3). .
 12. Fractional Integrals of the Potential Type
In many fields of mathematical analysis there frequently occur operators of
fractional integration with "constant limits of integration" (they admit a natural
extension to the case of many variables). From analogies in mathematical physics

214
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
such operators came to be called potential type operators. In this section we shall
not be concerned with the corresponding forms of fractional differentiation, i.e. we
will not consider the construction of operators inverse to potentials. They will be
framed in S 30.4. We begin the consideration of potentials by studying first the
case of functions given on the whole axis R1.
12.1. The real axis. The Riesz and Feller potentials
We consider the integral
00
[ 01 - 1 f cp(t)dt Re 0  1
cp - 2r( a) cos( a 11' /2) It _ x11-0I ' a > I a.,.. 1 3 ,5,...
-00
(12.1)
the choice of normalizing factor being clarified below - see (12.23) and (12.24).
The integral [OIcp is called the Riesz potential. We shall consider side by side with
(12.1) the following modification
00
H OI 1 f sign (x - t) d
cp = 2r(a)sin(a1l'/2) Ix _ t11-0I cp(t) t,
-00
Rea> 0, a 1= 2,4,6, .. . I (12.2)
It is evident that
[01 = [2cos(a1l'/2)t1{l+ +[),
H OI = [2sin(a1l'/2)]-l(I+ -,[),
(12.3 )
(12.4)
where [ are operators (5.2) and (5.3). Thus operators [01 and HOI with
o < Rea < 1 are defined on functions cp(t) E Lp(R1), 1  p < I/Rea, and in
the case 1 < p < I/Rea are bounded from L,(Rl) into Lq(R1), provided that
q = p(1 - pRea)-l.
It will be shown in S 30.4 that operators inverse to [01 and H OI may be
constructed in the form
00
([01)-1 f = 1 f f(x - t) - f(x) dt
2r( -a) cos(a1r/2) It1 1 +01
-00
00
_ 1 f f(x - t) + f(x + t) - 2f(x) d
- / t,
2r( -a) cos(a1l' 2) t 1 +01
o
(12.1')

 12. FRACTIONAL INTEGRALS OF THE POTENTIAL TYPE
215
00
01 -1 1 f f(x - t) - f(x) .
(H) f = 2f(-a)sin(a1r/2) It1 1 +01 slgntdt
-00
00
= 1 f f (x - t) - f (x + t) dt
2f(-a)sin(a1r/2) t 1 +01
o
(12.2')
(compare these with (5.57) and (5.58) for the Marchaud fractional derivative).
Let
00
Scp = !. f cp(t)dt .
1r t-x
-00
(12.5 )
Lemma 12.1. Operators 1 01 and HOt are connected by the relations
100cp = SHOIcp,
( 12.6)
(12.7)
HOIcp = -SlOtcp,
where 0 < Rea < 1, cp E L,(R1), 1 $ p < I/Rea.
Proof. It is sufficient to prove (12.6) since (12.7) is derived from (12.6) immediately
using (11.2). As for (12.6) we obtain it by summing up (11.10) and (11.11) and
taking (12.3) and (12.4) into account. .
Corollary. Fractional integration operators l are represented via potential type
operators 1 01 and H OI by the equations
( a1r . a1r )
I:i: = 1 01 cas 2 E =F sm 2S ,
(12.8)
( . a1r a1r )
I:i: = H OI :i:sm 2 E + cos 2S ,
(12.9)
where 0 < Rea < 1 and E is the identity operator.
Proof. To verify (12.8) and (12.9) it is sufficient to apply (12.6) and (12.7) and
use then the relations (12.3) and (12.4). .
Equations (12.3) and (12.4) actuate the natural generalization of the operators
1 01 and H OI in the form of an arbitrary linear combination of the operators It. and
I:
M:,vcp = ul+cp + vlcp,
(12.10)

216
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
u and v being arbitrary constants. The operator (12.10) may be written down as
00
MOl =  f Cl + c2 si g n (x - t) ( t ) dt
u,vCP r(a) Ix - tl l - OI cp
-00
(12.11)
with Cl = (u + v)/2 and C2 = (u - v)/2 so that
a 11'" . a1l'"
M: vCP = (u + v) cos _1 01 cp + (u - v) sm _HOIcp.
, 2 2
(12.12)
We shall call the operator (12.10) the Feller potential. Such operators in the
form
00
1 01 1 f sin{ a [11'" 12 + 6 sign (t - x)]} ( )d
6CP= . cpt t
r(a) sm a 11' Ix - tl l - OI
-00
(12.13)
were first introduced by Feller [1] who proved that they satisfy the semigroup
property If If cp = I;+f3. Clearly, Ifcp may be obtained from (12.10) under the
choice u = sin(0I:,,/2-0I6) and v = sin(0I.1I'/2+0I6) . We also observe that
sm 0111' sm 0111'
l:cp = cos a6I OI cp - sin a6H OI cp.
( 12.14)
Remark 12.1. The operator M:,v has a rather simple form of inverse operator
which is similar to the Marchaud fractional derivative. The inverse operator is
especially simple in the case of operators 1 01 and HOI. We give the inverse of the
potential type operator M::,v in S 30.4 below.
In Theorem 12.1 below we shall show that a composition of two operators
of the form (12.10) is again an operator of such a type. We need the following
preliminary lemma.
Lemma 12.2. Let 0 < a < I, 0 < (3 < I, a + /3 < 1 and cp(x) E L,(R l ), 1  p <
1/( a + (3). Then
sine a + /3)1r P+l cp = sin a 11'" I:+ f3 cp + sin /311'" I+f3 cp,
sine a + /3)11'" I Icp = sin /311'" I:+ f3 cp + sin a1r 1+f3 cpo
(12.15)
(12.16)

 12. FRACTIONAL INTEGRALS OF THE POTENTIAL TYPE
217
Prool. Using equations (11.10) and (11.11), we have
1Icp = cos a1rl:+ f3 cp + sina7rSl:+ f3 cp,
I+' l cp = cos a7r 1+f3 cp - sin a1r S 1+f3 cpo
Hence by (12.3) and (12.4)
(I I + I I)cp = 2 cos a1r cos a ; {3 7r 10l+f3 cp + 2 sin a7r sin a ; {3 1r S H OI +f3 cp,
( 101 1f3 _ lOt If3 )cp = 2 cos a1r sin a + {3 7r H OI +f3 cp + 2 sin a1r cos a + {3 1r S 1 00 +f3 cp
-+ +- 2 2'
Using Lemma 12.1 we obtain the relations
f3 f3 a-{3
( 101 1 + 101 I )cp = 2 cos - 7rl Ol + f3 cp
+ - - + 2 '
( 101 If3 _ 101 1f3 )cp = 2 sin {3 - a 1r H OI +f3 cp
- + + - 2 '
which yield (12.15) and (12.16) after simple transformations. .
Theorem 12.1. Let M:1,Vl and Mf'J,tJ'J be operators of the form (12.10) and let
o < a < 1, 0 < {3 < 1 and a + (3 < 1. Then
MOl Mf3 - M OI +f3
Ul,tJl U'J,tJ2CP- U,tJ cp
(12.17)
where cp E L,(R 1 ), 1  p < 1/(a + (3), and
UI V2 sin a1r + VI U2 sin {37r
U = UI U2 + . ( IJ ) ,
sm a + fJ 1r
UI V2 sin {31r + VI u2 sin a7r
V = VI V2 + . ( IJ ) .
sm a + fJ 1r
Prool. Since any of the compositions 111cp is defined for cp E L,(RI), 1  p <
1/(a + (3), we have
M:1,VIMf'J,tJ'Jcp = UIU21:+fJcp + VIV21+f3cp + UIV21+.Icp + VIU21Icp (12.18)

218
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
by the semigroup property (5.15). Equations (12.15) and (12.16) allow us to
represent the compositions It. I, I I via I+f3 and I+f3, which change (12.18)
into (12.17) after simple manipulations. .
Corollary 1. If 0 < a < 1, 0 < P < 1, a + ,8 < 1 then
]lJ/ If3 = [a+ f3 ,
( 12.19)
(12.20)
(12.21)
H a Hf3 = _ [a+f3 ,
[a Hf3 = H a +f3.
Corollary 2. If 0 < a < 1, 0 <,8 < 1, a +,8 < 1, then
I a [ f3 - [ a+ f3
() 6 - () .
(12.22)
(12.22) is easily obtained using (12.14) and equations (12.19)-(12.21).
We note finally that (12.3) and (12.4) allow various other properties of the
operators I to be easily extended to operators la, H a and M;:,tJ' The following
theorem is in particular true.
Theorem 12.2. Let 0 < a < 1 and rp(x) E L1(R 1 ). Then
(:F[arp)(x) = Ixl-a(x), (12.23)
(:F Harp)(x) = isignxlxl-a(x), (12.24)
(F M:'.'I')(z) = [(u + v) cos "2" + i( u - v) sin "; signz] i;: . (12.25)
The proof is easily derived from (7.1) and Theorem 7.1 using relations (12.3),
(12.4) and (12.10).
12.2. On the "truncation" of the Riesz potential
to the half-axis
Similarly to Lemma 11.1 we ask the question: if f(x) = [arp, is it true that the
function f+(x), which is equal to f(x) for x > 0 and is zero for x < 0, is also

 12. FRACTIONAL INTEGRALS OF THE POTENTIAL TYPE 219
representable by the Riesz potential? That is
8+(x)(Iacp)(x) = (I a 1jJ)(x), -00 < x < 00,
8:i:(x) = (1:i: signx)/2,
( 12.26)
(12.27)
and how may one find the function 1jJ(x) from cp(x)?
Theorem 12.3. Given a function cp(x) E L,(R 1 ), 1 < p < 1/0:, there exists a
function 1jJ(x) such that (12.26) holds. It has the form
1 0:1r
1jJ(x) = 8+ (x)cp(x) + 2tg TNacp,
( 12.28)
where
00 Itlasign t _ 1
Nacp =  1 Ixlasignx cp(t)dt.
" t-x
-00
( 12.29)
Proof. It follows from (11.36) that 8_(x)(I.+cp)(x) = (I.+cpd(x), where
00
CPl(X) = 8_(x)cp(x) + sin 0:1r 8+(x) 1 1  l a cp(t)dt .
1r x t-x
-00
Then by (5.9)
8+(x)(Icp)(x) = (ICP2)(X),
( 12.30)
where
00
( ) _ 8 ( ) ( ) sinO:1r (} ( )1 ta cp(t)dt
CP2 x - + x cp x - - _ x --.
1r Ixl o t - x
o
Summing (11.36) and (12.30) and using (12.3) we obtain the equation
8+(z) Ia 'P = 2 COS("/2) (1+'P3 + e'P2),
(12.31)
where by CP3(X) we have denoted the function given in (11.37). Using (12.8) in

220
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
(12.31) we establish the equation
O+(x)/acp = [a [CP2 + CP3 + tg 0:2 1r S(cp2 - CP3)] .
Thus the desired function 1/J(x) in (12.26) is equal to
1 1 0:1r
1/J(x) = 2(CP2 + CP3) + 2 tg T S (CP2 - CP3)'
In order to evaluate the singular integral S(cp2 - CP3) here, it is necessary to
apply the Poincare-Bertrand relation (11.9) for interchanging the order of singular
integration. After simple manipulations this finally reduces the function 1/J(x) to
the form (12.28).
It remains to show that the function 1/J(x) belongs to Lp(Rl). This follows
from the boundedness of the operator Na in Lp(Rl), 1 < p < 1/0:, the latter being
obtained by passing to the half-axes Rl and using Theorems 5.5 and 11.3. .
Note that (12.26), in which 1/J(x) is defined by (12.28), may be checked directly
by the immediate interchange of the order of integration in the composition [a N a,
if we take into account that the inner integral that arises can be evaluated in
elementary functions:
00
f signt dt 0:1r signxsign(x - y)
= 1rctg
Itlalt - xI 1 - a (t - y) 2 Iylal x - yll-a
( 12.32)
-00
- see for example Prudnikov, Brychkov, and Marichev [1, 2.2.6.26, 2.2.6.27].
C II L t P _ { cp(x), x E [a, b], rJ"J.
oro ary. e (JbCP - . .L neR
0, X f/. [a, b].
a a ( 1 0:1r a )
P(Jb I cP = I P(JbCP + 2 tg T N(JbCP ,
( 12.33)
where
00
N:bCP =  f [( t - a ) a _ ( t - b ) a ] cp(t) dt, a = 1lasign.
" x-a x-b t-x
-00

 12. FRACTIONAL INTEGRALS OF THE POTENTIAL TYPE
221
12.3. The case of the half-axis
Potentials (12.1) and (12.2) may be considered on the half-axis as well:
00
1 01 - 1 f ep(t)dt
o ep - 2r(Q) cos(Q1I'/2) Ix _ tl l - OI ' x > 0,
o
( 12.34)
00
H OI 1 f sign (x - t) ()d
oep=2r(Q)sin(Q1I'/2) Ix-tp-Olept t,
o
x> 0,
( 12.35)
hence clearly
01 1 ( 01 0 )
10 = 2cos(Q1I'/2) /0+ + 1_ ,
H o Oi = 1 ( 101 _ 101 )
2 sin( Q1r /2) 0+ -'
Also relations of the type (12.6) and (12.7) are valid as well:
Hoep = -1 0 SOI/2ep = -S-0I/2 1 0ep,
( 12.36)
(12.37)
loep = H o S(l+0I)/2ep = S-(l+0I)/2 H Oep,
where the designation
00
S =! f ( !. ) 'Y ep(t)dt ,
'Yep 1r X t - x
o
x> 0,
( 12.38)
is used. Equations (12.36) and (12.37) may be obtained from (12.46) and (12.47),
proved below, by passing to the limit as b --+ 00. They may also be proved
independently in a fashion similar to the derivation of (12.46) and (12.47).
We prove now an important property of the Riesz potential 1 01 ep and the
integral HOIep: their represent ability by a composition of fractional integrals, which
is as follows.

222
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Theorem 12.4. Let 0 < a < 1 and rp(x) E Lp(R), 1 $ p < l/a. Then
[ 01 _ 1 01/2 [ 01/2
o cp - - 0+ rp,
( 12.39)
a-I 
Hgcp = 1 10+ rp,
( 12.40)
and more generally the formula
01  a - (3 OI +  . /3 - a OI + 
1 IfJ II') = cos -1rl fJII') + sm -1rH fJII')
- 0+ T 2 0 T 2 0 T
= sin a1r 101+13 sin /31r 101+13
sin( a + (3)1r - rp + sin( a + (3)1r + rp
(12.41)
is valid under assumptions that rp(x) E Lp(R), 1 $ p < 1/(a + (3), a + (3 < 1.
Proof. We shall prove (12.41), the equation (12.39) and (12.40) being obviously
obtained from (12.41). Interchanging the order of integration we have
00
1lg+rp = f(a)l f (/3) f J(x, y)rp(y)dy,
o
( 12.42)
where
00
J(x, y) = f (t - y)13-1 (t - x)0I- 1 dt.
max(z,y)
From equation (1.71) we obtain the relations J(x, y) = B(a, 1- a - /3)lx - yI0l+13-1
for x > y and J(x, y) = B({3,1 - a - /3)lx - yI0l+13-1 for x < y. Thus (12.41) is
derived from (12.42) by easy steps. .
We note finally that (7.11) and (7.12) yield the following relations for the
cosine- and sine-Fourier transforms (cf. (12.23) and (12.24»:
:Fc1grp = x-OI:F C rp, :F 8 Hgrp = X- 0I :F 8 rp.
( 12.43)
12.4. The case of a finite interval
In the case of a finite interval a $ x $ b operators similar to (12.1) and (12.2) will
be denoted by

 12. FRACTIONAL INTEGRALS OF THE POTENTIAL TYPE
223
b
A a - 1 f ep(t)dt
ep - 2r(a) cos(a1f'/2) Ix _ tp-a ' a < x < b,
o
( 12.44)
b
B a 1 f sign (x - t) ( )d b
ep ept t a<x<.
= 2r(a)sin(a1r/2) Ix - tl 1 - a '
o
( 12.45)
They admit relations analogous to (12.36) and (12.37). That is, the following
theorem holds:
Theorem 12.5. Let 0 < a < 1. Then
Aaep = B a S(a+l)/2ep = S_(a+l)/2 Ba ep,
( 12.46)
( 12.47)
Baep = _A a Sa/2ep = -S_a/2 Aa ep,
where, unlike (12.38),
b
S - 1 f [(t - a)(b - t»)'1 ep(t) d
'Yep - t,
1r[(x - a)(b - x)]'Y t - x
o
(12.48)
(12.46) and (12.47) being valid for example for functions ep(t) E L,(p), 1 < p < 00,
p(x) = (x - a)#J(b - X)II, where J.l,1I < p - 1 in (12.47) and in the first part of
(12.46), while J.l,1I < la p - 1 in the second part of (12.46).
Proof. Equations (12.46) and (12.47) are derived from the fundamental relations
(11.16)-(11.19). We shall illustrate this by proving the first of the equations in
(12.47) as an example. Replacing ep(x) by the function epl = COS a 2 1r ep + sin a 2 1r Sa/2ep
in (11.17) we have
a ( a1r . a1r S )
10+ cos Tep + sm T a/2ep
= I b _ (cos a1f' E - sin a1f'r;;a Srh) (cos a 2 1f' ep + sin a 2 1f' Sa/2ep) ,
where E is the identity operator and rbep = (b-x)ep(x). Applying here the Poincare-
Bertrand relation (11.9) and using (11.5) we derive by simple manipulations the

224
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
following symmetrical relation
1 01 ( a1r . a1r 8 ) 1 01 ( a1r . a1r 8 )
a+ cos -cp + sm - 0I/2CP = b COS -cP - sm - a/2CP .
2 2 - 2 2
Hence cos 0I 2 1r (1:+ - I b _)cp = - sin 0I 2 1r (1:+ + Ib_)801/2CP, which coincides with the
first of the identities in (12.47). By a similar way other relations are proved.
The above arguments are applicable to sufficiently good functions. Justification
of (12.46) and (12.47) for functions cP E L,(p) is easily obtained by means of
Theorem 1.7 provided we take into account the following: 1) the density of "good"
functions in L,(p); 2) the boundedness in Lp(p) of operators involved in (12.46)
and (12.47): this follows from Theorems 3.7 and 11.3, which show the boundedness
of the operators AOI, B OI and 8(01+1)/2 for J.l,1I E (p(a + 1)/2 - l,p - 1), of the
operators AOI, 8-(01+1)/2 and BOI for J.l,1I E (ap-l, p-l), and of the operators
B OI , AOI, 801/2 and 8-01/2 for J.l,1I E (ap - l,p - 1); 3) imbed dings in the spaces
L,(p), p(x) = (x - a)#J(b - x)/, in respect to parameters J.l and II. .
 13. Functions Representable by Fractional
Integrals on an Interval
This section adjoins S 6, where we characterized functions which are fractional
integrals of functions in L,(R1) We give here a similar characterization in the case
of an interval (and half-line) as well as a characterization of fractional integrals of
weighted Holder functions. The latter is obtained as a theorem on the mapping of
the space Ht(p) onto H;+OI(p) (Theorem 13.13).
As a preliminary we consider an analogue of the Marchaud fractional derivative
on an interval. We also call the reader's attention to the theorem in S 13.3 on the
continuation and sewing of fractional integrals. It is assumed everywhere below in
this section that 0 < a < 1. One may also admit 0 < Rea < 1.
13.1. The Marchaud fractional derivative on an interval
We now transform the Riemann-Liouville fractional derivative V<:+J to a form
similar to (5.57). For this purpose we consider the function J(x) to be at first
differentiable. Integrating in (2.24) by parts, we have
V':+f = r(I  a) { (/)a + !(z - Wad[f(t) - f(Z)]}
{ Z }
1 J(x) I . J(t) - J(x) J(x) - J(t) d
= + 1m + at.
r(I - a) (z - a)a t_< (z - t)a ! (z - t)'+a (13.1)

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 225
The middle term here vanishes for f(t) E C 1 , and we denote
x
D a - f(x) a f f(x) - f(t) dt
(J+f - r(1 - a)(x - a)a + r(l- a) (x - t)l+a '
(J
(13.2)
so that 'D+f == D+f for sufficiently good (differentiable) functions in view of
(13.1). The result in (13.2) may be called an analogue of the Marchaud fractional
derivative in the case of an intenJal [a, b], -00 < a < b  00.
One may come to (13.2) by another way if one continues the function f(x) by
zero beyond the interval [a, b] and applies the usual Marchaud fractional derivative
on the whole real line. Namely, let
f.(x) = { f(x), axb,
0, x rt. [a, b].
(13.3)
Then direct calculation shows that D+f. is for a < x < b exactly the same as the
right-hand side of (13.2), namely
(Df.)(x) = (D+f)(x), a < x < b.
(13.4)
In connection with (13.4) we observe that many results of this section, in
particular Theorems 13.1-13.4, may be deduced as corollaries of the corresponding
theorems in S 6 by means of (13.4). However we prefer to give their independent
and direct proof, avoiding continuation to the whole axis, since the questions of
fractional integra-differentiation on a finite interval are of more practical interest
in comparison with the case of the whole axis.
The right-hand sided Marchaud fractional derivative is introduced similarly,
thus
b
D a f - 1 f(x) a f f(x) - f(t) d
- + t,
b- r(l- a) (b - x)a r(l- a) (x - t)l+a
x
(13.5)
It is clear that the right-hand side of (13.2) is defined not only for differentiable
functions, but for example, for functions f(x), satisfying the Holder condition of
order A > a, as well.
Is it true that the Riemann-Liouville and Marchaud derivatives coincide with
each other for all functions, for which they are defined? We shall see below
(see Corollary of Theorem 13.1) that they coincide for functions representable by
fractional integrals of summable functions.
We emphasize that the integral in (13.2) will be understood, in general,
as conventionally convergent. Correspondingly let us introduce the truncated

226
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
fractional derivative similarly to (5.59):
a 1 J(x) a
D(J+.£J = f(1 _ a) (x - a)a + f(1 _ a) 1/J£(x),
(13.6)
where
:1:-£
/. ( x ) = f J(x) - J(t) dt
0/£ (x _ t)1+a ' C > O.
(J
(13.7)
Writing this down we assume that x  a + c. To introduce 1/J£(x) for a  x  a + c
we have to define the function J(t) for t < a. Two variants are possible:
1) to introduce 1/J£(x) for a  x  a + c by (13.7) considering J(x) to be
continued by zero beyond the interval [a, b]; then
:1:-£
1/J£(x) = J(x) f (x - t)-l-adt = J(x) [ ..!... _ ( 1) ] ,
a c a X - a a
(J
( 13.8)
a  x  a + c;
2) to set 1/J£(x) == 0 for a  x  a + c.
For functions J(x), which are not "very good", the fractional Marchaud
derivative (13.2) will be understood as
D+f = lim D+,J = r(1 f() ) + r(l'" ) lim ,p,(z),
£-0 - a x - a a - a £-0
(13.9)
where the mode of passage to a limit will be defined by the functions in hand.
In particular it will be in the norm of the space Lp when considering fractional
integrals of functions in Lp (see Theorem 13.1). Both variants of the definition of
the function 1/J£(x) for a  x  a + c will be used. We denote
1/J(x) = lim 1/J£(x).
£-0
(13.10)
We note that variant 1) to define the function 1/J£ (x) for a  x  a + c has the
advantage that it is connected with (13.4), i.e. the truncated fractional derivative
(13.6) coincides with Marchaud truncated derivative D+.£J.:
(D+,£J-)(x) = (D+.£J)(x), a < x < b,
(13.11)

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 227
where J.(x) is the same as in (13.3), provided that 1/Je(x) is defined for a  x  a+g
by the variant 1.
First of all let us verify that D+ is indeed the left inverse operator of the
operator of fractional integration within the frames of the spaces L,.
Theorem 13.1. Let J = I:+cp, cp E L,(a, b), -00 < a < b  00, 1  p < 00,
o < Q < 1. Then
D:+J = lim D:+ eJ = cpo
e-O '
(Lp)
Proof. Since
z-a z-a
1 f - 1 f -
J(x) - J(x - t) = r(Q) yO 1cp(x - y)dy - r(Q) (y - t)O 1cp(x - y)dy,
o t
using (6.10) we have
z-a
f(x) - J(x - t) = t o - 1 f k () cp(x - y)dy.
o
So for a + g  x  b we obtain
z-a y/e
t/>.(z) = f <p(z Y- y) dy f k(s)ds.
o y/(z-a)
Here the function K:(t) (see (6.12» arises, so
z-a
Q 1/Je(x) = f cp(x - y) [ !K: ( 11. ) - K: ( -1L )] dy.
r(1 - Q) g g X - a x - a
o
Since K: ( -1L ) = sinoll' ( -1L ) 0-1 by ( 6.7 ) we have
z-a 11' z-a I
Q
r(1 _ Q) ,pe(x) =
(z-a)/e
f J(x)
K:(y)cp(x - gy)dy - r(1 _ Q)(x _ a)o '
o

228
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
For convenience we consider the function rp( x) to be continued by zero beyond the
interval [a, b). Then in view of (6.8)
00
D+,£I - <p(x) = f K:(y)[rp(x - €y) - <p(x»)dy, a + €  X  b.
o
(13.12)
As for a  x  a + € from (13.8) we have
D a 1 ( ) I(x)
a+,£ - rp x = €ar(1 _ a)
z:
_ sin a1r f rp(x - t + a)dt
- 1r€a (t - a)l-a .
a
( 13.13)
Applying the generalized Minkowsky inequality (1.33), we obtain
00
IID+,£I - rpIlLp(a,b)  f K:(y)lIrp(x - €y) - rp(X)IILp(a,b)dy
o
1
+ €ar(l _ a) IIfIlLp(a,a+£).
(13.14)
The first term in the right-hand side tends to zero by Theorem 1.2 and Lemma
1.2. As for the second term, we have
a+£ { a+£ } ;
II/IILp(a,a+£) < sin a1r f dt f I (x - t + a) I P dx
€ar(1 - a) - 1r€a (t - a)l-a rp
a t
( 13.15)
 cllrpIlLp(a,a+£) --+ 0
as € --+ O. .
Corollary. For functions 1 = [:+rp, rp E L 1 (a, b) the Riemann-Liouville
derivative VC:+I and the Marchaud derivative D+I coincide almost everywhere:
(VC:+/)(x) == (D+/)(x) = rp(x).
Indeed, D+I = rp by Theorem 13.1 and V+f = rp by Theorem 2.1.
Since absolute continuity of a function f(x) provides that I(x) E [a(Ld (see
Corollary of Lemma 2.1), we obtain that the Riemann-Liouville and Marchaud
fractional derivatives coincide (almost everywhere) in particular for absolutely

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 229
continuous functions. In a non-direct form this assertion is already intrinsic in
(2.14).
Remark 13.1. In Theorem 13.1 we considered D+,£I to be defined for
a  x < a + € by introducing the function 1/J£(x) via (13.8). If we set 1/J£(x) == 0 for
a  x  a + €, Theorem 13.1 remains valid except for the case p = 1. The proof is
the same with the only difference that the second term in (13.14) is to be replaced
by r(la) II (!:o II ' which tends to zero for 1 < p < 00 only, see (5.46).
Lp(a,a+£)
13.2. Characterization of fractional integrals of functions
in L'P
In terms of the function 1/J£(x), defined by (13.7) and (13.8), we derive here, as in
S 6.3, the necessary and sufficient conditions for the represent ability of a function
I(x) by a fractional integral of a function in L,(a,b). The corresponding theorem
will be given in two variants.
Theorem 13.2. In order that a function I(x) be representable as 1 = P;+cp,
cp E L,(a, b), -00 < a < b < 00, where 0 < a < 1, 1 < p < 00, it is necessary and
sufficient that 1 E L,(a,b) and there exists lim1/J£(x) in L, where 1/J£(x) is the
£-0
(L p )
function (13.7)-(13.8), in the case 1  p < 00 these conditions being sufficient.
Proof. Necessity. Let 1 = I:+cp, cp E L'P. The necessity of the condition 1 E L, is
trivial. By Theorem 13.1 D+,£I converges to cp(x) in L'P(a, b), which was desired.
Sufficiency. Let 1 E L'P and lim 1/J£(x) = 1/J(x). We consider the function
£-0
1 I(x) a
cp£(x) = r(1 _ a) (x _ a)a + r(1 _ a) 1/J£(x).
( 13.16)
In view of (13.7) and (13.8) we can directly verify that CP£ E L'P( a, b). Since
the sequence {CP£} is fundamental we have CP£ (x )!:!.cp( x) E L'P' where cp( x) =
J :I: )  L h h 1 I a I . f h .. f
r(l-a (:I:-a)o + . et us s ow t at = a+CP. n VIew 0 t e contInUIty 0
the operator 1:+ in L,(a,b) it is sufficient to prove that 1 = lim P;+cp£. Taking
£-0
(13.7) and (13.8) into account we have
:I: a+£
I a sin a1r { f I(y)dy 1 f I(y)dy
cp -- +-
a+ £ - 11' (x - y)l-a(y - a)a €a (x - y)l-a
a+£ a
:I:
+af
a+£
1/-£
dy f f(y) - I(t) dt }
(x - y)l-a (y - t)l+a
a

230
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
in the case a + €  X  b. Hence by simple transformations we obtain
x x x
[a - sina1r {  1 f(y)dy _ a I t dt I dy }
0+1;'£ - 1r €a (x - y)l-a f( ) (x - y)l-a(y - t)l+a .
o 0+£ t+£
(13.17)
The result
b
I (y - a)IJ-l(b - y)"-l (b - a)IJ+II- 1 B(JJ, II)
dy-
(y - c)IJ+1I - (b - c)IJ(a - c)" '
o
( 13.18)
c < a < b, ReJJ > 0, Rell > 0,
is valid. This is proved by the change of variable y-c = (b-c)(a-c)[(b-a)t+a-c]-l.
Using this equation in (13.17) we find that
[a £ = sin a1r {I X f(y)dy _ I x -£ f(t)(x - € - t)a dt } .
0+1;' 1r€a (x - y)l-a X - t
o 0
Hence it is easily shown that
(x-O)/£
[:+1;'£ = I lC(t)f(x - €t)dt, a + €  X  b,
o
(13.19)
where lC(t) is the function (6.7).
As for the case a  x < a + €, by (13.8) we have
x
[a I;' - sin a1r I f(y)dy a  x < a + €.
0+ £ - 1r€a (x _ y)l-a'
o
( 13.20)
Having obtained the representations (13.19) and (13.20), it is now easily shown
similarly to transformations in (13.12)-(13.14) that [:+I;'£ f. This completes
the proof. .
Remark 13.2. If p = 1, then for a function f(x) = [:+1;', I;' E Ll(a, b) the
statement f(x)(x - a)-O E Ll(a, b) is in general untrue (see the counterexample
(5.41». In this case lim.,p£ f/. Ll in general, and the necessity condition does not
£-0

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 231
hold in Theorem 13.2 for p = 1. We may, however, include the case p = 1 into the
necessity part if instead of convergence in L 1 (a, b) of the sequence 1/J£ we require
L 1 (a, b)-convergence of the sequence (13.16). We introduce also a modification of
the function 1/J£(x). Namely we set
x-£
.7. ( x ) = f J(x)-J(t) dt a<x<b ,
'1"£ (x - t)1+a '
-00
(13.21)
assuming that the function J(x) is continued by zero beyond the interval [a, b]. It
is easily verified that £(x) == [a/r(1 - a)]-ll;'£(x), where I;'£(x) is the function
(13.16). So, if we replace 1/J£(x) by £(x) in Theorem 13.2, it would be valid for
1  p < 00, 0 < a < 1 both in the necessity and sufficiency parts taking Theorem
13.1 into account.
We recall that the case p = 1 was considered earlier when we gave the
characterization of fractional integrals 1:+1;', I;' E L 1 (a, b), in Theorem 2.1 in other
terms.
Note that one can formulate in an obvious way the analogue of Theorem 13.2
for right-hand sided fractional integrals I b _ 1;', directly following from Theorem 13.2
in view of (2.19).
Further, the following theorem similar to Theorem 6.2 is valid.
Theorem 13.3. In order that J(x) = 1:+1;', -00 < a < x < b < 00, with
I;' E L,(a,b), 1 < p < 00, it is necessary and sufficient that J(x) E L,,(a,b) and
sup IID+ £fll, < 00;
£>0 '
the latter being equivalent to the condition sup II£II" < 00, where £(x) as the
£>0
function (13.21). Besides
a -
I;'(x) = r(1 ) lim t/J£(x).
- a £-0
(L p )
The necessity is evidently contained in Theorem 13.2, while the sufficiency part
is obtained from the representations (13.19) and (13.20) by the same arguments
as those used in proving the sufficiency part of Theorem 6.2. - see considerations
following (6.24).
We note also a variant of Theorem 13.3 which is convenient, due the fact that
it uses information about the function 1/J£(x) for a + c  x  b only, and covers the
case of the half-axis.

232
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Theorem 13.4. In order that 1 = 1:+cp, -00 < a < x < b  00, where
cp E L,(a, b), 1 < p < 00, with b < 00 and 1 < p < 1/0: with b = 00 it is necessary
and sufficient that I(x)(x - a)-a E L,(a, b) and
b
sup f l1/Je(x)/" < 00.
e>O
a+e
( 13.22)
This theorem is obtained from direct analysis of the proof of Theorem 13.3.
Now we shall make use of the notations 1:+(L,) and 11:_(L,,) for the images
of fractional integration operators on L,. We established above (see Corollary 1 of
Theorem 11.4, Corollary 1 of Theorem 11.5 and (11.27)-(11.30» that
1:+(L,,) = 11:_(L,,), 1 < p < 1/0:,
where L, = L,(a, b), -00  a < b  00. Consequently, the characterizations of
the spaces 1:+(L,) and I b _(L,,) are equivalent in the case 1 < p < 1/0:. We shall
denote
Ia[L,(a, b)] = I:+(L,) = I{L(L,,), 1 < p < 1/0:.
( 13.23)
The space (13.23) becomes a Banach space if for / = I:+cp, cp E L,(a, b),
-00 < a < b < 00, we introduce equivalent norms similar to (6.17)
II /lIr: + [Lp(a.b)] = IIcpll" 1< "' < II/II, + Ulim o D:+,e/ll" 1< "' < II/II, + sup IID:+,elll".
_, 00 e- , 00 e>O
The space l a [L,(a,b)] has been characterized in Theorems 13.2 and 13.3. By
Theorem 13.2 it may be treated as a Sobolev type space of functions in L,(a, b)
which have a fractional derivative D+I E L,(a, b). In this connection we observe
that when dealing with fractional integra-differentiation, the space Ha"(a, b),
which is the restriction of Bessel potentials onto an interval [a, b] is often considered
in the literature. This will be discussed in S 18.4 below. Now we indicate only that
l a [L,(a, b)] = HQ'''(a, b), 1 < p < 1/0:,
in the case -00 < a < b < 00, see the proof in S 18.4.
Now we give a simple condition which is sufficient for representing a function

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 233
f( X) by a fractional integral of a function in L,. Let
{ b } 1/,
w,(f, h) = sup f If(x) - f(x - t)I'dx ,
Itl<h
a
(13.24 )
where it is assumed that f(x) is continued by zero beyond the interval [a, b).
b-a
Theorem 13.5. If f(x) E L,(a,b) and J t- 1 - a w p (f,t)dt < 00, then f(x) E
o
Ia[L,(a, b»), 1 < p < I/Ot.
This theorem follows directly from Theorem 13.4, since
if. 1t/>.(Z)I'dZ} II,  Trl-aw,U,t)dt.
A rather simple sufficient condition for a function f( x) to be represented by a
fractional integral of a function cp E L 1 (a, b) will be given in S 14.5 in terms of the
weighted absolute continuity of the function f(x).
One of the simplest conditions for a function I(x) to be a fractional integral
is the assumption that f(x) E H>' with A > Ot. Then the convergence of 1/Je(x)
is obvious and Theorem 13.2 is immediately applicable. Clearly this condition is
redundant: it gives f = I:+cp, where cp is not only in L" but also Holderian, see
Lemma 13.1 below. The following statement is of more interest.
Theorem 13.6. Let f(x) = (x - a)-#Jg(x), where g(x) E H>'([a,b», -00 < a <
b < 00, A > Ot, -Ot < JJ < 1. Then
I(D+,ef)(x)1  c/(x - a)#J+ a ,
where c does not depend on x and c. Besides f(x) E I:+(L,) if JJ + Q < IIp,
1  p < 00.
The proof follows on the same lines as that of Theorem 6.6, with the
corresponding simplifications due to the absence of the infinite point.
Corollary. If g(x) E H>'([a, b)), A > Ot, -Ot < JJ < -Ot + IIp, then (b - x)-#Jg(x) E
Ia[Lp(a,b»), 1 < p < I/Ot.
Indeed, it is sufficient to refer to the coincidence of the transforms (13.23).
Theorem 13.6 will be extended in the following subsection to the case of the
N
weights n Ix - al: l#Jk. - see Theorem 13.12.
1:=1

234
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
We observe that the assertion f(x) E I:+(L,) of Theorem 13.6 follows also
from the fact that (x - a) -IJ E 1:+ ( L,) with p(J.l + a) < 1 and from the following
theorem which is similar to Theorem 6.7.
Theorem 13.7. The space 1:+ [Lp(a, b)], 1  p < 00, 0 < a < 1, is invariant
relative to multiplication by functions a(x) E H>'([a, b]), A > a, and
lIafllr:+(L p )  cliallH" IIfllr:+(L p )'
This theorem is proved analogously to Theorem 6.7 with certain simplifications.
The following theorem is also valid.
Theorem 13.8. The space Ia[Lp(a, b)], 1 < p < 1/0:, is invariant relative to the
weighted singular operators
b
Saf = f ( x - a ) a f(t) dt,
t-a t-x
a
(13.25)
b a
Sbf = f ( b - x ) f(t) dt,
b-t t-x
a
a.e. if f = I:+,cp, cP E L" then also Saf = I:+1/J, 1/J E Lp, and similarly for Sbf.
The statement of this theorem follows from (11.18), (11.19) and (13.23).
13.3. Continuation, restriction and "sewing"
of fractional integrals
Let f(x) E I\[Lp( a, b)] on a given interval [a, b]. We put the following questions:
1) does the function f(x), continued by zero beyond the interval [a, b], belong to
IA+[L,(A, B)] on a larger interval [A, B] ::::> [a, b]? 2) does its restriction onto the
smaller interval [c, d] C [a, b] belong to I+[Lp(c, d)]? Since fractional integrals of
functions in L, are continuous functions in the case p > 1/0: and equal zero at one
end-point, the answers in general are negative for p > 1/0:. If P < 1/0: the answers
will be positive. - see Corollaries of Theorems 13.9 and 13.10. Theorems 13.9 and
13.10 will also lead to a very useful theorem on "sewing" of fractional integrals.
Theorem 13.9. Let f(x) E Ia(L p ) = [a[Lp(R 1 )], 1 < p < 1/0:, and let
-00  a < b  00. Then restriction of the function f(x) onto [a, b] belongs to the

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 235
,pace p;w[L,(a, b»):
J(x) = (1:+1/J)(x), x> a, 1/J(x) E L,(a,b),
( 13.26)
where
a
1/J(x) = I;'(x) + sin a 11' f ( a _ ) a I;'() , I;' = D+/.
11' x-a x-
-00
( 13.27)
Theorem 13.9 is a convenient paraphrase of Theorem 11.6, in which we showed
that the truncation of a function J(x) E [a[Lp(R 1 ») by zero beyond a given interval
[a, b) leaves the function in the space [a[Lp(R 1 »). Indeed, in the case x > 0 (11.36)
coincides with (13.26). Without any loss of generality we may take a = O.
Remark 13.3. Equation (13.26), i.e. the identity
(1+.1;')(x) = (1:+1/J)(x), x > a,
( 13.28)
with the choice of the function 1/J( x) by the rule (13.27) is valid not only for
I;' E L,(Rl) but evidently for functions I;'(x) E L,(-oo,b) as well, if a < x < b.
Corollary. Let I;'(x) E L,(c, b), c < a < b, 1 < p < l/a. The representation
(1:+I;')(x) = (I:+1/J(x), a < x < b,
is valid, where
a
1/J(x) = I;'(x) + sin a 11' f ( a -  ) a I;'( d E L,(a,b).
11' x-a X-,-
c
Indeed the corollary immediately follows from (13.28) if in (13.28) we choose
the function I;'(x) E L,( -00, b), which is equal to zero for x < c.
We emphasize that the corollary of Theorem 13.9 gives an answer to the well
known question in fractional calculus about the existence of a relation between the
fractional integrals [:+1;' and 1:+1;' with different lower limits of integration - see
the Historical Notes in S 17.1 (note to S 13.3).

236
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
The next theorem concerns the functions J(x) given on [a, b), which are
continued by zero beyond this interval. Let
J.(x) = { J(x), x E [a, b),
0, x f/. [a, b).
( 13.29)
Theorem 13.10. Let J(x) = 1:+ cp , a 5 x 5 b, where cp(x) E L,(a, b),
1 < p < I/Ot. Then
J.(x) = (I+cpd(x), x E R 1 ,
where CP1(X) E Lp(R 1 ) and
( 13.30)
! 0, x < a,
( ) cp( x ), a < x < b,
CP1 X = b
a J J(tdt - ( ) b
- r(l- a) a ( z - t 1 + or - 9 x, X > .
(13.31 )
Proof. We first show that g(x) E L,(b, 00). Indeed
b b
Ot sin Ot1r f d f dt
g(x) = - 1[' cp( r) r (x _ t)1+a(t _ r)1-a .
a T
(13.32)
The inner integral here is evaluated by means of the change of variable t =
x - (x - r)/ which gives
b
g(x) = _ sinOt1[' f ( b - r ) a cp(r)dr .
1[' x-b x-r
a
(13.33)
So g(x) E L,(b,oo) by the Hardy-Littlewood Theorem 1.5 on operators with a
homogeneous kernel. In the case considered transferring the point b to the origin
and reflecting onto the positive half-axis we obtain that the kernel k( x, t) in
Theorem 1.5 is (t/x)a(t + x)-l. It remains to check (13.30), which is evidently
valid for x 5 b, while for x > b we need to verify the equation
b z
o - f cp(t)dt f g(t)dt
- (x - t)1-a + (x - t)1-a'
a b

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 237
The latter is established directly: we have to substitute here get) from (13.33),
interchange the order of integration and then apply (11.4). The theorem is then
proved. .
Corollary. Let -00  a  c < d  b < 00. If f(x) E P:r[L,(c, d)], 1 < P < l/n,
then
. { f(x), x E [c,d] 01
f (x) = [ b] \ [ d] E [ [L,(a,b)].
0, x E a, c,
If f(x) E [OI[L,(a,b»), 1 < p < l/n, then
f(x )lzE[c,d] E [OI[L,( c, d)].
The corollary is obtained by direct application of Theorems 13.9 and 13.10
provided that the function f(x) is continued by zero beyond [a,b] or [c,d].
Theorem 13.11. (on "sewing"). Let functions hex) and 12(x) be given on [a, c]
and [c, b] respectively, -00  a < c < b  00 and let
f(x) = { hex), a  x  c,
12(x), c < x  b.
[f h(x) E [01 [L,(a, c)], f2(X) E [OI[L,(c,b)], 1 < p < l/n, then f(x) E [OI[Lp(a,b)].
This theorem is easily derived from the above corollary since f( x) =
fi (x) + f2 (x), where f; (x) is the continuation by zero of the functions fA: (x)
beyond the intervals where they are defined.
Now we demonstrate an application of Theorem 13.10.
Theorem 13.12. Let -00 < a = Xl < X2 < ... < X n -l < X n = b < 00. Then
f(x) 01
g(x) = n E [ [L,(a, b)], 1 < p < l/n,
n Ix - xl:l#k
1:=1
(13.34 )
if f(x) E H>'([XI:,XI:+l]), k = 1,2,...,n - 1, A > n, and p(PI: + n) < 1,
k = 1,2,. . . , n.
Proof. For x E [x 1:, x 1:+ 1] the function g( x) has the form g( x) = (x-
XI:)-#k91(X) + (Xl:+l - X)-#k+1g 2 (X), where gi(X) E H>'([Xt,XI:+l]), i = 1,2. Then
g(X)lzE[Zk,Zk+l] E [01 [L,(xl: , xl:+d] by Theorem 13.6 and its corollary. Therefore
g(x) E [OI[L,(a, b)] as well, by the Theorem 13.11 on sewing. .

238
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
13.4. Characterization of fractional integrals of Holderian
functions
We continue now the investigations started in S 3.1 and 3.2, where it was shown
that fractional integration improves the Holderian properties of functions by an
order Q (see Theorems 3.1-3.3). In this subsection we shall prove the statements
of the converse character by showing that fractional derivatives are defined on all
Holderian functions of order A > Q (with a weight) and are themselves Holderian
functions of order A - Q. After this we shall prove the main assertion of this
subsection which states that fractional integration realizes an isomorphism, i.e. a
continuous one-to-one mapping, of the weighted Holder space H(p) onto H+OI(p).
Namely, let
n
p(x) = II Ix - xkl Jlk .
k=l
(13.35 )
-00 < a  Xl < X2, . . . < X n  b < 00.
The following theorem holds.
Theorem 13.13. Let p(x) be a weight (13.35) with Xl = a. If
A + Q < 1, 0  ILl < A + 1,
A + Q < ILk < A + 1, k = 2, . . . , n,
( 13.36)
then the operator I+ isomorphically maps the space H(p) onto the space H+OI(p).
A similar assertion is valid for the operator I) I if X n = band
A + Q < 1, 0  Iln < A + 1,
A + Q < Ilk < A + 1, k = 1,2,.. . ,n - 1.
( 13.37)
The proof of the theorem is comparatively easy in the case of the weight
p(x) = (x - a)Jl and rather cumbersome for an arbitrary weight of the form (13.35).
To make the presentation simpler we split it up into several stages, providing a
series of intermediate lemmas. The theorem will be first proved for the weight
p(x) = (x - a)Jl.
We recall that the space H(p) consists of functions f E H>'(p) such that
p(x)f(x) equals zero at the points Xk, k = 1,2,..., n.

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 239
Lemma 13.1. If f(x) E H>'([a, b]), a < A  I, then
o f(a) 1
(Da+f)(x) = r(1 _ a) (x _ a)o + 1/J(x) ,
where 1/J(x) E H>'-O([a,b)) and 1/J(a) = 0, besides that 111/JIIH"-or  cllfllH'"
Proof. In view of (13.2) and (1.16) it is sufficient to show that
x-a
1/Jl (x) = f t-1-0[f(x) - f(x - t)]dt E H>'-o.
o
We have
x-a
1/Jl(X + h) -1/Jl(X) = f [f(x) - f(x - t)][(t + h)-O-l - t-O-l]dt
o
o
f f( x + h) - f( X - t) dt
+ (t + h)1+o
-h
( 13.38)
x-a
f f(x+h)-f(x)
+ (t + h)1+o dt = It + 1 2 + 13.
o
It is easily seen that
00
lIt I  c f t>'l(t + h)-o-l - t-O-lldt = clh>'-o,
o
00
where Cl = c J t>'l(t + 1)-0-1 - t-O-lldt < 00, and
o
o
11 2 1  c f (t + h)>,-o-ldt = C2 h >'-0,
-h
00
11 3 1  ch>' f(t + h)-O-ldt = C3 h >'-0.
o

240
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
It remains to note that 1Pt(a) = 0, which follows from the estimate l1Pl(X)1 =:;
z-:o
C f t>'-0I-1dt. .
o
By means of Lemma 13.1 we shall consider the case A + a > 1 in Theorem
3.1 which remained unproved until now. Let f E I:+cp, cp E H>'([a, b]), 0 < A =:; 1,
o < a < 1, A + a > 1. It is sufficient to consider the case cp( a) = O. In
correspondence with Definition 1.6 we have to show that (d/dx)I:+cp E H>'+0I-1,
i.e. that 1)+OIcp E H>'-(1-OI). Since A > 1 - a we see that cp(x) is representable by
a fractional integral of order 1 - a in view of Theorem 13.5. Thus the Riemann-
Liouville fractional derivative 1)+0I cp coincides with the Marchaud derivative - see
Corollary of Theorem 13.1. Since cp(a) = 0, Lemma 13.1 yields what is required.
Our next step is an extension of Lemma 13.1 to the case of weighted Holderian
functions.
Lemma 13.2. Every function f(x) E H;+OI(p), 0 < A < 1, 0 < A + a < 1, p(x) =
(x - a)IJ, 0 =:; p. < A + 1, is representable by a fractional integral f = I:+cp, where
cp E H8(p) and
IIcpIlH;(P)  cllfIIH;+o(p)'
( 13.39)
Proof. Employing Theorem 13.4 we shall show first that f = I:+cp with cp E L,
for some p > 1. Then cp = D+f by Theorem 13.1 and (13.39) may be verified
directly. Let us denote g(x) = (x - a)IJ f(x) E H;+OI. Verifying the assumptions
of Theorem 13.4 we see that f(x)/(x - a)OI = g(x)/(x - a)OI+IJ E Lp(a,b) provided
that p(p. - A) < 1. Further, for a + c  x  b we have
z-o
1/Je(x) - 1 f g(x)-g(x-t) dt
(x - a)IJ t 1 +01
e
z-o
f [ 1 1 j g(X-t) d
+ (x - a)IJ - (x - a - t)IJ t 1 +01 t
e
=Ae(x) + Be(x).
Since Ig(x) - g(x - t)1  clt>'+01 and Ig(x - t)1  C2(X - t - a)>'+01 we obtain the
inequalities
IAe(x)1  C3(X - a)>'-IJ,
IB£(x)1  C4(X - a)>'-IJ

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 241
where the constants
1
C3 = C1 f t>'-ldt,
o
1
C4 = C2 f [(1 - t)-IJ - 1](1 - t)>'- IJ t- 1 - a dt
o
do not depend on €. Therefore (13.22) is satisfied if p(J.l - A) < 1, thus f = I:+tp,
tp E L,. It remains to verify (13.39). The function tp(x) has the form (13.2).
The first term in (13.2) is in H6(p) and admits an estimate of the form (13.39)
in view of the property (1.16) of Holderian functions. It remains to show that
z-a
1/J(x) = f J(z)!z-t) dt E H8(p) and that 111/JIIH"(p) $ cllfIlH"+ar(p). We set a = 0
o
for the sake of simplicity. Let us denote 1/Jo(x) = x IJ 1/J(x). To estimate the difference
1/Jo(x + h) - 1/Jo(x) we represent it in the form
8
1/Jo(x + h) -1/Jo(x) = L AI:(x),
1:=1
where
z+h
A (x ) = [ 1- (  ) IJ ] f g(x + h) - g(y) d
1 X + h (x + h - y)1 +a y,
o
z+h
f (x + h)-IJ - y-IJ
A2(X) = [(x + h)IJ - x IJ ] (x + h _ y)1+a g(y)dy,
o
z+h
A3(X) = (  ) IJ f g(x + h) - g(y) dy,
x+h (x+h-y)1+a
z
z+h
f (x + h)-IJ - y-IJ
A4(X) = x IJ (x + h _ y)1+a g(y)dy,
o
z
A5(X) = f [g(x) - g(y)n(x + h - y)-l-a - (x - y)-l-a]dy,
o
z
A6(X) = x IJ f g(y)(x-IJ - y-IJ)[(x + h - y)-l-a - (x - y)-l-a]dy,
o
1 ( X ) IJ
A7(X) =; x + h [g(x + h) - g(x»)[h- a - (x + h)-a],

242
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
As(x) = a-IxIJg(X)[(X + h)-IJ - x-IJ][h- a - (x + h)-a].
Estimation of everyone of these terms leads to the inequality
l1/Jo(x + h) -1/Jo(x)1  cll/llh"+O(p)h>'.
We omit the calculations which are not difficult but take too much space; they are
to a great extent similar to the steps in the proof of Theorem 3.3. The condition
1/J(a) = 0 is easily checked. The lemma is thus proved. .
The following corollary results from Lemma 13.2 and the corollary of Theorem
13.1.
Corollary. The fractional Riemann-Liouville and Marchaud derivatives coincide
lor functions I(x) E H+a(p) under the assumptions of Lemma 13.2.
Comparison of Lemma 13.2 with Theorem 3.3 shows that Theorem 13.13 is
proved for the weight p( x) = (x - a)IJ .
The transformation a + b - x -+ X in Lemma 13.2 leads to the following lemma.
Lemma 13.2'. Each function I(x) E H+a(p), p(x) = (b - x)IJ, 0 < A + a < I,
o  Il < A + 1, is representable as I = Ib_cp where cp(x) E H(p) and IIcpIlH"(p) 
cIl/IlH"+O(p) .
Before we extend Lemma 13.2 to the case of an arbitrary weight of the form
(13.35), we prove two auxiliary assertions.
Assertion 1. If I = Ib_cp, where cP E H(p), A + a < 1, A + a < III < A + 1,
A < Ilk < A+l, k = 2,3,...,n, p(x) is a weight (13.35) with Xl = a, then
1= I:+1/J, 1/J E H(p) and 111/JIIH"(p)  cllcpIlH"(p)'
This assertion follows immediately from (11.16) if we take into account
boundedness of the singular operator S in the space H(p) (see Theorem 11.1).
Besides 1/J = cos a1rcp + sin a1rr;a Srcp.
Assertion 2. (On zero continuation of fractional integrals). Let I(x) = I:j+cp
on the interval [Xj,Xj+I], where cp E H(rj) on [Xj,Xj+l], rj(x) = Ix - xjlIJjlx-
xj+IIIJJ+l, j = 1,2, . . . ,n. Then the function
J.(X) = { J(x),
0,
Xj  x  Xj+l,
a  x < Xj, Xj+l < X  b,
is representable as J.(x) = I:+CPl, where CPI (x) E H(p) on [a, b] with the weight
(13.35) if 0  Illc < A + 1, k = 1,.. ., j - I, A + a < JLj+l < A + I, A < Ilk < A + I,
k = j,j + 2,..., n, p(x); besides
IIcpdIH"(p)  cIlCPIlH"(rj)'
( 13.40)

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 243
A similar assertion has already been proved for the case of fractional integrals
of Lp-functions. - see Theorem 13.10. Assertion 2 is proved following the same
lines as in the proof of Theorem 13.10. We only indicate that similarly to (13.31)
! 0,
cp( x ) ,
CPl(X) = Xj+l
- r(la) L f(t)(x - t)-l-adt,
x < Xj,
Xj < x < Xj+l,
x > Xj+l,
and it is necessary to use Theorem 11.1 while verifying (13.40).
Lemma 13.3. Each function f(x) E H;+a(p), A + a < 1, where
n
p(x) = II Ix - Xt IIJk, a = Xl < .. . < X n  b,
k=l
0JJ1<A+l, A+a<JJk<A+l, k=2,...,n,
is representable as f = I:+cp with cp(x) E H6(p) and IIcpIlH"(p)  cllfIlH"+cr(p).
Proof. Let c to be an arbitrary point in the interval (a, X2). Similarly to the proof
of Theorem 3.3' we set f(x) = h(x) + f2(X),
h(x) = { f(x), x  c,
f(c), x  c,
( ) { 0, x  c,
J2x=
f(x) - f(c), x  c,
(13.41)
so that h(x) E H;+a(Pa), J2(x) E H;+a(po), and
IIhIlH"+cr(p.)  cllfIlH"+cr(p),
IIf2I1H"+cr(po)  cllfIlH"+cr(p),
( 13.42)
n
where we denote Pa(x) = (x - a)IJ1, po(x) = (x - a)>'+£ TIlx - xkl IJk , 0 < g < 1.
t=2
We note that introduction of the functions (13.41) is in a sense the main
factor in the proof both of Theorem 3.3' and of the present theorem: it allows us

244
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
to separate the point Xl = a, which is the lower limit of integration, from the other
points Xi, k = 2,3,..., n, where singularities are admitted.
Everywhere below Xn+l = b in the case X n < b.
By Lemma 13.2 the representation h{x) = I:+CPl is valid with CPl(X) E
H6(Pa) C H6(p) and
IICPIIIHA(p)  CIlCPIIIHA(p.)  cllhIlHA+ar(p.)
 cII/IIH:+ar(p)'
To consider the function 12{X), we introduce the following notation
rl:(x) = Ix-xl:l,6ktx-xl:+d,6k+l, k = 1,...,n-l;
PI:{X) = Ix-xl:l,6k, k= 1,...,n,
where P1 =  + €, € > a, PI: = PI:, k = 2,... ,no Let us show that
J2(X) = (I:k+.,pk)(x), XI:  X  XI:+1,
( 13.43)
where ,pI: E H6(rl:) on [XI:, XI:+d, k = 1,2,..., n - 1, .,pn E H6{Pn) on [xn, b] (if
X n < b) and
1I.,pI:IIHA(rk)  cIl/2I1HA+ar(rk)'
lI.,pnIlHA(p,.)  cllf2I1HA+ar(p,.).
For f2(x) on [xn, b] this fact follows from Lemma 13.2. For the intervals
[XI:, XI:+l], k = 1,. . . , n - 1, let us arbitrary choose a point CI: E (XI:, XI:+1) and set
f ( ) j. (l) ( ) j. (2) { ) h
2 X = 2,1: X + 2,1: X , were
(13.44)
(1) { ) _ { f2{x), XI:  X  Ck,
1 2 I: X -
, h(Ck), Ck  X  XI:+1,
(2) { ) _ { 0, XI:  X  Ck,
1 2 I: X -
, 12(X) - 12(CI:), Ck  X  XI:+1'
Since IJ2{x) E H+OI{pl:) on [xl:,xl:+d we have 1.12{x) = I:k+.,pl) by Lemma
13.2, where .,pl){x) E H6{plc) C H6{rl:), and
1I.,p1)IIHA(rk)  cll/t2I1HA(Pk)  cllf2I1HA+ar(rk)'
(13.45 )

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 245
Further, since Ji(x) E Ht+Ot(pl:+d on [XI:,XI:+1] we obtain Ji(x) = I:Ic+l_1/Ji 2 )
by Lemma 13.2', where 1/Ji 2 )(x) E H(pl:+d C HS(rl:), and
111/J2) IIH" (rlc)  cIlJi IIH"+O(PIc+l)
 CIlJ2I1H"+O(rlc)'
(13.46)
Applying Assertion 1 (see above) to the function Ji we find that J = I:Ic+ ;P2),
where ;P2) E H(rl:) on [XI:, xl:+d and
1I;P2)IIH"(rlc)  clltPi 2 )IIH"(rlc)'
( 13.47)
Thus we arrive at (13.43) with tPl: = tPl) + .J;2). The norm inequality (13.44) is an
immediate consequence of the estimates (13.45)-(13.47).
Further, in view of Assertion 2 the function 1 2 1:(x), which is the continuation
,
by zero of the function h (x) beyond the interval [x I: , x k+ 1], is representable by a
fractional integral I; I: = I:+c{)I:, where c{)1: E HS(po) on [a, b] and
J
1Ic{)I:\lH"(po)  constlltPl:\lH"(rlc)' k = 1,..., n - 1;
\lc{)n\lH"(po)  const IItPI:IIH"(p,,).
( 13.48)
n
Hence 12 = I:+tp2 with tp2 = L: c{)1:(x) E H(po) and by (13.48), (13.44) and
1:=1
(13.42) we obtain the inequality
IItp2\1H"(po)  const II/\lH"+o(p)'
Let us show now that tp2(X) E H(p) on [a, b] and
\ltp2I1H"(p)  const IItp2\1H"(po)'
(13.49)
Noting that 12(x) == 0 for x  c and recalling the equation tp2(X) = D+/2 we
see that tp2(X) == 0 for x  c (see (13.2) for the Marchaud fractional derivative).
Hence (13.49) follows, which in view of (13.48), leads in its turn to the inequality
\ltp2\1H"(p)  constll/\lH"+o(p). The lemma is thus proved.
The statement of Theorem 13.13 follows now from Theorem 3.4 and Lemma
13.3. .

246
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
13.5. Fractional integration in the union of weighted Holder
spaces
Theorem 13.13 characterizes mapping properties of fractional integration in Holder
spaces with a power weight in the case when the Holderian exponent A and
exponents of the weight are fixed. Besides this, another result is of interest in
applications which reveals mapping properties of fractional integration in spaces of
Holderian functions with non-fixed exponents, i.e. in the union of all spaces H>'(p).
We mean unification both in respect to A and to weight. Let us give corresponding
notations for such unions.
By H- = H-(a, b) we denote the space of functions which are Holderian (of any
order) in the open interval (a, b) and have integral singularities at the end points
of the interval. More exactly, this space is defined by the following definition.
Definition 13.1. The space H- = H-(a, b) is a set of all functions f(x), for which
there exist numbers A, 0 < A  1, and C1 > 0, C2 > 0 such that
f-(x)
f(x) = (x _ a)1-£I(b _ X)1-£2 '
( 13.50)
where f-(x) E H>'([a, b]).
The space H- is well known as a class of functions widely used in the theory
of singular integral equations - Muskhelishvili [1] - and is sometimes called the
Muskhelishvili class.
For convenience we introduce also the notation
H;(£1, £2) = {f : f(x) = (x - aYl-1(b - XY2-1g(x),
g(x) E H\[a,b]), g(a) = g(b) = O}
(13.51)
for the familiar weighted Holder space with fixed exponents. By simple arguments
the equation
H- = U H;(£1,£2) = U H;(£1,£2),
0<>.<1 0<>'<>'0
£i>O O<£i<d(>')
( 13.52)
is established, where 0 < AO  1 and d(A) is an arbitrary positive number.
We shall need also the following union of functions, which are Holderian of
order more than a:
H: = U H;(£1,£2) = U H(£1'£2),
a<>.<1 a<>.<>.o
£i>O O<£i<-d(>')
( 13.53)

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 247
where a < AO < 1, d(A) > O.
We shall use the spaces H* and H: in S 30 concerning applications to integral
equations of the first kind. We shall give now the equivalent characterization of
the space H: and show that fractional integration maps H* onto H:. First we
need to introduce auxiliary spaces by the following definition.
Definition 13.2. We say that f(x) E Ho, if f(x) E C([a, b]) and f(x) is Holderian
of order A > a beyond the end points x = a and x = b:
If(x) - f(x + h)1  c(x)lhl\ A > a, x, x + h E (a, b),
( 13.54)
where c(x) may grow as x --+ a, x --+ b, but so that c(x)  const(x-a)-O(b-x)-o.
Definition 13.3. We say that f(x) E; H 0' if f(x) is Holderian of order A > a
within the open interval (a , b) and for x --+ a has the form
f(x) = f(a) + g(x)(x - a)-O,
( 13.55)
where g(x) E H>'([a, b]), A > a, and g(a) = 0, similar behaviour being assumed for
x --+ b.
Lemma 13.4. The spaces H 0 and H 0 coincide with each other.
I
Proof. . Let f(x) E H o. Taking a = 0 and f(a) = 0 we find for f(x) =
g(x)(x - a)-O with 9 E H>' that
If(x) - f(x + h)1 (x + h)-Olg(x + h) - g(x)1
+ Ig(x)lx-O(x + h)-O[(x + h)O - X O ].
Hence the estimate If (x + h) - f(x)1  cx-olhl>' is obtained by simple steps if
we take into accout that Ig( x) I  ex>'. Similarly the required estimate is obtained
for x --+ b, so that Ho  Ho. Conversely, let f(x) E Ho. Then for x --+ 0 we have
If(x) - f(x + h)1  cx-olhl>'.
( 13.56)
Letting here h --+ -x, we have
If(x) - f(O)1  cx>'-o.
( 13.57)

248
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Then
g(x) = xa[f(x) - f(O)] E H>'([a, b]). (13.58)
Indeed, g(x + h) - g(x) = [(x + h)a - xa][f(x + h) - f(O)] + [f(x) + h) - f(x)]x a .
Hence by (13.56) and (13.57) we obtain the inequality Ig(x + h) - g(x)1 =:;
c(x + h)>.-al(x + h)a - xal + ch\ h > 0, which gives the assertion (13.58) after
simple estimates. This rtion together with a similar consideration of the case
x -+ b shows that Ha C H a' .
We shall give in terms of the space H a = H a an equivalent characterization of
the space (13.53). The following lemma is valid.
Lemma 13.5. The space H: consists of functions of the form
f*(x)
f(x) = (x _ a)1-0-£1 (6 - X )1-0-£ '
(13.59)
where 0 < £1 < 1 - a, 0 < £2 < 1 - a and f*(x) E HO/.
The roof is obtained by direct verification on the basis of the definition of
the space H a.
Finally, the mapping properties of the operators I+, If_ of fractional
integration in the space H* is thoroughly characterized by the following theorem.
Theorem 13.14. The fractional integration operator of order a, 0 < a < 1, maps
H* one-to-one onto the space H::
I:+(H*) = If_(H*) = H.
(13.60 )
Proof. By Theorem 13.13 we have
1:+ [H; (£1 ,£2)] = H;+O/(£I' £2),
provided that A + a < 1, 0 < £i < 1 - a - A, i = 1,2. Then
I+[
u
H;(£l' £2)] = U Ht;(£I, £2),
0<>'<1-0/
O<£i<l-a->'
0/<14- 1
O<£i<I<14
which in view of (13.52) and (13.53) is nothing else but equation (13.60) for the
operator I+. By similar arguments the case of the operator If_ is considered.

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 249
13.6. Fractional integrals and derivatives of functions with a
prescribed continuity modulus
The investigation carried out in S 13.4 can be significantly developed if, instead of
Holderian functions, we admit spaces of functions such that
de!
w(cp, h) = sup sup Icp(x + t) - cp(x)) I  cw(h),
O<t<h z,z+tE[a,b]
where w(h) is a given continuous increasing function, with w(O) = O. We denote
the space of all such functions by HW = llW([a, b]) and equip it with the norm
IIcpllHw = IIcplic + supw(cp, h)/w(h). By H'O we denote the subspace in H W , which
h>O
consists of functions equal to zero for x = a. The space H>' of usual Holderian
functions corresponds to the power case w(t) = t>'. The function w(t) is sometimes
called a characteristic function or the characteristic of a generalized Holderian
space H W .
What are the mapping properties of the fractional integration in the space
H W ? Theorem 13.13 states that I:+(H>' + 0) = H+OI, A + a < 1. May one obtain
a similar assertion
1 01 ( H W ) - HWOI
a+ 0 - 0
(13.61)
with a priori assumption that wOl(t) = tOlw(t), and for what kind of characteristic
w(t) is (13.61) true? We shall answer these questions and outline the more general
weighted situation as well.
The idea of an almost decreasing function f(t) which is used below, means
that f(td  cf2(t2) for all t1  t2, where c does not depend on t 1 and t2' Similarly
an almost increasing function is defined.
The following two theorems give estimates which might be called Zygmund
types by analogy with the Zygmund estimate known in the theory of singular
integrals and estimating the continuity modulus w(H cp, 11) of a conjugate function
Hcp. - see (19.22). - via the continuity modulus w(cp, h) of a function cp(x) itself.
- see for example Bari and Stechkin [1].
Theorem 13.15. Let cp(x) be continuous on [a,b] and let cp(a) = O.
fractional integral I:+cp, 0 < a < 1, the estimate
For a
b-a
( 01 h) h f w(CP,t) d
w la+CP, $ C fl-OI t.
h
( 13.62)
is valid.

250
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Proof. We use (3.4) for the difference f(x + h) - f(x) of the function f(x) =
I:+cp. Let us estimate the summands J 1 , J 2 and J 3 in (3.4). We have:
Ih I  cw(cp, x - a)l(x + h - aY - (x - a)exl. In the case x - a  h we have
Ih 1  chexw(cp, h). Let x - a  h. Then
IJ,I:S; cw(cp,z - a)(z - at [(1 + z: a f -1]
w(cp,x - a) h
< c )1 .
- (x-a -ex
( 13.63)
Since
b-a b-a b-a
f w(cp, t)dt > f w(cp, t)dt > W (cp x - a ) f 
t 2 -ex - t 2 -ex -, t 2 -ex
h x-a x-a
w(cp, x - a)
>c
- ( ) 1-ex '
x-a
b-a
it follows from (13.63) that Ih 1  ch J W:..tldt . Further,
h
h h
IJ 2 1  f(h - t)ex- 1 Icp(x + t) - cp(x)ldt  f(h - t)ex- 1 w(cp, t)dt
o 0
( 13.64)
1
ex f w( cp, h) ex ( )
=h (1_)1_exdchwcp,h.
o
1
with  = J(1- )ex-1. To estimate J 3 we distinguish the cases 1) x - a  hand
o
2) x - a  h. In the first case
1 x;;: °
IJ 3 1  hex f It'x-1 - (t + l)ex- 1 I w (cp, th)dt + ch cx f tCX- 2 w(cp, th)dt
o 1
b-a
 chexw(cp,'h) + ch f tex- 2 w(cp, t)dt.
h

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 251
Obviously in the second case
IJ31 $ chaw(cp, h).
(13.65)
Estimates for J l , J2, J 3 lead to (13.62) if we take into account the fact that
haw(cp, h) is dominated by the right-hand side of (13.62). The latter is easily
obtained in view of the monotonicity of the function w( cp, t). .
Theorem 13.16. Let J(x) be continuous on [a, b] and J(a) = O.
Jractional derivative D+J, 0 < a < I, admits the estimate
Then its
h
( D a J ) f w(J,t)
w a+,h c t l + a dt
o
( 13.66)
provided that the integral on the right-hand side converges.
Proof. We begin by noting that the function F(x) = J:-a)Sa) , 0 < a < 1, admits
the estimate
h
f w(f, t)
w(F, h)  c t 1 + a dt.
o
(13.67)
Let us prove (13.67). Taking h > 0, we have F(x + h) - F(x) = [J(x) -
J(a)][(x + h - a)-a - (x - a)-a] + (x + h - a)-a[f(x + h) - J(x)] = Al + A 2 . Hence
h
IA21 $ (x + h - a)-aw(J, h)  h-aw(J, h)  c J t-l-aw(J, t)dt; - here in the last
o
inequality we made use of the fact that the function t-lw(J, t) almost decreases,
see e.g. Guseinov and Muhtarov [1, p.50]. Further, again taking this decreasing
into account, for Al when x - a  h we have
IAll $ (x - a)-aw(J,x - a)
z-a
 c f t-l-aw(J, t)dt
o
h
 c f t-l-aw(J, t)dt.
o
When x - a  h, the mean value theorem yields the estimate IAll  ch(x -
h
a)-l-aw(J, x - a) $ ch-aw(J, h) $ c J t-l-aw(J, t)dt. Gathering estimates for Al
o
and A 2 we obtain the inequality (13.67).

252
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
To prove the theorem, it is sufficient in view of (13.67) to consider only the
second summand in the expression (13.2) for the Marchaud fractional derivative,
i.e. the function (13.10). For this function we have
r(l- Q) [,p(x + h) - ,p(x)]
Q
z-a
= / [/(x + h) - /(x + h - t) - /(x) + /(x - t)]t-I-adt
o
+ %7-"[I{'" + h) - I{'" + h - t)]t-1-adt = Bl + B 2 .
z-a
z-a h
If x - a $ h, then IBII $ 2 J w(/, t)t-I-adt $ 2 J t-I-aw(/, t)dt. If x - a  h,
o 0
we have
h x-a
I B 1 < 2 / w(J,t)dt 2 / w(J,h) dt
1 - t 1 + a + t 1 + a
o h
h
$ 2 / t-1-aw(/, t)dt + 2Q-l h-aw(/, h)
o
h
$ (2 + 4/Q) / t-1-aw(/, t)dt.
o
z+h-a
As for B2' we have IB21 $ J t-1-Otw(J, t)dt. If x - a $ h then IB21 $
x-a
2h h
J t-1-aw(/, t)dt $ 2 1 - a J t-1-Otw(/, t)dt. If x - a  h, after the substitution
o 0
t =  + x - a and taking into account the (almost) decreasing nature of the function
t-1w(/, t), we have
h
IB 2 1 < / w(/,x - a +) 
- (x - a + )l+Ot
o
h
w(/, h) / d w(J, h)
<c- h ( C ) <c- h .
- x-a+,Ot- Ot
o

 13. FUNCTIONS REPRESENTABLE BY FRACTIONAL INTEGRALS 253
'Gathering estimates for Bl and B 2 we arrive at (13.66). The theorem is thus
proved. .
To formulate an assertion of the form (13.61) we shall introduce a space of
functions in terms of which we shall give conditions for the admissible characteristic
function w(t).
Definition 13.4. We say that
w(t)E, pO, 60,
( 13.68)
if
1) w(t) is continuous on [0, b - a], w(O) = 0 and w(t) almost increases;
t ( ) 6 
2) f t w    cw(t),
o
b-a ( ) /J
3) f t wn d  cw(t)
t
The space  may be called a two-parameter space of Bari-Stechkin type
(compare with Bari-Stechkin class /J' see e.g. Guseinov and Muhtarov [1], p.78).
It may be shown that the space  is empty for 6  /3, so we assume that
o < 6 < p.
Theorem 13.17. Let 0 < a < 1 and w(t) E -a. Then the operator 1:+
maps the space H/f isomorphically onto the space HOl with the characteristic
wa(t) = taw(t).
Theorem 13.17 is deduced from the Zygmund type estimates used in Theorems
13.15 and 13.16 and from the fact that the functions f E HOl under the above
assumptions on w(t) are representable by fractional integrals f = I:+tp of functions
tp E H/f. The latter is proved by means of Theorem 13.2 or 13.3. The proof is
easy and is left to the reader. Consider also the similar Theorem 19.8 concerning
the periodic case where such representability in an analogous situation is shown in
more detail.
Finally we observe that there is an extension of the above results to the case
of weighted spaces. Namely, let H/f(p) be a space of functions f(x) such that
p(x)f(x) E H, IIfIlH(p) = IIpfIlH'
For p(x) = (x - a)#J, 0  JJ < 2 - a, the Zygmund type estimate (13.62) in the
case of functions tp(x), such that p(x)tp(x) satisfies the assumptions of Theorem
13.14, is replaced by the estimate
h b-a
W (p l a l/'J h ) < cha+'Y- 1 f w(ptp, t)dt + ch f w(ptp, t) dt
a+ r , - t'Y t 2 - a '
o h
( 13.69)
where l' max( 1, JJ). The following theorem is also valid.

254
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Theorem 13.18. Let 0 < a < 1, p(x) = (x - a)#J, 0  p. < 2 - Q. If
w(t) E t-a with 6 = max(p. - 1,0), then the operator 1:+ maps the space
H:J (p) isomorphically onto the space HOI (p) with the same weight and with the
characteristic wa(t) = taw(t):
I:+[H:J(p)] = HOI(p).
(13.70)
The proof of this theorem may be found in Murdaev and Samko [1-3], where
the case of the weight p(x) = (x - a)#J(b - x)" is also considered - see also Samko
and Murdaev [1].
 14. Miscellaneous Results for Fractional Integro-
Differentiation of Functions of a Real Variable
In this section we shall consider various aspects of fractional calculus in real
variable function theory, which were not treated in the previous sections. The
main aspects are: 1) the mapping properties of fractional integration in the
spaces of functions which satisfy the Holder condition in L,-norm; 2) fractional
differentiability of functions which are absolutely continuous with a weight; 3)
the Riesz mean value theorem and the Kolmogorov type inequality for fractional
integrals and derivatives; 4) the connection with the summation of series and
integrals.
14.1. Lipschitz spaces H; and iI:
Let w,(f, 6) be the integral continuity modulus (13.24) of a function f(x), which is
given on [a, b] and is continued as identical zero beyond the interval [a, b].
Definition 14.1. We say that f(x) E H; = H;([a, b)), where 0 < A  1, if
f(x) E Lp(a, b) and w,(f, 6)  c6>'. If as well w,(f, 6) = 0(6)') as 6 - 0, we say
that f(x) E h; = h;([a, b)).
The spaces H; and h; are usually called Lipschitz spaces, the space H; being
sometimes designated as lip(A,p».
We consider some properties of functions in H;. The Hardy-Littlewood
imbedding theorem in the space H; will be stated. Initially we interpret Definition

 14. MISCELLANEOUS RESULTS
255
14.1 by stressing that it implies the estimates
b-6
f If(x + 6) - f(x)IPdx  c6>'P,
a
(14.1)
a+6
f If(x)IPdx  c6>'P
a
b
and f If(x)IPdx  c6>'P,
b-6
6> O.
(14.2)
The Lipschitz space might be defined by (14.1) only, i.e. without worrying
about zero continuation of a function f(x) beyond [a, b]. Such a space will be
denot.d by if;:
{ b-6
if; = I(z): I(z) E Lp(a, b), ! I/(z +6) - l(z)IPdz  c6'P,
6 > 0 }, (14.3)
so that H; C if;. Similarly the space it; is introduced, if 0(6)'P) in (14.3) is
replaced by 0(6)'').
The spaces if; and H; are equipped with the norms
{ b-6 } l /P
IIfllk" = IIfll, + sup 6->' f If(x + 6) - f(x)IPdx ,
p O<6<b-a
a
(14.4)
{ ( a+6 b ) -' } IIp
IIfIlH" = IIfllb" + SUp 6->' f + f If(x)IPdx ,
p p O<6<b-a
a b-6
(14.4')
so if; and H; become Banach spaces.
We note that H>' --+ if; for 1  p < 00 and H>' --+ H; for 1  p  1/>". The
imbedding H --+ H; is also valid for all 1  p < 00, where H = H([a, b]) is the
space of Holderian functions vanishing at the points x = a, x = b. (We denote a
continuous imbedding of Banach spaces X and Y by X --+ Y). It may be verified
directly that (x - a)#J E H; and (x - a)#J E if; if and only if p.  >.. - l/p in the
case 0 < >.. < 1, and p. > >.. - l/p in the case>.. = 1.
A natural question arises. If f(x) E H;, does the smoothness of the function
f(x) of the "order" >.. lead to the fact that f(x) E Lr with r> p? And if this is so,
does f(x) have a smoothness in Lr-norm, i.e. is an imbedding H; --+ H possible?
The answer to this question is given by the following Hardy-Littlewood theorem,
though its proof is omitted, see references in S 17.1 (note to  14.1).

256
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Theorem 14.1. If >.p  1, then H; -+ Ht' and iI; -+ fit' in the case
1  p  r < q, where q = p/(1 - >.p) and p. = >. - l/p + l/r. If >.p > 1, then
H; -+ H;-l/, and fI; -+ H>.-l/,.
Here imbedding in the space of continuous functions in the case >.p > 1 is
understood as usual up to the equivalence of functions.
We note that Theorem 14.1 yields the following corollary.
Corollary. Functions f(x) E H; or fI; are integrable to the power p with a
weight:
b
j If(x)I'dx
(x - a)II'(b - X)II'  cllfll n ;,
a
(14.5)
where II < >. for f(x) E H; and II < >.  l/p for f(x) E iI;.
Indeed, if >.p  1, by imbedding fI; -+ Lr, p  r < p/(1 - >.p), we have
j b If(x)I'dx , {j b dx } (r_,)/r
a (x - a)II'(b - X)II'  IIfll Lr a [(x - a)(b - x)]lIr,/(r-,)
 cllfll"
p
under the additional choice p/(II- lip) < r < p/(1 - >.p). If >.p > 1, by Theorem
14.1 we see that If(x)1  c(x - a)>'-l/, as x -+ a and similarly as x -+ b, so that
the validity of (14.5) becomes evident.
14.2. Mapping properties of fractional integration in H;
We shall show that fractional integration 1:+ maps the space H; into H;+OI in
the case >. + Q < 1 if, roughly speaking, we step aside from the left end-point
x = a. If we wish to consider what occurs right up to the point x = a, then
1:+ : H; -+ H;+OI-t, c > 0 - see Theorem 14.3. We first consider the simpler case
>. = 0, when we can prove a mapping theorem for 1:+ from H = L, into H; on
the whole interval [a, b] (cf. Theorem 3.6).
Theorem 14.2. Operators 1:+ and I_, 0 < Q  I, are bounded from L,(a, b)
into iI;([a, b]) for any p  1 and into H;([a, b]), if 1  p < I/Q.
Proof. Let f(x) = r(Q)I:+rp, rp E Lp(a, b). Applying the generalized Minkowsky

 14. MISCELLANEOUS RESULTS
257
inequality (1.33), we obtain
(TI/(z+ 6) _ I(Z)I'dZ) II,
6 b-o (14.6)
 1I<p1I, 1(6 - t)a-ldt + 1I<p1I, 1 [t a - l - (t + 6)a-l]dt
o 0
 e6 a 1I<p1I"
so IIfllk:  ell<pII,. To estimate the norm IIfIlH: for 1  p < 1/0: we need to verify
conditions (14.2). We have
( 0+6 ) 11, ( 6 Z ' ) 11,
! I/(z)I'dz = [dZ [ta-I<p(Z + a - t)dt
6 ( 6 ) 11,
:5[ t a - I ! l<p( Z + a - t)dt I' dz dt,
from whence
( 0+6 ) 11,
! II (z ) I' dz :5 6 a '" -III <pII"
1  p < 00, 0: > 0
(14.7)
- see Theorem 17.2 in connection with this estimate. Further, for J(6) =
b
f If(x)I'dx by the Hardy inequality (3.18) we have
b-6
b
J(6) < 1 If(x)I'dx < e ll II '
6 a , - (b - x )a, - <P,
b-6
in the case p > 1. If p = 1, then
b-6 b b b
J(6)  1 1<p(t)ldt 1 (x - t)a-ldx + 1 1<p(t)ldt 1 (x - t)a-ldx  e6all<pII"
o b-6 b-6 t
which completes the proof. .

258
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
0+6
Remark 14.1. If f(x) = I:+cp, a > 0, cp E L" then J If(x)IPdx  c6 01 ' for all
o
6
1  p < 00. Similarly, if f = Ir_cp, a > 0, cp E L" then J If(x)I'dx  c6 01 ' for
6-6
all 1  p < 00. These estimates for 1:+ in the neighborhood of the point b and for
Ir_ in the neighborhood of the point a hold for 1  p < 1/ a only. This is justified
by taking cp(x) == 1.
Remark 14.2. In view of the imbedding iI; --+ HOI-l/, given by Theorem 14.1
for a > l/p, Theorem 14.2 is a strengthening of Theorem 3.6, which states that
1:+ : L, --+ HOI-l/, for a > l/p.
Theorem 14.3. Let 1  p < 00, A + a < 1. Then the operator 1:+ realizes the
following continuous mappings:
[01
- >. -+ - >.+
Hp ([a, b» ---+ Hp 01 ([a 1 , b]), al > a,
1i; ([a , b»  iI;+OI-e ([a, b», 1  p  1/ A.
(14.8)
(14.9)
Also if cp E H; and the estimate
6
f
0+6
Icp(x)I'dx < c6(>'+0I-l),
(x - a)(1-o)p -
(14.10)
holds as well, then I:+cp E iI;+OI([a, b» and satisfies the former of the conditions
(14.2).
Proof. To estimate the difference f(x + 6) - f(x) with f(x) = r(a)I:+cp, we shall
use representation (3.4): f(x + 6) - f(x) = J 1 + J 2 + J 3 , where the summands
J l , J 2 and J 3 are given by their expressions in (3.4) with g(t) being replaced by
cp(t) and h by 6. We have
{ 6-6 } 11, 6 { 6-6 } IIp
! IJ 2 1"dz  jc6 - W-1dt ! Icp(z + t) - cp(z)I'dz
6
 cllcpllJi" f t>'(6 - t)OI-ldt  c 1 6>'+0IIIcpIlJi'"
p p
o

 14. MISCELLANEOUS RESULTS
259
Further
{ 6 } llP { 6-0-6 z P } llP
! IJ.I'dz = / dz /[<P(Z + 0 - I) - <p(Z - 0)][1 0 - 1 - (I + W- 1 ]dl
6-0-6 { 6 } lip
 f [t a - l - (t + 6)a-l]dt f Icp(x - t) - cp(x)IPdx
o +t
b-o
 cllcpllii: f t>'[t a - l - (t + 6)a-l]dt
o
(6-0)16
= cllcpllii,,6>.+a f t>'[ta-l - (t + l)a-l]dt  c 1 6>.+allcpIlH'"
p p
o
Complications connected with the left end-point are caused by the summand J l .
We have
{ 6-6 } llP { 6-6 } llP
! 1Jd"dz :s c ! 1<p(z)I'(z - 0)0' [(1 + z  0 ) 0 - f dz
{ 6 } IIp
 C16 ! 1<p(z)I'dz  C111<pII,6,
(14.11)
where Cl does not depend on 6 but depends on ai, and Cl --+ 00 as al --+ a. Thus
(14.8) is proved.
To obtain (14.9) we shall give such estimates in (14.11), which cover the left
end-point x = a. Using the inequality (1 + y)a - 1  cy(1 + y)a-l, we obtain
{ 6-6 } llP { 6 } llP
f IJ IPdx < c6 f Icp(x)IPdx
1 - (x - a + 6)(1-a)p
o 0
Since (x-a+6)(1-a)p  (x_a)(>.-e)P6(1->.-a+e)p, where 0 < c < min(, 1--a),

260
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
by (14.5) we have
{ b-6 } l/P { b } l/P
! IJd'dz :5 c6'+a-< ! {{::)p :5 C,6'+a-<lIl"lIb:"
Let now <fJ E H; and let (14.10) be satisfied. The summand J 1 is estimated as
follows:
{ b-6 } l/P { O+6 } l/P
! IJ,I'dz :5C! II"(z)I'[{z-a+6t-{z-a)a)'dz
J b } lh
+ C tL II"{z)I'[(z - a + W - (z - a)a]'dz
f b } IIp
:5 c6>+aIlI"IIH: + c6 tL II"{z)I'(z - aj<a-')'dz
The application of (14.10) completes the estimate, so that I:+<fJ E H;+a([a, b]).
As for the former of the conditions (14.2) for f = I:+<fJ, this is easily verified
directly. .
Remark 14.3. Similarly to Corollary of Theorem 3.6 it may be shown that
Theorem 14.3 holds for the spaces h; as well. It may be shown also that Theorem
1:+
14.2 holds in the form Lp(a, b) --+ h;([a1' b]), a1 > a.
It follows from Theorem 13.5 that functions in the space H;, A > a, are
representable by fractional integrals of the order a of Lp-functions:
H; C Ia[Lp(a, b)], A > a, 1 < p < l/a.
( 14.12)
Verification of the condition (13.22) more accurately than in Theorem 13.5
J b } IIp b-o { b-t } IIp
tL loP<{z)I'dz :5 ! t;;a ! I/{z + t) - f{z)I'dz ,

 14. MISCELLANEOUS RESULTS
261
shows also that, by Theorem 13.4,
->.
Hp C p;r[Lp(a, b)], A > a, 1 < p < l/a.
(14.13)
We make imbedding (14.13) more exact by giving the following theorem.
Theorem 14.4. Let f(x) E iI;([a, b]), A > a, 1 < p < l/a. Then f(x) =
I:+tp, where tp = D+f E iI;-a([a, b]), if A $ l/p and cp(x) - ra) (z_la)O E
ii;-a([a, b]), if A > l/p.
Proof of this theorem imitates estimates in the proof of Lemma 13.1 with the
only difference that the summands It, 12 and 13 in the representation (13.38) are
to be estimated now not in the uniform norm, but in Lp-norm by means of the
Minkowsky inequality (1.33). The corresponding steps are not difficult and will
be omitted. We should take into account the fact that (x - a)-a E iI;-a only if
A $ l/p.
Corollary. Equation (2.64) of fractional integration by parts as valid for the
functions
f(x)EiI;, 9(x)EH;, A>a, l/p+l/q$l+a.
14.3. Fractional integrals and derivatives of functions
which are given on the whole line and belong to H;
on every finite interval 
We consider now functions tp(x), given on the whole real line, and study the
behaviour of fractional integration It. in terms of the spaces H; in the case when
information on H;-behaviour of functions cp and f is local, while the fractional
integral under consideration has an infinite limit of integration. Such a statement
of the question allow us to avoid" the influence of end-points of the interval -
compare Theorems 14.6 and 14.7 with Theorems 14.3 and 14.4. The fractional
integral will be considered to be conventionally convergent:
z
a 1 I . J cp(t)dt
I + tp= - 1m
r(a) N-oo (x - t)l-a'
z-N
(14.14)
This interpretation of the integral is necessary when we admit functions tp(t), which
do not necessarily vanish at infinity (in particular they may be periodic). Thus
existence of the limit (14.14) in Theorem 14.5-14.7 below will be postulated. Below,

262
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
in  19, we shall see that (14.14) does exist and is conventionally convergent on
2'11'
27r-periodic functions under the choice N = 21rn, n = 0, 1,2,..., if J cp(t)dt = 0).
o
Theorem 14.5. Let 0 be an arbitrary interval with a length I and let cp(x) satisfy
the condition
f Icp(x)IPdx  c, 1  p  00,
(1
(14.15)
where c = c(/) does not depend on location of 0, and let /(x) = It.cp, 0 < a < 1,
exist as the limit (14.14) for almost all x. Then
b
f I/(x) - /(x + h)IPdx = o(h OP ), h  0,
a
(14.16)
for every interval [a, b] with a length Ib - al  I.
x x x-dh x-Nh
Proof. We have f(a)/(x) = J cp(t)(x - t)O-1dt = J + J + J = /1 +
-00 x-dh x-Nh -00
/2 + /3, where d> 0 and N > 0 are constants, which will be chosen later on. Let
/)./ = /(x) - /(x + h). It is sufficient to show that
b
f l/).f2l P dx = o(h OP )
a
(14.17)
for all fixed d and N and that
b
f 1/)./dPdx  chop,
a
b
f 1/)./3I P dx  chop
a
(14.18)
for all 0 < h  1 and sufficiently small d and sufficiently large N. We have
x-dh
/).12 = f (x - t)o-1[cp(t) - cp(t + h)]dt
x-Nh
x-dh
= O(h o - 1 ) f Icp(t) - cp(t + h)ldt.
x-Nh

 14. MISCELLANEOUS RESULTS
263
Hence
b b z-dh
f IAhlpdx $ ch Olp - l f dx f Icp(t) - cp(t + h)IPdt
a a z-Nh
b-dh t+Nh
$ ChOl,,-l f Icp(t) - cp(t + h)I"dt f dx
a-Nh t+dh
(14.19)
b-dh
= clh OlP f Icp(t) - cp(t + h)IPdt.
a-Nh
It follows from (14.15) that cp(t),cp(t + h) E L,,(a - Nh,b - dh) for fixed d and N,
and sufficiently small h. Then (14.19) yields (14.17) in view of (1.34).
To obtain the former of the inequalities (14.18), it is sufficient to prove that
b
f Ih(x)IPdx $ chOlp and similarly for h(x - h). We have
a
b IIp dh b 11
(f lf1(z)I'dz)  f ta-1dt (f I/(z - t)I'dZ) ·
a 0 a
b
 (d)a ( f I/(t)I'dtr / .  c1/'h a
a-dh 
b
in view of (14.15), d being chosen sufficiently small. The integral f Ih(x - h)IPdx
a
is estimated similarly. Finally,
00 Nh+h
Af3 = f cp(x - t)[t Ol - l - (t - h)OI-l]dt + f cp(x - t)tOl-ldt
Nh+h Nh
= 1 1 + 1 2 ,
Taking (14.15) into account we have
( b ) IIp 00 ( b ) IIp
f 111lPdx 5 f [(t - h)OI-l - tOl-l]dt f Icp(x - t)IPdx
a Nh+h a

264
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
00
 ch Oi f [(t - 1)01-1 - tOl-l]dt  cl/Ph Ol ,
N+l
if N is chosen sufficiently large. Further,
( b ) lip ( b Nh+h ) 1/1'
! Ihl'dz :5 (N ht- 1 hl-l / . ! dz l''P(Z - t)l"dt
( b-Nh
 NOI-lhOl-l/'P f
-Nh-h
t+Nh+h ) lip
Icp(t)IPdt f dx
t+Nh
( b-Nh ) IIp
= N OI - l hOi f Icp(t)IP dt
-Nh-h
b
Hence, using (14.15) we obtain the estimate (J Ihl'Pdx)l/'P  cNOI-lh Oi  cl/Ph Ol , if
a
N is sufficiently large, which completes the proof. .
Theorem 14.6. Let f(x) = I+cp exist as the limit (14.14) for almost all x and let
cp(x), cp(x - h) E Lp(a,b) for some a and b. If
b
f Icp(x) - cp(x - h)I'Pdx  ch>'P,
a
( 14.20)
1  p  00, 0 < A < I,
then for the same a and b
b
f If(x) - f(x - h)I'Pdx  Ct h(>'+OI)'P , A + a < 1,
a
(14.21)
with Cl depending only on c.

 14. MISCELLANEOUS RESULTS
265
Proof. For Af = f(x) - f(x - h) we have
00
r(a)Af = f [cp(x) - cp(x - t)][(t - h)a-1 - t a - 1 ]dt
h
h
- f [cp(x) - cp(x - t)]t a - 1 ]dt = J 1 + J2
o
Simple estimates lead to
( b ) l/p 00 ( b ) l/p
! IJd"dz 5, c /[(1 - h)a-' - ,a-'Id, ! l<p(z) - <p(z - 1)I"dz
00
 c f t>'[(t + h)a-1 - t a - 1 ]dt = C1 h >.+a.
o
Similar ly
( b ) l/p h ( b ) l/p
! IJ 2 1"dz 5, I ,a-'d, ! l<p(z) - <p(z - 1)1" dz 5, ch).+a,
which proves the theorem. .
In the following theorem the Marchaud fractional derivative is treated as a
limit in the space Lp(a, b):
D.! = l im o (D+.ef)(x), a  x  b,
e-
(14.22)
where D+.ef is the truncated Marchaud fractional derivative (5.59).
Theorem 14.7. Let f(x), -00 < Z < 00 satisfy the condition (14.15). If
b
f If(x) - f(x - h)IPdx  ch>'P, 0 < a < A < 1,
a
( 14.23)

266
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
then the limit (14.22) exists. If besides this
b-d
f If(x) - f(x - h)IPdx $ ch>",
II-d
( 14.24)
for all sufficiently small d > 0 with a constant c, not depending on d, then
b
f Icp(x) - cp(x - h)I'dx $ ch(>'-Ot)" cp = D+.f.
II
(14.25 )
Proof. First we note that D+,£f E Lp(a, b), which may be easily obtained by
using (14.15) applied to the function f, and (14.23). Let us show the sequence
cp£ = Dt-I£f to be fundamental in L,(a, b). For £1 < £2 we have
b b ( £2 ) P
! Icp.,(z) - cp.,(z)IPdz  ! dz ! I/(z) - I(z - t)lr a - 1 dt
= i dz (J I/(z) - I(z - t)lr a -'-l/ p t H/P' dt ) P
< (J r ' +'P' dt ) p-1 (i dz ll/(Z) - I(z - t)IPt-pa-1-p'dt) , (14.26)
where 6 > 0; in view of (14.23) we see that (14.26) is dominated by
£2 b £2
0(1) f t- pa - 1 - p 'dt f I/(z) - I(z - t)IPdz = 0(1) f t p (>-a-')-ldt = 0(1)
£1 II £1
under the choice 0 < 6 < A-a. Thus existence of the limit (14.22) is established.

 14. MISCELLANEOUS RESULTS
267
Further, for cp = D+ f we have
00
r(1 - a) f
a [cp(x) - cp(x - h)] = [!(x - t) - f(x - h)][(t - h)-a-l - t-a-l]dt
h
00
+ f[!(x) - f(x - h)]t-a-ldt
h
h
+ f [!(x) - f(x - t)]t-a-ldt
o
=h + 1 2 + 13.
Simple estimates with application of (14.20) give
( b ) IIp 00 ( b ) IIp
! IId'dz  / [(t - h)-I-a - Cl-aldt ! I/(z - h) - I(z - t)I'dz
00
 c f [(t - h)-l-a - t-l-a](t - h)>'dt = clh>.-a.
h
The estimate for 12 is evident, while for 13 we obtain
( b ) IIp h ( b ) IIp
! II31'dz  ! t-I-adt ! I/(z) - I(z - t)I'dz  ch'-a
by (14.23), which completes the proof. .
Remark 14.4. We have restricted ourselves to O-form in Theorems 14.6 and 14.7.
It is not difficult to show that they are true in o-form too.
14.4. Fractional derivatives of absolutely continuous functions
It was established earlier - Lemma 2.2 - that absolutely continuous functions
f(x) have fractional derivatives of order a E (0,1) almost everywhere and admit
equations (2.24)-(2.25). Here we shall extend these assertions to a wider space of
functions:
f*(x)
f(x) = (x _ a)#J(b _ X)II ' Il, v E [0,1 - a),
( 14.27)

268
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
where f.(x) E AC([a, b».
Theorem 14.8. Functions of the fonn (14.e7) are representable by a fractional
integral f = I+tp or f = If_cp of a summable function tp E L}(a,b).
Prool. For the sake of simplicity we restrict ourselves to the case v = 0, the general
:&
case being reducible to this. Since f.(x) E AC([a,b]), then f.(x) = f.(a)+ f ,p(t)dt,
a
where 1/J(t) ELI. Since the function f.(a)(x - a)-#' is representable as a fractional
integral of the function const (x - a)-II-a E Ll it remains for us to show that the
:&
function (x - a)-#' f 1/J(t)dt is representable as it is required in the theorem. We
a
:&
set cp(x) = f 1/J(s)A(x,s)ds, where
a
:&
A(x, s) = :x !( - a)-#'(x - )-ad
,
:&-8
- ( ) - #' ( ) - a ! d
- s - a x - s - p. (x _ a _ )l+#,a '
o
and verify the equality
:& :&
(x - a)-#' ! 1/J(t)dt = sin a1r ! ( <p(li, dt.
1[' x - t -a
a a
( 14.28)
b b
We make sure first that cp(x) E Ll(a, b). Indeed IIcpllL l ::S f 11/J(s)lds f IA(x, s)ldx.
a 8
b
We show that fIA(x,s)ldx::S const. Since A(x,a) = (1-p.-a)B(I-p.,I-
a
a)(x - a)-#,-a and :8 A(X, s) = a(s - a)-#'(x - s)a-l > 0, then A(x, s) > ° for all
a < s < x. So
b b b :&
! A(z,B)dz=(s-a)-" !(z-s)-adZ-I-'! dZ! ({-s)a(z+-a-{)'+"
a 8 8 8

 14. MISCELLANEOUS RESULTS
269
b
(S - a)-IJ(b - sF-a f (b + s - a - )-IJ - (s - a ) -IJ
- + 
1 - a ( - s)a
8
b
f 
- < const.
( - 8)a(b - a + s - )IJ -
8
To verify (14.28), we have:
:& :&:&
f cp(t)dt = f 1/J(S)ds f A(t,s) dt.
(x - t)1-a (x - t)1-a
a a,
It remains just to evaluate the inner integral
:& :& t
f A(t,s)dt 1 11' f dt f 
(x - t)1-a = (s - a)IJ sin a1r - p (x - t)1-a (t - )a( - a)1+IJ
, '8
:&
1 1r f  11'
= (8 - a)IJ sin a1l' - p ( - a)1+IJ sin a1r
,
11'
= --;--(x - a)-IJ,
sm a1l'
which gives (14.28) and thereby proves the represent ability of the function f(x) by
a fractional integral. .
Theorem 14.9. Fractional derivatives of functions of the form (14.27) admit
formulae
:&
va f = 1 f (1 - a)f(t) + (t - a)f'(t) dt,
a+ r(1 - a) (x - a) (x - t)a
a
(14.29)
o < a < 1,
b
Va f = 1 f (1 - a)f(t) - (b - t)f'(t) dt.
b- r(1 - a)(b - x) (t - x)a
:&
( 14.30)
Proof. We denote cp = V:+f. By Theorem 14.5 cp E L 1 and f = I:+cp. We

270
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
introduce also the function rpl (x) = (x - a )rp( x) and consider the fractional integral
z Z
01 1 f (t - x)rp(t) 1 f (x - a)rp(t)
1 0 +rpl = f(a) (x _ t)I-OI dt + f(a) (x _ t)1-OI dt
o 0
= -a1:1 1 rp + (x - a)(1:+rp)(x).
Using semigroup property of fractional integration we obtain
z
f d
(1:+rp})(x) = -a f(t)dt + (x - a)f(x) = hex).
o
(14.31)
The right-hand side here is an absolutely continuous function on [a, b - 6], 6 > O.
This is evident for the first term and it follows for the second term from the equality
(x - a)f(x) = (x - a)I-II(b - X)-II f.(x), all factors being absolutely continuous on
[a, b - 6], 6 > O. Then, by Lemma 2.2, (14.31) is solvable relative to rpl and we
obtain
z
1 f fl(t)dt
rpl(X) = f(1 _ a) (x - t)OI '
o
This is (14.29). Equation (14.30) is proved similarly. .
Remark 14.5. Equations (14.29) and (14.30) are extended to values a > 1:
Z
1)01 = 1 f (t - a)OI[(t - a)n-OI f(t)](n) dt
o+f f(n - a)(x - a)n (x _ t)OI-n+l '
o
(14.32)
where n = [a] + 1, a 1= 1,2,3,..., this being justified similarly to Theorem 14.9.
14.5. The Riesz mean value theorem and inequalities for
fractional integrals and derivatives
Let a function rp(x) be given on the half-axis x  a. The following theorem is
known as the Riesz mean value theorem for fractional integrals.
Theorem 14.10. Let 0 < a < 1, rp(x) E L 1 (a,b) and let
f(x) = (1:+rp)(x) E C([a, b]), f( a) = O.
(14.33)

 14. MISCELLANEOUS RESULTS
271
Given x > b, there exists T E [a, b) such that
b r
f (x - t)a-1rp(t)dt = f (T - tt-1rp(t)dt
a a
( 14.34)
or (which is the same)
(I:+rp)(x) - (It:+rp )(x) = (I:+rp)( T).
(14.35)
The latter assumes that rp(t) is integrable beyond [a, b].
Proof. The identity
b b
f a-1 (x - b)a f f(u)du
rp(t)(x - t) dt = r(1 _ a) (x _ u)(b _ u)a '
a a
x> b,
(14.36)
1S valid. It is verified directly by substituting I:+rp into the right-hand side,
interchanging the order of integration and then evaluating the inner integral by
means of (11.4). Since the function (x - u)-l(b - u)-a does not change its sign for
u E [4, b], by the first mean value theorem for integrals - Nikol'skii [7, p.363] - we
)
obtain from (14.36) the equation
b
1 f -
r(a) rp(t)(x - t)a 1dt = f(TdM(x),
a
a  T1  b,
( 14.37)
where
b
M(x) = (x - b)a sin a1r f du
11' (x - u)(b - u)a
a
11'
(b-a)/(z-a)
f
o
(14.38)
sm a1r
ds
sa(1- s)1-a
=
after the substitution b - u = (x - b)s(1 - S)-l. Since 0 < M(x) < 1 and f(x)
is continuous, there exists a point rp E [a, TiJ such that f( T1)M(x) = f( T). Thus
(14.34) is proved, and this leads to (14.35). .

272
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
Corollary 1. Let rp(t) satisfy the assumptions of Theorem 14.10. Then
6 
! (x - tYJ/-lrp(t)dt 5 max ! ( - t)Ot-lrp(t)dt, x > b.
e[a,6]
a a
(14.39)
Corollary 2. Let rp(t) E LI(a, b). Then
6 
! (x - t)Ot-lrp(t)dt 5 esssup ! ( - t)Ot-lrp(t)dt, x > b.
E[a,6]
a a
( 14.40)
Indeed, (14.39) follows immediately from (14.34), while (14.40) is derived from
(14.36) without the additional conditions (14.33) because
6 b
! IM'(t)ldt = - ! M'(t)dt = M(a) < 1.
a a
We apply now Theorem 14.10 to obtain certain inequalities for fractional
integrals.
Theorem 14.11. Let rp(x) E L 1 (a, N) for every N > a. If
Irp(x)15 V(x), 1(1:+rp)(x) I 5 W(x), x > a,
(14.41 )
where V(x) and W(x) are non-decreasing functions, then for each /3, 0 < {3 < a
the inequality
1(I+rp)(x)15 c[V(x)P-fjIOt[W(x)]fjIOt,
( 14.42)
holds, where c does not depend on x and /3.
Proof. We start by letting 0 < a < 1. We have
 x
r({3)(I:+rp)(x) = !(x - t)fj-lrp(t)dt + !(x - t)fj-lrp(t)dt
a 
( 14.43)
= II + 1 2 ,

 14. MISCELLANEOUS RESULTS
273
where we choose the point  as follows:
{ X - [W(x)/V(x)p/a,
=
a,
if x > a + [W(x)/V(x)p/a,
if x < a + [W(x)/V(x)p/a.
(14.44)
So 1 1 = 0 for x < a + [W(x)/V(x)p/a. We note that the following always holds:
0< x -  [W(x)/V(x)p/a.
(14.45)
By the monotonicity of V(x) we have
:&
1121  f V(t)(x - t){j-1dt  P- 1 V(x)(x - )(j

and then due to (14.45)
1121 < !V(x) [ W(X) ] {j/a = ![V(x)p-{j/a[W(x)]{j/a
- P V(x) P .
( 14.46)

Further, It = f <p(t)(x - t)a-1(x - t){j-adt and since (x - t){j-a increases on [O,]
o
for fixed x, by the second mean value theorem for integrals - Nikol'skii [7, p.368] -
we have

It = (x - ){j-a f <p(t)(x - t)a- 1 dt, 0  u  .
(14.47)
u
Applying (14.40) to (14.47), we obtain
11 1 1  (x -){j-ar(a)esssup I(I\<p)(y)1
YE[u,]
 r(a)(x - )(j-aw().
Hence, taking into account (14.44) and the monotonicity of the function W(), we
see that
[ W( ) ] ({j-a)/a
11 1 1  r(a) V(:) W(x)
= r( a)[V(x)p-{j/a[w(x )]{j/a.
( 14.48)

274
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
So in view of the estimates (14.46) and (14.48) we derive inequality (14.42) from
(14.43) with the constant c = prcp) + H . This is dominated by a constant not
depending on (3 owing to the properties of gamma-function.
Let now a > 1. Let us choose an integer n such that aln < 1/2 and let
an = akIn, k = 1,2,..., n - 1. We denote Ip(x) = max 1 (1:+ rp)(t) 1 and apply
a<t<z
the theorem already proved for the case 0 < a < 1, replaing V, 1:+ rp and W by
laic_I' laic and laic + 1 :
laic (x) :5 c J 1 001c-l (x)IOIIc+l (x), k = 1,2,..., n - 1
with 1 01 0 = V, 101.. = W. We raise these inequalities, corresponding to numbers
k = 1,2,...,1- 1, I, I + 1,..., (n - 2), (n - 1), to the powers of the orders
(n - I), 2(n -I),..., (1- l)(n -I), l(n -I), I(n -1- 1),...,21, I
respectively. Then multiplying them all together we obtain
1011 (x) :5 cdV(x)P- OI d Ol [W(x)] OI d Ol ,
(14.49)
and thus (14.42) is proved for the values /3 = ai, . . . ,an-i. Since for each {3 E (0, a)
there exists an 1 = 1,2,. . . , n - 1 such that 0 < /3 - a, < 1, then 1:+ rp = 1:+ 011 1:+ rp.
Hence once more applying the theorem proved for small a, w easily derive the
theorem for an arbitrary {3 E (0, a). .
Corollary. Let rpOl( x) denote the monotone majorant 01 a fractional integral:
rpOl(X) = sup 1(1+rp)(t)l, rp(x) E L 1 (0,/), a  O.
O<t<z
Then
rpp(x):5 c[rpo(x)]'-P/OI[rpOl(X)]P/OI, 0 < /3 < a,
( 14.50)
with c not depending on rp(x).
Remark 14.6. Theorem 14.11 is true in o-form: if relations rp(x)/V(x) --+ 0
and 1(1\rp)(x)1 :5 W(x) or Irp(x)1 :5 Vex) and (I+rp)(x)/W(x) --+ 0 are valid as
x --+ 00, then lor 0 < {3 < a
(1:+rp)(x)/[V(x)P-P/OI[W(x)]P/Oi --+ O.
(14.51 )

 14. MISCELLANEOUS RESULTS
275
The proof of this assertion is given by Riesz [1].
We give now an important restatement of Theorem 14.11.
heorem 14.11'. Let f(x) E I\[Ll(a, N)], a > 0, for each N > a. If
If(x)1  W(x), I(V:+f)(x)1  V(x), x > a,
where W(x) and V(x) are non-increasing functions, then in the case 0 < 'Y < a
I(V+f)(x)1  c[V(x)p1a[W(x)p-'Yla.
(14.52)
The proof of this theorem is obtained from Theorem 14.11 after rewriting
IC:+ tp = f and (3 - a = 'Y.
Corollary. Let f(x) E I<f+[L 1 (0, N)] for each N > O. If f(x) and V+f are
bounded on the half-axis R = (0,00), then in the case 0  'Y  a
Ilv;r+fIlC(R)  Kllfll()IIV+fIlI:(R)'
( 14.53)
Inequalities estimating intermediate derivatives via a higher derivative and
the function itself are usually called Kolmogorov inequalities. Thus (14.53) is a
Kolmogorov type inequality for fractional orders.
We shall not dwell here on the question about the sharp constant in (14.53).
Some additional information about this and other inequalities for fractional
integrals and derivatives, including sharp constants, may be found in S 17.2 (notes
14.5-14.7) and in S 19.8.
14.6. Fractional integration and the summation of series and
integrals
There is a close connection between the generalized summation of series, in terms
of ''fractional" means known as Riesz means, and fractional integration. Let us
00
consider a series L: c n , which is not necessarily convergent. Various summation
n=l
methods are known for divergent series - e.g. Hardy [3]. They are based on the
fact that instead of partial sums
[z]
C(x) = LC n
n=l
( 14.54)

276
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
one or another of their averages is considered and the limit of the average, if it
exists, is called the sum of the divergent series. One such way uses averages of the
form
[z]
Ca(X) =  "'(x - n)a cn , a > 0,
xa L.J
n=l
(14.55)
the number
s = lim Ca(x)
z-oo
(14.56)
00
being called the "sum" of the series L: Cn. The means (14.55) are called Riesz
n=l
normal means. It is known - Hardy [3, p.1l5] - that this method of summation is
regular, i.e. it sums up a convergent series to its usual sum. The equality
z
Ca(x) =  f C(t)(x - t)a-1dt
x a
o
(14.57)
is valid, where C(t) is the partial sum (14.54). Equation (14.57) is verified using
integration by parts:
z z [z]
a f C(t)(x - tt- 1dt = f(x - t)a-1dC(t) = L:(x - n)a cn .
o 0 n=l
So the Riesz means (14.55) is, up to the factor x- a r(1 + a), the fractional
integral of a partial sum of the series.
Equation (14.55) can be similarly used for the summations of integrals
00
f f(t)dt
o
(14.58)
which may diverge at infinity, by defining the value of this integral as the limit
z
lim  f F(t)(x - t)a-1dt, a> 0,
z-oo x a
o
( 14.59)
t
where F(t) = J f(s)ds.
o

 15. THE GENERALIZED LEIBNIZ RULE
277
Analogously the integrals
a
1 f(t)dt, f(t) E L 1 (£, a), 0 < £ < a,
o
( 14.60)
which are divergent because of a non-integrable singularity at the point t = 0, may
be summed up. If the limit of the following integral exists
:& a
lim ax-a l (x - t)a- 1 dt 1 f(s)ds
:&-0
o t
(14.61)
for some a > 0, it is called the value of the integral (14.60).
This method of evaluating divergent integrals as defined in (14.59), (14.61) is
called the (C, a)-method or Cesaro-Lebesgue integration.
There are a number of investigations concerning the role of fractional
integration in the theory of the summation of series and integrals. We note a paper
by Hardy and Riesz [1], 1915, which is fundamental to the theme, and the paper by
Bosanquet [5] devoted to properties of Cesaro-Lebesgue integrals (14.60), (14.61).
See also the books by Hardy [3], by Chandrasekharan and Minakshisundaram [1]
and the bibliography given in U 17.1 and 17.2.
 15. The Generalized Leibniz Rule
In this section the classical Leibniz rule
(/g}(a) =  (:) fa-n)g(n), ,,= 1,2, ...,
(15.1)
is generalized to the case of differentiation and integrc;ttion of fractional order.
Together with the generalization in the form of an infinite series - see (15.12) - an
integral analogue is also considered - (15.17).
15.1. Fractional integro-differentiation of analytic functions
on the real axis
We first prove some preliminary statements on the possibility of term-by-term
fractional integration and differentiation of a functional series.
00
Lemma 15.1. If the series f(x) = L: fn(x), fn(x) E C([a, b]), as uniformly
n=O

278
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
convergent on [a, b], then its term wise fractional integration is admissible:
00
00
(1:+ L In)(x) = L(I:+ln)(x), a> 0, a < x < b,
n=O n=O
(15.2)
the series on the right-hand side being also unilormly convergent on [a, b].
The proof is accomplished by simple estimates of the quantity
1 1 .:'+/ -1.:'+ CO In) I taking the uniform convergence of the series into account.
Lemma 15.2. Let the fractional derivatives V:+ln exist for all n = 0,1,2,.. .
00 00
and let the series L: In and L: V:+fn uniformly converge on every sub-interval
n=O n=O
[a + €, b], € > O. Then, the former series admits termwise fractional differentiation
using the formula
(1':+  In) (z) = (1':+ln) (z), ,,> 0, a < z < b.
(15.3)
Proof. Since (V:+/)(x) = (d/dx)[a]+l(I+{a}J)(x), Lemma 15.2 is reduced to
Lemma 15.1 as term-by-term application of the operator (d/dx)[a]+l is possible
following a known theorem from mathematical analysis.
We recall that V:+ means the fractional integral I;;;, if a < 0, see (2.34).
Let us prove two lemmas on the represent ability by series of a fractional
derivative of an analytic function in an interval (a, b); i.e. a function which is
expandible into a power series in this interval.
Lemma 15.3. If I(x) is an analytic function in an interval (a, b), then
(V: I)(x) = t ( a ) (x - a)n-a [<n)(x), X E (a, b),
+ n=O n f(n+l-a)
(15.4)
() being the binomial coefficient (1.48).
Proof. Let a < 0 and
x
(1':+/)(z) = r( ,,) J (z - W a - I I(t)dt.
a

 15. THE GENERALIZED LEIBNIZ RULE
279
Since f(t) is an analytic function, it can be represented as a convergent power
series:
f(t) = f: (-l)n f,(n)(x) (x - tt.
O n.
n=
According to Lemma 15.1 termwise fractional integration is possible here which
leads to (15.4). In view of (1.48) the case a = 0 in (15.4) corresponds to the
equality f(x) = f(x). Now let a > O. Since (V+f)(x) = (d/dx)[a]+l(v}-l)f(x),
then, by (15.4),
( d ) [a]+l L: oo ( {a} -1 ) (x - a)n-{a}+lf(n)(x)
( va f )( x ) - -
a+ - dx n=O n r(2-{a}+n)'
Carrying out term-by-term fractional differentiation (Lemma 15.2 shows that it is
possible due to the uniform convergence of corresponding series) we have
a _ 00 ( {a} - 1 ) ( d ) [a]+l (x _ a)n-{a}+l f(n)(x)
(Va+f)(x) -  n dx r(2 - {a} + n) .
By the Leibniz rule (15.1) and (1.49) we find
a _  ( {a} - 1 ) [l ( [a] + 1 ) (x - a)n-a+1: f(n+I:)(x)
(Va+f)(x) -  n  k f(n - a + k + 1)
= f: f: ( {a} - 1 ) ( [a] + 1 ) (x - a)n-a+1: f(n+ic)(x) .
n=OI:=O n k f(n-a+k+l)
Introducing a new variable of summation j = n + k and changing the order of
summation we obtain
(Va f)(X)=f: ( t ( {a}-I )( []+I )) (x-)j-af(j)(x) . (15.5)
a+ j=O n=O n J - n f(J - a + 1)
Hence, by (1.53) we deduce (15.4). .
Lemma 15.4. If the junction f( x) has an expansion of the form
00
f(x) = (x - a)#J L: cn(x - at,
n=O
(15.6)

280
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
in the neighbourhood of the point a, then its fractional derivative V<:+I IS
represented by the formula
(V+/)(x) = (x - a)l'-a g (x),
(15.7)
where
( ) _  enr(n + Jj + 1) ( _ ) n
gx-L.J ( ) x a,
n=O r n - a + J.l + 1
and the radii 01 convergence of the series (15.6) and (15.8) coincide with each
o th e r.
( 15.8)
Proof. According to Lemmas 15.1 and 15.2, we can apply termwise fractional
integro-differentiation to the series (15.6). Therefore, the assertions (15.7) and
-(15.8) of the lemma follow from (2.44). The coincidence of the radii of convergence
is checked via their evaluation by the usual Cauchy-Hadamard formula taking the
relation
cnf(n + Jj + 1) a (15.9)
,.y cnn n --+ 00
r( n - a + Jj + 1) ,
into account, the latter following from (1.66). The lemma is thus proved. .
Corollary. If I(x) is analytic in (a, b), it is also trne for (V<:+/)(x) with a E R i .
We note that the semigroup property (2.65) discussed for arbitrary functions
in S 2, holds under wider assumptions on the parameters a and /3 in the case of
analytic functions. Actually, Lemma 15.4 shows that if a function I is analytic in
the right neighbourhood of the point a, then
-na -nP I - -n a+P I R i {.l 1
v a + v a + - V a + ' a E , fJ < .
( 15.10)
15.2. The generalized Leibniz rule
We realize an extension of the Leibniz rule (15.1) to fractional values a in two
forms.
Theorem 15.1. Let I(x) and g(x) be analytic on [a,b]. Then
V':+(fg) = 'E () (V,::;' f)g<'), "E HI,
V,:+ (f g) = f (k: fJ) (V,::;fJ -. f)(:' g),
k=-oo
(15.11)
( 15.12)

 15. THE GENERALIZED LEIBNIZ RULE
281
where (1:,6) is given by (1.50), 0:,{3 E R 1 and 0: f; -1, -2,..., if {3 is non-integer.
Proof. By (15.4) we have
00 ( ) ( ) 1:-01
01 _ 0: X - a (1:)
Vo+(fg) - f; k r(k + 1 - 0:) (fg) .
Applying the usual Leibniz rule (15.1) we obtain after the interchange of the order
of summation the result
VOl (fg) =  g(I:)(x)  ( 0: ) ( k + j ) (x - a)l:+j-OI f(;)(x). (15.13)
0+ f:'o  k+j k r(k+j+l-o:)
Since (I:j) et j ) = (:) (OIjl:), then (15.13) takes the form
00 ( ) 00 ( k) ( ) 1:+ 3 '-01
0: 0: - X - a (I:) (')
V+(fg) = f; k f; j r(k + j + 1 _ 0:) 9 (x)f 3 (x).
Hence, applying (15.4) again we obtain (15.11).
Further, by means of the simple renumeration {3 + k = j, the C3$e of integer
{3 in (15.12) is reduced according to (1.49) to (15.11). If {3 is not an integer it is
sufficient to consider the case (3 < lowing to the same renumeration. We rewrite
00
(15.11) as V+(gf) = L: ()f(I:)V+l:g. Applying the operator V:+,6 to both
1:=0
sides of this equality and taking the semigroup property (15.10) into account we
have
V:+(/g) = V:+fJ(+(/g» = V:+ fJ ( ()l)V+g) .
(15.14)
Carrying out termwise fractional differentiation (the possibility of doing this is
justified at the end of the proof of the theorem) and using (15.11) and (15.10)
again we obtain
V+(fg) = f f ( : ) ( 0: -. (3 ) v:+,6-j+1: fV+Ic+j g.
1:=0 j=O J

282
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Changing the order of summation by setting j - k = n we get
V+(fg) =   (:) (:  :)(v:+ f3 - n f)(V:ng)
+ t t (:) (:  :)(v+f3-n f)(V:ng).
n=-oo k=-n
(15.15)
(1.48), (1.72) and (1.77) give the relations
 ( {3 ) ( 0: - (3 ) r(o: - (3 + 1)
 k n+k = r(o:-{3-n+ 1)n! 2 F l(-{3,{3-0:+n;n+ 1;1)
_ r(a + 1)
- r( n + 1 + (3)r( 0: - (3 - n + 1) ,
and
t ( {3 ) ( 0: - (3 ) f(o: + 1)
_ k n + k - f(n + 1 + (3)f(o: - (3 - n + 1)'
k--n
Substituting these equalities into (15.15) and taking (1.48) into account we arrive
at (15.12).
To complete the proof of Theorem 15.1 we will justify the termwise fractional
differentiation of the series in (15.14). By Lemma 15.4 and (15.12) the initial and
00
differentiated series can be represented as (t - a)-{jgl(t) L: ()(t - a)k and
k=O
-a  f(o: + 1)
(t - a) g2(t) L..J f(n + 1 + (3)f(o: - (3 - n + 1)
n=-oo
respectively, functions gl(t) and g2(t) being analytic in some neighbourhood of the
point a. The latter series converge uniformly with respect to t E [a, x], and the
convergence of the second one is easily shown by means of the integral test for
convergence if we observe that
00
J f(o: + l)dr - 2 a
f(r + 1 + (3)r(o: - (3 - r + 1) -
00
( 15.16)
- Prudnikov, Brychkov and Marichev [2,2.2.2.4]. Therefore, termwise differentiation
of the series in (15.14) is possible in view of Lemma 15.2. .

 15. THE GENERALIZED LEIBNIZ RULE
283
An integral analogue of the generalized Leibniz rule (15.12) is given by the
following theorem.
Theorem 15.2. If functions 1 and 9 are analytic in some neighbourhood of the
point a, then the formula
00
V:+(fg) = J C: p)V:';:T-1l fvtll gdT,
-00
(15.17)
is valid if a, (3 E R 1 , a I -1, -2, . .. in the case of a non-integer /3 and (T:,6) as
defined by (1.48).
Proof. Since functions 1 and 9 are analytic in some neghbourhood of the point a
we have
00 I(n) { a )
/(x) = " (x - at,
 n!
n=O
00 g(m){a )
g{x) = L , {x-a)m.
m.
m=O
( 15.18)
By virtue of Lemmas 15.1 and 15.2 and (2.44) we obtain
00 I (n) { )( ) n-a+T+,6
(Va-T-,6 I){x) = " a x - a
a+  r{ T - a + /3 + n + 1) ,
00 (m) { )( ) m-T-,6
(VT+,6 g){x) = " 9 a x - a .
a+ ';;:0 r{ m - T - (3 + 1)
Substituting these relations into the right-hand side of (15.17) we get
00
1 = J (T: p)(V:';:T-1l f)(Vtllg)dT
-00
(15.19)
J oo 00 00 r{a + 1)/(n){a)g(m){a){x - a)n+m-adT
=-00 ; f(T+ p+ l)f(or - T- p+ l)f(T - or + p+ n + l)f(m - p - T + 1)"
Direct estimates demonstrate that the double series converges uniformly with
respect to T, encompassing the whole real axis. Therefore, its term-by-term

284 CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
intep;ration is possible, and, hence, (15.19) takes the form
00 00
1= L L f{a + l)f(n){a)g(m){a){x - at+ m - a
n=O m=O
00
xf
dr
f{ r + f3 + 1) f( a - r - (3 + 1) f( r - a + f3 + n + 1) f{m - r - f3 + 1) .
-00
It is known that the latter integral is equal to
00
f dr
f{ a + r)f{f3 - r)f{, + r)f{ 6 - r)
-00
f{ a + f3 + , + 6 - 3)
- r{a + f3 - l)f{a + 6 - l)f(, + f3 - l)f(, + 6 - 1)'
Re{a+f3+,+6) > 3.
( 15.20)
- Prudnikov, Brychkov and Marichev [2, 2.2.2.9]. Therefore, we have
I =   f(n){a) g<m){a) r{m + n + 1) (x _ a}n+m-a.
  n! m! f { n + m - a + 1 )
n=O m=O
Hence, according to (2.44) we obtain
00 00 f (n) { ) (m) ( )
1= L L n! a 9 m!a 'D+«x - at+ m )
n=Om=O
( 00 f (n) ( ) 00 (m) { ) )
= V::+  n! a (z - at I g m! a (z - a)m = V::+(fg),
which completes the proof of the theorem. .
We note that (15.12) and (15.17) are extended to the left-hand sided fractional
differentiation 'D b _ by a simple change of variable.
In conclusion we would point out that there are many generalizations and
modifications of the Leibniz rule (15.1). They touch upon the formulae both for
Liouville form of fractional integro-differentiation and some other ones - see S 17.2.

 16. ASYMPTOTIC EXPANSIONS OF FRACTIONAL INTEGRALS 285
 16. Asymptotic Expansions of Fractional
Integrals
The present section deals with the asymptotic expansions of the fractional integrals
:&
(I+f)(z) = r(lQ) J (z - t)a-l f(t)dt, Q > 0,
o
(16.1)
as x --+ 0 or x --+ +00, if an asymptotic expansion of f in the neighbourhood of
these points is known. Besides, the asymptotic expansions involving power terms
and the logarithmic and exponential ones are also considered.
16.1. Definitions and properties of asymptotic expansions
Let us list some necessary definitions. One can find them in detail in Sidorov,
Fedoryuk and Shabunin [1, Sect. 42], Fedoryuk [2, Ch. 1] and Olver [1, Ch. 1].
Definition 16.1. Let M be some point set on the real axis and let a be its limiting
point. The sequence {rpn(x)}, n = 0,1,2,..., defined for x E M, is called an
asymptotic sequence (as x --+ a, x EM) if for any n
rpn+l (x) = o( rpn(x» (x --+ a, x EM).
(16.2)
Here are some examples of asymptotic sequences:
1) rpn(x) = x lJa , X --+ OJ 2) rpn(x) = x- lJa , X --+ ooj
3) rpn(x) = (In x)lJa, X --+ OJ or x --+ 00, where {J.ln} is any increasing sequence
such that lim J.ln = 00.
n-oo
Definition 16.2. Let {rp(x)} be an asymptotic sequence (as x --+ a, x EM). A
00
formal series L: anrpn (x) with constant an is called an asymptotic expansion, or
n=O
an asymptotic series of the function f(x), if for any integer N  0
N
f(x) - L anrpn(x) = O(rpN(X» (x --+ a, x EM).
n=O
(16.3 )
To denote the connection between the asymptotic expansion and the function

286
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
f, we use the following symbol
00
f(x) """ L anCPn(x) (x -+ a, x EM).
n=O
(16.4)
An asymptotic expansion with respect to a power asymptotic sequence as in
examples (1) and (2) will be referred to as a power asymptotic series.
Further we shall omit mentioning the set M. We also note that although
convergent series are asymptotic ones, the term "asymptotic series" is usually used
for series which are not convergent in general.
Let us give some properties of asymptotic expansions. The most important
of them is the uniqueness property which means that the asymptotic expansion of
the given function with respect to a given asymptotic sequence is unique. However,
00
different functions may have the same asymptotic expansions, e.g. e- Z """ L: O.x- n
n=O
00
and 0 """ L: 0 . x- n as x -+ +00.
n=O
One may treat power asymptotic series in the same way as convergent power
series. Namely, we can add, multiply, integrate and sometimes differentiate term by
term such series. Definition 16.2 can be used to prove the following propositions.
Lemma 16.1. Let {Iln} and {lIn} be increasing sequences, lim Iln = lim IIn =
n-oo n-oo
+00 and let the asymptotic expansions
00
f(x) """ L anx- IJ ",
n=O
00
g(x) ,.y L b n x- lI "
n=O
hold as x -+ 00. Then
00
f(x) + g(x) """ L cnx->''',
n=O
00
f(x)g(x) """ L cnx- u "
n=O
as x -+ +00 where the increasing sequence {An}, lim An = +00, is obtained by
n-oo
regrouping the members of the sequences {Iln} and {lI n }, while the increasing
sequence {un}, lim Un = +00, is obtained by regrouping the products Ilkllj
n-oo
according to their increasing values. In particular, Ao = min(llo, 110) and Uo = ilOilO.
Lemma 16.2. Let {Iln} be an increasing sequence, Iln > 1 and lim Iln = +00. If
n-oo

 16. ASYMPTOTIC EXPANSIONS OF FRACTIONAL INTEGRALS 287
J(x) is continuous in (c, +00) and satisfies
00
J(x)  LOnX-IJn, as x --+ +00,
n=O
(16.5)
then this series can be integrated term by term
00 00
f J(t)dt  L x-IJn+l as x -:-+ +00.
x n=O J1.n - 1
Remark 16.1. Propositions similar to Lemmas 16.1 and 16.2 are true for
asymptotic series with respect to the sequence xIJJLas x --+ 0 i.e.
00
f(x)  LOnxIJn, as x --+ O.
n=O
(16.6)
We also need the following Watson lemma.
Lemma 16.3. Let a > 0, /3 > 0 and let J(t) be continuous for 0  t  0 and
infinitely differentiable in the neighbourhood of t = O. Then the asymptotic equality
a 00 (n)
f ,_%to t fJ - 1 f(t)dt   L r ( n  11 ) f nO) .,-
o n=O
(16.7)
is trne as x --+ +00.
One may find the proof of this lemma, for example, in Sidorov, Fedoryuk and
Shabunin [I, p .408].
16.2. The case of a power asymptotic expansion
Let {J1.n} be an increasing sequence and lim J1.n = +00. We shall look for
n-oo
asymptotic expansions of the fractional integrals Io+J provided that the function
J has an asymptotic representation of the form (16.5) or (16.6).
The simplest result is obtained from (16.6). Namely, the following statement
is true.
Theorem 16.1. Let {J1.n} be an increasing sequence, J1.n > -1 and lim J1.n = +00.
n-oo
If J(x) satisfies the condition {16.6}, then the fractional integral (I+J)(x) has the

288
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
asymptotic expansion as x --+ 00
(1 01 f)( x) "'J  anr(Jjn + 1) xIJ.. +01.
0+  f( a + Jjn + 1)
(16.8)
N
Proof. By (16.6), we have f(t) = L: ant IJ .. + RN(t). Therefore, in view of (2.44)
n=O
we obtain
1
(11:+/)(z) = r:) J (1- tt- 1 f(zt)dt
o
(16.9)
N 1
= L anr(Jjn + 1) x IJ ..+ OI +  J RN(xt)(1 - t)0I-1dt.
n=O r(a + Jjn + 1) f(a) 0
By (16.4) we see that RN(t) = o(t IJ ..) as t --+ O. Then for sufficiently large N we
have RN(t) = t IJN a(t), where a(t) is an infinitely small function as t --+ O. Hence,
1
we deduce the relation J RN(xt)(l- t)0I-1dt = o(x IJN ) as x --+ 0 and the conclusion
o
of the theorem follows from (16.9). .
The case when f(t) has an asymptotic expansion (16.5) is more complicated.
Three main approaches to find asymptotic expansion of the fractional integral
(16.1) are known. They are: the method of successive expansions - Riekstyn'sh
[1] and [2, Vol. 1, Sec. 11]; the method based on the representation of I+f as the
Parseval equality (1.116) for the Mellin transform (1.112) - Riekstyn'sh [2, Vol. 3,
SS 31.1-31.3], Handelsman and Lew [1, 2] and Wong [2]); and the method based on
the theory of distributions - McClure and Wong [1]. We shall give the asymptotic
representation for (16.1) by the method of successive expansions which is the most
elementary approach. For simplicity, we suppose that f(t) has the asymptotic
expanSlOn
00
f(t) "'J L ant-n-/J, 0 < /3  1, as t --+ +00.
n=O
( 16.10)
It should be noted that the asymptotic expansions for I+f will be different
in the cases when 0 < /3 < 1 or /3 = 1. If the first case also gives us a power
asymptotic expansion then in the second case we obtain a power-logarithmic one.
Theorem 16.2. Let f be a locally integrable function on (0, +00) satisfying (16.10)

 16. ASYMPTOTIC EXPANSIONS OF FRACTIONAL INTEGRALS 289
with 0 < /3 < 1, and let
l1(t) = t P I(t) - ao,
(16.11)
In+l(t) = tln(t) - an, n = 1,2,...
If Kn = sup Itl n + 1 (t)1 is finite for each n  0 then the fractional integral
tE(I/2,oo)
(Ig+ 1)( x) has the asymptotic expansion
( Ia I)( )  r(1 - n - /3) a-p-n  b n a-n ( 16 12 )
0+ x f'OoJ  r(1 + a _ n _ /3) anx -  r(1 + a _ n) x, .
as x --+ +00 where
00
(-I)n f -P
b n = (n _ I)! In (t)t dt,
o
n = 1,2,...
(16.13)
Proof. Since I(t) = t-P[ao + 11 (t»), where l1(t) is given by (16.11) then by (2.44)
we have
a _  aor(1 - /3) a-p
(lo+/)(x) - r(a) I(x) + r(1 + a _ /3) x, (16.14)
1
where I(x) = x- P J(1 - t)a- 1 /1(xt)t- 4 dt. Denoting 91(t) = (1 - t)a-l - 1 and
o
taking (16.11) into account we obtain
1 1
x P I(x) = f 11(xt)gl(t)t- P dt + f 11 (xt)t-Pdt
o 0
( 1 1 ) 1
=  f h(zt) 91t) t-Pdt +°1 f 91t) rPdt + f f1(zt)r P dt.
000
We continue this process. We introduce the notations
gn(t) = (1- t)a-1 -  (1 t)' t',
k=O
( 16.15)
9:(t) = t-ngn(t), n = 1,2,....

290
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Then x fJ I( x) can be rewritten as
1 N 1
x fJ I(x) = x J fN+l(xt)giv(t)t- fJ dt + E :: J g:(t)t-fJdt
o n= 1 0
1
N-l(l_a) J
+ L n!x n n fn+dxt)t-fJ dt.
n=O 0
( 16.16)
We note that (16.10), (16.11) and (16.15) yield the relations
00
fn+l (t)  L an+tt- t , t --+ +00, n = 0,1,2,.. .,
k=1
(16.17)
- ( t ) =  (go)(k)(O) t k - n =  (1- a)k t k - n
gn L...J k! L...J k! '
k=n k=n
(16.18)
_ ( 0 ) = (1 - a)n
gn "
n.
n = 1,2,....
In accordance with (16.17) and Lemma 16.2 we have the estimate as x --+ +00
1 00 00
J fn+l (xt)t- fJ dt = x fJ - 1 J fn+l (t)t- fJ dt - x fJ - 1 J fn+l (t)t- fJ dt
° 0 x
00 00
 ",P-l J /n+l(t)I-Pdl - E (k +  1)",>' n = 0,1,2,...
o k=1
(16.19)
1
Now we estimate J fN+l(xt)9N(t)t- fJ dt. According to the conditions of the
°
theorem IfN+l(t)1 $ KNt- I , t  1/2, for any N  O. In view of (16.15)
LN = sup Igiv(t)/ is also finite and hence we have the estimate /giv(t)1 $ LN,
tE(O,I/2)
o $ t $ 1/2. Therefore, we obtain
1 1/2 1
J fN+l (xt)giv (t)t-fJdt  LN J IfN+l(xt)lt- fJ dt + J(NX- 1 J Igj..r(t)lt- 1 - fJ dt
° 0 1/2

 16. ASYMPTOTIC EXPANSIONS OF FRACTIONAL INTEGRALS 291
00 1
 L N x 13 - 1 f I/N+1{t)lt- 13 dt + KNX- 1 f Igiv{t)lt- 1 - 13 dt.
o 1/2
Hence, we deduce the estimate
1
f IN+1(Xt)giv(t)t- 13 dt = O(x13-1) as x --+ +00.
o
( 16.20)
Taking (16.19) and (16.20) into account we have the relation for I(x) as
x --+ +00
N ( 1 n-1 )
_ an . _ 13 (1 - a) 1c
1(.,) -  .,n+p I gn(t)t dt -  k!(n + P - k - 1)
N1 00
 (1 - a)n f -13 ( 1 )
+ L...J ,(n+1) In+1 (t)t dt + 0 N+1 '
O n.x x
n= 0
and by (16.15), (1.72) and (1.77) we find
N
1(x) = '" a n f(a)f(1 - n - /3) x-n-13
 f(l-n-13+a)
N 1 00
 (1 - a)n f -13 ( 1 )
+ L...J 'n+1 In+1{t)t dt + 0 N+1 .
n.x x
n=O 0
Substituting this into (16.14) and taking the relation
(1 - a)n = (-I)nf(a)/f(a - n)
(16.21)
into account we obtain the desired result (16.12). .
Theorem 16.3. Let 1 be a locally integrable function on (0, +00) satisfying (16.10)
with 13 = 1, and let
h(t) = I(t), In+1(t) = tln(t) - a n -1 n = 1,2,...
( 16.22)
If Kn = sup Itln+1 (t)1 is finite for each n  0, then the fractional integral
tE(I/2,oo)

292
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
(I+ 1)( X) has the asymptotic expansion
00 ( l ) n 00
(101 I)(x) f"J In x "" - an x Ol - n - 1 + "" c n x Ol - n - 1
0+  r ( a-n ) n! 
n=O n=O
(16.23)
as x --+ +00, where
Co = ao[1/J(I) -1/J(a)] + d 1 ,
an  (1- a)A; (-I)nd n + 1 (16.24)
C n = r(a) t'o k!(k - n) + f(a _ n)n!' n = 1,2,...,
Ien
1 00
d n + 1 = f In+1(t)dt + f (In+1(t) - a tn ) dt, n = 0,1,2,..., (16.25)
o 1
with 1/J(z) being given by (1.67).
This theorem is proved in the same way as Theorem 16.2 above. Using the
notation (16.15) and (16.22) we obtain the following representation
1 N 1
r(a)x-OI(I+/)(x) = x f IN+1(xt)gN(t)dt + L a:1 f g:(t)dt
o n=1 0
(16.26)
N-1 1
"" (1 - a)n f
+  n!x n In+1(xt)dt
n=O 0
instead of (16.16). Here the estimate (16.17) is replaced by
00
In+1(t) f"J Lan+Ie-1t-k, t --+ +00, n = 0,1,2,...
1e=1
( 16.27)
1
Using (16.25) we rewrite the integral J In+1 (xt)dt as
o
1 :t: [ 00 ]
f In+1(xt)dt =  f In+1(t)dt =  an In x - f (In+1(t) - n ) dt + d n + 1 .
o 0 :t:

 16. ASYMPTOTIC EXPANSIONS OF FRACTIONAL INTEGRALS 293
Then, according to (16.27) and Lemma 16.2 we have
1 ( 00 )
1 at+n
f fn+l(xt)dt f'OoJ;- an In x - E kxl: + d n + 1
o 1:=1
 Z -+ +00. Substituting this estimate into (16.26) and taking (16.20) with (3 = 0
into account we obtain the following equalities as x -+ +00
N 1 N-l
x-Otr(Q)(I+f)(x) = L a:: 1 f g:(t)dt + L (!:nt (an In x + dn+d
n=1 0 n=O
N-2 ( ) N-n-l ( )
_  1 - Q n  ak+n + 0 In x
L.J n!xn+l L.J kxl: xN +1
n=O 1:=1
N-l ( 1 )
(1 - Q)n an 1 .
=Inx L 'n+l + - a O f gl(t)dt+d l
O n.x x
n= 0
( 1 )
N-l 1 . n-l (1- Q)t
+  .,nH [an f gn+l(t)dt -  k!(n - k)
n_l 0 t-o
(1 - Q)n ] ( In x )
+ d n + 1 n! + 0 x N +1 .
According to (16.15) and the formula 2.2.4.20 from Prudnikov, Brychkov and
Marichev [1] we have
1 1
f f (1 t)Ot-l 1
g(t)dt = - t - dt = 1jJ(I) -1jJ(I- Q),
o 0
1 n-l 00
f ."" (I-Q)t "" (I-Q)1:
gn+l(t)dt- L.J k!(n-k) = L.J k!(n-k)'
o t=o 1:=0
I:n
Hence, in view of (16.21) we derive the desired result (16.22). .

294
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
16.3. The case of a power-logarithmic asymptotic expansion
Let f(t) have a power-logarithmic asymptotic representation
00
f(t) f'OoJ L ant IJ .. (In t)m.. as t --+ +00,
n=O
(16.28)
where {J.ln} is an increasing sequence, lim J.ln = +00 and m n are arbitrary real
n-oo
numbers.
In this case asymptotic expansions of the fractional integrals Ig+f can be
found in the same way as it has been done in S 16.2 by means of the method of
successive expansions. We shall not dwell on this here (see S 17.1, note to S 16.3)
and consider only a case of a power-logarithmic asymptotic representation of the
form
00
f(t) f'OoJ t- P L an(In t)'Y-n as t --+ +00,
n=O
( 16.29)
where {3 is a nonnegative real number and, is an arbitrary one. In this case the
asymptotic expansion for Ig+f can be obtained in two ways: by a method based
on the representation of Ig+f as the Mellin convolution (1.114) - Bleinstein [1],
and by a method of direct estimates - Wong [1]. We shall use the latter approach.
Just as in the case of the power asymptotic expansion (16.10) in Theorems
16.2 and 16.3 the asymptotic expansions for Ig+f will be different in the cases
when 0 $ {3 < 1 or (3  1. We consider the former case first.
Theorem 16.4. Let f(t) be a real-valued nonnegative and locally integrable
function on [0, +00) satisfying (16.29) with 0  /3 < 1. Then
00
(Ig+f)(z) f'OoJ za-p E Bn(ln z)'Y-n,
n=O
(16.30)
as z --+ +00 where
1 n ( , + k - n )
Bn = Bn(a) = f(a) E an-k k h(a, /3),
k=O
(16.31)
1
Ok(a,{3) = 1(1- t)a-1t-P(Int)kdt.
o
( 16.32)

 16. ASYMPTOTIC EXPANSIONS OF FRACTIONAL INTEGRALS 295
Proof. We rewrite the fractional integral Ig+f as
..;x x
la x -  f f(t)dt  f f(t)dt
( o+f)( ) - r(a) (x - t)l-a + r(a) (x - t)l-a
o Vi
( 16.33)
= J 1 f + J 2 f.
Let us prove that hf is asymptotically small in comparison with J 2 f. We choose
t such that 0 < c < 1- {3. Since (lnt)'Y = O(t£) as t --+ +00, it follows from (16.29)
that f(t) = O(t-,6+£) as t --+ +00 and hence,
Vi
f f(t)dt = 0(x(1-,6+£)/2) as x --+ +00.
o
(16.34)
Vi
We set Ma = max (1- t)a-1. Then r( a)hf  Maxa-1 J f(t)dt and by (16.34)
Ot 1/2 0
we obtain the estimate
(hf)(x) = 0(x a -,6-6) as x --+ +00,
( 16.35)
where 6 = (1 - c - (3)/2 > O.
Now we estimate J 2 f. By (16.29) we have
N
f(t) = t-,6 E an (In t)'Y-n + RN(t),
n=O
( 16.36)
RN(t) = 0(t-,6 (In t)'Y-N-1) as t --+ +00.
Hence
N
(J2I)(z) =  rc:) L(a. 11. 'Y - R; z) + TN(Z).
where the notation
( 16.37)
x
L(a,{3,-Yi x) = f (x - t)a- 1 t-,6(1nt)'Ydt,
Vi
( 16.38)
x
TN(Z) = rta) f (z - t)O-1 RN(t)dt
Vi

296
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
is used. Substituting t = xr into (16.38) we get
1 
L(",p,1;Z) = za-p(lup J (1- r)a-1r- p (1 + ::: ) dr.
:e-1/'J
(16.39)
 00 1c
Since x- 1 / 2  r  1 then Ilnr/lnxl  1/2 and hence, (1 + :; ) = L: n) (I; ) .
1c=O
Inserting this expression into (16.39), carrying out the term by term integration
and using (16.32) and the directly verified estimate
:e-1/'J
J (1 - r)a- 1 r-,6(ln r) 1c dr = O(x- 6 ), 6 > 0, as x --+ +00,
o
we obtain the asymptotic expansion
L(",p, 1; z)  za-p  (r)n>(",p)(Inz P -> as z -> +00.
( 16.40)
According to (16.36) there exist constants k > 0 and c > 1 such that IRN(t)1 
kt-,6(ln tp-N-l for t  1. Hence, we find that
:e
TN(Z)  rt,,) J (z - tt- 1rP (ln t),-N- 1 dt.
.;z
Taking into consideration (16.38) and (16.40) we get rN(x) = O(x a -,6(lnxp-N-l)
as x --+ +00. Substituting (16.40) into (16.37) and taking the last estimate into
account, after regrouping the terms we obtain the asymptotic expansion of the
form
00
(J2/)(X)  x a -,6 L Bn(lnxp-n as x --+ +00,
n=O
where Bn is given by (16.31). Hence, according to (16.33) and (16.35) we conclude
the proof of the theorem. .
The case /3 = 1 is characterized by
Theorem 16.5. Let / be a real-valued, nonnegative and locally integrable function

 16. ASYMPTOTIC EXPANSIONS OF FRACTIONAL INTEGRALS 297
41R (0,+00) satisfying (16.29) with /3 = 1. Then
:I: 00
r(a)x1-a(I+f)(x)  / f(t)dt + L Cn(ln Xp-n,
o n=O
(16.41)
as x --+ +00, where
n ( , + k - n )
C n = Lan-I: k AI:(a),
1:=0
( 16.42)
1
A.(n) = / (1 - 1);-1 - 1 (In t)'dt.
o
( 16.43)
The proof of this theorem follows the same lines as in Theorem 16.4 after
putting down the integral I+f in the form
a-1 / :1: 1 / :1:
(I+)(x) = ;(a) f(t)dt + f(a) [(x - t)a-1 - x a - 1 )f(t)dt
o 0
( 16.44)
and representing the second term in the right-hand side as a sum of two integrals
over the intervals (O,JX) and (y'X,x) (see (16.33».
The case of {3 > 1 in (16.29) is investigated by expanding (x - t)a-1 into the
Taylor series of powers of x and by using the representation
a_1[,8]-1 ( ) :1: ( ) 1:
(1+)(x) = ;(n)  (-1)' n  1 / J(t)  dt
1:_0 0
:I: [ [,6]-1 1: ]
+ f(1nJ (x - W- I - x a - I  (-1)' (n  1) () J(t)dt
o 1:-0
( 16.45)
instead of (16.44). The result is, however, more complicated than (16.30) and
(16.41), so we shall not dwell on its formulation.
16.4. The case of a power-exponential asymptotic expansion
In order to find an asymptotic representation for the fractional integral I+ f
when f(t) has a power-exponential asymptotic representation we have to use the

298
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
Watson Lemma (Lemma 16.3). Here we consider only the case of the simplest
power-exponential asymptotic expansion near zero when
00
f(t) f'OoJ e- 1 / t L ant n as t --+ +0.
n=O
( 16.46)
Theorem 16.6. If f(t) satisfies the condition (16.46), then
00
(Ig+f)(x) f'OoJ e- 1 / x L H n x n + 2a ,
n=O
(16.47)
as x --+ +0, where
Hn = Hn(a) = r(a + n + 1)  (-1t- kr (n - k + a)aA; .
r(a)  (n - k)!r(a + k + 1)
( 16.48)
Proof. According to (16.1) and (16.46) the relation
00 x
(la f)(x) f'OoJ   f (x - ty-le-l/ttndt
0+  r( a) 0
(16.49)
as x --+ +0 is true. We consider the integral
x
Ja,n(x) = f (x - t)a- 1 e- 1 / t t n dt.
o
( 16.50)
Changing the variable t = x/(1 + r) in (16.50) we have
00
Ja,n(x) = xa+ne- 1 / x f e- T / x r a - 1 (1 + r)-a-n-1dr.
o

 16. ASYMPTOTIC EXPANSIONS OF FRACTIONAL INTEGRALS 299
Taking Lemma 16.3 into account we find
J ( ) f"J 2a+n -1/x  (-I)k(a + n + l)kf(k + a) k
a,n X X e L...J k! x ,
k=O
(16.51)
as x --+ +0. Setting this expression into (16.49) we obtain the desired result
(16.47). .
16.5. The asymptotic solution of Abel's equation
In connection with some applied problems it is often sufficient to seek not for a
solution cp of Abel's equation
x
a_I f cp(t)dt
(Io+CP)(x) = f(a) (x _ t)1-a = J(x),
o
o < X < +00,
(16.52)
but its asymptotic expansion near some point (usually near zero or infinity) if the
asymptotic expansions of J near this point is known. In this connection we use the
idea of the asymptotic solution.
Definition 16.3. Let {tP(x)} be an asymptotic sequence as x --+ a, and let the free
term J(x) of (16.52) have the asymptotic expansion
00
J(x) = L bntPn(x) as x --+ a (b n E C, n = 0,1,2,...).
n=O
( 16.53)
If there exists an asymptotic sequence {Xn(x)} as x --+ a such that
00
cp(x) f"J L dnXn(x) as x --+ a (d n E C, n = 0,1,2,...).
n=O
(16.54)
and
00
(I<f+cp)(x) f"J L bntPn(x) as x --+ a,
n=O
then the asymptotic expansion (16.54) will be called asymptotic solution of (16.52)
as x --+ a.
We note that the uniqueness of an asymptotic solution follows from the
uniqueness property of the asymptotic expansion of the given function with respect

300
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
to the given asymptotic sequence. However, the existence of the solution itself does
not follow in general from the existence of its asymptotic solution.
If we know asymptotic expansions for the fractional integral (16.1), we can
find an asymptotic solution of Abel's equation (16.52). For example, the following
propositions which characterize the asymptotic solutions of (16.52) as x --+ 0 and
x --+ +00 are corollaries of Theorems 16.1 and 16.4.
1) If the free term J(x) has the asymptotic expansion
00
J(x) f"J L anx IJ .. as x --+ 0,
n=O
(16.55)
where fln is an increasing sequence, JLo > a-I and lim fln = 00, then the
n-oo
asymptotic solution of (16.52) is given by the formula
cp(t) f"J f: f(fln + l)a n t IJ ..- a as t --+ O.
n=O f(fln - a + 1)
( 16.56)
2) Let 0 < a < 1 and let J have the asymptotic expansion
00
J(x) f"J x-/3 L bn(ln x)'Y-n as x --+ +00,
n=O
(16.57)
where 0  {3 < 1 - a and, is an arbitrary real number. Then the asymptotic
solution of (16.52), considered to be real, nonnegative and locally integrable on
[0, +00), is representable as
00
cp(t) f"J t- a -/3 L cm(ln t)'Y-m as t --+ 00.
m=O
( 16.58)
Here the constants C m are found (by the known b n ) via the formulae
n ( )
1 ,-n+m
b n = f(a) E C n - m mm= Ok(a, a + {3),

 17. ADDITIONAL INFORMATION TO CHAPTER 3
301
where h(a,,8) are given by (16.32). For example,
Co =
r(1 - ,8)
r(l_a_,8) b o ,
r(1 -,8)
r(1 _ a _ ,8) {b l + b O 'Y[1/J(1 - ,8) -1/J(1 - a - ,8)]}
Cl =
etc.
fi 17. Bibliographical Remarks and
Additional Information to Chapter 3
17.1. Historical notes
Notes to  10.1. The second set of relations in (10.4) and (10.5) are sometimes called the first
index law - Love [5] (1972). They characterize the semigroup property of the operators 1+ and
1&_ considered usually in L,(a, b), 1  'P < 00. The case p = 1 is specially treated in Theorem
2.5, in  4.1 (note to  2.7), and in the papers by Love [2, p.174], [3, p.l058, 1060] (1967). The
final statement of Theorem 10.1 was proved by Marichev [13, Theorem 1] (1990).
Relations (10.4) were dealt with by many authors. In the case Rea > 0 and ReP > 0 they
were considered by Love and Young [1] (1938), Hille [1] (1939) and Kober [1] (1940). Kober [2]
(1941) investigated also the case Rea = ReP = 0, a = 0 for functions in a certain subspace of
L,(O, b), p  2. Riesz [6, p.12] (1949) extended these relations to the case when Rea or Rep
may be negative, but I e C'P([a,b]), p> max(-Rea,-Re(a + P)). The relations in (10.4) for
all values of a and P in various spaces of distributions were considered by Gel'land and Shilov
[2] (1959) and McBride [2] (1975), [4] (1977), [9] (1983); the latter author used the spaces F',IJ
and Fp,1J mainly -  8.4. In the space Ll (a, b) and in its subspaces defined by existence of the
cOlTesponding fractional derivatives these relations were treated in detail by Love [5] (1972) for
all values of a and p.
As regards (10.6) and (10.7) - see the notes to  10.3 below.
The particular case al - P1 = 1 of (10.12) was given in Chen [I, p.309] (1959), although
this expression apparently seems to be known earlier. In the case of analytic functions it was
treated in the thesis by Tremblay [I, p.292] (1974).
Theorems 10.2-10.3 and Lemmas 10.1 and 10.2 were proved by Marichev [13, Theorems
2, 3 and Lemmas I, 2] (1990). Theorem 10.4 in the given fonn is published for the first
time. In the case p = 1 and Rea> 0 they were proved by Love [2, p.181, 184], [3, p.l063,
1070] (1967), who dealt with the weighted spaces Qq = {/(x) : x q I(x) e Ldo, d], d < oo} and
R,. = {/(x) : x r I(x) e LICa, 00), a > o}. In these papers the operators (10.18), (10.19) and
(10.20), (10.21), respectively, were studied in detail and the decompositions (10.22)-(10.29) were
given for them for the values of the parameters, guaranteeing the validity of decompositions for
all I(x) in Qq or Rr. The values of parameters guaranteeing the validity of these relations
only by the condition of the representability I(x) = 1+ ,p(x), ReJ.l. > 0, 'fjJ e L,(o, d), were not
considered in the cited papers.
Notes to  10.2. Theorem 10.5 is published for the first time. However, the fonner
of the relations (10.38) in another fonn was obtained by Love [7) (1985). He called this
equation the third index law and investigated in detail the conditions of its validity in the space
LIC(O, oo)jXIJ(x + 1)11) or its subspaces defined by the representability conditions (see above)
which were firstly used by the author in the cycle of his papers.

302
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
We observe that the inversion of the operator (10.39) for a function J(t) such that J(t) = 0
for t > a > 0 was obtained by Ahiezer and Shcherbina [1] (1957) -  39.2 (notes 35.7 and 35.8).
Notes to  10.3. Theorems 10.6 and 10.7 were proved by Marichev [13, Theorems 4, 5]
(1990). However, the relations (10.42), (10.43) and (10.6), (10.7), characterizing the second index
law (in Love's tenninology - Love [5], 1972) arose long ago. The paper by Widder [1, p.17,
Lemma 3.11] (1938) was apparently the first which contained the particular case of the relation
(10.6) of the fonn V k x2k-1Vk-1 J(x) = xk- 1 V2k-1xk lex), V = dldx, obtained by means of
the Leibniz fonnula. These relations in tenns of the Kober operators Ia and K;;'a (see (18.5)
and (18.6» were established by Kober [1] (1940) for a > 0, (3 > 0, see (18.16). Equations of the
type (10.6) and (10.7) oCCUlTed also in Higgins [4, p.7-8] (1965) and Love [2, (3.9)] (1967). As
well, Love [5] (1972) investigated (10.6) in details in the space Qq for all values of parameters and
obtained the results of the type of Theorem 10.6. The study of the second index law in certain
spaces of distributions was Wldertaken by Erdelyi [15] (1972) and McBride [1], [2], [4], [6], [9]
(1975-1983), the latter dealing with the operators Ig+ ixm and I;xm as in (18.41), in the spaces
Fp,/J' Fp,/J -  8.4.
The operator (10.47) in the spaces Qq and Rr was considered by Marichev [I], [2] (1972-
1974), while the operator (10.48) with m = 0 was treated by McBride [7] (1982) in the space Fp,/J
and by Dimovski and Kiryakova [2] (1985), and also Kiryakova [I, 2] in the case n = O.
Notes to  10.4. Theorems 10.8 and 10.9 are published for the first time. However, the
n
compositions n I:ie>'jX lex) were considered long ago. They can be fOWld in the papers by
;=1
Letnikov [9] (1888) and Nekrasov [3] (1888) where the solutions of nth order differential equations
with two-term coefficients were looked for in such a fonn. Such compositions with n = 2 and their
inversions occurred in Davis [2, p.137] (1927), but their systematic investigation was Wldertaken
much later by Prabhakar [1] (1969), [2] (1971) in connection with the problem of inverting the
operator (10.55). The operators (10.55) and (10.56) as particular cases of the operator (10.63),
and certainly its analogue were considered by Prabhakar [5] (1972) and [6] (1977), respectively.
The operator in the right-hand side of (10.62) in the case .x = 1 was studied by Belward [1]
(1972), who obtained its inversion via the fractional integrals which use the composition structure
(10.62) of the operator.
Notes to  11.1. Theorem 11.1 is due to Plemeli (1908) and Privalov (1916) in the
non-weighted case (the proof and references are given in Muskhelisvili [1]) and to Duduchava
[I], [2] (1970) in the weighted case. Theorem 11.2 is obtained from Theorem 11.1 by simple
arguments. For example, by mapping real axis onto a circumference of a circle. Theorem 11.3
is due to Riesz (1927) in the non-weighted case and to Hardy and Littlewood [6] (1936) and
Hvedelidze (1957) in the weighted case. The proof and references may be seen for example in
Gohberg and Krupnik [4].
Notes to  11.2 and 11.3. The first paper, which contained at least implicitly the idea
of connection between left- and right-hand sided fractional integration via a singular operator,
was the work by Zeilon [1] (1924). There is a formalism in this paper in the case of a half-axis,
which in fact gives the connection (11.30) or one of the relations (12.8). This is seen in p.9 and
7 of the cited paper. It should be stressed that this article, although executed fonnally and
having some errors in expressions including the afore mentioned p.9, contains interesting and
original ideas evidently influenced by T. Carleman, but not noticed either by contemporaries or
by later investigators. This was the first paper that contained consideration of the generalized
Abel equation which involves both left- and right-hand sided fractional integration. This equation
is considered in  30. This interesting work remained unknown to the authors of the present book
also, who only discovered it by chance in 1985.
Relations (11.10)-(11.12) were obtained independently by von Wolfersdorf [1] (1965) and by
Samko [4] (1967), the identity (11.10) being also given by Kober [5] (1967). Corollaries 1 and 2 of
Theorem 11.4 and Remark 11.2 are given in the same paper by Samko. Relations (11.16)-(11.19)
were established by Samko [1], [2] (1967), [3], [5] (1968); (11.18) and (11.19) were also obtained
by yon Wolfersdorf [2] (1965). We refer also to the papers by Kuttner [1] (1953) and Chen (1961)
where an attempt was made to find the relation of the type (11.17), but the explicit connection of

 17. ADDITIONAL INFORMATION TO CHAPTER 3
303
such a type was not discovered there. Relation (11.19) was rediscovered later on in Juberg [1]-[3]
(1972-1973).
Modifications of (11.16) and (11.17) for values 0 < cv < 2 in the form (11.23) and (11.24)
were obtained by Kilbas [5] (1977) (see also (7)) and Rubin [10] (1977). Modifications of (11.18)
and (11.19) in the form (11.25) and (11.26) were given by Rubin [11] (1980).
Notes to  11.4. Theorem 11.6 and Corollary 1 were in fact obtained by Rubin [1] (1972).
Invariance of the space [a(L p ) relative to multiplication by the characteristic function of the
half-axis by another way was shown in Samko [15] (1974). Theorem 11.8, proved by Samko, has
not been published elsewhere.
Notes to  12.1. The wide use of the operator [a originated in the Riesz's papers [2]
(1936), [4] (1938), [5] (1939), where this operator was introduced in the multidimensional case.
The potential Hal{) as an independent object for investigation appeared in Okikiolu [1] (1965),
[2] (1966), although the more general potential [<t, containing both the one-dimensional Riesz
potential [al{) and the potential Hal{), was introduced earlier by Feller [1] (1952). Relations (12.6)
and (12.7) were proved in Okikiolu [2] (1966) and Samko [4] (1968). We refer also to [13] (1971)
although at least the former of these relations was known already to Thorin [I, p.37) (1948).
Lemma 12.2 was proved in Kober [6] (1968) and Samko [13] (1971). Theorem 12.1 is given in
Samko [13]. Equations (12.19)-(12.21) were earlier noted in Kober [6]. The statement (12.22)
on the semi group property of operators 1 was obtained by Feller [1] (1952). Potentials [al{)
and Hal{) in the form of Stieltjes integrals were considered in the book by Butzer and Trebels
[2, p.31-35] (1968), see also [1].
Note to  12.2. The results are obtained by Samko and have not been published elsewhere.
Note to  12.4. Relations (12.46) and (12.47) are obtained by Samko [1] (1967), see also
[3] (1968).
Notes to  13.1 and 13.2. Theorem 13.1 was obtained by Rubin [1] (1972). In connection
with the Corollary of Theorem 13.1 we observe that the coincidence (almost everywhere) of
Riemann-Liouville and Marchaud derivatives for absolutely continuous functions was noted by
Tamarkin [I, p.222, Lemma 1] (1930). The characterization of fractional integrals of fWictions
in L,(a, b) given in Theorem 13.2 was obtained in Rubin [1] (1972), [3] (1973). Statements of
Theorems 13.3-13.5 have not been given earlier elsewhere. A statement close to that of Theorem
13.5 is contained in Marchaud [I, p.385] (1927). Theorem 13.6 and more general Theorem 13.12
were given by Samko not being published earlier. Theorem 13.7 is proved in Samko [14] (1973),
the case p = 1 in this theorem being earlier obtained by Dzherbashyan and Nersesyan [6, p.17]
(1968) Wider the assumption that a(x) e H 1 ([a, b)). In connection with Theorem 13.7 we note
that multipliers in the weighted space x-a [g+[L,(R; p)], p = x"', were considered by Penzel
[I, p.I8-23], [2] (1986-1987).
Notes to  13.3. The results here are obtained in Rubin [1] (1972) and are presented
here with some supplements. In connection with the representation [+ I{) = 1:+ 1/1, given in the
Corollary of Theorem 13.9, we note that Prof. E.R. Love put the question at the Conference
on Fractional Calculus held in New Haven (1974) about the existence of the connection between
fractional integrals with different linear limits of integration - Osler [9, p.376] (1975). The answer
to this question was in fact known in 1972 in Russian investigations.
Notes to  13.4. Results here are also obtained by Rubin [7] (1974). We note that Theorem
13.13 and Lemmas 13.2 and 13.3 are valid for all JJl < 1 + .x - Rubin [22] (1986). Similarly one
may take JJl < .x + 1 also in (13.37) and in L 13.2'.
Notes to  13.5. The spaces Ha and Ha and Lemma 13.4 were contained in the paper by
Samko [27,  9] (1978). Theorem 13.4 was also proved there. A statement close to Theorem 13.14
was implicitly contained in von Wolfersdorf [2] (1965).
Notes to  13.6. The generalized Holder space HW apparently appeared first in Stechkill
[1] (1951). As for Zygmund type estimates for singular integrals we refer to Bari and Stechkin
[1] (1956) and the book by Guseinov and Muhtarov [1]. Theorems 13.15-13.17 were proved
by Murdaev [1] (1985). We note that there is a generalization of Theorem 13.16 to the
multidimensional case - Samko and Yakubov [2] (1985). The functional class () was defined in
Murdaev and Samko [1] (1985) and also [2]. This class is known (for (3  1) in the case S = 0,

304
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
e.g. Guseinov and Muhtarov [1]. In the case of integer (3 = k this class was introduced by Samko
and Yakubov [2] (1985). It was used implicitly in Samko and Yakubov [1] (1984). The estimate
(13.69) and Theorem 13.18 were obtained by Murdaev and Samko [1] (1985), see alao [2], [3]
(1986) and Samko and Murdaev [1] (1987).
Notes to  14.1. Theorem 14.1 was established by Hardy and Littlewood [4] (1928) by
complex analysis methods. Its proof by the methods of real analysis may be found in D'in [1]
(1959).
Notes to  14.2 and 14.3. Theorems 14.5-14.7 were given by Hardy and Littlewood [3]
(1928) in the case of periodic functions. We gave these theorems in a more general setting than in
[3]. Theorems 14.2-14.4 are their modifications. In the recent paper by K06tometov [1] (1990) the
sharp version of these Hardy-Littlewood's results was obtained: the operator 1:+ maps the space
H;([a, b» onto H;+a([a, b]), 0 < a < I, one-to-one provided that these spaces are defined in the
unsymmetrical way, i.e. via the continuation by zero to the half axis (-00, G) while preserving the
space and without trouble about the possibility of such a continuation to the right of the point b.
Notes to  14.4. Equations (14.29) and (14.30) were established by Letnikov [4, p.20 and
58] (1874) in the case of sufficiently good functions f(x). They were given by Moppert [I, p.149]
(1953) on continuously differentiable functions. Theorem 14.9 on the validity of these results on
weighted absolutely continuous functions and Theorem 14.8 were proved by Samko and were not
published earlier elsewhere.
Notes to  14.5. The mean value theorem 14.10 for fractional integrals was proved by
Riesz and was first published in Hardy and Riesz [1] (1915) and afterwards in Riesz [1] (1922).
The proof given here is also due to Riesz. It was presented in the book by Chandrasekharan
and Minakshisundaram [I, Lemma 1.41]. The estimate (14.40) under the only assumption that
<pet) e L1(a,b) was observed by Verblunsky [1, p.173] (1931).
Theorem 14.11 and the assertion of Remark 14.5 were proved by Riesz [1] (1922). The
Kohnogorov type inequality (14.53) for functions f(x) e Ig+(LOC) is an immediate corollary of
the Riesz Theorem 14.11. Under other assumptions and on the whole line such an inequality
was proved in the papers by Bang [1] (1941), and Geisberg [3] (1968). See also bibliographical
information concerning some other inequalities for fractional integrals and derivatives in  17.2
(notes 14.1-14.7) and in  19.8.
Notes to  14.6. The relation of fractional integration to the summation of series and
integrals briefty sketched here was observed by Hardy and Riesz [I, p.21] (1915), who developed
00'
the method of summation based on Riesz normal means for Dirichlet series L: an e->' ,,8. We refer
n=l
to other papers using fractional integration in summation theory in  17.2 (notes 14.10-14.15).
Notes to  15.1 and 15.2. The Leibniz formula (15.11) for fractional differentiation
first appeared in the paper by Liouville [2, p.118] (1832) where the fractional differentiation had
been defined via the expansion of a function in an exponential series. As for the case of the
Riemarm-Liouville definition of fractional differentiation V+, the Leibniz formula was proved by
Hohngren [I, p.I2-13] (1865-1866) and was given with a remainder in the integral form. A year
later the Leibniz formula appeared in a paper by Grunwald [1, p.466] (1867) and afterwards in
Letnikov [I, p.58] (1868), see also Letnikov [4, p.83] (1874). Grunwald proceeded neither from the
Liouville nor from the Riemann fonn of fractional differentiation, using instead an approach to
the definition of Va f via the limits of the difference quotient. This approach is presented in detail
in  20, as Griinwald-Letnikov differentiation. The Leibniz formula for fractional differentiation
was also considered by Sanine [2, p.35] (1872).
A rigorous proof of the Leibniz formula (15.12) in terms similar to modem approaches was
given by Y. Watanabe [I, p.12] (1931) for analytic functions. The proof of (15.12) in Theorem
15.1 follows the ideas of Y. Watanabe.
A series of papers by Osler [1], [2], [4], [5], [7], [8], (1971-1973) is devoted to generalizations,
specifications and other forms of the Leibniz fonnulae (15.12) and (15.17) for fractional derivatives.
Fractional differentiation in these papers is considered in the complex plane and is defined as a
generalization of the Cauchy integral formula. Unlike most of the previous works Osler gave the

 17. ADDITIONAL INFORMATION TO CHAPTER 3
305
exact domain of convergence for the series on the right-hand side of (15.12). Integral analogues
of the Leibniz formula were proved in Osler [6], [7] (1972), (15.17) being their particular case.
Notes to  16.2. Theorem 16.1 is well known, e.g. Riekstyn'sh [2, v.3, p.271]. Theorems
16.2 and 16.3 are particular cases of results by Kilbas [13, Remarks 2, 3] (1988), see also [14,
Remarks I, 2], [15], who used the method suggested by Riekstyn'sh [1] (1970). The latter author
gave asymptotic results for the convolution integral
:&
O(x) = f .r(x - t)J(t)dt as x - +00
o
(17.1 )
under the assumption that the functions .r(t) and J(t) have a general asymptotic form more
general than (16.10) - see the survey in  17.2 (note 16.1). We followed the latter paper
while proving Theorems 16.2 and 16.3. McClure and Wong [1] (1979) obtained the asymptotic
expansions (16.12) and (16.23) by the methods of distribution theory while Berger and Handelsman
[1] (1975) were first who applied the Parseval-Mellin equality representation to integrals (more
general than 1t:+J) to obtain their asymptotic forms. We refer also to Riekstyn'sh [2, vol. 3,
31.1-31.3], Wong [2] and  17.2 (note 16.2).
Notes to  16.3. Riekstyn'sh [1] (1970) observed the possibility of finding an asymptotic
fonn of the convolution integrals (17.1), which are more general than fractional integrals, by a
modified method of successive expansions under the assumption that functions F(t) and J(t) in
(17.1) admit power-logarithmic asymptotic forms (16.28). Other methods to obtain asymptotic
expansions of the integral (17.1) may be found in the monograph by Riekstyn'sh [2. Subsections
11.3, 15.1, 22.8, 32.3].
Theorems 16.4 and 16.5 were proved in Wong [1] (1978). It was also shown there that
asymptotic expansions of fractional integrals (16.1) with a density having an asymptotic form
(16.29) might be obtained by the method developed in Bleistein [1] (1975). This method, based
on the Parseval equality (1.116) for Mellin transforms (1.112), allows one to construct asymptotic
expansions for integrals of the form
a
lex) = f k(xt)J(t)dt as x - +00
o
(17.2)
provided that CI = +00 and the functions k(t) and J(t) have a logarithmic asymptotic form at the
origin.
Notes to  16.4. Theorem 16.6 is a particular case of a more general result of Riekstyn'sh
[I, T,heorem 5], which allows One to obtain asymptotic expansions of the convolution integral
(17.1) as x - 0 under the assumption that the function F(t) has an asymptotic form (16.6), while
J(t) may have certain asymptotic relation generalizing (16.46).
17.2. Survey of other results (relating to  10-16)
10.1. Grin'ko and Kilbas [1] investigated mapping properties of the operators (10.18)-(10.21) in
the Holder weighted spaces H(p) with the power weights (3.12) or (5.31) and gave conditions for
such operators to realize an isomorphism between the Holder spaces H6(p) and H+Ot(p). Grin'ko
and Kilbas [2] found conditions on the parameters a, b and c in (10.18)-(10.21) sufficient for all
compositions of the operators (10.18)-(10.21) with special power weights to be the operators of
the same form. In particular, some analogues of the semigroup property (2.21) for these operators
were obtained.
11.1. Relations (11.16)-(11.19) and (11.23)-(11.26) between left- and right-hand sided

306
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
fractional integration may be extended to the case of an arbitrary a> O. Namely, the identity
l a l a. l a 1 S a-n P,
b-<P = cos a1l" a+<P + sma1l" a+  ra <P + n<P,
ra
(17.3)
is valid (Rubin [11], p.919), where n = [a],
n
Pn<p = E Ck(X - a)n-t,
k=l
b
Ck = (_I)n-k[(n - k)!r(k + a - n)]-ll (t - a)a+k-l-n<p(t)dt.
a
11.2. Let x> O. Representations for the compositions Vff x a lx-a and xaVx-a lff+ via
the singular operator S and certain operators of the fonn (1.44) with a homogeneous kernel were
obtained by Kober [4, p.450]. These representations may be derived by means of (11.27)-(11.29)
and (3.15)-(3.16). It was also shown in the cited paper (p.449) that the operators x-a Iff+ and
lx-a have the same range if they are considered on the space Lp(R).
12.1. The semigroup property (12.17) for Feller potentials is valid in the presented general
form - for M::1,tJl' M,v with arbitrary coefficients ui, Vi - in the case a + {3 < 1 and is in
general meaningless if a + {3 = 1. Under the special choice of U2, V2 (with given Ult vd the
semi group property holds in the case a + (3 = 1 as well:
:& 00
M::,vM-;:<p = M,_b<P = a I <p(t)dt - b I <p(t)dt,
-00 :&
where 0 < a < I, <pet) e LlCR1) and a = u 2 + uvcosa1l", b = v 2 + uvcosa1l". Hence the inversion
formula
<p(x) = sina1l" [ u 1 :& f(t)dt _ v 1 00 f(t)dt ]
11" dx (x - t)Q dx (t - x)a
-00 x
for the Feller potential M::,v<p = f follows provided that <p(x) e LdRl). See aJso (30.68) and
(30.78). Let us denote
:& 00
Wj (x) = u I f(t)(x - t)-adt - v I f(t)(t - x)-adt.
-00 :&
The following theorem holds.
Theorem 17.1. A function f(x) is repre8entable by the Feller potential f = M:: v<P of a
function <p e Ll(R n ) if and only if Wj(x) e AC(Rl) and (u 2 + uvcosa1l")wj(-oo)' + (v 2 +
uvcosa1r)wj(+oo) = O.
Here AC(Rl) is the space of functions absolutely continuous on the whole axis - see
Definition 6.1. Compare also Theorem 17.1 with Theorem 6.3. The results proposed here were
obtained in Samko [13].

 17. ADDITIONAL INFORMATION TO CHAPTER 3
307
12.2. Okikiolu [4] considered potentials 1 01 and HOI (see (12.1) and (12.2)) with a power
weight:
00
1 01 - IxI IJ - Il - Oi f t II <p(t)dt
IJ,II<P - 2r(a) C06 a1l"/2 II It _ xI 1 - a '
-00
00
IxI IJ - II - a f signet - x)
H:,II<P = 2r(a) sin a 11"12 It I II It _ xl 1 - a <p(t)dt
-00
and introduced their modifications differing from them by a one-dimensional operator:
00
I I IJ-II-a f
l a II<P = X Itlll[lt _ x1 0l - 1 _ ItI 0l - 1 ]<p(t)dt
IJ, 2r( a) cos a 11" 12
-00
00
Ha IxI IJ - II - a f II [ signet - x) signt ] d
IJ,II<P = 2r(a)sina1l"/2 It I It _ xl 1 - a - Itl 1 - a <pet) t
-00
(the notation has been somewhat changed in comparison with the cited paper). The main results
are composition relations of the type (12.19)-(12.21):
-{J 01 _ -OI+{J
I q ,II+OI-IJ IIJ,II - I IJ+q,II'
if{J 01 _ - 01+ (J
o,II+OI-IJHIJ,1I - -IIJ+q,II'
1/J H OI - iI{J 1 01 - ifOl-{J
q,II+OI-IJ IJ,II - q,II+OI-IJ IJ,II - IJ+q,1I
and the boundedness of the operators 1:,11 and if:,11 from L,(Rl) into Lr(Rl) provided that
,,> 1, 0 < J.l.  a < I, or 0 < a < 1 if J.l. = 0, and -1 + III' < v < -a + III', l/r = -J.l. + III' > o.
The inversion of the operators 1::,11 and H::,II was also given in this paper by means of
. f th d _ 1 1 - 01 d d H - 1-01
COnstlUCtlOns 0 e type tIF q ,II+OI-IJ an tIF q,II+OI-IJ'
In Okikiolu [5] a connection between the weighted Riesz potential IIJ and the weighted
Fourier transforms
00
II) <P = Ixlll+q f Itlll eixt<p(t)dt
-00
was studied. It was shown in particular that .r<+0I) I <P = .r<)<p, 0 < a < I, -1 < J.l. < -a,
,IJ
and 1II+qII)<p = 1I-OI)<p, 0 < a < 1, for sufficiently good functions <p(x).
12.3. The operator (12.44) is positively defined:
b b
r(a) cos(a1l"/2)(AOI<p,<p) = f fix - yI 0I - 1 <p(x)<p(y)dxdy > 0,
a a
where <p e L2(a, b), <p(x)  0 (Tricomi [1]).

308
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
12.4. The operators in (12.44) have the property
1
fix - yla-l(1 - y2)-J c,;r (y)dy = 'xnC,;r (x)
-1
where Ixl  1,0 < a < 2, 'xn = 8in(:}ji()n! and C(y) is a Gegenbauer polynomial (P6lya
and Szego [1]).
12.5. Kokilashvili [7]-[9] investigated the Riesz potential operator along a curve
00
f f(q)dq
(Kaf)(s) = Iz(s) _ z(q)ll-a '
o < a < 1,
-00
where z(s), s e Rl, defines a curve r, which is regular in the following sense: given any circle
D(z,r), a measure of the set D(z,r) n r is less than cr, c not depending on r > 0 and z e C.
He proved in particular a theorem which states that Ka is bounded from L,(R l ) into Lq(Rl),
1 < p < , t = t - a, if and only if r is regular. He also gave the weak type estimate for p = I,
investigated in [9] conditions on a curve in the case   t - a and considered in [8] the case of
Lorentz space on curves. The weighted case with Muckenhoupt-type weight functions on curves
can be found in Gabidzashvili and Kokilashvili [1] and Kokilashvili [8].
12.6. The Riesz potential [a<p was considered by Love [11] in a setting when no integrability
condition was imposed on <p other than the existence of fa <p as a Lebesgue integral. If 0 < a < 1
and (fa<p)(x) exists for one value of x then it exists for ahnost all x, and [a<p is locally integrable
in R l . Love showed that if (fa <p)( a) exists then f = fa <p has average continuity at the point a in
the sense that
h
 f If (a f: s) - f(s)lds - 0 as h - +0.
o
Love also conjectured that a similar result is valid in the n-dimensional case. This conjecture
is true. - see  29.2 (note 25.19).
13.1. We dwell now on local properties of fractional integrals f(x) = [:+ <p of functions
<p(x) from L,(a, b). To specify their behaviour in Lq(a, b), q = p/(1 - ap), 1 < p < l/a, or in
Ha-l/'(a, b), p> l/a (see Theorems 3.5 and 3.6) we introduce the following quantities
( £ ) llr
Xr.O (J. oj = ;! 1/("') I r d", .
( c+£ ) llr
Xr,.(f. oj = :.. 1. 1/("')1' dz
if 0 < c < e, 0 < e < e
taking a = 0 and b = e for the sake of simplicity. Special attention will be paid to the limiting
case p = l/a which was already discussed in  4.2 (note 3.3). Results given here were obtained
by Karapetyants and Rubin [3], [5].

 17. ADDITIONAL INFORMATION TO CHAPTER 3
309
Theorem 17.2. Let f = 10+ 'P, 'P E Lp(O, e), 0 < a < 1, 1  'I'  00, Va = V(1 - av)-l. Then
( E: ) IIp
X.,D(f,.)  0.. 0 - 1/ . ! 1<p(x)I'dx
(17.4)
where 1  r < Va if V = Ij 1  r  Va if 1 < V < l/a; 1  r < 00 if V = l/a; 1  r  00 if
lla < V  00, and
Xr,c(J, e)  CrWa,'P(e)II'PII'P
(17.5)
where
J .:=:/.
wo,p(') = 1 1+ Iln.I)I/.'
if 1 < V < lla,
1  r  Va,
1  r < Va,
1  r < 00,
1  r  00.
if V = 1,
ifV= lla,
if lla < V  00,
We note that the constant C r in (17.4) does not depend on e, while in (17.5) it does not
depend on c E [0, e). For V < lla the constant C r in (17.5) does not depend on e as well. In
the case of V  lla the constant C r in (17.5) cannot be chosen independent of e. In the case of
V = lla the constant C r in (17.4) and (17.5) admits an asymptotic estimate
C r = O(r 1/ 'P') as r - 00.
(17.6)
In the case 1  V  lla the estimate (17.5) may be made more exact: Xr,c(J, e) = o(wa,'P(e))
as e - o. Hence a local characterization of fractional integrals follows. Namely, let 1  V < 00,
then Wlder the assumptions of Theorem 17.2 the statements
. If(x)1
es smf o <x<6- 1 1 = OJ
x a - p
. If(x)1
essinf 1x _ cl <6 (I I) =0,
wa,'P x - c
1  V  l/a.
(17.7)
are valid for all S > o. They give simple necessary conditions for a fWlction f(x) to belong
to the space 10+(Lp(0,e)) of fractional integrals of fWlctions in Lp(O, e). For example, if f(x)
is continuous in (0, e) and lim x1/p-a f(x) '# 0, then f(x) f. 10 + (Lp(O, e)). Also for the same
x-o
reason f(x) = wa,p(lx - cl) f. 10+(Lp(0,e)).
We note that (17.4) and (17.5) are exact in a sense. For example, in the case V = lla,
the fractional integral f = 10+'P6 of the fWlction 'P6(x) from (4.14), S > 0 being small, admits
[ -1 ] -6+1/'1"
the estimate f(x)  ! In ( - x) for lie < x < 2/e - see  4.2 (note 3.3)). So for
c = 2/e we have
c
1 f '1 ( 1 ) -6+1/P'
x1,df, e)  4e [In(c - x)-lr 6 + 1/ 'P dx  4 In-;
C-E:
which demonstrates the accuracy of estimate (17.5).
Further, let m[ = ih J If(x) - frl dx , 1 C (a, b), see (4.16), and let IIfll* be the nonn (4.15)
r
in the space BMO ( a, b).

310
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
Theorem 17.3. Let f = 1:+"" '" e Lp(a,b), 1  p  00,0 < cv < 1 + lip. Then
1
sup I/lp-a mlf  cll",lIp'
Ic(a,b)
(17.8)
The inequality (17.8), verified directly in the case cv = lip, leads to the boundedness of the
operator 1:+ from Lp(a,b) into BMO(a,b) and even into VMO as was observed in  4.2 (note
3.3).
We note a refinement of Theorem 17.3 for cv = lip which allows for the behaviour of f(x)
in the neighbourhood of the end-point x = a. Let us denote
BMO(a,b) = { f: f e BMO(a,b),A = sup ! a j +tf(X)dX < oo } .
O<£<b-a e
a
The space BMOg(a,b) is similarly introduced with the cOlTesponding condition for the right
end-point. One can show that the spaces BMO  (a, b) and BM 0 g (a, b) are maximal subspaces in
BMO (a, b), which consist offunctions continuable by zero to a function in the spaces BMO (-00, b)
and BMO (a, 00), respectively.
Theorem 17.4. In the case cv = lip the operator8 I+ and I_ are bounded from Lp(a,b),
1 < p < 00, into BMO (a, b) and BMOg(a, b), respectively_
Since BMO (a, b) C n Lr(a, b) by the John-Nirerberg inequality - see Kashin and Saakyan
rl
[I, p.2IO] - we obtain the relation
IIflir = OCr), r - 00,
(17.9)
for f e BMO (a, b).
The constructions in  4.2 (note 3.3) may be generalized by considering the spaces
X-Y,IJ = {f: ..pJ(r) = r--Yllfllr e LIJ(I,oo)}, 'Y  0, 1  p.  00,
introduced by Karapetyants and Rubin [1]. They showed that I? (L p ) C X l/p,oo' Let us
define somewhat more narrow spaces based on the behaviour of local characteristics Xr o(J,e)
and Xr,c(J,e) as r - 00. We denote w(I) = 1 for I C (0,1), if I = (0, e), e > 0, and
w(l) = (1 + IIn 1111)>', .x  0, for any other interval. Let
Y-Y,IJ,>.(O,I) = {f : Ilfll = IIr--Y sup w(I)Xr(J,1)IIL (1 (0) < oo}
IC(O,l) II I
where 'Y > 0, 1  p.  00, Xr(J, I) = Xr,O(J,e) for 1= (O,e) and Xr(f,/) = Xr,c(J,e) for
1 = (c - e, c + e). By the inequality xr(f, I)  X8(j, 1), r  s, it is necessary that 'Y > p.-l in the
definition (17.10) if p. < 00. It can be shown that Y-y II',>' are Banach spaces. We observe also that
the space Y-Y,IJ,>'(O, 1) coincides with Loo(O, 1) in the case .x = 0.
(17.10)
Theorem 17.5. Let cv = lip, 1 < p < 00. The operator I+ i8 bounded from Lp(O, 1) into the
'paccs X-Y,IJ(O, 1) and Y-Y,IJ,>.(O,I), if .x = lip', 'Y> lip' + lip. when 1  p. < 00 and 'Y > lip
when p. = 00.
It is of interest to clarify the interconnections between the spaces Y-y ,1',>' and BMO. Without

 17. ADDITIONAL INFORMATION TO CHAPTER 3
311
entering into details we observe only that for A = "Y = l/p', p. = 00
Y-y ,IJ,>' (0, 1) rt. BMO(O,I), BMO(O,I) rt. Y-Y,IJ,>'(O, 1).
The fonner of these assertions may be directly checked by means of the example Yl(X) = {O,
l/p'
as ° < x < 1/2 and (In Z_ 1 1/2 ) as 1/2 < x < I} f. BMO (0, 1). The latter is valid for
arbitrary admissible values of "Y, p. and A. In fact otherwise for 1 E BMO (0,1) we would have
sup Xl,O(J, e) < 00 from whence 1 E BMOg(O, 1). This is not always possible.
0<£<1
Filtally, we note that the spaces X-Y,IJ(O, 1) unlike Y-Y,IJ,>" admit the imbedding BMO (0,1) C
X-Y,IJ(O,I) in view of (17.9), if "Y > 1 + 1/p.. (Prof. B. Muckenhoupt called the authors' attention
 this fact).
13.2. As was shown in  13.2, the ranges 1+(Lp) and 11:_(L p ) of fractional integrals
coincide with each other if 1 < I' < 1/01, see (13.23). They coincide also with the range of the
Riesz potential Aalp on the interval [a,b] defined by (12.44):
1:+(L p ) = 11:_(L p ) = Aa(L p ), 1 < p < 1/01.
The latter of these inequalities is derived from (12.46) or (11.16)-(11.19). In the case I' > 1/01 the
relation
1:+ (L p ) e R l = 11:_(L p ) e R l
holds (Rubin [11]); see (17.3) in this coIUlection). In the case a = 1/1' neither of these coincidences
takes place. This may be shown by means of the local estimates of Theorem 17.2. For I' > 1/01
we also note the following imbeddings
1:+(L p ) C Aa(L p ), 0< 01<2/1',
1:+(L p ) C Aa(L p ) e PI, 2/1' < a < 1,
Aa(L p ) C 1:+ (L p ) e PO, 1/1' < a < 1,
where Pj is a space of polynomials of order j, j = 0,1. These imbeddings are derived from the
connections of the operators 1:+,11:_ and A a with each other via a singular operator. - (11.16)
(11.19), (12.46) and (12.47). Karapetyants and Rubin [3] gave another proof of the first and
second of these imbeddings by means of the representations
A a _ 01/2 [ 01/2 -01 1 01/2 01/2
- X 0+ x 1- x ,
[a _ 01/2[01/2 -01101/2 01/2
0+ - x 0+ x 0+ x .
13.3. A generalization of Theorem 13.6 to the case of the weighted Lp-space8 was
proved in Samko [27,  8, Theorem 8.6] and [5, p.306] (under more strict assumptions). Let
p(x) = (x - a)1'(b - x)6. The following theorem is valid:
Theorem 17.6. If l(x) E 1:+[L p (p», 1 < I' < 00, 0< a < 1, then also
(x - a)IJ(b - x) II l(x) E 1:+[L p (p})],
p(x)
pICx) = (x _ a)IJP(b - X)IIP ,
provided that -1 < p. < 1, 01-1 < v < 1 - 01, "Y < I' - 1 + min(O,pp.), (01 + v)p - 1 < 6 <
p - 1 + min(O,pv).

312
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
13.4. Theorem 13.18 was extended to the case of the weight which is fixed to both
end-points of an interval [a, b]. Let p(x) = (x - a)#J(b - x)/, J.l.  0, v> 0, J.l. + a < 2, J.l. + v < 2.
The following theorem is valid (Murdaev and Samko [I], [2] and Samko and Murdaev [1]).
Theorem 17.7. The fractional integration operator 1:+,0 < a < 1, isomorphicaU, maps the
weighted generalized Holder space Ho(p) onto the 'pace H:for (p):
1:+ [Ht (p)] = n:J'OI(p), wa(c5) = c5 a w(c5),
if w(cS) e ()-l, "y = m&x(I, J.l., v), (3 = min(1 - a, v - a), where ()-1 i, the Junction class
(19.68).
13.5. Theorem 13.17, stating that 1::+ maps the generalized Holder space H'O onto the
space OI, wa(h) = haw(h), was extended by Karapetyants, Murdaev and Yakubov [1] to the
case of spaces H'P with a generalized Holder condition being considered in Lp-norm on the
interval [a, b]. The authors gave the cOlTesponding isomorphism theorem, firstly discovering the
Lp-analogue of the property for a function to vanish at the end-point x = a, which is well-suited
the required isomorphism. For this see also the reference to Kostometov [1] in  17.1 (notes to
 14.2 and 14.3) in the case w(h) = h>'.
13.6. Theorems on restrictions and continuation by zero as well as "sewing" theorems for
fractional integrals, presented in  13.3, were extended by Linker and Rubin [1] to the case of
convolution operators with a power-logarithmic kernel.
14.1. The Riesz mean value theorem in the form of (14.40) is valid in the case of an infinite
limit of integration:
 (0: - t)a-t",('}dt  es;p  ({ - t}a-t",(t)dt, 0 < a < t,
only under the assumption that the integral in the left-hand side converges (Isaacs [1]).
14.2. There is a generalization of the Riesz mean value assertion (14.40) as follows. Let
Ip(x) e Ll(O,I), I> o. ITo < a < 1 and.x $ 1- a, then
1
x>' f
essmino<t<lt>'(1+Ip)(t) $ - ( Ip(t)(x - t)a-1dt $ esssuPt>'(I+Ip)(t)
- r a) O<t<1
o -
provided that the right (left)-hand side here is non-negative (non-positive), respectively (Bosanquet
[7]).
Steinig [1] used the generalized Riesz mean value theorem to answer the following question:
if a fractional integral x>'(1+ Ip)(x) with weight x>' has local extrema, what effect does it have
upon changing the signs of the function Ip(x) itself? This is a fractional analogue of a known
theorem about a function which changes its sign at the points of local extrema of its primitive.
We note also that there is an extension of the Riesz mean value theorem to the cases a > 1,
see Tiirke and Zeller [1].
In the paper by Bosanquet [7], cited above, a generalization of the Riesz mean value theorem
is given for the case when the power kernel (x - t)a-l is replaced by a more general one G(x - t)
of Sonine type. See  4.2 (note 2.4) concerning the latter.
14.3. The Riesz mean value theorem (14.40) was extended to functions which are integrable
in the Denjoy-Perron sense - Verblunsky [1] and Burkill [1]. The latter, as well as the papers
by Sargent [1]-[3], concern also Cesaro-Perron integration of fractional order, the so-called
Cap-integration.

 17. ADDITIONAL INFORMATION TO CHAPTER 3
313
14.4. Bosanquet [3] proved a discrete analogue of the Riesz inequality (14.40) as follows
I fA:all l  Om I tA:all l '
11=0 11=0
where Ak = et U ), n  m  0 and all is an arbitrary sequence. The sum above may be
-considered as a discrete analogue of a factional order difference.
Certain analogues of the "interpolation" Riesz inequality (14.42) for discrete differences of
fractional order were proved in the paper [6] of the same author.
14.5. The Marchaud fractional derivative (D+. I)(x), x E Rl, 0 < cv < I, admits the
estimate of Hadamard type:
IID+./lle  kll/ll-Q/211f'1I12
where II/lIe = sup I/(x)1 and IIgllHl = sup Ig(x) - g(y)I/lx - yl, the constant k
zeRl z,yeRl
22-cv/2(21/(1-cv) - l)cv-l/r(3 - cv) being sharp - Geisberg [1].
14.6. The Kohnogorov type inequality
:z:.J!. I!.=.!!.
III/IILp(Rl ) $ kIlI/lIl- ( oRl ) III/lIl- ( oRl ) ' 1 < p < 00,
+ p + p +
where -A < cv < (3 < "Y < A and k = k(p, A), was proved by Hardy, Landau and Littlewood [1].
A similar inequality for the fractional integro-differentiation Ig+ which generalizes (14.53) was
proved by Hughes [1]. The analogous inequality
IIV:Jlle(R)  kll/ll(R)lIln)II(R)
for fWlctions I E L2(R), which have a generalized derivative I(n)(x) E L2(R), was obtained
by Magaril-D'yaev and Tikhomirov [1], who also gave the exact boWlds of its validity: cv E
(-1/2, n -1/2), "'1 = n-l(n - cv - 1/2), "'2 = 1- "'1 together with the sharp value of the constant
k.
[ 2 r( cv+l )P -"Y/o
As for the best constant in (14.53), it was found by Arestov [1] as r(cv-'Y+ l ) Wlder
the asswnptions that Marchaud-type derivatives are used in (14.5) and 0 < "Y $ I, "Y $ cv < 2.
An extension of (14.53) to fractional power of operators in Banach spaces is due to Trebels
and Westphal [1] and has the fonn
1I(-A),6 III  kll/ll l -,6hll(_A)'Y III,6h, 0 < {3 < "Y,
w here A is a generator of a strongly continuous semigroup of operators, (- A),6 being gi yen by
(5.85).
14.7. The following "Opial type" inequality
b b
f 1(1:+J)(x)IPI(I\/)(x)lqx'Y-QP-,6q-lw(x)dx $ c f I/(x)l r x'Y-lw(x)dx,
a a
is valid, if p > 0, q > 0, p + q = r  I, "Y < r, 0  a < b $ 00, and w(x) decreases and w(x) > 0
as a < x < b (Love [9]).

314
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
We note alao the inequality
b b
f ( ) [ 1(1:+I)(X)I ] d < fa( ) [ IU)(x)l ] d
8 x!/J (1:+g)(x) x _ x!/J g(x) x,
a a
where a = gI b _ ( 1:+9 )' !/J(u) is a positive convex increasing function and 8(X)  0, g(x) > 0 -
Godunova and Levin [1]. There are generalizations of this inequality in Rozanova [1] and Sadikova
[1].
14.8. Askey [1] gave interesting connections of fractional integration with methods of
deriving various inequalities for trigonometric and algebraic polynomials.
[t] 00
14.9. Let c(t) = L: Cn. IT a series L: Cn is summable to the sum 8 in the sense that
n=l n=l
:&
a = lim :& f c(t)(x - t)a- 1 dt for some a> 0 (see  14.6), then it is summable to the same sum
:&-00 0
in a similar way for any a ' > a - Hardy and Riesz [1, p.29].
14.10. Let 0 < 'xO < 'xl < "', 'xn - 00, c(x) = L: Cn and let
>'..;5:&
:&
Ca(x) = rta) f (x - t)a-1 c (t)dt = rea 1+ 1) L (x - 'xn)a cn , a > 0,
o >'..;52
be a generalization of the Riesz mean (14.55). It is shown in Bosanquet [4] that if lim >'11 > 1
and Ca(x) = o(x'Y), 'Y > -1, as x - +00, then CI3(x) = o(xl3- o +'Y) provided that 0  {3  a.
The case 'Y = a, {3 = 0 was earlier considered by Hardy and Riesz [1].
14.11. Fractional integration may be used to investigate a limit of a function similar to its
application in summation of series (see  14.6). Namely, let !/J(t) be given for t > O. One may say
:&
that !/J(t) - 0 in the sense of the (C, a)-method, if lim x f !/J(t)(x - t)a- 1 dt = O. Let now
j-o :&-0 0
a function f(x) be such that there exists the limit

25 = a lim 2.. f U(x o + t) + f(xo - t)]( - t)a- 1 dt, a > O.
 -0 a
o
The number 5 is called a generalized value of the function f at the point xo. Denoting
!/J(t) = f(x+ t) + f(x - t) - 25, let us say that f(x) has the value 5 at the point Xo in the seWie of
the (C, a)-method if !/J(t) - 0 as t - 0 in the sense of this method. An investigation of functions
which have finite values in the above sense was given by Verblunsky [1].
14.12. Bosanquet [2] gave sufficient conditions in tenns of fractional integration for a
Fourier series to be A-summable. A series L: an is called A-summable to a sum 5, if L: anX n
converges for Ixl < 1 to a sum A(x) which has the finite limit lim A(x) = 5 and a bounded
:&-1-0
variation on [0,1]. Let f(x) e L1(0, 211") and 2!/J(t) = f(xo + t) + f(xo - t) - 25. It was proved in
the cited paper that the Fourier series of a function f(x) is A-summable to a sum 5 at the point

 17. ADDITIONAL INFORMATION TO CHAPTER 3
315
o, if there exist such a > 0 and e > 0 that
£
1 t-1-al(I<f+Ip)(t)ldt < 00.
o
1
A weaker condition J t- 1 - a (1g+ Ip)(t)dt = 0 (log}), a  0, was used for the summation of
t
the Fourier integral by Mohapatra [1]. See also Bosanquet [2a] , Bhatt and Kishore [I], Kishore
and Hotta [I], where the bounded variation of t (Ig+ Ip)(t), Ip(t) = ![J(x+ t) + f(x- t)] was used
as a sufficient condition for the generalized summability of Fourier series of the function f(t).
14.13. Beekmann [I, 2] showed that the integral transfonn
:&
(Calp)(x) =  I (X - t)a-llp(t)dt, x> 0,
x a
o
is "perfect" in the sense that the set of all functions in certain linear spaces B, having a finite
limit lim Ip(x), is dense in the set of all functions in B, having lim (Calp)(x). More specified
:&-00 x-oo
formuJations may be seen in the cited papers.
00
14.14. Cossar [1] investigated the summability of integral J f(x)lp(x)dx by (C, a)-method,
o
00
which leads to the convergent integral J(1<f+f)(x)(VIp)(x)dx, a > 1. This is the fonnula of
o
fractional integration by parts, where V1p is the C06sar derivatiye (9.1).
14.15. The connection of fractional integration with methods of summation of series
sketched in  14.6 holds not only in the case of Riesz mean method, but in the case of other
methods as well. Thus, Kuttner and Tripathy [1] investigated such a connection with Hausdorff
summation methods. We note also that B06anquet and Linfoot [1] used "generalized" Riesz means
with a power-logarithmic kernel, which lead to "fractional integrals" with a power-logarithmic
kernel.
In connection with fractional integration in summation theory we mention finally the papers
by Flett [1], [2], Sulaxana K. Gupta [I], Hardy and Rogosinski [I], Minakshisundaram [1], Mikolas
[1]-[3], [7], [8], Wang [I], [2], Paley [1] and Lao [1]. We note also that fractional integration
arose naturally in the &<>-called Hausdorff inclusion problem for Hausdorff and Cesaro methods of
summation, see Garabedian, Hille and Wall [1].
We also mention the paper by Yogachandran [I], which deals with the problems of the
asymptotics f(x) - Ix P as x - 00 treated in the Cesaro sense.
14.16. Connections are lmown between the differentiability properties of a function and the
rate of its approximation by algebraic or trigonometric polynomials - see for example, the book by
Timan [3]. The first paper containing investigation of such a connection in the case of fractional
differentiability was the work by Montel [1]. He proved a generalization of Bernstein's theorem in
the case of fractional derivatives. This states that the rate of approximation of a function f(x) by
algebraic polynomials given by the inequality If(x) - Pn(x)1  An--Y, -1  x  I, 'Y > 0, implies
the existence of all fractional derivatives (V:+J)(x), a = -1 of orders a < 'Y. We note also that
Montel [1] gave a generalization even to the case of two variables - see fractional derivatives of
functions of many variables in  24 and the mentioned generalization in  29.2 (note 24.8).
There are many investigations concerning trigonometric polynomial approximation of
functions, which have derivatives of fractional order, see  23.2 (note 19.6). Such investigations in
the non-periodic case may be found in Ibragimov [I], [2], Nasibov [1] and Kofanov [I], where the
approximation of fractional integrals by algebraic polynomials was considered, the results of the

316
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
latter paper being extended to the case of functions of two variables in the work [2] of the same
author. His paper [1] contains a result which is final in a sense. Let Wf = 1:+ (Lt> denote the
space of fWlctions representable by the fractional integral of a function f(x) in L 1 (-I, 1). Let
En (Jh = inf{lIf - PIIL I } be the best approximation of f(x), the infimum being taken over the
set of all polynomials of a degree  n, and let En (Wfh = sup{En(J) : f e Wf}. Then, as was
shown by Kofanov
1
En(Wfh = sup r ( 1 ) I (X - t)a-lsign sin(n + 2) arcc06xdx
te[-l,l] a
t
if a  I, n  [a] - 1. Finally, we note that Starovoitov [1]-[4] investigated the approximation of
fractional integrals by rational fWlctions.
15.1. Besides the Leibniz fonnula (15.11) for a fractional derivative the following Leibniz
type expression with a remainder:
n-l
a ( ) '" ( a ) a-k (k) R
'D a + uv = L.J k 'D a + uv + n,
k=O
x x
Rn = (_I)n I (x - t)-a-lu(t)dt l (X _ {)n-lv(n)({)d{
r( -a)(n - I)!
a t
(17.11)
holds (Holmgren [I, p.12], Y. Watanabe [I, p.16], AI-Bassam [1]). In comparison with (15.11)
(17.11) has the advantage that it does not require the infinite differentiability of the fWlction vex).
15.2. The Leibniz result (15.11) was proved by Post [2, p.755] for generalized fractional
derivatives (20.10) -  23.2 (note 20.6). Y. Watanabe proved the relation
00
'D a a +( fg ) = '" ( a ) 'Da-IJ-n f'DIJ+n
L.J {3 + n a+ a+ 9
n=-oo
(17.12)
more general than (15.12), by expanding f and 9 in power series.
15.3. Osler [I], [5], [7], [8] gave a series of generalizations of the Leibniz roles (17.12)
and (15.17) via generalizations of the Cauchy integral formula for fractional derivatives on the
complex plane (see (22.4». In Osler [1], [8] (17.12) was extended to fractional derivatives (and
integrals) of a fWlction f by a function <p, i.e.
x
'Df(x) = r(a) I f(t)[<p(x) - <p(t)]-a-l<p'(t)dt
tp-I(O)
- see  18.2. The generalization
00
'Df(x,x) = L ({3: h)'D:((x)f(x, t)lt=x'
n=-oo
(17.13)
was also given, where 'DaJ.f denotes mixed fractional derivative of f(x, t) by a function <p with
tp,."
respect to the first variable and by a function 1/1 with respect to the second one, of orders a and {3,
respectively. In Osler [5], [8] a connection of (17.12) with the Parseval equality in Fourier series

 17. ADDITIONAL INFORMATION TO CHAPTER 3
317
theory was discussed which led to further generalizations of (17.13). The results were applied
in the above papers to expand certain special functions in series. Arora and Koul [1] used the
integral analogue of the Leibniz role (15.17) with (3 = 0 to obtain the integral representations for
some special functions. In Osler [7] a general integral analogue of the Leibniz role was proved,
(15.17) and (17.12), (17.13) being particular cases. We add that Lavoie, Tremblay and Osler [1]
proved the relation generalizing (17.12):
00
'D + (xP+9/(x»g(x) = '" C ( cv ) 'Dcv-13-cn(xp l(x»'D13+cn(x9g(x»,
L.J en + {3 a+ a+
n=-oo
0< e  1,
and gave its integral analogue by means of the evaluation of fractional derivatives via the
Pochhanuner integral (see  22) under weaker assumptions than in the papers by Osler cited
above.
15.4. Walker's expression
N
'DN(JN uv] = L W(N,n),
n=O
W(N,n) = (N)'DN-n(JN-nv)'Dn-l(Jnu l )
n
and the Cauchy relation
N
'DN-l[/N'D(uv» = L C(N, n),
n=O
C(N,n) = (N)'DN-n-l(JN-nvl)'Dn-l(Jnu l ),
n
which are generalizations of Leibniz formula, were extended from the values N = 1,2,3,... to
arbitr8l')' values Nee by Osler [10].
15.5. AI-Salam and Verma [2] and Agarwal [3] extended Leibniz role (15.11) to the case of
the so-called q-derivatives -  23.2 (note 18.5).
15.6. PolIcing [1] established an analogue of Leibniz rule for the non-linear multidimensional
differentiation operator of fractional order 'D:,9' 0 < cv < 1, 1  P  00, which is defined as
[ 00 ( ) 91p ] 1/9
('D:,9/)(x) = f f I I(x + p;l- I(x) r dy d: '
o YIl
1  q < 00,
( ) l
('D:,ool)(x) = sup p-cv f I/(x + py) - l(x)IPdy ,
O<p<oo
YIl
q = 00.
15.7. Gaer and Rubel [2] obtained the Leibniz rule (15.1) and its generalizations for a
fractional derivative which is specifically defined in the complex plane in terms of a certain entire

318
CHAPTER 3. PROPERTIES OF FRACTIONAL INTEGRALS
function of exponential type - see  23.2 (note 22.9).
15.8. Manocha [3], Manocha. and Sharma [I], Arora and Koul [1] and Srivastava [2] applied
the generalized Leibniz role to obtain some relations between certain functions of hypergeometric
type and the Fox H-function.
16.1. Riekstyn'sh gave asymptotic expansions for the convolution integral (17.1),
generalizing the fractional integral (16.1) in the following cases: 1) for x - +0 under the
00
assumption that the functions F(t) and J(t) have asymptotic expansions J(t) - e- a/tOl L: antI'''
n=O
as t _ +0. This generalizes (16.6) and (16.46), provided that a  0, ev > 0 and I'n is an increasing
sequence such that IJO > -1 and lim I-&n = +00; 2) for x - +00 when F(t) and J(t) have
n-oo
asymptotic representations
M 00
J(t) - E b'" + E an t - n -,6, 0 < (3  1, 0  .xo < .xl < ... < .x n < ...
m=O n=O
(17.14)
generalizing (16.10). We remark that Riekstyn'sh used a modification of the method of successive
expansions, firstly applied in Tihonov and Samarskii [1], in order to obtain an asymptotic
expression of the integral (17.2) as x - +00 - see also the book by Riekstyn'sh [2. v. I,  11].
16.2. Berger and Handelsman [1] obtained asymptotic expansions of a fractional integral
(Ig+;zpJ)(x) by a function x P - (18.41), in the following two cases: a) for x - +0 under the
assumption that J(t) has an expansion (16.6) as t - +0; b) for x - +00 provided that J(t) has
an asymptotic expansion
00
J(t) - e- at L an t - IJ .., a  0,
n=O
(17.15)
where IJn is an increasing sequence such that lim ReIJn = +00. The investigation uses the
n-oo
methods developed in Handelsman and Lew [1], [2] and is based on the Parseval equality (1.116)
for the Mellin transform (1.112). These methods allow one to find asymptotic expansions of the
integrals (17.2) if the functions k(t) and J(t) have a power asymptotic expansion as t - +0 and
t _ +00. See also the book by Riekstyn'sh [2, v. 3,  31.1-31.3]. In Berger and Handelsman [1]
the following results were obtained: 1) if J(t) has the asymptotic expansion (16.6), then
00
(lev . pJ)(x) _ "" anr(1 + IJn/P) xIJ..+pa;
O+,z L.J r(I + ev + I-&n /p)
n=O
(17.16)
as x _ +Oi 2) if J(t) allows the asymptotic expansion (17.15) as t - +00, then for x - +00:
00
(la . pJ)(x) _ "" (-I)nrot{J;p(n + I)} xP(a-n-l), if a> 0,
O+,z L.J n!r(ev - n)
n=O
(17.17)
rot being Mellin transform (1.112), and
00
(Ia. J)(x) _ "" (-I)n9Jl{JiP(n + I)} xp(a-n-l)
O+,zP L.J n!r(ev - n)
n=O
(17.18)
00
+ "" amr(I - Ilm/P) x pa - IJ "' ,
L.J r(I + ev - I-&m/P)
m=O

 17. ADDITIONAL INFORMATION TO CHAPTER 3
319
if a = 0 and J.l.m '# pen + 1) for all n, m = 0,1,2,.... In particular, if J has the asymptotic
expansion (16.10) with 0 < {3 < 1, it follows from (17.18) that the fractional integral (16.1) has
the following asymptotic expansion as x - +00: namely
00
(lOt J)(x) _ '" anr(1 - (3 - n) xOt-{j-n
0+ L.J r(1 + Ot - (3 - n)
n=O
(17.19)
00
+ '" (-I)nrot{J; n + I} xOt-n-l
L.J n!r(Ot - n)
n=O
which is equivalent to (16.11) - see also Wong [2]. McClure and Wong [1] obtained this result by
a method based on the theory of distributions, when J has power asymptotic expansions (16.12)
or (16.23). Although powerful, the method does not seem to lead to the construction for the elTOr
tenns in the expansions (17.17) and (17.18), such constructions being important in applications,
in particular, for the approximate calculation of integrals. Kilbas [14] used a modification of the
method of successive expansions (see  16 and  17.1 (note to  16.2) for obtaining the complete
asymptotic expansions with explicit error terms for (Ig+;xpJ)(x) where J satisfies (17.15) with
a = O. The results differ in the cases of integer and non-integer values of J.l.m/P. The fonner
case leading to logarithmic asymptotic results was pointed out by Berger and Handelsman [1]
who gave only the first tenn of the asymptotic expansion. Kilbas [13], [15] also used this method
to obtain complete asymptotic expansions and explicit error tenns for the Erdelyi-Kober-type
operators (18.1) with a = 0 when J satisfies (17.15) with a = O.
Berger and Handelsman [1] and Kilbas [13], [15] applied also the above results to find
the asymptotic representations for solutions of boundary value problems of the Euler-Poi880n-
Darboux equation and of generalized axially symmetric potential theory - see (41.23), (41.24),
(41.42) and (41.44).
16.3. In the case when J(t) is oscillating as t - 00, one of the methods, given in the book
by Riekstyn'sh [2, v. 3, n. 32.2], may be used to find asymptotic representations for a fractional
integral.
16.4. Erdelyi [16] obtained asymptotic representations for the integrals
+00
f e-x(t-a)(1'+>' J)(t)dt,
a
a
f e-xt(1'+>' J)(t)dt, 0 < a < +00,
o
as x _ +00, where 1)+>' J is a fractional derivative (2.22), 0 < .x < 1.
16.5. Wong [1] found the five first terms of asymptotic expansions for fractional integral
(Ig+!P)(x) as x - +00 under the assumption that !pet) has a power-logarithmic asymptotics of
the fonn
{j [ InInt (Inlnt)2 ]
!pet) - t- (In t)'Y Co + Cl};t + C2 (In t)2 +...
ast-+oo
where 0  {3  1 and "Y e R 1 (d. (16.29)). This result and Theorems 16.4 and 16.5 were 8i)plied

320
CHAPTER 3. PROPEFUIES OF FRACTIONAL INTEGRALS
to obtain asymptotic solutions Ip(x) of non-linear integral equations
:&
';;Ip(x) = f (x - t)-1/2[J(t) - Ipn(t)]dt, n = 1,2,...,
o
(17.20)
under the assumption that the known function J(t) admits a power asymptotic expansion (16.5)
as t _ +00.
We note that the first term of the asymptotic expansion for solutions Ip(x) of the equation
(17.20) in these and other cases were earlier given in Handelsman and Olmstead. [1], and Olmstead.
and Handelsman [1], where an application to a problem in heat conduction theory was also
demonstrated.
16.6. A problem of finding an asymptotic expansion for fractional integrals I+ Ip using
a given asymptotic expansion of a function Ip, may be set in the case of a fixed number of
asymptotic terms. Namely, let Ipn(X), x E R, be an asymptotic sequence and let a function
J(x) satisfies (16.3) as x - +00 for a certain fixed N but not necessarily being satisfied for
N + 1. What is then an asymptotic expansion for I+ Ip as x - oo? In the case Ipn (x) = x-n
the question on asymptotics in such a setting was considered by Betilgiriev [1], [2] in connection
with an investigation of a convolution type equation on the half-axis - the Wiener-Hopf equation
- the symbol of which has real roots of fractional order.

Chapter 4. Other Forms of Fractional
Integrals and Derivatives
This chapter completes the presentation of the theory of one-dimensional fractional
calculus. Here various types of fractional integrals and derivatives of real variables,
not considered previously are studied and fractional calculus in the complex plane
is presented (S 22). Some of the new forms under consideration are modifications
or direct generalization of Riemann-Liouville fractional integrals and derivatives,
which were studied in the previous chapters. Others are based on quite different
approaches. Nevertheless many different forms coincide with one another on certain
spaces of functions, and in some cases even the domains of definition of a priori
different forms coincide. This is shown in this chapter. We draw special attention
to an interesting approach via differences of a fractional order, which goes back to
Grunwald and Letnikov. This approach is developed in S 20 both in the periodic
and non-periodic cases.
The end of the chapter (S 23.3) contains a collection of answers which are
given in this book to some open questions put at the First Conference on Fractional
Calculus in New Haven, 1974.
 18. Direct Modifications and Generalizations
of Riemann-Liouville Integrals
Various modifications and generalizations of classical fractional integration
operators are known and are widely used both in theory and applications.
In this section we dwell on such modifications as Erdelyi-Kober type operators,
fractional integrals of a function by another one, Hadamard fractional integrals and
derivatives, Bessel type fractional integra-differentiation, Chen fractional integrals
and Dzherbashyan generalized integral.

322
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
18.1. Erdelyi-Kober-type operators
In investigations of dual integral equations (see S 38) and in some other applications
the following modifications of Riemann-Liouville fractional integrals and derivatives
are widely used:
:&
(1'X-O'( a+'1) f
]a J( x ) = ( xO' _ to' ) a- 1 tO''1+0'-l J( t ) dt
0+;0','1 r( a) ,
o
a> 0,
(18.1 )
]a. J( x ) = x-O'(a+'1) ( d ) n xO'(a+n+'1) ]a+,n J ( x )
0+,0','1 (1'xO'-ldx 0+,0','1'
a> -n,
(18.2)
b
0''1 f
]a_, J ( x ) =  ( to' - xO' ) a-l t O'(l-a-'1)-l J ( t ) dt
b ,0','1 r(a) ,
:&
a> 0,
(18.3)
] a J( ) - 0''1 ( d ) n O'(n-'1) ]a+n J( )
b-'O' '1 X - X 0' ld x b-.O' '1 - n x,
, , (1'X - X ' ,
a> -n,
(18.4)
where 0  a < x < b  00 for any real (1' or -00  a < x < b  00 for integer (1'. In
particular, if a = 0, b = +00 and (1' = 1, the integrals (18.1) and (18.3) are
:&
It-al(x) = 1+;I.,/(x) = x;(' f (x - tt-It'/(t)dt,
o
a> 0,
(18.5)
00
K;;'al(x) = 1::'_;I.,/(x) = r:) f (t - x)a-Ir,-a I(t)dt,
x
a> O.
(18.6)
The operators (18.1) and (18.3) with a = 0 and b = +00 are called Erdelyi operators
while the integrals (18.5) and (18.6) are called Kober (or Kober-Erdelyi ) operators.
It is thus natural to call the operators (18.1)-(18.4) Erdelyi-Kober-type operators.
We shall also use the following notations:
]OO+jO','1 = ]+';0','1'
]a _ fa
+00- jO','1 - - ;0','1'
(18.7)
if a = -00, b = +00, and
]+;2,'1 = ]'1,a,
]j2,'1 = K'1,a,
( 18.8)
if (1' = 2. The operators (18.8) are often called Erdelyi-Kober operators.

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 323
The operators (18.1) and (18.3) (in particular, the operators (18.5) and (18.6»
are the operators with a homogeneous kernel of degree -1. So, the fact that it is
bounded in Lp(a, b), 1  p < 00, with a > 0 and b < +00 follows from Theorem 1.5.
When a = 0 and b = +00 the operator (18.1) is bounded in Lp(O, 00), 1  p < 00,
if 17 > -1 + I/pu, and the operator (18.3) is bounded in Lp(O, 00), 1  p < 00, if
17 > -l/pu. In particular, the operators (18.5) and (18.6) are bounded in Lp(O, 00),
1  p < 00, if 17 > -lip', and 17 > -lip respectively.
After the change of variables ,l' = y, to = T (18.1)-(18.4) are reduced to the
usual Riemann-Liouville fractional integrals and derivatives (see S 2):
l:+;o,,,J(x) = y-Ot-"(l:,,+cp)(y), cp(y) = y" J(x), XO = y, (18.9)
Ib-;o,,,J(x) = y"(lb..-_1P)(Y), 1/J(y) = y-Ot-" J(x), X O = y,
(18.10)
and therefore, if a = 0 we set
l+;o,,,J(x) = J(x), I_;o,,,J(x) = J(x).
(18.11 )
Let us note that the property
( d ) n ( d ) n
- x 1 - o x o - 1
ux o - 1 dx - udxx o - 1
( 18.12)
allows us to rewrite (18.2) and (18.4) in the equivalent form
lOt. J ( x ) = x 1 - 0 (1+Ot+,,) ( d ) n x o (1+0t+n+,,)-110t+,n J ( x )
0+,0,,, udxx o - 1 0+,0,,, ,
a> -n,
( 18.13)
lOt J ( x ) = x 1 + 0 (,,-1) ( d ) n xo(n-,,+l)-l I Ot + n J ( x)
b-;o,,, udxx o - 1 b-jO,,,'
a> -no
(18.14)
Relations (18.9) and (18.10) enable us to extend the known properties of
the Riemann-Liouville fractional integrals 1:+ and Ib_ to Erdelyi-Kober-type
operators. Let us give the main properties in order to use them in S 38 when
solving dual equations:

324
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
a) shift formulae
l:+;o,'1xof3 I(x) = x0f31:+;o,'1+f3I(x),
1 b _;o,'1 xof3 I(x) = xof31b_;o,'1_f3I(x);
(18.15)
b) composition formula,e
1:+;0''11:+;0,'1+al(x) = l::i,'1I(x),
( 18.16)
1b-;o''11t-io,'1+al(x) = l:!;,'1I(x),
which hold in the corresponding spaces of the functions I if /3 > 0, a + /3  0 or
{3 < 0, a > 0 or a < 0, a + /3  0 (see Theorem 2.5);
c) factorization formulae
n
1:+;ol'1l(x) = n- a II 1:7no'('1+1c)/n-1/(x),
k=1
(18.16')
n
1b_;0,'1I(x) = n- a II 1:!o'('1+k-1)/nl(x);
k=1
d) expressions lor inverse operators
(1:+;0,'1)-1/(x) = r;o,'1+al(x),
(Ib_;o,'1)-l/(x) = Ib_o,'1+al(x);
(18.17)
e) formula 01 fractional integration by parts
b b
J zO-1 !(z)I::+,o.,g(z)dz = J zO-lg(z)Ir_;o.,!(z)dz,
a a
( 18.18)
Let JII(z) be the Bessel function of the first kind (1.83). We define the operator

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 325
of the modified Hankel transform 8",0I;q by the formula
00
S.,o;af(z) = "oz-oa/2 J ,-oa/2+ a -l 1,'+0 ((zW/2) f(t)dt,
o
u > o.
( 18.19)
After the corresponding changes of variables and functions (18.19) can be reduced
to the form of the usual self-dual Hankel transform (Erdelyi, Magnus, Oberhettinger
and Tricomi [4, 8.1(1)]) from whence it is easy to obtain by reverse changes the
following representation for the operator inverse to (18.19):
8;',qf(x) = 8,,+0I,-OI,qf(x), Re(27] + a)  -1/2.
(18.20)
For (18.19) the following composition formulae hold
Ig+;q,,,+p8,,,p;qf(x) = 8",0I+{J;qf(x), I;q,,,S"+OI,p;qf(x) = S",OI+p;qf(x);
(18.21 )
8"+0I,p;u8",0I;q f(x) = I;:,,,f(x), S",OI;qS,,+OI,p;u f(x) = 11/"f(x); (18.22)
8"+0I,p,qlg+;q,,,f(x) = 8",0I+p;qf(x), S",OI;ql;q,"+OIf(x) = S",OI+p;qf(x).
( 18.23)
All of them are proved by means of the definitions of the involved operators, by
interchanging the order of integration and by evaluation of the inner integrals
under some convergence conditions of the type a > 0, {3 > 0, a + {3 > O. In these
calculations it is convenient to use 2.12.4.6, 2.12.4.7 and 2.12.31.1 from Prudnikov,
Brychkov and Marichev [2]. These relations can be extended to other values of
the parameters a, {3 and 7] by analytic continuation but in more narrow functional
spaces where the corresponding operators are defined.
18.2. Fractional integrals of a function by another function
Erdelyi-Kober operators used the power (xq - t q )OI-l instead of (x - t)OI-I.
Developing this idea we may introduce a fractional integration of the form
:&
1 01 1 J <p(t) ' ( )
o+;g<P = r(a) [g(x) _ 9(t»)I-0I 9 t dt,
o
a> 0,
-00  a < b  00, (18.24)

326
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
defined for every function <p(t) E Ll(a,b) and for any monotone function g(t),
having a continuous derivative. The integral (18.24) is usually called a fractional
integral of a function <p( x) by a function g( x) of the order a.
If
g' (x) f; 0, a  x  b,
(18.25)
then the operator 1:+;g is easily expressed via the usual Riemann-Liouville
fractional integration after the corresponding changes of variables:
1:+;g = Ql+Q-l<p, c = g(a),
( 18.26)
where Q is a substitution operator: (Qf)(x) = f[g(x)]. So, many properties of the
fractional integral (18.24), in particular the semigroup property
[ a I p - [ a+ p
o+;g o+;g<P - o+;g<P,
( 18.27)
follows under assumption (18.25) directly from the corresponding properties of the
Riemann-Liouville fractional integral [+. It follows also from (18.26) and (2.44)
that, if gp(x) = [g(x) - g(a)]P-l, then the equality
a r()
(l o +;ggp)(x) = r(a + ) ga+p(X), a,p> 0,
( 18.28)
holds.
In view of (18.26) we may introduce the corresponding fractional differentiation
V:+;gf so that V+;gf = Q1)+Q-l f. Simple calculations lead to the equality
x
( Va )( ) lid f f(t) ' ( )
o+;gf x = r(1 _ a) g'(x) dx [g(x) _ g(t)]a g t dt,
o
O<a<1.
(18.29)
The expression (18.29) may be called a Riemann-Liouville fractional derivatives
of function f(x) by a function g(x) of the order a (0 < a < 1). Derivatives of the
higher orders are defined by a relation similar to the former of (2.30).

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 327
We may easily transform (18.29) to the Marchaud form of the type (13.2):
('D:+;g)f(x)
1 f(x)
r(1 - Q) [g(x) - g(a)]a
:&
Q f f(x)-f(t) ,
+ r(1 - Q) [g(x) - g(t»)1+a g (t)dt,
a
( 18.30)
O<Q<1.
To show this, it is sufficient to apply the operators Q and Q-l to the operator
D+, c = g(a), defined by (13.2), from the right and from the left respectively.
We prove now the following theorem, which goes back to Erdelyi [9].
Theorem 18.1. The space of functions representable by a fractional integral
1:+;gtp, 0 < Q < 1, of a function tp E Lp(a, b), 1  p  00, -00 < a < b < 00, does
not depend on the choice of a function g( x):
1:+;g(L p ) = I:+(L,,),
provided that g(x) E C 1 ([a,b», g'(x) E H>'([a,b» and that (18.25) holds. Besides
this
I:+;gtp = 1:+.,p,
:&
f o(x, s)
.,p(x) =[g'(x)]atp(x) + ox tp(s)ds E Lp,
a
(18.31)
:&
where (x, s) = sin 1r a1r g'(s) I(x - u)-a(u - s)a- 1 h(s, u)du and
8
h(s, u) = [ g(u = ;(S) r a = [j g'(u + (s - u){)d( r- I
Proo/. Let us show first that
1:+;g(Lp)  I:+(Ld.
( 18.32)

328
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Let f(x) = 1:+;gtp, tp E Lp. In accordance with Theorem 2.3 we shall show that
It-OI 1+0I f = 1+0I 1:+ jg tp E AC([a, b]), It-OI(a) = o.
(18.33)
We have
:& u
1 f du f g'(s)tp(s)ds
It-OI(X) = r(a)r(l- a) (x - U)OI [g(u) - g(S»)1-OI
a a
:&
= f (x, s)rp(s)ds.
a
(18.34)
Since
1
sin a1l' f -1
(x, s) = ----;-g'(s) OI (1 - )-OIh(s, s + (x - s))d,
o
( 18.35)
:&
we see that (s,s) = [g'(S)]OI and so (x,s) = [g'(S)]OI + f 8.:''') du. Substituting
,
this into (18.34) and interchanging the order of integration we arrive at the equality
:&
ft-OI(X) = f 1jJ(t)dt,
a
(18.36)
where 1jJ(t) is a function (18.31). It is easily derived from assumptions on g(t) that
1 8hU) 1  c(u-s)>'-1, so we see from (18.35) that 1 8.:") 1  c(x-s)>'-1 too.
Consequently 1jJ(t) E Lp(a, b), and then both conditions (18.33) hold by (18.36).
Thereby the conditions (18.32) is proved which gives the imbedding
1:+;g(L p )  1:+(L p ),
(18.37)
since 1/J E Lp.
Equation (18.31) is a Volterra integral equation with respect to a function
rp(t). In view of (18.25) this equation is solvable in L" for any function 1jJ(x) E Lp
- see for example, the book by Kolmogorov and Fomin [1, p.461] for p = 2 and the
paper by Zabreiko [1] for arbitrary p > 1. This means that (18.37) in fact gives the
relation 1:+ jg (L p ) = 1:+(L p ). The theorem is thus proved. .

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 329
We shall single out the case g(x) = X O (as in Erdelyi-Kober operators), taking
b = 00, and denote
00
I a " =....!!...- f to-lcp{t)dt Rea> O.
-jX cp f( a) (to - x0)l-a'
x
( 18.38)
So using (18.3) and (18.7) we may write down the equality
I a - oa l a ( )
-"x" cp - x -"0 -aCP X .
, I ,
( 18.39)
The operator of fractional differentiation, inverse to (18.38), has the form
00
u 1 - n ( d ) n f to-l J ( t ) dt
Va " = x oa I-a X =
-jX J -jo,a!() r(n _ a) xo-1dx (to _ xo)a-n+l '
x
n = [Re a] + 1.
( 18.40)
The operators
I+ jx " , I:+ jx '" I b - jx '" Vt.jX'" V+jX'" V b - jx "
(18.41 )
are similarly written down in terms of (18.1)-(18.4). We note the analogues of the
formulae (18.16'):
I:+jx"J(x) = n-axoa+on(x-oa-o I:':x"..t J(x),
Ib-ix"J(x) = n-axoa+on(x-oa-o I:!"..t J(x).
(18.41')
18.3. Hadamard fractional integro-differentiation
Riemann-Liouville fractional integra-differentiation is formally a fractional power
(djdx)a of the differentiation operator djdx and is invariant relative to translation
if considered on the whole axis.
Hadamard [1] suggested a construction of fractional integra-differentiation
which is a fractional power of the type (x d ) a. This construction is well suited
to the case of the half-axis and is invariant relative to dilation. Thus Hadamard

330
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
introduced fractional integrals of the form
x
01 1 1 cp(t)dt
+cp = rea) 0 t (In f)1-0I' x > 0, a> O.
( 18.42)
The integrals
00
01 1 1 cp(t)dt
-cp = rea) x t (In )1-0I' x> 0, a > 0,
( 18.43)
may be similarly defined. It is convenient to represent the operators 3'%:cp in the
form (see the notation (5.5»
1 1 00 ( 1 ) 01-1
%cp = r(a) c 1 In t % cp(xt)dt.
o
( 18.44)
We shall call the expressions in (18.42)-(18.44) Hadamard fractional integrals. If
n.s! = f(6x), 6 > 0, is a dilation operator, then obviously
II.s% = %II.s.
(18.45)
It is clear that (18.42) is a fractional integral of the form (18.24) with the
function get) = lnt, so that Hadamard fractional integration (18.42) is a ''fractional
integral of a function cp by a function get) = In t". However, the condition of
the existence of a continuous derivatives g'(t), which was assumed in S 18.2, does
not hold in this case. So the integrals (18.42)-(18.43) need to be independently
considered. (The continuity of g'(t) would be satisfied if we take the lower limit of
integration to be equal to a > 0 instead of zero, but then the property of invariance
of the integral relative to dilation would be broken.)
It is easily seen that operators 3'%: are connected with the well familiar Riemann-
( 01 x 1 J x cp ( et ) dt
Liouville operators I via change of variables: +cp)( e ) = r(OI) (x':'t)l-a and
-00
similarly for cp. So the relation
%cp = A -1 IAcp, (Acp)( x) = cp( eX),
( 18.46)
is valid.
The connection (18.46) allows us to extend various properties of operators I

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 331
to the case of operators 3'%. Thus, observing for example that
IIACPIlLp(R) = IIcpll.cp(R)'
( 18.47)
where
£p(R) = {(t) : l'(t)'P t < oo} ,
( 18.48)
we conclude from (18.46) and the Hardy-Littlewood Theorem 5.3 that the operators
3'% are bounded from .cp(R) into .cq(R) with q = p/(1 - ap), if 1 < p < 1/a.
We see also from (18.46) that operators 3'% admit the semigroup property
3'%3'icp = +,6 cp
( 18.49)
under appropriate assumptions on a function cp and exponents a and /3.
We may also consider Hadamard fractional integrals of the form
x
a 1 f cp(t) dt
3' cp - X > a > _ 0,
0+ = r(a) 0 (In f)l-a t '
( 18.50)
b
a 1 f cp(t) dt 0 < x < b,
b-CP = r(a) x (In )l-a t '
(18.51 )
so that 3'g+ = 3'+ and 3'- = 3'. Similarly to (18.46) we have
3'+cp = A- 1 [:l+Acp,
'l:'a cp = A- 1 [a A cp
Ub- b 1 -,
( 18.52)
with al = Ina and b 1 = Inb.
By direct verification we obtain the properties
d 'l:'a+l 'l:'a
X dx Uo+ = Uo+'
_x'l:'a+l - 'l:'a Rea> O.
dxUb- - Ub-'
( 18.53)
The Hadamard fractional derivative, introduced similar to the Riemann-Liouville

332
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
fractional derivative, has in accordance with (18.53) the form
( ) [a]+l
'D+I'=1 x d -{a} I
( ) [a]+l
= -{a} X  I,
a> 0,
(18.54)
where [a] is an entire part of a, and {a} = a - [a], and fractional derivatives 'D/,
'D+ and 'D b _ being written down similarly. In particular, if 0 < a < 1, we have
x
a 1 d f cp( t ) dt
'D+I = f(1 _ a) x dx (In f)a T'
o
(18.55)
The derivative (18.55) is easily transformed to a form, analogous to Marchaud
fractional derivative (5.57). Namely, operating as in (5.56) or using (18.46), we
may transform (18.55) to the form
x
'D a - a f I(x) - I(t) dt
+1 - f(1 - a) 0 (In f)l+a t
1
a f I(x) - /(tx) dt
- f(l-a) IIntj1+a t'
o
( 18.56)
provided that I(x) is a sufficiently good function. As for the fractional integral
(18.43), its corresponding fractional differentiation is
00
'D a I = a f /(x) - I(t) dt .
- f(l-a) x (In)l+a t
( 18.56')
If a  1, an analogue of the Marchaud fractional derivative (5.80) is to be
introduced by the formula
'D'tf = x(a, ,) t( -1)' G)f(t'X) Ilnt+at' ,> a.
o k-O
( 18.57)
As for the fractional derivatives '1)+/ with a > 0, in the case 0 < a < 1 we

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 333
have
:&
a _ J(x) a f J(x) - J(t) dt ( )
a+J - r(1 _ a) (In !:)a + r(1 - a) ( In !: ) l+a t 18.58
a a t
instead of (18.56) (cf. (13.2».
Finally we note that the action of Hadamard fractional integra-differentiation
on power function is given by the formulae
+(x#') = fl-a X #', fl> 0,
( 18.59)
 (x#') = Ifll-ax#', JL < 0,
( 18.60)
where -00 < a < 00,  being understood as %, if a < O. These formulae are
obtained by simple calculations.
18.4. One-dimensional modification of Bessel fractional
integro-differentiation and the spaces H.,p = L;
Operators of Riemann-Liouville fractional integration, considered on the whole
axis, have as we know, a certain disadvantage: they do not keep the spaces
Lp (Rl) invariant and they are not defined within the frames of these spaces for
all values of a. In some questions, for example in the theory of Sobolev type
spaces of fractional smoothness it would be convenient to deal with such fractional
integration operators which are defined in Lp(Rl) for all a > 0 and are bounded
in Lp(RI) for all p such that 1 $ p  00. The way to define such operators
is suggested by the picture in Fourier transforms. We introduce a convolution
operator
00
Gacp = f Ga(x - t)cp(t)dt,
-00
(18.61 )
which is defined in the Fourier transform by the equality
- 1
Gacp = (1 + IxI 2 )a/2 (x),
Rea> 0
( 18.62)
- (7.1) and (12.23), (12.24). The function Ga(x), whose Fourier transform
is (1 + IxI2)-a/2, is evaluated in terms of Bessel functions. That is why the
operator (18.61) is called an operator oj Bessel fractional integration (or Bessel
potential). This operator is discussed in more detail in S 27. Such an operator
plays an essential role in the theory of fractional differentiation of functions of
many variables. It may be used in the case of function of one variable, but the
specific character of the one-dimensional case allows us to define similar fractional

334
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
integrals of simpler nature. Specifically we shall introduce modifications of the
Bessel potential (18.61)-(18.62) which are defined by the equalities
G a 1  ( ) R 0
%cp= (1 =F ix)aCP x, ea>,
( 18.63)
instead of (18.62). It is implied that a branch of the power function in (18.63) is
chosen as in (5.27). By (7.9) we find that the operators G may be represented as
convolutions with elementary functions:
00
G a - 1 TX a-l _  J e-tcp(x =F t) dt
%cp - r(a) e x% * cp - r(a) t1-a '
o
( 18.64)
so that
x
1 J e-(x-t)cp(t)
Gt. = r(a) (x _ t)l-a dt,
-00
(18.65)
00
G a =  J ex-tcp(t) dt.
- r(a) (t-x)l-a
x
The operators (18.65) are defined by Theorem 1.4 on functions cp(x) E Lp(Rl),
1  p  00, if Rea > O.
We observe that (18.62) and (18.63) yield the relation
G/2G/2 = Ga.
( 18.66)
For fractional operators G simple differentiation formulae
d
:i: G a - G a - 1 G a Rea> 1,
dx % - % - % ,
( 18.67)
are valid. They are generalized in the form
(E:!:  r G% = G%-n, Rea> n,
( 18.68)
E being the identity operators. The validity of these formulae is easily seen using
Fourier transform and may be also established by direct differentiation of the
integrals (18.65).

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 335
In view of the representations
G% = e=fx Ie:J:x,
(18.69)
it follows from (5.15) that
G%Grp = G%+f3rp, rp E Lp(R 1 ), 1  p  00,
(18.70)
where Rea > 0 and Re,8 > 0, so that each of the families of operators {Gf.}a>O
and {G }a>O forms a semigroup in Lp(R 1 ). It is continuous, which may be verified
as in Theorem 2.6.
Let us introduce the corresponding fractional derivatives as operations inverse
to Bessel fractional integration, i.e. to (18.65). Since (Gn-l = e=fx(l;n-1e:J:x by
(18.69), the realization of the operators (G)-l becomes evident. Let 0 < a < 1.
Operators (I:i)-l which admit both Liouville and Marchaud form, are defined
correspondingly
x
a de! e- x d f etrp(t) dt
(E+V) f= r(l- a) dx (x _ t)a
-00
x
1 ( d ) f e-(x-t)rp(t)
= E+- dx
r(l-a) dx (x-t)a
-00
(18.71)
and similarly for (E _v)a I, and
00
(E:i: D)a 1':2 a f f(x) - e- t I(x  t) dt.
r(1 - a) t 1 + a
o
(18.72)
The designations in the left-hand sides of (18.71) and (18.72) have their obvious
origin in (18.68). Fractional derivatives (E:i: v)a I and (E:i: D)a I coincide (under
the same choice of signs) with each other for "sufficiently good" functions I(x).
It is easily seen that the "Bessel fractional derivative" (18.72) is connected
with the Marchaud derivative DI by the relation
(E:i:D)al= D%I+a:!: *1,
(18.73)
where
a 1 - e=fx 1
a:J:(x) = r(l _ a) x+a E L1(R ).

336
CHAPTER4. OTHER FORMS OF FRACTIONAL INTEGRALS
We consider now the spaces
GOt(Lp), Gt.(L p ), G(Lp), 1  p < 00,
which consist of functions representable by the integrals (18.62) and (18.65)
respectively of functions cp E Lp(RI). They coincide with each other
GOt(L p ) = Gt.(L p ) = GC:(L,,), 1 < p < 00,
by the Corollary of Theorem 1.6. Sometimes other notations are adopted for the
space G(Lp), e.g. HOt,p or L;, which we shall also use, so that
L;(R 1 ) = HOt'P(Rl) = GOt(L,,), 1 < p < 00.
These spaces are known as the spaces of Bessel potentials. Later on in S 27 we shall
go into details while considering these spaces in the more general case of functions
of many variables. Such spaces represent a widely known version of the spaces of
differentiable functions with a fractional smoothness. Many investigations concern
such spaces - see for example the books by Nikol'skii [6] and Besov, Il'in and
Nikol'skii [1], the references in these books, and the Bibliographical Notes to S 27
in S 29 below. In this Section we give a simple characterization of the space L; (Rl )
in the one-dimensional case, which is contained in the following theorem and its
important corollary.
Theorem 18.2. Let 0 < a < 1 and 1 < p < 00. Then f(x) E L;(Rl) if and only
if f E Lp(Rl) and (E + D)Ot f E L,,(Rl), where the "Bessel fractional derivative"
is considered as an integral conditionally convergent in L,,-norm:
00
(E + D t f = a lim f f( x) - e - t f( X - t) dt.
f(1 - a) £-0 t 1 + Ot
(Lp) £
(18.74)
The proof of this theorem is quite similar to those of Theorem 6.1 and 6.2 with
the corresponding simplifications due to the exponentially decreasing factor; we
note for example that instead of (6.6) we obtain the equality
00 00
f I(z) - ;l:!(z - t) dt = f e-'t.qt)(z - £t)dt, I = G'i-,
£ 0
(18.75)

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 337
where K(t) is the kernel (6.7).
Corollary. Comparing Theorem 18.2 with Theorem 6.2 and taking into account
the connection (18.73) between Bessel and Marchaud differentiation, we see that
Lp n ]a(L p ) = H a ,P(R 1 ), 1 < p < 1/0:.
(18.76)
We observe that Theorem 18.2 may be extended to all values 0: > 0, if
(E + D)a is realized as
001
(E + ntf = K( I) !J 1:,(-1)' G)e-"f(X - kt) ta
, (L p ) £ k=O
(18.77)
in correspondence with the formula (5.80) and the relation (E + D)a = e-zD+e t .
Sometimes, the spaces H8,P([a, b]) are considered in the literature. These are
defined as the spaces of restrictions of functions in H8'P(Rl) on the interval [a, b]
with the norm IlfIlHe,p([a,bD = inf IlgIIHe,p(Rl), the infimum being considered on all
functions 9 E H"P(Rl), which coincide with f(x) on [a, b]. From the point of view
of presentation in this book it is important to remark that the spaces H8,P([a, b])
coincide in general with the spaces ]a[Lp(a, b)] of fractional integrals of functions
in Lp(a, b) that were studied in S 13. Specifically the following theorem holds.
Theorem 18.3. Let 0 < 0: < l/p and 1 < p < 00. Then
Ha,P([a, b]) = ]a[Lp(a, b)]
(18.78)
up to the equivalence of norms, -00 < a < b < 00.
Proof. Let f(x) E Ha,P([a, b]). Then there exists g(x) E H a ,P(R 1 ) such that
g(x) = f(x) as a  x  b. By (18.76) we see that g(x) E ]a(L p ). But then
f(x) E ]a[Lp(a, b)] according to Theorem 13.9. Conversely, let f(x) E ]a[Lp(a, b)]
and let f. (x) be the continuation of this function by zero beyond the interval [a, b].
By Theorem 13.10 we have: f.(x) E [a(L p (Rl». Since obviously f.(x) E Lp(Rl),
by (18.76) we see that f.(x) E H a ,P(R 1 ). .
Remark 18.1. Analysis of the proof of Theorem 18.3 shows that in the case
l/p < 0: < l/p + 1 the relation
H,P([a, b]) = [\[Lp(a, b)]
(18.79)

338
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
is valid, the space HJP ([a, b]) consisting of functions f(x) E Ha,P([a, b]) such that
f(a) = O. It may be shown that in the bounding case a = lip (18.78) does not
hold and even the space Ha,P([a, b]) is not imbedded into I+[Lp(a, b)]. The reason
for this is in the unboundedness of the operator (13.27) in Lp when a = lip. In the
case of an arbitrary a > lip, but a - lip 1= 1,2, .. ., (18.78) is true provided that
the assumptions f(l:)(a) = 0, k = 0, 1,... ,[a - lip]' are added to the definition of
the space H,P([a, b]).
18.5. The Chen fractional integral
The points +00 and -00 do not play an equal role for Riemann-Liouville operators
Icp considered on the whole real axis. This reveals itself, for example, in the fact
that the fractional integral Icp keeps in general, decrease of the function cp(x) at
=Foo and does not keep it at :1:00 (see (7.8». In this subsection we consider -
following Chen [2] - a modification of Riemann-Liouville integration, for which the
left and the right infinite points are symmetric.
We fix an arbitrary point c E R 1 and set
I rla) j cp(t)(x - t)a- 1 dt, x > c,
(I:cp)(x) = C c
rla) J cp(t)(t - x)a- 1 dt, X < c,
x
( 18.80)
where a > O. We shall refer to (18.80) as the Chen fractional integral. The
expression in (18.80) has an evident advantage in comparison with It. cp or I cp:
it is applicable to a function with any behaviour at infinity, and it may also be
considered in an interval [a, b] as well, with a < c < b, the end-points a and b
having equal rights.
Introducing the functions
{ cp( x ) ,
Pc+cp = CPc+(x) =
0,
x> c,
x < c,
(18.81)
{ 0,
Pc-Cp = CPc-(x) = (
cp x),
x> c,
x < c,
we may write down (18.80) as
(Icp)(x) = (It.CPc+)(x) + (Icpc-)(x)
( 18.82)

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 339
or in operator form
I: = It.Pe+ + IPe- = Pe+It. Pe + + Pe-IPe-'
(18.83)
Using (2.44) and (2.45), from (18.82) we obtain the formula
pX(lx _ clfj-l) = f(,8) Ix _ cla+fj-l.
e f(a+,8)
(18.84)
For the functions rp(t) with a "sufficiently good" behaviour at infinity, for
example, for rp(t) E Lp(Rl), 1  p < l/a, the integral Irp may be represented as
00 00
a 1 f rp(t)dt 1 f sign(x - t) .
Ie rp = 2f(a) Ix _ tp-a + 2f(a) Ix _ tl1-a rp(t)slgn(t - c)dt.
-00 -00
(18.85)
It is easily derived from (18.83) that the operators I admit the semigroup
property
I: I rp = I:+fj rp (18.86)
for any locally integrable function rp(t) and all a > 0 and ,8 > O.
Let us consider the question as to what operator is inverse to I, i.e. what is
the corresponding form of fractional differentiation to be. We have formally
(Vj)(X)12(I:)-lj= { (V+)(x), x> c,
(Ve_)(x), X < c.
(18.87)
In particular
! d j j(t)(x - t)-adt, x > c,
(V j)(x) = 1 e e
f(1 - a) _ d J j(t)(t - x)-adt, x < c,
x
( 18.88)
if 0 < Q < 1. In the case a  1 it is necessary to use (2.30) and (2.31). We observe
that obviously
(V j)(x) = [sign (x - c)r j(n)(x), n = 0,1,2,...
(18.89)
Justification of this inversion, i.e. the relation V Irp == rp(x), a > 0, for
rp(x) E Loc (Rl), follows from Theorem 2.4 immediately.

340
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Recalling the expression for fractional derivatives in Marchaud form, - (13.2)
and (13.5) - we come from (18.88) to
max(z,c)
(V: 1)(20) = r(l- \i _ cia + " f f(x) - f(t) dt
f(1 - a) Ix - tl 1 + a
min(z,c)
(18.90)
in the case 0 < a < 1. The passage from (18.88) to (18.90) is possible with
"sufficiently good" functions - see S 13.1. Let us denote the right-hand side in
(18.90) by (D f)(x), so that for "sufficiently good" functions we have
(Vf)(x) = (D f)(x).
(18.91 )
This right-hand side (D f)(x) may be transformed to
00
(D a f)(x) = a f f(x) - fc+(x - t) - fc-(x + t) dt
c r(1 - a) t 1 + a
o
= DPc+f + DPc-f,
( 18.92)
where D is Marchaud fractional differentiation (5.57)-(5.58) and the notations
(18.81) are used. We may write down the derivative (D fHx) in the form similar
to (18.92) in the case a  1, if we use (5.80), which leads to the representation
00 I I
( D a f )( x ) =  f (Lltfc+)(x) + (Ll_tfc-)(x) dt
c x(a, I) t 1 + a '
o
( 18.93)
where I > a and x(a, I) is a normalizing factor (5.81).
We now put a natural question: what is the influence of the choice of the
point c upon Chen fractional integration? Do the operators fcp and flcp differ by
their smoothness properties? More concretely, if a function f(x) is representable
by a fractional integral f(x) = Icp of a function cp in a certain space, e.g. Lp, is it
representable as f(x) = Il1/J with 1/J E Lp? The answer to this question is given in
Theorem 18.3 below. As a preliminary the next lemma clears up the possibility of
the representation of I cp as I cp = If. 1/J.
Lemma 18.1. Let f(x) be representable as f(x) = (Icp)(x), where cp(x) E

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 341
Lp( -00, b), 1 < p < 1/0:, for some b. Then
f(z) = I+1Pc,
c
sin 0:1r f cp(t)
1Pc(z) = Nccp = vc(z)cp(z) + - -dt,
1r t-z
-00
-00 < z < b,
( 18.94)
where vc(z) = 1 for z > c and vc(z) = coso:1r for z < c and 1Pc(z) E Lp( -00, b). If
cp E Lp(Rl), then 1Pc(z) E Lp(Rl) too.
Proof. Recalling (11.10) and (11.12) we replace I in (18.82) by means of these
identities. We obtain relation P:cp = 1+ (Pc+Cp + COS 0:1r Pc-Cp + sin O:1rSP _cp), which
coincides with (18.94). The operator N c = P c + + cos 0:1r Pc- + sin 0:1r SP c - is
bounded in Lp(Rl) by Theorem 11.3. This completes the proof. .
Lemma 18.2. The operator
N c = Pc+ + COS 0:1r Pc- + sin 0:1r S Pc- ,
(18.95)
is invertible in the space Lp(R 1 ), 1 < p < 1/0: and
N;- 1 = Pc+ + COS 0:1r Pc- - sin 0:1r SPc- ,
( 18.96)
where S is a singular operator of the type (11.39):
00
Scp = .!. f I t - c l a cp(t)dt .
1r z-c t-z
-00
( 18.97)
Proof. For the sake of simplicity we take c = 0, and denote P c % = P% and
S = Sa. The operators (18.95) and (18.96) have the form No = P+ + AP_ and
N 0 1 = P++BP_, where Acp = cosO:1rcp+sinO:1rScp and Bcp = cosO:1rcp-sinO:1rSacp.
Both are bounded in Lp(R 1 ) by Theorem 11.3. We are to show that
(P+ + AP_)(P+ + BP_)cp = (P+ + BP_)(P+ + AP_)cp = cp
( 18.98)
for cp( t) E Lp (R 1 ). Meanwhile Lemma 11.1 states that operators A and Bare

342 CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
reciprocally inverse on the half-axis, i.e.
P_AP_BP_cp = P_BP_AP_cp = P_cP,
cP E Lp(R 1 ), 1 < p < l/a.
( 18.99)
Multiplying in (18.98) and using (18.99) we see that (18.98) is reduced to the
relations P+BP_ +P+AP_BP_ = 0 and P+AP_ +P+BP_AP_ = O. Substituting
here the expressions for A and B, we transform these equalities to
sina1rP+SP_S a P_ = cosa1rP+SP_ - P+SaP_,
(18.100)
sina1rP+S a P_SP_ = P+Sp_ - cosa1rP+SaP_.
(J8.101)
Their validity is checked by direct calculation of left-hand sides by means of the
Poincare-Bertrand formula (11.9). In fact, we have already obtained (18.101)
earlier - (11.47); we also remark that (18.100) follows from (18.101) because
raP_SP_SaP- = (P+S-aP_SP_)r a , where raCP = Ixlacp(x). .
Theorem 18.4. Let J(x) be representable as f(x) = (Icp)(x), c E Rl, where
cp(x) E LOC(Rl), p> 1, p 1= l/a, 0 < a < 1. Then f(x) is representable as
J(x) = (Il1/J)(x), 1 < p < l/a,
(18.102)
J(x) = J(d) + (I:1/J)(x), p> l/a,
( 18.103)
whatever the point d is, where 1/J(x) E LOC(Rl). ffcp E Lp(Rl), 1 < p < l/a, then
1/J E Lp(Rl) also. The function 1/J(x) is evaluated by the formulae
d a
de! s ina7r Jl t-d l cp(t)
1/J(x) = N C dCP= lI cd(X)CP(X) + - - d -dt,
1r X- t-x
c
l<p<l/a,
(18.104)
. d a-l
- de! sma1r Jl t-d l cp(t)
1/J(x) = N cd CP= lI cd(X)CP(X) + -J-tcd(X) - d -dt,
- 1r x- t-x
C
p> l/a,
(18.105)

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 343
where
( { cos a 1r , x E (e, d),
IIcd x) =
1, x ft (e, d),
Jjcd(X) = sign [(x - e)(e - d)].
Proof. Let I;'(x) and 1/J(x) be arbitrary functions in Lp(R l ), 1 < p < l/a, at first.
By Lemma 18.1 we have 1<;1;' = I+Ncl;' and I:t1/J = 1+ N d 1/J, where the operators
N c and Nd are given by (18.94) or (18.95), which is the same. Consequently, the
desirable relation Ie:I;' = 1:t1/J will be achieved, if Ncl;' = Nd1/J. Since the operator
N d is invertible in Lp(Rl) with 1 < p < l/a by Lemma 18.2, we are to verify the
relation ,p = Nil Ncl;'. Let us calculate the composition Nil N c . We have
Nil Ncl;' = I;' + Nil(N c - Nd)l;'.
(18.106)
It is evident that N c - Nd = (cosa1r - I)Pcd + sin a1rSP c d, where
p _ { 1;'( x), x E (d , e),
cdI;' - 0, x ft (d,e)
(we take d < e to be definite). So
Nil(N c - Nd) =(cos a1r - I)P cd + sin a1r Pd+SPcd
+ sin a1r cos a1rP d -SP cd - sin 2 a1rS:Pd_SPcd'
(18.107)
The calculation of a composition sgPd-SP cd by applying the Poincare-Bertrand
formula (11.9) and then using (11.46), gives after simple transformations
sina1rS:Pd_S P cd = Pd+SPcd + cosa1rPd_SP cd - S:Pcd.
Substituting this into (18.107) we arrive at the relation Nil(N c - Nd) =
(cosa1l'-I)Pcd + sin a 1I'sgPcd. Thereby from (18.101) we have
C 01
-1 sin a1r f I t - d I I;'(t)dt
,p=N d Ncl;'=I;'+(cosa1l'-I)Pcdl;'+- - d -,
1r x- t-x
d

344
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
which coincides with (18.104). So the identity
I:cp = Id Nedcp,
(18.108)
is proved. This gives (18.102) and (18.104) for functions cp(x) E Lp(RI),
1 < p < I/Q. Since the operators I: and Id have variable upper and lower limits
of integration, (18.102) uses the values of the function cp on a finite interval only
(via (18.104». Realizing that each function cp E Loc coincides on each interval
with some function in Lp(Rl), we see that (18.102) with the function (18.104) is
valid for cp E Loc too.
It remains to consider the case p > 1/ a. In virtue of the imbedding Loc C Loc,
r < I/Q, the representation (18.102) with the function in (18.104) holds again but
we cannot state that ,p E Lp, because the operator Ned is unbounded in Lp, when
a > l/p. So we transform the function in (18.104) to a new form. We take d < c
and for t > d we have
I t-d I Oi. I t-d I 0l - 1 It-dIOl-l
x-d =slgn(x-d) x-d + Ix-dl o (t-x),
which is directly confirmed. Therefore
e
,.., sin a1r 1 J 1
1/J(x) = 1/J(x) +  Ix _ dl o (t - d)OI- cp(Odt.
d
e
Observing that f(x) is continuous in the case p > l/a and so J(t - d)0I-1cp(t)dt =
d
f(d)r(a), we see that
IN = I:t;f + r({)a/ :t(l:r - dl-<> = I:t;f + fed)
by (18.84). Thereby we have transformed (18.102) to (18.103). It remains
to note that the singular operator in (18.105) is bounded in L", p > l/a by
Theorem 11.3. .
18.6. Dzherbashyan's generalized fractional integral
We consider the case of the interval [0,1]. The Riemann-Liouville fractional
integral may be written down as

 18. GENERALIZATIONS OF RIEMANN-LIOUVILLE INTEGRALS 345
1
(IO+CP)(X) = r:) f cp(xt)(l- t)a- 1 dt, X > O.
o
Let us generalize (18.109) replacing the function (1t-1 by an arbitrary
(integrable) function, but neglecting the factor x a . Special)y, following Hadamard
[1] and M.M. Dzherbashyan [4], [5] we introduce the operator
(18.109)
1
(L(w)cp)(x) = - f cp(xt)w'(t)dt,
o
(18.110)
where the function w(x) E C([O, 1]) is supposed to satisfy the following assumptions:
1) w(x) is monotone,
2) w(O) = 1, w(l) = 0, w(x) 1= 0 as 0 < x < 1,
3) w'(x) E L 1 (0, 1).
Condition 3) implies that the operator (18.110) is well defined on bounded
functions cp(x).
If w(t) = ) ' then obviously (L(w)cp)(x) = x-a(Io+cp)(x).
It is evident also that integrating by parts in (18.110) we arrive at (L(w)cp)(x) =
1
cp(O) + x J cp'(xt)w(t)dt in the case of continuously differentiable functions cp(x).
o
The operator in (18.110) is well suited to power functions:
L(W)(x lJ ) = Au (JL)x lJ , JL > -1,
(18.111)
1
where w(JL) = - J tlJw'(t)dt, which in the case JL > 0 may be replaced by
o
1
Aw(JL) = JL f t lJ - 1 w(t)dt.
o
( 18.112)
So an applications of the operator L(w) to functions cp(t), representable by
p ower serIes
00
cp(t) = L al:tA:,
1:=0
( 18.113)
is straightforward:
00
(L(w)cp)(x) = L Au (k)al:xl: .
1:=0
(18.114)

346
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
The following lemma generalizes Lemma 15.4.
Lemma 18.3. The series (18.113) and (18.114) have equal radii of convergence.
Proof. To prove the lemma we shall show that
lim V'L\w(k) = 1.
k-oo
(18.115)
By (18.112) we have
1 1
k(1 - £)k-1 f w(t)dt  L\w(k)  k f w(t)dt, k = 1,2,...,
1-£ 0
whatever £ > 0 is. Using assumptions 1) and 2) we see that 1-£  k lim tI L\w(k) 
-00
1 then, which yields (18.115). .
Relation (18.111) shows how the generalized differentiation, M(w), inverse to
L(w) : M(w) L(w)cp = L(w) M(w)cp, is to be introduced. Namely, the operator M(w) is
to be defined in a such way that M(w)(x#') = ",1(#,) x#'.
We restrict ourselves to a statement, proved by M.M. Dzherbashyan [5], that
each function wet) admits a non-decreasing bounded function aw(x) such that
1
f zP dar w (z) = 8 w 1 (I-') '
o
1
(M(w)cp)(x) = f cp(xt)daw(t).
o
Remark 18.2. M.M. Dzherbashyan [4], [5] considered the operators L(w) 10 a
more general form
L(w)<p = -  {z I <p(zt)dp(t) } ,
1
pet) = t f w(x) dx
x 2
t
under weaker assumptions than given in 1)-3). For simplicity we dealt with the
operator L(w) in the form (18.110).

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
347
fi 19. Weyl Fractional Integrals and Derivatives
of Periodic Functions
The usual forms I+ or I b _ for the Riemann-Liouville fractional integral prove to
be inconvenient in the theory of trigonometrical series which deals with periodic
functions. It is natural that the operation of fractional integra-differentiation is to
be defined in such a way that it transforms periodic functions into periodic ones.
Riemann-Liouville fractional integra-differentiation does not have this property.
So, for periodic functions another definition of fractional integra-differentiation,
suggested by Weyl, is used. It will be thoroughly treated in this section.
19.1. Definitions. Connections with Fourier series
Let rp(x) be a 21r-periodic function on R 1 and let
00
rp(x) "-J L rplee- ikx ,
le=-oo
2'11'
rpk =  f e-ikxrp(x)dx,
21r
o
(19.1 )
be its Fourier series. Throughout this section, while dealing with fractional
integration, we shall consider functions having zero mean value:
211'
21rrpo = f rp(x)dx = 0,
o
(19.2)
i.e. we "throwaway" constants while considering fractional integrals of periodic
functions. Let us recall that a convolution
2'11'
(Arp)(x) = 2 f a(t)rp(x - t)dt
o
(19.3)
of two periodic functions is represented by the Fourier series
00
(Arp)(x) "-J L ale rplee ikx ,
le=-oo
(19.4)
where ale and rple are Fourier coefficients of the functions a( x) and rp( x). The
sequence {ale }r=-oo is sometimes called a Fourier multiplier of the convolution
operator A.

348
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
00
Since <pCn)(x) f'OoJ L: (ik)n<pkeikz, we define fractional integration, following
k=-oo
H. Weyl, such that
00
lOt){f) f'OoJ "'"' eib
% T L.J (:i:ik)Ot
k=-oo
(19.5 )
where (19.2) is taken into account. Similarly fractional differentiation is defined:
00
1JOt)<p f'OoJ L (:i:ik)°<pkeib
1:=-00
(19.6)
- compare with formulae (7.1) and (7.4). Thus the requirement that fractional
integrals and derivatives of 21r-periodic function are again 21r-periodic function is
fulfilled. In (19.5)-(19.6) (:i:ik)Ot = IkIOte%signk in correspondence with (7.3).
Owing to (19.3)-(19.4) the definition (19.5) may be interpreted as
211'
IiOt)<p =  f <p(x - t)w%(t)dt, a > 0,
21r
o
(19.7)
where
00 ikt 00 (k / )
W O = ",",'  = 2 "'"' cos t 1= a1r 2
% L.J ( :i:ik ) Ot L.J k Ot
1:=-00 1:=1
(19.8 )
The dash indicates that the term k = 0 is omitted. The right-hand side in (19.7)
will be called the Weyl fractional integral of order a. It is known - Zygmund
[6, p.201) - that the series (19.8) convergence for all t E (0, 21r), if a > O.
The functions 1/J may be expressed in terms of the generalized Riemann zeta
function (1.87):
r(a)w%(t) = (21r)Ot«I- a,:i:t/21r), 0 < t < 21r,
(19.9)
according to the Hurwitz formula (1.88). Equation (19.9) is contained implicitly
in the representations (19.11) and (19.12) proved below. In the case of an integer
a = 1,2, .., the functions w(t) may be represented by (1.89) as
w m ( t ) = - (:i:21r)m B (  ) 0 < t < 21r,
% m! m 21r '
(19.10)
Bm(t) being the m-th Bernoulli polynomial.

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
349
In view of (19.5) it is clear that Ia) fl)<p = Ia+{j)<p under the same choice
of signs. We observe that the case of integer Q = 1,2,3,... corresponds to usual
integration. This implies the choice of a primitive having zero mean value over the
period. Thus, in fact, it is sufficient to study mainly the case 0 < Q < 1.
We concentrate in detail on the ''left-hand sided" fractional integral Ia) <p
noting that w (t) = w+( -t).
Lemma 19.1. The function w+(t) in the case 0 < Q < 1 has the form
,T.a ( ) 21r 01-1 ( )
't!'+t = r(Q) t+ +ra t ,
-21r < t  21r,
(19.11)
where the function
ra(t) =  lim [ 21r  (t + 21rm)a-l _ (21rn)a ]
r ( Q ) n-oo L..J + Q
m=1
(19.12)
is infinitely differentiable for t E (-21r, 21r].
Proof. We write
21r [ n n a ]
G(t) = r(Q) nl l;o(t + 21rm)+-1 - (21r)a-l '
(19.13)
and we have to show that G(t) == w+(t) for -21r < t  21r. For this purpose it is
sufficient to show (in view of (19.8» that the Fourier coefficients of the function
G(t) coincide with (ik)-a:
211'
Gle =  f G(t)e-iletdt = (ik)-a.
21r
o
(19.14)
To show this we shall prove the formula
00
G(t) = L g(t + 21rm) ,
(19.15)
m=-oo

350
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
where the function g(t) is defined by the equation
[ 2(m+l)1I' ]
21r a-l 1 a-l
g(t) = r(a) t+ - 21r f s+ ds
211'm
as 21rm  t < 2(m + 1)1r, m = 0, :1:1,...
Equation (19.15) is verified directly if we take into account that (21r)a-l n a /a =
2nll' 2(n+l)1I'
2 J ta-1dt and that J ta-1dt --+ 0 as n --+ 00.
o 2nll'
The function g(t) is absolutely integrable over R 1 . In fact
00 2(m+l)1I' 2(m+l)1I'
f Ig(t)ldt = r:) f; f t a - I - 2 f rIds dt
-00 m-O 2mll' 2m 11'
2(m+l)1I' 2(m+l)1I'
= r(l,,) f: f f [t a - I - .a-I]ds dt.
m=O 2mll' 2m 11'
Applying here the mean value theorem for m  1 we obtain
00 2(m+l)1I' 2(m+l)1I'
f Ig(t)ldt:5 CI + C2 f: f f {a- 2 dsdt
-00 m=l 2mll' 2m 11'
00
< "" a-2
_ Cl + C3 L.J m < 00,
m=l
since  = cp E (2m1r,2m1r + 21r). The absolute integrability of g(t) and
00 211' 00
the equation L: J Ig(t + 2m1r)ldt = J Ig(t)ldt show that (19.15) absolutely
m=-oo 0 0
converges for almost all t. We use the known fact that the convergence of the series
b
L: J Ifm(t)ldt implies the almost everywhere convergence of the series L: Ifm(t)l; -
a
Zygmund [5, p.49].
It is known - Zygmund [5, p.1l6] - that Fourier coefficients for the sum
of a series (19.15) coincide with Fourier transform of the function g(t) at the
points k = 0, :1:1,..., i.e. G le = g(k), where g(x) is the Fourier transform (1.104).

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
351
Consequently
G. = 2 _Z e-;"g(t)dt = r(1,,) moo 2 1 '). [t a - , - 2 2 1 1). S't- 'dS ] e-;"dt
00
=  ! ta-ie-iktdt k  0
f(a) , T'
o
since the integral of e- i1ct over the period is zero, if k 1= O. Using the value of the
above integral, which was calculated in (7.6), we arrive at (19.14).
It is left to state that the infinite differentiability of the function ra(t) is easily
seen from (19.12) and rP{t) = Cj f: (t + 2m1r)a-i- j , Cj being evident constants,
m=i
with j  1.
Corollary. Let It I  1r. The functions W%(t) and their derivatives admit the
estimates
I dj I .
dtj w%(t)  cltl a - i -"
j=O,I,2,...
(19.16)
By Lemma 19.1 and its corollary the definition of the Weyl fractional integral
by (19.7) is correct in the sense that it is applicable to any integrable function and
the integral in (19.7) exists almost everywhere and also gives an integrable function
(it is clear that Icp is continuous, if cp(t) is).
By (19.11) we have
z 2
(a) _ 1 ! cp(t)dt 1 !
I+ cp - f(a) (x _ t)l-a + 21r ra(x - t)cp(t)dt,
o 0
o < X  21r.
The latter of these integrands is infinitely differentiable if 0 < x < 21r, so the Weyl
fractional integral (19.7) differs inessentially from the Riemann-Liouville fractional
integral I%cp (the first summand) by differentiability properties, inner points of
the interval (O,21r) being in mind. The behaviour of the second integrand at the
end points x = 0 and x = 21r is in general the same as the first.
It is natural to define the Weyl fractional derivative in the case 0 < a < 1 by
the equality
1)( a) f = :!:.!!.- I(i-a) f
% dx %
(19.17)

352
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
in complete correspondence with (19.6). Equation (19.17) may be called the
Weyl-Liouville derivative in comparison with the expression
211'
Da) f = 2 j [f(x - t) - f(x)]  wi-a(t)dt,
o
( 19.18)
which might be called the Weyl-Marchaud derivative, cf. (5.57) and (5.58). The
right-hand side of (19.18) is obtained from that of (19.17) by formal differentiation
under the integral sign and integration by parts.
Lemma 19.2. Fractional derivatives (19.17) and (19.18) coincide on functions
f(x) E H>'([0,21r]), A > 0::
1Ja) f == Da) f.
Proof. Let F(x) be any primitive of a function f(x). Then
211'
1Ja) f = 2  j W-a(t)d[F(x) - F(x - t)]
o
=  { w1-a(t)[F(x) - F(x _ t)]1 2 11'
21r dx + 0
211'
- j[F(z) - F(z - t)]  w-a(t)dt}
o
211'
= j[F'(X) - F'(x - t)]  W-a(t)dt = Da)f,
o
where all the transformations are easily justified for f(x) E H>', A > 0:, by means
of (19.16).
19.2. Elementary properties of Weyl fractional integrals
We begin with the following lemma which is of importance throughout this section
for understating the nature of the Weyl integral.
Lemma 19.3. Let I;'(t) E L 1 (0, 21r) and be 21r-periodic, and let (19.2) be satisfied.
Then the Weyl fractional integral coincides with the Riemann-Liouville integral on

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
353
the real line:
:&
[(a) cp = [a cp =  f cp(t)dt 0 < a < 1 ,
+ + r(a) (x_t)l-a'
-00
(19.19)
provided that the integral on the right-hand side is understood as conventionally
convergent:
:& :&
f cp(t)dt = lim f cp(t)dt
(x - t)1-a n- (x - t)1-a .
nEZ :&-2mr
( 19.20)
-00
The representation by an absolutely convergent integral
Ia)1" = r(lCJt) i 1"(0: - ,+ , - a - (2,. [ 2 ' ,. ]) a_I} dt
(19.21)
is also valid.
Proof. Since the limit in (19.12) is uniform in t E [0,21r] from (19.11) we have
211' [ ]
1 n 2 a-I a
Ia)1" = r(CJt) l!...rr,J 1"(0: - I) 2)1 + 2..W- 1 - ( ,.) CJt n dl.
o k=O
Using (19.2) and periodicity of the function cp(t), we have
n 2(k+l)1I'
a)cp =  lim '" f cp(x - t)t a - 1 dt
+ r(a) n-oo L.J
k = 0 211' k
2(n+l)1I'
= r(lCJt) nl!...n.:. f 1"(0: - 1)la- 1 dl,
o
which gives (19.19) in accordance with (19.20). To show (19.21), it is sufficient to
2(k+l)1I'
note that f cp(x - t)(21rk)a- 1 dt = 0 for all k = [t/21r] = 0,1,2,... .
211'k
We should like to stress that the interpretation of the integral in (19.19) in
terms of (19.20) is caused by the nature of the problem and its convergence is
connected with vanishing of mean value of a function cp(t) - see (19.2).

354
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
There exists a connection of the type (11.10) and (11.11) between the fractional
integrals 10I)<p and 10I)<p. To formulate it we introduce the singular integral with
the Hilbert kernel
211'
1 f x-t
H<p = 21r <p(t)ctgdt
o
(19.22)
taken in the principal values sense which connects conjugate Fourier series -
Zygmund [5, p.88] -
00
<p f'OoJ L <P1c eilcz ,
1c=-oo
00
H <p f'OoJ -i L:' signk<p1c eib .
Ic=-oo
( 19.23)
Theorem 19.1. Let <p(t) E Lp(O, 21r), 1 < p < 00. Then
10I)<p = cos a1rIOI)<p + sin a1rH 10I)<p,
(19.24)
1(0I){f) = cos a 1rI(OI) (f) - sin a1rH I(OI){f)
+ T - T - T'
( 19.25)
Proof. It is known that the operator H is bounded in Lp(O, 21r), 1 < p < 00, by
the Riesz theorem - Zygmund [1, pA04]. The operators 11 01 ) are also bounded in
L,,(O, 21r), 1  P < 00 (since w(t) E Lt). So it is sufficient to verify equations
(19.24) and (19.25) on the set of infinitely differentiable functions which is dense
in Lp(O, 21r). Fractional integration may be written as (19.5) on such functions,
.. ffi . t h . 1 - COSOlll' sinOlll' ( -isi gn 1c ) . . f
so It IS su clen to prove t e equatIOn (-i1c)'" - (ik)'" + (ilc)'" m view 0
(19.23). This equality is obvious. Similarly (19.25) is verified. .
19.3. Other forms of fractional integration of periodic
functions
We may introduce an analogue of the Riesz potential (12.1) for periodic functions.
Starting from (12.23) we must define it in such a way that
00
I(OI)<p f'OoJ L:' <pic e ib
k=-oo Ikl Oi .
( 19.26)

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
355
Since Ikl a = 2C08("-/2) [(ik)a + (-ik)a], we may introduce [(a)cp as a convolution
2,.-
[(a)cp = J.. f cp(x - t)wa(t)dt
211'"
o
(19.27)
where
wa(t) = w+(t) + W(t) = 2 f: cos kt . (19.28)
2 cos( a 11'" /2) k=1 k a
Similarly to Weyl fractional integration (19.7) the operator (19.27) of the
Riesz potential type satisfies the semigroup property [(a) [(fJ) = [(a+ f3 ) , which is
evidently seen from (19.26).
Equations (19.24) and (19.25) after simple transformations give the following
relations between the operator [(a) and fractional integration [a):
[(a)cp = cos(a1l'"/2)[a)cp + sin(a1l'"/2)[a) Hcp,
( 19.29)
[a)cp = cos(a1l'"/2)[(a)cp - sin(a1l'"/2)[(a) Hcp,
where cp(x) E Lp(O, 2pi) , p> 1.
We may go on and call any convolution operator (19.3) with
( 19.30)
c
ak  Ikl a as Ikl -+ 00
(19.31)
a fractional integration operator of order a. The constant c in 19.31) may be
allowed to have different values as k -+ +00 or k -+ -00. [a and [(a) are
examples of such operators. They satisfy the semigroup property but have the
drawback that they are convolutions with non-elementary functions. Conversely,
we may construct convolutions with elementary functions with the property
(19.31) but without the semigroup property. Thus, for example the operator
2,.-
[a)cp = r() f [sin(x - )]+-1cp() with t+- 1 understood as usual, see (5.5), has
o
an expansion
00
[a)cp  L akCPkeikz,
k=-oo
e-iV + (-I)l:e iV r ()
ak = 21+a r ( k++a ) .
(19.32)
by formula 2.5.12.36 from Prudnikov, Brychkov and Marichev [1]. The condition
(19.31) is satisfied then in view of (1.66). However, the semigroup property is not
valid as may be seen from (19.32).

356
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
We call attention to a two-parameter family of fractional integration operators
211'
IOI)rp = f KOI,IJ (x - t)rp(t)dt,
o
where
K ( ) _  cos(kx -J.l.1r/2)
OI,IJ x - L....J k Ol .
k=l
Such operators are used in the approximation theory of periodic functions -
Bibliographical Notes in S 23.1 (notes to S 19.3) and S 23.2 (note 19.6). Simple
transformations give the formula
KOI (x) = sin [(Jj.+ a)1r/2] w(0I)(x) + sin[(a. - Jj)1r/2] W0I)(X).
,IJ sm a1r + sm a1r
So the operator 10I) is a linear combination of Weyl fractional integration operators:
1(01) = sin[(Jj + a)1r,/2] lOl) sin[(a - Jj)7(/2] lOl)
IJ rp sin a1r + rp + sin a1r - rp.
Here one may evidently see the analogy with Feller potentials which we considered
in S 12 for a non-periodic case - see (12.10) and (12.13). It is obvious that
li Ol ) = 1+ and I':2 = 10I).
19.4. The coincidence of Weyl and Marchaud fractional
derivatives
It follows from Lemma 19.3 and definition (19.17) that the Weyl fractional
derivative (19.17) coincides with the Riemann-Liouville derivative
:&
1)(01) - 1  f f(t)dt
+ f - r(1 - a) dx (x - t)O'
-00
o < a < 1,
( 19.33)
211'
provided that J f(t)dt = 0 and the integral in (19.33) is treated in the sense
o
of (19.20). The conventional convergence of the integral in (19.33) at infinity
is essential. We shall show that the use of the Weyl-Marchaud form (19.18) of
fractional differentiation allows us to avoid conventional convergence. Let us prove

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
357
the formula
211'
2 j [f(x - t) - f(x)]  W-a(t)dt
o
00
= a j f(x)-f(x-t) dt
- f(1 - a) t 1 + a
o
(19.34)
for 21r-periodic functions f(x), i.e.
D (a) = D a f
+ - +.
(19.35)
, Although both integrals in (19.34) are to be interpreted in general as conventionally
'onvergent at t = 0, the chief thing to stress now is that the integral on the
right-hand side is "good" at infinity: it converges absolutely as t --+ 00 and this
211'
does not require the condition J f(t)dt = O.
o
For the exact interpretation of (19.34) we introduce the corresponding
"truncated" derivative
211'
Df = 2 j[f(x) - f(x - t)]  w-a(t)dt
e
 (19.36)
and let D+,ef be the truncation of the right-hand side of (19.34), which is familiar
by (5.59). We observe that D+.,ef is a 21r-periodic function, if f(x) is such a
function.
Lemma 19.4. Let f(x) E L p (0,21r), 1  p < 00. The trnncated fractional
derivatives Df and D+,ef converge (for almost all x in Lp(O, 21r») simultaneously
and
lim Daf = lim D+. ef.
e-O ' e-O '
( 19.37)
Proof. The following equation is valid:
00
Df == D+,ef - aef(x) + j be(t)f(x - t)dt,
o
( 19.38)
00
where the constant a e is the sum of the senes a e = r(la) L: [(2m1r)-a -
m=l
(21rm + c)-a] and the function be(t) is defined as be(t) = r(la) t-l-a when

358
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
2m1r  t < 21rm + c, m = 1,2,... and b£(t) = 0 beyond these intervals. Equation
(19.38) is obtained from (19.36) by direct manipulations which are similar to those
in the proof of Lemma 19.3, and by using
W-a(t) =
dt
21ra  ( ) -a-l
r(l-a)  t+2m1r , 0 < t < 21r.
The latter is derived from (19.11) and (19.12).
00
It is clear that a£ --+ 0 as c --+ O. Further, b£(t) E L 1 (0,00) and J b£(t)dt =
o
a£ --+ O. So the second and the third terms on the right-hand side of (19.38) tend
to zero - both almost everywhere and in Lp(O, 21r) - which completes the proof. .
Remark 19.1. Limits (19.37) do exist on functions I(x) E H>', A > a (or
I(x) E H;, A > a, see S 19.7). We note also that all considered forms of fractional
differentiation coincide with each other on such functions:
1)a) 1 == Da) 1 == D/.
( 19.39)
The latter of these equalities follows from Lemma 19.4, the former was established
earlier in Lemma 19.2.
19.5. The representability of periodic functions by the Weyl
fractional integral
Let I( x) be a 21r-periodic function, x E R 1 . We shall show that the convergence in
L p (0,21r) of the truncated Marchaud derivative
00
( D a 1 )( ) = a f I(x) - I(x - t) d
£ X r(1 _ a) t 1 + a t,
£
o < x < 21r,
( 19.40)
is equivalent to the represent ability of a function I( x) by the Weyl fractional
integral of a function in L,,(0,21r). Namely, the following theorem is true.
Theorem 19.2. Let 0 < a < 1. If 1 E X 2 '11'1 where X 2 ,.. = L,,(0,21r), 1  p < 00,
or X 2 ,.. = C(O, 21r), then the following statements are equivalent:
1) there exists a function I;'(x) E X 2 '11' such that IID+ £1 - I;'llx.. --+ 0;
, £-00
2'11'
2) I(x) = 10 + Ia)l;', where I;' E X 2 '11' and fo = 2 1 '11' J I(x)dx. In the case
o
X 2 ,.. = Lp(O, 21r), 1 < p < 00, these assertions are also equivalent to
3) IID+,£/lIp  c, where c does not depend on c.

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
359
Proof. Without loss of generality we may take fo = O. Let 2) be satisfied. Then
f(x) = It.cp by Lemma 19.3, the integral It.cp being conventionally convergent. We
shall use the arguments of S 6, where it was shown that the difference f(x) - f(x-t)
of a function It.cp admits the representation (6.9) with the integral absolutely
convergent. In view of this representation (6.6) is valid:
00
(D+,ef)(x) = f K(t)cp(x - ct)dt
o
(19.41)
with the kernel K(t) E L1 (0,00). Although this formula was derived for
cp(x) E L1(R1), it is easily seen to be true for the 21r-periodic function cp(x) E X2,..
since K(t) E L1(R1). Using (6.8) we see that (19.41) yields statements 1) and 2) of
the theorem.
00
Let 1) be satisfied. We use the representation (6.23): AhD+,ef = J K(t)[f(x-
o
€t) - f(x - h - €t)]dt where Ah is the operator (6.22). Again it is easily established
that this equality holds in our 21r-periodic case. Following the same arguments as
in S 6.3 - see the reasonings after (6.23), where the Lq(R 1 )-convergence is to be
replaced by X 2,..-convergence - we obtain
f(x) - f(x - h) = lim[(It.cpe)(x) - (It.CPe)(x - h)], where CPe(x) = D+,ef.
e-o
Here the integral It. CPe of the periodic function CPe does exist in the sense of (19.20).
2,..
We take into account the equality J CPe(x)dx = O. Since two 21r-periodic functions,
o
having identically coincident differences, may differ only by a constant, we have
f(x) = lim It.CPe = lim Ia)CPe = Ia)(lim CPe),
e-O e-O e-O
( 19.42)
because of the assumption fo = O. The limit in (19.42) is taken in X2,... Equation
(19.42) yields statement 2). The validity of 3) in the case 1) is obvious.
In the case when 3) is satisfied, arguments are also similar to the proof of
Theorem 6.3 in the part concerning the condition (6.19). .
It is time to observe that Ia)[Lp(O, 21r)] = Ia)[Lp(O, 21r)] in the case 1 < p < 00
and Q > O. This follows from (19.24) and (19.25), since the operator H commutes
with Ia) and Ia) and is bounded in Lp. It follows also from (19.29) and (19.30)
that Ia)(Lp) = I(a)(L p ) for p > 1. So we have
I(a)(L p ) = Ia)[Lp(O, 21r)] = Ia)[Lp(O, 21r)], 1 < p < 00, Q > O.
( 19.43)

360
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Remark 19.2. Theorem 19.2 gives a characterization of the space I(a)(L p ) in
terms of left-hand sided truncated Marchaud fractional differentiation in the case
o < a < 1. It may be extended to the case of an arbitrary a > 0 by means of the
construction (5.80) of the Marchaud derivative. Namely, let
00 1
D a f =  f (tf)(x) dt 0
+,£ x(a, I) t1+a ,a > ,
£
( 19.44)
where I > a and the constant is given in (5.81). Then Theorem 19.2 is valid for all
a > 0, if D+,£ in its formulation implies (19.44) with the choice I > a. We omit
the proof but observe that a non-periodic analogue of this statement will be proved
for functions of many variables in S 27 (see Theorem 27.3).
We conclude this subsection with the following theorem.
Theorem 19.3. A function f(x) E L p (0,21r), 1  p < 00 (or C(0,21r») IS
representable by a fractional integral:
f(x) = fo + Ia)cp, cp E Lp(O, 21r) (or C(0,21r»,
( 19.45)
if and only if there exists a function g(x) E L p (0,21r) (or C(0,21r) respectively)
such that
(ik)a fA: = gkJ k E Z,
and then g(x) == cp(x). In case p > 1 (19.46) is equivalent to
( 19.46)
Ikl a fA: = 1/Jb k E Z, 1/J(x) E Lp(O, 21r)
( 19.47)
but with 1/J(x) f; cp(x).
Proof. Let (19.45) be satisfied. Then a direct calculation of the Fourier coefficients
in (19.45) gives fk = (Ia)cp)1: = (ik)-acpk in view of (19.45). Conversely, let g E Lp
(or E C) exist such that (19.46) holds, i.e. f" = gk/(ikt. Then fk = (1+g)" by
(19.5). By the uniqueness theorem for Fourier series we have
f(x) = fo + Ia)cp.
To obtain (19.47) it remains for us to observe that, if p > 1, the representability
a function f(x) by a fractional integral Ia)cp, cp E Lp, is equivalent to the
representability by the Riesz potential (19.26) and (19.27) in the case p > 1, see
(19.43).

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
361
19.6. Weyl fractional integration and differentiation in the
space of Holderian functions
We consider 21r-periodic functions, continuous on the whole line, thus rp(O) = rp(21r).
Let, as is usual, w( rp, t) = sup Irp(x + h) - rp(x)1 denote the continuity modulus of
zeR l
Ihlt
a function rp(x). The following theorem, similar to Theorem 13.15 clarifies how the
operator IOI) improves the smoothness properties of a function rp(x).
Theorem 19.4. Let f(x) = (IOI)rp)(x), where rp(x) is a continuous 21r-periodic
function and 0 < a < 1. Then
11'
If(x + h) - f(x)1  ch f Wt':) dt,
h
( 19.48)
11'
If(x + h) - 2f(x) + f(x - h)1  ch 2 f Wt<:-') dt.
h
( 19.49)
Proof. Let us represent the difference f(x + h) - f(x) as
11'
f(x + h) - f(x) = 2 f [rp(x - h) - rp(x)][w+(t + h) - w+(t)]dt,
-11'
( 19.50)
11'
where the periodicity of rp(x) and the equation J w+(t)dt = 0 were used. Hence
-11'
f(x + h) - f(x) = J + J = A + B since one may take 0 < h < 1r/2. For the
Itl<2h Itl>2h
first term we have
2h
IAI  c f w(rp,ltl)(lw{t + h)1 + Iw(t)l)dt.
-2h
Since w(rp, It I)  w(rp, 2h)  2w(rp, h) - Timan [3, p.ll1], by the estimate (19.16)

362 CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
with j = 0 we obtain
3h
IAI  cw(cp, h) f Iw+(t)ldt
-3h
3h
 c}w(cp, h) f ta-1dt  C2 haw (cp, h).
o
(19.51)
As for B, by the mean value theorem and (19.16) with j = 0 we obtain
IBI :S ch f w(, It I) I (  +) (t +Oh)1 dt
2h<ltl<1I'
 c1h f w(cp, ItlHltl- h)a- 2 dt.
2h<ltl<1I'
Here Itl- h  Itl/2, so
11'
IBI  ch f w(cp, t)t a - 2 dt.
2h
( 19.52)
11'
Since haw(cp, h)  ch J w(cp, t)t a - 2 dt, the estimate (19.48) follows from (19.51) and
2h
(19.52).
To prove (19.49) we obtain, similarly to (19.50), the equation
f(x + h) - 2f(x) + f(x - h)
11'
= 211f f [cp(x - t) - cp(x)][w+{t + h) - 2w+{t) + w+(t - h)]dt
-11'
= 211f f + 2 1 h f = Al + BI'
Itl2h Itl2h
Exactly in the same way as above we have
IAII  chaw(cp, h).
( 19.53)
Let us estimate BI' Equation (5.75) for finite differences and (19.16) with j = 2

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
363
yield
Iw+(t + h) - 2w+(t) - W+{t - h)1  ch 2 t Ot - 3 ,
(19.54)
if It I  2h. Note that non-centered differences were considered in (5.75), while
we use a centered difference. It is easy to show what the modification of (5.75)
should be in the case of centered differences. In view of (19.54) we see that IBll is
dominated by right-hand side of (19.49). Then (19.49) is established since (19.53)
has already been obtained. .
Theorem 19.5. For any continuous 27r-periodic function the estimate
h
w(D) f, h)  c f w(f, t)t-1-Otdt, 0 < a < 1,
o
( 19.55)
is valid provided that the integral on the right-hand side converges.
Proof. Let rp(x) = (DOt) f)(x). From (19.18) we obtain the representation
11'
rp(x + h) - rp(x) = 1/(21r) f L\(x, h, t){d/dt)w-Ot(t)dt, h > 0,
-11'
where A(x,h,t) = f(x+h-t)-f(x+h)-f(x-t)+f(x). Obviously, 1L\(x,h,t)1 
2w(f, It!), and IA(x, h, t)1  2w(f, h). So, by (19.16) we have
Irp(x + h) - rp(x)1  c ( f + f ) w(f, t)ltl-1-Otdt
tlh hltl211'
:s; c (I w(f, 1)1-1-adt + w(f, t)h- a ) .
( 19.56)
Since w(f, h)/h $ 2w(f,t)/t - Timan [3, p.l11] - the second term in (19.56) is
dominated by the first one which leads to (19.55). .
Corollary. Let f( x) be a continuous 27r-periodic function and
00
rp(x) = f[f(X) - f(x - t)]t-1-Otdt, 0 < a < 1.
o

364
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Then
h
W( cp, h)  c f w(J, t)t- 1 - a dt.
o
(19.57)
Proof. It is sufficient to recall Lemma 19.4.
We consider now a generalized Holder space Ha([O, 211"]), consisting of 211"-
periodic functions, which are continuous on the real line, have zero mean value
over the period, and are such that w(cp, t)  cw(t), where w(t) is a given continuous
function. This space is equipped with the norm similar to S 13.6. We point out
that the lower zero index in the designation of the space Ha([O, 211"]) implies that
211'
J cp( t)dt = 0, unlike the non-periodic case.
o
Ao([0,211"]) denotes the space of continuous 211"-periodic functions with zero
mean value such that
IJ(x + h) - 2/(x) + J(x - h)1  cw(h), h > 0,
and is named a generalized Zygmund space. Let
IIJIIAIi' = IIJlle + sup IJ(x + h) - 2J(x) + f(x - h)l/w(h).
x,h
We suppose that everywhere below
w(t) E C([O, 211"]), w(O) = 0, w(td  CW(t2), t1  t2.
( 19.58)
The next theorem is an immediate consequence of Theorem 19.4.
Theorem 19.6. Let w(t) satisfy conditions (19.58). The operator Ia), ° < a < 1,
of Weyl fractional integration is bounded Jrom Ha([0,211"]) into H'" ([0,211"]),
wa(t) = taw(t), if
11'
f () I-a wt) dt  a.J(h),
h
and from H'O ([0,211"]) into A'" ([0, 211"]) under the weaker assumption
(19.59)
11'
f ( ) 2-a wt) dt  a.J(h).
h
( 19.60)

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
365
Corollary. The Weyl operator Ia), 0 < a < 1, is bounded from H([O, 21r]) into
H+a ([0, 21r]), A + a < 1, and from H([O, 21r]) into the Zygmund space
A+a = {f(x) : If(x + h) - 2f(x) + f(x - h)1  ch>.+a, fo = O},
(19.61)
if A + a < 2.
Indeed, if w(t) = t>', (19.59) and (19.60) are satisfied provided that A + a < 1
and A + a < 2 respectively.
Remark 19.3. If one considers 21r-periodic functions which are in H'O on [0,21r],
but are not necessarily continuous on the whole line, i.e. rp(O) = rp(21r) may not be
true, then the statement of Theorem 19.6 is to be modified as follows
Ia)(rp - rp.) E H W o([0,21r]),
where rp.(x) is a 21r-periodic function equal to [rp(0)-rp(21r)](rp-x)/(21r) on [0, 21r].
Remark 19.4. The corollary of Theorem 19.6 is exact in the sense that
Ii a ) rp 'I. H 1 , in general, for rp E H->'. The corresponding example is the
Weierstrass function - Zygmund [6, p.207].
h
Theorem 19.7. LetO < a < 1, letw(t) satisfy (19.58) and letft- 1 w(t)dt  cw(h).
o
Then the operator Da) of Weyl-Marchaud fractional differentiation, as well as the
operator D+ of Marchaud fractional differentiation, is bounded from H';o ([0, 21r])
into H'O([0,21r]).
This theorem follows immediately from (19.55).
Corollary. If f(x) E H>'([O, 21r]), 0 < a < A  1, then Da) f E H-a([O, 21r]).
In the case A = 1 a somewhat stronger assertion, that stated in Corollary, is
D(o)
valid: A  H-a, where A is the space (19.61) with A + a = 1. We omit the
proof, but refer to Zygmund [6, p.206].
Finally, unifying Theorem 19.6 and 19.7 we come to the following theorem in
which we use the function class , defined in (13.68).
Theorem 19.8. Let w(t) E -a' Then operator Ia), 0 < a < 1, of Weyl
fractional integration maps the space H'O([0,21r]) isomorphically onto H';o([0,21r]).
Proof. If follows from assumption about w(t), that Ia) is bounded from H'O into
H';o by Theorem 19.6, while Da) is bounded from H';o into H'O by Theorem 19.7.

366
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
It only remains thus to show that every function J(x) E HW OI is representable by the
Weyl fractional integral J(x) = [a)cp of a function cp in H'O. Since H'O C C([O,21r]),
by Theorem 19.2 we see that the function J(x) will be representable by the Weyl
fractional integral of a function cp E C, if D+,£J converges in C-norm. Since
 £
Da D a a J(x)-J(x-t)d
1 +..,1 - +..,11  f(1 - "') f tHo t
£ £
< C f wa(t) dt = C f w(t) dt  0
- t 1 + a t '
£1 £1
the convergence of D+,£ in C becomes evident. Then J = [a)cp, cp E C([O,21r]), by
Theorem 19.2. We know that Da) [a)cp == cp, cp E L 1 ([O,21r]), so by Theorem 19.7
cp is not only in C([O, 21r]), but also in H'O. .
Corollary. Weyl fractional integration isomorphically maps the Holder space
H([O, 21r]), 0 < A < 1 - a, onto the space H+a([o,21r]).
Indeed, w(t) = t>' E _>., if 0 < A < 1 - a.
We give also a "multiplier" paraphrase of Theorem 19.8.
Theorem 19.8'. Let J(x) be a continuous 21r-periodic function and let [(a) be
an operator (19.26) defined by its Fourier multiplier Ikl- a . The operator [(a)
isomorphically maps the space H'O([O, 21r]) onto HOI ([0, 21r]), if w(t) E -a'
Proof. We recall (19.28) and (19.30). The singular operator H with Hilbert kernel
is bounded in H'O([O,21r]) under the stated assumption on w(t), which follows from
the known Zygmund estimate for the conjugate function - Zygmund [5, p.199].
Then it follows from (19.29) and (19.30) that [(a)(Hw) = Ia)(HW). Therefore the
statement of Theorem 19.8' follows from Theorem 19.8. .
In conclusion of this subsection we observe that the Hardy-Littlewood
Theorem 3.5 and 3.6 on the mapping properties of fractional integration in L'P are
valid in periodic case for Weyl integrals too:
11
Lp(O, 21r) ---+ L,(O, 21r), q = p/(l - ap) if 0 < a < l/p (1 < p < 00),
( 19.62)
1(01)
L'P(O,21r)  H a - 1 /P([O,21r]) if l/p < a < l/p+ 1 (1  p < 00). (19.63)
Besides this Ha-l/'P ([0, 21r])may be replaced by the space ha-1/'P ([0, 21r]) (cf.

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS
367
Corollary of Theorem 3.6). In the case a - IIp = 1 we may state that
reo)
L,(O, 211")  A([O, 211"])
(cf. Theorem 3.6), where A is the function space defined similarly to the space
(19.61) with O(h) being replaced by o(h).
We omit the proof of these periodic analogues (19.62) and (19.63) of
Theorem 3.5 and 3.6, referring to Zygmund [6]. We observe, however, that (19.62)
may be derived directly from Theorem 3.5 in view of (19.16) with j = O.
19.7. Weyl fractional integrals and derivatives of periodic
functions in H:
We give here the Bardy-Littlewood theorem on mapping properties of Weyl
fractional integration and differentiation in the spaces of periodic functions
satisfying the integral Holder condition.
By H;([O,211"]) we denote the space of 211"-periodic functions rp(x) E L,(O, 211"),
satisfying the condition
211'
f Irp(x) - rp(x - 6)I"dx  c6>'
o
( 19.64)
(cf. (14.1) and (14.2», while h;([O,211"]) is the space defined similarly with (19.64)
being replaced by
211'
lim ;>. f Irp(x) - rp(x - 6)I"dx = O.
6-0v
o
(19.65)
Theorem 19.9. Let 1  p < 00, 0 < a < 1, 0 < A < 1, .\ + a < 1 and let Ii a ) and
Da) be Weyl fractional integro-differentiation (19.7) and (19.8). Then
r(o)
L,(O, 211")  h;([O,211"]),
( 19.66)
r(Ot)
H;([O, 211"])  H;+a([o, 211"]),
(19.67)
D(Ot)
H;+a([o, 211"])  H;([O,211"]).
(19.68)

368
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
This theorem is an immediate corollary of Theorem 14.5 and 14.6 in view of
the periodicity of the functions in hand. It may be shown that (19.67) and (19.68)
are also valid in o-form:
1(01)
h;([0,211"]) .!. h;+a([0,211"]),
(19.69)
D(O)
h;+a([o, 211"])  h;([0,211"]).
(19.70)
From (19.67) and (19.68) we conclude that the operators fi a ) map H;
onto H;+a,  + Q < 1, one-to-one. Initially the problem of representability
by the fractional integral may be solved by showing that H;+a C f(a)(L p ) via
Theorem 19.2. Analysis of the proof of Theorem 14.6 and 14.7 makes sure that the
mapping
Iia)(H;) = H;+a
(19.71)
is an isomorphism.
19.8. The Bernstein inequality for fractional integrals of
trigonometric polynomials
Let
n
Tn(x) = L ak eih
k=-n
(19.72)
be a trigonometric polynomial. The inequalities
IITlIc  nllTnllc, IITlIp  nllTnllp, 1  p < 00,
(19.73)
due to Bernstein, are well known Nikol'skii [6. p.94]. The following theorem
presents their analogues for the case of Weyl fractional derivatives (19.6).
Theorem 19.10. Any trigonometric polynomial Tn(x) admits an estimate
111)a)Tnllp  c(Q)naIlTnll p , 1  p  00, 0  Q  1,
(19.74)
where c(Q) = 2 1 - a jr(2 - Q).

 19. FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS 369
Proof. Initially let p = 00. By (19.39) the fractional derivative may be taken in
Marchaud form, so that
00
V(a)r. = a f Tn(x) - Tn(x =F t) d
:i: n f(l-a) t 1 + a t.
o
Hence
2/n
IV(a) 1< a f ITn(x) - Tn (x =F t)l dt
:i: n - r(1 _ a) t1+a
o
00
2 a llTnilc f dt
+ f(l - a) t 1 + a '
2/n
(19.75)
Since ITn(x) - Tn(x =F t)1  tIlTlIc, by the first of the inequalities given in (19.73)
we obtain from (19.75)
( ) 2 1 - a
IIV:i: a Tnllc  r(2 _ a) nallTnllc.
(19.76)
211'
Let 1  p < 00. We introduce the convolution operator Anrp = J Tn(x -
o
t)rp(t)dt. It is easily seen that (Anrp)(x) is a trigonometrical polynomial of the
same order n. With (19.76) already proved and using the Holder inequality we
have
IVa) Anrp)(x)1  c(a)naIlAnrplic  c(a)naIlTnllpllrplipi.
(19.77)
211'
On the other hand (Va) Anrp)(x) = J (Va)Tn)(x - y)rp(y)dy, and then
o
IVa) Anrp)(x) I  IIVa)Tnllpllrpllpi.
(19.78)
The inequalities (19.77) and (19.78) hold for all functions rp E Lpl and the
latter is sharp since it turns into equality as rp(x) = Ig(x)IP-1signg(x) E Lpl and
g(x) = Va)Tn' Then IIVa)Tnllp  c(a)naIlTnllp. .

370
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
We note that (19.74) with 1  p < 00 might be obtained along the same lines
as in the case p = 00 by realizing (19.75) and (19.76) in Lp-norm via the Minkowski
inequality, if we use the latter of the inequalities in (19.73). The proof presented
above used only the first of the inequalities in (19.73).
Remark 19.5. If Tn is a trigonometrical polynomial, then Va)Tn is again such
a polynomial of the same order. Therefore, it follows from (19.74) and (19.73)
that the estimate of the form (19.74) is valid for all a > 0 with the constant
c(a) = 2 1 - N - a jr(2 + N - a), where N is the greatest integer less than a. This
constant is not sharp in the case a > 1. For such an a the inequality
IIVa)Tnllp  nallTnllp, 1  p  00, a  1,
(19.79)
holds with the sharp constant equal to 1. This was proved by Lizorkin [3] in a
n
more general context of the "trigonometrical integral" Tn (x) = J e ixt dO'( t).
-n
Remark 19.6. The Bernstein inequality for fractional derivatives of almost
periodic functions of the form
m
I(x) = L akei>'"x
k=l
( 19.80)
is valid in the following presentation:
IIvr) IIiLoo  c( a )naIl/IlLoo'
n = max IAkl,
lSkSm
(19.81)
with the same constant c(a) = 21-a jf(2 - a) - Bang [1, p.21-22], 1941 - for
O<a<1.
The following Favard type inequality
IIIa) IIiLoo  [min Ak lJa II/IILoo, a > 0,
(19.82)
for almost periodic functions (19.80) is close in a sense to the Bernstein type
inequality (19.81); the constant c depends on a, but does not depend on I(x) -
Bang [1].

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
371
 20. An Approach to Fractional Integro-
Differentiation via Fractional Differences
(The Griinwald-Letnikov Approach)
It is well known that a function f(x) which is differentiable up to order n admits
the formula
f (n) ( ) - r (f)(x)
x - h h n '
(20.1)
where (Ai:f)(x) is a finite difference (5.72) of a function f(x). This equality may
be used to define a fractional derivative by direct replacing n by a > 0 in (20.1), if
we can properly interpret the difference of the fractional order.
This approach to fractional differentiation and integration via fractional order
differences, presented in this section, is less used in mathematical analysis in
comparison with other definitions (Riemann-Liouville, Marchaud and others).
However, this approach is natural from the point of view of the development of
mathematical analysis and was suggested long ago by Grunwald [1] and Letnikov
[1]. It has recently attracted attention both from the point of view of function
theory, Westphal [4], Butzer and Westphal [1], Bredimas [1]-[5], Neugebauer
[I], Wilmes [2], Bugrov [1], [2] and Burenkov and Sobnak [1], and also from
its convenience in computational mathematics, Zheludev [1] and Lubich [1], for
example. In S 20.2 our presentation is essentially based on the paper by Butzer
and Westphal, cited above.
While considering fractional order differences of a function f(x) it is natural
to assume it to be given on the whole real line. If we deal only with a left- or
only with a right-hand sided translation, the case of the half-line may be also
considered. The case of a finite interval is specifically discussed in S 20.4. We shall
treat separately the periodic and non-periodic cases.
Everywhere in this section X = X(R 1 ) denotes any of the spaces Lp(Rl),
1  p < 00, or C(R 1 ) and in the periodic case X 2 ,.. = X(O, 21r) denotes any of the
similar spaces on [0, 21r].
20.1. Differences of a fractional order and their properties
Given a function f(x) on the whole line, we define
(dU)(z) = (E - Th)O f = (-1)' () f(z - kh), ,,> 0,
(20.2)

372
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
stating from (5.72). Here (:) are the binomial coefficients (1.48). Since
e(a)I(;)I<oo
(20.3)
by (1.51), the series (20.2) converges absolutely and uniformly for each a > 0 and
for every bounded function; if f(x) E X (X2,.. ), it converges in X (X2,.. )-norm. We
note that c(a) = 2 a for an integer a. The series (20.3) may be always represented
as the finite sum
[a]
e(a) = (1 + (_l)H[<>]) (;).
(20.3')
00
Indeed, since E(-I)I:(:) = 0 and (:) = (_I)I:-[a]-ll(:)1 for k  [a] + 1 by
1:=0
00 00
(1.48), we see that E (-1)1:(:) + (_I)l+[a] E 1(:)1 = 0, which yields (20.3').
1:=0 I::[a]+l
It is easily derived from (20.3') that
c( a) == 2 if 0 < a  1,
2[a]  c(a)  2[a]+1 if a > O.
The difference (20.2) will be called a left-hand sided one, if h > 0, and a
right-hand sided one, if h < O.
Remark 20.1. The difference in (20.2) is not in general defined in the case a < 0,
since the series may prove to be divergent. This is the case for example if f( x) == 1.
Actually
t(-l)i (  ) = (_I)n ( a  1 )
J:O J J
1 f(n + 1 - a)
f(1-a) f(n+l)
(20.4)
- Prudnikov, Brychkov and Marichev [1, 4.2.1.5] - which diverges as n --+ 00, if
a < 0 - see (1.66). Therefore (20.2) with a < 0 is evidently unacceptable in the
periodic case. In the non-periodic case the series in (20.2) may converge for a < 0,
if f(x) has a "good" decrease at infinity, for example If(x)1  c(1 + Ixl)-IJ, P > lal.
We give some elementary properties of the differences defined in (20.2).
Property 1. (L\h(Af»(x) = (L\:+{j f)(x).

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
373
Property 2. If I E X (X2,.. ), then lim 1Il:lllx(x...) = o.
h-oo
Property 3. IIA:+,6 IlIx(x...)  c(a)IIfllx(x...).
Property 1 is established by direct verification. It immediately yields
Property 3 in view of (20.3). Property 2 is proved by the methods of functional
analysis: since the operator AI: is bounded uniformly in h, it is sufficient to check
Property 2 on some set "good" functions dense in X(X 2 ,..).
In the non-periodic case the Fourier transform of AI: I is given by the formula
(£f/)(x) = (1 - eixht J(x),
(20.5)
provided that I(x) is in L1(Rl), for example, while in the periodic case a similar
formula may be written for the Fourier coefficients, namely
(AI:I)1e = (1 - e-ileh)a lie,
2,..
lie =  f I(t)e-iletdt.
21r
o
(2,0.6)
Equations (20.5) and (20.6) are established by direct verification.
Starting from (20.1) we introduce the function
f a) ( x ) = lim (Ahl)(x) 0
::I: h-+O h a ' a > ,
(20.7)
where the nature of the limit may be different depending on the question in hand:
i.e. for each x, for almost all x or in the norm of X or X 2,... The function defined
in (20.7) will be referred to as the Griinwald-Letnikov fractional derivative. If the
limit in (20.7) is taken in the sense of X(X 2 ,..)-convergence, (20.7) may be called a
strong Griinwald-Letnikov derivative in X(X2"')'
In Theorems 20.2 and 20.4 below we shall show that the Griinwald-Letnikov
derivative coincides with the Marchaud derivative
00
n a 1= lim a f I(x) - I(x  t) dt, 0 < a < 1.
::I: £-0 r(1 - a) t 1 + a
£
(20.8)
The form (5.80) should be taken instead of (20.8) in the case a  1. Moreover,
we shall see that both the Griinwald-Letnikov and Marchaud derivatives have the
same domain of definition so that convergence in (20.7) implies convergence in
(20.8) and vice versa.

374
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
We might suggest a difference approach using the following symmetric way:
rJ/)(x) = 1 lim (Af)(x) + (Ahf)(x)
2 cos( a1f' /2) h-O Ihl a
(20.7')
instead of (20.7). Such a form would coincide with an operation (12.1'), inverse to
the Riesz potential, i.e. with
00
fa)(x) = a f 2f(:£) - f(x - t) - f(x + t) dt.
2r(1 - a) cos(a1r/2) t 1 + a
o
(20.8')
instead of (20.8). So we might call the limit (20.7') the Griinwald-Letnikov-Riesz
fractional derivative. We do not elaborate on this since it would follow along the
same lines as studying derivatives (20.7) in this section.
Remark 20.2. Equation (20.7) may be used to define the fractional integral by
taking a < 0 in (20.7). According to Remark 20.1 such a definition will be proper
for functions which have a sufficiently rapid decrease at infinity. In S 20.4 the
definition of a fractional integral via (20.7) will be considered in the case of a finite
interval.
Generalized differences may be introduced which are generated by an analytic
function a(E - Th) instead of (E - Th)a. Thus, let a function a(), 0    2,
satisfy the following assumptions:
1) it is analytic in the neighbourhood of the point  = 1:
00
a() = Lak( - l)k, ak = a(k)(I)/k!,
k=O
(20.9)
the series being convergent as I - 11  1;
2) the convergence at the points  = 0,  = 2 is absolute:
00
del"
c(a) = L..J lakl < 00;
k=O
3) a() equals zero at the point  = 0:
00
a(O) = E(-l)l: ak = O.
k=O

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
375
de! 00 00
We define a(E- Th)= E(-I)l:aI:T: = E(-I)l:aI:Tl:h' This operator generates a
1:=0 1:=0
generalized difference
00
dq  I:
(a - Ah/)(x) = a(E - Th)1 = L.J( -1) al:/(x - kh),
1:=0
which is evidently coincident with (A/)(x) in the case a() = a.
The property a - Ah(b - Ah/) = ab - Ahl is easily proved, were ab - Ahl is
a generalized difference generated by the product a()b(). This is an extension of
Property 1 for the differences Aial.
Following Post [2] we may introduce generalized differentiation a(1)/, defining
it similarly to (20.7) by the equation
( E - Th )
a(1)1 = 1 a -,;- I,
(20.10)
where
a ( E  ,.,. ) f =  ai ( E  ,.,. - I Y f.
(20.11)
We might give another definition:
. a - A I
a[1)]1 = lim (h) '
h-O a
(20.12)
Equation (20.10) has some shortcomings in comparison with (20.12) as formally
the series in (20.11) may prove to be divergent for small h, if the series in (20.9)
diverges for large . So if the function a() is not analytic for all , equations
(20.10) and (20.11) may be made meaningful using restrictions on the function
I(x), which guarantee the convergence of the series in (20.11). Nevertheless,
(20.10) is more natural then (20.12), because this is precisely the result which leads
to the formula
[a(1)/]1: = a(ik)j",
(20.13)
for periodic functions I(x), where II: . are the Fourier coefficients of the function
I. Note that (20.13) itself assumes the function a() to be defined at last on the
whole imaginary line.
We shall not elaborate further on generalized differentiation a(1) as well as
on any other generalization of such a kind. We note only that applying (20.10)
and (20.11) formally we obtain the following representation in the case of infinitely

376
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
differentiable functions I(x), namely
00 00
a('D)1 = E( -11 aj(E - V)j 1= E( -1)j bj'D j I,
j=O j=1
'D = d/dx,
bj = f:(-l). (  ) all'
11=3 J
20.2. Coincidence of the Griinwald-Letnikov derivative with
the Marchaud derivative. The periodic case
We shall show first that the existence of the Griinwald-Letnikov derivative (20.7)
for a function I( x) is equivalent to this function being represented by the Weyl
fractional integral up to a constant term - Theorem 20.1. We shall first establish
some auxiliary lemmas for the fractional order differences h I of functions I(x),
representable by a fractional integral.
Let ka(x) = r() x+-1 and
Pa(x) = (fka)(x)
00 ( )
_ 1 j a . a-1
- r(a) [;(-1) j (x - J)+ .
(20.14)
We shall also need functions
21r  ( X + 21r j )
Xa(x;h) = h. Po h
J=-OO
= 2; E Pa ( Z+h2"j ).
- f;; <j <00
(20.15)
Lemma 20.1. The function Pa(x), a > 0, has the properties
00
1) Pa(x) E L 1 (R 1 ), J Pa(x)dx = 1;
-00
2) P;;(x) = ( 1:;;r ) a, the principal value of the power function being chosen so
that P;;(O) = 1;
3) pa(x) == 0 as x > a in the case a = 1,2,...

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
377
Proof. We begin with the simplest property 3). In the case of an integer Q we
have for x > Q
Pa(Z) = f(la) (-lY (;)<z - no-I
1 A 01 01-1
= r(Q) UIX .
But (APm)(x) == 0 as I > m for any polynomial Pm(x) of order m - (5.75).
Consequently Afx OI - 1 == 0 and then POI (x) == 0 as x > Q.
2) is obtained by direct application of the Fourier transform to POl(X) and by
use of (7.6).
The most difficult part in the proof of the lemma is a demonstration of the
first of the statements in 1). Let W = F(L 1 ) be the ring of Fourier transforms
of functions in L1(Rl). We have thus to show that ( 1:;;a: )01 e W. Let p(x) be
a smooth step function, i.e. an infinitely differentiable function on R 1 such that
p(x) = 1 as Ixi  2, p(x) = 0 as Ixl  3 and 0  p.  1 as 2  Ixl  3. We have
( ' I_.e iZ ) OI = p.(x) ( 1_.e iZ ) OI + (1- e iz )0I 1 - (x) .
-IX -IX (-IX)OI
(20.16)
The first term here is an infinitely differentiable finite function and so undoubtely
belongs to W. It is sufficient to refer to the known fact that any function in the
Schwartz space S is a Fourier transform of a function in the same space (see S 8.2).
Further, let us show that
1 - p(x) w:
(-iX)OI e .
(20.17)
We shall use the following known fact (see, for example, Bochner [1, p.271]):
if feL1(R 1 ) and jeL1(R 1 ),
then few.
(20.18)
Hence, integrating by parts the Fourier integral j, we easily derive the following
simple sufficient condition for a function f to belong to W:
if feL1(R 1 ) and f'eL2(R1),
then feW.
(20.19)

378
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
We see that (1- ix)-a E W for any Q > 0 by (7.9). Then [1- p(x)](I- ix)-a E W.
It is easily seen that the difference [1 - p(x)][( -ix)-a - (1 - ix)-a] satisfies the
assumptions of (20.19) and is therefore in W. Thus (20.17) is proved.
To show that the second term in (20.16) belongs to W, we expand (1 - eiz)a
in the binomial series and notice that
(1- eiz)a(x) = .,'/J(x),
00
where 1jJ(x) = E(-I)"(:)cp(x - kh), 111/Jlh  c(Q)lIcplh, which gives what IS
"=0
required.
So we have obtained the first part 1). As for the latter it is readily seen from
2). .
Lemma 20.2. Let Q > 0 and h > O. Then
211'
1) f Xa(Xj h)dx = 21r;
o
2) IIXa('j h)IIL 1 (O,2W') 5 M < 00, with M not depending on h;
211'
3) lim f Xa(x; h)dx = 0 for 6 > 0;
h-O+ 6
4) The Fourier coefficients of the function Xo(x; h) are equal to
( l_e- ikh ) a
(Xa('j h»" = ikh
(20.20)
with the natural interpretation (Xa('; h»o = 1.
Proof. Properties 1 )-4) are corollaries of known arguments in harmonic analysis
connected with the Poisson summation formula. Namely, it is known and is easily
00
verified that if G(x) = E g(x + 27rj), then
j=-oo
g(x) E LI(Rl) => IIGIIL 1 (0,2W')  IIgIIL1(Rl), G" = g(k).
(20.21)
We have already used this device while proving Lemma 19.1. Using (20.21), we

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
379
obtain
211' 00
f Xa(X; h)dx = 21r f Pa(x)dx,
o -00
211' 00
f IXa(x; h)ldx  21r f IPa(x)ldx,
6 6th
(20.22)
[Xa('; h)JA: = Pa(k),
where Pa(k) is the inverse Fourier transform of the function Pa(x). By Lemma
20.1 the relation in (20.22) yield the properties 1)-4). .
We add to Lemma 20.2 the equation
X ( X' h ) = (L\l:w+)(x) + 1
a , h a '
(20.23)
which clarifies why we need the function Xa(x; h); here w+(x) is the kernel (19.8) of
the Weyl fractional integral (19.7). To prove this equation it is sufficient to show the
coincidence of the corresponding Fourier coefficients: (Xa('; h»1: = h-a(L\l:w+)I:,
k f; O. The latter follows from (20.20), (20.6) and (19.8).
We are now ready to prove our main statements. We recall that in Theorem
20.1 and 20.2 the space X211' meant any of the spaces L,(O, 21r), 1  p < 00, or
C(0,21r).
Theorem 20.1. Let f(x) E X211'. A(strong) Griinwa/d-Letnikov derivative (20.7)
ezists in X211' if and only if there exists a function rp%(x) E X211' such that
211'
f(x) = Ia)rp-l: + fa, fa = 2 f f(x)dx,
a
(20.24)
and then rp-l:(x) = f::)(x).
Proof. For definiteness we choose a left-hand sided variant of the fractional
differentiation, namely the sign +. Let the derivative in (20.7) exist as convergent

380
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
in X2,... Then
2,..
(fa»1e =  f e-ib lim h-a(Ll a f)(x)dx
+ 21r h-+O h
o (X:l tr )
2,..
=  lim h-a f (a/)(x)e-ibdx.
21r h-+O h
o
Hence (/»1e = (ik)a lie in view of (20.6). Since I)(x) E X 2 ,.., we see that the
representation (20.24) is true by Theorem 19.3.
Conversely, let (20.24) be satisfied. Then
2,..
(/)(x) = 2 f cp+(t)(wJ(x - t)dt.
o
Applying (20.23), we have
2'11'
(/:!tYl(Z) = 2 f cp+(t)X,,(z - t, h)dt.
o
(20.24')
This equation allows us to justify the limiting process II J - cp+IIX:ltr --+ 0 by
Theorem 1.3. .
Theorem 20.2. Let I E X 2 ,... Then the Griinwald-Letnikov fractional derivative
(20.7) exists simultaneously with the Marchaud derivative (19.34) and they coincide
with each other:
00
I . (A%"/)(x) _ a I . f f(x)-/(xt) dt
1m - ) 1m 1 .
h-+O h a r(1 - a £_0 t +a
(Xtr) (X 2 ...) £
(20.25)
If X2,.. = L,,(O, 21r), 1 < p < 00, the Griinwald-Letnikov and Marchaud derivatives
exist simultaneously even under the different choice of signs + or -.
Proof. Theorem 20.2 follows immediately from Theorems 20.1 and 19.2, the
statement on the simultaneous existence of derivatives with different signs being
derived by means of (19.24) and (19.25). .

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
381
Theorem 20.2 is extended to values of a  1 under the proper interpretation
of the Marchaud derivative - Remark 19.2.
It was shown in Theorem 19.2 that the condition IID+.,efIlL p (O,2,..>  c of
uniform boundedness of truncated Marchaud derivatives is necessary and sufficient
for existence in Lp of the Marchaud derivative or, which is the same, for a function
f(x) to be represented by a fractional integral of order a of a function in Lp. The
following theorem gives a similar statement for the Griinwald-Letnikov derivative.
Theorem 20.3. Let f(x) E Lp(O, 21r), 1 < p < 00. Then the Griinwald-Letnikov
derivative f>(x) or fc:.>(x) exists if and only if
IIAh fIlL p (O,2,..>  cho, h > 0,
(20.26)
where c does not depend on h.
Proof. The only thing which needs to be demonstrated is to show that (20.26)
provides the existence of the limit lim h-oLlhf in Lp. It is known that bounded
h-+O
sets in Lp are weakly compact, i.e. any bounded set in this space has a weakly
convergent sequence - Dunford and Schwartz [1, p.314]. So there exists a sequence
h m -+ 0 and a function g( x) E Lp such that
2,.. 2,..
2oo f h;'O(A h ...f)(x)1/J(x)dx = f g(x)1/J(x)dx
o 0
(20.27)
for all functions 1/J(x) E L,,'(O, 21r). Let us choose 1/J(x) = te-ib. Then (20.27)
turns into the equation lim h (Ll fh: = gk. Hence by (20.6) we see that
m-oo... ...
(ik)O fie = 9k, g(x) E Lp. Then Theorem 19.3 asserts that f(x) may be represented
as in (20.24), which in view of theorem 20.1 is equivalent to existence in Lp of the
Griinwald-Letnikov derivative. .
Remark 20.3. In the case of functions f(x), representable in the form (20.24),
the inequality (20.26) may be written more exactly,
IIAhfllx..  choIlD> fllx..,
2,..
C = sup f IXo(x; h)ldx
h>O
o
(20.26')
which is derived from (20.24').

382
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
20.3. Coincidence of the Griinwald-Letnikov derivative with
the Marchaud derivative. The non-periodic case
In the case of the whole axis R 1 fractional integration does not keep the space
Lp invariant. So dealing with the Griinwald-Letnikov fractional derivative I:> or
the Marchaud derivative DI convergent in the space L p (R 1 ), we do not require
that the functions themselves be in Lp(Rl). Otherwise it would be an evident
restriction of the initial setting of the problem, compare Theorem 20.4 with the
similar Theorem 20.2.
Theorem 20.4. Let I(x) E Lr(Rl), 1  r < 00, and let D+.,el be the truncated
Marchaud derivative (5.80'), D,el being similarly written from (5.80). Then the
limits
fa) ( x ) - lim
::I: - h-+O
(Lp(R 1 »
(L\%hl)(x)
h a
(D%/)(x) = lim (D% e/)(x)
e-O '
(Lp(R 1 »
(20.28)
exist simultaneously and coincide (under the same choice of signs) for any
p E [1,00) and any a > 0, the values of p and r being independent.
Proof. For the sake of brevity we consider the case 0 < a < 1, 1 < p < l/a only.
We refer the reader to the paper by Samko [34] for the proof applying to all values
of a and p.
Let the second of the limits (20.28) exist. Then I(x) = I+.cp, cp = D+.I E Lp
by Theorem 6.2. Applying the fractional order difference to the integral I = I+.cp
we have
00 00
(dU)(z) = J (t) 8(-1); (;)ko(z - t - hj)dt,
-00 J-
(20.29)
where ka(x) = rla) x-l, the termwise integration being easily justified. After the
change x - t = hr in (20.29) we obtain
00
(d:!(Z) = h (dI'i-)(z) = J Po( T)(Z - Th)dT,
-00
(20.30)
where Pa( r) is a function (20.14). In view of Lemma 20.1 we see that Theorem 1.3
may be applied, so the right-hand side in (20.30) converges to cp(x) in Lp(Rl) as
h --+ +0. So from (20.30)
lim (L\I:I)(x) = cp(x) = (D a I)(x).
h-+O h a +
(Lp(R 1 »

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
383
Conversely, let the Griinwald-Letnikov derivative I:)(x) exist. Let us prove
the identity
00
I'i- ( f ) == J Pa(t)f(z - ht)dt
-00
(20.31 )
under the theorem's assumptions on I(x). Equation (20.31) is immediately reduced
to (20.30) in the case of "good" functions, since It. and 1: commute when applied
to "good" functions. The problem is to justify (20.31) in the situation when It. is
applicable to 1:I, but is not applicable termwise to the series A1:I. To obtain
(20.31) we use a method which has already been demonstrated in the proof of
Theorem 6.2. Let A be the operator in (6.22). We have (on "good" functions for
the present),
AdA1:I/ha) = h- a [(I+A1:I)(x) - (I+A1:f)(x - )]
= h- a [(A1: I+/)(x) - (1: I+/)(x - )].
Applying (20.30), we obtain
00
Ad A 1:I/h a ) = J Pa(t)/(x-th)dt
-00
00
- J Pa(t)/(x -  - th)dt.
-00
(20.32)
Since the operators both on the left- and right-hand sides are bounded in Lr'
(20.32) is valid not only for "good" functions, but for the whole space Lr. Equation
(20.32) means that
dP'i- Ua df) = dpa Ua ) * f,
(20.33)
where A is a difference of order one. Functions having the same finite difference
may differ by a constant at the utmost. Since the functions considered are in
Lr(Rl), this constant is zero, so (20.33) leads to (20.31).
Equation (20.31) being obtained, we let h -+ O. Then by the properties of
the kernel pa(t) and by Theorem 1.3 we arrive at I(x) = It. ( lim J I ) = It.cp,
h-+O
cp = I). SO we have shown that the existence of the Griinwald-Letnikov derivative
for a function I(x) implies that I(x) may be represented by the fractional integral

384
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
of this derivative. Then by Theorem 6.1 the function f(x) has a Marchaud
fractional derivative, coinciding with <p = f). .
The following is a non-periodic analogue of Theorem 20.3.
Theorem 20.5. Let f(x) E Lr(RI), 1 < r < 00. Then f(x) E [Ot(L p ),
1 < p < l/a, if and only if
lIL\h flip  ch Ot , h > O.
(20.34)
Proof. The necessity of the inequality in (20.34) follows from (20.30). To
show that it is sufficient we shall make use of (20.31), which is valid under our
assumption on f(x). Since the space Lp(Rl) is weakly compact, the uniform
boundedness of the functions implies the existence of a sequence h m --+ 0 and a
function <p(x) E Lp(Rl) such that h;;.OtL\mf weakly converges in Lp(Rl) to rp(x).
Since the right-hand side of (20.31) strongly converges in Lr to f(x), it so much
the more weakly converges to Lr. Then there exists a weak limit (w-lim) in Lr on
the left-hand side:
:!im [+(Amf /hc:n) = f(x).
(Lr)
(20.35)
Furthermore, since h;;.Ot Ahmf weakly converges in Lp and the operator [+ is
bounded from Lp into Lq, q = p/(I- ap), we conclude that the limit
( L\Ot f ) ( L\Ot f )
w-lim [Ot  = [Ot w-lim  = [Ot rp.
m-oo + hOt + m-oo hOt +
(L.,) m (L p ) m
(20.36)
exists too. Since weak limits in Lq and in Lr of the same sequence are to
coincide with each other almost everywhere, from (20.35) and (20.36) we see that
[+<p = f(x) almost everywhere. .
Remark 20.3'. Similarly to (20.26') we note that (20.34) can be made more
exact:
00
lIL\hfll p  chOtIlD+fll p , c = f IpOt(x)ldx, h> 0,
-00
(20.34')
in the case of the functions f(x) E [Ot(L p ), 1 < p < l/a. This follows from (20.30).
The inequality
I(Ahf)(x)1  ch Ot sup I(D+f)(x)l, h > 0,
:&
(20.34")
is also valid for such functions f(x).

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
385
Remark 20.4. In the non-periodic case we have restricted ourselves for the sake
of brevity to functions given on the whole line. It is not difficult to see that
Theorem 20.4 remains valid on the half-line R for the right-hand sided fractional
differentiation f) and Df. For this purpose it is sufficient to define f(x) == 0 for
negative x. The following theorem presents a modified version of Theorem 20.5 for
the case of a half-axis.
Theorem 20.5'. Let f(x) E Lr(R), 1 $ r < 00. Then f(x) has the Marchaud
fractional derivative Df = lim D £f in Lp(R), 1 < p < 1/0:, if and only if
£-0 '
(L p )
IIAhfllp  ch Ol , h > 0, with c not depending on h.
Remark 20.5. Theorem 20.5' is formulated for p E (1,1/0:), but it valid for
p E (1,00), which requires other means - Samko [34].
20.4. Griinwald-Letnikov fractional differentiation on a finite
interval
The definition itself of the fractional order difference Af by (20.2) assumes that
the function f(x) is given at least on the half-axis. When the function f(x) is
given on a finite interval [a, b] only, the natural way to define the difference A f
is connected with continuation of the function f( x) as a vanishing function beyond
the interval [a,b]. So, for a function f(x), given on [a,b], we introduce as a
definition
(Ahf)(x)(Ahf.)(x) = f)-I)j (  ) f.(X - jh),
;=0 J
(20.37)
f.(x) = { f(x),
0,
x E [a, b],
x 'I. [a, b].
It is clear that (20.37) may be rewritten in terms of the function f(x) itself,
avoiding its continuation as a vanishing function:
[T]
(Ahf)(x) = L (-I)j (  ) f(X - jh), x> a,
j=O J
(20.38)
[¥]
(Ahf)(x) = L (-I)j (  ) f(x + jh), x < b, .
j=O J
where we have assumed that h > O. After this fractional derivatives f: and f:l
of the Griinwald-Letnikov type are introduced in the same way as in the case of
(20.39)

386
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
the whole line:
[ z;;_ ]
f:(x) = lim _ h i "" (-I)j (  ) f(x - jh),
h-+O a  J
3=0
(20.40)
[]
f:(x) = lim _ h i "" (-1); (  ) f(X +jh).
h-+O a  J
3=0
(20.41)
In (20.38) and (20.39) one may choose a variable step h, depending on x, e.g.
II = (x - a)/n in (20.38) and h = (b - x)/n in (20.39). Then (20.40) turns into
f: (II:) = ( 1) 1im n a  (-1); (  ) f ( X - j :...=.!!. )
x - a a n-oo L..J J n
;=0
(20.42)
and similarly for f:l.
We note that it was in just this way that Griinwald and Letnikov introduced
their fractional differentiation.
Starting directly from (20.40) and (20.41) one may develop an independent
theory of fractional differentiation. There is no necessity in such a development,
however, because Griinwald-Letnikov derivatives coincide with other forms used in
the case of an interval as well, e.g. with the Marchaud derivative. The following
theorem holds.
Theorem 20.6. Let f(x) E L,(a, b), 1  p < 00. Then the limit (20.40) exists in
the sense of Lp(a, b)-convergence if and only if there exists tile Marchaud fractional
derivative (13.9) in the same sense. Both limits, if they exist, coincide with each
other:
:&
fa) f(x) a f f(x) - f(t) d
a+(x) = f(1 _ a)(x _ a)a + f(l - a) (x _ t)l+a t,
a
(20.43)
O<a<1.
This theorem is easily derived from (20.40), (20.37), Theorem 20.4 and (13.2)
and (13.4).
Based on the coincidence shown in (20.43) of two different definitions of
fractional differentiation, one may obtain various properties of derivatives (20.40)

 20. AN APPROACH VIA FRACTIONAL DIFFERENCES
387
or (20.41). We note for example simple formulae following from (20.43) such as
f(x) == 1 => f:1(x) = r(1 _ a:(x _ a)a
(20.44)
although (20.44) may be also established directly by means of (20.4); and
[T]
lim -.!... '" (-IY (  ) (x - jh - a),6
h-+O h a  J
3=0
=
r(l +,8) ( ) ,6-a
x-a
r(1 + ,8 - a)
(20.45)
(cf. (2.44» and so on. In particular, (20.45) yields the equation
lim N a t(-IY (  ) ( 1- 1. ) ,6 = r(1 + ,8)
N-oo. J N r ( 1 + ,8 - a )
J=O
for all ,8  0, 0 < a < I, although its validity may be shown for all a > O.
We concentrate now on the Griinwald-Letnikov definition of fractional
integration - see Remarks 20.1 and 20.2. Starting from (20.40) we introduce
[T]
J:+ <p = lim h a '" (-I)j ( - ) <p(x - jh)
h-+O  J
J=O
[..--]
1 1 . h a  r(j + a) ( . h)
= r(a) h':O f;:o r(j + 1) <p x - J .
(20.46)
Similarly J:_ <p is defined. Let us demonstrate that (20.46) is just the same as the
Riemann-Liouville fractional integral.
Theorem 20.7. Let a > 0 and let <p(x) E L1(a,b). Then the limit (20.46) exists
for almost all x and
x-a
J:+", = f(l,,) J ",(z - t)t a - 1 dt.
o
(20.47)
Proof. Since the function ta-1<p(x - t) is integrable for almost every x, the
right-hand side of (20.47) may be written as a limit of the integral sum

388
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
rf;;y E<p(z _j);-IXj, Xj ::; j ::; Xj+l. We may choose in particular Xj = jh,
j ::; [(x - a)/h] and j = Xj. Then
z-o [T]
r(I",) J ta-1cp(z - t)dt = r() hO h a  r-1cp(z - ;h).
o J-O
This coincides with the limit in (20.46), since Ija-l - gt I ::; r/-OI by (1.66)
[T]
and lim h a E ja-21<p(x - jh)1 = 0 for almost all x.
h-+O j=O
One may prove also a statement similar to Theorem 20.7 for functions given
on the axis or half-axis provided that the functions decrease sufficiently rapidly at
infinity.
 21. Operators with Power-Logarithmic Kernels
One of the direct generalizations of the fractional integrals 1:+ <P and If_ <P on a
finite interval [a, b) of the real axis has the form
z /3 ..:L-
a,/3 _ 1 J In z-t
(1 0 + <p)(x) - r(a) (x _ tp-O <p(t)dt,
o
z /3 ..:L-
( a,/3 )( ) _ 1 J In t-z ( )
I b _ <p X - r(a) (t _ xp-a<P t dt,
o
(21.1)
a > 0, {3  0, "'I > b - a,
containing both logarithmic and power singularities. We shall call these
constructions operators with power-logarithmic kernels. Such integrals arise
when investigating integral equations of the first kind with power-logarithmic
kernels (see U 32 and 33).
The inversion of the operators in (21.1) will be obtained in S 32.3. In
this section we shall consider the mapping properties of operators with power-
logarithmic kernels in Holder spaces and in the spaces of summable functions. The
results stated and proved here for the integrals I:: <P generalize similar ones for
the fractional integrals I:+<p = 1:+ 0 <p from S 3. We note also that the space H>',1c
(see Definition 1.7) now plays an essential role for the operators (21.1), unlike the
case of fractional integration operators for which this space arises only in separate
cases (see Theorems 3.1 and 3.6).

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
389
21.1. Mapping properties in the space H>'
The following theorem shows how operators with power-logarithmic kernels improve
the Holder exponent for a function rp( t) beyond the point x = a (cf. Theorem 3.1).
Theorem 21.1. Let rp(t) E H>'([a, b]), 0  A  1. If 0 < a < 1 and /3  0, then
u':f<p)(x) = ;: p,o(x) + .p(x),
(21.2)
where
x-a
,6,a(x) = f t a - 1 In,6 fdt,
o
(21.3)
and 1/J(x) E H>.+a,,6 if A + a f; 1 and 1/J(x) E H>'+a,,6+1 if A + a = 1, and the
estimate
11/J(x)1  A(x - a)>.+a In,6, A > O(x  a) (21.4)
x-a
holds. The function ,6,a(x) is infinitely differentiable beyond the point x = a and
has the power-logarithmic behavior:
..... ( ) = ( _ ) a E m (-I)1c(-m)1c I m-1c 
.....m a X x a L +l n
, a x-a
1c=O
(21.5)
if (3 = m is an integer. If /3 is not an integer, then for any N = 1,2,. " we have
 a(x) = (x - a)a  (_I)1c( -(3)1c In,6-1c  + rN ( x)
,6, L.J a1c+l x - a
k=O
(21.6)
where
( (x- a)a )
rN(x) = 0 N+l-,6 ,
(In)
xa.
Proof. By Theorem 3.1 it is sufficient to consider the case (3 > O. If we set
x
.p(x) = rta) f (x - I)O-lln P x  I [<p(I) - <p(a)]dt
a

390 CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
we obtain (21.2). The relations (21.5) and (21.6) are derived from (21.3) by
successive integration by parts. For 1/J( x) we have
x
11/J(x)1  r-1(a)llcpIlH" f (x - t)a-l(t - a)>'ln,6 dt
x - t
a
and after the change of variable t = a + r( x - a) we find
1
11/J(x)1  c(x - a)>.+a f ra-l(1- r)>' ( In  + In 2 ) ,6 dr.
x - a r
o
Applying the known estimate
(a + b)"  2 max (/,1)(a" + b/), a  0, b  0, II> 0 (21.7)
- for example, see Bari [1, p.31] - we obtain the inequality
1.p(z)1  (z - a)A+a [c111n P z  a 1+ C2]
which yields (21.4).
Further we consider 1/J(x) and suppose for simplicity that a = 0, b = 1, and
/ > 1. We denote
g(x) = cp(x) - cp(O) , Ig(x)1  IIcpIlH'\X\
(21.8)
and observe that
Ig(x) - g(y)1  IIcpIlH" Ix - yl>'.
(21.9)
Let 0 < h < 1/2; x, x + h E [0,1]. We investigate first the case A + a  1. We
have

 21. OPERATORS WITH POWERrLOGARITHMIC KERNELS
391
f(a)[1jJ(x + h) -1jJ(x)]
z+h 0
=g(x) 1 ta-llnfj 2dt + I (t + h)a-1lnfj ---2-[g(x - t) - g(x)]dt
t t + h
x -h
I z [ Infj..:L- In fj 1. ]
+ 0 (I + ;ro - Il- w(z - I) - g(z)]dt
=h + 12 + 13'
x+h
According to (21.8) lId $ IIrpIIH"x>' J ta-1ln fj tdt. If x  h, then
z
substituting t = hr and taking the inequality (21.7) into account we find
2h 2 fj
1 Infj 1. 1 I ln 1 + In 1. 1
I I I < ch>.+a ----1..dt = ch>.+a h T dr
1 - t 1 - a rl-a
o 0
:S h Ho (c1ln P k + C2) :S ch'+o In P k.
As for the case x > h, the change of variable t = xr and the use of (21.7) gives
l+h/z
1 ( In 1. + In l ) fj
Ih I  cx>.+a T r 1 - a z dr
1
:S CZ'+O ( In P  7'% To-1dT + 7'< To-11n P dT) .
Since ra-1ln fj   In fj /, we obtain
I h l < c1h ( Infj ! + 1 ) < ch>.+a lnfj !.
- xl-a->. X - h
for x > h.
Further, when we replace t by h(r - 1) in 12 and take (21.7) and (21.8) into

392
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
account we find
o
/ It I>' In,6 m
1121  c (t + h)1tOt dt
-b
/ 1 ( In! + In 1. ) ,6 ( I - r ) >.
< ch>'+Ot h T dr
- r 1 - Ot
o
1
 ch >'+01 In,6 "h:
Finally, we estimate 13' We have
/ z I ln,6 ..:L- In,6 1. 1
[. < c t+h t dt
I 31 - 0 {t + h)1-Ot - t 1 - Ot
z/h
= ch>'+Ot / r>.lr Ot - 1 In,6 :r I dr.
o
If x  h, then by (21.7) we obtain
1 ,6
11.1  c1hHa / T aH - 1 (In  + In ) dT
o
1
< C h>'+Ot In,6 -
- h'
If x > h, then
( 1 Z/1& )
11.1  ch+a ! + I T IT a - 1 In P {T - (T + l)a-1In P h( T: I) I dT = 1., + 132.
For 1 31 we find similarly to previous arguments that 11 31 1  ch>'+Ot In,6 (1/h). To
estimate 1 3 2 we use the relation
I r Ot - 1 In,6 r - (r + I)Ot-lln,6  I < cr Ot - 2 In,6 r r> 1.
r r+ 1 - r'-
(21.10)

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
393
which follows from the mean value theorem. Then by (21.7) we have
z:/h fj
1 1 I <Ch>.+a j (lnk+ln) dr
32 - r 2 - a ->.
1
( z:/h z:/h fj )
>.+a fj 1 dr In 
 ch In h j r 2 - a ->. + j r 2 - a ->. dr .
1 · 1
00
Therefore, if A + a < 1, then owing to the convergence of the integrals J r a +>'-2dr
1
00
and J In fj ra+>'-2dr, we have 11 32 1  ch>.+a In fj k. If A + a = 1, then
1
11 32 1 :5 h>+a (c1n ll  In X + c,ln llH X + c 2 )
1
 ch>.+alnfj+l h'
Collecting the estimates for 1 1 , 12 and 13 we obtain the proof of the theorem in
the case A + a  1.
Let now A + a > 1. We have to prove that 1/J'(x) = (djdx)lg:t 9 E H>'+a-l,fj.
We transform d Ig:t 9 to
r(a) d (lgf g)(x)
z:
= xa-1lnfj 1 9 (x) + j  ( ta-1lnfj 1 ) [g(x - t) - g(x)]dt
x dt t
o
= Gdx) + G 2 (x).
(21.11)
If 9 is a continuously differentiable function, then (21.11) is confirmed by direct
differentiation and later integration by parts. If 9 is a Holderian function, the
coincidence of the left- and the right-hand sides of (21.11) is proved in the same
way as the coincidence of the Liouville fractional derivative and the Marchaud
derivative on an interval (see the Corollary to Theorem 13.1 and Corollary of
Lemma 13.2).

394 CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Let us show that G1(x), G 2 (x) E H>'+0I-1,p. We have
G 1 (x + h) - G 1 (x) =(x + h)0I-1In P -2- h [g(x + h) - g(x)]
x+
+ [ (x + t)0I-1In P -2- - x Ol - 1 In,8 r ] g(x)
x+h x
=G ll + G 12 .
Taking (21.9) into account we obtain for G ll that IGlll :$ ch>.+0I-1In,8 t. By
(21.8) we find for G 12 that
IC 12 1 s Ax Ix a - 1 ln P ; - (x + h)a-lln P xl h i.
From here we have
I G I < 2Ax>.+0I-1I n P r < ch>.+0I-1In P .!.
12 - X - h
if x :$ h. Let x > h. Then, according to (21.10), we obtain that
I G I < cx>.+0I-2h In P r < ch>.+0I-1In P .!...
12 - X - h
We introduce the notation
K(t) =  (t Ol - 1 In P f), IK(t)l:$ ct Ol - 2 1n P f.
(21.12)
Then
o
G 2 (x + h) - G2(X) = f K(t + h)[g(x - t) - g(x + t)]dt
-h
:& :&
+[g(x + h) - g(x)] f K(t + h)dt + f [I«t + h) - K(t)][g(x - t) - g(x)]dt
o 0
= G21 + G2 2 + G 23 .

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
395
We estimate G 21 first. By (21.9), (21.12) and (21.7) we have that
o
IG 21 1  c j {t + h)>.+a- 2 In13 t  h dt
-h
1 13
 ch Ha - 1 j T Ha - 2 (In k + In:;) dT
o
1
 ch>.+a- 1 l n 13 X.
Further we find that
z
I G 221  ch>' j (t + h)a- 2 In13 t  h dt
o
z/h
= ch>.+a-l j (T + 1)a- 2 In13 / dT
(T + l)h
o
( z/h z/h )
 ch>+a-l In ll k ! (T + l)a- 2 dT + ! (T + I)a-2In ll T: 1 dT
1
< ch>.+a- 1 l n 13 -
- h
00
in accordance with the convergence of the integral J (T + 1 )a- 2 1n13 ;trdT for /3  o.
o
Lastly, taking (21.12) and (21.9) into account we have
1 j z I ln13-1: ..:L- In 13 -1: 1. 1
< c t+h t t>'dt
IG 23 1 -  I: (t + h)2-a - t 2 - a
1:_0 0
1 z/h 13-I::L In 13 -1:  I
- h >.+a-l L j 10 Th _ ( T+l ) h >. d
- Cl: 2 ( )2 T T.
T -a T + 1 -a
1:=0 0
From here if x  h, we find
1 ( 1 I ln13-1: :L I 1 I ln 13 -1:  I )
>.+a 1 Th (T+l)h
I G 231  h - L: c. j T2-a-> dT + j (T + l)2-a T -> dT
1:=0 0 0

396
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
< h>.+a-l ( c In,6! + c In max (O,,6) ! )
- 1 h 2 h
1
$ h>.+a- 1 l n ,6 h'
If x > h, then in view of (21.10) and (21.7) we obtain
1 ( 1 X/1& ) [ ,6 1c:L I ,6-1c  ]
IG I < h>.+a-l '" c f + f In - Th _ n ( T+l ) h r>'dr
23 - L.J 1c r2-a (r + 1 )2-a
1c=O 0 1
( 1 x/h ( 1 1. ) ,6_1c )
< h>.+a-l cln,6.!. + '" c f In h + In T dr
- h L.J 1c r3-a->.
1c=O 1
1
< ch>.+a- 1 l n ,6 -
- h'
which completes the proof. .
Corollary 1. The operator
x
 f cp(t) - cp(a) In ,6 --.:L dt 0
( ) , < a < 1, /3  0,
r a (x - t)l-a X - t
a
is bounded from H>', 0  A  1, into H>.+a,,6 if A + a f; 1, and into H>.+a,,6+1 if
A+a=1.
Corollary 2. The operator I::t is bounded from C = HO into H a ,,6.
Remark 21.1. The operator I::t is bounded even from Loo into H a ,,6 I which is
easily seen from the analysis of the proof of Theorem 21.1.
21.2. Mapping properties in the space H;(p)
As in S 3 we begin our investigation with the case of the simplest weights
p(x) = (x - a)#J and p(x) = (b - x)".
Theorem 21.2. Let 0 < A < 1, A + a < 1. Then the operator r::t, (3  0, is
bounded from H(p) into H;+a,,6(p) ifp(x) = (x-a)#J, p. < A+l orp(x) = (b-x)",
II > A + a.
Proof. Firstly, let p(x) = (x - a)#J. By Theorem 3.3 it is sufficient to consider the

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS 397
case (3 > O. Let a = 0, b = 1 for simplicity. Let rp(t) E H6(p), so rp(t) = g(t)t- IJ
where g(t) E H>', g(O) = O. We are to prove that
:&
G(x) = 1 (  ) IJ Infj ---2- g(t)dt E H>.+a,fj
t x - t (x - tp-a 0
o
and IIGIIH"+Q.  cllgIlH'" We represent G(x) as
:& R :&
1 In""..:L- 1 IxIJ t IJ I
G(x) = ( :&) g(t)dt + (- p_ ln fj ---2- g (t)dt
x - t a tIJ X - t a X - t
o 0
= (x) + w(x).
(21.13)
By Theorem 21.1 we obtain that (x) E H;+a,fj. For w(x) we have, similarly to
(3.7),
:&+h
1 fj 'Y (x+h)IJ-t IJ
w(x + h) - w(x) = In h (h P g(t)dt
x + - t tIJ x + - t -a
:&
:&
+ [(x + h)IJ - x IJ ] I lnfj 'Y (x + h - t)a-1 g(t)dt
x + h - t tIJ
o
1 :& [ I fj --2..- I fj..:L- ]
X IJ - t IJ n +h t n t
:& - :&- t dt
+ 0 tIJ (x + h - t)l-a - (x - tp-a g()
=J 1 + J2 + J 3 .
(21.14)
We estimate J 1 first. Let fl  1. Since Ig(t)1  IIgIlH"t>' then in view of (3.9) and
(2.17) we find that
:&+h
1 (x + h - t)a fj 'Y
IJ 1 1  IfllllgllH" ( p_>. In h t dt
t-x x+ -
:&
1 1 ra ( 1 'Y ) fj
 IfllllgllH" h>.+a (1 _ r)l->' In h + In;; dt
o
1
 cllgIlH"h>.+a ln fj h'

398
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
If JJ > 1, then, according to (3.8), we have that
z+h
IJ 1 1 < IJJllIgIIHA(x + h)IJ-1 f (x  h  t)a In,6  dt
- IJ- x + - t
z
z+h
h a In P 1. f dt
<c h
- (x + h)1-IJ tIJ->'
z
h a ln P 1.
= c h [(x + h)>'-IJ+1 _ x>'-IJ+1]
(x + h)1-IJ
< ch 1 + a In,6 ! ( x + h ) >'-l < ch>.+a In,6 .!:...
- h - h
We estimate J2. If x =:; h, then we obtain
z
f t>,-IJ
IJ21 < ch IJ ( h )1 In,6  dt
- x + - t -a X + - t
o
z
< ch IJ + a - 1 In,6 ! f t>.-IJdt
- h
o
< ch IJ + a - 1 In,6 !x>'-IJ+1 < ch>.+a In,6 .!:..
- h - h'
if JJ  0, and
f z t>,-IJ
J < cx IJ In,6 'Y dt
1 21_ (x + h - t)1-a X + h - t
o
z
< cx IJ h a - 1 In,6 ! f t>'-IJdt
- h
o
< ch a - 1 In,6 2x>'+1 < ch a +>'ln,6 .!:..
- h - h
if fl < O. If x > h, then by applying the inequalities (3.9) and (21.7) we find that
z
f t>,-IJ
I J 21 =:; Ifll hxlJ - 1 ( h )1- In,6  dt
x+ -t a x+-t
o

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
I z In.8 ..:L-
< ch z-t dt
- t 1 ->'(x - t)1-a
o
399
1
ch 1 (In; + In - h 1 ),8
- dt
- xl->.-a (1 - r)>'r 1 - a
o
h In,8 1 1
< c h < ch>.+a In,8 _
- xl->.-a - h'
if JJ  1. Let JJ > 1. Then by (3.8), we have
z+h ,8
11215, I'h(z + h)"-' 1 t"->;: :)'_a dl
z
1
= J.l h 1 (1- r)>'-#ra-l ( In -1- + In .!. ) ,8 dr
(x + h)1->.-a x + h r
o
h 1 ,8 'Y >.+a,8 1
< c ( h)l >. n - < ch In _ h '
- x + - -a X + h -
For J 3 the change t = 8X gives the estimate
1 1 1,8 'Y 1 ,8..:L- 1
>.+a 1 - s# n 1-,+h / x n 1-,
IJ.I:5 IIgliHAZ I 8"-> (1- 8 + hjz)'-a - (1- 8)'-a d8.
If x  h, then IJ21  cllgllHA h>.+a. If x > h by (21.10) we have
1
>. 1 1 1 - s# ,8 'Y >.
IJ 3 1  IIgIIH" X +a- h 8#->'(1 _ S )2-a In 1 _ s ds  cllgllH" h +a.
o
Collecting the estimates for Jl, J2 and J 3 , we obtain
1
Iw(x + h) - w(x)1  cllgIlH"h>.+a In,8 h,'
Hence taking (21.13) and the inequality IG(x)1  cllgllH" x>.+a In,8 ;. into account
we see that G(x) E H>.+a,,8. So we have proved the theorem for the case
p(x) = (x - a)#, JJ <  + 1. When p(x) = (b - X)II, V >  + Q, the proof of the

400
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
theorem is carried out in the same way as the proof of Theorem 3.3 using the
inequalities (3.8), (3.9), (21.7) and (21.10). .
The following statement analogous to Theorem 3.3' follows from Theorem
21.2.
Theorem 21.2'. Let 0 <  < 1,  + a < 1. Then the operator 1:+ is bounded
from H(p) into H;+OI,/J(p), p(x) = (x - a)IJ(b - X)II, Il <  + 1, II >  + a.
The proolof Theorem 21.2' is carried out in the same way as the proof of
Theorem 3.3' using the following inequality for functions in H;,/J (p): IIgIlHA,(p) 
max(lIgIlHA,(p.), IIgIlHA,(p,,» where pa(x) = (x - a)IJ, Pb(X) = (b - X)II.
To extend Theorems 21.2 and 21.2' to the case of the general weight (1.7) a
statement of the type of Lemma 3.1 for integrals with the kernel
(x + t)OI-llnlJ, x > 0, t > 0,
x +t
is required. It has the following form.
Lemma 21.1. Let a function rp(x), 0 < x  I, have the estimate Irp(x)1  kx-'Y,
a < 'Y < 1. Then
I
f In IJ ...::I...-
f(x) = (x + t)i'.-OI rp(t)dt E HOI+/J,IJ([O, I]; x'Y+/J)
o
for any P  0, 'Y > 21 and {3  0 such that a + (3  1, and IIfIlH"'+."(z"'+)  ck
where c does not depend on rp( x).
The prool of this lemma is carried out in the same way as the proof of Lemma
3.1 using the inequalities (21.7) and (21.10).
Theorem 21.3. Let p(x) be the weight (3.12),  + a < 1 and let the following
conditions be satisfied:
1) PI <  + 1;
2)  + a < Pfc <  + 1, k = 2,3, " ., n - 1;
3)  + a < Iln <  + 1 if X n < b and  + a < JLn if X n = b. Then the operator
I:: is bounded from H(p) into H;+OI,/J(p).
The proof of Theorem 21.3 is carried out in the same way as the proof of Theorem
3.4 using Theorems 21.2 and 21.2' and Lemma 21.1.
Remark 21.2. The condition +a < Ilfc, k = 2,3,... ,n, in Theorem 21.3 may not
be weakened: if +a  Ilfc, then Theorem 21.3 is not true. Indeed, if we put p(x) =

 21. OPERATORS WITH POWE&.LOGARITHMIC KERNELS
401
(b - x)#J, P. $ A + a and <p(x) = (x - a)>'(b - x)#J, then direct evaluation shows that
(p I OI ,P p -1 l /'J )( x ) = (botr J z (t-a)" ( X - t ) 0I-1In P ..:L-dt .../.. 0 ( b - X ) >.+ OI l n P -1- )
a+ TrOt (b-t),,- A z-t r b-z
a
as X --+ b. Hence, we obtain that f = 1:/ p-1<p 'I. H;+OI,P (p) - cf. remark 3.2.
21.3. Mapping properties in the space Lp
By means of the generalized Minkowsky inequality (1.33) it is easily proved that
the operator 1:/ is bounded in Lp = Lp(a, b), p  1:
Ill:: <pIIL p  cll<pIIL p '
The following two theorems contain more precise statements.
Theorem 21.4. If 0 < a < 1, {3  0 and 1  p < l/a, then the operator 1:: IS
bounded from Lp into L 8 , 1 $ S < q = p(1 - ap)-l.
This theorem follows from Theorem 3.5 since
11:/ <pI  cI:;£ (I <pI)
(21.15)
for any small c > 0 and 0 < a < 1.
Theorem 21.5. Let p > 1 and l/p < a  1 + l/p. Then the operator 1:/ is
bounded from Lp into H OI - 1 /p,P if a < 1 + l/p and into H OI - 1 /p.P+1/p if a = ; + 1,
and
(I:/<p)(x) = 0 ( x - a)0I-1/p In P  ) as x --+ a.
x-a
(21.16)
Proof. We set a = 0, b = 1 for simplicity. The estimate (21.16) follows from the
Holder inequality (1.28) in view of (21.7):
( z ) IIp ( z ) IIp'
II'P <pI <  1 1<p(t)IPdt l (x - t)(O-l)pl In Pp ' -Ldt
+ - r( a) x - t
o 0
( 1 ) IIp' ( z ) IIp
= r(lOt) ",a-I/' ! r(a-I),' (In; + In ;J'dr ! 1",(t)I'dt
( z ) IIp
 c",a-I/'ln P ; ! 1",(t)I'dt
(21.17)

402 CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Let x, x + h E [0, 1] and 0 < h  1/2. We have
r(a)[(Ifcp)(x + h) - (Ifcp)(x)]
z+h
= f (x + h - t)a- 1 1n,6 'Y cp(t)dt
x+h-t
z
z
+ f [ (x + h - t)a- 1 In,6  - (x - t)a- 1 In,6  h] cp(t)dt
x+ -t x-
o
=1 + 2.
(21.18)
Applying the Holder inequality and taking (21.7) into account we find
( Z+h ) IIp
IcJ1d :<; IIIIL. ! (x + h - tj<a-I),'ln P ,' x +  _ , dt
( 1 ,6' ) IIp'
= IIIIL. h a-II, ! T( a-I),' (In  + In k) , dT
1
 cllcpliLpha-1/p In,6 h'
( zlh ) IIp'
121  IIcpliL h a - 1/p f I ra-1In,6 :L - (r + It- 1In ,6 'Y I P'dr
p rh (r+l)h
o
Hence
IcJ1 2 1 :<; 211IIL.ha-I/' (i T(a-I),' (In  + In ) p,' dT) II"
 CllcpliLpha-1/p In,6 *,

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
403
if x  h. Let x > h. Then we apply the inequalities (21.7) and (21.10) and have
{( 1 Zlh ) } lip'
121  1I<pIIL hOt-lip f + f I rOt-1InP l_ (r + I)Ot-ll n P 'Y I P'dr
· rh (r+l)h
o 1
[ ( zlh ) liP' ]
 III"IIL.ha-!/. c!ln ll  + C2 I T(a-2).'ln ll .' :h dT .
Then
_ ( 1 ) 1
121  1I<pIILph Ot IIp C1 lnP h + C2  cll<pIILphOt-l/p In P h
00 ,
in the case a-I < IIp owing to the convergence of the integral J r(Ot-2)p' In,8p dr.
1
If a-I = IIp, then
[ 1 1 ' 1 1 ' ]
zlh P (z-a)lh, P
1 , 1 dr In,8p 1-
121  III"IIL.ha-!/. c!ln ll h + C2 nll' h I  + c. ( I dT)
 cll<pIILphOt-1/p In P + 1/p ' .
Collecting the estimates for 1 and 2 we complete the proof. .
We note that the statement on the boundedness of I::t from Loo into HOt,P
observed in Remark 21.1 corresponds to the case p = 00 in Theorem 21.5.
Remark 21.3. In the case of the pure logarithmic kernel (a = 1) and 0 < {3 < IIp'
the statement of Theorem 21.5 can be improved: if 0 < {3 < IIp', then the operator
It is bounded from Lp into HP.
Let us estimate 2 in (21.18). Using the inequality laP - bPI  la - bl P ,
o < {3 < 1, and the Holder inequality we have
( a:-. ) IIp'
121  III"IIL.h!/.' [Inll" t: 1 dt

404 CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
 III"IIL. h 1fp' [c + C1 (/. t2P' ) l/ P ]
 III"IIL. h l/p' [c + C1 (., : a r -l/ P ']  CII 1"11 L. h P .
21.4. Mapping properties in the space Lp(p)
As in S 21.2 we consider first the weight p( x) = (x - a)#. We have to distinguish
the cases 1 < p < 1/0: and p > 1/0:.
Theorem 21.6. If p > 1, p. < p - 1 and 0 < 0: < l/p, then the operator I::t is
bounded from Lp(p), p(x) = (x - a)#, into L8(r) where 1  s < q = p(l - o:p)-I,
r(x) = (x - a)#8/p.
This theorem follows from Theorem 3.7 in view of (21.15).
Theorem 21.7. Let 1 < p < 00 and l/p < 0:  l/p + 1. Then the operator I::
is bounded from Lp(p), p(x) = (x - a)#, p. < p - I, into HOI-I/p,P (pl/P) if 0: < l/p
and into HI,P+I/p' (p) if 0: = l/p + 1 and
(I:: cp)(x) = 0 ( x - a)OI-(I+#)/p In P  ) , x --+ a.
x-a
(21.19)
Proof. We set a = 0 and b = 1 for simplicity. Let rp(x) = X-II g(x), V = p./p,
9 E Lp. The estimate (21.19) is obtained by the same arguments as in (21.17).
We shall prove that
:&
G(x) = f (  ) II (x - tt-Iln P -Lg(t)dt E HOI-I/p,P,
t x - t
o
(21.20)
IIGIIHO-l/P,fl  cllgllL p '
if 0: - l/p < 1 and
G(x) E H I ,P+l/ p ', IIGIIH1,fl+1/P'  cllgllL p '
(21.21)
if 0: - 1/ p = 1.

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
405
We represent G(x) as
G(x) = (x) + W(x),
(21.22)
where (x) and w(x) are the same as in (21.13) but with p. being replaced by
v = p.lp; note that v < 1 due to assumptions of the theorem. Let 0 < h < 1/2. By
Theorem 21.5 we have
{ ChOl-llp In,6 tllgllLp
x+h -x < ,
1 ( ) ()I - ch In,6+1/p kllgllL p
if a - lip < 1,
if a-I I p = 1.
We represent the function w(x) in the form (21.14): w(x+ h) - w(x) = J l + J 2 + J3
where J l , J 2 and J 3 are the same as in (21.14) but with p. being replaced by
v = p.lp.
Let p. > 0 and hence v > O. Using the Holder inequality, (21.7) and Holder
property of x" we obtain
( z+h ) IIp'
(x + h - t)(1I+0I-1)p' ,6' 'Y
IJd :5 IIgIlL.! (I _ z)"P' In p z + h _ 1 dl
(1 r(II+0I-1)p' ( 1 'Y ) ,6p )
= IIgIlL.h O - 1/p 0 ' (1 _ T)"P' In h + In; 'dT lip'
1
 cllgIlLphOl-l/P In,6 h'
Applying again the Holder inequality to J 2 we have
z IIp'
IJ 2 1  [(x + h)" - x"] IIgllL p (1 (z + h - ,!(O-l)P' Inllp'  dl )
t"p x + - t
o
Hence if x  h, then
( z ) IIp'
Ihl :5 Ch"+O-lln ll  IIgIlL. I r"p' dl
1
 ch Ol - llp In,6 hllgllLp'

406 CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
If x > h, then using (3.9) and (21.7) we find
( z: ) lip'
(o-l)p' ,
I J 21  IIhxll-lllgllL p J (x - t) , In,6p --1-dt
tllp x - t
o
1
 ch o - llp In,6 hllgllLp'
Finally we estimate J3. Applying the Holder inequality and changing "the variable
t = 8X we obtain
( 1 , I ,6 ,6 I P ' ) I / P'
o-llp 1 - 8 11 P In 1-8+h(z:-a)-1 In -2:;
IJ31lIgIiHAZ ! (---;;-) [1- s + h(z _ a)-I)1-a - (1- s)l-a ds .
If x  h, then IJ 3 1  chO-lIPlIgIILp' If x > h, then by (21.10)
IJ 3 1  xl+P-O IIgllL p  chO-l/PlIgIILp'
Collecting the estimates for J l , J2 and J 3 we have
1
Iw(x + h) - w(x)1  ch o - llp In,6 hllgllLp
Hence, according to (21.22) and (21.23) we obtain (21.20) and (21.21), which
proves the theorem in the case p. > O. If J.l = 0, then the theorem coincides with
Theorem 21.5.
Let now J.l < 0 (II < 0). Then
z:+h
IJll  2 J (x + h - t)O-Iln,6  Ig(t)ldt
x+ -t
z:
 ch o - l/p In,6 *lIgIlL p .
For J2 we have the estimate
1
IJ 2 1  21Igf gl  cx o - l/p In,6 -lIgllL p
x
 ch o - l/p In,6 kllgllLp

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
407
if x $ h. If x > h, then we obtain the same estimate by the arguments similar to
those in the case II > O.
We apply the Holder inequality to J 3 and obtain
( Zlh ) IIp'
IJ 3 1  211gliL h a - 1/p f I ra-1In,6 l - (r + 1 )a- 1 In,6 / I P ' dr
p rh (r + l)h
o
If x  h, then we have
IJ 3 1  411911L. h a-II. (! T( a-I ).' (In  + In ;: ) fJ.' dT) II.'
$ cllgllLpha-1/p In,6 *.
If x > h, then taking (21.10) into account we find
IJ 3 1  liYIIL.h a - l/ . [Clln fJ  + C2 (T T(a-2).'ln fJ .' :h dT) II,']
{ cllgllLpha-1/p In,6 k, if a -  < 1,
< I '
- cllgllLph In,6+1 p k, if a -  = 1.
Whence (21.20) and (21.21) with p. < 0 follow, which completes the proof. .
Corollary. Under assumptions of Theorem 21.7 the operator 1:/ is bounded
from Lp(p), p(x) = (x - a)IJ, into h-1/PI,6(p1/p) if a < lip + 1 and into
h-1/PI,6+1/p' (p1/p) if a = lip + 1.
This corollary is proved in a similar way to that of Theorem 3.8.
For the weight p( x) = Ix - dl IJ , a < d < b, the result is the following.
Theorem 21.8. If 1 < p < 1/a, p. < p - 1, a < d  b, then the operator
I::t is bounded from Lp(p), p(x) = Ix - dl IJ , a < d  b, into L8(r) where
1  s < q = P(1 - ap)-l, r(x) = Ix - dill and II is given by (3.21) with q = s.
According to (21.15) this theorem follows from Theorem 3.9.
Theorem 21.9. Let 1 < p < 00, lip < a $ lip + 1, 0 < p. < p - 1 if a < d < b or
p > 0 if d = b.

408 CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
1) Let Q - IIp < 1. Then the operator r:f is bounded from Lp(p),
p(x) = Ix - dl IJ , a < d  b, into Hr;in(a-l/p,IJ/p),P (pl/p) if p. :/= QP - 1 and into
Hg-I/p,P+I/p' (pl/p) if P. = QP - 1.
2) Let Q - IIp = 1. Then the operator I:f is bounded from Lp(p),
p(x) = Ix - dl IJ , a < d < b, into H/p,P(pl/p) and it is bounded from Lp(p),
p(x) = (b - dY, into H/P,P(pl/p) if p. < P or into H,P+I/P' (pl/p) if p.  p.
Proof. We prove this theorem following the scheme of proof of Theorem 3.11. Let
v = p.lp, g(t) = It - dl"cp(t) E Lp(a, b),
z:
f(x) = J I x - d d I II (x - t)a-Iln P g(t)dt.
t- x-t
a
We are to prove that if Q - IIp < 1, then
IIfIlH"'  cllgllL p ' A = min(Q - IIp,p.lp) for p.:/= QP - 1,
(21.24)
IIfIlH"'-l/p.+l/p'  cllgllL p ' for p. = QP - 1,
and if Q - IIp = 1, then
IIfIlH,./p.  cllgllL p ' if d < b or d = b, p. < p,
IIfIlHl.+l/P'  cllgllL p ' if d = b, p.  p.
(21.25 )
We fix the point al E (a, d). By Theorem 21.7 we have f E H a - 1 /p,P(a, ad if
Q - IIp < 1 and f E H 1 ,P+1/p' (a, ad if Q -lIp = 1. We are to show that f satisfies
the estimates (21.24) and (21.25) on [ai, dj. We represent f as f =  + W where
 and Ware the same as in (21.13) but with x IJ being replaced by Id - Xi". Then
by Theorem 21.7  E H a - 1 /p,P if Q - IIp < 1 and  E Hl,P+I/p' if Q - IIp = 1.
Now we represent the function W as W = J1 + J 2 + J 3 where J1, J2 and J 3 are
the same as in (21.14) but with x IJ being replaced by Id - xl lJ . Let 0 < h  1/2,
a1 < x < x + h < d. Then for it we have
( z:+h , ) 1/1"
l(d-x-h)"_(d-t),,1 P / l'
IJ"  11911L. ! I (d _ t)"(z + h - t)l-a In :r + h _ t I dt

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
409
II II I fj / ( :& f +h (d - t)-II P 'dt ) IIp'
 c 9 Lp n X :& (X + h _ t)(l-a-lI)p'
$ ch a - llp lnfj lIgIlLp.
For J2 we find
[ ( :& , ) l IP' ]
II fj / (d - t)-IIP dt
IJ 2 1 :5 ch In hllgllL. c+ CI ( x _ I)(I-a).' .
Hence by the estimate for (3.27) with II = /lIp we obtain
{ ch>'lnfj k IIgllL p ' A = min ( a - ,  )
IJ 2 1  ,
ch Infj+llp tllgllL p '
if /l f; ap - 1,
if /l = ap - 1.
We observe that if a - lip = 1, then /lIp < 1 = a - lip for a < d < b (by the
assumption /l < P - 1 of the theorem). Therefore, only one case /l < ap - 1 = p is
realized here. Having fixed an arbitrary point 6 E (a, al) we can rewrite J 3 as
J = (f 6 f OI f :& ) [(d - x)1I - (d - t)lI]
3 + + (d-t)1I
o 6 01
x [ (X + h - t)a-ll n fj / - (X - t)a-ll n fj  ] g(t)dt
x+h-t x-t
= J 31 + J 32 + J 33 .
It is obvious that IJ 31 1 $ chllgllLp' For J 3 2 we have
( 01 , ) IIp'
IJ 32 1 $ cllgllL f I (x - t)a-llnfj  - (X + h - t)a-ll n fj  I p dt
p x-t x+ -t
6
1
$ ch a - llp In fj xllgllLp'

410
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Finally, for J 3 3 we have
( X, ,  111"
IJ",IcllgIlL. f ( ==: r l(z-W- 1 In ll zt -(z+h-W- 1 Inll z+:-l dt)
(  ) 111"
cha-l/plIgIlL J ! ra-lln,6l_(r+l)a-lln,6  I P' dr
p rh h+rh
o
Hence we obtain
1 111"
IJ",I  2ch a - 1 / P lIgIlL. (/ T(a-l)p' In llp ' :h dT)
 ch a - l/p In,6 lIgllLp
if x - al  h. As for the case x - al > h, we find
[ (  ) 1/1" ]
IJ",I  h a - 1 / P lIgIIL. cln ll  + cl i T(a-2)p'ln llp ' :h dT
< { cha-l/Pln,6 kllgllL p ,
- ch In,6+1/p' k IIgllL p '
if a - lip < 1,
if a - lip = 1.
Having collected the estimates for h, J 2 and J 3 we see that f satisfies the
estimates (21.24) or (21.25) on [all d]. The validity of the same on [d, b] with d < b
is proved similarly. Since we have proved the estimates (21.24) and (21.25) on the
intervals [a, ad, [al, d] and [d, b] with d < b, they can be extended to [a, b] too. By
arguments similar to those given above we can prove that f(d) = O. This completes
the proof. .
In conclusion of this subsection we remark that on the basis of Theorems
21.6-21.9 one may formulate the corresponding theorems for the case of the general
power weight (1.7) - for example, the theorems similar to Theorems 3.10 and
3.12. We also note that in the case of the pure logarithmic kernel (a = 1) and
o < /3 < lip' the assertions of Theorems 3.7 and 3.9 can be improved - see remark
21.3.

 21. OPERATORS WITH POWE&.LOGARITHMIC KERNELS
411
21.5. Asymptotic expansions
Asymptotic representations of the integrals Igf cp with power-logarithmic kernel
can be found by the methods presented in S 16 for fractional integrals. In the
cases when cp has the power asymptotics (16.5) or (16.6) or the power-logarithmic
asymptotics (16.28) we can use the method of successive expansion or the method
based on the Parse val equality (1.116) for the Mellin transform (1.112). If cp has
the power-logarithmic asymptotics (16.29) two other ways can be also used: the
method based on the representation of Igt cp as the Mellin convolution (1.114) and
the method of direct estimates; see S 16 and S 17.1 (notes to U 16.2-16.4).
Let us obtain, for example, the asymptotic expansion of the integral
:&
(J.;" <P)(x) = r(I",) J (x - W- 1 In"(x - t)<P(t)dt,
o
° < Q < 1, v = 0,1,2,.. .
(21.26)
as z -+ +00 if cp has the power-logarithmic asymptotics (16.29). As in S 16.3 we
use the method of direct estimates.
Theorem 21.10. Let a function cp be locally integrable on [0, +00) and
00
cp(t) f'OoJ t-fj L an(lntp-n as t -+ 00
n=O
(21.27)
where ° < {3 < 1, -00 < "( < 00 are any fixed numbers. Then
00
(Jg: cp)(z) f'OoJ ZOl-fj L bn(lnzt+'Y- n
n=O
(21.28)
as z -+ 00 where
n m ( ) ( )
1 v "(-n+m
b n = b n (Q,{3,v, "() = r(Q) o a n - m  k m _ k h,m-I:(Q,{3),
n = 0, 1,2, . . .
(21.29)

412
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
and
1
Ok,m(a,{3) = 1(1- r)a- 1 r-,6ln k rlnm(l- r)dr
°
k 81c+m
= (-1) 8am8{3k B(a,{3),
(;) and c:;.m) being the binomial coefficients (1.48).
Proof. We represent (21.26) as
( .fi z-.fi Z )
(J:<p)(z) = r() 1 + 1 + 1 (z- t t- 1 In"(z-t)<p(t)dt
° .fi z-.fi
= J 1 CP + J2CP + J 3 cp.
The integrals J 1 cP and J 3 CP are asymptotically small in comparison with J 2 cp:
hcp = O(X a -,6-Pl), hcp = O(X a -,6-P2), as x --+ 00
where P1 > 0 and P2 > 0 are fixed numbers. We prove the estimate for J 3 cp. We
choose € so that 0 < € < a/3. Since In'Y t = O(t£) as t --+ 00 we have from (21.27)
that cp(t) = O(t-,6+£) as t --+ 00. Substituting x - t = xr into J 3 CP we have the
following inequalities for sufficiently large x:
z-1/2
J 3 CP :5 c 1 (xr)a- 1 ln ll (xr)[x(l - r)t,6+£ xdr
°
.fi
:5 cM_,6+£x-,6+£ 1 t a - 1 ln ll tdt
°
where Ma = max (1 - r)a-1. We have t a - 1 ln ll t = O(t a - 1 +£) as t --+ 00 and
05:T5:1/2
fi
hence J t a - 1 1n ll tdt = O(x(a+£)/2) as x --+ 00. Taking this fact into account we
°
obtain the estimate for J 3 CP with P2 = (a - 3€)/2. The estimate for J 1 CP is proved
similarly.

 21. OPERATORS WITH POWER-LOGARITHMIC KERNELS
413
We estimate J2'{J. By (21.27) we have
N
'(J(t) = t- P L anOn tp-n + RN(t), RN(t) = O(t- P (In tp-N-l)
n=O
(21.30)
as t -+ 00. Therefore
N
"" an
J 2 '{J =  f(a) L(a, {3, v, / - n; x) + rN(x),
(21.31)
where
z-..Ji
L(a,{3,v,"Yi x ) = f (x-t)a- 1 1 n ll(x-t)t- P ln ll tdt,
..Ji
(21.32)
z-..Ji
1 f -
rN(x) = r(a) (x - t)a Iln ll (x - t)RN(t)dt.
..Ji
We replace x - t by TX in (21.32) and have
l-Z-1/
L( a, (3, v, "y; x) =x a - p Inll+'Y x f (1 - T)a-I T - P
z-l/
x [I + In\  T) r [I + ::: r dt.
Since 1 1 + In(1-T) 1 < ! and I !!:!..!. I < ! for x- 1 / 2 < T < 1- x- 1 / 2 we have
In z - 2 In z - 2 - -
[ 1 + In(1 - T) ] II [ 1 + In T ] 'Y
In x In x
= fo [() (m  k) Ink (1- T)ln m - k T] In- m z.
Substituting this expression into (21.33), carrying out the term-by-term integration

414
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
and using the estimates as x --+ 00
:&-1/
f (1 - r)a- 1 r-,6ln m rink (1- r)dr = 0(x- 61 ), 6 1 > 0,
o
1
f (1 - r)a- 1 r-.8ln m r Ink (1 - r)dr = 0(x-6), 62 > 0,
1-:&-1/
which are checked directly, we find the asymptotic expansion as x --+ 00
L(a,P,I/,r;"')  ",a-{J  [() (m k)nt,m-t(a,p)] (Inz)"+1- m ,
(21.34)
At last, substituting (21.34) into (21.31) and taking into account the estimate
rN(x) = 0(x a -.8ln ll +'Y- N - 1 x) as x --+ 00 which follows from (21.30) we obtain
(21.28) and (21.29). The theorem is proved. .
Remark 21.4. Theorem 21.10 generalizes Theorem 16.4 and can be used to find
the asymptotic solution of (21.26); see S 16.5 and S 34.2 (note 32.4).
 22. Fractional Integrals and Derivatives
in the Complex Plane
Our previous discussions have concerned functions of a real variable. We are now
going on to consider ideas and notions connected with the fractional calculus of
functions of a complex variable. It is worth-while emphasizing that the fractional
calculus was developed in the complex plane from its very origin. - Liouville,
Grunwald, Letnikov, Sonine and others.
The following approaches to fractional calculus in the complex plane are
known and widely used.
I. Fractional integro-difJerentiation of analytic functions, represented by
exponential series (Liouville's approach) or power series (Hadamard's approach).
These are based on termwise integra-differentiation of the series. The equation
1)a e oz = aaeoz in the former case and the e q uation 1)OI ( Z - Z ) 1' = 1'(1+1') ( z-
o 1' ( 1+IJ- a )
zo)lJ- a in the latter case serve in fact as definitions, a, a and p. being arbitrary
numbers.
For functions f(z) = f(re itp ), analytic in a disc, Hadamard's approach is in
fact Riemann-Liouville integra-differentiation with respect to the variable r.

fi 22. FRACTIONAL INTEGRALS IN THE COMPLEX PLANE
415
II. Extension oj Weyl fractional integro-differentiation to junctions analytic in
the disc by the rule
00 00
j(z) = "" b:z le => 1(01) j = "" zle
L...J L...J ( ik ) 01
1e=0 1e=1
(22.1)
- Hardy and Littlewood. This is indeed the Weyl fractional integro-differentiation
of a function j(rei'P) with respect to the angular variable cpo
III. Direct introduction of Riemann-Liouville integro-differentiation in a
complex plane:
Z
01 1 f f(t)dt
(l zo j)(z) = r(a) (z - t)1-OI '
Zo
Rea > 0,
(22.2)
('D:oj)(z) = :; (1;:-01 j)(z), m = [Rea] + 1, Rea> 0,
(22.3)
with integration along the straight line interval connecting points Zo and z as a
rule. In general, one may use integration in (22.2) along a curve connecting Zo and
z and lying in the domain of definition of the function j(z). The situation when
the function is given on a certain curve only may be also admitted (see e.g. S 23.1
(note 22.1».
We add that if a function j(z) is considered in a domain, then (22.2)-(22.3)
with integration along the segment [zo, z] implies that the domain is to be starlike
relative to zoo The latter means that if a point z is in a domain, then the same is
true for the whole segment [zo, z].
IV. A definition based on a generalization of the formula for differentiating the
Cauchy type integral:
j(OI)(Z) = r(1 + a) f f(t)dt
21ri (t - z )1+01
I:.
(22.4)
- Sonine, Laurent, N ekrasov and others. It ought to be stressed that this approach
is to be applied only to analytic functions.
We emphasize that any work with definitions (22.2)-(22.4) requires precision
aimed to single out a branch of the multivalued function. It is usually achieved
by means of a cut which goes from the branching point to infinity or by fixing
arg(t - z) in one or another way. Different choices of a cut, which fixes the branch
of the function (t - Z)l+OI in (22.4), and of the curve L gives different values of
j( 01) (z) in general.
We add finally that there exist various generalizations of these main approaches
I-IV - see e.g. S 22.3.

416
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
For the sake of simplicity we consider a to be real throughout this section,
although all the presentation is easily extended to the case when a is complex with
Rea f; O.
22.1. Definitions and the main properties of fractional integro-
differentiation in the complex plane
Let a function f(z) be defined in a certain domain G in the complex plane. We
shall adopt a direct extension (22.2) of the Riemann-Liouville integral as a basic
definition in our presentation, considering the others to be conjecturally derived
from (22.2). To make the integral in (22.2) exist for all z E G we consider the
domain G to be starlike relative to the point Zo. We have the multivalued function
(z - t)l-a in (22.2). We fix the point z and select the principal value of the
function (z - t)l-a to interpret the integral (22.2) uniquely. By this we mean
the following. Since the point t lies in the interval [zo, z], we can choose from all
possible values of the arg(z - t) a value which coincides with
arg(z - t) = arg(z - zo).
(22.5)
This, of course, requires us to fix arg(z - zo). Everywhere below we assume that
o  arg(z - zo) < 21r.
(22.6)
Then proceeding from (22.5) we have
(z - t)l-a = Iz _ tI1-Oei(1-a)arg(z-zo).
(22.7)
The integral
z
a 1 f f(t)dt
( l zo f)(z) = f(a) (z - tp-a '
Zo
a> 0,
(22.8)
with integration over the rectilinear segment [zo, z] and with principal value (22.7)
will be called a fractional integral of the function f(z) of order a. The condition
given in (22.6) means that we consider the fractional integral (l:Of)(z) in the
complex plane with the cut along the ray which is parallel to the real axis and goes
from Zo to the infinite point +00 + iIm Zo.
From (22.8) and (22.7) we may write down l:Of also as
. r
e,a", f .
(l:Of)(z) = f(a) pa-l f(zo + (r - p)e'''')dp
o
(22.9)

 22. FRACTIONAL INTEGRALS IN THE COMPLEX PLANE
417
with <p = arg(z - zo), r = Iz - zol. It is obvious that there is no ambiguity now,
concerned with the choice of a branch. If we denote f.(p) = f(zo + pe irp ), then
r
(l: o f)(z) = e iOirp r(l a ) 1 pOl-1 f.(r - p)dp.
o
(22.10)
So the fractional integral l:of is a Riemann-Liouville fractional integral (IC+f.)(r)
with respect to the radial variable r = Iz - zol up to the factor e ia arg(z-zo).
Since t = Zo + (z - zo) with 0    1, after the obvious change of variable,
we obtain from (22.8) the following equation
1
(If)(z) = (z »a 1(1- oa-1 1[(1- Ozo Hz],
o
(22.11 )
where
(z - zot = Iz - zoIOieioarg(z-zo). (22.12)
We have used the fact that [(z - zo)(1 - )]O-l = (z - zo)0I-1(1 - )0I-1. Let
us stress that, in general, (uv)0I-1 1= U Ol - 1 V Ol - 1 , the validity of equality being
dependent on the choice of a branch of the power function. However, if u > 0,
(ZU)0I-1 = ZOl-lu Ol - 1
(22.13)
whatever branch of the power function is chosen.
We note finally that the fractional integral (1:of)(z) is defined at any point
z E G (or almost everywhere) if f(z) is continuous (locally integrable).
We shall call the expression in (22.3) a fractional derivative of order a.
Similarly to (22.11) we may write in the case 0 < a < 1:
1
( Va f )( ) = 1 d [( _ ) 1-01 1 f«l - )Zo + z)  ]
Zo z r( 1 _ a) dz z Zo (1 _ )OI .
o
(22.14)
As usual we define V:ofl';oOl f in the case a < O.
Lemma 22.1. Let a function f(z) be locally integrable (continuous) in a domain
G. Then for almost all (for all) z E G the semigroup property holds
(l: o l: o )(z) = (1:/ fJ fHz), a  0, {3  O.
(22.15)

418
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
The proof is similar to that in the real variable case - Theorem 2.5 - and is
obtained by direct interchange of the order of integration in the left-hand side of
(22.15), which is possible by the Fubini theorem, and then by evaluation of the
integral
z
f dt ( IJ )( ) a+,6-1
(t _ ()l-a(z _ t)I-,6 = B 0:, fJ Z - ( .
,
The latter result is easily obtained after the change of variable t = (+ s(z - (),
(22.13) being taken into account.
Let us consider now fractional integration corresponding to the case 0/ the
infinite point zoo We denote T = e iS , -1r  8 < 1r, and introduce the operator
(22.16)
z
1 f /(t)dt
(1+,s/)(z) = f(o:) (z - t)l-a
ei'.oo
(22.17)
with integration over the ray which goes from infinity to the point z parallel to the
vector T = e iS and with the principal value of (z - t)l-a. Here Zo = e iS . 00 formally
and we would like to stress that the choice of 8 in the interval [-1r, 1r) corresponds to
the agreement in (22.6) on the choice of arg(z - zo) = arg(z - e iS . 00) = arg e i (s+1I')
in the interval [0, 21r).
Equation (22.17) assumes the domain G, where the function /(z) is given, to
be starlike relative to the point e iS .. This means that any point z belongs to
G together with the whole ray (z, e iS . 00). It is clear that such a domain is a
half-strip, parallel to the vector T = e iS . In particular, it may be a half-plane with
a curvilinear boundary and containing the infinite point e iS . 00. The function /(z)
must have "good" behaviour at infinity to guarantee convergence of the integral.
We define also a "right-hand sided" fractional integral
e" '00
a 1 f /(t)dt
(1_,,/)(z) = f(o:) (t - z)l-a
z
(22.18)
with integration from the point z along the ray (z, e iS . 00) and with the power
function taking its principal value.
It is easily seen that
00
( fa / )( z ) = e i (S+1I')a  f f(z + pe iS ) dp
+,S f( 0:) pI-a '
o
(22.19)

 22. FRACTIONAL INTEGRALS IN THE COMPLEX PLANE
419
00
(I I)(z) = e i8a  1 I(z + pe i8 ) dp (22.20)
,8 r( a) pi-a
o
under the afore mentioned choice of the branch of the power function. We see from
(22.19) and (22.20) that 1-,81 == e ia ,.. 1t..81. Nevertheless, it may be of benefit to
use both constructions, because the operators 1t.,8 and 1.8 can be considered for
different values of 8, which implies different domains of definitions for a function
I(z), in general.
Similarly to (23.3) we define operations
(1)1,,f)(z) = ( :i: :. r (1;:; a f)(z), m = [Rea] + 1, Rea > O.
(22.21 )
Now we single out the case T = =Fl, i.e. 8 = -1r and 8 = O. Then in (22.17) and
(22.18) we integrate along the ray which is parallel to the real axis. In this case,
preserving the designation (5.2), we shall write It..-,..I = Ii.! and I.ol = I/, so
that
00
(101 1Hz) =  1 I(z =F p) dp,
% r(a) pi-a
o
according to (22.19) and (22.20). The corresponding fractional derivatives have the
form
(22.22)
(1)%/)(z) = ( :i: :. r (J-a f)(z)
due to (22.21). Fractional integra-differentiation ('D%/)(z) implies that a function
I(z) is defined in a horizontal half-strip with a curvilinear "vertical" boundary, i.e.
left-hand or right-hand respectively for the signs + and -.
In the case 0 < Rea < 1 the fractional derivative in (22.21) may be represented
in Marchaud form as
00
a _ ae- i (8+,..)a 1 I(z) - I(z + pe i8 )
(1)+,8/)(z) - r(1 _ a) pl+a dp
o
(following the same lines as in S 5, see (5.56) and (5.57». One may analogously
write ('D (/)(z) also. In the case Rea  1 we have to use considerations of the
type (5.80):
-i(8+,..)a 1 00 ( A' . I)( z )
('Dol 1)( z) = e pea. dp,
+,' x( a, I) pl+a
o
where (Ae../)(z) is a finite difference (5.72) with step pe i8 , the constant x(a,l)
being given by (5.81). The passage from (22.21) to (22.21') may be achieved in
(22.21')

420
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
the case of sufficiently good functions in a similar fashion to the transformations in
S 5.5. We would like to emphasize that the fractional derivative may exist in the
form (22.21') in the case when it does not exist in the form (22.21), as has already
been seen in the real variable situation - S 5. Hence the domain of definition of the
operator generated by the right-hand side of (22.21') is in general larger than that
of the operator (22.21). Thus it might be expedient to denote it with a different
designation (D+,sf)(z), as in (5.80).
22.2. Fractional integro-differentiation of analytic functions
Riemann-Liouville fractional integra-differentiation (22.8) and (22.3) of an analytic
function gives an analytic function again only in the case of integer a, yielding
functions which have Zo as a branch point in the case of non-integer a (this is
immediately seen from (22.11) and (22.14». Thus in desiring to preserve analyticity
we have to modify the definition of fractional integra-differentiation so that it does
not yield branching. We might, for example, give up the factor (z - zo)a in (22.11)
or (z - zo)-a in (22.14). This is equivalent to consideration of (z - zo)-a l:Of and
1':o[(z - zo)a f(z)] instead of l:Of and 1':of respectively. Very often this way is
used in consideration of (generalized) fractional integra-differentiation of analytic
functions, see the next subsection. Now in preserving the initial definitions (22.8)
and (22.3), we, on the contrary, enlarge the set of functions in hand by allowing
them to have Zo as a branching point from the very ou tset. More precisely 1 we
assume that the function f(z) has the form fez) = (z - zo)#'g(z), P. E R 1 , where
g(z) is analytic in some neighbourhood of the point ZOo Then
00
f(z) = L CI:(z - zo)I:+#',
1:=0
g(1:)(zo)
Ck = k! .
(22.23)
Any branching that arises here will be removed by the choice of the principal
value such that (z - zo)#' is analytic in the complex plane with a cut along the ray
(ZOI Zo + 00) and
(z - zo)#' = Iz - Zo I#'e i #' arg(z-zo), 0  arg(z - zo) < 21r.
Lemma 22.2. Let f(z) be a function as given in (22.23). Then for all a E RI
(V a f)( ) _ ( ) #'-a r(k+p.+l) ( ) 1:
Zo Z - Z - Zo L..J r(k 1) CI: Z - Zo ,
1:=0 + JL - a +
p. # -1,-2,-3,...,
(22.24)

 22. FRACTIONAL INTEGRALS IN THE COMPLEX PLANE
421
the radii of convergence of the series (22.23) and (22.24) being coincident with each
other.
A similar statement was earlier proved in the case of analytic functions
on the real axis (see Lemma 15.4). Lemma 22.2 is proved analogously if we
take into account the possibility of integrating and differentiating (22.23) term
by term and using the formula Vo[(z - zo)P] = rrJ:l) (z - zo)p-a. The
latter is established directly in view of (22.16). Equation (22.24) connects the
initial definitions of fractional integra-differentiation given in (22.2) and (22.3)
with Hadamard's approach. As for Liouville's approach, it is connected with
the fractional integra-differentiation defined in (22.17), (22.18) and (22.21). The
formula
V+,Se OZ = aaeoz if Re(ae is ) < 0
(22.25)
is derived immediately from (22.21) and (22.19) in view of (7.5). The right-hand
side of (22.25) depends in fact on 8: the value of () influences the choice of branch
for aa. Namely, in (22.25) aa denotes aa = lalaeiaArgo, 0 < 8 + Arga < 21r
(which in fact may be specified as 1r/2 < 8 + Arga < 31r/2 due to the condition
Re ( ae iS ) < 0). In particular
V+e OZ = aaeoz, if Rea > 0,
(22.26)
(22.27)
VeOZ = (_a)Qe OZ , if Rea < 0,
with the principal value of (:i:a)a. It follows from (22.26) and (22.27) that functions
00
f(z) representable by Dirichlet series f(z) = L: akeAkZ, where all ReAl: > 0 or all
k=1
ReAl: < 0, admit the formulae
00
(V+f)(z) = L aI:Ake>'k%, ReAl: > 0,
1:=1
00
(Vf)(z) = Lal:(-AI:)ae>'kz, ReAl: < 0
1:=1
00
(Liouville wrote (va f)(z) = L: al:Ake>'k z by definition, whatever AI: were).
1:=1
We now pass to one of the most important questions in the theory of fractional
differentiation of analytic functions: that is the extension of the Cauchy formula
fCn)(z) =  f f(t)dt
21ri (t - z)n+1
J;

422
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
to non-integer values of n. Here the multi valued function (t - z)-a-l arises. To
single out its one-valued branch we cut the plane by the ray which passes from z
through Zo to infinity (we recall that we are now considering functions of the form
(22.23» and deal with the single-valued function (z being fixed)
(t - z)-a-l = It _ zl-a-le-(l+a)arg(t-z)
(22.28)
in the plane with such a cut. It is assumed that the principal value of arg(t - z)
is chosen, i.e. arg(t - z) = 0, if t - z > O. Since the cut may prove to be parallel
to the real axis and lie to the right of the point z, we specify this choice by the
condition
arg(t - z)ltEc+ E (-21r, 0],
(22.29)
where C+ is the edge of the cut, indicated at Fig. 1.
Fig. 1. Contour of integration
Theorem 22.1. Let j(z) = (z - zo)#Jg(z), where p. > -1, g(z) is analytic in a
domain G and the principal value of (z - zo)#J is chosen. Then
(va j)(z) = r(l + a) f j(t)dt
Zo 21ri (t - z)l+a
£.0
(22.30)
for all a E R 1 , except a = -1, -2, .. ., where the principal value of the function
(t - z)-a-l is fixed by (22.28) and (22.29), while the closed contour £zo lying in
the domain G is anyone that passes through the point Zo and goes round the point
z in positive direction. If, in particular, we choose the circle It - zl = Iz - zol as
£zo, we have
211'
r ( l + a ) f . 8 . 8
(va )(Z) = e- ,a j(Z + Iz - zole ' )dO.
Zo 21rlz - Zo la
o
(22.30')

 22. FRACTIONAL INTEGRALS IN THE COMPLEX PLANE
423
Proof. Let a < 0 first. Since the integrand in (22.30) is analytic in the domain G
with the cut along the ray from z to zo, then by the Cauchy integral theorem we
have
r(1 + a) f f(t)dt r(1 + a) [ . f . f . f ]
2' (t)1 + - 2' hm +... + hm +... + hm ...,
11'1 - Z 01 11'1 'y+-c+ 'Y--C- 'Y-{z}
/:'.0 'Y + 'Y - 'Y
where 1': are straight lines parallel to the edges of the cut and l' is a circle of
radius € enveloping the point z (see Fig. 1). So by the jump of (t - Z)-l-OI at the
cut we have
z
r(l + a) f f(t)dt = r(1 + a) (1 _ -2011ri) f f(t)dt
211'i (t - z)1+01 211'i e (t - Z)1+OI
/:'.0 Zo
(22.31 )
2,..
r ( 1 + a ) f .,
+ . lim€-OI f(z+€e"P)e-10I'Pdcp.
21r1 £-0
o
With condition (22.29) we see that (t - Z)-l-OI = e-(l+01),..i(z - t)-l-OI, where
(z - t)-l-OI coincides with the value chosen in (22.7) with 1 - a being replaced by
-1 - a. Therefore from (22.31) we obtain
r(1 + a) f f(t)dt = (1-01 f)(z),
211'; (t - z )1+01 Zo
/:, .0
(22.32)
which proves (22.30) for a < O. Starting from (22.3) if a > 0 we arrive at (22.30)
after differentiating (22.32), with a - [a] - 1 being written in place of a, the
corresponding number of times under the integral sign. .
We remark that the right-hand side of (22.30) is often adopted as an initial
definition of a fractional derivative of any order a, a 1= -1, -2.. ..
Remark 22.1. It is convenient to represent (22.30) as
( 'DOl f)( ) = r(1 + a) f f(t)dt
Zo z 2 . ( ) 1+01
11'1 1 _ Z-Zo ( t _ z ) 1+01
/:'.0 t-zo 0
the contour /:'zo being the same. The convenience is in fact that we deal with
standard cuts: it is assumed that the principal value of the function (1 - W)l+OI is

424
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
chosen, considered in the plane cut along the rays (-00,-1) and (1,+00), while
the principal value of (t - zo)l+a is treated in the plane cut along the ray from z
through Zo to infinity. If, in particular Zo = 0, we have
( Va f )( z) = r(1 + a) f f(t) dt
o 21ri (1_!.)l+ at l+a'
£0 t
(22.33)
where [,0 goes around z passing through the origin.

C
Fig. 2. Pochhammer loop
Remark 22.2. The restriction p. > -1 in Theorem 22.1 may be replaced by
p. f; 0, :i:l, :i:2,... if one takes a so-called Pochhammer loop C = (z+, zo+, Z-, zo-)
instead of the contour [,zo and changes the coefficient before the integral, thus
a _ e- IJ1ri r(1 + a) f f(t)
(1J zo f)(z) - 41r sin P.1r (t - z)l+a '
C
f(t) = (t - zo)IJg(t).
(22.33')
- Lavoie, Tremblay and Osler [1].
We shall also discuss (22.30) corresponding to the infinite initial point
Zo = e iS . 00, -1r  (J < 1r. Let a function f(z) be analytic in a curvilinear half-plane
Gs containing the infinite point e iS . 00. Similarly to Theorem 22.1 we obtain for
all a E Rl, a f; -1, -2,..., the equation
( Va f )( z ) = r(1 + a) f f(t)dt
+,S 21ri (t - z)l+a'
£,
(22.33")
with any Hankel contour [,s = [,,(z), which envelopes the ray (z, e iS .00) in the
positive direction (Fig. 3). It is assumed that the principal value of the function
(t - z)l+a = It - zll+aei(l+a)arg(t-z), analytic in the plane cut along the ray
(z, e iS . 00) is fixed by the condition arg(t - z )ltEc+ E (-21r, 0] so that
{ (J,
arg(t - Z)ltEC+ =
(J - 21r,
-1r  (J  0,
o < (J < 21r.

 22. FRACTIONAL INTEGRALS IN THE COMPLEX PLANE
425
Fig. 3. Hankel contour
In conclusion of this subsection we give without proof the Hardy-Littlewood
theorem of the mapping properties of fractional integration
z
( 101 f)( z ) -  f f(t)dt
o - r(Q) (Z - t)1-OI
o
1
ZOI f
= r(Q) (1 - )OI-l f(z)
o
(22.34)
in Hardy spaces Hp. We recall that the Hardy space Hp, 0 < p < 00, consists of
functions f( z) analytic in the unit disc and such that
211'
IIfll p = sup f If(reitp)IPd<p < 00.
r>O
o
Theorem 22.2. Let 0 < p < 00, 0 < Q < IIp, 'Y > -1. The operator
1
Z-"(-OI 10 Z7 f(z) =  f 7(1- )OI-l f(z)
r(Q)
o
is bounded from Hp into H" l/q = IIp - Q.
The prool may be found in the original papers by Hardy and Littlewood [5], [7]
or in the book by Zygmund [6, p.209] - compare Theorem 22.2 with Theorem 3.5
and 3.7.

426
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
22.3. Generalization of fractional integro-differentiation of
analytic functions
Let the function
00
f(z) = Lh:zl:
1:=1
(22.35)
be analytic in the unit disc so that by (22.24)
( OI f)( ) -01  r( k + 1) , I:
V o Z =z r(k_a+l)Jl:z.
(22.36)
A natural way to generalize the (22.36) is to replace the factor r(k + 1 )/r(k - a + 1)
in (22.36) by a more general one. We begin with the generalization which is known
as generalized Gel10nd-Leont'ev differentiation. Starting from the simple fact that
usual differentiation d/dz corresponds to the factor r(k + 1)/r(k) in (22.36), we
introduce, following Gel'fond and Leont'ev [1], the operation
00
Vn(a; f) = E al:- n fl:zl:-n,
al:
I:=n
(22.37)
00
where a(z) = L: al:zl:. The function a(z) is assumed to be entire of order p and of
1:=0
type uf;O (as regards these ideas, see for example the book of Leont'ev [1]). Let
also
lim k 1 / p  = (uep)l/'P.
1:-00
(22.38)
We remark that (22.38) is always satisfied in the case of the upper limit
lim l:_oo - Leont'ev [1, p.13]. From (22.38) we see that there exists the limit
lim k- v" lal:_n/al:l = 1, so the series in (22.37) has the same radius of convergence
1:-00
as (22.35). The operator in (22.37) is called the Gel10nd-Leont 'ev operator of
generalized differentiation. It is obvious that Vn(a; f) = cr f /dz n in the case
a( z) = e Z .
The operator
00
In(a; f) = L ak+n fkZI:+n,
k=O ak
(22.39)
which is the right inverse to (22.37), will be called the Gel10nd-Leont 'ev operator
of generalized integration.
We obverse that although operators v n and In are introduced as direct
generalizations concerning integer order n of integra-differentiation, they contain

 22. FRACTIONAL INTEGRALS IN THE COMPLEX PLANE
427
that for fractional order as well. To show this, let us consider the following special
case:
00 1c
a(z) = Ea(Z) =  r(,, + 1) ' ,,> 0,
(22.40)
Le. the case of Mittag-Leffler function (1.90). It has order p = 1/0:, type u = 1 and
satisfies (22.38). The corresponding operator for generalized integration of order
n=lis
( rr 1)( )  I l (E . I) =  r(ok + 1) , 1c+l
va Z 1/01' L...Jr ( o:k+O:+l ) J1cz.
1c=O
(22.41 )
Lemma 22.3. The operator of generalized integration defined in (22.41) admits
the following integral representation
1
(JOII)(z) = r(lo:) 1 (1 - t)OI-l f(ztOl)dt.
o
(22.42)
The proof of the lemma becomes clear if we observe that
1
r(o:k + 1) = B(o:, o:k + 1) =  1 (1 _ t)0I- 1 t Ol1c dt.
r(o:k + 0: + 1) r(o:) r(o)
o
Lemma 22.3 allows us to extend the definition of the integration type operator
(22.41) from analytic functions I(z) to continuous functions (and even integrable
ones), given in a domain starlike in respect to the origin.
We also observe that
1
1 1 01-1 t-1
(JOII)(z) = r(o:) (z: - t) f(t)7 dt ,
o
where we have integration along the rectilinear interval connecting points 0 and z,
the principal values of multivalued fanctions being chosen in a proper way. Thus
the operator JOI may be also interpreted as a fractional integral operator of order
0: of a function I(z) by the function g(z) = ZI/OI - see S 18.2. Based on (18.29), we

428
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
can construct the operator VOl which is the (left) inverse to .101:
z
lid f I(t)g'(t)dt
(VOI/)(z) = f(l- a) g'(z) dz [g(z) - g(t)]OI '
o
where g(z) = ZI/OI. Simple transformations lead to
1
1 ( 1 - a d ) f I( ztOl)dt
(VOI/)(z) = f(1 _ a)  + a dz (1 _ t)OI .
o
(22.43)
The operator VOl corresponds to the expansion
 f(ka + 1) 1:-1
(VOI/)(z) =  f(ka + 1 _ a) lk Z .
(22.44)
We can extend our generalizations further, replacing r(:21) in (22.36) by
an arbitrary sequence bl: satisfying certain assumptions. Thus, let the function
00
b(z) = L bkZI:
k=O
(22.45)
be analytic in the unit disc. Let us consider the operator
00
V{b; I} = b 0 1= L bklkZI:.
1:=0
(22.46)
This expression is known as the Hadamard product composition of functions b(z) and
I(z). Of course (22.46) is a very wide generalization of the integra-differentiation
notion. It will indeed generalize differentiation if bl: -+ 00 as k -+ 00. Equation
(22.46) is invertible if bl: f; 0, k = 0, 1,2, . .. and we shall denote
00
b.(z) = L zl: /b k .
k=O
(22.47)

 22. FRACTIONAL INTEGRALS IN THE COMPLEX PLANE
429
in this case. The corresponding operator
00 /,
I{b;f} = V{b-;f} = L b le Zle
Ie=O Ie
may be called a generalized integration under the assumption that b le -+ 00. We
observe that the functions b( z) and b- (z) are usually called associated (with each
other), - Smirnov and Lebedev [1, p.168].
Lemma 22.4. Let the series in (22.45) and (22.47) converge in the unit circle. If
g(z) = b 0 f, where f(z) is analytic for Izi < I, then
1 f ( z ) dt
g(z) = 21ri b t f(t)t'
Itl=r
(22.48)
1 f ( z ) dt
f(z) = 21ri b- t g(t)t'
Itl=r
Izi < r < 1.
(22.49)
Equations (22.48) and (22.49) are obtained by the expansion of f(t), g(t),
b(z/t) and b-(z/t) in corresponding series and by termwise integration.
Choosing various functions b(z), we shall obtain integro-differentiation
operations of various types. Let us consider the examples.
1. Let
b(z) = r(1 + a) =  r(1 + a + k) zle Izi < 1.
(1 - z)l+a f:'o r(1 + k) ,
(22.50)
00
Then V{b; f} = L: r¥(i1)1e) flezle, which coincides with
Ie=O
V{b; f} = vg[za f(z)]
(22.51)
by (22.24), or with
V{b;f} = r( Q) f f(t) dt
(1 - f) 1 +a t
Itl=r
(22.52)
by (22.48), cf. (22.30) and (22.33). Therefore, the choice (22.50) of the function b(z)
gives Riemann-Liouville fractional differentiation of the function za f(z). Equation

430
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
(22.51) allows us to obtain the operator I{b;f} inverse to V{b;f} in the form
1
I{b; f} = z-a(I1f f)(z) = f(l",) J (1- t)a-l f(zt)dt.
o
(22.53)
By (22.49) it may be also written as
1 J ( z ) dt
I {b; f} = 21ri I 1,1; 1 + a; t f(t)T'
Itl:r
(22.54)
with Izi < r < I, the hypergeometric function 2Fl(l,l; 1 +a; z) = b.(z) arising here
as the associated kernel.
If we took b(z) = ;.::  instead of (22.50), we would obtain
00 f( a + k)
V{b; f} = {; f(k) !kzk = zvg[za-I fez)].
(22.55)
This latter construction is sometimes called the Ruscheweyh fractional derivative.
00
2. Let b(z) = L: (ik)azk. Then
k:1
00
V{b; f} = L( ik)a fk Zk
k=1
(22.56)
is the fractional differentiation by Weyl: it coincides with Weyl differentiation
Va)f, - see (19.6) - of a function fez) = f(rei'P) with respect to the angular
variable <po
3. Finally, let us choose b(z) so that V{b; f} coincides with the generalized
integro-difJere ntiation
1
(L(w) fHz) = - J f(zt)w'(t)dt
o
(22.57)
of Dzherbashyan - see S 18.6. We observe that (22.57) may be regarded as the
result of applying the operator (18.110) to the function fez) = f(rei'P) with respect

 23. ADDITIONAL INFORMATION TO CHAPTER 4
431
to the radial variable r. Starting from (18.111)-(18.114) we introduce the function
00
b(z) = L Aw(k)zl:.
1:=0
Then it is obvious that V{b; f} = (L(w) f)(z).
As regards other generalizations, see subsection 23.2 (notes 22.3, 22.5, 22.8,
22.15, 22.16 and 22.18).
 23. Bibliographical Remarks and
Additional Information to Chapter 4
23.1. Historical notes
Notes to  18.1. The operators (18.5) and (18.6) were introduced by Kober [1] (1940). The
operators (18.1) and (18.3) with a = 0 and b = +00 were put forward by ErdeJ.yi [6] (1950),
see also [13], and the operators (18.2) and (18.4) in the fonn (18.13) and (18.14) were given by
Lowndes [2] (1971) cv > -Ii [6], cv > -no
The operators (18.8) are also known to be associated with the names of Erdelyi and Kober;
it appears that Sneddon was the first who named them in this way. We note, however, that the
operator
:e
Vi f t2'1+1 f(t)dt
(AP f)(x) = reI + p)r(1/2 _ p) (x2 _ t 2 )p+1/2 '
o
(1/2) :e 'JQ+'J"
such that (A -01 J)(x) = 2r ( 3/2-0I) 1'1.OIf(x) is also known as Sonine's operator, see
commentaries in Sonine's book [6, p.208] and Levitan [1, p.129] (1951). In the latter paper the
operator A (p) was considered as transfonning trigonometric functions into Bessel functions in
connection with expansions in Bessel functions series. We must add that the idea connected with
a fractional differentiation "by the function x 2 " or "by the function ...;x" may already be found
in the papers by Liouville as for example, [1, p.lO] (1832). .
Equations (18.15), (18.16) and (18.18) for the operators (18.5) and (18.6) were obtained
by Kober [1] (1940). Relations (18.15) and (18.18) for the operators (18.1) and (18.3) with
a = 0 and b = +00 were given by Erdelyi [6] (1950), (18.17) are due to Lowndes [2] (1971), [6]
(1980). Equations (18.16') and (18.41') with a = 0 and b = +00 were established by McBride
[10, p.243-244] (1984) who gave conditions for their validity in the space F p ,,.,, 1  P  00. For
the latter, see subsection 8.4. The case q = 1, a = 0 in the first relation in the (18.16') can be
found in Buschman [5] (1964). In the case q = 1 the representations (18.20) and (18.22) were
obtained by ErdeJ.yi [4] (1940) for the truncated modified Hankel and Kober transforms. These,
however, were more general, than (18.19), (18.5) and (18.6) (see  23.2 (notes 18.2 and 18.3». In
the case q = 2 (18.15)-(18.23) for the operators (18.8) were given in Sneddon [2] (1962).
Notes to  18.2. The fractional integral of a function by another function is an idea
already known to mathematicians of the past century. It was in fact introduced by Hohngren
[1, p.l0] (1866), although indications of such an idea were in embryo suggested by Liouville
(7) (1835). Hohngren's paper contained a detailed investigation of compositions of the form
D:(:e)h (x)D:(:e) ... D::(:e/n(x)u(x), where fj (x) denoted the operation of multiplication

432
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
by the fWlction /j(x), and n;:(:I:) was the fractional differentiation "by a function 8j(x)".
The composition of operators was applied to u(x). The modem papers, where the notion of
differentiation of a fWlction by another one appeared again, are those of Erdelyi [9], (1964),
Talenti [1] (1965) and also Erdelyi [14] (1970). The case of integration of integer order was treated
by Shelkovnikov [1] (1951). Some simple properties of fractional integrals of a fWlction by another
were coIlBidered by Chrysovergis [1] (1971). We also note that such integrals can be fOWld in an
implicit form in Sewell [2, ch. 3, s. 14] (1937) in the proof of the invariance property of fractional
differentiability of functioIlB given on curves Wlder conformal mappings.
Fractional integrals of a fWlction by another in the complex plane were investigated by
Osler [1], [2] (1970), [5], [7] (1972). Fractional differentiation of a function by another in
Griinwald-Letnikov form (which was coIlBidered in  20.4) was studied by Krasnov [2] (1977).
Theorem 18.1 was proved by Erdelyi [9] (1964) the proof, however, not being complete. The
represent ability by a fractional integral was not justified.
Notes to  18.3. Fractional integra-differentiation (18.42) was introduced by Hadamard
[1] (1892). Modification of Hadamard fractional differentiation by the Marchaud-type form
(18.56)-(18.58) was not introduced elsewhere.
Notes to  18.4. Bessel fractional integration (18.61) has been known for a long time.
It was widely investigated in the papers by Aronzajn and Smith [1] (19tH), Aronzajn, Milla
and Szeptycki [1] (1963) and Aronzajn [1] (1965) which were devoted to multidimensional Bessel
potentials. Fractional integra-differentiation (E  1')0, considered in  18.4, was known, as for
example, in Livennan's book [1, p.28-31] (1964) where it may be found in the fonn (18.71)
adapted to the case of the half-axis. The forms (E  D)O of this operator, used here, are
contained in Karapetyants and Samko [1] (1975). The relation (18.73) was proved there as well.
In this paper the operators (E  D)O were used to investigate the normal solvability of singular
integral equations by convolutions (having discontinuous symbol). We note also the paper by
Karapetyants [1] (1977), where the operators (E D)O were applied to solve Wiener-Hopf integral
equations (with a symbol which has a vanishing of fractional order.)
The statement of Theorem 18.2 has to be considered as known, although it was not
apparently given elsewhere in an obvious form. The statement (18.78) of Theorem 18.3 on the
coincidence of H-,P([a, b]) with [O[Lp(a, b)] in the case 1 < p < 1/01 is also known, being an
immediate corollary of results by Rubin [1] (1972) on the continuation and restriction of fractional
integrals (Theorems 13.9 and 13.10 in this book). In its direct fonn Theorem 18.3 is in the fact
contained in the paper by Biacino [2, Theorem 2.1] (1984).
Notes to  18.5. The constructions of fractional integration which are considered here
were introduced by Chen [2] (1961» where Lemma 18.1 and Theorem 18.4 were also proved. Here
we have given proofs different from those in Chen [2]. We also remark that there is an error in
the proof of Theorem 18.4 in this paper.
Notes to  18.6. The constructions which are under consideration were introduced by
Dzherbashyan [4] (1967), [5] (1968).
Notes to  19.1-19.3. The definition of fractional differentiation of periodic functions,
considered in Section 19, is due to Weyl [1] (1917). A beautiful presentation of the main results
for Weyl fractional integra-differentiation can be found in the book by Zygmund [6, Ch. XII,  8,
9]. We have used the ideas of this book while elucidating some results in Section 19. Fractional
integra-differentiation of almost periodic functions was studied in the papers by Love [1] (1938),
Nagy [2] (1939), Takahashi [1] (1940) and Bang [1] (1941). The first of these papers dealt also
with arbitrary bounded functions given on the whole line. - see  9.2 (note 5.3).
The representation (19.9) of the function (I+, (x) via the Riemann zeta-function was observed
by MikoIas [1] (1959). The connection with the zeta-function was noted in different way by Hardy
[2] (1922).
The representation of the function (I+(x) in the form (19.11) is due to Weyl [1, p.300] (1917),
the proof of Lemma 19.1 follows the presentation in the book by Zygmund [6, Ch. 12,  8]. Some
properties of the Weyl kernel (I+(x), which are of importance in the theory of approximation by
trigonometric polynomials, were obtained by Nagy [1] (1938), and see also Dzyadyk [1] (1953).
The construction (19.18) has not been dealt with before. The assertion (19.19) was observed by

 23. ADDITIONAL INFORMATION TO CHAPTER 4
433
Weyl [1, p.300] (1917). The representation of the Weyl fractional integral in the fonn (19.21) was
suggested by Mikolu [I, p.80] (1959).
Relations (19.24) and (19.25) have apparently not been noted elsewhere. The operators OI)
(convolutions with the kernel KOI,#J(x» were widely used in the approximation theory of periodic
functions, as in the papers by Nikol'skii, Efimov and Telyakovskii, cited in  23.2 (note 19.6).
Notes to  19.4. A formal coincidence (19.35) of the Weyl fractional derivative with the
Marcha.ud derivative can be in fact found in Weyl [I, p.301-302] (1917). Lemma 19.4 is new in a
sense.
Notes to  19.5. Theorem 19.2 was proved in Samko [33] (1985). The equivalence of
(19.46) and (19.47) in Theorem 19.3 was shown by Butzer and Westphal [I, p.129] (1975).
Notes to  19.6. The estimates (19.48) and (19.49) and Theorem 19.6 were proved by
MW'Claev [2] (1985). The statement of the Corollary of Theorem 19.6 in the case .x + 01 < 1 was
obtained by Hardy and Littlewood [3, p.589] (1928), where the statement of Remark 19.4 was
alao given; the case 01 + .x = 1 is due to Zygmund [2, p.53] (1945). Theorems 19.7 and 19.8 were
proved by Murdaev [2] (1985). The statement of the Corollary of Theorem 19.7 and the assertion
(19.62) are due to Hardy and Littlewood [3, p.576 and 591] (1928), see also [2] (1926).
Notes to  19.7. The results presented here, except the isomorphism (19.71), are due to
Hardy and Littlewood [3, p.592-604] (1928).
Notes to  19.8. An extension of the Bernstein inequality for trigonometric polynomials to
fractional derivatives in 0([0, 21r])-nonn was given firstly by Civin [1] (1940), [2] (1941). In fact
in other terms it can be found in Sewell [2, p.lll] (1937). We remark that a similar inequality
for fractional derivatives of algebraic polynomials, given on a finite interval was in fact obtained
by Montel [I, p.170] (1918). An extension to the case of Lp(O, 211")-nonn, 1  P  00 is due to
Ogievetskii [3], [4] (1958). We note that Civin considered a more general case of entire functions
of exponential type, but the constant in the inequality was rough. The sharp constant equal to
1, as in the classical Bernstein inequality, was obtained by Lizorkin [3] (1965) for all 01  1 in the
case of entire functions of exponential type as well. The simple proof for 0 < 01 < 1 presented in
Theorem 19.10 yields the constant given in (19.74). This proof follows Geisberg [2] (1967). In
a more general form (19.81), the Bernstein inequality for fractional derivatives was obtained in
another way by Bang [1, p.21-22] (1941). Wilmes [I], [2] obtained the Bernstein inequality (for
functions of exponential type) with the sharper constant 21-01,0 < 01 < 1.
The Favard type inequality (19.82) for fractional integrals of periodic functions lex) =
00
L: ai eib was obtained first by Nagy [I, p.123] (1938).
k=m
Notes to i 20.1. Relations representing fractional differentiation as a limit of a difference
quotient, appeared first in Liouville [2, p.l07-110] (1832), It was used by Liouville in [6, p.224]
(1835) to formally derive the Fourier expression for fractional differentiation by his approach, and
alao in [2, p.136] (1832) to evaluate fractional derivatives of the functions e Clx sin bx and e Clx cos bx.
These cases were not developed in Liouville's papers. In 1867 Griinwald's paper [1] appeared in
which an approach to fractional integro-differentiation via (20.7) or, more precisely, via (20.42) was
developed. However, his arguments were not quite strict. The correct construction of a complete
theory of fractional calculus on the basis of this approach appeared a year later in the paper by
Letnikov [1] (1868). To illustrate the independence of Letnikov's investigation of Griinwald's work
we cite here Letnikov's contemporary Sludskii [1] (1889). He wrote: "A.V. Letnikov's research
was to serve as a dissertation for a Master's degree. It was already almost completed when there
appeared in one of the last issues of Schlomilch jOW'T1al in Moscow the paper by Dr. Grunwald.
Letnikov, to his great surprise, found in this paper results which he had obtained by rather
different methods. For this reason he at once came to decision not to present his work for his
Master's degree, but also not even to publish it. Only due to the insistence of A.Yu. Davydov,
who undertook to declare at the viva that the most important results of Letnikov's work were
obtained before his acquaintance with Griinwald's paper, was this decision repealed".
We add that certain discussions of the difference quotient approach may be also found in
Most [1] (1871).

434
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
It should be noted that both Grunwald and Letnikov dealt with researches in the complex
plane, taking an increment h in (20.7) to be in fact complex with a fixed direction defined by the
initial point a, see (20.42). Their approach fell into disuse for a long time, since Riemann-Liouville
forms of a fractional integro-differentiation were considered as evidently more preferable. Many
years later there appeal"ed the papers by FelTar [1] (1928), Stuloff [1] (1951), Moppert [1] (1953),
Mikolas [4] (1963), [5] (1964), where the Griinwald-Letnikov approach was presented from a more
modem point of view. Examples were considered and connections with other forms were retraced
and so on. We add that Letnikov's approach was also presented by Ju.L. Rabinovich [1]. This
approach found a new life with the publication of the papers by Westphal [1] (1974), and Butzer
and Westphal [1] (1975), where it was interpreted from the point of view of modem function
theory, and where various classical questions of fractional calculus were connected with modem
problems of fractional analysis and function theory. Further development can be found in the
papers by Butzer, Dyckhoff, Gorlich and Stens [1] (1977), Bugrov [1] (1985), [2] (1986), Samko
[33] (1985), [34] (1990). We note finally that the approach via fractional differences within the
framework of generalized functions was developed by Bredimas [1]-[5] (1973-1976).
The representation (20.3') for the constant (20.3) and the identity C(a) = 2 for 0 < a < 1
were noted by Burenkovand Sobnak [1] (1985).
The generalized differentiation a(V) as in (20.10), was introduced by Post [2, p.726] (1930).
We note that the generalized differentiation Va log V or <p(V) log V was investigated by Davis
[4, p.7S-85] (1936) on the basis of another approach.
Notes to  20.2. We used here constructions from the papers by Westphal [3], [4] (1974),
and also Butzer and Westphal [1] (1975). In particular, the functions (20.14) and (20.15) were
introduced there. Lemma 20.1 was proved by Westphal [3, p.560], see also Butzer and Westphal
[I, p.127-128]. We have given another proof of the first statement in this lenuna, based on
the technique of the Wiener ring of absolutely integrable Fourier integrals. We note that this
statement was known to Bosanquet [5] (1945), see Westphal [4, p.562]. Lenuna 20.2 and Theorem
20.1 were obtained by Butzer and Westphal [1, p.128-129]. Theorem 20.2 is new. Theorem 20.3
was proved by Butzer and Westphal [I, p.I33].
Notes to  20.3. Theorem 20.4 was proved by Samko [33] (1985), [34] (1990), cf. also [35].
A new version of Theorem 20.4 is contained in Westphal [4, p.568] (1974) in the general context
of fractional powers of operators, but under the restriction that functions and their fractional
derivatives belong to the same space - see  23.2 (note 20.1). In the case of an infinite axis such
an assumption is essentially restrictive. Theorems 20.5 and 20.5' are analogues of Theorem 20.3
and have not been noted elsewhere; a close version of Theorem 20.5 (with r = p and 1 < 'P < 00)
can be found in Bugrov [I, p.62, 64] (1985).
Notes to  20.4. Definitions (20.42) and (20.46) go back to Griinwald and Letnikov. The
coincidence (20.47) of the integral (20.46) with the Riemann-Liouville integral had already been
shown by Griinwald [I, p.455-458] (1867), and Letnikov [1, p.19] (1868) in the case of sufficiently
good functions. Theorem 20.6 is new.
Notes to  21.1. Theorem 21.1 for the operators I:f with a power-logarithmic kernel
was proved by Kilbas [1] (1975) for integer {3 and'Y = 1. In the general case this theorem was
obtained by Kilbas also, the proof was not published earlier.
Notes to  21.2. Theorems 21.2 and 21.3 were obtained by Kilbas [6] (1978) for integer (3
and 'Y = 1 in the case of the weight p(x) = (x - a)#' with 0 $ #' < .x + 1 or the general weight
(3.12) with 0  J.l.l < .x + I, .x + a < J.l.1c < .x + 1, k = 2,3,... ,n. These theorems in the general
case were proved by Kilbas, and were not published earlier.
Notes to  21.3 and 21.4. The results presented here are new being obtained by Kilbas.
The generalized power-logarithmic Holderian property of purely logarithmic integrals
:&
(1f <p)(x) = f In,6 x  t <p(t)dt, <p E Lp(a, b), {3 > 0,
a
observed in Remark 21.3, was shown by Kilbas and Samko [1] (1978) in the case p > l/a.

 23. ADDITIONAL INFORMATION TO CHAPTER 4
435
Notes to  21.5. The asymptotic expansion of the integral/g: <p with an integer power of
logarithm given in Theorem 21.10, follows Kilbas [8] (1982), where besides the case 0 < {3 < 1 in
the asymptotics (21.27), the cases {3 = 1 and {3 > 1 were considered. We note that in the cited
paper there is a misunderstanding: instead of the integral
:&
f (x - t)a- 1 ln ll (x - t)<p(t)dt, 0 < a < I, -1 < v < +00,
o
which may have complex values, the integral
:&
f (x - t)a-1Iln(x - t)11I <p(t)dt
o
must be considered.
Notes to  22.1 and 22.2. Fractional calculus from the very first was considered in the
complex plane. It is sufficient to refer, for example, to the papers by Liouville [1]-[8] (1832-1837)
with the starting definition of fractional differentiation of functions representable by series of
exponential functions, or by Griinwald [1] (1867), Letnikov [1] (1868) and Sonine [2] (1872). The
approach connected with the name of Hadamard goes back to his paper [1] (1892).
Weyl fractional differentiation (22.1) (approach II) in the complex plane first appeared
in Hardy and Littlewood [5] (1932). Approach III, i.e. Riemann-Liouville fractional integro-
differentiation (22.2)-(22.3) with the integration along the rectilinear interval [ZO, z], can be fOWld
in the first papers on fractional calculus. Relation (22.11) is already contained in the paper by
Hohngren [I, p.l] (1865-1866). Griinwald defined fractional integra-differentiation via fractional
differences (see Section 20) in the complex plane and showed the reducibility of this definition to
that of Riemann-Liouville.
Riemann-Liouville fractional integra-differentiation with the integration along a CW'Ve in
the complex plane was investigated in detail by Sewell [1] (1935), [2] (1937) in connection with
questions of polynomial approximation in the complex plane - see  23.2 (note 22.1).
As for approach IV, we note that the generalized Cauchy fonnula (22.4) appeared first in the
paper by Sonine [2] (1872) who in particular proved the coincidence (22.32) of Cauchy fractional
derivative with the fractional integral/a in the case a < O. This fonnula was treated later by
Laurent [1] (1884), Nekrasov [I, p.87] (1888) and Krug [1] (1890), and was effectively used by
Sint80V [1] (1891), to study Bernoullian functions <Pp,8 = D [( e:l ) 'P (e:&z -1)] Z=o with a
non-integer.. The Cauchy-Sonine approach was used by Montel [I, p.167] (1914) and Hardy and
Littlewood [5] (1932). Relation (22.4) was used in Blumenthal [I, p.490] (1931), where fractional
integration was treated from the point of view of composition theory developed by Volterra.
Construction (22.30) was firstly given by Letnikov [3, p.428] (1872).
Fractional integra-differentiation as given in (22.17), (22.18) and (22.21) was apparently not
considered elsewhere except for the cases 1+',_" ,1)+,_" and 1+,,0,1)+,0 which can be fOWld in
the papers by Nishimoto [1]-[5], [7] and others (1976-1984), see also Owa and Nishimoto [1]
(1982).
Theorem 22.2 is due to Hardy and Littlewood [5] (1932), [7] (1941).
Notes to  22.3. The representation of the fractional integral/ in the fonn (22.34) was
used by Hadamard [1] (1892). This served to him together with the integrals (18.44) as the
1
starting point to suggest consideration of more general constructions J V(t)j(zt)dt. However,
o
this idea was not achieved by him in any completed fonn. The realization of this idea with
sufficiently rich content was given by Dzherbashyan [4] (1967), [5] (1968).

436
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
The generalized integra-differentiation of Gel'fond-Leont'ev goes back to the paper by
Gel'fond and Leont'ev [1] (1951). There is a wide literature containing generalizations and
development of ideas connected with this theory. We refer in this connection to the papers
by Korobeinik [1], [2] (1964), [3] (1965), [4] (1983), where the idea of generalized integro-
differentiation of analytic functions was developed in the most general form. Korobeinik [4]
alao introduced and investigated generalized differentiation and integration operators 'Dc and J c
defined on an arbitrary number set C = {Ca}aeM, where M is not necessarily a countable set.
We note the paper by Nagnibida [1] (1966), which contains the investigation of certain questions
connected with Gel'fond-Leont'ev operators. The operator (22.41) was noted as an example in
Gel'fond and Leont'ev [1] (1951). Its representation in the form (22.42) was given in Dimovaki
[1] (1981), see also [3, p.l05] (1982). A construction dose in a sense to (22.43) can be found in
Dimovski [2], [3, p.l06] (1982).
The passage from the Hadamard composition (22.46) to the convolution (22.48) is well
known, as for example, the book by Smimov and Lebedev [I, p.169]. The integral operators
(22.48) in a more general situation were investigated by Korobeinik [1]-[2] (1964), [3] (1965), -
see  23.2 (note 22.18). In the examples 1-3 after Lemma 22.4 we follow the paper by Belyi [6]
(1977), where the generalized differentiation in (22.46) was considered in connection with integral
representations for functions analytic in an annulus - see also Belinskii and Belyi [1] (1971). The
variant (22.55) can be also found in Owa [3] (1981), [6] (1982), [13] (1985).
The generalized integro-differentiation given in (22.57) was introduced and used in some
problem of analytic function theory in the papers by Dzherbashyan [4] (1967), [5] (1968).
23.2. Survey of other results (relating to  18-22)
18.1. Let Ig+iO",'1 and 1,0",'1 be the operators (18.1) and (18.7) and let rot be the Mellin
transform (1.112). IT J e Lp(O,oo), 1  p  2, then Ig+;O",'1 J , 1,O",'1J e Lp(O,oo) and
a r(1+t1- s /O")
(rotI O + i O",'1 J )(s) = r(1 + t1 + a _ s/O") (rotJ)(s), Re(t1 - s/O") > -I,
(23.1 )
a r(t1+ s /O")
(rotl_ 0" '1J)(s) = r( / ) (rotl)(s), Re(t1 + s/O") > O.
" t1+a+sO"
(23.2)
for s = l/p + iT, -00 < T < 00. IT 2 < p < 00 and
J e rot p = {g : 9 = rot- 1 <p, <p e Lp/(O"p-1 - ioo, O"p-1 + ioo), l/p + l/p' = I},
then Ig+iO",'1 J , 1,O",'1J e rot p and (23.1) and (23.2) hold (Erdelyi [6, Lemma 5]). In the case
0" = 1 these statements for the operators (18.5) and (18.6) were proved first by Kober [I, Theorems
5a and 5b]. Rooney [4] studied mapping properties of the operators Ig+;O",'1 and 1,0",'1 considered
on the weighted space Lp(R;x#J), 1  p < 00; see also  9.2 (note 5.6).
let rot, L, L-1 be the operators of the Mellin transform (1.112) and Laplace transfonns
(1.119) and (1.120), and let l:'a and K;j,a be Kober operators (18.5) and (18.6). Let also
a> 0, t1 > 0; J(x),x- 1 / 2 J(x) e L(O, 00); res) = rot{J(t);s} e L(I/2 - ioo,I/2+ ioo);
and x- 1 / 2 (1:'aJ)(x), x- 1 / 2 (K;j,aJ)(x) e L(O,oo). Then the representations
(1:'aJ)(x) = x- a -'1 L -1 {C a L[T f1 J(T); t]; x},
(K;j,aJ)(x) = {y1-a-'1 L -l{C a L[T'1- 1 J(I/T); t]; y} }y=Z-l

 23. ADDITIONAL INFORMATION TO CHAPTER 4
437
are valid (Fox [7]).
Srivastava [2] showed that the compositions If oJ! and K= OIK; of the Kober
." .. ,01 .".. ,01
operators (18.5) and (18.6) may be represented as compositions of two operators of the form
 
f(xt)'YwIJ,II(xt)/(t)dt, where WIJ,II(X) = x 1 / 2 f T- 1 J II (T)J IJ (X/T)dT is the Watson function,
o 0
JII(x) is the Bessel function (1.83).
Let RI(x) = x-I I(X- 1 )i ReIl,Re(II+201) '# -2, -4,..., and let TOI = (1+ '2 )-1 RI+ / 2 in
II ,01 II ,01
the case ReOi > 0, and TOI = (K- / 2 + )-1 RK- / 2 + in the case ReOi < o. IT Ie L2(O, 00),
II 01,-01 II 01,-11 -

then TOIl e L2(O,oo), and T1 I = TOIl, Tol = R/, lim TOII () = f J II (2.;xt)/(t)dt
OI-+ 0
where JII(x) is the Bessel function (1.83) (Erdelyi [5]).
18.2. let 5'1,01 == 5'1,01,1 be the modified Hankel transform (18.19) and let 5'1 = 5'1,0, IT
Re'1 > -1/2, the Kober operators (18.5) and (18.6) admit the following representations
1:;01 = 52'1+20152'1+01,01 = 52'1+01,0152'1'
K:;OI = 52'1+01,0152'1+201 = 52'152'1+01,01'
(23.3)
1:;01 = 52'1+0I+/J,0I-/J52'1+/J,/J' 0  ReJ1  ReOi.
Kq,OI = 52'1+/J,/J 5 2'1+ 0I +/J,0I-/J
within the frames of the space L2(O, 00) (Kober [1]), d. (18.22).
ErdeIyi and Kober [1] established relations which connect Kober operators (18.5) and (18.6)
with the truncated Hankel transforms:

m5111(x) = f J II ,m(2.JXi)/(t)dt, Ie Lp(O, 00), 1  p  2,
o
 (_1)1: (z /2)11+21:
JII,m(Z) = L.J k!r(1I + k + 1) ' Rell/2 + l/p' '# 0, -1,-2,...,
I:=m
where JII,m(z) is the truncated Bessel function (cr. (1.83».
Joshi [1] used Kober operators (18.5) and (18.6) to find connections between the following
generalized Hankel transforms

f ("'t)+ 1/2 J [ ( ",:,2 r] J( t )dt,
o

f (xt)" J[(xt)IJ]/(t)dt,
o
IJ ( )  ( _z ) k
where J>. x = L.J 1:!r(I+>'+ 1:) ' J.l. > 0, is the Bessel-Maitland function (see Marichev [10,
k=O IJ
11.63]) and indices .x may have different values.

438
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Bhise [2] applied Erdelyi-Kober operators to the investigation of certain properties of the
integral transform
00
f 21 ( 1 ,,-m-(1I+1)/2. m-t+CII+I)/2 )
(Kf)(x) = G 24 xt 11/2. 1I/2+2m. -11/2, -1I/2-2m f(t)dt
o
with the Mejer G.function in the kernel. This is known as the generalized Hankel transform. It
reduces to Hankel transform in the case k + m = 1/2.
18.3. Braaksma and Schuitman [1] suggested a modification of the Erdelyi-Kober-type
fractional integral (18.1) alike the Erdelyi construction in (9.3):
:&
- O'x-O'( a+'1) f (
]a Ip(x) = [ (xO' _ to')a-1tO' '1+ l )-llp(t)dt
0+.0','1 rea)
o
(23.4)
m-l 00 k
- L: (-1)" (a  1) f (;) 0' xO'Ca-l)tO'C'1+ l )-IIp(t)dt]
"=0 0
giving the analogous modification 1 0' 'l ip of the operator (18.3) with b = +00. In the cited paper
. ,
the operators Ig+.O'.'1 and 1.0','1 were studied in the special spaces T('x, IL):
T('x.IL) = {Ip e COO (0, 00) : sup Itc+PIpCp)(t)l, p = 0,1,2,.... n,
t>O
>'..5 c <JI..
n=0,1,2....; 'xn<ILn, 'xO>'xl>'''' lim 'xn='x;
n-oo
ILO < ILl < ..., lim ILn = IL}
n-oo
of test functions (different from those presented in Section 8) as well as in the corresponding dual
spaces T ' (1 - IL, 1 -,X) of generalized functions. The relat.ions of such modified fractional integrals
with the modified Hankel transform mSllf (see n. 18.2 above) were also given.
18.4. More general than in (18.5) and (18.6), the operators
:&
-'1 -1 f
(Rf)(x) = x rea) 2Ft (1 - a,{3 + mi {3i t/x)t'Y f(t)dt,
o
(23.5)
00
x 6 f
(K f)(x) = rea) 2 F l (1 - a, {3 + mi {3i x/t)t- 6 - l f(t)dt,
:&
(23.6)
with the Gauss hypergeometric function (1.72) were introduced by Saxena [5]. He gave analogues
of (23.1) and (23.2) for the Mellin transform and of (18.18) for fractional integration by parts.
Nieva del Pino [1] found Vanna transform (9.6) and Mejer and Hankel transforms (see  1.4) of
the operators defined in (23.5) and (23.6). Representations of these operators via the Laplace
operators L and L-l were given by Kumbhat and Saxena [1]. Generalizations of the ErdeIyi-
Kober operators (18.5) and (18.6) of the type (23.5)-(23.6) with the Gauss hypergeometric

 23. ADDITIONAL INFORMATION TO CHAPTER 4
439
function in the kernel were also considered by Kalla and Saxena [1] and Saxena and Kumbhat
[1]. There are also generalizations with the Wright hypergeometric function pFq in the kemel-
Malovichko [1]i with the Bessel function - Lowndes [1]; with the Meijer G-function - Parashar [1]
and Kalla [6], [12] and Kiryakova [1], [2]; with the Fox H-function - Kalla [4], [12], Saxena and
Kumbhat [2], Goyal and Jain [1]; Kiryakova [3], Kalla and Kiryakova [1], and with an arbitrary
function of a certain class - Kalla [7]. See also Section 35.1, 36, 37.2, 39.2.
18.5. Let Fp,.. be the space of test functions defined in  8.4, let Fp,.. be the dual space and
let
:z:
1 (xm - tm)C-l ( X m )
Hf(x) == r(c) 2 F l a, bj Cj 1 - t m mt m - 1 f(t)dt
°
(23.7)
where m > 0, Rec > 0, a,b,E C, and 2Fl(a,biciz) is the Gauss hypergeometric function (1.72).
McBride [1] showed that the operator H:h f is representable as
H c f( ) I c-6 -ma 1 6 ma f( )
m X = O+,m x O+,mx x ,
x> 0,
with the Erdelyi-Kober-type operator Ig+,m == Ig+;m,O given in (18.1)-(18.2), and is continuous
from Fp,.. into Fp,,..-mc and from Fp,.. into F;,,..+mc' Similar investigations were carried out for
three other operators which are obtained from H:hf by interchanging x m and t m in the function
2Fl' or by the same interchange in the whole kernel and replacing the interval of integration
(0, x) by (x,oo), see also  39.2 (note 36.2).
Certain properties of the ErdeIyi-Kober-type operators (18.1) and (18.4) with a = 0 and
b = 00 in the spaces F p ,,.. and F p ,,.. were studied by McBride [2], [4]. Mapping properties
of Hankel transfonns (see  1.4) and of modified Hankel transfonns (18.19) with q = 2 from
F p ,,.., F;,,.. into Fp,2/p_,.._l' F;,2/P_,.._l' respectively, were considered by McBride [5]. He also
established various relationships between these transforms and the Erdelyi-Kober-type fractional
integration operators (18.1) and (18.3) and, in particular, proved (18.20)-(18.23) with q = 2.
We remark that operators of the type (23.7) with m = 1 were earlier studied by Love [2], [3]
in other function spaces Qq and Rr as in  17.1, Notes to  10.1. An operator of the type H:hf
but with lower variable limit of integration was also considered by Higgins [3] -  39.2 (notes
36.1-36.2).
18.6. Saigo [1]-[3] (and also [6], [7]) introduced the integral operators
:z:
(1:::'" f)(x) = (x - ;1O-/J I(X - t)o-12Fl (a + (3, -77jai : = : ) f(t)dt, Rea> 0,
a
(1:::'" f)(x) = ( dd x ) n (1:t n ,/J-n,,,-n f)(x) , Rea  0, n = [-Rea] + Ii
b
/J (b - x)-a-I' 1 1 ( t - X )
(/' ,,, I)(x) = ( ) (t - x)a- 2Fl a + {3, -77j aj - f(t)dt,
- ra b-x
:z:
Rea> 0,
(1!'" I)(x) = (- :x ) n (1!n,l'-n,,,-n f)(x) , Rea  0, n = [-Rea] + Ii
00
R 1 (t - x)a-l ( X )
(/.'fJ'" f)(x) = rea) t-o-/J 2 F l a + {3, -77; a; 1 - t f(t)dt,
:z:
Rea > 0,

440
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
(J'''''' 1) (:c) = (-  ) n (1+n,"-n,,,-n I)(x), Rea 50, n = [-Rea] + 1;
which reduce to Riemann-Liouville or Kober fractional integrals and derivatives in the cases
{3 = -a or {3 = 0, respectively. Various properties of these operators were investigated in the
cited papers. In particular, the paper by Saigo [1] contains composition representations of the
type (10.18)-(10.29) for the operators f'" and I'P,", validity conditions for the fonnula
00 00
f g(:c)(:'" 1) (:c)dx = f 1(:c)(I'P," g)(x)dx,
o 0
of integration by parts, and validity conditions for the following analogues of the semi group
property (2.65):
la,p,,, l'Y,6,a+" 1 = I a +'Y,P+6''' 1
a+ a+ a+ '
la,p,,, I'Y,6,"-P-'Y- 6 1 = I a +'Y,P+6,"-'Y- 6 I
a+ a+ a+ '
l'2"6,a+,, l ,p ,,, I = I+'Y ,P+6,,, I,
l'2"6,,,-P-'Y- 6 l'P," I = I+'Y,P+6,"-'Y-6 I.
The first two of these relations remain true after replacing a+ by b-. Mapping properties of
the operators l;f ,,, and t.'P ,,, in the spaces Lp « 0,00) j x'Y), 1 5 p 5 00, were also investigated
in this paper. Saigo [2], [3], [6], [7] also showed that the above equations yield the following
expressions for the inverse operators:
(1::,")-1 = l;.:,-p,a+". (1:':'p,")-1 = Ib_a,-p,a+",
(l,P,")-1 = l:a,-p,a+ 11
and studied Holder properties of the functions I::'" I and I:':'p'" I in the case I(x) e H>'([a, b]),
o < .x < 1. The isomorphism between the weighted Holder spaces realized by these operators was
obtained by Grin'ko and Kilbas [1]; see  17.2 (note 10.1).
Srivastava and Saigo [1] considered the multiplication of the operators l;f'" and ll:!''''
and applied their results to obtain the explicit solution of some boundary problems for the
Euler-Poisson-Darboux equations in tenus of Appel and Kampe de Feriet functions of two
variables -  43.2 (note 41.1). Saigo and Glaeske [1], [2] investigated mapping properties and
product rules of the operators 1:'" and IIP," in the spaces F p ,,,. - subsection 8.4.
18.T. Developing an idea. of Erdelyi [3], as in  4.2 (note 2.6) and  9.2 (note 5.4), Akopyan
and Nersesyan [1] used (18.18) of fractional integration by parts for Erdelyi-Kober operators
l:+jU,,, with U = 2 to construct a new biorthogonal syst.ems:
«In(x) = (  ) a x"+a J II + a (jn,II X ),
In,V
1
" m( x ) = 2 d f( t 2 x2 ) -atl-1l J II(' " m II t ) dt ,
J2 ( . ) r ( 1 - ) d - ,
11+1 'm,1I a x
:&

 23. ADDITIONAL INFORMATION TO CHAPTER 4
441
where jn,II, n = 1,2,..., are the roots of the Bessel fWlctions JII(z) and 0  a < 1. In the case
v = 1/2 the generalized Schlomilch biorthogonal system is obtained from this system.
18.8. Erdelyi-Kober operators (x 1 - q dldx) a were used by Klyuchantsev [1] to obtain the
transformation operator Fr for the differential operators
d" b 1 d r - 1 b r
B r =-+-- I +"'+-' r=I,2,...,
dx r x dx r - x r
i.e. such an operator Fr that Fr- 1 BrFr = d r Idx r . On functions v satisfying certain parity
conditions, the operator Fr has the form
r-l
Fr v = II x O "
k=1
( d ) c"
xr- 1 dx v,
the exponents ak and Ck being determined by the coefficients bl, . . . , b r .
The paper by Trimeche [1] contains a generalization of the Erdelyi-Kober operators in the
form
:e
(Xlp)(x) = f K(x, t)lp(t)dt,
o
00
('XIp)(x) = f K(t, x)A(t)lp(t)dt
:e
with a certain kernel K(x, t). This generalization serves as a transformation operator: Xf ==
X{l.sf for the differential operator  = A(:e) /; (A(x)/;) - q(x).
18.9. Rooney [1] investigated mapping properties of the Erdelyi-Kober-type operators
/8+'1 _ = x- 0I --lg+x 8 and 1'I' = x8/x-0I-8 (d. (18.1) and (18.7» in Lorentz spaces
L(p;q'). For the latter we refer tC: fr example, O'Neil [1]. It was shown that
11 101 I < reI + 1 - lip)
0+;1,81 £(P,9) - rea + a + 1 - lip) '
11101 II < rea + lip)
-;1,8 £(P,9) - rea + a + lip)
if 1  p < 00, q  p, a > 0 and a > -lip' in the former case, and a > -lip in the latter case.
This was applied to study mapping properties of the Vanna integral transform (9.6) and of the
00
integral transform f (xt)-m-l/2e-:et/2 Mk,m(xt)f(t)dt with the Whittaker fWlction Mk,m(z)
o
(Erdelyi, Magnus, Oberhettinger and Tricomi [1,6.9]) in the spaces L(p, q).
18.10. Let Ig+;q,,, and I;q,,, be the left- and right-hand sided Erdelyi-Kober operators -
(18.1) and (18.7). Their interconnections similar to those in (11.27)-(11.30) were given by Rooney
[2] in the form
ljq,,,Hl = Ig+;q,' Re{3  Rea,
Ig + . q tH2 = Ig_. q f'I' Rea  Re{3,
, ,\, , ,.,
where HI and H2 are operators bOWlded in the weighted spaces Lp(R,xIJ), which was obtained
by means of theorems on Fowier-multipliers. These interconnections were used in this paper to
obtain imbedding theorems for the ranges Ig+;q,,,[Lp(R;xIJ)] and 1;q,,,[Lp(Ri x IJ )] into each
other or a theorem on their coincidence - Rooney [3]. These results were applied to prove imbedding
theorems for the ranges Bp,>',7(Lp(R;xIJ», where Bp,>',7 = S "+"Y!P-l ,1_P,2S,1_p,2 is a
composition of two modified Hankel transforms (18.19) - Rooney [4].

442
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
18.11. Fox [1] showed that the Erdelyi-Kober fractional integration conserves, Wider certain
conditions, the 80 called chain property of integral transforms.
18.12. Hadamard fractional derivatives and integrals (18.42), (18.43), (18.56) and (18.56')
were used by Mamedov and Orudzhev [I], [2] to define certain spaces of fWictions on the half-axis
with a fractional differentiability, which are well suited for the application of the Mellin transform.
00
18.13. Let lex) e L2(R, Pa), where pa(x) = e- z x a , a  0, and let lex) = L: en Lh a ) (x),
n=O
where La)(x) = [rcn+a+l)/rcn+l)]-1/2 Lh a ) (x) is the nonnalized Laguerre polynomial system
00
(see  9.2, note 5.4). In order that L: c nil < 00, 0 < v < I, it is necessary and sufficient that
n=O
I (x) is representable as
00
lex) = f ez-t(t - x)II-1<p(t)dt, <pet) e L2(R,Pa+lI)
z
(Rafal'son [I».
18.14. Chen fractional differentiation given in (18.87) was applied by Skorikov [2] to
the characterization of the space L([a, b» = Ha,P([a, b» of Bessel potentials on an interval,
-00  a < b  00. The original definition of the space L([a, b]) in this paper was given in terms
of Chen fractional differentiation. This definition in the case 0 < a < 1, 1 < p < l/a has the form
L;([a,b)) = {I: I e Lp, D Ie Lp},
(23.8)
where D I is the Chen-Marchaud derivative (18.91)-(18.93), a  c  b. It was shown that this
definition does not depend on the choice of the point c and that L([a,b)) coincides with the
space of restrictions of Bessel potentials on the whole line onto [a, b]. These assertions develop
Theorem 18.3, which corresponds to the case c = a or c = b. In the case p > l/a (23.8) was
modified in [2] by taking into account the continuity of the functions in L in this case.
A generalization of the semigroup property (18.86) for Chen fractional integration (18.80)
to the case of different "initial" points was given by Nahushev and Salahitdinov [1]. Let
z
a R sign (x - c)lx - cl a - 1 f ( I t - d I Sign(Z-C» )
I 'fI<p= F l-a,I;I+p; - <p(t)dt.
c,d r(a)r(1 + p) x - d
a
In particular, it was shown in [1] that
1: I <p = If «t - a),6 <p(t» + (x - a)a-,6+11': «t - a),6-1<p(t»
I
if a > 0 and p > 0, the composition I k I: being also considered.
18.15. In a number of papers, AI-Salam [1], [2], AI-Salam and Verma [1], Agarwal [2], [3]
and Sharma [1] and Upadhyay [1] some objects refelTed to as q-integrals and q-derivatives were
introduced and investigated. We note that their definition, which does not contain the passage
to a limit is not a definition of integrals and derivatives in the usual sense, but is an extension
to fractional exponents of the idea of the q-derivative, namely, (Dql)(x) = /(:.:r) which
goes back to Jackson [1] and [2], and plays an important role in combinatorial analysis. Such
constructions of fractional order were first introduced by AI-Salam [1], [2] in the form

 23. ADDITIONAL INFORMATION TO CHAPTER 4
443
00
(ql:" f)(x) = q-I(I+I)/2 x " (1 - q)" 2:( _1)1: [-;] ql:(l:-l)/2 f(xq-II-I:),
1:=0
-00 < v < 00, 0 < q < I,
h [ 01 ] [ 01 ] [ 01-1 ] ... [ 01-1:+1 ] d [ ] \_qor Th h  f . . a1s
w ere I: = [1] [2] ... [1:] an a = _q' ere are ot er lOrms of ractlonal q-mtegr
and q-derivatives given in AI-Salam [1] and Agarwal [2], [3], where some applications of these
ideas were also given. Reference may also be made to Khan [1] and Khan and Khan [1]. The
book by Exton [I], where "q-theory" can be found in detail is also relevant.
18.16. Fractional differentiation was extended to the case of functions on a local field K
(Onneweel' [1]-[3]). Within the framework of the spaces Lp(K), 'P  I, strong fractional derivatives
were defined similar to the Riesz derivatives colTesponding to the case when K is a real line, and
the colTesponding spaces of Bessel-type potentials were in particular studied. The relationship
between these spaces and Lipshitz- Taibleson-type spaces over K can also be noted.
19.1. Let X21r be the same as in Theorem 19.2. We say that a 211"-periodic function
f(x) e X21r has a ,trong Weyl fractional derivative in X27f if there exist a function <p(x) e X21r
n
and a sequence of trigonometrical. polynomials Tn(x) = L: an,l:eil: z such that IIf - Tnllx..,
k=-n
II<p - TOI)lIx.. - 0 as n - 00. Then by definition <p = 'DOI) f is the strong Weyl derivative.
n
Here TOI) = L: (ik)OI an ,l:e i I: Z is the Weyl fractional derivative of a polynomial Tn(x). The
I:=-n
following theorem is valid - Malozemov [IH4].
Theorem 23.1. The e:.nstence of a strong fractional Weyl derivatives in X21r of a function
f(x) e X 21r i, equivalent to ita re'Presentability by the Weyl fractional integral: f(x) = fo + IOI) <p,
<p e X 211" a > O.
Comparing this theorem with Theorem 19.2 we see that the existence of a strong Weyl
derivative in X 211' is equivalent to the convergence of the truncated Marchaud derivative in X 21r .
19.2. The following is close in a sense to Theorems 19.3 and 19.2 (with Remark 19.2) which
characterize functions f(x) representable by the fractional integral of a function in Lp. Let a > 0
and let m be the least integer such that m  a. Katsaras and Liu [1] introduced a modification
(d. (19.17» of the Weyl fractional derivative in the fonn
V (OI) f li h -l h -l A A r (m-OI) f
+ = m 1 ... m h ... h ,
ha- O I m
(23.9)
where hf = f(x+ h) - f(x), hl ... hm f = hl (h ... hm J) and I(m-OI) f is the fractional
integral (19.26)-(19.27). They proved that a function f e Lp has the limit (23.9) in Lp, 1  p < 00,
if and only if there exists a function g(x) e Lp such that gn = (isignn)mlnia fn.
19.3. The following statement (Kudryavtsev [1]) generalizes the Szuz theorem known in
the theory of Fowier series - Bari [I, p.647].
Theorem 23.2. Let f(x) e Ll (0, 211") and let 'DOI) f e L p (O,211") ezist in the sen,e that there
e:.nsts a function <p = ('DOI) f) e Lp with Fourier coefficients <Pn = (in)OI fn, 1  p < 2, a > O.

444
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Then lor any 'Y> (a + 1 - l/p)-l the estimates
00
L I/nP < 00,
n= -00
00
L I/nP = o(N 1 -'Y(1+a-1/p»
Inl=N+1
(23.10)
a re va lid.
19.4. Kudryavtsev [1] introduced a generalization of Weyl fractional integra-differentiation,
defined by the expansion
00
'Da,p Ie L (in)a ln P Inle inz
n= -00
and called it a fractional-logarithmic derivative 01 order (a, (J). The statement, extending that
of theorem 23.2 to Da,pl and proved in [I), is as follows: if Da,pl e Lp under assumptions of
Theorem 23.2, then instead of (23.10) we have
00
L I/n 1'1 = o(N 1 -'Y(a+1-1/p) In-P'Y N).
Inl=N +1
19.5. In Esmaganbetov, Nauryzbaev and Smailov [1] theorems were proved which connect
the existence in Lp of a Weyl fractional derivative 'Da) I of a periodic function I(x) with the
behaviour of Lp-nonns of fractional derivatives of partial swns of Fourier series. The estimates
[ 00 ] 1h
1I'Da) Ilip  c E n-'Yr-111'D(a+r)SnU)II
were, in particular, proved, where a  0, r> 0, 'Y = miner, p) and so on.
19.6. We refelTed to the application of methods of approximation theory in questions of
fractional calculus in  17.2, note 14.16. In the periodic case we have to cite the fundamental
investigations by Favard [I), Nagy [I), Nikol'skii [2], [4], Kralik [I), Ogievetskii [I), [2], [5] and
Timan [2]. In particular, the estimate En(Ia) J)  cn-aw1cU,l/n) was proved by Nagy [1],
Nikol'skii [4] and Timan [2] showed En U) as being the best approximation of I in Lp by
trigonometric polynomials, and w1cU,I/n) as being the continuity modulus in Lp, 1  p  00 of
order k. We may add the papers by Xie Ting-fan [1], Nessel and Trebels [I), Taberski [1]-[4], [6],
Butzer, Dyckhoff, Gorlich and Stens [1] and Esmaganbetov [1], [3], and also Butzer and Nessel [I,
Ch. 11]. There are very many papers devoted to various problems of approximation of fractionally
differentiable periodic functions by all manner of trigonometric sums, such as Pinkevich [1],
Nikol'skii [1]-[3], Timan [1], Dzyadyk [1], Izumi and Sato [I), Stechkin [2], Efimov [1]-[4], Sune
YWl-Shen [1], Telyakovskii [I), [2], Pokalo [I), Rusak [1]-[3], Zhuk [I), Stepanets [I), [2] and others;
there are also references in these papers. In Babenko, V.F. [1] the Kolmogorov diameters were
estimated for spaces of periodic functions representable by fractional integrals Ia)Ip, Ip e L 1 or
Ip e Loo.
19.7. The assertion (19.62) of the Hardy-Littlewood theorem becomes false in the case
p = 1. It has to be replaced by the following result. IT 0 < a < 1, q = 1/(1 - a), then
2'11'
IIIa)IpIlL,,(O,2"')  A f I/(x)l(ln+ I/(x)l)l- a dx + A,
o

 23. ADDITIONAL INFORMATION TO CHAPTER 4
445
the constant A depending on cv only - Zygmund [1]. This result was generalized by O'Neil [3] to
211'
the case when f I/(x)l(ln+ I/(x)I)"dx < 00, 8> O.
o
00
19.8. Let lex) - L: an cosnx. Izumi [1] gave various sufficient conditions on coefficients
n=l
11'
an such that the existence of the finite limit lim J(t - x)CV- 1 /(t)dt would guarantee the
z-Oz
00
convergence of the series L: n -cv an, 0 < cv < 1. In the case cv = 1 necessary and sufficient
n=l
conditions of convergence were given by Matsuyama [1] in close terms.
19.9. The expression V(cv) I = d:) , g(x) = (1(m-cv) I)(x), m - 1 < cv < m, for the
Weyl fractional derivative of a periodic function lex) may be modified by treating differentiation
(dg/dx)m in the Peano sense. This implies the existence of a polynomial Pz(t) such that
9(X + t) - Pz(t) = o(ltl m ). Fractional derivatives modified in such a way were considered by
Zygmund [3], who investigated the connection of the property I(x + t) - Pz(t) = O(ltl m +.6)
with fractional differentiability of the function lex). For the non-periodic case we refer to Stein
and Zygmund [1]. The development of the ideas of these papers to functions I(x), which have
lacunary Fowier series, was made by Welland [1], while extension to the case of many variables
can be found in WeIland [2].
19.10. Let a kernel K:(x,() have the Weyl fractional derivative K:a)(x,() of order cv > 0
with respect to x and
211' 211' .....L
J (J 1.o;;a)(".{)IPd") p-' d( < 00.
o 0
Hille and Tamarkin [2] showed that the characteristic values >'n of the integral equation
211'
y(x) - ,X J K:(x, ()y()d = 0, 0 < x < 211",
o
satisfy the condition l'xnln- a - 1 + 1 / p - 00 as n - 00.
00
19.11. Let 1(8) '" L: anP,Ncos8) be the ultraspherical expansion of a function lex),
n=l
x e [0,11"]. Muckenhoupt and Stein ([1], p.75) introduced the fractional integration lal which
reduced to division of an by n a. Thus it matches well to ultraspherical expansions as the Weyl
operator does to the usual Fourier expansions. This fractional integration was realized in terms
of the convolution structure for ultraspherical expansions. They proved a Hardy-Littlewood-type
theorem on mapping property of this operator from Lp into Lr with l/r = 1/1' - cv/(2'x + 1) and
the norm in Lp with respect to the measure Idm>.81 = I sin612>'d8. Later, Bavinck [1] considered
a more general case of Fowier-Jacobi expansions. Along with a statement of the above type he
alao gave theorems on mapping of his fractional integration operator within the framework of
Lipshitz spaces Lip(T,p) under the appropriate definition of the latter. This is an extension of
Theorem 19.9.
19.12. For periodic (unctions Alexitz and Kralik [1] considered generalized fractional
integration of the type L: A(k)/lce ib with an arbitrary function A(k) such that the integral
00
J A(x)  converges, and they gave applications to some approximation theorems. Recent
1
investigations by Stepanets [1], [2] are also relevant. He gave the classification of periodic

446
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
functions with respect to such a generalized differentiability, with applications to approximation
theory problems.
20.1. The definition (20.7) of fractional differentiation is naturally extended to the case of
fractional powers of operators. For the concept of infinitesimal operators we refer to  5.7. The
idea of constructing fractional powers of operators via fractional differences is due to Butzer and
it was realized by Westphal [4]. Let Tt, A and X be the same as in  5.7. Let us introduce the
fractional power (_A)a, a> 0, by the relation
(_A)a f = lim t-a(E - Tt)a f,
t_O+
(23.11 )
00
where (E - Tt)a f = L: (_I)k ()Tkt similar to (20.2), and where the limit is in the nonn of the
k=O
space X. The following theorem is valid - Westphal [4].
Theorem 23.3. The domain D«-A)a) of the operator (_A)a i, den,e in X and D«-A)a) c
D« -A).6) if a> {3 > 0 and (_A)a i, a closed operator.
The following theorem due to Westphal [3], and see also  23.1, notes to  20.3, adjoins in a
sense to Theorem 20.4.
Theorem 23.4. For f e X the limit (23.11) eri,t, in X if and only if there ezi.ts the limit
00
 lim f Ca-l(E - Tt)' fdt, I> a,
x(a,l) £-0
£
in X and 60th limits coincide - ,ee the value of x(a, I) in (5.81).
The idea of fractional differentiation by Grunwald and Letnikov allows another wide reaching
generalization. Replacing the translation operator Thin (h f)(x) = (E - Th)a f by one or another
generalized translation operator one can obtain various forms of fractional differentiation. T his
idea was re alized by Butzer and Stens [1]-[3] with (ThJ)(X) = (1/2)[f(xh + '1" (1 - x 2 )(1 - h 2 » +
f(xh - '1" (1 - x 2 )(1 - h 2 »], -1 < x < I, -1 < h < 1 which corresponds to the fractional
derivative
v(a) f = lim (h f)(x) ,
h-l-0 (1 - h)a
00
(hf)(x) = (_I)[a] L(-IY ()(J)(x)
j=O J
called the Chebyshev derivative in these papers. The limit here is taken in the nonn of the space
X = C or X = Lp with weight (1 - x 2 )-1/2. In particular, v(1) f = (1 - x 2 )f"(x) - xf'(x) and
V(1/2) f = - Iz(H f)(x) where H f is the Hilbert singular operator defined in (19.22).
FUnctions f(x) eX, which have fractional derivatives (vCa) f)(x), were characterized as those and
only those functions which are representable by the "fractional integral" f = <P* .,pa, where f * <p =
1
(1/11") f (Tzf)(t)<p(t)(1 - t 2 )-1/2dt and .,pa(X) is such a function that a(k) = (_I)[a]k- 2a .
-1
1
Here f(k) = (1/11") f f(t) cos(k arccos t)(1 - t 2 )-1/2dt are Fourier-Chebyshev coefficients.
-1
20.2. A relationship between Griinwald-Letnikov and Riemann-Liouville fractional
differentiability is given by the following theorem (Westphal [3]), where X = Lp (R), 1  p < 00,
or X = CO, the latter being the space of bounded unifonnly continuous functions f(x) on R
with f(O) = O.

 23. ADDITIONAL INFORMATION TO CHAPTER 4
447
Theorem 23.5. Let I(x) e X and ev > O. Then the following datements are equivalent to each
other:
i) The Grinwald-Letnikov derivative e:J:ist, in X: lim IIh- a h I - la)lIx=o;
h-O+
ii) £r;(l:l-a I)(x) e ACloc' k = 0,1,..., n, where n = [ev], and vg+J eX.
20.3. An interesting way of introducing "differences of fractional order" was suggested by
Bosanquet [4, p.240]:
h
a)J(x)fev f(h - t)a-l(vg+J)(x+ t)dt
o
so that a) J(x)/h a -+ (vg+J)(x) as h -+ O.
20.4. The following is a generalization of the Bernstein-type inequality (19.74), and is due
q
to Lizorkin [3, p.llS]. Let g(x) = J ei:J:tdw(t), where wet) has a bounded variation in [-O',ol
-(1
Then the fractional derivative
(1
g(a)(x) = f (ix)aeixtdw(t), ev  I,
-(1
of a function g(x) allows the estimate IIg(a)lIp  qallgll p , 1  p  00.
20.5. The following inequality for the Weyl fractional derivative of a trigonometric
polynomial Tn (x) given by the following
IIVa)Tnllp  ( 2sin(:h/2» ) a lIhTnllp,
211"
0< h < -,
n
(Taberski [3]), generalizing the Stechkin-Nikol'skii inequality, and the inequality for the Griinwald-
Letnikov fractional derivative for an entire function G(x) of exponential type O'a, namely
IIG(a)lIp  O'a[2sin(O'h/2)]-allhGllp, 0 < h < 211"/0'
(Wilmes [I], [2] and Taberski [5, p.I33]) both adjoint the Bernstein-type inequality (19.74). Here
 h J is a "centered" fractional difference.
20.6. For the generalized differentiation a(V)1 by Post (see (20.10» the generalized Leibniz
role
00
a(V)[u(x)v(x)] = "" 2.. u (I:)(x)a(k)(V)v(x)
L..J k!
1:=0
is true - Post [2, p.755]. It yields, in particular, the Leibniz fonnula (15.11) in the case a(x) = x a .
20.7. Differences of fractional order are naturally used to define continuity moduli of
fractional order; thus wa(J,h) = wa,X(J,h) = sup lIfJllx, where X is a Banach space.
Itl<h
Properties of such continuity moduli were investigated in the papers by Butzer, Dyckhoff, Gorlich
and Stens [I], Taberski (3], [8], Gaimnazarov [I], [2], Esmaganbetov [2] and V.G. Ponomarenko
[1], [2]. One may also see some modified (averaged) differences of fractional order in Drianov
[1]-[3]. Most of the cited papers dealt with the methods of best approximation theory. Samko
and Yakubov [3] considered the generalized Holder space Hlp,a = Hk,a defined by the fractional

448
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
continuity modulus:
H""a = {f(x): f(x) eX, sUPWor(J,S)/"P(S) < oo}
6>0
where X = Lp(O, 211"), 1  p < 00, or X = C(O, 211"). It was shown that H""a = H"" for
"P(S) e «t& (see (13.68» if (3  a > o. The case "P(S) = S>. was proved earlier by Butzer, Dychkoff,
Gorlich and Stens [1]. One should see also the space H""a in the multidimensional case in Samko
and Yakubov [4].
One can consider more general continuity moduli in cOlTespondence with Post's ideas -
(20.11). Far reaching generalizations of the continuity modulus can be found in the papers by
Boman II], Boman and Shapiro [1] and Shapiro [I, p.219].
20.8. Taberski (7) introduced a modification [a f = «t a * f of fractional integration and
considered its connection with Griinwald-Letnikov differentiation. The kernel «ta(x), x e R 1 , of
00
this modification was defined as «ta(x) = '* J "Por(t)eixtdt, where "Pa(t) = (-it)aPa(t) and
-00
PaCt) is a non-negative even function such that Por(t) = 1 if t  c > 0, Pfr(t)  0 for all t  0,
PaCt) = O(t a + 2 ), Pfr(t) = O(t a + 1 ) and P:;(t) = O(t a ) as t - +0. See also Taberski [8].
20.9. Lubich [2] considered an approximation of the Riemann-Liouville fractional integral
(I+f)(x) by the following discrete convolution sum
n ,
(Ih f)(x) = h a L Wn_j f(jh) + h a L Wn,j f(jh), x = nh,
;=0 ;=0
with fixed 8. This can be regarded as a development of ideas connected with the Griinwald-
Letnikov approach. Let f(x) = x,6-1g(x), {3 > 0, with 9 smooth. Given the convolution
weights Wn_j and p a positive integer, "starting" weights Wnj are shown to exit such that
(Ih f)(x) - (Ig+J)(x) = O(h P ), provided that certain assumptions, connecting Wj' (3, a and p
hold.
21.1. Kilbas [9] proved the generalized Holder property for certain multidimensional
potential type integrals over a bounded domain in the Euclidean space Rn which contained
integrals with power-logarithmic kernels as particular cases.
21.2. Riekstyn'sh [1] demonstrated the possibility of obtaining an asymptotic expansion as
x - +00 of the convolution integral (17.1) and of generalizing the fractional integral (16.1) by
a modified method of successive expansions provided that the functions .r(t) and f(t) in (17.1)
have a power-Iogaritlunic asymptotic expansions as t - +00. The reader may find other methods
of obtaining asymptotic expansions for integrals of the form (17.1) and (17.2) in this case in
Riekstyn'sh [2].
22.1. Fractional integro-differentiation in the complex plane may be considered in the case
when functions, which are not necessarily analytic, are given on a certain curve only. Certain
properties of fractional integra-differentiation in such a setting, such as existence, index laws and
Taylor-type expansions (see  4.2, note 2.8), were considered by Fabian [1]. An investigation in the
situation of fractional integra-differentiation (22.2)-(22.3) with integration along a curve, on which
a function is given was C&lTied out by Sewell [1], [2] in connection with function theory problems.
In particular, he singled out a class of rectifiable curves over which the fractional integral of
fez) = 1 converged absolutely uniformly in z. For curves of this class the following results were
obtained: a) the lower derivatives (V,6 f)(z) in the case when (va J)(x) exist, for (3 < a; b) the
semigroup property holds; c) invariance of fractional differentiability under conformal mappings
is valid. (We note, however, that Sewell's argwnents implicitly contain the concept of a derivative
of a function by another function -  18.3; d) the Hardy-Littlewood theorems, similar to Lemma
13.1, on the existence of a fractional derivative Vol f on the curve for functions fez) satisfying
Holder condition on this curve of order .x > a; e) the Bernstein-type inequality IPa)(z)1  Ana

 23. ADDITIONAL INFORMATION TO CHAPTER 4
449
beyond the integration "starting point" zo, Pn(z) being a polynomial. As well as this some other
theorems connected with polynomial approximation in the complex plane were obtained.
A number of developments and generalizations of Sewell's results is contained in the papers
by Belyi [1]-[6], Belyi and Volkov [I], where various applications of the fractional calculus to
approximation theory in the complex plane can be found. The problem of the best approximation
by linear polynomial methods of analytic functions, which have a bounded generalized derivative
of the type (22.46) was treated by Dveirin [1].
In connection with the fractional calculus on curves we also refer to Fabian [2], where the
fractional integral (derivative) (VoJ)(z) was considered along a curve Lzo starting from the point
%0 and going to infinity, with a being any complex number. The behaviour of z-#J(VoJ)(z), as
% _ 00 along the curve, was studied provided that Z-II J(z) - A = const as z - 00 along the
curve. The results obtained were applied in [2] to the summation of series and integrals. Other
papen by Fabian [4], [5] are concerned with the character of the branching of the derivative Vo J
taken along a similar curve, at the point Zo and at the points where singularities are admitted for
J(z).
A paper by Peschanskii [2] has appeared recently. It gives a new version of fractional calculus
in the case of functions given on a closed smooth curve r. The author introduced Marchaud type
fractional differentiation of such functions in the form
DaJ = D7;J + DJ, J = J(t), t e r (O),
where 0 < a < I, D-;J = lim D% J and
e-O
D/ J =  f TJ(T) - tJ(t) dT
211"1 (1 - t/T)1+cr T 2
r Ir(')
D- J =  f J(T) - J(t) dT
211"1 (1 - T/t)l+cr t
r Ir(')
(d. (22.52», re(t) is a part of r which remains after removing the arc (t(a - e), t(a + e» with
tea) = t e r, a being the arc parameter. The corresponding fractional integration Fa<p = F<p
was defined via
Fj <p =  f 1 ( 1,1; 1 + a; .: ) <peT) dT
211"1 T T
r
- 1 f ( T ) dT
Fa <p = --: -#'1 1,1; 1 + a; - <p(T)-
211"1 t t
r
(d. (22.54». The main result in Peschanskii [2] states that J = Fa<P with <p e Lp(r), 1 < p < 00,
if and only if J e Lp(r) and D% converge in Lp(r) as e - +0, under the appropriate choice of
branches in all the integrals. These conditions being satisfied, then <p = Da J.
Finally we recall that a situation with fractional integration along a curve has already
occurred above once (see  17.2, note 12.5).
22.2. Carleman [2, p.42-47] proved the following theorem. Given a function J(x), x e R 1 ,
A
such that f IJ(x)ldx increases not faster than IAI'Y for some "y > 0, there exist functions J+(z)
o

450
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
and f-(z) analytic in the upper and lower half-planes C+ and C_, respectively, such that
:e" x"
lim I [J+(X + iy) - f-(x - iy)]dx == 1 f(x)dx
y-+O
:e' x,
unifonnly in x', x" e [a, b] whatever interval [a, b] is chosen. The result on the uniqueness of
the functions f%(z) was given by Carleman in terms of fractional integrals. Namely, he proved
the uniqueness of the functions f%(z) up to polynomials in the class of functions satisfying the
condition
I (I'?of%)(z) I
sup < 00
ZeC:i: (z - Zo)a(z - zo)p
with a  0, (3  0 and zo e C%, under the respective choice of signs.
00
22.3. Let f(a)(z) == L: (ik)a h:z lc be the Weyl fractional differentiation of a function
Ie=l
00
fez) = L: fle zle analytic in the unit disc. Hardy and Littlewood [7, p.232] proved the estimate
k=O
2 l/p 2 l/p
(I IjCa) (re;')IPdO ) 5, e(1 - r)-a (I If(rl/2';')IPdO) ,0 < r < I,
o 0
as well as a similar estimate for the fractional different.iation vg f of the factor r- a arising on
the right-hand side in this case.
Flett [5] used the following modification of Weyl fractional integration:
00
m a f = " fk Z k ,
;Q L..J ka I JO = 0,
1e=1
its version
00
Ja f " fie Ie
= L..J (k + l)a z
k=O
(23.12)
being treated in Flett [6]-[8]. These operators have the representation
z
23 a 1 1 Ip(t)dt
Ip == rea) [In(z/t)]l-at'
o
z
Jalp ==  1 Ip(t)dt ,
r(a)z On(z/t)]l-a
o
i.e. they are Hadamard-type constructions (18.42). We note, in particular, the estimate
1(23 a f)(re i8 )1 $ Ar(1 - r)-a(O), 0 $ r < 1,
obtained in Flett [5], where (8) = sup If(z)1 and 5 11 (0) is the part of the unit disc bounded
zeS,,(8)
by the two tangents from the point e i8 to the circle with center 0 and radius T1, with the longer
arc of this circle between the points of contact.

 23. ADDITIONAL INFORMATION TO CHAPTER 4
.51
In Flett [7, Theorem 6] one may find mapping properties of tbe operator Ja in the spaces
Bp,q,r defined by the nonn
1 2'11' i..!.
II/lIp,M = {1(1 - r)n- 1 { 1 I/(r';"')IPd"'} · dr} ·
o 0
One of Flett's results is the estimate IIJalpllp,q,'Y- a  cllcpllp,q,'Y with 0 < p  00,0 < q  00 and
"Y > cv > O.
A specification of the Hardy-Littlewood Theorem 22.2 for the operator Ja considered as an
operator from HP, 0 < p < 00 into Hq, q = "Yp/h - cv), under the additional assumption that
fez) = 0«1 - Izl)-'Y), 0 < cv < "Y  lIP, was given by Kim Hong Oh [1]. A certain development
of some results of the latter paper can be found in Kim Yong Chan [1]. In Kim Hong Oh [2]
Theorem 22.2 was extended to the case of functions fez, w) of two variables, (z, w) E C 2 , Izi  1,
Iwl  1.
00
Cohn [1] used the fractional differentiation (a f)(z) = L: (k + l)a h:zk, inverse to (23.12),
k=O
to study properties of certain "star-invariant" subspaces of the spaces H'P and BMO in the unit
disc. Jevtic [1] estimated such a fractional derivative (a f)(z), cv > 0, in the case of a function
fez) = exp (- ) known as atomic inner function in the unit disc. He showed in particular
211'
that J 1(af)(reilp)I'Pdlp  c(l- r)!-a ifp > 1/2cv, the power being replaced by 10g(l- r)-l
o
if p = 1/2cv. In this connection see also Ahern and Jevtic [1].
22.4. Hardy and Littlewood [5] proved results similar to Theorems 14.6, 14.7 and 19.9 on
fractional integro-differentiation in Lipshizian spaces of fWlctions fez) analytic in the unit disc
Izl < 1 and satisfying the condition
11' IIp
(1 I/( re;(B+h» - I(re;') IP dO )  clhl'. P  1.
-11'
These results were extended by Gwilliam [1] to the values 0 < p < 1.
22.5. In the paper by Pekarskii [3] the following modification of the fractional derivative
(vg f)(z):
r ( l+ ) 1 t-l-[a]f ( t )
f(a)(z) = . cv dt = Va[za-[a] fez)].
211"1 (1 - z/t)l+a 0
Itl=p
(23.13)
was suggested (d. 22.33) and (22.52)) with cv > 0 and Izl < p < 1. This modification corresponds
00
to the expansion f(a)(z) = L: r(ji» fkZk-[a] (d. (22.36)). It has the advantage
k=[a]
in the theory of analytic fWlctions that like operation za(vg f)(z) and operation (23.12), it
maps analytic fWlctions into analytic ones. This paper [3] contains Bernstein-type inequalities
for the fractional derivatives (23.13) of rational fWlctions. Investigations in [3] as well as in the
papers [I], [2] of the same author, were connected with the approximation of analytic functions
in Hardy-Besov spaces by rational fWlctions. We note that in [2] fractional differentiation of the
fonn (23.12) was used.
00
22.6. Let fa(z) = L: (ik)-a fkZk be the Weyl fractional integral (22.1). A number of
k=l

452
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
papers - Hirshmann [I], Flett [2], [4] - dealt with the Littlewood-Paley-type functions
I Ilk
9',a(8) = {f(1 - p).-a.-IIP-I/a_l(p';')I'dP}
o
and some of their modifications. Estimates of the L p ( -1I",1I")-nonn of gk,a(O) by the nonn of f in
the Hardy space Hp, 0 < p < 00 were given in particular. In this connection see also Sunouchi [1]
and Koizumi [1].
22.7. Lavoie, Tremblay and Osler [1] and also Lavoie, Osler and Tremblay [1] treated
fractional integro-differentiation Va of analytic functions of the fonn t fJ (In t)6 f( t) (S = 0 or S = 1)
with a power-logarithmic branching via a modification of the Cauchy integral fonnula (22.4). This
modification deals with integration along the "Pochhammer loop" (z+, 0+, Z-, 0-), (see Fig. 2),
which aims to guarantee a unified. fonn of definition of fractional integro-differentiation for all a
and {3 (except a = -I, -2,... and (3 = 1, 2,...). For such a fractional integro-differentiation
taken as a definition, the semi group property and the Leibniz role for Va(uv) were proved.. The
latter fonnula in the complex plane was earlier treated. in detail in the papers by Osler [1], [4], [5],
[7], [8], with the initial definition of fractional integro-differentiation via (22.30). As regards the
Leibniz rule, we refer also to  17.2, note 15.3.
The reader can also find in Osler [1]-[3], [5], [6] a number of representations for special
functions in the complex plane (such as the hypergeometric, Bessel, Struve and others) using
elementary functions, or other special functions by means of fractional integrals. The case of
hypergeometric functions and various of its particular cases may be seen also in Campos [2]-[4].
This case, as well as the case of hyperspheric functions, was in fact already known to Letnikov [7],
[8], [10]. The representations for the Riemann-Gurwitz zeta-function can be found in the paper
by Mikolas [33].
The role which the choice of an integration contour (the "Prochhammer loop") plays in the
case of functions having a branchpoint was also elucidated by Campos [1]. This author dealt with
the Cauchy type fonnula (22.33") using the Hankel contour £8 with 0 depending on z: (J = arg z,
and a number of his papers [2]-[8] was based on this fonn of fractional integro-differentiation.
In [2] in particular it was shown that both the Liouville and Riemann approaches (5.20) and
(2.26) are unified in the complex plane under the author's definition with the Hankel contour £6'
(J = arg z. We also especially note [5], where a useful concept of the so-called branch-operator was
introduced.. This concept arises, for example, if the order of integration in a composition of two
integro-differentiation operators is interchanged so as to make one path go through a branchpoint.
22.8. Osler [2] studied the derivative of a function by another function (see  18.3) defined
to be
Va fez) = reI + a) f f(t)h'(t)dt
h(z) 211"i [h(t) - h(z)]l+a
L
with the cut for [h(t) - h(z)]-I-a being a straight line going through t = z and t = h-I(O) and
the contour L going through h-I(O) and enveloping the point z.
The relation
{ [ ] l+a }1
a a f(z)g'(z) h(z) - hew)
V g(z)f(z) = Vh(z) h'(z) g(z) - g(w) w=z
in particular was proved., w = z being substituted after the evaluation of Vh(z) {. . .}.
Campos [8] gave extensions of Taylor and Laurent expansions of constructions of the type
Vh(z)f(z) in power of h(z). See also similar results in Osler [6] and Lavoie, Osler and Tremblay
[1] for the case h(z) = z - a.

 23. ADDITIONAL INFORMATION TO CHAPTER 4
453
22.9. Gaer [1] and Gaer and Rubel [1], [2] developed a peculiar approach to fractional
integra-differentiation via investigating it as an entire function of the parameter a. The space G
of functions analytic in the vicinity of the real line R1 and vanishing at infinity was considered.
Showing that for each t e R1 and any function f e G there exists a unique entire with respect
to z function F(z, t) of exponential type with order of growth along the imaginary axis less
than 11", such that k f(n)(t) = F(n, t), the authors defined the fractional derivative j<z)(t) as
f(z)(t) = r(1 + z)F(z, t). On the basis of this definition a systematic investigation of integra-
differentiation f( z) (t) was undertaken in the cited papers. There is also a development of some of
these results in Gaer [2], including an extension to the case of fractional powers of linear bounded
operators in a Banach space.
22.10. In the paper by Kober [7], fractional integration in the Liouville fonn (and in Riesz
fonn too) was considered in the Hardy spaces of functions fez) analytic in a half-plane or a strip.
22.11. Fractional order differences with the fixed step h = 1 in the complex plane were
considered by Diaz and Osler [1] in the fonn
00
OI fez) = L(-I)1: ()f(z + a - k), z e C,
1:=0
(c!. (20.2». The main result was the relation
OI fez) = r(a + 1) f f(t)r(t - z - a) dt
211"i r(t - z + 1)
C
where the contour C enveloped the ray L = {t : t = z + a -,   O} in tbe positive direction.
A function fez) was assumed to be analytic in a domain, containing the ray L, and such that
If(z)!  MI(-Z)OI-PI, p > O. A Leibniz-type fonnula for these differences OIf(z) was also
obtained.
22.12. A wide range of applications of fractional integro-differentiation in the theory of
analytic and meromorphic functions can be found in M.M. Dzherbashyan's papers. In his works
[I], [2, Ch. IX,  1-3], [3] fractional integra-differentiation was used to characterize some new spaces
of meromorphic functions in the disc and to obtain their parameter representations (factorization
theorems). In particular, a generalization ofthe Jensen-Nevanlinna expression, known in the theory
of mer om orphic functions, was given in terms of the function VOl (pe itp ,z) = p-OI Ig+ log 11-pe itp /zl,
a > -I, where the integro-differentiation is taken with respect to the variable p. The generalized
operator L(w) - (18.110) - was used for the same purpose by Dzherbashyan [7a]. Similar
results for functions meromorphic in a half-plane, but with the application of Liouville fractional
integra-differentiation, can be found in Dzherbashyan, A.M. [1].
In Dzherbashyan [6], [7] the fractional integra-differentiation V was used to generalize the
concept of quasi-analicity of functions. This generalization is based on replacing the integer order
derivatives in (implication) f(n)(xo) = 0, n = 0, 1,... => f(x) = 0 by fractional derivatives of
the fonn (VOI I)(xo), 0 < a < 1. The papers by Dzherbashyan and Nersesyan [3], Dzherbashyan
and Martirosyan [1] and Martirosyan and Ovesyan [I], concerning an applications of fractional
differentiation in theory of quasi-analytic functions are also relevant.
Dzherbashyan and Nersesyan [4], [5] used Riemann-Liouville fractional integro-differentiation
to COnstlUct and investigate expansions in biorthogonalsystems related to Mittag-Lemer functions
EOI,p(z). They showed that these systems contain eigenfunctions of certain mutually conjugate
boundary value problems for differential equations of fractional order. In this connection
Dzherbashyan and Nersesyan [5], [6] gave a detailed investigation of such boundary value
problems. These questions were further developed by Dzherbashyan [8]-[12].
22.13. Let fez) be a function analytic in the half-plane Rez > o. Results conceming the
behaviour at infinity can be found in Komatu [1]. HRef(z) > 0, then zOl-1(V: o f)(z) - c/r(2-a)
as z _ 00, I arg zl  a < 11"/2, where c does Dot depend on fez) and a, a e R1, provided that

454
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
Rezo > 0 and 1Yfof is the integro-differentiation (22.3). A certain analogue of this assertion for
the disc was also obtained.
For functions fez) analytic in the unit disc and satisfying conditions Ref(z) > 0 and
f(O) = 1 Komatu [2] proved the estimate IIl fIlLp(lzl=r)  r-alllkIlLp(lzl=r) where k(z) is a
fixed function, Ot> 0 and 1 f is the fractional integral (22.2) with zo = O.
22.14. Riemann-Liouville fractional integro-differentiation (V:oJ)(z), a e Rl and its
modification (22.55) have wide applications in studying the properties of univalent, convex
and starlike functions, questions connected with Biberbach problem on coefficients on univalent
functions, estimates for coefficients, distortion theorems and 80 on. In the last decade many
papers on these topics were published. We should like to emphasize the role of Komatu, Owa
and Srivastava in the investigations on these themes. We refer, for example, to Komatu [1]-[7],
Owa [1]-[18], Srivastava and Owa [1]-[7], Owa and Nishimoto [2], Owa and Shen [I], Reddy and
Padmanabhan [1], AI-Amiri [I], Owa and Ahuja [I], Sekine, Owa and Nishimoto [I], Srivastava,
Sekine, Owa and Nishimoto [I], Owa and AI-Bassam [I], Owa and Obradovic [1], Owa and
Sekine [I], Zhong zhu Zo [1], Cho, Lee, Kim and Owa [I], Owa and Ren [I], Fukui and Owa [I],
Nunokawa and Owa [1], Saitoh [1], Sekine [1]-[2], Shanmugan [1] and Sohi [1]. See also Owa,
Saigo and Srivastava [1] and Srivastava, Saigo and Owa [I], where Saigo's generalization (see note
18.6 above) of fractional integro-differentiation, extended to the complex plane, was used.
22.15. Dimovskii and Kiryakova [1] considered the weighted generalization (V a ,lJ f )(z) =
00
L: rf) ftzle-l of the special Gel'fond-Leont'ev operator (22.41) and the right-hand
le=l
00
sided inverse operator of integration (J o ,lJf)(z) = L: r ra1J fle zle + 1 . In particular, they
k=O
obtained the integral representation of the type (22.42):
1
(Ja IJf)(z) = -=-- ) 1 (1 - t)o-lt lJ - I f(zta)dt, J.I.  1
I rea
o
in the space H(G) of functions analytic in a starlike domain with the topology of compact
convergence. All linear continuous operators T: H(G) -+ H(G) commutiIig with Ja,l' were also
characterized.
Linchuk [1] extended this characterization to the case of an arbitrary complex J.I. and obtained
a simpler form of the characterization.
22.16. Operators (22.46) of generalized integra-differentiation in the form La f =
00
L: a k fle zle were considered by Komatu [3] from the semi group point of view and were
le=l
shown to admit, unlike (22.48), the representation
1
La f = 1 t- 1 f(zt)dO'a(t),
o
00
fez) = z+ Efkzle,
le=2
Izl < 1.
Here the measure dO'a(t) is determined by a and {ale}k:I' The case ale = Ilk was singled out,
and this corresponded to the operator o of Hadamaa'd fractional integration - note 22.3. In this
case the measure ua(t) was constructed effectively. The case 0'1 (t) = to, a > 0, when the density
of the measure O'a(t) is r(:) t o - I (log t) a-I, can be found in Komatu [6]. Some inequalities
for the operators La may be found in Komatu [4]. Komatu [5] investigated the oscillation of
Re [fLa fez)] on the unit disc in the case ak = k, the oscillation osc Reg(t) being defined as
Itl=r
max Reg(t) - min Reg(t), 0 < r < 1.
Itl=r Itl=r

 23. ADDITIONAL INFORMATION TO CHAPTER 4
455
22.17. Let a e R 1 and let H(a) be the space of functions f(z) analytic in the domain
S(a) = {z e C : Ihnzl < a, 0 < a  +oo} and satisfying the following condition. Given
t, 0  t < a, there exists A(t)  0 such that for each z = x + iy e S(t) the inequality
If(z)1  A(t) exp{x2/2 - Ixl(t 2 - y2)1/2} holds. Rusev [1] showed that the Erdelyi-Kober-type
operator
1
(I-1/2,Ot+1/2f)(z) = 2[r(Ot + 1/2)]-1 f (1 - t 2 )Ot-1/2 f(zt)dt
o
(d. (18.8)) with -1/2 < Ot < 1/2 realizes the isomorphism of the topological vector space H(a),
equipped with the topology of unifonn convergence on compact subsets of S(a), onto itself. This
was used to demonstrate that a necessary and sufficient condition for an analytic function fez) to
be representable in a strip Sea) by a series of generalized Laguerre polynomials {LOt)(z2)}=o
- ErdeIyi, Magnus, Oberhettinger and Tricomi [2, 10.12] with Ot e R1, Ot '# -I, -2,... - is that
fez) is an even fWlctions in H(a).
22.18. Let
1 f ( Z ) dt
(Pf)(z) = -: b - f(t)-
211"1 t t 2
C.
(23.14)
be an operator of the type (22.48) with a rectifiable closed Jordan curve C z enveloping the point
z and lying in the domain G of analyticity of fez). Also let the function b(z) be analytic in the
domain Iz - 11 > 0 and vanish at infinity with order not less than two. Korobeinik [I], [2] proved
that the operator (23.14) gives the general representation of a linear operator continuous in the
space H(G) of functions analytic in G. In the case 0 e G it coincides, for z sufficiently close to 0,
with some Gel'fond-Leont'ev operator of generalized differentiation. Moreover, the operator and
its powers admit also the representation by differential operators of infinite order:
00 
(pm f)(z) = L :m zl:-m f(I:)(z), m = 1,2,...,
I:=m
where the numbers I: m are determined by b(z) and satisfy the condition lim VII: ml = 0,
I 1:-00 '
m1.
23.3. Answers to some questions put at the Conference on
Fractional Calculus (New Haven, 1974)
We complete the part of the book related to the case of one variable by answering some questions
which were put at the conference, mentioned in the heading (Osler [9]). We single out below
those questions from Osler [9], which have an answer.
1. The question from Erdelyi. Let f(x) be continuous for x  0 and let S be the set of all
those nonnegative Ot for which (vg+f)(x) exists and is continuous or locally integrable. Does S
have the largest element?
2. The question from Lee Lorch. Does the mean value theorem of the differential calculus
possess an analogue which connects differences of fractional order with derivatives of the same
fractional order?
3. The question from Love. Are there any theorems known connecting fractional integrals
1:+ <p and 1b+ <p with different lower limits of integration?
4a. The question from Ross. The operation V:+ V+ might be said to be a measure of
deviation from the law of indices - the semi group property. What are some theorems that can
define this and of what significance can this be?

456
CHAPTER 4. OTHER FORMS OF FRACTIONAL INTEGRALS
7. The question from Lew. Let {/a}ao be a family of operators in Ll(O, 1) or L2(O,I). Do
t
the conditions: "1 0 f = f, II f = J f(u)du, [a [f3 = la+f3, and la is continuous in some operator
o
topology, la f  0 for f  0" determine the family {[a}aO uniquely?
The answers
1. The answer to the first question is in general negative: for f(x) = x f3 lnx, (3 > 0, we have
S = [O,(3) and S = [O,{3 + 1) respectively to the cases of continuity or integrability of (Vg+f)(x).
In the latter caae we have S = [0, (3 + 1) for f(x) = x f3 , {3 > 0, too.
2. In the case of functions given, for example, on the real axis, the answer is readily given
00
in the form (h f)(x) = ha(D+f)h (x), where Iph(X) = J pa(r)lp(x - 'Th)d'T denotes an average
-00
of a fWlction Ip(x) with the kernel (20.14) - (20.30).
However, it is not clear whether it is possible in the case of the continuous derivative to
write the equality (hf)(x) = ha(D+f)({), { E Rl. The latter is valid in the case of integer
a = I which follows from the expression
h h
(f)(x) = f... f f(l)(x + tt +... + t,)dtt,. .dt"
o 0
3. The answer is given in Section 13, see the Corollary of Theorem 13.9.
4a. The answer is derived from the Corollary of Theorem 13.9: (1+[+Ip)(x) = (1::t f3 t/1)(x) ,
x > a > c, where t/1(x) is the same as in the mentioned Corollary. See also note 18.14 in this
Subsection, the last part of which is also relevant to this question. As for the significance of
such connections, we can refer to the paper by Nahushev and Salahitdinov [1] stimulated by
applications to non-local bOWldary problems for differential equations.
7. The answer was given in  4.2, note 2.12.
We refer also to  17.2, note 12.6 and  29.2, note 25.19 for answers to some questions put
by Prof. E.R. Love at the 3rd International Conference on Fractional Calculus (Tokyo, Nihon
University).

Chapter 5. Fractional Integro-
Differentiation of Functions
of Many Variables
In this chapter we consider fractional integration and differentiation of functions
of many variables. In the multidimensional case there arise first of all partial
fractional derivatives , mixed fractional derivatives :OIl+;J and so on, as
. Zl Z
well as the corresponding fractional integrals. This approach is developed in
S 24. However, another approach is possible, that is to introduce fractional powers
(_A)a/2 for example where A =  + ... + k. Such an approach is developed
in U 25 and 26. It can be naturally generalized by considering the fractional
power [p(v)]a of a differential operator P(V) in partial derivatives with constant
coefficients. We do not dwell on such a generalization in this book, dealing only
with some special cases of the operator P(V) in SS 27 and 28. The reader can find
references to more general cases in  29.1, notes 28.2, 28.4 and 28.6.
The following notations are adopted throughout this chapter. Rn denotes
the n-dimensional Euclidean space, Rn is the compactification of R n by the
unique infinite pointj x = Xl,X2,...,X n ), t = (tl,t2,...,t n ), etc, are points in
Rnj Ixi = x + x + ... + Xj x . t = xlh + ... + xnt n is the scalar product in
R!'j x 0 t = (X it!,..., xnt n ) is the vector in Rn j dt = dtl ... dtn. We denote by
Rt....+ = {x : x E Rn, Xl  0,..., X n  O} the region in Jl!1 with nonnegative
coordinates. Sn-l will designate the unit sphere in Rn centered at the origin,
ISn-d = 21rn/2r-l(n/2) being its area. By j = (h,h,... ,jn) we denote any
multi-index so that xj = x{l  . . ... and Ijl = h + h + . . . + jn. This is not to be
confused with the designation for distance in Rn. Let V = ( {Jl "'" (J.. )' Then
vj = {Jbl
azl ...az" .
Writing a > 0 in the case a = (al," ., an) means that ak > 0, k = 1,2,..., n.
As usual, Lp(Rn) is the space of functions f(x) = f(Xl,...,Xn) such
that 11/11. = {j. I/(z)IPdz } 1/. < 00; CIf' = CIf'(R") is the space of infinitely
differentiable finite functions.

458 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
fi 24. Partial and Mixed Integrals and
Derivatives of Fractional Order
The following is a direct extension of Riemann-Liouville fractional integra-
differentiation operations to the case of many variables, when these operations
apply independently in each variable or in a part of them. Besides Riemann-
Liouville integra-differentiation, other forms will be also dealt with such as the
Marchaud or Griinwald-Letnikov approaches, Weyl-type definitions for periodic
functions of many variables and polypotentials.
24.1. The multidimensional Abel integral equation
We begin with a generalization of (2.1) to the case of many variables. Let
rp(x) = rp(Xl,"',X n ), f(x) = f(Xl,...,Xn) be functions of n variables and let
a = (a 1, . . . , an) be a fixed point in R n . We consider the equation
ZI Z..
1 f f rp(t)dt
r(a) ... (x - t)l-a = f(x),
al a..
X> a,
(24.1)
where
a = (at,..., an), (x - t)a-l = (Xl - tl)a 1 -l ... (xn - tn)a..-l,
r(a) = r(ad... r(a n ), dt = dtl ... dtn
(24.2)
and writing x > a denotes that Xl > aI, . . . , X n > an.
Assuming that 0 < ak < 1, k = 1,2,..., n, we operate on each variable
following just the same lines as in (2.2) and (2.3). This gives the relation
] ... 7 (8)d8 = [(1  Q) ]... 7 (a
al a. al a..
the notations of (24.2) being used here. Differentiating this with respect to
Xl,.." X n we obtain
ZI Z..
1 on f f f(t)dt
rp(x) = r(1 _ a) OXI ... oX n ... (x - t)a '
al a..
(24.3)
Thus, if (24.1) has a solution, it is unique and is given by (24.3).

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
459
24.2. Partial and mixed fractional integrals and derivatives
Starting from the one-dimensional definition (2.17), we can naturally define the
partial Riemann-Liouville fractional derivative of the order al: with respect to the
k-th variable by the relation
Zle
( rJ/1e )( ) =  f rp(Xl"" ,XI:-I,, XI:+1,... ,x n ) 
ale+rp X r(al:) (XI: _ )l-ale '
ale
(24.4)
where al: > O. This definition assumes functions rp(Xl,"', x n ) to be given for
XI: > al:. Introducing the notation el: = (0,... ,0,1,0,.. .,0) for the k-th unit
---..---
1:-1
vector, we can rewrite the fractional integral (24.4) in the following shorter form
zle-ale
( Iale )( ) f rp(x-el:)d.
..+ z = r(.) {i-a. "
o
Further, the expression
(I:+rp)(x) = (1: 1 \ ... I::+rp)(x)
Zl Z..
1 f f rp(t)dt
= r(a) '" (x-t)l-a ' a>O,
a1 a..
(24.5)
defined for functions rp( x), given for X I: > ak, k = 1,2,.. . , n, will be called a left-
hand sided mixed Riemann-Liouville fractional integral of order a = (al," ., an).
The mixed fractional integral may be applied to only some of the variables, i.e. the
values al: = 0 may be admitted. In such a case we set 1::+1,0 == 1,0 in (24.5) and
taking al: = 0 for simplicity for k = m+ 1,...,n and al: > 0 for k = 1,2,..., we
have
Xl z...
a 1 f f rp(r,x")
(Ia+rp)(x) = r(ad... r(a m ) ... (x' _ r)l-a t dr,
a1 am
(24.6)
m
with X' = (x},..., x m ), x" = (Xm+l"" , x n ), (x' - r)l-a = n (XI: - rl:)l-a le and
1:=1
1 - a ' = (1 - aI, . . . , 1 - am)'
The right-hand sided mixed fractional integral Ir_rp is defined in a similar way
for functions rp(x), given in the region x < b. A more general variant is indeed
possible when integration is left-hand sided on some of the variables and right-hand

460 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
sided for others, for example,
ZI Z'" 6"'+1 b..
01 1 f f f f cp(t)dt
(Ia+.6_CP)(x) = r(a) ... ... (x' - t')1-OI'(t" - X,,)l-OI" '
al am Z"'+1 z..
(24.7)
where x' and x" are the same as in (24.6), 1 - a" = (1 - am+l, . .. , 1 - an), and
t = (t',t"), t' = (it,.. .,t m ), t" = (t m + l ,... ,t n ), 1 ::; m::; n.
Starting from (24.4) and (2.22) we introduce a partial Riemann-Liouville
fractional derivative of order aI:, 0 < al: < 1, in the k-th variable as
Z,.
(VOl. f)(x) = 1 0 f f(XI,...,XI:-I,,XI:+I,...,Xn) 
a,.+ r(1 - al:) OXI: (XI: _ )OI.
a.
z.-a,.
= r(1  "'1) 0: 1 f Ca. fez - e1).
o
In the case of differentiable functions f(x) this may be written similarly to (2.24)
as
(VOl. f)(x) = 1 [ f(XI"",Xk-l,al:,Xk+I,...,xn)
a.+ r(1 - al:) (XI: - al:)OI.
z.
f 8(XI,...,XI:-I,,XI:+I,...'Xn) ]
+ () d.
XI: -  01,.
a.
(24.8)
In view of the inversion (24.3) valid for 0 < al: < 1, we spall call the right-hand
side in (24.3) a mixed Riemann-Liouville derivative of order a = (at,..., an).
Similarly to (24.6) a mixed fractional derivative may be applied to some of
the variables. Specifically let al: = 0 for k = m + 1,.. ., n, and 0 < al: < 1,
k = 1,2,.. . ,m, as in (24.6). The expression
ZI Z'"
01 1 om f f f(r, x")
(Va+f)(x) = r(1 _ a) OXI .,. oX m ... (x' _ r)OI' dr
al a",
(24.9)
will be called a mixed Riemann-Liouville fractional derivative of order a =
(al,'" ,am,O,... ,0).
We note that the order of differentiation in (24.3) or (24.9) is essential. Thus,
the function f(XI' X2) = (X2 - a)OI-lg(xd. 0 < a2 < 1, where g(xt) is assumed to
be continuous but nothing more, has the derivative V+ f == 0 with the order of

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
461
differentiation used in (24.3), but has no such a derivative if one takes 02/0X20Xl
in (24.3).
If the function f(x) is differentiable up to order m, the form (24.9) may be
transformed to an expression similar to (24.8) by means of differentiation under
the integral signs. We limit ourselves by the corresponding result in the case of
two variables, x = (Xl, X2):
(1'(al,a) f)(x) _ 1 [ f(al,a2)
(al,a)+ r(1 - adr(l- (2) (Xl - ad a1 (X2 - a2)a
z
1 f of(al, t2) dt2
+ (XI - ad a1 Ot2 (X2 - t2)a
a
ZI
1 f of(tl,a2) dt l
+ (X2 - a2)a at l (XI - td a1
al
ZI z
f f (j2 f(ll, t 2 ) dtldt2 ]
+ atlOt2 (Xl - t l )a 1 (X2 - t2)a .
al a
In the case aA: > 1 we define mixed fractional derivatives, following (2.30), by
the equality
1':+f = -pj [+a f, a  0,
(24.10)
where j = ([al] + 1,... ,[am] + 1,0,... ,0).
One may obviously introduce integra-differential operators [:+, a = (al,"',
an), with the orders aA: of different signs, i.e. such operations which imply
fractional integration for some of the variables and fractional differentiation for
others.
The validity of the semigroup property
[ a [ {j _ [ a+ {j
a+ a+<P - a+ <p,
a  0, {3  0,
(24.11 )
is verified as in the proof of (2.21), a + {3 being the sum of vectors a and (3. The
function <p(x) is assumed to be integrable in any bounded part of the region X > a.
The reader may directly check that any monomial x fJ - I = Xfl-1 ... x..-I has
partial or mixed fractional integrals or derivatives (24.4), (24.5), (24.9) or (24.10)
with a choice a = (0,... ,0) evaluated by the formula
-na ( (j-l ) _ f(,B) {j-a-l
£'0+ X - f({3-a)x ,

462 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where {3 = ({3I,. .. , (3n) > 0, Q = (QIt ... , Qn) is arbitrary, :zo;. = 10+011 in the case
Qi < 0 and r({3) = r({31)'" r({3n) (cf. (2.44». ·
In the case of functions given on the whole space R n we may consider
Liouville-type fractional integrals
1t....+cp = r(I Q ) f tOl- 1 cp(X - t)dt,
R+...+
(24.12)
Jl!...._cp = r(I Q ) f tOl- 1 cp(X + t)dt,
R"
+...+
(24.13)
where Rt....+ is the region {t: tl  O,...,t n  O}, t Ol - 1 = trl-1...t,,-I. One
may also define fractional integration of the type l...%cp similarly to (24.7) with
an arbitrary choice of signs + and -.
Liouville partial derivatives 1)k j are introduced by the relations
00
0 0110 j'@1)OIkj'@ 1 0 f f(Xl,...,xk-l,xk-,xk+l"..,xn) d
OXk + r(1 - Qk) OXk OIk '
o
(24.14)
where mixed Liouville derivatives
1).,.+j, 1)..._j
(24.15)
are defined similarly to (24.9), (24.10), e.g.
(-1 )n+[ 0I 1]+"'+[0I,,] o[ 0I 1]+'''+[0I,,]+n f
1)..._j = n [ 01 1]+1 [01,,]+1 tOl- 1 cp(X + t)dt. (24.15')
n r[[Qk] + 1 - Qk] oX I . . .ox n R"
k=1 +...+
We note also the validity of the "fractional integration by parts" formula
f cp(x)(lt....+ 1P )(x)dx = f 1P(x)(l..._cp)(x)dx,
R" R"
(24.16)
which can be derived directly from (5.16).

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
463
24.3. The case of two variables. Tensor product of operators
We shall specially consider the case of two variables Xl and X2 in this subsection.
It is convenient to use the concept of the tensor product of operators, introduced
by the following definition.
Definition 24.1. Let Al U and A 2 v be linear operators defined on functions U(Xl)
and V(X2) of one variable. The tensor product of operators Al and A2 is an
operator Al @ A2, which is defined on functions of the form
I;'(Xl, X2) = L Ui(Xl)Vi(X2)
(24.17)
by the relation
(AI @ A 2 )1;' = LA I U i A 2 V i.
(24.18)
In the case of concrete classes X of functions I;' (for example, X = Lp(R2» the
functions of the form (24.17) generate, as usual, a dense set in X, so the operator
Al @ A 2 if continuous, is uniquely extended from the functions (24.17) to the whole
space X.
It follows from Definition 24.1 that the mixed fractional integral 1:+1;',
Q = (Ql, (2), is the tensor product of one-dimensional fractional integrals:.,
f a - f a 1 10\ f a
0+1;' - 01+ '<Y 0+1;',
Ql  0, Q2  O.
(24.19)
the same being true for fractional differentiation
1):+f = 1)::+ @ 1)::+f, Ql  0, Q2  O.
In the case when Ql and Q2 have different signs we write
fa - V- a1 10\ fa
0+1;' - 01+ '<Y 0+1;',
Ql < 0, Q2  O.
Liouville forms (24.12) and (24.13) of fractional integration for functions
1;'( X I, X2), defined on the whole plane, are
00 00
a al f a 1 II I;'(XI =Ftl,X2 =F t 2) d d
1.%%1;' = f% @ % I;' = r(Qd r (Q2) tl-altl-a tl t2.
o 0 1 2
(24.20)

464 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
One may also consider the operators
[ 01 _ [ Oil 10\ I OI
::I:T<J' - ::I: '<Y T cP,
Ql  0, Q2  0,
(24.21)
under the respective choice of signs. The case Ql = 0 or Q2 = 0 corresponds to
partial fractional integration
I ( 011,0) - l Oll 10\ E [ (O.OI) - E 10\ 1 01
::1:::1: -::I: '<Y , ::1:::1: - '<Y::I:,
(24.22)
E being the identity operator.
Let Q1CP = cp(-Xl,X2), Q2CP = cp(Xl, -X2) and Q12CP = CP(-Xl,-X2) so that
Ql = Q@E,
Q2 = E@Q,
Q12 = Q1Q2 = Q@Q
in correspondence with definition (24.18), Q being the operator defined in (5.9).
The evident relations
QII++ = I+ Ql,
Q2 1 ++ = 1_Q2,
QI2 1 ++ = 1_Q12
are true, cf. (5.9).
Fractional differentiation can also be written In a form similar to (24.19)-
(24.22), for example
00
'Da"o) I = ('D%' 0 E)I = :!: r(1  ad a1 J tr' -1/("'1 - tl, "'2)dt,
o
(24.23)
V::I:f = Vll @ V% f and so on.
The concept of tensor product of operators is easily extended to the case of n
variables, but we shall not dwell on such an extension.
24.4. Mapping properties of fractional integration operators
in the spaces L,(Rn) (with mixed norm)
To simplify the exposition we shall consider the case of two variables Xl and X2
only. Our aim being a natural extension of the Hardy-Littlewood theorem 5.3, we
S h all deal with O p erators 1(011,0) 1(0,0I) and I(0I1.0I) not from L ( R 2 ) into L (R 2 )
::I: ,%::1:::1: p q,
but within the frames of the spaces of functions integrable in respect to each
variable with in general the different powers PI and P2. Specifically let us define

 24. PAFUIAL AND MIXED INTEGRALS AND DERIVATIVES
Ute space Lp(R2), P = (Pt,P2), of functions /(Zl,Z2) with the finite norm
465
{ J [J ] P'J/PI } I/P'J
II/lip = 1/(zl, z2)I Pl dz i dZ2 < 00,
Rl Rl
(24.24)
where, as usual, 1  Pi < 00, i = 1,2. The space Lp(R2) is known as the mixed
Bonn space. It is clear that L p (R 2 ) = Lp(R2) in the case PI = P2 = p.
The main theorem below is preceded by the following auxiliary lemma.
Lemma 24.1. Let Al be an arbitrary linear operator bounded from L P1 (Rl) into
L q1 (Rl) and let A 2 be a convolution operator
00
A 2 <P = J k( )<p( Z2 - )d
-00
with a non-negative kernelk()  0, which is bounded from Lp'J(Rl) into Lq'J(Rl),
1  Pi < 00, 1  qi < 00, i = 1,2. Then the operator Al @ A 2 is bounded from
L,(R2) into L,(R 2 ), P = (Pit P2), q = (q}, Q2).
Proof. The operator Al @ E is bounded from Lpl,q'J(R2) into Lql,q'J(R2) (for any
q2) if the operator Al is bounded from L P1 (R I ) into L q1 (R 1 ), this being easily
verified by means of Fubini's Theorem 1.1. Since Al @ A 2 = (AI @ E)(E @ A 2 ),
it is sufficient then to show that the operator E @ A 2 is bounded from L p1 ,p'J (R 2 )
into Lpb9'J(R2). We have
00
(E @ A 2 )<p = J k(t 2 )<p(z}, Z2 - t2)dt 2 ,
-00
and applying the generalized Minkowsky inequality (1.33) we obtain the estimate
{ J } l/Pl J .
I(E @ A2)<pI P1 dz l  k(t2)1I<p(', Z2 - '2)lIpl dt2,
Rl Rl
(24.25)
the norm being taken with respect to the first variable. Since 1I<p(.,t 2 )lI p l E Lp'J(Rl)
and A 2 is bounded from L p2 (R 2 ) into Lq'J(R2), after estimating (24.25) under
Lf'J(Rl )-norm with respect to Z2, we have
II( E @ A 2 )<pllpl,q'J = IIII(E @ A 2 )<pllpll1q'J  cll<pllpl'P'J'

466 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
which was what is required. .
Theorem 24.1. Let 1  Pi < 00, 1  qi < 00, i = 1,2. The operator Ii allo ) of
partial fractional integration is bounded from LP1,P'J(R2) into Lqllq'J(R2) if and only
if
1 < PI < I/ Ot 1, 1  P2 < 00, l/q1 = I/p1 - Ot1, q2 = P2. (24.26)
Operators 4 a 1,a'J) of mixed fractional integration are bounded from LplIP'J(R2) into
Lql,q'J(R2) if and only if
1 < Pi < I/ Ot i, l/qi = I/pi - Otl, i = 1,2.
(24.27)
In view of (24.22) and (24.20) the sufficiency part of this theorem follows
directly from Lemma 24.1 and the one-dimensional Hardy-Littlewood Theorem
5.3. The verification of the necessity of (24.26) and (24.27) is realized in the same
way as in the case of functions of one variable in Theorem 5.3, if we introduce the
dilatation operator in each variable:
116rp=rp(6ox), 6ox=(6 1 x},...,6 n x n ), 6>0
for which 111I 6 rpll" = 6- 1 /"llrpll" with 6- 1 /" = 6;1/ P1 .. . 6;1/ p ".
Remark 24.1. Let us note that the behaviour of operators Al !g) E and E!g) A 2 in
the space Lfi(R2) is unequal because of integration (24.24) being taken first in Xl
and then in X2. The operator A 1 !g) E is bounded from LplIp'J(R2) into Lql,P'J(R2)
for any P2, and any operator A 1 , bounded from L P1 (R 1 ) into L q1 (R1), by Fubini's
theorem, while the fact that the operator A 2 is bounded from L q1 (R 1 ) into Lq'J(R2)
implies that of the operator E!g) A 2 from L p1 ,ql (R 2 ) into Lpl,q'J(R2) not for all
Pl. (We refer to Krepkogorskii [1] for examples of such bounded operators A 2 in
L 2 (RI) that E!g) A 2 is bounded in Lp1I2(R2) not for all pd.
24.5. Connection with a singular integral
For functions of two variables we consider the operator
00 00
N ( ) a1 f rp(t1,X2) d a2 f rp(X1,t2) d
rp =aorp Xl, X2 + - t1 + - t2
7r t1 - Xl 7r t2 - X2
-00 -00
00 00
a12 f f
+-;2"
-00 -00
(24.28)
rp(t 1 , t2)dt 1 dt 2
(t1 - X.)(t2 - X2) ,

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
467
known as a bisingular integral operator. The coefficients ao, al, a2 and a12 are
taken to be real. Using the designation
00
Scp = .!.. f cp(t)dt
7r t-x
-00
for the one-dimensional singular operator (which is familiar from S 11), we can
represent the bisingular operator (24.28) in terms of tensor products:
N = aoE@ E+ a1S@ E+ a2E@ S + a12S@S,
Lemma 24.2. The bisingular operators S @ E, E @ Sand S @ S are bounded in
the space
Lfi(R2), 1 < Pi < 00, i = 1,2.
In this lemma that S@E is bounded follows from Lemma 24.1. We do not give
the proof for E @ S, noting only that it can be obtained by means of a theorem of
J .T. Schwartz [1] on singular operators with values in a Banach space - Lizorkin [7].
As for S @ 5, that it is bounded follows from the relation 5 @ S = (S @ E)(E @ S).
We designate by
NOll cp = cas al7rcp + sin al1rScp
the one-dimensional singular operator arising in (11.10) and (11.11) which connects
fractional integration operators It. and I with each other. We introduce the
similar bisingular operator
NOII'OI'JCP = NOll @ NOI'Jcp,
(24.29)
which allows us to write analogous connections between the operators It. + , It.-
andI_.
Theorem 24.2. Let cp(Xl,X2) E Lfi(R 2 ), 1 < Pi < l/ai, i = 1,2. The following
identities are valid
It.+cp = I+NOII'OCP,
(24.30)
(24.31 )
(24.32)
It.+cp = It.- No,OI'J cp,
It.+cp = I_NOII'OI'JCP, I_cp = It.+N-OII,-OI'JCP,

468 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Prool. Equations (24.30)-(24.32) are easily derived from the corresponding
equations (11.10) and (11.11) for functions of one variable. Indeed, (11.10) and
(11.11) show directly the validity of (24.30)-(24.32) for functions of the form
<P(Xl, X2) = L: U,(XI)V,,(X2)' u,,(xd E LPI (R 1 ), V,,(X2) E Lp(Rl).
.
Such functions are dense in L,(R 2 ) and we observe that it is sufficient to consider
u,,(xd and V,,(X2) to be step functions only. Thus the validity of the required
equations follows from the boundedness of the operators on the left- and right-hand.
The latter is given by Theorem 24.1 and Lemma 24.2. .
24.6. Partial and mixed fractional derivatives in the Marchaud
form
In the case of "good" functions we can represent partial Liouville fractional
derivatives (24.14) in the Marchaud form according to (5.57):
00
VOile 1 = Qt 1 I(x) - I(x =F et)  0 < Qt < 1,
% r(l - at) 1+0I1e '
o
(24.33)
t-l

where et = (0,...,0,1,0,.. .,0). A similar passage in the case at  1 is made as
in (5.80), so that
VOile 1 _ 1 1 00 (leele 1)( x) 
+ - x(at, It) 1+0I1e '
o
(24.34)
where the integer It is chosen to be Ik > a", the finite difference leele applies in
respect to the variable Xt and x(at, I,,) is the constant (5.81). Hence it is readily
seen that for mixed fractional derivatives V<J:...:i;/, a = (al,' .. ,an), we obtain
1 1 (At/)(x) dt
V%...%I = x( a, l) t 1 +0I '
R+...+
a> 0,
(24.35)

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
469
instead of (24.34). In (24.35) the mixed finite difference
(L\f)(x) = L\ [L\ . .. (A:f)](x)
= L (-1) Ij 1 (  ) f (x - jot)
Ojl J
(24.36)
of the vector order 1= (/1,... ,In) and with a vector step t = (tl,'" ,tn) is used.
Here j 0 h = (hh l ,..., jnhn) and lie are integers such that 0 < ale < lie and
n
C) = n e k ). The normalizing constant x(a, I) in (24.35) is equal to
) Ie=l)k
n
x(a,/) = II x(ale,IIe).
Ie=l
We shall denote the right-hand side in (24.35) by the new notation D%...:i:f,
taking into account the fact that it may exist when 'D...:i:f does not exist.
In (24.35) various kinds of fractional differentiation of the type 'Dt.--+... for
example, may be considered, which correspond to various choice of signs: we deal
with the left-hand sided differentiation for some of the variables and with the
right-hand sided for others. In this case the step of the difference is chosen of the
type (tl, -t2, -t3, t4, .. .), which corresponds to the selected distribution of signs +
and -.
Let e = (el,"', en) > O. We shall call
00 00
01 1 f f (Af)(x)
D+...+,Ef = x(a, I) ... tl+OI dt.
El E..
(24.37)
a truncated Marchaud fractional derivative.
It was assumed in (24.35) and (24.37) that a > O. The reader can easily
write down. the corresponding constructions in the cases when ale = 0 for some
k = 1,2,....
We shall briefly consider the notion of fractional integro-differentiation in
a given direction. Various types of definition are possible. The Marchaud
form (24.33) of partial fractional differentiation in the k-th variable admits of a
direct generalization to the case of differentiation in a given direction. Thus, let
'JJ = (WI,. .., w n ), Iwi = 1, be a vector in defining a direction. Starting from (24.33)

470 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
we shall call the expression
00
( Va f )( ) = a 1 f(x) - f(x - w) d
w x r(1 _ a) l+a .
o
(24.38)
the fractional derivative in the direction w of order a (E R I ), 0 < a < 1. Starting
from (24.34) we shall define the fractional derivative of arbitrary order a E R in
a given direction by the formula
( Va f ) (x ) =  1 00 (L\wf)(x) 
w x(a, I) l+a
o
(24.38')
using the finite difference taken along the direction W, Iwl = 1. This definition
is well suited to functions f( x), defined in R n , or at least in the infinite domain
containing with each its point and also the ray starting from this point and having
the same direction as the vector -w. As for functions defined in a bounded domain
n in R n , it is possible to introduce an analogue of Marchaud fractional derivative
(13.2) for the interval. Specifically, let a = (a I,... I an) be a fixed point in the
domain n. Starting from (13.2) and (24.38) we introduce the expression
I x-a 1 f ( ) f ( c x-a )
f(x) a 1 x - x - , rz=ar
D+f = r(1 _ a)lx _ ala + r(l _ a) l+a d,
o
o < a < 1,
(24.38")
and call it a fractional derivative of order a in the point x in the direction from the
point a. A similar variant with D_f might be called a derivative in the direction
to the point a.
We may also define the fractional integral in a given direction as
00
I;:f = r(l",) 1 a-l f(x - w)d.
o
(24.39)
In the case of sufficiently good functions f(x) we can write the fractional
derivative (24.38) in the form
Va f = ll-a f 0 < a < 1 ,
w dw W I

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
471
Iw denoting the usual differentiation in the direction w. The operation (24.38') is
the inverse to fractional integration (24.39): V:J:CP == cp in the case of sufficiently
good functions cp(x). This may be verified directly, but it is much easier to refer to
(24.48') for the Fourier transform of fractional integrals and derivatives. Relations
(24.48') yield also the semigroup property I:Icp == I:+fJcp. S 29.2, note 24.3 gives
further information of fractional integra-differentiation in a given direction.
24.7. Characterization of fractional integrals of functions in
L,( R2)
We deal with functions I(Xl, X2) of two variables in this subsection. The results
obtained here are similar to those of S 6.2 and may be considered as their direct
extension to the case of partial and mixed fractional integra-differentiation of
. to'-I_(t_l)O'-1
functions of two variables. Following (6.7) we denote K:a(t) = slOf(af( + t + ,
thereby noting the dependence of this kernel on a. Like in (6.6) we obtain the
relations
00
D (a 1 ,0) I (a 1 ,0) - f r (t) ( t )dt
+,£1 + cp - "'al cp Xl - Cl , X2 ,
°
(24.40)
00
D:::) I,a)cp = f K:a(t)CP(Xl,X2 - c2 t )dt,
o
(24.41 )
00 00
D++,£I++cp = f f K: a1 (t)K:a(t)CP(Xl - cltl, X2 - c2 t 2)dt l dt 2,
° °
(24.42)
where DO) = D£1 @ E is the truncated Marchaud fractional differentiation
(5.59) in the first variable, D:::) being the same in the second variable, while
D++,£ is the truncated mixed Marchaud fractional differentiation (24.37).
Theorem 24.3. Let I(Xl,X2) = Iallo)cp with cp(Xl,X2) E L,(R 2 ), 1  Pl < l/al,
1  P2 < 00. Then cp(Xl' X2) = lim D'O) I, the limit being taken in L,(R 2 )_
£1- 0 ,1
norm. Similarly, il I(Xl,X2) = I++cp with CP(Xl,X2) E L,(R2), 1  Pi < l/ai,
i = 1,2, then
cp(Xl, X2) = lim D++ £1.
£-0 '
(L,(R»
(24.43)
Theorem 24.3 is proved by means of (24.40)-(24.42) following the same lines
as in Theorem 6.1: the properties (6.7) and (6.8) of the kernels K:a.(t), i = 1,2, are
to be used.

472 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
We introduce similarly to (6.1) the space
1:l::i: (L p ) = {f : f = 1:l::i:tp, tp E L p (R 2 )}
(24.44)
of mixed fractional integrals of functions belonging to Lp(R2). This space is
properly defined if 1 =:; Pi < Qi, i = 1,2. By Theorem 24.2, and in view of the
boundedness of the bisingular operator Na1,a'J in Lp(R2) by Lemma 24.2, the space
(24.44) does not depend on the choice of the signs if 1 < Pi < I/Qi, i = 1,2, so we
denote
de! )
l a (L p ) = 1++(L p = 1_(L,) = 1+_(L,) = 1+(L,),
1
1 < Pi < -, 0 $ Qi < 1, i = 1,2.
Qi
(24.45)
In the theorem below we use the designation
_ ( Pl P2 )
Pa = , ,
1 - Q1Pl 1 - Q2P2
1
1 < Pi < -, i = 1, 2,
Qi
and notation D++,ef for the truncated Marchaud derivative (24.37).
Theorem 24.4. Let 0 < Qi < 1, 1 < Pi < ;i ' i = 1,2. Then f(x) E l a (L,) if and
only if
1) f(x) E Lpor(R2),
2) there exists lim D++ ef in L,( R2).
e-O '
Proof. The "only if' part follows from Theorem 24.3. The "if" part will be
obtained following the proof of the sufficiency criterion in Theorem 6.2. We are
to show that conditions 1) and 2) imply the representability f(x) = I++tp with
tp E Lp(R2). Let us denote g(x) = 1++D++f, where D++f = lim D++ ef E L,.
e-O I
Instead of the equality f(x) = g(x) we shall prove the coincidence
( (1,1» (1,1» ) )
A(h lJ h'J)f (x) = (A(h1,h'J)g (x
(24.46)
of finite differences of these functions. "'Ie mean here the mixed difference of the
first order in each variable, see (24.36). Further arguments are exactly the same as
in Theorem 6.2 after (6.21) and so the proof is left to the reader. We only note

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
473
two main points: a) instead of the operator (6.22) we have to use the operator
AhCP = f f ah(X - t)cp(t)dt, ah(X) = (Al,l)ka)(X),
R'J
where h = (h l , h 2 ), ka(x) = r(adlr(a'J) xrl-lx'J-l, if Xl > 0 and X2 > 0 and
ka(x) = 0 if Xl < 0 or X2 < 0; b) instead of (6.23) we arrive at the representation
00 00
AhD++,£1 = f f K,al (tdK,a'J(t2)(L\'11,t)/)(x - d)dt.
o 0
Theorem 24.4 allows us to obtain information on partial fractional derivatives
from information on the mixed fractional derivative. Specifically the following
Corollary is true.
Corollary. Let lim D++ £1 E Lp(R2), 1 E Lpor(R2). Then DaloO) 1 E Lpl.r'J(R2),
£-0 '
D,a'J) 1 E Lrl,P'J(R2), ri = pi/(1 - aiPi), i = 1,2.
Indeed, 1 = I++cp, cp E LploP'J' by Theorem 24.4 and then Dal'O) 1 =
D(a1,O) la1,a'J) cp - [a'J cp E L ( R 2 )
+ ++ - + Pl.r'J .
24.8. Integral transform of fractional integrals and derivatives
The whole of this section is concerned with functions of n variables defined on the
whole space Rn. Let
:Fcp = (x) = f eixotcp(t)dt
Rn.
be the multidimensional Fourier transform. Since the latter reduces to successive
applications of one-dimensional Fourier transforms to each variable, we conclude
from Theorem 7.1 that the following relations
:Flt....+cp = (-ix)-a<,O(x),
:FIo.._cp = (ix)-a(x),
(24.47)
(24.48)
are valid for Liouville fractional integrals (24.12) and (24.13), where (-ix)a =
(-iXl)a1...(-ixn)an., 0 < al: < 1, k = 1,...,n, (ix)a being defined in the same
way and cp(x) E Ll(Jl'&).
One may evidently write the relations for I..o%cp for all choices of signs + and

474 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
The results
:F(I:/) = (-ix 'W)-OIj(X),
:F(1):/) = (-ix . W)OI j(X)
(24.48')
for fractional integra-differentiation I I, 'DI in the direction w - see (24.39),
(24.38) and (24.38') - are also established by direct transformations.
Let
Lcp = J e-y.tcp(t)dt, Y = (Yl,.. ., Yn),
R+...+
(24.49)
be the multidimensional Laplace transform, Rt....+ being the reglOn {t : tl 
0,... , t n  O}. Similarly to (7.14) we obtain
(LIg+cp)(y) = y-OI(Lcp)(y),
(24.50)
where
ZI z..
Ig+cp = r(l a ) J... J(x - t)0I-1cp(t)dt.
o 0
(24.51)
The known result for differentiation of the Laplace integral
(-'D)I: Lcp = L[tkcp(t)], k = (k 1 ,..., k n ),
is extended to fractional integra-differentiation: thus
1)..._Lcp = L,p, ?jJ(t) = tOlcp(t), a  0,
1..._Lcp = L,p, ,p(t) = t- OI cp(t), a < 0,
(24.52)
(24.53)
which can be verified directly by interchanging the order of integration on the
left-hand side. The relation (24.53) is valid for a  1 also, if t- OI cp(t) is integrable.
For the multidimensional Mellin transform
(mtcp)(x) = J tZ-1cp(t)dt
R"
+...+
(24.54)

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
475
the relations
r(l-x-a)
(rolI+<p)(x) = f(1 _ x) (rol<p)(x + a),
01 f( X)
(rolI_..._<p)(X) = r(X + a) (rol<p)(x + a),
(24.55)
(24.56)
similar to (7.20) and (7.21), are valid under the usual assumption r(x + a) =
r(Xl + ad . . . r(x n + an) and so on.
The proof of the above relations is not dealt with in detail: it is easily derived
from the corresponding one-dimensional formulae in S 7 under the appropriate
assumptions on functions <p(t) = <p(t1, . . . ,t n ). For example in (24.53) it is sufficient
to assume that the function t-OI<p(t) is locally integrable and slowly increases at
infinity.
24.9. Lizorkin function space invariant relative to fractional
integro-differentiation
Following S 8.2 we define here the space  of functions of many variables which
will be invariant relative to partial and mixed fractional integra-differentiation.
As in S 8.2, the idea of constructing such a space is clear in Fourier transforms.
Namely, in view of (24.47) and (24.48) it is clear that the required invariance of
the space  holds if the functions cp(t) in this space have Fourier transforms, (t),
identically vanishing on the hyperplanes XI: = O. So we denote by W the subs pace
of Schwartz space S(R n ), which consists of functions vanishing together with all
their derivatives on the hyperplanes XI: = 0, k = 1,2,.. . ,n:
w = {,p(x) : ,p E S(R n ), (vi ,p)(Xl, .. . ,XI:-1, 0, XI:+1, .. . ,x n ) == 0,
Ij I = 0, 1, 2, . . ., k = 1, 2, . . . , n}.
..
-lzl- L: z;
The function 1/J(x) = e .=1 is an example of such a function in W.
We shall call the space
 = (Rn) = {<p : <p E S(R n ),  E W}.
(24.56')
of Fourier transforms of functions in  Lizorkin space. Since
00
(j)(x)lz.=o = i 1jl f eiz'.t' (t')j' dt' f <p(t',)j.,
R..-l -00

476 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where t' = (t1,...,tt-1,h+1,...,t n ), j' = (j1,...,h:-l,jl:+1,...,in), it follows
immediately from the definition that the space (Rn) consists of those and only
those functions rp(t) E S(Rn), which have all their moments along coordinate axes
equal to zero:
00
f rp( t 1, . .. , t I: -1,  , h+ 1, . . . , t n )m  = 0,
-00
= 0,1,2,..., k = 1,...,n.
We easily derive directly from the definition that if rp(t) E (Rn), then also any
Liouville fractional integral (derivative) [...:t.rp belongs to (Jr), -00 < a < 00.
Lemma 24.3. Relations (24.47) and (24.48) are valid for all rp(x) E (Rn) and
al:  0, k = 1,2,.. . ,n.
Lemma 24.3 follows from Lemma 8.1 if we take into account the fact that the
function rp(t1,...,tl:-1,,tl:+1,...,tn) belongs to the space (R1) for rp E (Rn),
t1, t2, . .. being fixed.
As in S 8.2 we might define the space '(Rn) of generalized functions dual
to (Jr) and consider fractional integrals and derivatives of such generalized
functions. We do not consider this in detail. We note only that one may
obtain justification for the formula of fractional integration of Dirac delta-function
6 = 6( x) in this approach:
[ a!: 1 t a - 1 0
+...+u= r(a) + ' a> ,
(24.57)
1 t a-1 t al-1 t a..-1 . ft 0 t 0 d t a-1- 0 . ft 0 . f
w lere + = 1 . .. n ,1 1 > ,..., n > , an + = , 1 I: < even 1
for a single k = 1,2, .. " n.
24.10. Fractional derivatives and integrals of periodic
functions of many variables
Let us consider periodic functions f(x) = f(xl"" ,x n ) of many variables. Let
A = {x : 0  Xi < 21r} be the period cube and let Cl: = (21r)-n J f(x)eil:,xdx,

k = (k 1 , . . . , k n ), be the Fourier coefficients of the function f(x). The Fourier series
of this function is
f(x) f'OoJ L cl:e ib
-00<11:1<00
00
= L
1: 1 =-00
00
""" c e i (1: 1 x 1 +...+k..x..)
L..J 1: 1 ...1:.. .
1:..=-00

 24. PAFUIAL AND MIXED INTEGRALS AND DERIVATIVES
477
The Weyl fractional integra-differentiation of a periodic function I(x) is
defined similarly to (19.5)-(19.6). While the one-dimensional fractional integral
was determined for all (summable) functions not containing the constant Co in the
Fourier series, we are now to exclude functions constant in each variable. Namely,
following (19.5) we define the multiple (mixed) Weyl integral of a periodic function
as
[(a) I = E' ( .t ) e i1c . z , (24.58)
-oo<ltl<oo I a
the dash denoting omission of all (!) the terms with multiindices k = (kl,'" ,k n )
such that ki = 0 even if for a single i = 1,..., n. Thereby we consider, in fact,
those functions which have C1c = 0 for such k, i.e.
2W'
J I(Xl,"., Xi-l,, Xi+l,... ,Xn) = 0
o
(24.59)
for i = 1,2,... ,n. Following Lizorkin and Nikol'skii [1] we call a periodic function
I( x) neutral on A, if it satisfies the condition (24.59) for all i = 1,2,. . . , n.
As for fractional differentiation, we define it by the expansion
1)(01) 1- E ck(ik)a e i1c'z.
-oo<lkl<oo
(24.60)
We have chosen, for definiteness, the variant of left-hand sided fractional
integra-differentiation with respect to each variable in (24.58) and (24.60). In
correspondence with (24.12) or (24.13) , for example, we might use symbols [!.+
and 1)?+, but we shall not consider any other forms except the left-hand sided
one for the periodic functions. The Weyl fractional integral (24.58) considered for
functions I(Xl,'" ,x n ), neutral and summable on A, is interpreted similarly to
(19.7) as
(a) - JI( )II n aa ( . )d
[ I - (211')n X - t. W + t, t, Qi > 0,
A ,=1
where wa(ti) are functions (19.8) of one variable.
We may admit the cases when Qi = 0 for some values of i. In these cases we
must omit the integration in (24.61) with respect to the corresponding variables.
For simplicity we assume that Qi > 0, i = 1,2,..., n.
In the case of sufficiently good functions we may write the Weyl fractional
differentiation (24.60) with 0 < Q < 1 as
(24.61 )
1)(01) I = on [(I-a) I,
OXI . . .ox n
(24.62)

478 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where 1 - a = (1 - al,..., 1 - an)'
Theorem 24.5. The operator 1(01) is bounded in the space Lp(A), 1  p  00.
If 1 < p < {3 = (m?-X ai)-l, then 1(01) is bounded from Lp(A) into Lq(A),
.
q = p/(1 - (3p).
n
The first assertion of the theorem is evident since n W:i(ti) E Ll(A), the
i=1
second is obtained by successive application of the corresponding one dimensional
statement (19.62) and use of the imbedding Lp(A) C Lr(A), p > r.
One can also obtain a statement similar to Theorem 24.1 on the boundedness
of the Weyl operator /(01) within the frames of the mixed norm spaces Lp. We
leave this to the reader.
Quite analogously to Lemma 19.3, using properties of the functions w:i(td,
we obtain the following theorem.
Theorem 24.6. The Weyl fractional integral (24.61) of 21r-periodic functions
rp(t) E Ll(A) with zero mean values (24.59) coincides with Liouville fractional
integral:
I(O)<p = fj...+<p = f(l,,) f <p(z - t)tO-1dt, 0 < " < 1,
R+...+
provided that the integral on the right-hand side as treated as conventionally
convergent at infinity:
2'11'ml 2'11'm..
f rp(x - t)ta-ldt = lim f . .. f rp(x - t)ta-ldt,
Iml-oo
R+oo.+ 0 0
m=(ml,...,mn)EZ n .
Other statements of S 19 for fractional integra-differentiation of one variable
can be extended similarly to the multidimensional case. We note for example,
by generalizing (19.34), that the Weyl fractional derivatives (24.62) coincide with
Marchaud derivatives (24.35):
1)(01) f =  f (Af)(x) dt 0
",(a,l) tl+a' a> .
R+".+
(24.63)

 24. PARTIAL AND MIXED INTEGRALS AND DERIVATIVES
479
We also give the formulation of the following theorem, which is proved in the
same way as Theorem 19.2.
Theorem 24.7. Let /(x) be a periodic function, neutral on A and let
f(x) E Lp(A), 1  p < 00. Then f(x) is representable by the mixed Weyl fractional
integral:
f(x) = I(a)tp, tp E Lp(A), Q > 0,
if and only if lim D+...+ £/ E Lp(A), the limit being taken under Lp(A)-norm.
£-0 '
24.11. Griinwald-Letnikov fractional differentiation
We define partial and mixed Griinwald-Letnikov fractional derivatives of functions
/(x) = f(Xl,' .. ,x n ) of many variables, following (20.7). Let h = (hl" .. ,h n ) be a
vector increment and let
(W(2O) = :E (-1)1;1 ()/(2O - j 0 h)
O\jl<oo J
(24.64)
be a difference of a fractional vector order Q = (Ql,"', Qn), Q  0, i = 1,2,..., n,
with a vector step h. In (24.64) j denotes a multiindex and j oh = (jlh b .. .,jnhn),
(j) = (j:) .. . (j:), so (24.64) turns into (24.36) in the case when all the Qi are
integers. We define now the Griinwald-Letnikov fractional derivative of order
Q = (Ql, . . . ,Qn) by the relations
r;.) (x) = lim (AI: f)(x)
+...+ h-+O h a '
(24.65)
f?._(x) = lim (L\f)(x) ,
h-+O a
(24.66)
where h a = hr 1 . . . hlI., hi > 0, i = 1,2, . . . ,n.
We now give without proof certain extensions of the results exhibited in S 20
to the multidimensional case. First we formulate the generalization of Theorem
20.2 stating that the mixed Griinwald-Letnikov derivative (24.65) of a periodic
function /(x) = f(x1l'" ,x n ) exists simultaneously with the Marchaud derivative.
Let A = {x : 0  Xi < 21r}.
Theorem 24.8. Let a periodic function f(x) belong to the space Lp(A),
1  p < 00. The mixed Griinwald-Letnikov derivative (24.65) exists simultaneously

480 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
with mixed Marchaud derivative and they coincide:
00 00
lim (Af)(x) =  lim f '" f (Af)(x) dt.
h-+O h a x(a, I) £-0 t l + a
(L p (6» (L p (6»£1 £..
The prool is similar in a sense to that of Theorem 20.2 and needs Theorem 24.6
and the results of S 20 to be taken into account.
The corresponding result for a non-periodic case is more difficult. We
concentrate on the non-mixed case, when Q is a scalar. Let Af be a fractional
order difference with a vector step h = (hi,"" h n ), defined as in (24.64) but
with scalar j = 0, :1:1,... and j 0 h = jh. The following theorem shows that the
Griinwald-Letnikov approach via the quotient ;! yields fractional differentiation
(24.38') in the fixed direction .
Theorem 24.S'. Let f(x) E Lr(R n ), 1 :::; r < 00, or f(x) E Cekn). Then the
limits
r (A f)(x)
Ihio Ihl a '
(X)
00 1
I . 1 f (Ah,f)(x)
Im-
(xf x( Q, I) £ l+a ,
where X = Lp(Rn), 1 :::; p < 00, or X = C(k n ), exist simultaneously and coincide,
the values of p and r being independent.
Certain connections of fractional differentiation in a given direction with
fractional differences is also demonstrated by the following theorem.
Theorem 24.S". Let f(x) E Lr(Rn), 1 < r < 00. Then the fractional derivative
(24.38') exists as an integral convergent in Lp(), 1 < p < 00, if and only if
IIALfli p :::; ct a , t > 0,
c being independent on t.
The prool of Theorems 24.8' and 24.8" can be found in Samko [34].
24.12. Operators of the polypotential type
Starting from (12.1) we introduce the operator of the Riesz potential type in each
variable
K,a - 1 f cp(t)dt
cp - A n'
RIO n IXk - tk II-a"
k=l
(24.67)

 24. PAFUIAL AND MIXED INTEGRALS AND DERIVATIVES
481
n
where at > 0, at f; 1,3,5,..., k = 1,..., n; A = 2 n n r(at) cas at1f'/2.
k=l
We shall call (24.67) the operator of the Riesz polypotential type. The operator
1{0I =.!.. f ( t ) rr n sign(xt - tt) dt
<P B <P IXt - h Il-Olk '
R" t=l
(24.68)
n
with at > 0, at f; 2,4,6,..., B = 2 n n r(at) sin at1f'/2, can also be defined as a
t=l
generalization of (12.2).
The operators K,OI and 1{0I are well defined for example, for functions
<p(t) E Lp(flR), 1 $ P < n(1/at), which can be shown as in the one-dimensional
case (see S 5.1). We can prove directly by the one-dimensional Hardy-Littlewood
Theorem 5.3 (by its successive application to each variable) that in the case
al = a2 = ... = an the operators K,OI and 1{0I are bounded from the space Lp(Rn)
1 < P < l/al, into the space Lq(Rn), q = p/(1 - alP).
It is of much more interest to have information on the mapping properties of
polypotential type operators in Lp for different values of at. In this case it is
natural to consider the space Lp with a vector p = (Pl, . . . , Pn) - see S 24.4. Thus
generalizing (24.24), we consider the space L,(Jl!1) of functions with the mixed
norm
{ f{ { f [ Ii ] P'J/Pl } P3/P'J } P../P..-l } l/p..
IIfll,= ... Jlf(xl,...,xn)IPldxl dX2 ... dXn <00.
Rl Rl Rl
Theorem 24.9. The polypotential type operator K,OI, a = (al,"" an), at > 0
is bounded from L,(Rn) into Lf(Rn) with p = (Pl,...,Pn), ij = (ql,...,qn),
1 5 Pt < 00, 1 5 qt < 00, if and only if
I<Pt<l/at, qt=pt/(I-akPt), k=I,2,...,n.
Theorem 24.9 is proved following arguments developed in the proof of Theorem
24.1: the "if' part is obtained by the successive application of Hardy-Littlewood
Theorem 5.3 to each variable, while the "only if' part needs the use of the
dilatation operator.
We might easily extend various results for one-dimensional operators in S 12
to the case of polypotential type operators. We outline only connections between
the operators K,OI and 1{0I via the polysingular operator. In the one-dimensional

482 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
case such a connection was given by (12.6) and (12.7). It is readily seen that these
identities yield the following relationships
K,OIcp = S1t Ol cp,
1t Ol cp = -SK,O/ cp,
(24.69)
for the operators K,OI and 1t 0l , where S is the polysingular operator
Scp = f n cp(t)dt
R" n (tl: - XI:)
1:=1
(treated in the principal value sense). It is assumed that al: > O. If ak = 0 for
some k, polypotentials and the polysingular operators are taken for those variables
for which al: f; O.
It is also easily proved by means of (12.19)-(12.21) that
K,OI K,/J = K,OI+/J,
1t 0l 1t/J = _K,OI+/J,
K, 0I 1t/J = 1t 0l +/J ,
(24.70)
where a = (al,'" ,an), /3 = (/31,'" ,/3n), a + /3 = (al + /31"", an + /3n).
The Fourier transform of the polypotential K,OIcp is evaluated in view of (12.23)
by the relation
n
()(x) = II IXI:I-OIkcp(X)
1:=1
(24.71 )
(cf. (24.47) and (24.48».
One can also define operators of Bessel type potentials:
n
GOIcp = f II G Oik (XI: - tl:)cp(t)dt, a = (at, . . . , an)
R" 1:=1
(24.72)
(see (18.61», where the kernel GOIk(XI:) has the Fourier transform, in the variable
XI:, equal to (1 + IXI:12)-0I1r/2. The modification of the polypotential (24.72)
generalizing (18.64):
f n e- tk
G OI - x-t&
+cp - II f( al:)t 1 - OIk cp( )
R" 1:= 1 I:
+...+
(24.73)

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
483
can also be mentioned. Let
GOt(L p ), Gt.(L p ), 1  p  00,
denote the spaces of functions representable in the form (24.72) and (24.73)
respectively with <p E Lp(R n ). By means of the Corollary of Theorem 1.6 and
(18.62) and (18.63) it is not difficult to show that
GOt(L p ) = Gt.(L p ), 1 < p < 00.
(24.74)
The spaces GOt (Lp) and Gt. (L p ) may be characterized in terms of the existence in
Lp of mixed fractional derivatives, and thus they are called function spaces with a
dominating mixed derivative, see S 29.2, note 24.4. .
 25. Riesz Fractional Integro-Differentiation
We shall now study fractional integra-differentiation of functions of many variables
which is a fractional power (_A)Ot/2 of the Laplace operator. The idea of how to
define such a power is obvious in Fourier transforms: (-A)Ot/2f = F- 1 IxI Ot Ff in
the case of sufficiently good functions f, - see (25.6) below. The investigations
in this section are aimed at the effective construction of such a fractional power
and studying its properties. The negative powers (_fl)Ot/2, Rea > 0, will be Riesz
potentials
[Ot -  f cp(y)dy
<p - "Yn( a) Ix - yln-Ot '
Rft.
a 1= n, n + 2, n + 4,. . . ,
(25.1)
which have already been considered in Section 12 in the one-dimensional case; the
normalizing constant "Yn (a) is defined below. The positive powers of the Laplace
operator will be realized as the so-called hypersingular integrals DOt f, defined
below by (25.59). The operation
{-t:.y/2f= r ' l.r.1J = { !f,
Rea > 0,
Rea < °
(25.2)
where exact definitions are given below will be the object which we call a fractional
Riesz integro-difJerentiation.
The natural and convenient apparatus for the investigation of Riesz integra-
differentiation is the Fourier transform.

484 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
25.1. Preliminaries
Let
I(x) = (:Fcp)(x) = <p(x) = f cp(y)ei:t:'Ydy
R"
(25.3)
be the Fourier transform of a function cp(y) = cp(Yl,"', Yn) and let
1 - 1 f .
cp(x) = (:F- I)(x) = I(x) = (21r)n l(y)e-a:t:'Ydy
R"
be the inverse Fourier transform. It is well known that
:F(1Ji I) = (-ix}7 j(x),
j = (h, . . . , jn),
(25.4)
so that we have
:F(L\cp) = -lxI 2 :Fcp
(25.5)
for the Laplace operator A, or
-Acp = :F- 1 IxI 2 :Fcp.
(25.6)
It is also well known that the Fourier transform of the convolution
1 * cP = f I(x - y)cp(y)dy
R"
(25.7)
is given by the formula
:F(I * cp) = j . <p.
(25.8)
So, while considering Riesz potential (25.1) we must in any case know the Fourier
transform of the kernel Ixla-n. It will be evaluated in S 25.2 by means of the
Bochner formula for the Fourier transform of radial functions, a result proved in
Lemma 25.1 below. (A function cp = cp(lxl), depending only on lxi, is called radiaQ.
In the proof of the lemma below we shall make use of the following formula
1
f 2 (n-1)/2 f
I(x . CT)dCT = ; () l(lxlt)(1 - t 2 )<n-3)/2dt,
5"_1 -1
(25.9)

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
485
known in the integral calculus. Its proof may be found for example in Fikhtengolts
[1, p. 405-407] or in Samko [31, p. 42-43]). We recall that Sn-l is a unit sphere in
R n , centered at the origin, du being the surface element on Sn-l.
Lemma 25.1. The Fourier transform of a radial function is a radial function
again. Also the following relation is valid:
N
f e;%"(lyl)dy = 1f<:n;2 f <p(p)pn/2 I n / 2 - 1 (pi'" I)dp
lyl<N 0
(25.10)
if <p(lyl) is summable in the balllyl  N. Moreover
n/2 00
f i%" (lyl)dy = I;: 2)/2 f <p(p )pn/2 I n / 2 - 1 (pi'" I)dp
Ra 0
(25.11)
for any function cp(p) such that
00
f pn-l(1 + p)(1-n)/2Icp(p)ldp < 00
o
(25.12)
provided that the integral on the left-hand side of (25.11) is interpreted as
00
conventionally convergent. It converges absolutely if J pn-1Icp(p)ldp < 00.
o
Proof. Passing to spherical coordinates on the left-hand side of (25.10) we have
N
f e izoy cp(ly!)dy = f ifJ(p)pn-1dp f eipzoq du.
lyl<N 0 5a-l
By (25.9) and Poisson's formula (2.52) we have
f . (21r)n/2
e lZoq du = Ixln/2-1 I n / 2 - 1 (Ix!),
5a_l
(25.13)
which yields (25.10). Since IJII(p)1  c/..;p as p --+ 00, the limit of the right-hand
side in (25.10) exists as N --+ 00 provided that (25.12) is satisfied. This yields
(25.11). .

486 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
It is of particular interest in the context of this book to remark that the
n-dimensional Fourier transform of a radial function can be represented via the
one-dimensional Fourier transform by means of fractional (generally speaking)
integra-differentiation. For this purpose we introduce the designation
00
(.1'.<p)(r) = f eirt<p(t)dt, <p( -t) == <p(t)
-00
for a radial function <p(lxl).
Lemma 25.1'. Let (25.12) be satisfied. Then
 ( ) _ (n-I)/2 ( :F.l n - I )/2 )( )
<p x - 1r . _ ;r 2 <p r,
r= lxi,
(25.14)
<per) = 1r 1-;" (1);.})/2 .1';1 f)(r),
(25.14')
where fer) = (x)llxl=r with (x) denoting the n-dimensional Fourier transform in
x of a function <p(lxl) and 1);21)/2 and I;})/2 are integra-differentiation (18.38)
( ) . (n-l)/2 ( ' d ) (n-l)/2'. h
and 18.40 wIth 1) -jr 2 == - P 10 t e case of an odd n.
Proof. Since by Lemma 25.1 (x) = f(lxl) depends on the radial variable only, it
is sufficient to consider (x) for x = (lxi, 0,... 10) only. For such x we have
00
(x) = f eilxl6d6 f <p(Il)d6... dn'
-00 R,,-1
The passage to polar coordinates in the inner integral gives
00 00
.p(z) = I S n-21 f eilzl61 f cp ( V P 2 + l) pn- 2 dp
-00 0
00 00
= I S n-21 f eilxl6 1 f t<p(t)(t 2 - n(n-3)/2dt.
-00 161
In accordance with the notation of (18.38) this yields (25.14). Inverting the
one-dimensional Fourier transform.1'. and the integral I;21)/2, we obtain (25.14').

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
487
We shall need the Lizorkin space of test functions which is adapted to
Riesz integra-differentiation. This spC),Ce, unlike its variant for partial and mixed
fractional Liouville derivatives (see S 24.9), may be defined as consisting of functions
with Fourier transforms vanishing not on the coordinate planes, but at the origin
only. Namely, let
w = {,p(x) : ,p E S(R n ), (1Y ,p)(0) = 0, Iii = 0,1,2,...}.
(25.15)
(Compare this with the definition of this space in S 24.9, which is considerably more
restrictive than (25.15». The function 1/J(x) = exp( -lxl 2 - Ixl- 2 ) is an example of
a function in the space (25.15). Let us consider now the space , consisting of
Fourier transforms of functions in W:
 = .1'(w) = {cp(x) : cp E S(R n ), cp =.,'/J, 1/J E w}.
(25.16)
This allows a simple characterization: the space  consists of those and only those
Schwartzian functions cp, which are orthogonal to polynomials:
f xi cp(x )dx = 0, Iii = 0,1,2, . . .
R"
(25.17)
Indeed, it is known - Gel'fand and Shilov [2, p. 208] - that the Fourier transform
maps the Schwartzian space S onto itself and
f xi cp(x)dx = i-iii f (ix}'cp(x)ei:l:.odx = i- ljl (1Y <;)(0) = 0
R" R"
according to (25.15).
Thus  is the subspace of Schwartzian functions, for which all the moments
equal zero.
The space  may be equipped with the topology of the space S( Jl!1), which
makes  a complete space.
We shall consider generalized functions on  (and on W as well) in order to
justify some of our operations. We recall that the Fourier transform of a generalized
function f (E ') is defined as a functional j = .1' f, introduced by the rule
(j,,p) = (f,,p), ,p E W.
(25.18)
This definition is correct, since F(w) =  and the Fourier transform is a continuous
operation in the topology of the Schwartz space S(Rn) - Gel'fand and Shilov [2,

488 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
p. 208]. If 9 is a functional in w', the relation
(9, cp) = (g, ), cp E ,
(25.18')
serves similarly as a definition of the Fourier transform for a functional 9 E w'.
The function lxi-a as an element of the space W' generates the regular
functional (lxi-a,,,) = f Ixl-a,,(x)dx for all a E C 1 since (1);,,)(0) = 0,
Rft.
Ijl = 0,1,2,... However, it is a nonregular functional as an element of S' or ' if
Rea  n. For such a it will be understood in the sense of regularization achieved
by the analytical continuation of the functional (lxi-a, 1/J) from the half-plane
Rea < n. The latter is given by means of the equation
C:la ') = 1
Izl<1
cp(x)- L: (1);cp)(O)x j /j!
Ijlm dx
Ixl a
(25.19)
+1
Izl>1
(m/2]
cp(x)dx + L Ci . (L\icp)(O),
I x l a . n - a + 21
1=0
a - n f; 0,2,4,...
where cp E S(Rn», m > Rea - n -1 and Ci = 1rn/221-2i[i!f( i+ n/2)J-1. This yields
the direct realization of the analytic continuation of the functional (lxi-a, cp) in the
half-plane Rea < m + n + 1. We have already dealt with such a regularization in
the one-dimensional case - see (5.68).
Equation (25.19) is derived in a standard way by subtracting the Taylor sum
and taking the relation
(m/2]
L , (-pjcp)(O) 1 x-1lxl-adx = L Ci 2' (L\icp)(O)
. J. . n - a + 1
IJIm Ixl<1 1=0
into account. The latter can be verified directly by use of the relation
L\m = ""' m! 1)2j.
L.J .,
J.
Ijl=m
(25.20)

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTrATION
489
In the cases a - n = 2k, k = 0,1,2,..., excluded in (25.19), we set
C.,I;+2" cp ) = aJ2' [ C.,l,a ' cp ) + a _:' _ 21: (d'CP)(O)]
(25.21)
by definition.
The regularization (25.19) may be represented in a simpler form as
cp(x) - L: (1)j cp)(O)x j
(  ) = f IjIS[Rea]-n J. dx
Ixl a ,CP Ixl a
R"
(25.22)
with m = [Rea] - n chosen here (if Rea> n); this, however, has to exclude the
values Re a = n, n + 1, n + 2, . . . .
Remark 25.1. The functional defined by (25.19) is an analytic function with
respect to a for all a E C 1 (since it is possible to choose m as large as required; it is
evident that the right-hand side of (25.19) does not depend on this choice) except
the points a = ak = n + 2k, k = 0,1,2,... As for the points ak, the functional
(25.19) has a first order pole there and admits the representation
( 1 ) ( Ck k
_ I l a'CP = ga,CP) + -(A cp)(O),
x a - ak
(25.23)
where (ga, cp) is a function analytic in the neighbourhood of the point ak and
liffi(ga,cp)= ( - l l l ,cp ) .
a-a. X a.
(25.24)
25.2. The Riesz potential and its Fourier transform. Invariant
Lizorkin space
Arguing formally for the moment we see that operation (25.2) is to be realized
as a convolution (in a generalized meaning) of a function f with the function
F-1(lxl- a ). Let us evaluate this function first, treating the Fourier transform in
the sense of generalized functions. It is convenient to use the Lizorkin space (25.16)
as a test function space.
Lemma 25.2. The Fourier transform of the junction lxi-a, interpreted according

490 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
to (25.18'), is given by the relation
{ Ixla-n,
F(lXI- a ) = (21r)n Ixl a - n In fh,
"Yn(a) (-A)-a/26,
a f; n + 2k, a f; -2k,
a = n + 2k,
a = -2k,
(25.25)
where 6 = 6(x) is the Dirac delta-function, k = 0, 1,2,. . ., the constant "Yn( a) being
equ al to
{ 2a1rn/2r () /r ( n;a ) ,
"Yn(a)= 1,
(_I)(n-a)/2 1r n/22 a - 1 ( a;n )!r (),
a 1= n + 2k, a f; -2k,
a = -2k, (25.26)
a=n+2k.
Proof. Let Rea < n. Then lxi-a is a locally integrable function. We shall use
Bochner's formula (25.11) to compute the Fourier transform of this function. Let
us assume first that (n + 1)/2 < Rea < n, so that the condition (25.12) is then
satisfied for the function cp(p) = p-a. Relation (25.11) yields
00
F(lxl- a ) = Ixl a - n J, where J = ( 21r t/ 2 f p-a+n/2 I n / 2 - 1 (p)dp.
o
We use the equation
00
f pP J.(p)dp = 211r( v + : + 1 ) jr ( v - : + 1 )
o
(25.27)
(known as Weber integral and obtained by substituting the Poisson integral (2.52)
into the left-hand side of (25.27) and then interchanging the order of integration).
So J = (21r)n/22- a + n / 2 r ( n;a ) /r (), which means that we have proved the first
line in (25.25) in the case (n + 1)/2 < Rea < n. For the remaining values of a the
Fourier transform of the function lxi-a will be interpreted in the sense of (25.18').
In accordance with (25.18') we are to show that
(21r)n ( 1 ) ( 1  )
"Yn(a) Ixln-a 'CP = Ixla 'CP,
cP E, 0: f; n + 2k, a 1= -2k.
(25.28)

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
491
This is true for (n + 1) /2 < Re a < n since the first line has already been proved for
such values. The left-hand side in (25.28) is interpreted as (25.19) if Rea  O. The
right-hand side is defined and analytic for all a E C 1 since Ij> E W. The left-hand
side is analytic for all a E C 1 except perhaps the points a = -2k and a = n + 2k,
where it has removable singularities: we see that (Ixla-n, 1;') has a pole cancelled
by the zero of the function 1/ "Yn (a) in the first case and it has a zero cancelled
by the zero of "Yn(a) in the second case. So (25.28) follows from its validity for
(n + 1)/2 < Re a < n by the uniqueness theorem for analytic functions.
It remains to consider the cases a = -2k and a = n + 2k, corresponding
to removable singularities on the left-hand side of (25.28). The former of
these cases follows from (25.4) immediately. To consider the case of the values
Q = ak = n+2k we rewrite (25.28) for a near the points ak as (a-ak)(lxl-a,lj» =
(211')na(a)(lxla-n,I;'), where a(a) = (a - ak)/"Yn(a). Differentiating this equality
in respect to a, we obtain
 ( a - ak  ) = ( 211' ) n ( al(a) + a(a) In Ixl ) .
da Ixl a ' I;' Ix In-a' I;'
(25.29)
Owing to the representation (25.23) and the continuity property (25.24) we have
d ( a - ak  ) (  ) ( ) d (  ) ( 1 A )
da ' I;' = ga, I;' + a - ak da ga, I;' --+ Ixl ak ' I;'
as a --+ ak. Then (25.29) yields
C., a. ,v> ) = (2,. t a( a» Cln!. '  )
(25.30)
since (Ixl ak -n, 1;') = 0 by (25.17). Equation (25.30) gives the second line in (25.25).
The value of the constant in the form a(ak) = -[-Yn(ak)]-l is obtained by easy
calculations. .
Remark 25.2. If the Fourier transform in (25.25) is interpreted in the generalized
sense not over test function space  = (Rn), but over the Schwartz space S(R n ),
then 10  in (25.25) is to be replaced by In  + d k with
. d [ a-a k ]
d k = "Yn(ak) hm - d - ( )
a-ak a "Yn a
= In 2+ Hr'(l) H'( > )/r( » +  n a> = n+ 21:.

492 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
This may be easily seen from the arguments in (25.29) and (25.30).
Thus the operation F-1Ixl- a F is to be realized as the convolution with the
function (25.25). In the case Rea> ° this function is locally summable and
the corresponding convolution is the Riesz potential (21.1) except for the values
a = n, n + 2, n + 4,... In the excluded cases we are to take the logarithmic factor
into account according to (25.25). Thus we define the Riesz potential for all a,
Rea > 0, as the convolution
[acp = f ko(x - y)cp(y)dy,
R'"
(25.31)
where
1 { Ixla-n,
ka(x) = "Yn(a) Ixl a - n ln,
a - n 1= 0,2,4,6,. . . ,
a -n = 0,2,4,6,...,
(25.32)
and the normalizing constant is given by (25.26). The function ko(x) is known as
the Riesz kernel.
Passing to polar coordinates in the integral
[acp = ["Yn(a)r 1 f Iyla-ncp(x - y)dy, a - n 1= 0,2,4, ... ,
R'"
we have
00
[a cp =  f d(1' f cp(x - (1') d
"Yn(a) e- a '
5"'_1 0
(25.33)
i.e. we may interpret the Riesz potential as a result of fractional integration of
order a in the direction of the vector (1' (see (24.39») followed by integration with
respect to (1'. In the notation of (24.39) (25.33) has the form
r(1Z.9:)r() f
[acp = 2(n+l)/22 ([;cp)(x)d(1'.
5"'-1
(25.34)
The Fourier transform of the Riesz potential [ocp, Rea > 0, is reduced to
division by Ixl a : '
J:W 'P) = I ",a .,;( '" ),
(25.35)

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
493
for functions <p(x) E  at least. This was in fact justified in Lemma 25.2. Indeed,
(25.35) is equivalent to the equation
fk ( ) ( + )d 1 f i zoycp(Y) d
01 Y <p X Y Y = (21r)n e lylOI y,
R- R.
<p E,
(25.36)
which was proved in Lemma 25.2 for x = 0 - see (25.28) - and follows from (25.28)
by the invariance of  relative to the translation operator.
We conclude from (25.35) that
AIOI<p = _1 0I - 2 <p, <p E, Rea> 2.
Theorem 25.1. The Lizorkin space  is invariant relative to the Riesz potential
1 01 and IOI() =  with
1 01 IfJ <p = IOI+fJ <p,
<p E, Rea > 0, Re,8 > O.
(25.37)
Proo/. The invariance of the space  is readily seen from (25.35) and (25.36):
since lyl-OIcp(y) E W for any a and <p(y) E , the right-hand side of (25.36) belongs
to the space ,i.e. IOI()  . Every function of the space W being representable
in the form 1/J(x) = Ixl- OI 1/Jl (x) with 1/Jl(X) E W, we see that IOI() =. The
semigroup property (25.37) follows from (25.35r .
Remark 25.3. The Riesz potential does not necessarily vanish rapidly at infinity
even if <p(x) E S(R n ). Thus, for example, let <p(x)  0 and <p(x)  A > 0 for
Ixl < 1. Then (IOI<p)(x)  clxl Ol - n as Ixl --+ 00 (see (7.8) in the one-dimensional
case in this connection). Indeed, (J<<p)(x)  A[-Yn(a)t 1 J Ix-yIOl-ndy. Passing
Iyl<l
to polar coordinates here, we have:
1
(IOI<p)(x)  c f du f pn-l(p2 - pu, x + IxI2)(0I-n)/2dp
5._1 0
1
  f du f pn-l(p2 + IxI 2 )(0I-n)/2dp  cllxI Ol - n .
5._1 0
Remark 25.4. The choice of the Lizorkin space  is essential for writing the

494 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
semigroup property (25.37) for all a and (3 with Rea> 0 and Re{3 > O. In the
case of functions cp( x) of the Schwartz space S( Jl!1) (25.37) has meaning under the
restriction Re (a + (3) < n only - see Remark 25.3 above.
We note that the semigroup property (25.37) after interchanging the order of
integration on the left-hand side, yields the relation
f Ie - yla- n l y l,6-ndy = 'Yn(a)'Yn({3)/'Yn (a + (3),
R"
(25.38)
a > 0, {3 > 0, a + {3 < n,
for any unit vector e, lei = 1. We observe also that
([a(eiaoy»(x) = lal-aeia.z, a = (a1,..., On).
(25.39)
This result is a paraphrase of the assertion ka(a) = lal-a. The Riesz potential
of e iaoSf is a conventionally convergent integral if 0 < a < (n + 1 )/2. In the case
a  (n + 1)/2 it is to be interpreted as the analytic continuation in the parameter
a.
25.3. Mapping properties of the operator [a in the spaces
Lp(Rn) and Lp(Rn; p)
First of all we remark that the operator fa is defined for functions cp(y) E Lp(Rn)
if 1  p < n/ a, 0 < a < n. This is verified similarly to the one-dimensional case
(see S 5.1):
'Yn(a)[acp = f Iyla-ncp(x - y)dy + f Iyla-ncp(x - y)dy.
ISfI<1 lyl>1
The existence of the first integral here for almost all x E Jl!1 may be established by
showing that it belongs locally to Lp by using the Minkowsky inequality, while the
second one exists for all x, if 1  p < n/a, by the Holder inequality.
The extension of the Hardy-Littlewood Theorem 5.3 to the case of the
multidimensional Riesz integration [a is given by the following statement known
as the Sobolev theorem.
Theorem 25.2. Let 1  p  00, 1  q  00 and a > O. The operator [a is
bounded from Lp(Rn) into Lq(Rn) if and only if
o < a < n,
n
1 < p < -,
a
1
q
1 a
(25.40)
p n

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
495
We omit the proof of this theorem as well as of other statements of this
subsection. See the references in S 29.1. A simple way to reduce Theorem 25.2 to
the one-dimensional case may be found in S 29.2, note 25.2.
The number q = np in (25.40) is known as the Sobolev limiting exponent.
Later in S 26.7 we shall deal with the Riesz potential of functions cP E Lp(Rn) with
p  n / 0:, interpreting this potential in a generalized sense. Let Lp (J1:& ; p) be the
{ } l
weighted space with the norm IIfIlLp(R"jp) = 1.. p(x)lf(x)IPdx , where p(x) is
a non-negative function. The following is a generalization of Sobolev's Theorem
25.2 to the case of the weight which is a power of the distance Ixl.
Theorem 25.3. The operator [a is bounded from L p (J1:&; Ixlll) into L r (J1:&; IxIIJ),
if
0: > 0, 1 < p < 00, 1 < r < 00, o:p - n < , < n(p - 1),
1 0: 1 1
---<-<-
p n - r - p'
JJ+n ,+n
-=--0:.
r p
(25.41 )
We observe that Theorem 25.3 contains the "if part" of Theorem 25.2 under
the choice, = 0, JJ = 0 and r = np(n - o:p)-l. We note also a useful particular
case, = 0, JJ = -o:p, r = p in Theorem 25.3:
f Ixl- ap l(f a cp)(x)l.6dx  Allcpll, 1 < p < n/o:
R"
(25.42)
which is a generalization of the Hardy inequality (5.45).
In the case r = np(n - o:p)-l there is an extension of Theorem 25.3 to the
weights of a general nature. Namely, let p(x) satisfy the so-called Muckenhoupt-
Wheeden condition
( 1 f ) P/9 ( 1 f ) P-1
IQI p9/P(x)dx IQI pl/(l-p)(x)dx  c < 00,
Q Q
(25.43)
where Q is an n-dimensional cube, IQI being its Lebesgue measure.
Theorem 25.4. Let 0 < 0: < n, 1 < p < n/o: and q = np/(n - o:p). The operator
fa is bounded from Lp(Rn; p) into L 9 (J1:&; p9/P), if and only if p(x) satisfies the
condition (25.43) with q = np/(n - o:p).

496 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
The assertion of the Sobolev theorem does not hold in the case p = 1, but
a weaker statement is true which is known as a weak type estimate. Namely, let
J(t) be the distribution function for a function f(x), x ERn:
J(t) = m{x : If(x)1 > t}, t > O.
(25.44)
It is clear that J(t) ::; t-Pllfll with IIfll = J If(x)IPdx, so the estimate
R"
t!j./ (t)  ell/II"
(25.45)
is weaker than the estimate liT fllg ::; ell flip , giving boundedness of the operator T
from Lp into Lg. It is in terms of (25.44)-(25.45) that information on the Riesz
potential in the Sobolev theorem is changed. Namely, the estimate
m{z: I (I0cp)(z)I > t}  (illcplI')',
n
q --
- ,
n-a
(25.46)
is valid with 0 < a < nand c not depending on t > O.
We also outline the mapping properties of the Riesz potential operator in the
spaces of Holderian functions f(x), x E Rn, where R n is a compactification of R n
by the unique point at infinity. To guarantee existence of the Riesz potential [acp
of a Holderian function it is necessary for cp(x) to be zero at x = 00 and it will be
convenient to achieve this by means of a weight. Let
H>'(Rn;p) = {f(x) : p(x)f(x) E H>'(R n )}, 0 < A < 1,
where
>. . n _ { . . n C1h l >' }
H (R ) - f(x). f(x) E C(R ), If(x + h) - /(x)1  (1 + Ixl)>'(1 + Ix + hI)>'
(cf. (1.5) and (1.6». Let also
Ht(Rn;p) = {f: f E H>'(Rn;p), pflx=o = pflx=oo = O}.
The space H>'(Rn;p) is equipped with the natural norm, which makes it a Banach
space.
Theorem 25.5. Let 0 < a < 1 and A + a < 1. The operator [a maps the space
H>'(Rn; (1 + IxI 2 )(n+0I}/2) isomorphically onto the space H>'+OI(Rn; (1 + IxI2)(n-0I}/2)

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
497
and the space Htciln; Ixl.6(1 + X 2 )N/2) onto the space HtCk n ; IxI P (1 + x 2 )-a+N/2),
if a: + >. < p < n + >. and a: - P < x < n - >. - p.
We omit the proof and note only that it is obtained by mapping the space
R,n onto the unit sphere Sn C Rn+l via the stereographic projection. It is worth
observing that the Riesz potential [acp is transformed thereby into a similar
potential (up to the weights) over the sphere, i.e. over a compact set in the usual
metric, where the Holder property is established by direct estimation - see Vakulov
[1, 2].
We conclude this subsection by some simple relationships between Riesz
potentials and one-dimensional fractional integrals of the Liouville type. For this
purpose we need the Poisson and Gauss- Weierstrass integrals
(PtCP)(x) = f P(y,t)cp(x - y)dy, t > 0,
R"
(25.47)
(Wtcp)(x) = f W(y, t)cp(x - y)dy, t > 0,
R"
(25.48)
where
) cnt
P(x, t = (lx1 2 + t 2 )(n+l)/2 '
_ -(n+l)/2 r ( n + 1 )
C n -  2
(25.49)
is the Poisson kernel and
W(x, t) = (4t)-n/2e-lzl/(4t)
(25.50)
is the Gauss- Weierstrass kernel.
Theorem 25.6. The Riesz potential [acp, cp E Lp(Rn), 1 < p < n/a:, admits of
the representations
00
(Ia<p)(z) = rtn) f ta-1(p.<p)(z)dt
o
(25.51)
00
= r ( l!!) f t'f-'(W.<p)(z)dt.
2 0

498 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Moreover, the relationships
(PtIOtep)(z) = (1(PTep)(z»(t),
(WtIOtep)(z) = (1/2(WTep)(x »(t),
(25.52)
(25.53)
are valid, where the operators l and 1/2 apply with respect to the variable r.
Proof. Substituting the representations (25.47) and (25.48) for Peep and Weep
into (25.51) and then interchanging the order of integration, we obtain the inner
integrals which are easily evaluated and we obtain IOtep as a result. Equations
(25.52) and (25.53) are derived from (25.51) be applying operators Pt and We and
taking the semigroup properties PtP T = Pt+T and W t W T = W t + T into account. .
We observe that (25.51)-(25.53) are obvious from Fourier transforms if we note
that
P(.,t) = e- t1xl ,
W(., t) = e-tlxl ,
(25.54)
(25.55)
- see, for example, Stein, Weiss [3].
Comparing (25.51) with the Liouville fractional integral (5.4) or, more
generally, with the definition (5.86) of a negative power of an operator, we see
that the Riesz potential is a realization of (5.86) under the choice of the Poisson
operator as a semigroup Ttep = Ptep in (5.86).
25.4. Riesz differentiation (hypersingular integrals)
By Lemma 25.2 Riesz differentiation (_/).)01/2 f = .1'-llxI Ot .1,/, Rea > 0, is to be
realized as a convolution with the generalized function Ixl- Ot - n . Such a convolution,
i.e. an integral with the kernel Ix - yl-Ot-n, in contrast to the Riesz potential has
an order of singularity higher than the dimension of the space Rn and so it will be
called a hypersingular integral. Such an integral diverges and so our convolution
needs to be properly defined. Let at first 0 < a < 1 (or 0 < Reo: < 1). We can
guarantee convergence of the convolution of the function Ixl- n - Ot (with sufficiently
good functions) introducing it as
f f(y) - f(x) dy = - f f(x) - f(x - y) dy.
Iy - xln+Ot lyln+Ot
Ra Rft
(25.56)
This integral converges if 0 < a < 1 for bounded differentiable functions and may
be considered as a multidimensional analogue of the Marchaud derivative (5.58).
An extension to the case a  1 may be given either in terms of regularization,

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
499
using Taylor sums as in (25.19) and (25.22), or by taking finite differences. We
have already considered both approaches in the one-dimensional case while dealing
with the Marchaud fractional derivative of the order a  1 - see (5.68) and (5.80).
To realize the Riesz derivative we shall use the approach via finite differences as
more preferable although equivalent, generally speaking, to the former (see  26.5,
where the equivalence is shown for good functions in the more general case).
Let us define finite differences (Af)(x) of a function f(x) of many variables
with a vector step h. Let Th be a translation operator:
(Thf)(x) = f(x - h), x, hE R n .
We shall deal both with centered differences
(Af)(x) = (T-t - T)' f
= (-l)'G)f[Z+ G -k)h]
(25.57)
(with a vector step h and center x) and with non-centered differences
(Af)(x) = (E - Th)' f
= t(-l)' G)f(Z - kh).
k=O
(25.58)
To avoid complications in writing we use the notation (Af)(x) for both types of
differences, specifically mentioning the choice of difference when it is essential. We
consider both types of differences because it will not always be possible to use just
one type of difference - see  26.4 below.
Thus the realization of the operation (_A)a/2, a > 0, is expected to be given
in the form of the hypersingular integral
n a f =  f (Af)(x) dy,
d n ,'( a) Iyln+a
R"
(25.59)
where the normalizing constant dn,,(a) will be chosen so that n a f would not

500 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
depend on I, only if I > a. The construction
1 f (L\/)(x)
D I = d n ,,( a) Iyln+a dy
lyle
(25.60)
will be called a truncated hypersingular integral.
We find it convenient to have also the designation for the integral (25.59)
without normalizing constant:
T a l = f (A/)(x) d I
Iyln+a y, > a.
Rft.
(25.61 )
The next section is devoted to a more detailed investigation of the hypersingular
integrals (25.59) and of more general constructions. Here we dwell only on the
main fact that the hypersingular integrals (25.59) is indeed the Riesz derivative in
the sense that
-G""I = Ixla j(x)
(25.62)
under the appropriate choice of the normalizing constant dn,,(a) in (25.59).
Lemma 25.3. The Fourier translorm of the integral Tal with I > a is given by
the relation
:F(T a I) = dn,,(a)lxI Q j(x), I E ego,
(25.63)
where
dn,,(a) = f (1 - e itl )'Itl-n-adt
Rft.
(25.64)
in the case when (A/)(x) is a non-centered difference and
dn,,(a) = f (e iYI / 2 - e- iy I!2)'lyl-n- a dy = 2'-ai' f sin' ydyl-n-ady, (25.65)
Rft. Rft.
when the difference is centered.
Proof. Let Tea I be the integral (25.26) truncated as in (25.60). Let the difference

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
Af be centered. Then
501
F(T.' f) = t( -1)' G) J Iyf!+a J e'., f(" - 1:,1 )d"
k-O Iyl>£ R"
,
= i{:.) L:(-1)' G) J Iyl-n-ae;b.'d,l,
k=O Iyl>£
I.e.
 J (1 - e izoy ),
:F(T£Ot f) = f(x) lyln+Ot dy.
Iyl>£
It is easily shown that passage to the limit is possible here (in the sense of L2, for
example), so
(25.66)
J ( 1 iozllzl ) '
:F ( T Ot f) = Ixl Ot j ( x) - e 
1ln+Ot
Rft.
(25.67)
after the dilatation change of variables y = Ixl-l. We make also another change
of variables:
 = w z (7J),
x
wz( el) = j;!'
(25.68)
where w z (7J) is any rotation in Rn, which transforms the first coordinate vector
el = (1,0,...,0) into the unit vector x/lxl. It is clear that II = I'll and
 .  = 71 . e 1 = 711, so
J (1 - eioz/lzl)' = J (1 - e ifh )' d
1ln+Ot  17J1n+0t 71,
Rft. Rft.
which transforms (25.67) into (25.63)-(25.64).
Similar arguments are to be used in the case of a centered difference. .
Equation (25.63) yields (25.62) after division by dn,,(a). However, we have to
be cautious because of the possibility that the constant dn,,(a) is zero for some a.
In the case of the centered difference the question is easy: the normalizing constant
dn,,(a) equals zero identically (for all a) if 1 is odd and it is certainly different from
zero if 1 is even, this being seen from (25.65).
Thus the construction TOt is identically zero: TOt f == 0 in the case of a centered
difference of an odd order. Consequently, a centered difference is to be taken of
an even order I = 2,4,6, . " only and then the passage from (25.63) to (25.62) is
possible for all a, 0 < a < I.

502 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
The question regarding the non-vanishing of d n ,,( a) in the case of non-centered
difference is more difficult, and it will be considered in S 26, where the detailed
investigation of hypersingular integrals is discussed. It will be shown, in particular,
that dn,,(a) as a function of a vanishes at the integer points of the interval (0, I).
We formulate a corollary of Lemma 25.3.
Corollary. The normalizing constant being given by the formulae (25.64) or
(25.65) (in the case when it does not vanish), the hypersingular integral D a f does
not depend on the choice of I, I > a.
In conclusion of this subsection we give the relations
00
D a f =  I t-I-a(E - Pt)'fdt, 0 < a < I,
x(a, I)
o
(25.69)
00
D a f = 1 I t-l- ( E- W; ) ' f dt 0 < a < I '
x(,/) t,
o
(25.70)
similar to (25.51), where x(a, I) is the constant (5.81), while Ptf and Wtf
are the Poisson and Gauss- Weierstrass integrals (25.47)-(25.48). In the case of
"good" functions these relations are derived from (25.52) and (25.53) by inverting
the on_e-dimensional operators I and I/2 in terms of the Marchaud fractional
derivative (5.57) with further passage to the limit as t -+ O. Relations (25.69) and
(25.70) may be considered as realization of (5.85) under the concrete choice of a
semigroup Tt in (5.85). Relation (25.69) is herein a realization of the fractional
power ( y_A )a, while (25.70) gives the fractional power (_A)a/2.
25.5. Unilateral Riesz potentials
We call integral operators
a Cn-l I y:
I-l:cp = f(a) Iyln cp(x =F y)dy,
R"
+
f(n/2)
C n = 1rn/2 '
(25.71 )
unilateral Riesz potentials. Here x = (Xl, .. " x n ) E Jl!1 and integration is carried
out over the half-space R+ = {x : x E Jl!1, X n > O}. Since y:lyl-n  Iyla-n,
the operators (25.71) are similar in a sense to the Riesz potential [acp, being in
particular bounded from L" into Lq, q = np/(n - ap), with 1 < p < n/a. The
integrals Icp are immediate generalizations of Liouville fractional integrals (5.4),
coinciding with them if n = 1. That is why we keep to the same notation for these
objects. As in the one-dimensional case we shall call potentials It. cp and I cp left-
and right-hand sided respectively. We denote x' = (Xl,' .. ,xn-d and observe that

 25. RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
503
the constructions (25.71) represent an "interlacing" of one-dimensional fractional
integration and the Poisson integral:
00
I:;' 'P = r (lOt) J V:- I (P.. 'Ph Zn 'F Vn»( z')d1Jn ,
o
(25.72)
where
( ( »( ' ) J <p(X' - y',x n - Yn) 11
Py,. <p " X n - Yn X = Cn-lYn (IY'12 + y)n/2 d .
R,,-l
Let us evaluate the Fourier transform of the potentials I<p. As in S 25.2, it is
convenient to use Lizorkin spaces  and W from S 24.9 for this purpose.
Lemma 25.4. Let Q > 0, ,p E Wand <p E. Then
r(:) J (;I >b(z)dz = J (WI 'F i{n)-a",({),
R" R"
(25.73)
(Iicp )() = (lei =F in)-a<p().
(25.74)
Proof. We denote
k a ( x ) = Cn-l ( X ) a I x l -ne-elz,,1
%.e r(Q) n %
=  ( x ) a-le-elz"lp ( x' I x I)
r( Q ) n % , n ,
where P(x', IXn I) is the Poisson kernel (25.49):
P(x',lxnl} = c n _d x nl(lx'12 +x)-n/2.
00
By (25.54) we have kL() = (Fz..Fz,kte)() = rfa; J x-le-z,,(e+IeITi")dxn'
o
Relation (7.5) shows that the latter integral equals (c + I'I =F in)a under the
assumption that arg(c + I'I =F in) = 0 as n = O. Hence
C J (x )a e-elz,,1 A J
r(:) n jxln ,p(x)dx = (c + I'I =F in)-a1/J().
R" R"
(25.75)

504 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
This yields (25.73) as c -+ 0, which, in its turn, leads to (25.74). .
Corollary 1. Potentials I possess the semigroup property: II = I;+/J.
Corollary 2. The Riesz potential 1 01 is representable as a composition 01 unilateral
potentials:
1 01 = 1:/21/2 = 1/21:/2.
(25.76)
We find now the explicit form of the operators inverse to I. The desired
constructions are to coincide with the Marchaud derivatives D - see (5.57) in the
case n = 1. We apply formally the Fourier transform :Fzl in the variable x' to the
relation (I+<p)(x) = I(x), x = (x/,xn)' By (25.54) we obtain
(I+eYalel (:Fz1<p )(/, Yn»( x n ) = (:F z 1 I)(e, xn)eZal(/1
where the operator It. is applied with respect to the variable Yn' Inverting it by
means of the Marchaud derivative (5.57), we arrive at
Zalel ( ) I ) _ 1
e :Fz1<p ( ,X n - - r( -a)Al(a)
00 1
X / ((-l)' W(F..f)({',"'n - kyn)e(..-...)IE'I) Ja '
whence after multiplication by e-zalel and application of the inverse Fourier
transform :F;,1 we finally have
\0("') = n;) J f: (8f)(",)dy, I > n,
R+
where AI is the non-centered difference (25.58). So
(1 01 ) -1 1 = C n -l J y;OI ( AI 1)( x ) d y 12D OI I, I >
% x(a, I) Iyln u%Y :i: a.
R+
(25.77)
The integrals in (25.77) are hypersingular. It is not difficult to show that for
I E S( R n ) they converge (conventionally) in the sense that
D%I = lim D% e/,
e-O '

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 505
where
00
D a I Cn-l f f y;;a ( ' )( )
%,e = x(a,l) Iyln /).-J:yl x dy.
e R,,-l
The constructions DI are identical with Marchaud fractional derivatives if
n = 1. This was the reason for keeping the same notation as in the one-dimensional
case.
We also note that (Di/)() = (lei =F in)a j().
 26. Hypersingular Integrals and the Space of
Riesz Potentials
We now go on to a more detailed consideration of the hypersingular integral
D a I =  f (/)./)(x) dy, I> a,
d n ,'( a) Iyln+a
R"
(26.1 )
defined in S 25. In particular it will be shown that the hypersingular integral
generates a "true" inverse to the Riesz potential when the latter is considered
within the framework of the spaces Lp. We shall clarify what advantage the
centered or non-centered form of a finite difference has in the definition of
the hypersingular integral. Further, hypersingular integrals, more general than
D a I, will be considered. They prove to be a natural extension of partial
differential operators into the case of fractional orders, since any homogeneous
partial differential operator with constant coefficients may be represented as a
hypersingular-type integral, as will be demonstrated in S 26.6.
A special role in this section is assigned to S 26.7, where the space [a(L p ) of
Riesz potentials is investigated, the spaces L;,r(Rn), generalizing Sobolev fractional
spaces L;(R n ), are considered and the relationship between the spaces L;,r(R")
and the space [a(L p ) is established. We note the relation
L;,r(R n ) = Lr n [a(L p )
(26.2)
as one of the main resul ts.
26.1. Investigation of the normalizing constants dn,,(a)
as functions of the parameter a
As it was made clear at the end of S 25, we have first of all to answer the question
about the zeros of the function dn,,(a) used in (26.1). We find them in this
subsection and at the same time we give representations of the integrals (25.64)
and (25.65) via elementary functions of the parameter a.

506 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
We introduce the following functions of the parameter a:
! 1
AH a) = L: (-1 )1:-1 () k a
1:=0
A,(a) = [1/2]
A'(a) = 2 I:ld:o (_1)1:-1 () ( - k)a
in the case of the non-
centered difference,
in the case of the
centered difference,
(26.3)
in terms of which the constants dn,,(a) will be expressed. Here I = 1,2,.... The
former of these functions is familiar to the reader from the one-dimensional case,
where it arose in the process of the evaluation of the normalizing constant for
the Marchaud fractional derivative of order a  1 - see (5.73) and (5.80). We
considered there the Marchaud derivatives with non-centered differences only. The
latter of the functions (26.3) in the case of even I = 2,4, . " may be represented
also as
A1'(n) = (_l)H G) I - kl a (26.4)
The right-hand side here is identically zero in the case of an odd I and so it differs
then from A' ( a).
Lemma 26.1. The function AHa), a E R 1 , vanishes at the points a =
1,2,3,..., I - 1 and nowhere else. The function A'(a) has even integers
a = 2,4,6,..., I - 2 as zeros in the case of an even I and odd integers
a = 1,3,...,1- 2 in the case of an odd I.
Proof. We consider A, ( a) = AHa) first. Let us use the relation
1
, _ I f (1 - ')'-1
A,(a) - r(1 _ a) Ilnla '
o
a < I,
(26.5)
obtained in S 5 - see (5.81). It is readily seen from (26.5) that in the case a < I the
function AHa) vanishes only at the poles of the gamma-function: a = 1,2,. . . ,1-1.
If a = I, we have AHa) f; 0 by (5.74'). Let a > I. The recurrence relation
11 A+1(a + 1) = A+1(a) - AHa) holds, and this is verified by definition (26.3).
It is then not difficult to show by induction that for a > I, AHa) > 0 if I is odd
and AHa) < 0 if I is even. Thus A(a) 1= 0 for a > I also.
Let now A,(a) = Ana). Using (26.4) we conclude that if a is an integer and
has the same parity as I, then
A1/(n)=(-1)HG) G-kr

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 507
,
Since (pl/2 - p-l/2)' = L: (-1)11 ()p'/2-1I, then
11=0
-(p :p r ( p ,fiI ) t, = A:'(n)
(26.6)
for an integer a of the same parity as I. Hence the proof of Lemma for A'(a)
follows. .
Remark 26.1. Lemma 26.1 answers the question about the zeros of the function
A,(a) = AHa). In some problems related to hypersingular integrals (Samko
[31, p.140]) it is of importance to be sure of the absence of other zeros of the
quasipolynomial A,(z) in the complex plane except at the points z = 1,2,...,1- 1.
This question is open.
Theorem 26.1. The normalizing constants dn,,(a) in (26.1) are analytic functions
of the parameter a and are given by the relations
A, ( a )
dn,,(a)=Pn(a) . ( /2) '
sm a1r
(26.7)
1rl+n/2
Pn(a) = 2ar (1 + ) r ()
(26.8)
ezcept for the case of a centered difference of an odd order I, when dn,,(a) == O.
The constant dn,'( a) is different from zero for all a > 0 in the case of a centered
difference and of an even I, while in the case of a non-centered difference it vanishes
for a > 0 if and only if a = 1,3,5,... ,2[1/2] - 1.
Proof. We begin with the remark that A,(a) = 0 for an even integer a by
Lemma 26.1, thus A,(a)/sin(a1r/2) in (26.7), in the case a = 2,4,6,..., is to be
understood as
. A,() _  _ a/2
hm . ( /2) - (1) d A,(a). (26.9)
-a SID 1r 1[' a
E'valuating the integral (25.64) in the case of a non-centered difference, we
have
dn,,(a) = 1(1- eiYl)'dYl 1 (liil 2 + yn-(a+n)/2dii
Rl R..-1
= 1(1 - e iYl )'IYll-1-adYl 1 (1 + 17J1 2 )-(a+n)/ 2d 7J.
Rl R..-l
(26.10)

508 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
We find the value of the integral over Rn-l by changing to polar coordinates:
00
1 (1 + 17J1 2 )-(a+n)/ 2d 71 = I S n-21 1 pn-2(1 + p2)-(n+a)/2dp
Ra-l 0
00
= ISn-21 1 r(n-3)/2(1 + r)-(n+a)/2dp
o
1 ( n-l l+cr )
= 21Sn-21B '  .
Since
ISn-d = 2..(n-l)/2/r ( n ; 1 ),
(26.11 )
we have
1 (1 + 1'11 2 )-(a+n)/2d'1 = ..(n-l)/2r ( 1  ", ) Ir ( n  ", ) .
R..-l
So
00
1r(n-l)/2r (H,g) 1 (1 - eit)' + (1 - e- it )'
d ( cr ) - 2 dt
n,' - r ( na ) 0 t 1 + a
21r(n-l)/2r ( U:i! ) 1 00 , ( I )
= 2 t- 1 - a ""' ( _I ) 1c cosktdt
r ( !!.:f:!!) L..J k .
2 0 1c=O
,
Let 0 < cr < 1. Using the relation L: ( _1)1c G) = 0 and the equation
1c=O
1 00 ka-l
x-a sinkxdx = 2sin(:1r/2)r(cr)
o
(26.12)
derived from (7.6), we transform the remaining integral to
, ( ' ) 1 00 1 ' ( I ) 1 00
E(-I)1c k t-1-a(coskt -1)dt = ;; L(-I)k-l k k t-asinktdt
1c=O 0 1c=O 0
_ 1rA,(cr)
- 2r(1 + cr)sin(cr1r/2)

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 509
which gives the value of dn,,(a) presented in (26.7). For other values of a  1 the
required result is easily obtained by analytical continuation in a.
In the case of a centered difference and of an even I we obtain
00
_ '/2 2 '-a+l 1r (n-l)/2r (¥) f sin't
dn,,(a) - (-1) r (!!.:f:.!!) t 1 + a dt
2 0
(26.13)
similar to (26.10). We make use of the formula
'/2 ( 1 )
sin't = 2 1 -' (_I)I:-l 1/2 _ k (1- cos2kt)
which is simply obtained by expanding sin' t as a Fourier series in cos let on the
interval [0,1r]. Using this result and integrating by parts yields
00 '/2 00
f ,-'-0 sin' tdt = 2'-' )-1)'-' C/2 1 - ,,)  f,-o sin 2"t dt
o 1:=1 0
(26.14)
a-' ij2 ( )
= 1r2 -1 1:-1 I ka
r(1 + a)sin(a1r/2) () 1/2 - Ie
in the case 0 < a < 2. It is easily seen that
'/2 ( ) ) '/2
(-1)'-' IlL" "0 = (- Ann).
So (26.14) is transformed to
00 . , a-'-1
f SID t dt = _1'/2 2 A,(a)
t 1 + a () r(l+a)sin(a1r/2}'
o
(26.15)
Substituting this expression into (26.13) we obtain (26.7) after simple transforma-
tions. The required result for values a  1 is obtained by analytic continuation in
respect to Q. The identity dn,,(a) == 0 in the case of a non-centered difference and
of an odd 1 is readily seen from (25.65).

510 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
It remains to elucidate the question concerning the vanishing of d,.,,(a). The
case of a centered difference is clear in view of (25.65). So let the difference
be non-centered. By (26.7) the question is reduced to that of the zeros of the
function A,(a) = AHa), which were found in Lemma 26.1. Zeros of this function,
being a = 1,2,3,..., I - 1, are prime according to (26.5). So dn,,(a) f; 0 for
a = 1,2,3,... in correspondence with (26.9). .
We give also explicit values of dn,,(a) for even a:
(_I)a/21rn/221-a dA,(a)
dn,,(a) = r (1 + ) r () da ' 1 = 1,2,3,... ,
(26.16)
(_1)1+(a+1)/22 2 - a 1r n / 2 /! dA,(a)
dn,,(a) = r(I+)r() da ' 1=2,4,6,...,
(26.17)
a = 2,4,6,...,
in the cases of non-centered and centered differences, respectively.
Thus, the normalizing constant dn,,(a) being chosen in correspondence with
(25.64), (25.65) and (26.7), (26.1) properly defines the operators n a / according
to Theorem 26.1, in all cases except the case when the difference is non-centered
and a = 1,3,5,... In this case dn,,(a) = 0 and we come across the phenomenon of
"annihilation" of the construction (25.61):
T a / = 0, a = 1, 3,5, . . . < I,
(26.18)
if a non-centered difference is used.
So a non-centered difference has, on the one hand, an advantage in that its
order is not necessarily even. (The requirement to choose 1 even means that
we are in general to take 1 greater, which is undesirable). On the other hand
a non-centered difference leads to the non-desirable effect (26.18) in the case
a = 1,3,5,.. .. We shall consider this special case in S 26.2.
26.2. Convergence of the hypersingular integral for smooth
functions and diminution of order 1 to 1 > 2[a/2] in the case of
a non-centered difference
Let us show that the hypersingular integral (26.1) converges absolutely for functions
/(z), which have bounded partial derivatives of order [a] + 1. For definiteness we
consider the case of a non-centered difference.

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 511
The relation
. 1 1
(dJ)(z) = m ;E  J(1- U)m-l ;E(-l)m->kmG)(V'" J)(z -I.:uh)du,
bl=m 0 k=O
l m,
(26.19)
is valid, where hi = h{l ... h", We have given its proof in the one-dimensional
case - (5.75). In the case of many variables it is proved in a similar fashion by
means of the Taylor expansion for functions of many variables with the remainder
in the integral form.
We conclude from (26.19) that
I(Af)(x)1  clhl m sup I (vi f)(x)1 = ctlhlm, I  m.
zeR"
Ijl=m
(26.20)
We readily obtain from (26.20) that the integral (26.1) converges absolutely if
a function f(x) and all its derivatives (D; f)(x), Ijl = [a] + 1, are bounded.
We show now that the hypersingular integral (26.1) converges conventionally
for I > 2[a/2] in the case of a non-centered difference. In other words, the integer
I may be chosen not necessarily greater than a, but as the odd integer which is the
nearest to a. We consider I to be odd, since if I is even and I > 2[a/2], then I > a.
The following directly verified relation
1 ( 1 1 1 ) ) /+1
(r - 1) = p,(r) + :;: + ... + r(l-I)/2 + 2r(1+1)/2 (r - 1 ,
(26.21 )
is valid, where the function F}(r) = (r - 1)/(r + l)r-(1+1)/2 is anti-invariant
relative to the inversion, i.e. F}(r- 1 ) = -F}(r). Equation (26.21) generates the
corresponding identity in finite differences
(/-1)/2
(J)(z) = (P:J)(z) - ;E (d+l f)(z + ky) _ (d+1 f) (z + I  1 y),
k=1
x,y ERn,
(26.22)

512 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where the function (Pf)(x) is odd in y. Then the limit
lim f (Af)(x) d - (/-1)/2 f (L\+1 f)(x + ky) d
£_0 Iyln+a y - -  Iyln+a y
Iyl>£ k_1 R"
(26.23)
_! f (A+lf) (x + ¥y) d
2 Iyln+a y
R"
exists, since 1+1 > a, which yields the conventional convergence of the hypersingular
integral Da f.
We return now to the phenomenon of the annihilation (26.18) of the
hypersingular integral of odd order with a non-centered difference. Since I > 2[a/2]
is allowed, we may avoid this phenomenon by choosing I = a in the case
a = 1,3,5,.... Then A,(l) f; 0 and the definition of a hypersingular integral by
(26.1) with I = a = 1,3,5,. .. becomes proper. It should be stressed, however, that
conventional convergence in this case is essential even for the very good functions.
Everywhere below we assume that I > a and I is even in the case of a
non-centered difference and I > 2[a/2] in the case of a non-centered difference with
the obligatory choice I = a in the case a = 1,3,5,.... However, in S 26.4 while
generalizing the hypersingular integral (26.1) we shall allow odd I in the case of a
centered difference too.
The following lemma is close to (25.34).
Lemma 26.2. The Riesz differentiation (_L\)a/2 = Da can be interpreted as a
result of a fractional differentiation in an arbitrary direction u, lul = 1, with the
posterior integration in respect to u:
Daf=
r(-a)sin(a1l"/2) f (Vf)(x)du,
!3n(a)
5"_1
(26.24)
where !3n(a) is the constant (26.8), (Vf)(x) is the fractional derivative (24.38)-
(24.38') in the direction u and Da f is taken with a non-centered difference.
The proof is obtained by changing to polar coordinates.
26.3. The hypersingular integral as an inverse of a Riesz
potential
The hypersingular integral (26.1) generates an operator mverse to the Riesz
potential [a <p:
Da [a<p = <p
(26.25)

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 513
for sufficiently good functions, e.g. cp E (Rn), which is obvious via Fourier
transforms, see (25.35) and (25.62). We shall show that the inversion (26.25) is
true on the whole domain of the Riesz potential within the frames of Lp-spaces:
cp( x) E L" (R), 1  p < n / Ot.
It is important to stress here that the hypersingular integral DO applies
to functions [0 cp which are not "very good" and so it will not be, in general,
absolutely convergent. It will be interpreted as conventionally convergent in Lp:
DO f = lim DO f
£-0 £ ,
(L p )
(26.26)
where D f is the truncated hypersingular integral (25.60).
We introduce certain auxiliary functions, which are finite differences of the
Riesz kernel ko(x). Namely, let us define the function
Al,o(X, h) = (L\ko)(x)
(26.27)
where Ot > 0, 1= 1,2,3,... and single out the case h = el = (1,0,. .. ,0), setting
k"o(X) = Al,o(X, el)
(26.28)
so that
1
k.,a(z) = "Yna) (-1)' G) Iz - ked a - n
(26.29)
if Ot - n f; 0,2,4, ... and a non-centered difference is chosen. An important role in
our consideration will be played by the kernel
K:.,a(lzl) = dn,'()lzln J k"a(y)dy,
lyl<lxl
(26.30)
Since ko(x) = Ixl-o by (25.25), the Fourier transform of the function (26.27) is
given by the expression
{ Ixl-0(l- eixoh)'
.1'(Al,o(" h »(x) = .
Ixl-O( e,xoh/2 _ e-ixoh/2)'
for a non-centered
difference
for a centered
difference.
(26.31)
The function Al,o(X, h) may be expressed in terms of k"o(X) by means of the

514 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
rotation:
"a('" h) = IW-nk',a C w;l(h»)
(26.32)
where I > er - n in the case er - n = 0,2,4,.... Here Wz(h) IS the rotation
(25.68). Equation (26.32) is verified directly by means of the equality Ix - kell =
IlxleI - kg I, which yields the relation hw;l(h) - ke11 = Ihl- 1 Ix-khl if we take
I
into account that In Ihl L: (_1)1: G) Ix - khl a - n == 0 in the case er - n = 0,2,4,6,...
1:=0
and I > er - n.
Lemma 26.3. The function k"a(x) satisfies the estimate
Ik',a(x)1 $ c(l + Ixl)a-n-l as Ixl  1+ 1,
(26.33)
so that
k"a(x) E Lq(Rn), 1 - er/n < l/q  1
(26.34)
if I > er. Besides
f k"a(x)dx = O.
Rft.
(26.35)
Equation (26.35) is valid in the case 2[er/2] < I $ er too, if I is odd, the
difference in (26.28) is non-centered and the integral in (26.35) is interpreted as
lim f k"a(x)dx.
N-oo l I
z- el <N
Prool. We denote W(s) = ka(x + sed, s E Rl, so that k"a(X) = (A W)(O). By
the known relation
 
(A W)(s) = f... f w(l)(s - SI - '" - sl)ds 1 ... dSI
o 0
(26.36)
we obtain k"a(X) = W(l)( -8), 0 < 8 < I, whence the estimate (26.33) is easily
derived, the inequalities Ib(pl: Inp)1  cpl.:-l with p > 1 and I > k being taken
into account in the case er - n = 0,2,4,.... Since k"a(X) is locally integrable to
the power q, l/q > 1 - erin, we see then that (26.33) implies (26.34). Equation

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 515
(26.35), i.e. the relation k"a(O) = 0 follows from (26.31) if I > a. Further
J k',a(z)dz = J t(-l)" ()ka (Y + G - v)e,)d Y
Iz- !ell<N lyl<N 11_0
Yl>O
+ J t(-l)" ()ka(Y - G - v)e,)d Y .
lyl<N 11-0
Yl>O
Changing the index v in the second sum by I - v we see that it differs from the
first one by the factor (-1)'. So
J
k"a(x)dx = 0
(26.37)
Iz- !ell<N
for all N > 0 in the case of an odd I. .
Lemma 26.4. The estimates
{ Ixlmin(a-n,o),
1K:"a(lxl)l  c 10 1
IZT'
1K:"a(lxl)l  clxl a - n - 1 . ,
a f; n,
Ixl  1,
a= n,
(26.38)
Ixl  1,
(26.39)
are valid, where I- = 1 in all cases when 1 > a, and I- = 1 + 1 in the case
when 2[a/2] < 1  a and a non-centered difference is used in (26.38). So
K:',a(lxl) E Lq(Rn), 1- a/n < l/q  1.
Proof. The verification of (26.38) is obvious. To prove (26.39) we use the property
(26.35) and have
1K:"a(lzDI = 1:ln k:(Q) J k"a(Y)dyl  Izln:,-a '
lyl>lzl
If 2[a/2] < 1  a, so that 1 is odd, the estimate (26.39) may be obtained because
(26.37) enables us to carry out the integration not over the baIllyl < lxi, but over
the layer Ixl- 1/2 < Iyl < Ixl + 1/2. Namely by (26.37) and (26.33) we have for

516 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Ixl > , + 1:
1.t,.a(lzDI  Izlnld.,(a)1 f Ik',a(y)ldy
Izl- <IYI<lzl+
I( 1 ) 01-1 ( ' ) 01-1 1
<  Ixl + - - IXI - -
- IX In 2 2
< I l -n+Ot-l-1
_ C1 X .
We now prove the following theorem which is preparatory for the inversion of
the Riesz potential [Otcp and may be considered as a multidimensional generalization
of Lemma 6.1.
.
Theorem 26.2. Let f(x) = [Otcp, 0 < a < n, cp E Lp(Rn), 1 :$ p < n/a. Then the
truncated hypersingular integral D f admits the representation
(D f)(x) = f K"Ot(lyl)cp(x - cy)dy.
Rft.
(26.40)
Proof. Applying a finite difference to the Riesz potential, we have
(Af)(x) = f L\"Ot(, y)cp(x - )d
Rft.
(26.41)
where L\"Ot(, y) is the function (26.27). Hence
(n: f)(z) = dn.:(aJ <p(z - ) f 81i!:'..") dy.
R" lyl >£
The interchange of the order of integration is possible in view of the absolute
convergence in the case of c > O. We apply (26.32) and make then the change of
variables wi1(y) = z E Rn, where w;l(y) is the rotation inverse to the rotation
(25.68). Then Iyl = II/izi and
01 1 f cp(x - ) f
(D£ f)(x) = d n ,l(a) 1ln d k',Ot(z)dz.
Rft. Izl<II/£
The dilatation transformation  -+ c leads now to (26.40). .

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 517
Remark 26.2. The kernel K:"a(lyl) has an averaging property:
f K:',a(lyl)dy = 1.
Rft.
(26.42)
This relation, being a consequence of the choice of the normalizing constants, is
easily established indirectly: let <p E , then the passage to the limit is possible in
(26.40) as g -+ 0: D a I = <p(x) f K:"a(lyl)dy. Since D a I = <p for I = [a<p with
Rft.
<p E  by (26.25), hence (26.42) follows.
Corollary. The Fourier transform of the kernel K:"a(lXI) is given by the expression
A 1 f (1 - eizoy)'
K:"a(X) = dn,l(a)lxl a Iyln+a dy
lyl>l
(26.43)
being written lor the case of the non-centered difference in (26.27).
Indeed, taking I E  in (26.40), we have 6i1(x) = t"a(gx)n;;-/(x). Letting
g = 1 we obtain then (26.43) by (25.66) and (25.62).
Theorem 26.3. The operator Da I = lim D I is the left inverse to the Riesz
£-0
(L p )
potential within the frames of the spaces Lp(Rn):
D a [a<p == <p, <p E Lp(Rn), 1::; p < n/a.
(26.44)
The way to prove this theorem is provided by Theorem 26.2. Really, by
(26.42) we have
(D: I)(x) - <p(x) = f K:"a(lyl)[<p(x - €y) - <p(x)]dy (26.45)
Rft.
from (26.40). By the Minkowsky inequality we obtain
liD: 1- <pllp ::; f 1K:"al(lyl)w;(<p,€y)dy
Rft.
(26.46)
{ } 111'
where w(<p, h) = 1... l<p(x) - <p(x + h)IPdx . Hence IID 1- <pllp -+ 0 as € -+ 0

518 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
by the Lebesgue dominated convergence theorem, the application of which is
justified by the fact K:"a(lyl) E L 1 (R n ), as proved in Lemma 26.4.
Let us emphasize that we have demonstrated the Lp-convergence of the
truncated Riesz derivative (hypersingular integral) D: I in the range [a(L p ) of the
Riesz potential and have given an estimate of this convergence in (26.46). One
may show that the convergence also holds almost everywhere. We will not dwell
here on this and refer the reader to S 29.2, note 26.1.
In conclusion of this subsection we note that besides the hypersingular
integrals, considered above, constructions (25.69) and (25.70), defined in the
previous section, may be used to invert the Riesz potential [a cp with cp E Lp.
Namely, the following theorem is true.
Theorem 26.3'. Let I = [acp, cp E Lp(Rn), 0 < a < n, 1  p < n/a. Then
00
cp=_ ( I /) lim f t-l-a(E-Pt)'ldt, I>a,
x a, £-0
£
the limit may be taken under the Lp-norm or almost everywhere.
26.4. Hypersingular integrals
with homogeneous characteristics
Generalizing the hypersingular integral (26.1), we introduce the construction given
by
a f (/)(x)
(Do/)(x) = Iyln+a O(x, y)dy
Rft.
where we shall call the function O(x, y), not depending on I(x) a characteristic.
This follows the terminology accepted in the theory of multidimensional singular
integrals (see Mihlin [2]). We shall not consider here the constructions in such
a general form, but treat the case when O(x, y) does not depend on x and is
homogeneous in y: 0 = O(y'), y' = y/lyl so that
a 1 f (!)(x) ,
Dol = dn,,(a) Iyln+a O(y)dy, I > a
Rft.
(26.47)
We write the normalizing constant as before in spite of the presence of an arbitrary
homogeneous characteristic O(y') in order that Do! be independent of the choice
I > a. See Corollary 1 of the Theorem 26.4.
As regards more general characteristics, we refer to the references in S 29.1
(notes to S 26.4) and to S 29.2 (notes 26.3 and 26.4).

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
519
We note at once that the presence of a characteristic O(y') in the hypersingular
integral allows to use this integral with centered differences of an odd order.
We shall give a certain classification of hypersingular integrals (26.47) and be
concerned with following questions in this section:
A) What are the relations giving the Fourier transform of Dol?
B) Under what conditions on O(y') does the convergence in L" of the
hypersingular integral DO I with a constant characteristic imply the convergence
of the hypersingular integral with the characteristic O(y') and vice versa?
In S 26.6 we shall see more clearly what are the hypersingular integrals
(26.47). We shall show that the class of such integrals contains, in particular,
operators of the form :F- 1 a:F with a(x) = IxIOa(x'), x' = x/I, being a sufficiently
smooth homogeneous function of degree a. In the case of an integer a the class
of the operators (26.47) contains all homogeneous differential operators in partial
derivatives of order a.
First of all we state the following.
Definition 26.1. We shall call the integral (26.47) a hypersingular integral of the
neutral type, if it uses a non-centered difference, and a hypersingular integral of
the even (or odd) type, if it uses a centered diffe,rence of an even order (or odd)
respectively.
By (26.20) the integral (26.47) converges for I > a in the case of a sufficiently
good function I(x) if O(y') E L 1 (Sn-l). For a hypersingular integral of a neutral
type and with the even characteristic O(y') = O( -y') the order of the difference
may be lowered owing to conventional convergence to
I > 2[a/2]
(26.48)
with an obligatory choice I = a in the case a = 1,3,5,.... This may be shown
following the same lines as in S 26.2. We consider I everywhere below to be chosen
as it has been indicated.
The neutral type of a hypersingular integral has some advantages in comparison
with an even or odd type. This has already revealed itself in the possibility (26.48)
to lower the order I. It will prove to be more universal in the problems of inversion
of potential type operators. In particular, it makes sense to consider integrals
of the even (odd) type for even (odd) characteristics O(y') only. Namely, if a
characteristic is arbitrary:
O(y') = O+(y') + O_(y'),
O:i;(Y') = O(y'):f: O( -y')
2
then
Dol == Do+l,
Dol == Do_1
(26.49)

520 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
for integrals of the even and odd type respectively. I.e. integrals of the even or odd
types are annihilated in the case when the characteristic has an opposite parity.
These relations follow for example from (26.69)-(26.71) proved below. On the
other hand the hypersingular integrals of neutral type have their own "strangeness"
which is reflected in the following remark.
Remark 26.3. A hypersingular integral of the neutral type IS "annihilated"
identically:
f lyl-n-a(L\/)(x)O(y')dy == 0
R'"
for any Q = 1,3,5,..., whatever / and 0 were, if 1 > Q. This was established in
S 26.1 (see (26.18», for O(y') = const. For O(y') 1= const this annihilation will
be obvious in Fourier transforms as seen in (25.68) and (25.69) below. As for the
case 1 = Q, the hypersingular integral of the neutral type converges conventionally
if and only if O(y') is even. This may be seen from the transformations in S 26.2
by substituting the characteristic O(y') into the left-hand side of (26.23).
Remark 26.4. In the case of a hypersingular integral of the odd type the
normalizing constant dn,,(Q) has not been determined yet. The constant (26.65)
equals zero in the case of an odd I. Starting from (26.7) and taking into account
that A:'(Q) = 0 for Q = 1,3,5,... by Lemma 26.1, we set by definition
_ A:'(Q)
dn,,(Q) - ,Bn(Q) ( /2)
cos Q1r
(26.50)
for hypersingular integrals of the odd type.
We also note that for hypersingular integrals with a homogeneous characteristic,
the relation
DIU = - r(-Q:Q\Q1I"/2) f fI(u)(Vm(z)du
5"'-1
(26.51 )
is valid, where ('D: /)(x) is the fractional derivative (24.38)-(24.38') in the direction
u. This result generalizes (26.24) and is similarly obtained by passage to polar
coordinates in (26.47).
A) Evaluation of the Fourier transform for D o /'
Following the transformations in (25.66)-(25.67), we can easily show that the
relation in Fourier transforms
.1'(Do/) = 'Dn(x)j(x)
(26.52)

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 521
is valid, where the function 1'o(z) is given by the relations
! ( I e i 2: oy ) '
J 11 " + o o(y')dy
1 R"
Vo(z) = d ( a )
n,' J
R"
for a non-centered
difference
(26.53)
o(y')dy for a centered
difference.
We shall call the function Vo(z) the symbol of the hypersingular integral Dof.
This follows the terminology accepted in the theory of multidimensional singular
integrals - see Mihlin [2]. It is clear that the symbol Vo(z) is homogeneous of
order a:
1'o(z) = Izla1'o(z'), z' = z/izi.
(26.54)
Theorem 26.4. Let 0(0') E L1(Sn-d. The following representations for the
symbol are valid:
V(.,)= c:t:;2) / !1(u)(-i",,,Ydu, 0:/1,3,5,...,
5"-1
(26.55)
Vo(z) = wn(a) / o(O')lz, O'ladO',
5"-1
(26.56)
1'o(Z) = -iwn(a) / o(O')lz, O'lasign(z. O')dO',
5"_1
(26.57)
for hypersingular integrals of the neutral, even and odd type respectively, where
wn(a) = r«n + a)/2)[21r(n-I)/2r(I/2 + a/2)t l and the designation (i)a =
1lae-iTsign is used in (26.55). In the case of an even characteristic 0(0') (26.55)
and (26.56) coincide and are valid for a hypersingular integral of the neutral type
of order a = 1,3,5,...
Prool. In the case of a non-centered difference we have
a_I / / 00 (1 _ eipzou)'
1'o(z) - dn,,(a) o(O')dO' pl+a dp.
5"-1 0
(26.58)
As the lOner integral IS analytic 10 a, it IS sufficient to calculate it for

522 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
,
o < Q < 1. Using the identity L: (_1)1: () = 0 we obtain
1:=0
00 ,0000
/ p-l-a(l_.;p{)'dp = (-1). G) U p-l-a(cos kp{-l)dp+i / p-l-a sin kP{d P ] ,
o 1:_0 0 0
where  = x . u. Integration by parts in the first of the integrals and using (26.12)
give
/ 00 (1- eip)' 1rAHQ) . a
l+a dp = r(1 )' (-z), Q > 0, I > Q,
p + Q sm Q1r
o
(26.59)
which transforms (26.58) into (26.55) in view of (26.7).
In the case of a centered difference we arrive at the inner integral
00
I() = / p-l-a (ei-P _ e-p)' dp
o
, 00
= L:(-1)" G) / p-l-a [.;(!-")p -1] dp,
11=0 0
 = x.u,
similarly to (26.58), assuming that 0 < Q < 1 again. Applying (26.59) we obtain
I()= 1r (_IY-l ( / )[ _i( ( _v )] a
sin Q1rr(1 + Q)  v 2
_ 1r1la { It _1)11-1 ( I ) I  _ V i a
2r(I+Q) sin(Q1r/2) 1I=0( V 2
. sign E ' ( ) 11 1 ( I ) I I l a. ( I ) }
+ I -1 - - - V sIgn - - V .
cos( Q1r /2) 11=0 V 2 2
, ,
Noting that L: (_1)11 () I  - via == 0 if I is odd and L: (-lye) 14 - via X
11=0 11=0
. ( ' ) 0 . f/ ' th t I( C ) 'll'A;'(a)I{IO l' I d
sIgn 2 - v == I IS even, we see a ,= 2 r( I+a)sin(a'll' / 2 ) lor an even an
".iA" a 1IOsign l' d I . h . ( ) ( )
I() = 2r l+a cos a". 2 lOr an od , wluc gives 26.56 - 26.57 . .

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 523
Corollary 1. The hypersingular integrals (26.47) do not depend on the order I of
the finite difference under the suggested choice of the normalizing constant.
Corollary 2. The hypersingular integrals of the neutral and even type coincide in
the case of an even characteristic.
We also conclude from (26.55)-(26.57) that the hypersingular integral Dgf
of an integer order a is a homogeneous differential operator in partial derivatives
under the proper choice of the type of a hypersingular integral. Namely, the
following corollary is true.
Corollary 3. The symbol of a hypersingular integral of the neutral or even type
with a = 2,4,6,... or of the odd type and with a = 1,3,5, ... is a polynomial
7r ""' OJ .
vg(x) = 26{3 ( ) L.J -=tr'
n a 1 ' 1 J.
) =a
(26.60)
where 6 = 1 for a hypersingular integral of the neutral and even type and 6 = i for
a hypersingular integral of the odd type, OJ being spherical moments of the function
O( (1):
OJ = f O«(1)d(1.
5"'_1
(26.61 )
To obtain (26.60) from (26.55)-(26.57), the relation
""' m' . .
(x . (1)m = L.J -:fr' (1)
J.
Ijl=m
is to be used.
B) Convergence of hypersingular integrals with different
characteristics.
We wish to answer the following question. Let f( x) be such a function that
its hypersingular integral converges for a certain characteristic 01(Y'). Does then
the hypersingular integral of the same function converge in the case of another
characteristic 02(Y')? It is convenient to break up this question in the following
way: does the convergence of a hypersingular integral with a constant characteristic
(i.e. the convergence of the Riesz derivative (26.1» imply that of such an integral
with an arbitrary characteristic, and vice versa? In the first direction this question
will be solved positively in the sense that the convergence of D a f implies that of
Dgf for an arbitrary bounded characteristic. The boundedness condition may be
weakened. Clearly, since the convergence of Dof with the characteristic O(y') == 0
cannot imply the convergence of D a f, the inverse assertion will require a certain
"ellipticity" condition on the characteristic O(y').

524 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
For the function I(x) itself we make an a priori assumption that I(x) E Lr(R n ),
1 < r < 00, while the convergence of its hypersingular integral will be considered
in the space Lp(Rn), 1 < p < 00:
Dol = lim Do e/,
e-O ·
(26.62)
where
01 1 f (L\/)(x) ,
Do..el = dn,,(a) lyln+OI O(y)dy.
Iyl>e
We do not go into details here and give only the outline of our arguments.
The details of the proof may be found in Nogin and Samko [1], [3].
Let I( x) E Lr ( R n ) and let the Riesz deri vati ve DOl I exist in Lp ( Rn ). For such
functions one can directly verify that
DOl 1 = A D Ol i
o.,e e
(26.63)
where Ae is a convolution operator with the kernel
( ) 1 f L\"OI(X, y)O(y') d
a e x = dn.,(a) lyln+OI y
Iyl>e
(26.64)
A,. OI (X, y) being the function (26.27). This kernel is locally integrable and
lae(x)1  clxl- n as Ixl -+ 00. Using the properties of the function L\"OI(X, y) (see
S 26.3) one can show that the convolution operator with the kernel is bounded
in Lp uniformly in €: IIAe IILp-L p $ c, where c does not depend on €. This
fact allows us to justify the passage to a limit in (26.63) as € -+ 0 by the
Banach-Steinhaus theorem. The limiting operator A = lim Ae is an operator of
e-O
the form A<p = 00<P + B<p where 0 0 is a mean value of the characteristic over the
unit sphere and B is a certain Calderon-Zygmund singular operator, bounded in
Lp(R"), 1 < p < 00. Thus (26.63) gives convergence of Do,el in Lp(Rn). To
obtain the inverse assertion, the invertibility of the limiting operator A is required.
Calculating the Fourier transform of the kernel of this singular operator we can
show that the criterion of invertibility of the operator A has the form
f (-ix. O') OI O(O')dO' 1= 0, Ixl = 1,
5"-1
(26.65)
provided that hypersingular integrals of the neutral type are used (cf. (26.55».
If the characteristic is sufficiently smooth and satisfies the ellipticity condition

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 525
(26.65), then the inverse statement can be derived from (26.63), namely that the
convergence of Dol in Lp implies that of D a I in Lp, I E Lp. The above arguments
are justifiable provided that 1 < p < 00, 1 < r < 00 and l/p - o:/n ::; l/r ::; l/p.
26.5. Hypersingular integral with a homogeneous
characteristic as a convolution with the distribution
We shall show that the hypersingular integral (26.47) .may be represented as
a convolution with a generalized function of the form O(x')/Ixl n + a , i.e. the
hypersingular integral coincides with the regularization of a divergent integral.
Corresponding with (25.19) we define a regularization of a similar divergent integral
with the characteristic O(x') by the relation
.f. O(x') * 1 = f O ( , ) /(x - y) - (R-1 I)(x) d
p Ixl n + a y Iyln+a y
lyl<1
+ f O(y')/(x - y) d
Iyln+a y
lyl>1
(26.66)
1'1 . ,
+ L: (_:J (1Vf)(z)pJ. f Y'1}adY ,
Ij19-1 lyl<1
where
(R-1 I)(x) = L (-); (V; I)(x)
Ij151-1 J.
is the Taylor sum and I> 0:. Passing to polar coordinates, we easily obtain
f f 1I0(y') d _ { OJ /(Ijl- 0:),
p. . Iyln+a y - 0,
lyl<1
Ijl #= 0:,
Ijl = 0:,
in accordance with the definition 5.2. of a p.f.-integral, with OJ being the spherical
moments (26.61). So
.f. O(x') * 1 = f O ( ' ) I(x - y) - X(y)(-1 I)(x) d
p Ixln+a y Iyln+a y
Rft.
,,",' ( -1 )1; 1 0 . .
+  "(1'1_ :) (1)1 I)(x),
lil5 i - 1 J J
I> 0:,
(26.67)

526 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where the dash denotes omission of terms with Iii = a in the case of an integer a
and X(y) is the characteristic function of the ball Iyl < 1.
If we choose I to be the integer, nearest to a, I > a, i.e. I = [a] + I, then
(26.67) may be written as
f O(x') f - f f(x - y) - (Rf,a] f)(x) O ( ' ) d
p. 'Ixl n + a * - Iyn+a y y,
R"
(26.68)
a :F 1,2,3,.. .,
which is directly verifiable. The form (26.68) has an evident advantage in
comparison with (26.67), but it is not suitable for integer values of a. (In the case
of an integer a it may make sense at the cost of the restriction J O(u)du = 0,
5"_1
which excludes, in particular O( u) = const .)
Theorem 26.5. Let us suppose that O(u) E L.(Sn_.}, while f(x) E C'(Rn) and
is bounded. Then
( D n a f )( x ) = sin( a1r /2) p f O( x') * f
f3n(a) . ' Ixl n + a '
( D a f)( ) - _ sin(a1r/2) f O(x') + O( -x') f
n x - f3n(a) p.. 2lxln+a *,
( D a f)( ) = _ cos(a1r/2) f O(x') - O( -x') f
n x f3n(a) p.. 2lxln+a *,
a # 2,4,6,..., (26.69)
a:F 2,4,6,..., (26.70)
a:F 2,4,6,..., (26.71)
for hypersingular integrals of the neutral, even and odd type respectively. In
the excluded cases of integer a in (26.69)-(26.71) the hypersingular integral is a
differential operator in partial derivatives of order a:
(Dof)(x) =
(_I)[a/2]1r L : CD; f)(x)
2,Bn(a) 1;I=a J.
(26.72)
(a = 2,4,6,... for a hypersingular integral of the neutral and even type and
a = 1,3,5,... in the case of an odd type).
Proof. Result (26.72) follows from (26.60), but can be established directly avoiding

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 527
consideration of Fourier transforms. Let us prove for example (26.69). We have
(Df)(z) = dn,(Q) 1 1' {feZ) + t(-l)" G) [I«z - IIY) - R-;'(z)X(IIY)]
+ t( -1)" G) -;'(Z)X(I/Y) }d y .
Hence after simple transformations
(DOt 1)( ) = - AHa) f O( ,) /(x - y) - X(y)-l(x) d
{} x dn,l(a) y lyln+Ot y
Rft.
(26.73)
1 ( -1) Ij 1 Ij j
+ ;r-- ( ) L ., (V I)(x),
n,' a IjI5'-1 J.
where
Ij = f II 't(-l}"G) (I/Y)jX(lIy)dy,
Rft. 11_0
By direct calculation we find that Ij = m Oj, where AHa)/(a -Ijl) is to be
replaced by dAHa)/da in the case Ijl = a. So (26.73) turns into (26.69). Similarly
(26.70) and (26.71) are proved.
26.6. Representation of differential operators in partial
derivatives by hypersingular integrals
We have shown above that the set of hypersingular operators (26.47) with a
homogeneous characteristic contains some homogeneous differential operators, see
(26.72). Now we prove a stronger assertion, i.e. the inverse statement that all the
homogeneous differential operators in partial derivatives of order a with constant
coefficients may be written as a hypersingular integral Dol with a homogeneous
characteristic. But first we put a more general question, inspired by (26.52) in
Fourier transforms. Let a(x/lxl) be a given homogeneous function. Does there
exist a characteristic O(x/lxl) such that
Dol = .1'-la(x/lxl)lxI Ot .1'f,
(26.74)
and how can it be constructed for a given a(x/lxl}?
We observe that the expression itself in the right-hand side of (26.74) may be
considered as a generalization of the Liouville differential operation. It contains

528 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
partial or mixed Liouville differentiation as a particular case (see (24.14)-(24.15'»
under the choice a(x) = Ixl-lalxi1. ..x". Such a generalization of fractional
differentiation - a convolution with a homogeneous function of order -Q - n -
was considered by Sobolev and Nikolskii [1]. We want thus to show that this
generalized differentiation coincides with a hypersingular integral with a certain
characteristic.
For precision let us deal with hypersingular integral of the neutral type. By
(26.55), (26.74) is reduced to
C:t;}2) J fI( IT)( -;Z . IT) a dIT = a( Z),
5"'-1
(26.75)
Ixl = 1, Q 1= 1,3,5,.. . ,
which is an integral equation of the first kind relative to the unknown characteristic
0(0"). This equation can be solved by means of harmonic analysis on a sphere,
which is well presented in Stein [2] and Stein and Weiss [3] - see also Samko [31].
We recall that the Fourier-Laplace series of the function f(x), x E Sn-1, is the
series of the form
f(x)  L Ym(f,x)
m=O
(26.76)
where Ym(f,x) is a harmonical polynomial of order m confined to the unit sphere.
The harmonic component Ym(f, x) of order m is evaluated by the relation
d n (m) J
Ym(f, x) = I S n-11 f( 0") Pm (x . O")dO"
5"-1
(26.77)
where dn(m) = 2=:22 (m+';:-2) is a number of linearly independent harmonics of
order m. Pm(t) is the generalized Legendre polynomial:
Pm(t) = { (m+';:-3) -1 C:;';2 (t),
Tm{t),
if n  3
if n = 2
where C;. (t) is the Gegenbauer polynomial an d T m (t) = cos( m arccos t) is the
Chebyshev polynomial. The series in (26.76) converges to f(x) provided that this
function is sufficiently smooth.
Expanding the solution and the right-hand side of (26.75) in Fourier-Laplace

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 529
series, we arrive at the relation
CO:(:}2) t J (-i.,. u)aYm(n,u)du = t Ym(a,.,).
m=05,,_1 m=O
(26.78)
The relation
J -l r(m-a)
(-ix. O')aYm(O')dO' = - 1I"a r(1 + a) sin a 11" r () imYm(x)
5"'_1
(26.79)
is valid for any spherical harmonic Y m ( 0') if I x I = 1; in the case of an integer a we
have
{ cmYm(x),
sl (., . u)aYm(u)du = 0,
a - m = 0,2,4,...,
a - m = 1,3,5,.. . ,
a-m= -1,-2,-3,...,
(26.80)
with C m = 2 1 - a 1l"n/2r(1 + a) [r ( m+;+a ) r (1 + a-;m )]-l and Ixl = 1 (see, for
example, Samko [28, p. 163], [31, p. 91]). So in the case of a non-integer a from
(26.78) we obtain
( n + a ) ( a ) . a1l" 11" r ()
r  r 1+2" slDTYm(O,X)=- i m r(T) Ym(a,x)
resulting from the linear independence of spherical harmonics of different orders.
So the desired solution O(x) may be represented by the series
O(x) =
11" 00 r ( m+n+a )
"" 2 y: ( a X )
r ( nt a ) r (1 + ) sin a 2 1r o imr ( m;a ) m , ,
(26.81 )
a f; 1,2,3,...
It can be shown that this series converges and is really a solution to the equation
(26.75) if the function a(x), Ixl = 1, is sufficiently smooth on the sphere. We do
not elaborate on the proof of this. We observe also that the "slowly" convergent

530 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
series (26.81) may be summed up in terms of the divergent integral:
_ r ( n+;+l ) f a(u)du
O(x) - - 21r(n-l)/2r(1 + 0:/2) pJ. (ix. U)n+a '
5"-1
(26.82)
Ixl=l, 0:#=1,2,3,...
(see Samko [30, p. 37]; the integral on the right-hand side can be made meaningful
by regularization or by analytic continuation in respect to 0:).
The case of integer 0: requires special consideration. We give the corresponding
resul t in the following theorem.
Theorem 26.6. Let 0: = 1,2,3, . .. and let
aa(V) = E ajVj
Ijl=a
be a homogeneous differential operator of order 0: with constant coefficients aj.
There exists a homogeneous polynomial r2a(Y) of order 0: such that
1 f (f)(x) ( y )
aa(V)f = dn,,(O:) Iyln+a Oa 1Yi dy
R"
(26.83)
where a hypersingular integral of the neutral or even type is used if 0: = 2,4,6, . . .
and of the odd type if 0: = 1,3,5,..., the functions f( x) being assumed to be
sufficiently good. The characteristic Oa(Y') of the hypersingular integral may be
found by a given polynomial aa(x) via the relation
Oa(Y) = f aa(u)K(y. u)du, Iyl = 1,
5"_1
(26.84)
where
(-I)[a/2]r(n/2) [a/2] k ( n )
K(t) = ( !!) () L (-1) r 2" + 0: - k k!dn(o: - 2k)P a - 2 1:(t),
r 1 + 2 r 2 k=O
-1 < t < 1, dn(o: - 2k) and Pa-2k(t) are the same as in (26.77).
Proof. For clarity we shall deal with the case of even 0: = 2,4,6,. .. By (26.56)

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 531
for the Fourier transform of a hypersingular integral, the desired characteristic is
to be determined from the equation
f ( 1)a/2
Oa(O')(x, O')adO' = :n(a) aa(x),
5"'-1
IXI = 1.
(26.85)
We expand aa(x) and Oa(x) in spherical harmonic series as above in (26.76), only
these expansions are finite sums now. Substituting the expansions into (26.85), we
arrive at the relations
a f a _ (_1)a/2 a
L (x. 0') Ym(Oa, 0') dO' - wn(a) L Ym(aa,x).
m=05..._ 1 m=O
Hence, by (26.80)
a (-I)t a
L cmYm(Oa,x) = - ( ) L Ym(aa,X).
m=O W n a m=O
In view of (26.80) the summation on the left-hand side here is to be carried out
for m = 0,2,4,.. ., a only, which agrees with the fact that the polynomial aa(x)
contains harmonic components of even orders only. Since spherical harmonics of
different orders are linearly independent, we obtain
r ( m+n+a ) r (1 +!!.=.!!!)
Ym(Oa,x) = (_I)a/2 2 ( ) 2 Ym(aa,X)'
r 1 + 
Consequently,
( -1 )a/2 a ( m + n + a ) ( a - m )
Oa(X) = r(1 + a/2) O r 2 r 1 +  Ym(aa, x)
which is reduced to (26.84) by means of (26.77).
The case a = 1,3,5,. .. is treated similarly. .
By straightforward calculations the reader can show that in the case a = 2
n
when a2(x) = L: aijXiXj, (26.84) becomes very simple, namely
i,j=l
( n + 2 ) [ n + 2 1 ]
02(Y) = r  -a2(Y) + 2tra2

532 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
n
where tra2 = ?: aii = 15':11 J a2(O")dO". In particular we see that
1=1 5"'_1
{)21
{)Xi{)Xj
=
r«n + 4)/2) f (L\/)(x) titj d
d n ,,(2) Itln+a Itl 2 t,
R'"
if; j.
(26.86)
26.7. The space fa (Lp) of Riesz potentials and its
characterization in terms of hypersingular integrals.
The spaces L;,r(Rn)
We denote by
[a(L p ) = {I : I = [aep, ep E Lp(Rn)}
(26.87)
the space of Riesz potentials of functions in Lp (Jl'&). It is well defined if 0 < a < n
and 1  p < n/a. By Theorem 25.2 and the inequality in (25.42)
[a(L p ) C Lq(R n ), [a(L p ) C Lq(Rn; lxi-a,,),
1 < p < n/a, q = np(n - ap)-l.
Repeating arguments from S 6.1, we can easily see that [a(L p ) f; Lq(Jr&),
[a(L p ) f; Lq(Rn; Ixl-a p ). It can be shown that [a(L,,) does not coincide with any
of the spaces Lr (Rn ; p), 1 $ r  00, and thereby needs to be characterized.
The space [a(L p ) may be defined for p  n/a also, the case a  n being
admitted, if we interpret the Riesz potential in the sense of generalized functions
on the Lizorkin space (25.16):
([aep,w) = (ep,[a w ), wE <1>, ep E Lp,
which is proper by the invariance of the space <1> relative to [a. Of course under
such a definition this is a space of generalized functions. These functions might be
called "quasisingular" in the sense that their finite differences of sufficiently large
order I > a are usual functions. (The finite differences of generalized functions are
defined in a standard way: (L\/,w) = (I, L\'-hw». We have already dealt with
such a situation in the one-dimensional case - see Remark 8.2. More precisely, the
following lemma is true.
Lemma 26.5. Let I = [aep, ep E L", a > 0, 1  p < 00, I > a. Then
(A/)(x) E Lp(Rn) lor any h E R n if p < n/a. If p  n/a, this assertion

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 533
is replaced by the following: the functional t1.f E ' is regular and admits an
extension from the Lizorkin space  to the Schwartz space S:
(t1.f,w) = (g,w), w E S,
(26.88)
where 9 = g(x) = Al,a(" h) * <p E Lp(Rn).
Proof. From (26.32) and (26.33) we derive the estimate
IA"a(X, h) I  clhl'(lhl + Ixl)a-n-l
(26.89)
for Ixl  (I + l)lhl with c not depending on x and h. Thus t1."a(X, h) E Ll(R n )
for every h, and then it follows from (26.41) that t1.f E Lp(Rn). Let
p  nlOt. Then (26.41) is valid in the sense of generalized functions in :
(Af,w) = (A"a(',h) * <p,w), w E , <p E Lp, which is directly verified. The
right-hand side of the latter equation admits an extension to all w E S. Besides
this 9 = A"a(" h) * <p E L p (Jl!1) since t1. ' ,a(x, h) E L 1 (R n ), which was required. .
We define the norm in the space [a (L1') by the relation
II/lIrO(L p ) = II <p 111' , 1  p < 00.
(26.90)
Let us pass to characterization of the space [a(L p ). We shall prove this
characterization together with a consideration of the spaces
L;,r(R n ) = {I : IE Lr(R n ), DalE Lp(Rn)},
II/IIL;,r = II/lIr + IID a IlIp, Ot > 0, 1  p < 00, 1  r < 00,
(26.91 )
which arise naturally as a certain generalization of the space [a(L p )' We emphasize
that D a I in (26.91) is understood as convergent in Lp. - see (26.26). We begin
with the following auxiliary theorem.
Theorem 26.7. Let f(x) E L;,r(Rn), Ot > 0, 1  p < 00, and 1  r < 00. Then
the difference (t1.r I)(x) with m > Ot admits the representation
(Ar)(x) = f Am,a(x -, h)(D a I)()
R"
(26.92)
where Am,a(x,h) is the function (26.27).

534 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Proof. Let us suppose that D a 1 is constructed by means of a non-centered
difference, the difference Ah is centered and for simplicity let Q - n f; 0,2,4, . . . ,
other variants being considered in a similar fashion. We denote cpe = D 1 and
Bcp = Am,a(.,h) * cp. We have
B<p. = d...:(a) { J dm.a(z - Y. h)f(y)dy J
R" Iy-zl>e
dz
Iy - zln+a
+ t(-l)" G) va J dm,a(z-y,h)dy J IYZ:I::a }'
- Rft. ly-zl>£II
Since Am,a(z, h) E Lt(Rn) if m > Q in view of (26.89), the interchange of the
order of integration is possible. So
B = (-lt ( I ) J I()d J Am,a(z - z - IIY, h) d
CPe d n ,'( Q)  o II Z Z Iyln+a y
11_ Rft. Iyl>e
= rn(a)L(a) i(-1)'(7)+-z+ ( ; -k)+Z
x (-I)1I ( I ) J Iz - lIyla-n dy.
L.J II Iyln+a
11=0 Iyl>e
Hence after the substitutions z = cT, Y = cwz{), where wz() is the rotation
(25.68), we obtain
1 J (Ah I)(z - cT) J ' II ( ' ) I  l a-n
Bcpe = "Yn(Q)dn,,(Q) ITln dT (-1) II 1lel-lI d,
Rft. II<ITI 11-0
which owing to the equality 11lel - II/III = I - lie 1 I yields
J d m . a ( z - T. h )<p. ( T )dT = J (d'h f)( z - tT )JC'.a(lTl)dT.
Rft. Rft.
(26.93)
Hence (26.92) is obtained as c --+ O. Let us justify the passage to a limit in (26.93).
Since CPe --+ cp in Lp and Am,a(z, h) E Ll(R n ), the left-hand side converges in
Lp. On the other hand, since I(z) E Lr and K"a{ITI) ELl, the right-hand side

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 535
converges to (h I)(x) in Lr. The left- and right-hand sides in (26.93) coinciding,
their limits have to coincide also (almost everywhere). This gives (26.92). .
Remark 26.5. The representation (26.92) is an extension of (26.41) to the case
when information I(x) E [Ot(Lp) is replaced by the condition I(x) E L;,r(Rn).
Remark 26.6. Since (26.92) is obtained by the passage to a limit in (26.93),
it is clearly true not only for I(x) E L;,r(R n ) but also under assumption that
(DOt I)(x) E Lp(Rn), (h I)(x) E Lp(Rn), m > n, 1 $ p < 00.
The characterization of the space [Ot (L p ) is given by the following theorem.
Theorem 26.8. Let I(x) be locally integrable and lim I(x) = O. Then
Izl-oo
I(x) E [Ot(L p ), n > 0, 1 < p < 00, il and only if
1) I(x) E Lq(Rn), q = np/(n - np), and DOt IE L,,(Jl1I) ill < p < n/n;
2) (/)(x) E Lp(Rn) and there exists lim J Ihl-n-Ot(/)(x)dh il p 
. £-o lhl
(L p ) >£
n/n, where I > 2[n/2] is chosen as in  26.2. In the case p = 1 the condition
DOt I E L} holds in the "only if" part. Moreover
[Ot (Lp) n Lr = L;,r, n > 0 < 1 $ p < 00, 1 $ r < 00.
(26.94)
Proof. Consider the "only if" part. Let J(x) E [Ot(L p ). If 1 < p < n/n then
I E Lq by the Sobolev Theorem 25.2, and if 1 $ p < n/n, then DOt f E Lp by
Theorem 26.3. Let p  n/n. Then (/)(x) E Lp by (26.41) and the analysis of
the proof of Theorem 26.2 shows that (26.40) is valid for D: I and therefore there
exists lim D: I. We have thereby shown that JOt(Lp) n Lr C L p Ot r(Rn).
£-0 '
Consider the "if" part. Let I(x) E L;,r(Jl1I). Then we have (26.92). We
observe that the right-hand side in (26.92) is h [Ot DOt cp with [Ot DOtcp , where
DOtcp E Lp, being interpreted in the usual way if 1 $ p < n/n and in the sense
of the space , if p  n/n. Consequently, (26.92) implies h I == h [OtD Ot I in
the sense of '. We observe that functionals I and 9, which have identically
coincident finite differences, may differ themselves only by a polynomial. To
see this, it is sufficient to pass within the frames of S' to Fourier transforms:
(1- eiz.h)ffl(j - g) = 0, x, h E Rn, and to use the known theorem on a functional
supported at a point. Consequently
I(x) = [OtD Ot 1+ P(x)
(26.95)
where P(x) is a polynomial. Here I E Lr "does not contain" a polynomial. If
1 < p < n/n, then [OtD Ot f E Lq does not contain a polynomial either. So P(x) == 0

536 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
and then I(x) E [a by (26.95). As for the case p  n/o:, then (26.95) means the
same, since the presence of a polynomial agrees with the definition of the space
[a(L p ) in the case p  n/o:. Therefore, the imbedding L;,r(R n )  [a(L p ) n Lr is
proved, which together with the inverse imbedding, proved above, gives (26.94).
The relationship (26.94) yields the sufficiency of the conditions 1), if we take
r = np/(n - o:p). To obtain the sufficiency of the conditions 2), we have to repeat
arguments given above in the sufficiency part taking Remark 26.6 into account. .
Corollary 1. Let 1 < p < n/o:. Then
L;,q(R n ) = [a(L p ), q = np/(n - o:p).
Corollary 2. The spaces L;,r(Rn) do not depend on the type of the finite
difference defining D a I and on order I of this difference under the role for choice
of I, indicated at the end of S 26.2.
Corollary 3. The spaces L;,r( R n ) are complete.
In view of (26.19) Theorem 28.8 gives a simple sufficiency condition for
a function to belong to [a(L p ), 1 < p < n/o:: I/(x)1  c/(1 + Ixl)>', and
I(Dj 1)(x)1  c/{l + Ixl)#J, Ijl = m > 0:; A > n/q, p. > nip.
We present another useful variant of the characterization of the space [a (L,,)
which deals with Lp-uniform boundedness of the truncated Riesz derivatives D: I,
instead of their convergence.
Theorem 26.9. Let 1 < p < n/o:. Then I(x) E [a(L,,), if and only if
I(x) E Lq(Rn), q = np/(n - o:p) or (1 + Ixla)/(x) E L,,(Rn)
(26.96)
and there exists a sequence c1c --+ 0 such that
liD:" III"  c < 00,
(26.97)
where c does not depend on c1c.
The proof is based on the property of the weak compactness of the space Lp(R n ).
It is analogous to the proof of the similar one-dimensional Theorem 6.2 and is
therefore omitted, see the proof of Theorem 6.2 which is presented in full; it is
necessary to replace the operator (6.22) by the operator m,a(-, h) * cp, where
Am,a(x, h) is the function (26.27), we shall obtain then (26.95) instead of (6.23),
after which the arguments repeat the considerations in (6.24).

 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS 537
The space [01 (L p ) may also be characterized in terms of the convergence in
Lp of the hypersingular integrals (25.69) and (25.70), related to the Poisson and
Gauss-Weierstrass semigroups. Namely, the following theorem is valid.
Theorem 26.10. Let 0 < a < n, 1 < p < n/a. Then I(x) E [OI(L p ) il and only il
IE Lq, q = np/(n - ap), and
00
lim f t-l-OI(E - Pt)'ldt E Lp or
£-0
(Lp) £
00
lim f t-l-0I/2(E - Wt)'ldt E Lp.
£-0
(L p ) £
We also note the relation
lim D 1= lim t-OI(E - Pt)OI I,
£-+0 t-+O
(Lp) (Lp)
(26.97')
I E Lq(Rn), 1 < p < 00, 1 < q < 00,
which is valid provided that there exists one of these limits. If q = p this result
follows from Theorem 23.4 and Theorem 26.10. Its validity for independent values
of p and q may be seen in Samko [34].
We conclude this section by some simple estimates for the integral continuity
modulus of the Riesz potential. Let f = [OIcp, cp E Lp, 1 < p < n/a, and I > a.
Then
1IL\fllp  clhlOlllcpllp,
(26.98)
and
1IL\/lIp = o(lhl Ol )
(26.98')
as Ihl -+ O. Here c = IIk"OlIIi, where k"Ol(x) is the kernel (26.28). The estimate
(26.98) is derived from the representation
(A/)(x) = Ihl Oi f k"OI(Y)CP(x - Ihlwl:(y»dy
Rft.
(26.99)
where Wh(Y) is the rotation given in (25.68), which is obtained by the following

538 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
transformations
(A' /)(X) =   (_I)I: ( I ) f cp(x-y)dy
h 1'n(a)  k Iy - khl n - a
1:_0 Rft.
= ;::) f t( -1)' G) I - keda-n(z -Ihlwh(md,
Rft. 1:_0
the latter coinciding with (26.99). Since k"a(X) E L 1 (Rn) by Lemma 26.3, (26.99)
yields (26.98). It also gives (26.98') in view of (26.35).
 27. Bessel Fractional Integro-differentiation
In this section we consider a fractional integra-differentiation which is the fractional
power (E - A)-a/2, where E is the identity operator and A is the Laplace operator.
This integra-differentiation is reduced by Fourier transforms to multiplication by
the power (1 + IxI2)-a/2, cf. (25.35) and (25.62) where the powers (_A)a/2 were
considered. It will be realized as a convolution of the function Ga(x), which is the
original of the Fourier transform of the function (1 + IxI2)-a/2. This pre-image is
a non-elementary function in contrast to the case of Riesz integra-differentiation.
The function Ga(x) is expressed in terms of the modified Bessel function (see
(27.1) below). That is why the form of fractional integro-differentiation treated
in this section was so named. An essential virtue of Bessel fractional integration
is that it has behaviour better at infinity than the Riesz potential, being defined
for all functions in Lp(Rn) and representing a bounded operator in Lp(Rn) for
whatever p, 1  p  00, which is not true for Riesz fractional integration.
27.1. The Bessel kernel and its properties
The Fourier transform of the function (1 + IxI 2 )-a/2 may be evaluated by means
of the Bochner relation (25.11). We have
00 /2
r l [(1 + I z l-a/2] = (2.rn/2Izll-n/2 f p" (/1;j!zl) dp.
o
Here the integral converges if a > (n+1)/2. Assuming this condition to be fulfilled,
then by (1.86) we obtain
:1'-1 [( 1 + Ix I 2 ) -a/2) = 2(2-n-a)/2 ]{(n-a)/2(lxl) ':;!G ( x )
1("n/2f( a/2) Ixl(n-a)/2 a
(27.1)

 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
539
where KII(z) is the modified Bessel function (1.85). This expression has a meaning
for all a > O. The function Ga(x) is called the Bessel kernel. We give some of its
properties. First of all we shall clear up its behaviour at the origin and at infinity.
Lemma 27.1. The function Ga(x) is infinitely differentiable beyond the origin
admitting the estimate
GQ(Z)  !
r({n-a)/2) I l a-n
2 01 1(ft. ' 'Jr(a/2) x ,
1 I 1
2"-11(ft. ' 'Jr(n/2) n JXT'
r(a-n)/2)
2ft.1(ft. ' 'Jr(a/2) ,
if 0 < a < n,
if a = n,
if a > n,
as Ixi --+ 0 and the estimate
I x l(a-n-l)/2 e - 1 x l
Ga(x) f'OoJ 2(n+a-l)/2 1r (n-1)/2f(a/2)
as Ixi --+ 00.
The statements of the lemma follow immediately from the known properties
of the modified Bessel function: /(II(Z) f'OoJ 2"- 1 f(II)Z-II, 111= 0, /(o(z) f'OoJ In(l/z) as
% --+ 0 and KII(z) f'OoJ (1r /2z )1/2e- z as z --+ 00. - Erdelyi, Magnus, Oberhettinger
and Tricomi [2, 7.4.1 (1)]. In this connection we note the relation
1/2 II -z 00
K (z) = (1r/2) z e f e-ztt"-l/2(1 + t/2)"-1/2dt
II f(1I + 1/2) .
o
(27.2)
Corollary. Ga(x) E L1(Rn), a > O.
Now, employing properties of the fun ction G a (x) with firstly its rapid decrease
at infinity, it is not difficult to show by means of (25.11) again that
Ga(x) = (1 + IxI 2 )-a/2
(27.3)
for all a > 0, Re a > 0, in fact may be taken.
It follows from (27.3) that
(E - )Ga = G a - 2 , a > 2,
(27.4)
(27.5)
G a * G {j = G a+ {j , a > 0, /3 > 0,

540 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
f GOt(x)dx = 1, a > O.
Rft.
(27.6)
We mention also the relation
00
G ( x ) = 1 f (Ot-n)/2-1e--'Z'/(4).
Ot 2n1fn/2r(a/2)
o
(27.7)
Its proof may be seen in Stein [2, p. 155].
We now define the Bessel potential by the relation
GOtcp = f GOt (x - y)cp(y)dy.
Rft.
(27.8)
By Lemma 27.1 it is well defined for example for functions cp(y) E Lp(Rn),
1 $ p $ 00, and represents an operator bounded in Lp(Rn):
IIGOtcplip $ IIcpllp, 1 $ p $ 00, a > 0,
(27.9)
this inequality being implied by (27.6) and the positiveness of the kernel GOt(x).
Equations (27.3) and (27.4) immediately yield the following simple properties of
Bessel fractional integration:
GOtG/J cp = GOt+/J cp, a > 0, /3 > 0
(27.10)
(the semigroup property), and
(E - A)GOtcp = G Ot - 2 , a > 2.
(27.11 )
In contrast to the case of the Riesz potential (27.10) holds for functions in Lp with
all 1 $ p $ 00, a > 0, {3 > O. We write below GO = E when it is necessary to
admit the case a = O.
The operator GOt may be considered as a constructive realization of a negative
power of the operator E - A. It is of great interest to realize constructively positive
powers of this operator as was done in SS 25 and 26 for (_fl)Ot/2. Such a realization
will be given in S 27.4.

 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
27.2. Connections with Poisson, Gauss-Weierstrass and
metaharmonic continuation semigroups
541
The Riesz potential was seen to be connected with the Poisson integral by (25.52).
A similar connection of the Bessel potential with this integral is given in terms of
special functions. Thus, the following theorem is valid.
Theorem 27.1. Let rp(x) E Lp(Rn), 1  p  00. Then
(1-a)/2 00
(Cacp)(z) = 2 r(Q/2r f t(a-'}/2 J ";' (t)(P,cp)(z)dt
o
(27.12)
where Ptrp is the Poisson integral (25.47).
The proof of the theorem is obtained by interchanging the order of integration in
the right-hand side of (27.12), which is made possible by Fubini's theorem, and
applying then the relation
00
2(l-a)/2...;:i f (a-l)/2
r(Q/2) t J(a-l)/2(t)P(X, t)dt = Ga(x)
o
(27.13)
where P(x, t) is the Poisson kernel (25.49). As for (27.13), it is contained in (1.86)
according to (27.1).
Meantime there exists a modification of the Bessel potential (suggested by
Flett) which is connected with the Poisson integral in a simpler manner. This
modification is based on using the function (1 + Ixl)a instead of (1 + IxI 2 )a/2 in
Fourier transforms. We set
arp = f a(x - y)rp(y)dy
Rft.
(27.14)
where the kernel a(x) is the Fourier original of the function (1 + Ixl)-a. This is
connected with the Poisson kernel P(x, t) by
00
a(z) = r(IQ) f ta-'e-' P(z, t)dt
o
(27.15)

542 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
which may be easily verified by Fourier transforms:
00
(1 + Ixl)-a = r(l a ) f ta-1e- t P(.,t)dt,
o
(27.16)
the point denoting the variable in which the Fourier transform is applied. As for
(27.16), it follows from (25.54).
The kernel (!5a(x) may be written in the form
00
C f sae-,Izi
(!5a(x) = r(:) Ixl a - n (1 + s2)(n+l)/2 ds
o
(27.17)
where C n is the constant from (25.49), this result being a paraphrase of (27.15).
The following properties of the kernel (!5a are easily derived from (27.15) and
(27.17):
1) (!5a is infinitely differentiable beyond the origin and (!5a(x) > 0, X f; OJ
2) tCt. ( ) r((a+l)/2)r((n-a)/2) I l a-n 0 . f 0 tCt. ( )
\'la X ""J 2r(a)1I' ( "+1 )/  x as x  1 < a < n, \'In X ""J
 In  as x  0 and (!5a(x) is continuous in the point x = 0, if a > nj
3) (!5a(x) ""J aCnlxl-n-l as x  00, so that (!5a(x) E L1(R n ), besides this
f (!5a(x )dx = 1.
Rft.
It follows from the above properties that the potential (!5a cp is defined for
functions cp(x) E Lp(R n ), 1 :$ p :$ 00, and is a bounded operator:
II (!5a cpllp :$ IIcpllp, 1:$ p :$ 00.
(27.18)
The following theorem on the connection between the modified Bessel potential
(!5acp and the Poisson integral (cf. (25.51» follows immediately from (27.15).
Theorem 27.2. Let cp(x) E Lp(R n ), 1 :$ p :$ 00. Then
00
tCt. a cp =  f ta-le-t p, cp dt
\'l r( a) t.
o
(27.19)
We give also two other integral representations of type (27.12) and (27.15) for
the Bessel potential Cacp. The first one connects Cacp with the Gauss-Weierstrass

 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
543
integral:
00
GOt<p = r(/2) f tOt/2-1e-t(Wt<p)(x)dt.
o
The second one expresses GOt <p via metaharmonic continuation
(27.20)
M - 2t f I«n+l)/2( v' lyl2 + t 2 ) x - d
t<P - (21r)(n+l)/2 (v'lyI2 + t2)(n+l)/2 <p( y) y
R"
(27.21)
of a function <p by the formula
00
1 f -
GOt<p = r(a) tOt l(Mt<p)(x)dt
o
(27.22)
(compare the representation (25.51) for the Riesz potential). The following results
e-t(WtGOt<p){x) = (I/\e-TWT<P)(x»(t),
(MtGOt<p)(x) = (I(MT<P)(x»(t),
(27.23)
(27.24)
are close to relations (27.20) and (27.22) being similar to (25.52) and (25.53).
Equations (27.20)-(27.24) become obvious in Fourier transforms if we take into
account that
.1'( e-tWt<p) = e-t(l+lxl)(x),
.1'(M t <p) = e- h/l+lxl (x),
(27.25)
(27.26)
27.3. The space of Bessel potentials
We shall call the range
GOt(L p ) = {f : f(x) = GOt<p, <p E Lp(Rn)},
(27.27)
a > 0, 1  p  00
of the operator GOt the space of Bessel potentials. Sometimes this space is called
the Liouville space of fractional smoothness a. This is Banach space relative to the

544 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
norm
IIfIlGar(L p ) = IIcplip.
As we shall see below, this space is an extension of the Sobolev spaces L(R")
to the case of fractional order a. So the spaces defined in (27.27) are also called
Sobolev spaces of fractional order.
Remark 27.1. The spaces GOt(L p ) and (!5Ot(L p ), where (!5Otcp is the modified Bessel
potential (27.14), coincide:
GOt(L p ) = (!5Ot(L p ), 1  p  00.
One can prove this by means of the fact that the functions
(1 + Ixl)Ot _ 1 and
(1 + IxI2)Ot/2
(1 + IxI2)Ot/2
-1
(1 + Ixl)Ot
are Fourier transforms of functions integrable over R n (cf. the Corollary of Lemma
5.2 in Samko [31, p. 52]).
The characterization of the space GOt(L p ) presented in this subsection, has
much in common with that of the space of Riesz potentials given in terms of the
Riesz derivative. Among the spaces L;,r(Jl1I) defined in S 26.7, we consider now the
important case r = p. We find it convenient to introduce the special designation
L;(R n ) = L;,p(R n ) = {f : f E Lp, DOt f E Lp},
(27.28)
a > 0, 1  p < 00.
Theorem 27.3. The space of Bessel potentials consist of those and only those
functions f(x) E Lp(Rn) which have Riesz derivative of order a in Lp(Rn), i.e.
GOt(L p ) = L;(R n ), 1  p < 00, a > 0" cll1fllL  IIfIlGar(L p )  c2l1fllL p '
We precede the proof of this theorem with the following lemma.
Lemma 27.2. The set C{f is dense in
L;(R n ), 1  p < 00.

 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
545
Proof of the Lemma. 1. We show first that infinitely differentiable functions
in Lp(Rn) form a dense set in L;(R n ). This is achieved by standard means which
are used to show the denseness of "good" functions in various function spaces, i.e.
by the identity approximation
Im(z) = J O(Y)/(Z -  )dY, IE L;,
R"
where a(y) E C{f, f a(y)dy = 1. Then IIf - fmll" -+ 0 as m -+ 00 and it is
R"
readily seen that D: fm = (D: f)m. So there exists D a fm = lim D: fm and
£-0
D a fm = (D a f)m. Therefore IIDa(fm - f)lIp = II(D a f)m - D a flip -+ 0 as m -+ O.
We note that since a(x) E C{f, then 1)j fm E Lp(Rn) for allljl = 0,1,2,...
2. It remains to approximate the function f(x) in the space L;(Jl'&) by
C{f-functions under assumption that f(x) E coo(R n ) and (-pj f)(x) E Lp(Rn),
/jl = 0, 1,2, . .. Let J.l( x) be any function in Cgo which has the support in the ball
Ixl < 2, is identically equal to 1 for Ixl  1 and such that 1J.l(x) I  1. We have to
show that the "truncation"
de!
J.lN(x)f(x) = J.l(xIN)f(x) E cgo
approximates the function f(x) under the norm IIfll p + IID a flip as N -+ 00.
We denote v( x) = 1 - J.l( x) and VN (x) = v( xl N). It is sufficient to verify that
IIDa(VN f)lIp -+ O. We have
DO(VNf) = vNDoI = d':(",)  G)BN.d
where
BN,1cf = J lyl-n-a(A:vN )(x)(A-1c f)(x - ky)dy,
R"
(27.29)
k = 1,2,..., I, so that the passage IIBN,l:fll p -+ 0, N -+ 00 has to be justified.
Since A;VN == A;J.lN, k > 0, and I(A;J.l)(x)1  clyl1c 1(1 + lyl)1c according to (26.20),
by the Minkowsky inequality we obtain
liB N,> III. 5. cN-> J lyl>-n-o ( 1 +  ) ->lId-> III.dy.
R"
(27.30)

546 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
It follows from (26.19) that IIA-I: Ilip $ clyl'-I: L: l11)j Ilip = clyl'-I:, since
Ijl=I-1:
1)j IE Lp(Rn) for allljl = 0,1,2,... So from (27.30) we have
Cl f dy C2 f dy
IIBN,l:/lIp $ NI: Iyln+a-l + NI: n a-I: ( W ) I: .
lyl<l lyl>l Iyl + 1 + N
Hence simple transformations lead to IIBN,l:/lIp $ C3N-1: + C4N-a in the case
kIn; if n is an integer and k = n, then IIBN,l:/lIp  cN-a In N. .
Proof of Theorem 27.3. Let us prove the imbedding
Ga(L p )  L;(R n ).
(27.31)
Since Ga(x) E L 1 (Rn) by Lemma 27.1, then Ga(L p ) C Lp. Further, we have
1 _ 1 Ixla
(1 + IxI 2 )a/2 - Ixl a (1 + IxI2)a/2'
(27.32)
It is known (Stein [2, p. 157], see also Samko [31, p. 51]) that the function
Ixl a (1 + IxI 2 )-a/2 - 1 is the Fourier transform of a function in L 1 (Rn). So (27.32)
means in Fourier pre-images that .
Gacp = [a(E + A)cp
(27.33)
where A is a convolution operator with a summable kernel. By the boundedness
of the involved operators (27.33) is extended from the case of functions cp E COO to
cp E Lp(Rn), 1 $ p < n/n. The validity of (27.33) for cp E Lp(JlR) when p  n/n
is a corollary of the definition of [acp for such cp (see S 26.7). The identity (27.33)
implies then that Ga(L p )  Lp n [a(L p ). So the imbedding (27.31) is proved in
view of (26.94). As well, by (27.33), for I = Gacp we have
II/IIL; = II/lI p + IID a Ilip  IIcplip + II(E + A)cpllp
 cllcpllp = cIl/IlGO(L p )'
Let us prove the inverse imbedding. Let I E L;(Rn). We use the density of C{f
in L;(Rn), obtained in Lemma 27.2, and approximate f by functions 1m E C{f with
1 (1 + I Z I  r /
respect to the norm II/lIp + IID a Ilip. We have 1 = (l+lxl)o/ l+lz or (1 + Ixl a ).
Here (1i,';,):/ - 1 is the Fourier transform of a summable function (see Stein [2,

 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
547
p. 157-158], and also Samko [31, p. 52]. The latter relation yields the following
identity in Fourier originals:
1m = Ca(E + U)(/m + D a 1m)
(27.34)
where U is a convolution operator with a summable kernel. Since the operators
C a and U are bounded in Lp(Rn), 1  p < 00, after passage to the limit as
m --+ 00 we obtain from (27.34) that IE Ca(L p ), i.e. L;(Rn)  Ca(L p ), as well as
II/IIGar(L p ) = II(E + U)(I + D a I) lip  c(lI/lIp + IID a IlIp) = cll/IlL; which completes
the proof. .
Corollary. The operator Ga maps the space L(Rn) isomorphically onto the space
L;+f3(Rn), cr 2: 0, {32: o.
27.4. The realization of (E - )a/2, a > 0, in terms of
hypersingular integrals
Since (E - A)a/2 I = .1'-1 (1 + IxI 2 )a/2.1' I by definition, our discussion is in fact
about the inversion of Bessel potentials I = Cacp, cr > O. The function cp will
be first taken to be in the Schwartz space S which is invariant relative to the
operator C a , and then we consider the problem of inversion for cp E Lp. The
operator inverse to C a will be considered in terms of hypersingular-type integrals,
containing the remainder of the Taylor series for the function I(x) (see (26.68».
For I E Sand cr > 0, cr f; 2,4,6, . .. we set
T a 1= L C a ,j(1)j I)(x) + d a f [/(x - y) - (RLa] I)(x)] Iyl-n-a Aa(lyl)dy (27.35)
jEA ar Ra
where (RLa]/)(x) is the same as in (26.65), Aa is the set of multiindices with length
$ [cr] and even components, and
2 a
d a = 1("n/2r( -cr/2) ,
dar () r ()
c . - W'
a,J - j!21+a-ljl J'
Wj = f d(1',
5a-1
Aa(lyl) = 2 1 -(n+a)/2IYI(n+a)/2 I«n+a)/2(lyl)
(27.36)
00
= f -1+(n+a)/2e--IYI/(4)d.
o

548 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
We remark that r ( nliI ) Wj = r (¥) .. . r e ft : 1 ) in the case of even h, ... ,jn.
We note also that if 0 < a < 1, then
T a 1= I{z) - d a J '(i,:!a{Y) .\a{lyDdy.
Rft
(27.37)
It is easily verified that the characteristic Aa(lyl) of the hypersingular integral
in (27.35) and (27.37) stabilizes at the origin and at infinity as a Holderian
function; hypersingular integrals with such characteristics were studied in the
paper by Samko [20], the results of which will be used below.
Theorem 27.4. Let a > 0, a f; 2,4,6,... and let cp E S. Then Tacacp = cp,
where T a is the operator (27.35).
Proof. We note that the operator inverse to C a may be written for functions in
Sas
(C a )-l f = (.1'-1(1 + lyI2r/2, f(y + x))
(27.38)
where the Fourier transform .1'-1 (1 + lyI2)a/2 is interpreted in the sense of S'-
distributions. This is easily justified by means of the known Gel'fand-Shilov
theorem (Gel'fand and Shilov [1, p. 179]). So we are to show that
(.1'-1(1 + lyI2)a J 2, f(x + y») = (T a f)(x), a 1= 2,4,...
(27.39)
If Rea < 0, taking (27.36) into account, we have
(r'{l + lyl2)a l 2 .f(z + y» = d a J :II: I(Y+ z)dy,
Rft
(27.40)
Since the left-hand side is analytic with respect to a in the whole complex plane, we
may consider (27.40) to be valid for Rea> 0 also if the integral in the right-hand
side is interpreted as an analytic continuation. Such a continuation is realized by
subtracting a Taylor sum of the function f(x) under the integral sign. Carrying
out a direct evaluation we arrive at (27.39). .
1c
Remark 27.2. Since (1 + IxI 2 )1c = L: e) Ixl2i, then the operator inverse to C a in
i=O

 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
549
the case a = 2,4,6,. .. has the form
01/2
(G'T' f =  C'{2) (-8if.
(27.41 )
Now, we shall show that the operator T OI inverses the potential f = GOI<p for
cp E Lp as well, if it is interpreted as
T OI f'=2lim TeOi f
e-O
(L p )
(27.42)
where T: f is the construction (27.35) with the integration over Jl!1 being replaced
by that over Iyl > c. Derivatives 'D j f of a function f(x) E GOI(L p ) will be
understood in the sense of generalized functions. Existence of derivatives 'D; f for
f = GOI cp follows from the fact that
GOI(L p ) C L';, m = 0,1,..., [a].
Theorem 27.5. Let a > 0, a f; 2,4,6,..., cp E Lp, 1 < p < 00. Then
TOIGOI<p = <p, where T OI is the operator (27.42).
Proof. We are to show that
lim T:GOIcp = cpo
e-O
(Lp)
(27.43)
We first prove the uniform estimate
II TeOi G OI cpllp  cllcpll"
(27.44)
with c not depending on c. Its proof will use the idea of Fourier-multipliers in Lp
(Stein [2, p. 113]). Since lI'D j GOI<pllp  cllcpll" which is easily verified by means of
Theorem 3 from Stein [2, p. 114], then (27.44) will readily follow from the estimate
IIDt.;eGOIcplip  cllcpllp
(27.45)

550 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
with c not depending on c, where
D a . f = f f(x - y) - (Rl,a]f)(x) A (I I) d .
>'0.£ Iyln+a a y Y
Iyl>£
(27.46)
Let
= f ( iZ.y - '" iljlxjyi ) Aa(lyl) d
e L.J j! Iyln+a y
Iyl>£ IjI5[0]
so that :F(DOli£f) = (1'£(x)1(x), f E S. As for (27.45), it is a consequence of the
following lemma.
(27.46')
Lemma 27.3. The function (1 + IxI 2 )a/2(1'£(x), a > 0, belongs to the space Mp of
Fourier-multipliers in Lp, 1 < p < 00 (1  p < 00 if 0 < a < 1), besides this
11(1 + IxI 2 )a/2(1'£(x ) liMp  C
(27.47)
with c not depending on c.
Proof of the lemma. In the case 0 < a < 1 we refer to Samko [20] where
(27.47) was proved in this case for arbitrary stabilizing characteristics even in a
stronger form - in terms of absolute summability of the Fourier integral of the
function (1 + IxI 2 )-a/2(1'£(x) uniformly in c. So we give the proof in the case a  1.
We shall use Theorem 3 from Stein [2, p. 114], which states that to show (27.47) it
is sufficient to verify the estimates
Ixllll'l1)I1(1 + IxI 2 )-a/2(1'£(x)1  c, Ivl  [n/2] + 1,
(27.48)
with c not depending on c. By the Leibniz formula it is clear that (27.48) follows
from the inequalities
11)1I(1'£(x)1  clxIO-I/l I , v  (n/2] + 1,
(27.49)
c not depending on c again. Let us prove (27.49). If [a] + 1  v  [n/2] + 1, then
V" IT. (.,) = f Ira Aa( lyD.'" dy
Iyl>£
(27.50)
since differentiation under the integral sign is possible owing to the rapid decrease
of Aa(lyl) as Iyl--+ 00 which is seen from (27.36). \Ve represent the monomial (iy)1I

 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
551
[1111/2]
as (iy)1I = L: lyl2m .Pj1l1-2m(y), where .Pj1l1-2m(Y) are homogeneous harmonic
m=O
polynomials of orders 1111- 2m (see Stein and Weiss [3, p. 159]). Passing to polar
coordinates and applying the Funk-Becke formula, (Erdelyi, Magnus, Oberhettinger
and Tricomi [2, 11.4] or Samko [31, p. 43]), and the relation 2.12.2.2 from the
hand-book by Prudnikov, Brychkov and Marichev [2], we have
II _ [I] '11I1-2m ( x ) a-IIII 1 00 Aa () JIIII-2m-l+n/2()
'D O'£(x) -  I 11111-2m j;j Ixl a-llIl+n/2 '
m_O £Ixl
(27.51 )
If a -1111 + (n + 1) /2 > 1, then owing to the known behaviour of the Bessel function
as  -+ 0 and  -+ 00, we have
00
I'D II 0'£ (x) I $ clxl a - II L 1 11I1-Q-n/2IJIIII_2m_l+n/2()ld = clxl a - II .
o
As for the case a - 1111 + (n + 1)/2 $ 1, the estimates (27.49) follow from (27.51)
in view of the uniform boundedness of the integrals
00
1 Aa(/IXI)Jlllti2m;1+n/2() , m = 0,1,..., [ 111 2 1 ] ,
a- II +n
o
which is obtained owing to monotonicity of the function Aa()' Let now 1111 < [a]+1.
If a f; 1,3,5,..., we have
l'DIIO'£(x)1 c 1 Iyllill-a-nl(x. y)l[a]+l- llIl dy
lyl<lxl- 1
+ c L Ixll;l
Ijl[a]-II
1
ly>lxl- 1
Iylllll+ljl-n-ady
clxla-IIII.
If a = 1,3,5,..., by the Taylor formula with the remainder in the integral form,

552 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
we have
-nil ( ) _ '1111 '" (Q + 1 - IIII)! j
v u e X -I L.J ., x
J.
Ijl=a+l-IIII
1
X f (1 - )a-Illld f yll+j Iyl-n-a Aa(lyl)eiz.y dy.
o Iyl>e
Transforming the integral over Iyl > c in the same way as the integral in (27.50),
we obtain (27.49). .
The uniform estimate (27.44) being thus obtained, it remains by the Banach-
Steinhaus theorem to verify (27.43) on the Schwartz space S which is dense
in Lp. Since Taca<p = <p fOl <p E S by Theorem 27.4, it can be shown that
II Tea C a <p - <p1l2 ---+ 0 by means of the Parseval equality and the Lebesgue
e-O
dominated convergence theorem - employ the representation (27.46') at first. Let
us choose r > 1 so that p was between 2 and r and let 8 = 2(r - p)/p(r - 2). Then
we derive from the interpolation inequality II/lIp $ 1I/1I-811/11 and the uniform
estimate (27.44) that (27.43) holds for <p E S. .
Remark 27.3. It is easily verified that in the case Q = 2,4,6,... the operator
(27.41) is inverse to the potential Ca<p within the frames of Lp-spaces in the weak
sense:
(GO 1",  ("{2) (-);w ) = (I",w), I" E L" wE S.
In conclusion we give three other ways to effectively construct the operator
(E - L\)a/2, Q > O. We shall limit ourselves to giving very brief information, the
corresponding references may be found in S 29.1. The first way is connected with
the transformation of the operator (27.35) to the form
T a I = ha * I + Dol
(27.52)
where ha(Y) E L}(Rn) and Dol is the hypersingulal integral
D a = f (L\/)(x) O ( ) d
{} Iyln+a y y
Rft.
(27.53)
the characteristic O(y) of which is a polynomial of degree less than Q - see the
explicit expression for ha(Y) and O(y) in Nogin [7].

 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
553
Theorem 27.6. Let a > 0, a f; 2,4,6, .. ., 1 < p < 00 and <p E Lp(Rn). Then
TOt GOt<p = <p where
TOt f = hOt * f + lim Do £f
£-0 I
(L p )
Do,£f being the "truncated" flyl > g) integral, corresponding to (27.53).
We remark that a certain advantage of the construction (27.52) in comparison
with (27.35) consists in the fact that it contains a hypersingular integral of a
simpler form, constructed in terms of translations of the function f( x).
The second way is related to (27.20) and is similar in a sense to the statements
for the Riesz potentials in (25.70) and in Theorems 26.3' and 26.10. Let us denote
00
Ot f - 1 f t-l-Ot/2 ( E - e-t ) ' f dt 0 < a < 21
£ - x( a /2, I) t ,
£
(27.54)
where x(a/2, I) is the constant (5.81).
Theorem 27.7. Let <p E Lp(R n ). Then lim GOt<p = <p if 1  p ::; 00 and
£-0
(p.p.)
lim GOt<p = <p if 1  p < 00.
£-0
(L p )
Theorem 27.7 remains valid if we set
00
Ot f -  f t-l-Ot ( E - At[ ) ' f dt 0 < a < I ,
£ - x( a, I) t,
£
(27.55)
where Mt is the semigroup (27.21).
Finally, the third way is based on the idea of introducing hypersingular
integrals with the "weighted" differences
Ot f =  f (L\f)(x,p) d
, dn,,(a) lyln+Ot y,
Rft.
(27.56)
(f)(o:,p) =  G) (-1)' p(k, y)/(o: - ky).
For distinctness we consider the case of non-centered differences. We remark that
the "weighted" difference with the weight p(k, y) = e- lcy has already occurred in
consideration of the Bessel potentials in S 18.4, as in (18.77). We define the weight

554 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
p(k, y) in (27.56) by the relation
p( k, II) = 2 ' -( n+a)/2 ( r ( n  a ) ) -1 (kIIlD(n+a )/2 K (n+a )/2{ kiIlD.
If f E S, the integrals (27.56) with this weight have the following properties:
a) the integral (27.56) converges absolutely if I > a and conventionally if
2[a/2] < I  a; b) if a = 1,3,5,... and I > a, then T,a f == 0; c) :F(T,a f) =
d n ,,(a)(1 + IxI 2 )a/2f(x), where the constant dn,,(a) is the same as in S 26.
The statement of Theorem 27.7 remains true if we replace 1) f by the
truncated integral (27.56) with integration over Iyl > c (Rubin [24, 25]).
 28. Other Forms of Multidimensional Integro-
differentiation
Fractional integra-differentiation of functions of many variables defined in the
whole space Rn, which was considered in SS 25-27, was a realization of the fractional
power (_A)-a/2 or (E - A)-a/2, A being Laplacian. The direct generalization of
this approach is to consider fractional powers of an arbitrary differential operator
in partial derivatives. We do not consider this question in such a generality, but
focus only on fractional powers of the simplest differential operators
{p {J2 {J2 {j2
o --+...+----...-- lp<n
p - 8x 8x {JX+l {Jx '
(28.1)
(ultra-hyperbolic case)
8 2 8 2 8
-+...+---
8x 8X_l 8x n
(28.2)
(parabolic case).
Fractional integra-differentiation considered in SS 25-27 pertains, thereby, to
the elliptic case.
The realization of a negative fractional power of the former operator will lead
to Riesz hyperbolic potentials with Lorentz distance and of the latter to parabolic
potentials. These potentials are first considered, then the realization of a positive
fractional power of the parabolic operator (28.2) is given.
In the conclusion of this section we study the fractional integration which
adjoins in a sense to constructions in S 24 and differs from them by the fact
that integration in the corresponding fractional integrals is carried out not over
the octant or a parallelepiped with the opposite vertices x = (Xl,' . . , x n ) and
a = (ab . . . ,an) as in S 24, but over a pyramid with the vertex at the point x and

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
555
a base being a constant hyperplane not depending on x. We call such a fractional
integration 'pyramidal'.
28.1. Riesz potentials with Lorentz distance (hyperbolic Riesz
potentials)
Similarly to the definition of Riesz fractional integro-differentiation in the elliptic
case in S 25, the idea of introducing fractional powers is obvious, at least formally,
in Fourier transforms. In fact, since the application of the operator (28.1) is
reduced by Fourier transforms to multiplication by the quadratic form:
.1"( -Dpcp) = P( x).1" cp,
P(x) = x + ... + x; - X;+1 -... - x,
(28.3)
it is natural to introduce the fractional powers (-D p )>' as the operators, which are
defined by Fourier transforms by means of multiplication by the fractional power
of the quadratic form P(x). We note, however, that in contrast to the elliptic case,
the quadratic form P(x) has no definite sign. Of course, one might avoid raising
negative values to a power by considering IP(x)l>' or IP(x)l>'signP(x). But the
fractional powers obtained in this way do not contain the usual integer powers of
the operator -Dp, the former for odd A and the latter for even ones. So we shall
use the standard way of raising to a power with the choice of the "principal" value.
We introduce the standard notations:
p = { ("')I'
>. { o
P- = IP(x)l>'
if P(x) > 0,
if P(x)  0,
if P(x)  0,
if P(x) < 0,
and
(P:i: iO)>' = lim (P :i: ic)>'
£-+0
{ IP(x)I>',
e%>.1ri IP(x )1>',
P(x) > 0,
P(x) < O.
So
(P:i: iO)>' = P + e%i>.1r P. (28.4)
Starting from (28.3) we introduce two following forms of the fractional power
of the D 'Alembertian:
(-Dp)cp = .1"-1(p =F iO)>'.1"cp
(28.5)

556 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(cf. (7.4) in the one-dimensional case). The formula of the Fourier transform
of fractional powers of quadratic forms is known in the theory of generalized
functions. We give it in the form
:F [(P:i: iO)T] = e T T1ri"Yn (a)(P =F iO)-Sf
(28.6)
convenient for our purposes, where "Yn(a) is the familiar normalizing constant
(25.26) used for the elliptic Riesz potential. The proof of (28.6) may be found in
Gel'fand and Shilov [2, p. 349].
By (28.6) the operation (28.5) with A = -a/2 is to be realized as a convolution
with the function
%!!jL1ri 0 .. 1 a-. Of-a Of-' o-a
e (P:i:iO)-T- = _ ( e% T1rIp ---r +e%2.jL1r.p-r ) .
"Yn(a) "Yn(a) + -
(28.7)
We denote this convolution by
% i .!!.::Z. 1r
e  Of-n
Ip%iorp = () (P:i: iO) * rp
/n a
so that (_Dp)0/2 = Ip:!:io' By (28.7) this operator has the form
1°. =  [ e%!!jL1ri f rp(x - y)dy + e%1ri f rp(x - y)d Y ]
p:!:.orp "Yn(a) rn-o(y) Irn-o(y)1
K+ K_
(28.8)
where
r(y) = ../P(y) = V x +... + x - X+l - '" - x
and K % denote the following cones:
K+ = {x : x E R n , P(x)  O},
K_={x:xERn, P(x)O}.
The function r(y) is called the Lorentz distance and the cone K+ is known as the
light cone or characteristic cone.
The integrals in (28.8) converge in the case of sufficiently good functions if
a > n - 2. One call make sure that this is so by representing rn-o(y) as rn-o(y) =
II ' 1 2 I " 1 2 1 )n-o)/2 h ..) - ( ) E R p d " - ( ) E R n-p
y - y were y - Yl , . . . , yp an y - yp+ 1, . . . , Yn ,

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
557
by passing to repeated integration over RP and Jl'1-P and introducing polar
coordinates in the integral over Rn-1', which will give the condition (n - a) /2 < 1.
In the case a  n - 2 the construction (28.8) is not determined, but it admits
the continuation, analytical in respect to a which is well suited for a  n - 2. We
do not elaborate on such a continuation, see the references in S 29.1.
It follows directly from (28.5) that
[ a [ f3 _ [ a+f3
P::I:iO P::I:iO - P::I:iO'
(28.9)
We put the following question: is it possible to introduce an operator of
hyperbolic fractional integration of the type (28.8) with the integration over only
one of the cones K+ and K_ and such that the semigroup property (28.9) remains
true? It is clear that for this purpose we have to start not from the function
(P :i: iO)>', but from the functions P or P. In view of (28.4), (28.6) after simple
transformations yields relations
:F(p (a-n)/2 ) _ 'Yn(a) ( . P - a P -a/2 . P1r p -a/2 )
+ - . , ) /2 sm 2 1r + + SID 2 - ,
SID\n - a 1r
(28.10)
:F(p (a-n)/2 ) _ 'Yn(a) ( . n - P P -a/2 + . n - p - a P -a/2 )
- - . ( ) /2 sm 2 1r + sm 2 1r - .
sm n - a 1r
Hence
:F(p(a-n)/2) = (_I)1'/2-1'Yn(a) . sina1r/2 p-a/2
+ sm(n - a)1r/2 +
(28.11 )
if p = 2,4,6, .. .
and
:F ( pOl;a» ) = (-I)-I'Yn(a) . sina1r/2 p:!f
sm(n - a)1r/2
(28.12)
if n - p = 2,4,6, .. .
Therefore, the potentials
(_1)1'/2-1 f cp(x - y)dy
[a cp_
P+ - H(a) rn-a(y) ,
K+
p= 2,4,6,...,
(28.13)

558 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(-1)9- 1 f cp ( x - y)dy
[a _
P- cp - H(a) Irn-a(y)1 '
K_
n- p= 2,4,6,...,
(28.14)
with a > n - 2 and
, sin a1r /2
Hn(a) = "Yn(a) . ( ) /2
sm n - a 1r
(28.15)
admit the semigroup property
[a [/J _ [a+/J
P-J: P-J: - P-J:
(28.16)
in view of (28.11) and (28.12), provided that p = 2,4,6,... in the case of the sign
+ and p = n - 2, n - 4, n - 6,... in the case of the sign -.
We single out the case p = 1. The form (28.13) may not be applied in this
case, while (28.14) may be used for odd n:
[ a _ (_I)(n+1)/2 f cp(x - y) d
p-cp- H(a) Irn-a(y)1 y, p=l, n=3,5,7,...
K_
(28.17)
However, there is another variant m the case p = 1, namely that which was
introduced by Riesz. In the light cone K+ we consider the positive half-cone
Kt = {x : x  x +... + x, Xl  O}
( 28.18)
and designate
L a - 1 f cp(x - y) d
cp - - y,
o Hn(O:) rn-a(y)
K+
+
a > n - 2,
(28.19)
emphasizing the relation of the operator (28.19) to the D'Alembertian 0 =
a'J a'J a'J Th al . . H ( ) . h
fP - fP - . . . - 7Ji'r' e norm Izmg constant n a IS c osen as
Xl X'J ..
Hn(a) = "Yn(a)
2sin(n - a)1r/2
= 2 o - 1 ..-1+ n /2r () r ( ", +  - n )
(28.20)
which becomes clear below.

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
559
This is the potential (28.19) that is usually called the hyperbolic Riesz potential.
It may be also rewritten as
lOi 1 f
Dip = Hn(a)
K;(z)
Ip(y)dy
rn-OI(x - y)
(28.21)
where
K;(x) = {y: (Xl - Yl)2  (X2 - Y2)2 +... + (Xn - Yn)2, Xl - Yl  O}
is the negative light half-cone with the vertex shifted to the point x.
Let us show that the operator 10 admits the semigroup property
Iolglp = l+{j Ip, DI+21p = lolp
(28.22)
provided that the normalizing constant is given by (28.20). To prove this we first
calculate the Riesz potential of the exponential function e Zl . In this case we have
e Zl f
lOl(e Z1 ) = _ e-Y1rOl-n(y)dy
D Hn(a)
K+
+
and if we choose
Hn(a) = f e-Y1rOl-n(y)dy
K+
+
(28.23)
we obtain
lo(e Z1 ) = e Zl .
(28.24)
Calculating the integral (28.23) we have
00
Hn(a) = f e-YldYl f (y _112)(0I-n)/2,  = (Y2,'" ,Yn) ERn-i.
o 1I<Yl

560 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Hence
00
Hn(Ot) = 1 yr-le-YldYl 1 (1 - 17J1 2 )(a-n)/ 2d 7J
o 1,,1<1
1
= r(Ot)ISn-21 1(1- p2)(a-n)/2pn- 2 dp,
o
which coincides with (28.20) after simple transformations.
We observe that a more general equality
a a..:!: e a . x
Io( e ) = 2 2 2 /2 '
(a 1 - a 2 - . .. - an)a
(28.25)
a = (al,... ,an) E Kt, a. x = alXl +... + anX n
may be derived from (28.24) (cf. (25.39».
This is the result (28.24) which allows us to give a direct proof of the semigroup
property. (28.22). Taking Ot > n - 2 and (3 > n - 2, we have
lolg<p = Hn(o<)lH n (.8) 1 rO-n(x - y)dy 1 ,.p-n(y - ()<p({)d{.
K;(x) K+(y)
It is clear that K;(y) C K;(x) for y E I<;(x). So, interchanging the order of
integration we obtain
lolg<p = H n (o<;H n (.8) 1 <p({)d{ 1 ro-n(x - y)"p-n(y - ()dy (28.26)
K;(x) D(x,O
where D(x,) = K;(x) n I{t() (introduce the characteristic function of the cone
K; (y) while interchanging the order of integration). By means of the shift along
the surface of the cone and dilatation we see that the inner integral is equal to
ra+fj-n(x - )Bn(Ot,{3), Bn(a, (3) = 1 ra-n(y)rfj-n(el - y)dy,
D(O,el)
fl = (1,0,...,0),

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
561
where the integral Bn(O:, (3) is clearly a constant. So we conclude from (28.26) that
1 01 1{j Bn ( 0: , (3) H ( (3) 1 0l+{j
o 01;' = Hn(o:)Hn({3) n 0: + 0 1;'.
(28.27)
Taking here I;' = e Z1 , we see by (28.24) that the relation
Bn( 0:, (3) = Hn( o:)Hn({3)/ Hn( 0: + (3)
must hold (cf. (25.38», which immediately transforms (28.27) into the former of
the relations (28.22). As for the latter, it is obtained by direct checking after
differentiating under the integral sign.
Remark 28.1. The Fourier transform of the hyperbolic Riesz potential IDI;' is
given by the result
:F(IDI;') = Ir(:)IQ (x), XER n ,
(28.28)
where q = eTsignzl if r 2 (x) > 0 and q = 1 if r 2 < 0, r 2 (x) = x - x - ... - x.
We conclude this subsection by consideration of the Riesz hyperbolic potential
(28.21) in the two-dimensional case:
01 1 f
101;'= H2(0:)
Iz-yI<ZI-YI
l;'(y)dYl d Y2
[(Xl - yd 2 - (X2 - Y2)2]1-0I/2
or
01 1 f
101;' = H2(O:)
lyI<Yl
I;'(X - y)dYl d Y2
(y - y)1-0I/2 '
X = (Xl,X2), Y = (Yl,Y2).
(28.29)
The change YI + Y2 = 1, Yl - Y2 = 26 of variables transforms this potential to
Liouville fractional integration in each variable, which was considered in S 24:
00 00
1 01 _ 1 f f I;'(Zl + Z2 - 6 - 2, %1 - Z2 - 6 + 2) d 
01;' - r2(0:/2) C 1 -0I/2c 1 -0I/2 1 2
o 0 '1 '2
(28.30)
%1 %
1 f f I;'(Sl + 82,81 - 82) d8 1 d8 2
= r2(0:/2) (Zl - 81)1-0I/2(Z2 - S2)1-0I/2'
-00 -00

562 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where %1 = (Xl + x2)/2, %2 = (Xl - x2)/2.
Equation (28.30) can be represented in the operator form:
l Oi - A - l f Ol / 2 ,0I/2 A
Drp - ++ rp
(28.31)
where A is the operator of the linear change of variables: (Arp)(x) = <P(XI +X2,Xl-
X2) and f2,0I/2 is the operator of the Liouville fractional integration (24.20).
The following theorem is an immediate consequence of (28.31) and Theorem
24.1 on the Liouville fractional integration I:2,0I/2.
Theorem 28.1. Let n = 2, 1  p < 00, and 1  r < 00. The operator fg is
bounded from L,(R2) into Lr(R2), if and only if
o < a < 2, 1 < p < 2/a, r = 2p/(2 - ap).
28.2. Parabolic potentials
We consider negative fractional powers of the heat equation operator (28.2) in
this section. As usual, we find it convenient to single out the "time" variable
by redenoting it as t and the space variables as Xl," . , X n , i.e. to carry out the
investigation in the (n + I)-dimensional Euclidean space Jl'&+l of the points (x, t),
where X E R" and t E R l . We shall deal with the negative fractional power of the
operator
a
T = - +-
x at
(28.32)
where Ax is the Laplacian with respect to variables Xl,. . . , Xn. For the operator
(28.32) we have in Fourier transforms:
T"cp = (lxl2 - it)<p(x, t)
(28.33)
where the Fourier transform is applied in R n + 1:
(.1'rp) (X , t) = <p(X, t) = f eix.+itT rp(, T)ddT.
R"+l
(28.34)
So a negative fractional power T- 0I /2, a > 0, may be naturally introduced in
Fourier transforms via the function (lxl2 - it)-0I/2 with the choice of the principal

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
563
value: arg(IxI2 - it) E (-1r /2,1r /2). This function will be represented as the Fourier
transform of a function defined in terms of the Gauss-Weierstrass kernel
W( x, t) = (411t)-n/2e-lxr /(4t), t > O. (28.35)
Let
_ { t- 1 + a / 2 W(x, t), t> 0, (28.36)
ha(x, t) - r( a/2) 0, t < O.
Lemma 28.1. The Fourier transform (28.34) of the function ha(x, t) is equal to
ha(X, t) = (lxl2 - it)-a/2, a > O.
(28.37)
The proof of the lemma is obtained by direct calculation:
00
f . t + ' t 1 f Of-" 1 . t f  + . t
h ( T)e 1x '<" I T dT = T- e l T dT e- h' 'X'<"d.
a , (41r)n/2r( a/2)
R"+l 0 R"
The inner integral here is the product of the easily evaluated one-dimensional
integrals so that further verification of a (28.37) is easy in the case of small a > 0,
the general case being achieved by the usual analytic continuation with respect to
a..
Lemma 28.1 allows us to introduce the negative fractional power of the heat
equation operator T as the convolution
T- a / 2 cp = Hacp  f ha(, T)cp(X - ,t - T)ddT
R"+l
= r(/2) f T- 1 + a / 2 W({, T)(X - {, t - T)dT
R,,+l
+
(28.38)
or in the explicit form
Hacp = 1 f T-l+(a-n)/2e-II/(4T)cp(x -  t - T)dT (28.39)
r(a/2)(411')n/2 '
R+l

564 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where R++l is the half-space {(x, t) : x E R n , t > O}.
That is the operator (28.39) which is called the parabolic potential operator.
If we use the definition
(WTCP)(x, t) = (41rr)-n/2 f e-II /(4T)cp(X - , t)
Rft.
(28.40)
for the Gauss-Weierstrass operator, we can rewrite the operator HO/, as it appears
from (28.38), in the form
00
(HO/cp)(x, t) = r(/2) f r Ot / 2 - 1 (WTCP)(x, t - r)dr
o
(28.41 )
(cf. formula (25.51) for the Riesz potential).
The operator HOt is defined for any a > 0 on a rather large set of functions,
for example on functions cp(x, t) which are bounded and vanish in t as t --+ 00 so
that Icp(x, t)1  c(1 + Itl)-o, a > a/2. This may be verified directly. Existence of
parabolic potentials of functions in L,,(Rn+l) is clarified in the following theorem.
Theorem 28.2. The integral HO/cp, a > 0 with cp E L,,(Rn+l) converges absolutely
almost everywhere if 0 < a < n + 2 and 1  p < (n + 2)/a and the operator
HO/ is bounded from Lp(Rn+l) into Lq(Rn+l), where 1 < p < (n + 2)/a and
q = (n + 2)p/(n + 2 - ap).
We do not give the proof of this theorem, but see the references in S 29.1.
Similarly to the spaces [O/(L p ) of Riesz potentials or the spaces GO/(L p ) of
Bessel potentials, the spaces
HO/(L p ) = {f : f = HOtcp, cp E L,,(Rn+l)}
(28.42)
of parabolic potentials may be introduced. These spaces are well defined by
Theorem 28.2 in the case 1  p < (n + 2)/a. In the next subsection we give
a characterization of the spaces HOt (L,,) in terms of the convergence of certain
hypersingular-type integrals.
We note that the spaces of type (28.42) may be also considered in the case of
functions f(x) defined in the half-space R+l only.
Finally, we remark that similarly to what was stated above, we can consider
a negative fractional power of the differential operator E - z + a/at, where E is
the identity operator. In Fourier transforms we have then to deal with the function
(1 + Ixl 2 - it)-Ot/2
(28.43)

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
565
so that the fractional power
( {) ) -a/2
E-z+ at
(28.44)
defined in this way, that is the corresponding potential, relates to parabolic
potential, Hacp, already considered above in the same manner as the Bessel
potential Cacp relates to the Riesz potential [alp. The fractional power (28.44) is
realized as the convolution with the original of the Fourier transform (28.43) and
has the form
1{a", = r("/2)4".)n/2 J T .,- -le-IV,:'--T ",(z -{,t - T)d{dT
R,,+l
+
00
1 J
_ -1-a-/2 -T
- r(a/2) r e (WTCP)(x,t - r)dr
o
(28.45)
(compare (27.20) with the Bessel potential).
The operator (28.45) is also called a parabolic potential. Owing to the
factor e- T its domain in essentially larger than that of the potential (28.39) and,
in particular, it is bounded in any space Lp (Rn+ 1 ), 1 $ p < 00. For further
information about the operator (28.45) and references see S 29.2, notes 28.2-28.3.
28.3. The realization of the fractional powers (-z + :, ) a/2
and (E - z + :t ) a/2, a > 0, in terms of a hypersingular integral
In this subsection we construct effectively the hypersingular integrals T a f and CX a f,
a > 0, which are inverse to the parabolic potentials f = Hacp and f = 1{acp defined
by (28.39) and (28.45). So it is natural to call them the parabolic hypersingular
integrals. They will contain the non-standard finite differences, which reflect the
different behaviour of potentials with respect to the space variable x and the time
variable t.
We shall present first the scheme for the formal construction of the operators
r a and CX a , a > 0, inverse to the potentials H a and 1{a respectively., and then
justify the fact that they indeed invert the corresponding potentials. We shall use
the Gauss-Weierstrass kernel (28.35) below and we recall that it has the following

566 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
properties (Stein and Weiss [3, p. 16, 24]):
f W(x, t)dx = 1,
Rft.
(28.46)
f W(y,t)W(x - y,r)dy = W(x,t + r),
R"
(28.47)
(.1'z W)(, t) = e-tll
(28.48)
where
(.1'zcp)(,t) = f cp(x,t)ei.zdx
Rft.
(28.49)
is the Fourier transform in the space variables x.
Let f(x, t) = (HOtcp)(x, t). Applying the operator .1'z to both parts of this
equation and using (28.48), we obtain
I/2 e'1II (.1'zcp )(, 7])(t) = (.1'xf)(, t)etl12,
(28.50)
where I/2 is the operator of the one-dimensional fractional integration (5.4)
applied in respect to the variable 7]. Inverting the operator I/2 in (28.50) by
means of the Marchaud fractional derivative (5.80) we arrive at the relation
00,
etlI(.1: cp)( t) = 1 f '" ( / ) (_1)1:(.1'. f)( t - k7])e(t-k'1)II.-!!!L
z , x( 0:/2, I)  k x, 7]1+01/2 .
o 1:_0
(28.51 )
Now we multiply by e-tll and then apply the operator .1'-;1, which gives
00 ,
I"(z, t) = K(Q2, I) f [fez, t)+ (-1)' G) f f(z-y, t-k'1)W(y, k'1)d Y ] '11/2 '
o k-l Rft.
Hence after the substitution y  ../ky and simple transformations we have
( t ) f (L\,'1f)(x, t) W ( ) d d ':2T Ot f
I" z, = K( Q2, I) '1 Ha12 y, '1 Y '1
R+l
(28.52)

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
567
where
(d../)(z,t) = (-1)' W/(Z - ../ky,t -Ie,!).
(28.53)
Thus we have constructed the inverse operator T OI = (H OI )-l in the form of the
hypersingular integral (28.52).
Similar arguments applied to the equation 1{OIcp = f(x, t) yield the following
expression for cp(x,t) via the hypersingular integral
1 f ( A' I)( x t. e-" ) d J
y," " W ( ) d d ..! ( TOI /)( t )
cp(x, t) = x(Ot/2, I) 1]1+01/2 y,1] y 1] - x"
R;+l
(28.54)
where the weighted difference of the type (28.53):
(d../)(z, t; .-') = t G) (-1)' .-" I(z - ../ky, t -Ie,!)
k=O
(28.55)
is used. Weighted differences have already occurred earlier in the elliptic case, as
in (27.56).
It is not difficult to show that the integrals in (28.52) and (28.54) converge
conventionally for I E S(R n + 1 ) at any point (x, t), provided that they are
understood as
T OI 1= lim(Te Oi I)(x, t),
e-O
T OI I = lim('! f)(x,t),
e-O
(28.56)
where TeOi I and T I are the truncated integrals (28.52) and (28.54) with the
integration over the shifted half-space R+-:e 1 = {(y,1]) : y E R n , 1] > c}.
We show now that the hypersingular integrals TOI I and TOI I, interpreted as
suggested in (28.56), are real inverses to the corresponding potentials I = HOIcp
and I = 1{OIcp within the frames of the spaces  = (Rn+1) and S = S(R"+l)
which are invariant relative to the operators H OI and 1{0I respectively.
Theorem 28.3. Let I = HOIcp, Ot > 0 and cp E. Then T OI 1== lim TeOi I = cpo
e-O
Prool. We shall base this on the representation
TeOi 1= f W(y, 1])/,0I/2(1])cp(X - V€Y, t - c1])dyd1]
R,,+l
+
(28.57)

568 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where K"a(TJ) is the kernel (6.7'). We remark at once that by (6.8') and (28.46) we
have
W(y, TJ)K"a(TJ) E Ll(Rn+l),
1 W(y, TJ)K"a(TJ)dydTJ = 1.
R..+l
(28.58)
Let us prove (28.57). Substituting the expression for the finite difference
(d..f)(z, t) = E( -1)' G) 1 ha(z - Vky, ( - k'l)<p(z - z, t - ()dzd(,
k=O R..+l
ha(x, t) being the kernel (28.36), into the integral Tea f and interchanging the order
of integration, we obtain
1 1 [ 1 ( I ) 1 00 «( - k )a/2-1
T: f - f(Ot/2)K(0t/2, I) (_I)k k TJl;at2 dTJ
R..+l k-O e
x 1 W(y, '1)W(z - Vky,( - k'l)d Y ] <p(z - z, t - ()dzd(.
R"
Carrying out the change of the variable y --+ y/Vk, k f; 0, and observing that
W(y/Vk, TJ) = k n / 2 W(y, kTJ), by the property (28.47) of the Gauss-Weierstrass
kernel we have
1 1 [ 1 ( I ) 1 00 «( - k )a/2-1 ]
T: f - f(Ot/2)K(0t/2, I) (_I)k k TJl;at2 dTJ
R"+l k_O e
x W(z, ()cp(x - z, t - ()dzd(.
Hence, after the substitutions z --+ cZ, TJ --+ cTJ and simple calculations we arrive at
(28.57).
The properties (28.58) allow the possibility to pass to a limit as c --+ 0 under
the integral sign in (28.57). As a result we obtain lim Tea f = cp which was
e-O
required. .
Theorem 28.4. Let f = 1{acp, Ot > 0, and cp E S. Then Ta f = lim Tc; f = cpo
e-O
The proof of this theorem is similar to that of Theorem 28.3, and is obtained

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
569
from the representation
ex f = f W(y, TJ)I.(TJ)e-£'1cp(x - .jiy,t - cTJ)dydTJ
R++l
(cf. (28.57», which is proved analogously to (28.57).
The hypersingular integrals Ta f and ex a f are inverse to the potentials
f = Hacp and f = 1{acp for cp E Lp as well, if they are interpreted as limits in L'P
of the corresponding truncated integrals.
Theorem 28.5. Let 0 < a < (n + 2)/p, 1 < p < 00, and f = Hacp with cp E Lp.
Then
T a f == lim T£a f = cpo
£-0
(L p )
(28.59)
Proof. Equation (28.57), valid for cp E , is extended to the functions cp E L'P'
owing to boundedness of the operator T: Ha and of the operator involved in the
right-hand side of (28.57) from L'P into Lq, q = (n + 2)p/(n + 2 - ap). The latter
statement follows from Theorem 28.2 in view of the estimate 1I,a(Y, TJ)I  CTJ- l + a / 2 .
Passing to the limit in (28.57) in the space Lp we obtain (28.59).
Theorem 28.6. Let 1  p < 00, a > 0 and let f = 1{acp with cp E Lp. Then
ex a f == lim ex: f = cpo
£_0
(Lp)
The proof is similar (see also references in S 29.1).
28.4. Pyramidal analogues of mixed fractional integrals and
derivatives
In S 24 we introduced mixed integrals and derivatives of fractional order. The
domain of integration for such operators is a rectangular parallelepiped with
opposite vertices x = (Xl,'" , x n ) and a = (aI,.. ., an). In particular, it may be
an octant with the vertex x. The kernels of these operators have singularities on
those faces of parallelepiped which pass through the point x. Now the domain of
integration will be chosen to be a certain pyramid with a vertex x and with a basis
situated on the fixed hyperplane (which does not depend on x). As regards the
kernel, we assume it to have singularities on the hyperplane passing through the
point x. As a result the so-called pyramidal analogues of mixed fractional integrals

570 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
and derivatives arise which, as it will be shown below, have essential distinctions in
comparison with mixed fractional integral and derivatives considered in S 24 and
which are not reducible to them. See for comparison the domains of integration for
mixed fractional integrals and derivatives and for their pyramidal analogues in the
two-dimensional case in Figures 4 and 5.
I
,
o a l
,
.
xl
t l
t1

 -
X(Xl')
-
Figure 4. The domain of integration of the
integral 1:+ cp
1
Figure 5. The domain of integration of the
integral lAc cP
Later on we shall require some auxiliary terms and notations. Let A = lIajl:lI
(ajl: E RI) be a matrix of order n x n with determinant IAI = det A f; 0,
aj = (ajl,..., ajn) be its line vectors and ajl: be the elements of the inverse matrix
A-I. Without loss of generality we put IAI = 1. Let also A. X = (al . X,..., an . x),
(A.x)a = (al.x)a 1 ...(a n .x)a.. where a = (al,...,a n ) and b= (bl,...,b n ) ERn,
C = (Cl,'" ,cn) ERn, -00 < bj,cj < 00. We denote by
Ac(b) = {t E R n : A . (b - t)  0, c. t  O}
(28.60)
the n-dimensional bounded pyramid in l{!I with the vertex at the point b, with a
basis on the hyperplane C . t = 0 and with lateral faces situated on the hyperplanes
aj . (b - t) = 0, j = 1,..., n. In particular, if A = E = 116jl:II is a unit matrix and
C = 1 = (1,...,1) then (28.60) is the simplest model pyramid of the form
E 1 (b) = {t E R n : b  t, t . 1  O}.
(28.61)
If n = 2 it coincides with a triangular bounded by the lines tl = b l , t2 = b 2 and
tl + t2 = 0 (see Figure 6).
The Abel type integral equation (see (28.65» which will correspond to
the above mentioned pyramidal analogues of mixed fractional integrals is a
generalization of the equation considered by Mihlin [1, p. 48].
The next proposition contains conditions of non-emptiness and boundedness of
the pyramid (28.60) and the expression for interchanging the order of integration,
similar to the Dirichlet formula (1.32).

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
571
t 2
b(b},b 2 )
b 2
b} t}
Figure 6. The domain of integration of the integral I 1 cp
Lemma 28.2. The pyramid Ac(b) given by (28.60) is non-empty (bounded) in R n
if and only if A- 1 c. b > 0 (A- 1 c > 0, respectively).
The proof of this Lemma follows from the fact that the linear transform t --+ A -It,
b --+ A -1 b maps the pyramid Ac ( b) into the pyramid
Ad(b) = {t E R n : b  t, d. t  O}, d = A- 1 . c.
(28.62)
Lemma 28.3. If a function f(t, r) given on Ac(b) x Ac(b) is measurable then the
following expression for interchanging the order of integration
f dt f f(t, r)dr = f dr f f(t, r)dt,
Ac(b) Ac(t) Ac(b) U(b,T)
(28.63)
is true, where
u(b, r) = {t E R n : A . r  A . t  A. b}, (28.64)
under the assumption that one of the repeated integrals in (28.63) is absolutely
convergent.
The proof is obtained directly by Fubini's Theorem 1.1.
Now we consider the Abel-type integral equation on the pyramid Ac(b)
1 f cp(t)dt
r(a) (A. (x - t»)l-a = f(x),
Ac(z)
x E Ac(b),
(28.65)
where 0 < a < 1 (which means 0 < a1 < 1,...,0 < an < 1), r(a) = r(at)... r(a1)
and x < b (Xl < blt...,xn < b n ).

572 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
In particular if Ac(b) = El (b) (28.65) has the form
1 ! cp(t)dt
f(a) (x _ t)l-a = f(x), x E El(b),
El(Z)
(28.66)
which does not coincide with the Abel multi-dimensional equation (24.1). Therefore
(24.1) is not a particular case of (28.65). However the method which was used in
S 2.1 may be applied for inverting (28.65). Following this approach we replace t by
r and x by t, respectively in (28.65), multiply both parts of the resulting relation
by (A. (x - t»-a and integrate over the pyramid Ac(x). Applying Lemma 28.3 we
have
f(l Ot ) ! 'P( T)dT ! (A. (z - t))-a(A. (I - TW-1dl = ! (A. (z - t))-a f(l)dl
Ac(z) U(Z,T) Ac(z)
(28.67)
where the domain O'(x, r) is given by (28.64). To evaluate the inner integral in
(28.67) we introduce the new variables Sj = [aj . (x - t»)/[aj . (x - r»). Taking the
equalities 1 - Sj = [aj . (t - r»)/[aj . (x - r)] and (1.68) and (1.69) into account we
obtain
n 1
! (A. (z - t))-a(A. (I - T»a- 1 dt = g! s;a;(l_ Sj)a;-lds j
U(Z,T) J- 0
(28.68)
= f(a)f(1 - a).
Hence (28.67) may be rewritten in the form
! - I ! -a ':!!
cp(t)dt - f(1 _ a) (A. (x - t» f(t)dt = h-a(x).
Ac(z) Ac(z)
(28.69)
Here we change the variables
t = A - l .  , ... = A - l . !!. , X ( Xl xn )
d'" d d = d l ' . . . ' d n '
(28.70)
where d = (d l ,..., d n ) is given by (28.62). Then in accordance with Lemma 28.2

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
(28.69) is equivalent to
/ 1/J( r)dr = g(y),
El(Y)
where E1(Y) is the pyramid (28.61) and
n
1jJ(r) = cp (A- 1 . ), g(y) = It-a (A- 1 . ) II die.
1e=1
To invert (28.71) we rewrite it in the form
1
-(Yl +...+y..-l)
dTn 7'
-(Yl +"'+Y.._+T..)
1/J( r)dr = g(y).
dTn-l ... J
-(T+"'+T..)
573
(28.71 )
(28.72)
(28.73)
Successive differentiation of this expression with respect to Yn, Yn-1 , . . . , Y1 yields
the relation
8 n
1/J(y) = 8 8 g(y).
Y1'" Yn
Here we change the variable x = A-1 .  similarly to (28.70) so that
8 n a-:le 1 8
- = ""' ...1-_, k = 1,2,... ,n.
8YIe  die 8x'
3=1 3
Then finally we obtain the following inversion relation for (28.65):
1 n ( n 8 )/
<p(x) = r(1 _ Q) II ?: ajl: 8x' (A . (x - t»-a j(t)dt.
A:=1 3=1 3 Ac(Z)
In particular, if Ac(b) = E1(b) then (28.75) has the form
<P(z) = [(1  a) :z / (z - t)-a f(t)dt
E 1 (z)
where :z = 81 . . . 8:.. .
(28.74)
(28.75)
(28.76)

574 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Now we give a justification of the solvability of (28.65) in the space Ll(Ac(b».
For this goal we introduce the notation
IA.(LI) = {g: g(z) = f h(t)dt, h(t) E L1(A.(b»}
Ac(z)
(28.77)
and observe that if g E 1Ac(Ld then there exist partial derivatives of 9 up to order
n almost everywhere on Ac(b) and
n ( n {) )
g [; 0;. 8z; g(z) = h(z).
(28.78)
The following statement is an analogue of Theorem 2.1.
Theorem 28.7. The Abel-type equation (28.65) with 0 < a < 1 is solvable in
Ll(Ac(b» if and only if
J.-a(Z)d,1 r(l  a) f (A. (z - t))-a f(t)dt E IA.(L.),
Ac(z) .
(28.79)
n a
ft-a(x)lc.z=O = E ajn ax' ft-o(x)lcoz=o = ...
j=1 )
(28.80)
n ( n a )
= II E ajk ax. ft-a(x)lcoz=o = O.
k=2 j=1 )
These conditions being satisfied (28.65) has a unique solution given by (28.75).
Proof. In the model case Ac(b) = E 1 (b) the theorem follows from (28.71) and
(28.73). In the case of an arbitrary pyramid Ac(b) it is obtained from (28.71) and
(28.73) after the change of variables (28.70) with the property (28.74) being taken
into account. .
Corollary. Equation (28.66) is solvable in £1(E 1 (b)) if and only if
J.-a(z) = r(l  a) f (z - W a f(t)dt E LE,(L 1 ),
El(Z)
(28.81 )

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
575
h-a(x)lt-z=O = aa h-a(X) 1 = ...
x n 1.z=O
= a a ... a a h-a(x) ! = O.
x2 x n 1.z=O
(28.82)
These conditions being satisfied (28.66) has a unique solution given by (28.76).
Theorem 28.7 gives the criterion of solvability for the Abel-type equation
(28.65) in terms of the auxiliary function fl-a(X). Simple sufficient solvability
conditions in terms of the function f(x) itself are given in Theorem 28.8 and its
Corollary.
Theorem 28.8. Let the function f(x) have continuous partial derivatives up to
order nand
v l1 f(x)lc.z=o = 0, 0  1,81  n - 1.
(28.83)
Then the Abel-type equation (28.65) is solvable in Ll(Ac(b)) and its unique solution
is representable in the form
1 f n ( n a )
<p(X) = r(1 _ a) (A. (x - t))-a II ?: Oj1cl5"i: f(t)dt.
Ac(z) Ie:l 3=1 3
(28.84)
Prool. First we consider the model equation (28.66). Similarly to (28.73) we
represent h-a(x) as
ft-a(Z) - r(l  a) f (z - W a f(t)dt
E 1 (z)
z..
f (xn - tn)-a"dt n
r(l  a)
-(Zl +...+z..-d
Z.._l
X f (Xn-l - tn_d-a"-ldtn_l ...
-(Zl +'..+z.._:l+t..)
x
7
-(t+...+t..)
(Xl - td- a1 f(t)dt.
Carrying out the successive integration by parts and taking (28.83) into account

576 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
we have
1 f (x - t)l-a af(t)
ft-a(x) = r(1- a) (1 _ a) at dt.
E 1 (z)
The function (28.85) satisfies (28.81) and (28.82). Indeed, the former is clear,
while the latter is checked by successive differentiation of (28.85) with respect to
x n , Xn-l, . . . , X2. Hence according to the Corollary to Theorem 28.7 we obtain the
statement of the theorem for the model equation (28.66). The general equation
(28.65) is reduced to the model one by means of the change of variable (28.70).
The theorem is proved. .
(28.85)
Corollary. If the function f(x) has a continuous partial derivatives up to order n
and
VfJ f(x)h-z::o = 0, 0  1,81  n - 1
(28.86)
then (28.66) has a unique solution given by the relation
1 f -0 anf(t)
cp( x) = r( 1 _) (x - t) a 0 dt.
a tl . .. t n
E 1 (z)
(28.87)
Remark 28.2. Equation (28.71) is an example of an integral equation of the first
kind arising in the problems of integral geometry where the unknown function is to
be restored by its known integrals over some sets, see for example, Gel'fand, Graev
and Vilenkin [1, p. 111].
Proceeding from (28.65) and (28.75) we introduce the operators
1 f cp(t)dt
(ItCP)(x) = r(a) (A. (x - t»l-o '
Ac(z)
a> 0,
(28.88)
a 1 n ( n - 0 ) f f(t)dt
(V Acf)(X) = r(1 _ a) IJ  ajk ox; (A. (x _ t»a '
k_1 3- 1 Ac(x)
(28.89)
o < a < 1,
which we shall call pyramidal analogues of the Riemann-Liouville mixed fractional
integrals and derivatives of order a = (a1,"', an), cf. (24.5) and (24.9).
Expressions (28.88) and (28.89) are defined for functions given on Ac(b).
Theorems 28.7 and 28.8 contain conditions for the existence of the mixed
fractional derivative (28.89). In particular, if a function f has continuous partial

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
577
derivatives up to order nand (28.83) hold, then the mixed fractional derivative
(28.89) is representable in the form
1 f n n a )
(Vtf)(x) = r(1 _ a) (A. (x - t»-a II (?= ajle ax. f(t)dt.
Ac(Z) le=l 3=1 3
(28.90)
With the aid of (28.67) one may check the semigroup property
[a I/J cp = Ia+/J cp
Ac Ac Ac'
a  0, /3  0,
(28.91 )
for any function cp E LdAc(b». Here [ccp = cp, a + /3 = (al + /31,"" an + /3n).
The simplest operators of the form (28.88) and (28.89) are the following ones
1 f cp(t)dt
(IE1CP)(x) = r(a) (x-t)l-a '
El(z)
a> 0,
(28.92)
a 1 a f f(t)dt
(VElf)(x) = r(1 - a) ax (x - t)o '
E.(x)
O<a<1.
(28.93)
We shall call them the model pyramidal analogues of the Riemann-Liouville mixed
fractional integrals and derivatives of order a. In the case ale > 1, we introduce
the model pyramidal analogues of mixed fractional derivatives similarly to (24.10)
by the relation
( a ) [a]+l
(VEl f)(x) = ax (I:[a]-l f)(x),
a> 0,
(28.94)
where
[a] = ([ad,. .. ,[an]),
( !.- ) [0] = (  ) [ad. . . (  ) [aIL]
ax OXl aX n
Below only the operators (28.92)-(28.94) will be considered.
Example 28.1. Let cp(t) = (t 1 + .. . + t n )/J- 1 , a > O. Then
([ a )( ) _ r(/3) ( ) Ial+/J-l
El cp X - r(lal + /3) Xl + . . . + X n ,/3 > 0,
(28.95)

578 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
( VOl f)( ) - r(,8) ( ) p-Ial-l IJ I I
El X - r({3 _ laD Xl + .. . + x n , fJ > a,
(28.96)
where lal = al +... + an.
This example shows that the model pyramidal analogues of mixed fractional
integrals and derivatives (28.92) and (28.94) keep power behaviour invariant - i.e.
transform power functions into power functions - on the hyperplane Xl +.. '+xn = 0
which is the basis of the pyramid El(X). Unlike this, mixed fractional integrals and
derivatives (24.5) and (24.9) keep power behaviour invariant on the hyperplanes
X Ie = a Ie, k = 1,.. . , n, since
( 101 ( t - a ) /J-l )( x ) = r({3) ( X - a ) OI+/J- l 0 0
0+ r(a+{3) , a> ,,8> ,
(V+(t - a)p-1)(.,) = r(()n) (" - a)p-a-" n > 0, fJ > n,
(28.97)
(28.98)
Similarly to (24.6) and (24.9) we may introduce the model pyramidal analogues
of mixed fractional integrals and derivatives with respect to some of the variables.
Let
X = (x', x"), x' = (Xl,'" ,x m ), x" = (Xm+l,"', X n ),
a' =(al,...,am,O,...,O),
E ( X) = {t E R n : t'  X', t" = x", t' . 1 + x' . 1  O}.
(28.99)
Then we put
(I: )(.,) = r( ') f (.,' - t,)a' -1 (t' , .,II)dt ' ,
E (z)
a' > 0,
(28.100)
( ) [01']+1
(VE:f)(x) = :X (1:-[OI']-lf)(x),
a' > 0,
(28.101)
where [a'] = ([al],"', [am], 0,... , 0). In particular, if x" = (X2,"', x n ) then
(28.100) and (28.101) coincide with the one-dimensional fractional integrals (2.17)

 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
and derivatives (2.29):
579
(IE: cP )(x) = (I(z+...+z..)cp(t', x"»(x') ,
('DE: cp )(x) = ('D(z2+...+z..)f(t', x"»(x').
(28.102)
(28.103)
if x" is fixed.
The above arguments show that pyramidal analogues of mixed fractional
integrals and derivatives are specific constructions different from mixed fractional
integrals and derivatives. In particular, they may not be represented as a tensor
product of one-dimensional fractional integrals. However, some properties of
one-dimensional fractional integrals and derivatives may be transferred to such
operators. For example, the index law (28.91) is valid. An analogue of Hardy-
Littlewood Theorem 5.3 is also true. To formulate it we consider the case of two
variables just as was done in S 24.4.
We introduce the space L'lt,(El(b» of functions f(Xl, X2) equipped with the
norm
{ b [ bl ] ,/'P1 } l/,
IIfIlL;(E 1 ) = f f If'<xl,X2)I' ldx l dX2 < +00,
-b 1 -Z2
(28.104)
The following theorem is a corollary of Theorem 24.1.
Theorem 28.9. If 1 < Pj < I/Otj and l/qj = l/pj - Otj, j = 1,2, then the operator
of the pyramidal analogue (28.92) of the mixed fractional integral is bounded from
L'l,,(El(b» into LqlJq(El(b».
Proof. Let cp(t) E L'lJ,(El(b». We define the function cp(tl,t2) given on the
triangle E 1 (b) (see Figure 3) to be zero on the rectangle IT (b) = {(it, t 2) E R 2 :
-b 2 < tl < b 1 , -b 1 < t2 < b 2 }. Let XE1(b)CP = {cp, x E El(b); 0, x E IT(b) \ El(b)}
be the characteristic function of the triangle EI(b). Applying now Theorem 24.1
we obtain
II lEI CPIlLf(EI) = IIIb':blXE1(b)CPIlLf(El)
< II IOll'OI II
- -b,-bl XE1(b)CP Lf(n)
:5 cIlXE1(b)cpIlL,(n)
= cllcpIlL,(E 1 ).
The theorem is proved. .

580 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
 29. Bibliographical Remarks and
Additional Information to Chapter 5
29.1. Historical notes
Notes to  24.1. The solution of the multidimensional Abel equation (24.1) was probably
known long ago, even though it was not found in the early publications. It is natural to
suppose that it was already known to Hohngren [1] (1865-1866) - see the paragraph below. The
Laplace-transform method of solving (24.1) may be found in Vasilache [1] (1953) for n = 2 and in
Delerue [1] (1953) for an arbitrary n.
Notes to  24.2. There are many papers where partial or mixed fractional integra-
differentiation, in each variable or in a part of them, is "introduced". It is worth emphasizing
that these concepts arose long ago, and the first person who indeed introduced Riemann-Liouville
fractional integra-differentiation for functions of two variables was Holmgren [1, p. 14] (1865-
1866». These ideas were essentially used in problems of the function theory by Montel [1, p. 172]
(1918) yet.
Notes to  24.4. Details about the spaces Lfi with mixed nonn may be found in the
books by Besov, D'in and Nikol'skii [I, ch. 1] (1975) and in the paper by Benedek and Panzone
[1] (1961). The idea of considering fractional integration in the spaces Lp is contained in the
latter paper where it was realized for the Riesz potential operators. The boundedness from Lfi
into Lf, qi = Pi (1 - eviPi )-1, directly for the operators ll  I2 of mixed fractional integration,
and similarly for partial fractional integration, was noted by Skorikov [3] (1977), the case of
an arbitrary number of variables was observed by Magaril-ll'yaev [1] (1979), although it was
probably known earlier.
Notes to  24.5. The existence of relations between different fractional integration
operators via the bisingular operator was first noted by Pilidi [1] (1968). In particular, he gave
(24.30) for <p e Lp with 1 < P < min(evil, evil). Relations (24.30)-(24.32) for <p e Lp were given
by Skorikov [3] (1977).
Notes to  24.6. Partial and mixed Marchaud fractional derivatives (24.33)-(24.35) in
the case of two variables appeared in Marchaud [1] (1927). Fractional integra-differentiation in a
direction for functions of many variables was first introduced by Kipriyanov [1]-[6] (1951967) in
a fonn different from (24.38) and (24.39) (refer also to  29.2, note 24.3).
Notes to  24.7. Theorem 24.4 in a slightly weaker fonn was proved by Skorikov [3] (1977).
Notes to  24.8. The relations of this subsection are well known and are easily derived
from the corresponding one-dimensional results. There are a number of papers which contain
detailed proofs of these results, mostly in the case of two variables, as for example, Jain [1] (1970)
where (24.56) is contained. The expressions in (24.52) and (24.53) with n = 2 may be found in
Raina and Kiryakova [1] (1983).
Notes to  24.9. The space C), discussed here, was int.roduced and investigated by Lizorkin
[1] (1963). It was considered. also by Yoshinaga [1] (1964).
Notes to  24.10. Weyl-type partial and mixed fractional derivatives appeared first in
Bessonov [1] (1964). There, for functions f in Lp the author considered whether there exist in Lp
certain partial or mixed derivatives, if there exist in Lp some other such derivatives (see  29.2,
note 24.5). The detailed and complete investigation of f.'actional differentiability of periodic
functions of many variables was undertaken by Lizorkin and Nikol'skii [1] (1965) on the basis
of Lp-spaces. Refer also to Nikol'skaya [1] (1974), where the problem of the existence of mixed
Weyl fractional derivatives is related to the rate of convergence of partial sums of multiple Fourier
series. Theorem 24.7 is a slight modification of theorem 4 from Lizorkin and Nikol'skii [1].
Notes to  24.11. The definitions (24.65) and (24.66) seem not to have been introduced
elsewhere. Theorem 24.8 may be considered as new.

 29. ADDITIONAL INFORMATION TO CHAPTER 5
581
Notes to  24.12. The polypotentialS of type /COI were introduced by Okikiolu [5] (1969),
in a more general form with power weights, though. The connection between the polypotentials
/COI and 'H OI in the form (24.69) has not been noted earlier.
Notes to  25.1 and 25.2. Relation (25.11) is usually connected with the name of Bochner
[I, p. 263 and 315], whereas (25.14) and (25.14') for the Fourier transform of radial functions in
terms of fractional integration were given by Leray [1] (1953).
The space  defined in (25.16) firstly appeared in Semyarustyi [1] (1960). Afterwards
Lizorkin [1] (1963), and also [5] (1969), [8] (1972), gave a thorough investigation of spaces of such
a kind, together with application in the theory of functional spaces with fractional smoothness.
Since then such spaces have become objects of current research and are often used in mathematical
literature. We remark that this particular space appeared also in Helgason [I, p. 162] (1965). In
the book of Helgason [2, p. 20, 59, 62] (1983), who it appears was unaware of Lizorkin's papers,
the invariance of the space  relative to the Riesz potential operator was proved in connection
with its applications in the theory of the Radon transform.
Relation (25.25) is fomally known long ago, its justification in the sense of distributions on
S(R n ) was given by Schwarts [I, t. 2, p. 114] (1951). The result obtained there was different
from (25.25) by the polynomial summand in the second line, which is orthogonal to  (see
Remark 25.2). The interpretation of the Fourier transform of the function Ixl- OI in the sense of
distributions on () was suggested by Semyanistyi [1] (1960), and refer also to Lizorkin [8, p. 242]
(1972).
A potential with the kernel Ixi Ol - n first appeared in the thesis of Frostman [1] (1935)
devoted to the problem of the existence of the unique equilibrium potential of a compact set
in R n . This potential was introduced by Riesz, who was Fr06tman's teacher - Riesz [2] (1936)
and [4] (1938), and also [5] (1939) and [6] (1949). (In pointing out the role of this outstanding
mathematician we would like to mention here his obituary written by Girding [2] and the paper
by MikolAs [9] with brief remarks on Riesz's works). We do not concern ourselves here with the
connections of Riesz potentials with superharmomc functions but refer to Landkof [1].
The invariance of the space () defined in (25.16) relative to the Riesz potential was noted by
Semyanistyi [1] (1960) and Lizorkin [1] (1963).
Notes to  25.3. Theorem 25.2 is due to Sobolev [1] (1938). The proof of necessity
part follows Stein [2, p. 140]. The proof of Theorem 25.2 based on convexity theorems was
later suggested by Thorin [1], 1948. An elementary argument immediately reducing the Sobolev
theorem to the case n = 1, was suggested by du Plessis [2] (1955) (see  29.2, note 25.2).
Muckenhoupt and Stein [1] (1965) gave the proof based on the interpolation of linear operators.
This proof is given in Stein's book [2]. A rather simple proof which is well-suited for all n  1
was given by Hedberg [1] (1972). We refer also a development of Hedberg's idea in a more
general situation in Meda [1]. We also mention Yoshikawa [I], who obtained the statement of
the Sobolev theorem as a corollary of a general similar result for fractional powers of the fonn
00
A- OI f = r(0I)-1 f tOl-1Ttf, T, being a semigroup of operators satisfying certain assumptions.,
o
Theorem 25.3 was proved by Stein and Weiss [1] (1958). Similar theorems with weight
n
n IXiP' or (lx'1 2 + Ix"12)'1/2 with x' = (x},..., Xm) and x" = (x m +},..., Xn) were proved by
i=l
Nikolaev [I], [2] (1973). Theorem 25.4 is due to Muckenhoupt and Wheeden [2] (1974). A simpler
proof of this theorem can be found in WeIland [3]. A simple proof and even in a more general
case of the so called anisotropic Riesz potentials was given by KokiJashvili and Gabidzashvili
[1] (1985) - see also Kokilashvili [2, p. 36-54] (1985). There are also other generalizations (the
so-called two-weighted ones and others): see  29.2, notes 25.7 and 25.8. The weak type estimate
(25.46) for Riesz potentials was revealed by Zygmund [4] (1956). Theorem 25.5 was obtained by
Vakulov [1,2] (1986). Theorem 25.6 was proved by Stein and Weiss [2, p. 57] (1960) in the case of
the Poisson integral and by Johnson [1] (1973) in the case of the Gauss-Weierstrass integral.
Notes to  25.4. The realization of Riesz differentiation F-11IOIF<p in the form of the
hypersingular integral (25.62) appeared first in Stein [1] (1961) in the case 0 < 01 < 2. The general
case 01 > 0 was considered by Lizorkin [6] (1970) who introduced the hypersingular integrals

582 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(25.61) with a centered difference. Lizorkin [6] indeed defined an entity more general than (25.61),
which corresponds to the scalled anisotropic case. He characterized the space of anisotropic
Bessel potentials in terms of such anisotropic hypersingular integrals. The Fourier transform of a
hypersingular integral, that is (25.63), was studied by Lizorkin [6, p. 82] in the case of centered
differences, and by Samko [17, p. 1170] (1976) in the case of non-centered differences. Expressions
(25.69) and (25.70) for inverting Riesz potentials f = [OI<p, <p e Lp, were considered by Rubin [26]
(1987).
Notes to  25.5. The integral (25.71) seem to have first appeared in Eskin [1] (1973).
The term "unilateral potentials" was suggested by Rubin [19]-[21] (1984-1985), who investigated
other potentials of a similar kind as well. The representation of the operators inverse to (25.71)
in the form of hypersingular integral (25.77), and the justification of the inversion D%[%<p == <p
for unilateral potentials on functions <p e Lp(Rn) was given by Rubin [16], [18], [20].
Notes to  26.1. The presentation of the normalizing constants in this subsection follows
Samko [17] (1976) and [26] (1978), and also [28] (1980).
Notes to  26.2. The possibility of lowering the order I to I > 2[01/2] in the case of a
non-centered difference was noted by Samko [17] (1976). The case 0 < 01 < 2 when one may take
I = I, was known earlier - Stein [1] (1961) and Lizorkin [6, p. 85] (1970). The way to avoid the
phenomenon of the annihilation (26.18) by means of the choice I = 01 in the case of a non-centered
difference was suggested by Samko [17] (1976).
Fractional differentiation in a given direction was considered by Wihnes [1, 2], who defined
it in Fourier transforms without realization in Wltransformed fWlctions and gave (26.24).
Notes to  26.3. The ults of this subsection, except Theorem 26.3', were obtained by
Samko [17] (1976). The latter theorem was proved by Rubin [26] (1987).
Notes to  26.4. Hypersingular integrals with homogeneous characteristics appeared in
the papers of Wheeden [1]-[4] (1967-1969). The characteristics O(x,y) can be fOWld in Fisher
[7] (1973). However, hypersingular integrals with 01  2 were introduced in all these papers via
regularization (subtracting the Taylor sum), and not by taking finite differences. In the form
(26.47) hypersingular integrals were considered in Samko [21] (1977), [26] (1978) and [28] (1980)
in the case of homogeneous characteristics, and in Samko [18] (1976) and [20] (1977) in the case
of a non-homogeneous characteristics. It needs to be said that Wheeden and Fisher investigated
hypersingular integrals in the context of the theory of Bessel potentials, i.e. the hypersingular
integral DOl f was considered to be in Lp together with f. The approach taken in the cited papers
by Samko and in this book allows one to take f e Lr and DOl f e L" with different r and p which
includes both the "Bessel" and "Riesz" situation.
The classification of hypersingular integrals with a homogeneous characteristics both in type
and in order of the used finite difference was suggested in Samko [26, p. 235] (1978) and [28,
p. 198] (1980). Relations (26.55)-(26.57) were obtained by Samko in the same papers, the first of
these expressions in the case 0 < 01 < 1 being proved by Radzhabov [1] (1974).
The question about the simultaneous convergence of hypersingular integrals with different
characteristics was considered by Nogin and Samko [1] (1981) and [3] (1982). The same question
in the case of non-homogeneous characteristics was investigated by Nogin [1] (1980).
The papers by Horvath [1] (1978) and Ortner [2] (1985) conceming problems of analytic
continuation and compositions of convolutions with ''pseudfunctions'' k ( liT) Ix I>' , .x e R 1 , are
also connected in a sense with the context of  26.4.
Notes to  26.5 and 26.6. Theorem 26.5 was proved in Samko [26] (1978). The answer
to question (26.74) in the form (26.81) and (26.82) and Theorem 26.6 was obtained in Samko [28,
p. 205-207] (1980).
Notes to  26.7. The investigation of the space [01 (L p ) presented here follows the papers
by Samko [17], [18], (1976), [20, 23] (1977) and [31] (1984) and include consideration in the sense
of distributions for p  n/OI. The space [OI(L p ) had arisen earlier, but without constructive
characterization in terms of convergence of hypersingular integrals. We refer to, for example,
Maz'ya and Havin [1] (1972), where the authors showed that the closure of Co(R n ) in the norm
IIF-11(IOIF<p11P coincides with [OI(L p ) if 1 < p < n/OI. More exactly this statement is to be
Wlderstood in the sense that the closure coincides with [OI(L,,) after "throwing out" polynomials

 29. ADDITIONAL INFORMATION TO CHAPTER 5
583
of order less than a, which are acquired Wlder this closure. In particular, in Samko's papers there
are obtained both Lemma 26.5 ([31, p. 78]), the characterization of the space 100(L p ) presented
in Theorem 26.8 ([17], [18], (1976) and [31,  13], (1984», Theorem 26.9 ([23] (1977», and the
estimates (26.98)-(26.99) ([17) (1976». The spaces L;.r(R n ) arose as a natural generalization of
the spaces of Riesz and Bessel potentials coinciding with the former if r = np/(n - OIp), and with
the laUer if r = p. They were investigated in Samko (17) (1976) and [20] (1977).
Theorem 26.10 was proved in Rubin [23], [26], (1981987), (26.97') with different p and
q was proved in Samko [34]-[35] (1990) and the statements (26.98) and (26.98') were given in
Samko (17) (1976). .
Notes to  27.1. The fractional powers (E - )-0I/2 became named Bessel potentials
after the papers by Aronzajn and Smith [1] (1961), Calderon [1] (1961), Aronzajn, Mulla and
Szeptycki [1] (1963), Aronzajn [1] (1965) and Adams, Aronzajn and Smith [1] (1967).
The Bessel kernel as the Fourier original (in the sense of distributions) of the fWlction
(1 + Ixl)-0I/2 was considered in L. Schwartz [1, vol. 2, p. 116] (1951).
Notes to  27.2. Theorem 27.1 and the modification (!5011p of the Bessel potential were
given by Flett [6, p. 445-448] (1971), (27.20) and (27.23) were obtained by Johnson [1] (1973) and
(27.22), (27.24) and (27.26) were proved by Lizorkin [2] (1964).
Notes to  27.3. Theorem 27.3 is due to Stein (1961) in the case 0 < 01 < 2 and to Lizorkin
[6] (1970) in the general case.
In connection with the isomorphism between the spaces L;(R n ) realized by the operator of
Bessel fractional integration, we note that there is a number of investigations on the isomorphism
between fWlctional spaces by means of fractional powers of differential operators. We restrict
ourselves by references to the papers of Nikol'skii, Lions and Lizorkin [1] (1965) and Alimov [1]
(1972), and also  29.2, note 25.13.
Notes to  27.4. The results presented here were obtained by Nogin [3], [4] (1982) and (7)
(1985), except Theorem 27.7 and the statements following after this theorem, which were proved
by Rubin [23]-[26] (1986-1987). The approach to the inversion of Bessel potentials by means of
hypersingular integrals with weighted differences was given in Rubin [24], [25] (1986).
Notes to  28.1. The expressions for the Fourier transfonn of fractional powers of quadratic
forms used here were given in Gel'fand and Shilov [2, Ch. III,  3].
The potential (28.21) was introduced by Riesz [3] (1936) and [8] (1967(1939», and also [6]
(1949). The properties (28.22), (28.24) and (28.25) of potentials with the Lorentz distance were
in particular obtained in these papers. Riesz considered such potentials not only over the whole
cone Ki but alao over a domain D bOWlded by the cone surface p = 0 and a certain sufficiently
smooth other surface. For simplicity we dealt with the case when the domain D is the whole
cone Kt. Potentials with the Lorentz distance proved to be an effective means for solving the
Cauchy problem for hyperbolic equations, and this was demonstrated by Riesz [6] (1949) and [8]
(1967(1939», and refer alao to the book by Baker and Copson [1, p. 57-61] (1950). The various
applications of Riesz potentials with the Lorentz distance can be fOWld in Copson [1] (1943), [2]
(1947) and [4] (1956) and Fremberg [2], [3] (1946).
The Riesz potential 10 Ip defined in the case Re 01 > n - 2 admits an analytic continuation
to the whole complex plane except for a finite number of poles, and this was shown by Riesz.
Another proof of analytic continuation was given by Fremberg [1] (1945), and also [2] (1946). We
further point out the method of an analytic continuation given in the book by Baker and Copson
[I, p. 60] (1950) in the case n = 3.
Nozaki [1] (1964) considered the Riesz hyperbolic potential (28.19) in the general case (28.3)
with an arbitrary p = 1,2,.. . ,n instead. of p = 1. He evaluated the nonnalizing constant (28.20)
corresponding to an arbitrary p and verified the semi group properties (28.22) and the analytic
continuability in a. We refer to the case p '# 1 in the paper by Trione [2] (1987) or its reprint [3]
(1988).
Relation (28.28) was established by Schwartz [I, t. 2, p. 120] (1961). In connection with
the calculation of the Fourier (Laplace) transform of the power of the Lorentz distance, we
mention the book by Vladimirov [1,  30] (1963), and the book by Trione [1] (1980), the latter

584 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
being specially devoted to the Fourier-Laplace transform of fWlctions depending on the Lorentz
distance.
The statement of Theorem 28.1 should be considered as known, but the authors cannot find
the corresponding reference.
The realization of fractional powers of differential operators is closely connected with the
investigation of "generalized" functions IP( x) I >.., where P( x) is a polynomial. In this connection
we can refer to the following papers: Gel'fand and Graev [1] (1955), Fedoryuk [1] (1959), Bresters
[1] (1969), Bernstein and Gel'fand [1] (1969), Atiyah [1] (1970) and Palamodov [1] (1980), and
also the book by Gel'fand and Shilov [2, ch. Ill,  4], which contains some results for arbitrary
functions raised to the power .x.
Notes to  28.2. The parabolic potentials Ha<p and 'Ha<p, - (28.39) and (28.45) - were
introduced by Jones [1] (1968) in connection with investigations of the heat-type equations. The
further study of these potentials and of the spaces H a (L p ) and 'H a (L p ) was undertaken by
Sampson [1] (1968), Bagby [1] (1971) and [2] (1974), Gopala Rao [1] (1977) and [2] (1978),
Chanillo [1] (1981), Nogin [2] (1981) and [5] (1982) and Nogin and Rubin [I], [2] (1985) and [4],
[5] (1986).
The proof of Theorem 28.2 can be found in Gopala Rao [1] (1977).
Notes to  28.3. We followed here Nogin [2] (1981) and Nogin and Rubin [1] (1985) and
[4] (1986).
Notes to  28.4. The Abel-type integral equation (28.65) in the particular case n = 2,
A = II 1  II and "1 = "2 = 1/2 was lUst solved by MihIin in 1940 in connection with the
problems arising in investigations of wave reflection from a rectilinear bOWldary. For this we refer
to Mihlin [I, p. 48] and references in Preobrazhenskii [I, p. 9], and also applications of equations
of such a type to problems of supersonic flow over space angles in Fedosov [1]).
The solution of (28.65) in the case of an arbitrary natural number n and c =
k-l
.--""--.
(0, . . . ,0,1,0, . .. ,0) was obtained in Kilbas and Vu Kim Tuan [1] (1982). The other results of this
subsection, in particular, the definition and the properties of the pyramidal analogues of mixed
functional integrals and derivatives are published here for the first time.
29.2. Survey of other results (relating to 55 24-28)
24.1. Raina [3] dealt with the evaluation of the mixed fractional integrals I..._<p, a =
(ab"', an), of fWlctions of the form <pet) = exp( -Epiti)P(Eti), P being a polynomial, in terms
of some special functions. This is an extension of the one-dimensional result of H.M. Srivastava
[4], (see  9.2, note 5.5 in this connection). His result was corrected and further developed by
R. Srivastava [1].
We observe also that the mixed fractional integrals (24.6) were used by Kosdunieder [I],
[2] to derive some properties of the Lauricella hypergeometric function Fn) of n variables -
Prudnikov, BrycbJrov and Marichev [2, p. 745]. ,
24.2. Erdelyi-Kober-type modifications of fractional integration in the case of two variables
were given by Verma [1] and Mourya [1]. Namely, the operators Ig + ' l t  Ig + ' l ,I'l t  I'l f'I
J ,\. J,'I J,\' J J.'
(in the notation of  18.1 and 24.3) and some others were introduced. Mourya [1] established the
main properties of these operators, and gave applications to some special fWlctions. We observe
that there is a misprint on p. 173 and elTors on p. 175 of this paper. Venna [1] considered
the two-dimensional Mellin transform of mixed Erdelyi-Kober fractional integrals. Various
compositions of two operators of such a type were studied by Raina [2]. Mixed ErdeIyi-Kober
fractional integrals were used in Venna [2], [3] to study properties of the two-dimensional integral
transform generalizing the two-dimensional Hankel transform.
Kaul [1] introduced modifications of generalized fractional integration of the Saxena-type
(23.5) and (23.6) for the case of two variables, and gave the expression for their Mellin transform

 29. ADDITIONAL INFORMATION TO CHAPTER 5
585
and the expression of fractional integration by parts. These results were extended to the case of
more general multidimensional operators by Saxena and Modi [1] and Mathur and Krishna [1].
24.3. A number of papers by Kipriyanov [1]-[5] dealt with the fractional derivative fa)(x),
o < a < 1, of a function f(x) at the point x e Rn in the direction from the point a eRn. The
initial definition of such a derivative by Kipriyanov says that fa)(x) is a function satisfying the
relation
Ix-a I Ix-a I
f fa)(a + e)n-ld = 1 f f(a + e) - f(a) n-ld( (29.1)
r(1 - a) (Ix - al - )a
o 0
where e is the unit vector in the direction from a to x, i.e. e = (x - a)lIx - al. As was shown by
Kipriyanov [2]-[3] this definition yields the following relation for fa)(x):
Ix-al n 1
(a) _ a f f(x) - f(x - e) ( ( ) - r(n) f(x) - f(a) (29.2)
fa (x) - r(1- a) l+a 1 - Ix _ al + r(n - a) Ix _ aI1+a-n
o
(d. (24.38"».
In the papers [1]-[5] Kipriyanov studied various properties of such fractional derivatives
and investigated spaces of functions in the domain 0 C R n having fractional derivatives in all
directions. Relations of these spaces to the Sobolev spaces wt(O) were shown. We mention some
of these results, e.g. [3].
1. H fa)(x) exists, is continuous and bounded for (x, a) e 0 X 0, then f(x) e Ha(O); if
f sup Ifa)(x)ldx < 00 and 01 > lip, then f(x) e H;-l/P, p > 1.
o aeO
2. H 0 < 01 < .x  1 and f(x) e H>"(O), then fa)(x) exists for all (x, a) e 0 X 0 and is
continuous in respect to (x, a).
3. Let c(a)(o) be the space of functions continuous on 0, such that fa)(x) is continuous
with respect to (x, a) e 0 X 0 and let IIfllc(cr) (0) = IIflic + max Ifa)(x)l. Then
(x,a)eOxO
IIfllc(cr)(o)  Allfllw(o) if I> nip and 0 < 01 < min(l, 1- nip), wt being Sobolev space of an
integer order I.
4. A similar result holds for the spaces La)(O) with the norm IIfIlL.,(O) + IIfa)IILp(Oxo)'
Moreover, imbedding of the space wt into C( a) and L a) is compact under the usual assumptions
- Kipriyanov [5a].
We alao note that the operator
p
a f(x) = 1 d f (p2 _ 'T2)-a f(a + 'Tw)'Tn-1d'T (29.3)
p pn-l-ar(1 - a) dp
o
of fractional differentiation in a given direction w, Iwi = 1, with p = Ix - ai, a being a fixed point,
was investigated by Kipriyanov [7]. He showed that the operator (29.3) can be transformed to
p n 1
a f(x) = 2pl+a f f(x) - f(a + 'Tw) (  ) - d'T + r(n) f(x) , p = Ix - ell,
p r(l- (1) (p2 - 'T2)1+a p r(n - a) pa
o

586 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(c!. (29.2) and (24.38")) and the operator inverse to (29.3) can be given in the form
p
2 2-n f
....!!......- t n - 1 - a (p2 _ t2)a-I(a I)(x)dt.
rea) t
o
Moreover, the operator  was used in this paper to construct spaces of the La)(O)-type
of fractionally differentiable functions, which are compactly imbedded into the Sobolev spaces
wt(O) under the appropriate assumptions on p, I, q and a.
24.4. Liouville fractional partial and mixed derivatives (24.15), treated in the sense
of distributions: (D+.... I, <p) = (I, D..._<p), <p E (I, with (I being the space (25.46'), were
introduced by Lizorkin [. Lizorkin and Nikol'skii [1] introduced the spaces S;L(R n ) = S;(R n )
with dominating mixed derivatives. The space S;(R n ), a = (ab"', an), consists of functions
I E L p (Jr1) with the mixed derivative V+....+I E Lp(Rn) and with all support derivatives in
Lp(Rn). The support derivative for V+...+I is the derivative Vi...+1 where & is obtained from
a = (al,"', an) by replacing some of ak by O. It was shown in this paper that
S: = G a (L p ), a > 0, 1 < p < 00,
(29.4)
where Ga(L p ) is the space (24.74) of polypotentials (24.72). In the paper by Nikol'skaya [1]
the character of the approximation of a function I E S; (R n ) by its partial Fourier sums was
investigated.
Brychkov [1] considered the spaces S; for all a E Rn, defined as distributions, and proved
(29.4) under the appropriate interpretation of the left- and right-hand sides. The concept of
smoothness order of the generalized functions in S; with respect to a part of variables was
introduced in this paper (see also Brychkov [2] in this connection), and the relation of this order
to the existence of the trace of a generalized function on hyperplane was shown.
24.5. Let I( x) be a 211"-periodic function of many variables and let V( a) I, a = (aI, . . . , an )
be the Weyl fractional derivative (24.60), (24.62). The following statement was proved by
Bessonov [1]. Let I E Lp(),  = {x: 0  x; < 211"}, 1 < p < 00. IT there exist V(a) I E Lp()
and V({3) I E Lp(), ai  0, (3i  0, then V-yrl E Lp() for'Y = (Ja + (1 - (J){3, 0  (J  1. A
certain generalization of this statement was obtained by Nikol'skii [5], p. 238.
24.6. A similar question on the existence of some fractional derivatives V..._/, {3 =
({3b"', (3n) when there exist some other such derivatives of certain orders a i = (a1,..., a)
was considered in a general form by Magaril-ll'yaev [1] in the non-periodic case in the general
context of the spaces Lp(Rn) with mixed norm. This is the scalled problem of the intermediate
derivative. This question is connected with the multiplicative inequalities for fractional derivatives,
the validity of which was completely investigated in this paper. The development of the results
for the problem of intermediate derivative may be seen in the paper [2] by the same author, where
it was given in certain generalized terms.
24.7. Fractional differentiation on the manifolds M = Tn' X Jr1" i.e. on the product of
a certain number of circles and axes, was considered by Magaril-ll'yaev and Tikhomirov [2] in
the presentation of some questions of the hannonic analysis on such manifolds. It was defined
by using the analogue of the Lizorkin-type space (I adapted to circles and axes forming M. The
problem of the intermediate derivative, Bernstein-Nikol'skii and Favard inequalities for fractional
derivatives and other questions were considered in the mixed norm spaces Lp(M).
24.8. The Bernstein-type theorem for mixed fractional derivatives of a function I(x, y) of two
variables was already given by Montel [I, p. 187], in the following form: if I/(x, y) - Pn,m(X, y)1 
Am--Y + Bn- 6 , Ixl < I, Iyl < 1, 'Y > 0, S> 0, where Pn,m(X, y) is an algebraic polynomial, then
I(x,y) has all mixed fractional derivatives of order (a,{3) such that al'Y + f3IS < I, a> 0, (3 > O.
24.9. The connection between the existence in Lp of the mixed Weyl fractional derivative
Va+,a'J) I of a function I(XJ,X2) of two variables, and the behaviour of its partial Fourier sums
and mixed continuity modulus was considered by Esmaganbetov [2].

 29. ADDITIONAL INFORMATION TO CHAPTER 5
587
24.10. Let a function l(x), considered in the domain 0 C R n , be the restriction onto 0 of
a function g(x) which in the whole space R n belongs to the Bessel potential space L (Jr1) ( 27).
The characterization of such functions lex) in terms of partial Riemann-Liouville fractional
derivatives, or more generally, Chen derivatives (18.18), was noted by Skorikov [2] (see  23.2,
note 18.14 in the case n = 1). Some considerations in similar situations in the case n = 2 were
developed by Biacino and Miserendino [1]-[3], Biachino, Di Giorgio and Miserendino [1] and
Biachino [1]. We also mention Miserendino [I], [2] where for functions in L(O) with an integer
cv and a parallelepiped 0 in Rn the author investigated the problem of the existence of mixed
fractional derivatives at the points of the boundary 80 and their belonging to the space L2(80).
24.11. The definition of Griinwald-Letnikov differentiation given in  24.11 for functions
defined in the whole space R n , can be adapted to functions defined in a region. This is realized
in a similar fashion to  20.4, where Griinwald-Letnikov fractional differentiation on the finite
interval was defined. In the case of two variables the definition, generalizing (20.42), has the fonn
( N CVI N cv N 1 N
lcv)(x)= lim --L- ) ( -1- ) "''''(_I)i+j(CV.I)(CV)
a+ N 1 -00 Xl - al X2 - a2   I J
N-oo 1=03=0
( . xl - al . x2 - a 2 )
xl XI-I-'X2-J-
NI N2
(29.5)
where cv = (CV2, CV2), a = (ab a2) and Xl  ab x2  a2. Such a definition was used by Krasnov
[I], who gave some properties of the fractional derivatives (29.5).
The development of this definition for the case of analytic functions given in the region
G C Jr1 may be found in Krasnov and Foht [1]. Such an approach to fractional derivatives of
functions of many variables was considered in this paper in connection with the derivation of
certain integral estimates for solutions of elliptic differential equations. This was earlier studied
in Foht and Krasnov [1] in the case of the Laplace equation. In the latter paper the definition of
fractional differentiation U; of analytic functions was used, reduced to fractional differentiation
of monomials xi, j = (h,..., in), but the authors discarded all the monomials with h: < cv if
only for one k = 1,...,n.
24.12. Mixed fractional differentiation !Oo\ ... !oQ"" B CV ..+ 1 with the fractional power
vZ I vZ.. Z..+l
B::: 1 1 of the Bessel differential. operator By = -£.s + Iv, k > 0, was used by Brodskii [I], [2]
to define certain function spaces with fractionalsoothD.es in the half-space.
24.13. The estimate
IIDJI'"  [ 2oin[,P/2J IIwJI"',
211"
0< r < -,
p
for fractional differentiation (24.48') in the given direction w and the function lex) of the
exponential type p was given by Wihnes [I], [2] (see also  23.2, note 20.5). The case r = 1I"/p,
o < cv < I, yields the Bernstein inequality with the constant 2 1 - cv .
25.1. The Lizorkin space (25.16) consists of Schwartzian functions I e S with the Fourier
transfonn vanishing at the origin. The similar space «I, adapted to partial fractional derivatives
and integrals (see  24.9) is related to the vanishing of the Fourier transforms on the coordinate
hyperplanes. In some other problems a necessity arises to deal with the Lizorkin-type space «Iv of
Schwartzian functions such that (Vk<p)(x) = 0, Ikl = 0,1,2,..., X e V, where V is a given closed
set in Jr1. Such spaces were investigated by Samko [25], [29], who gave their characterization in
the case when V is a cone in Rn and proved the denseness of «IV in Lp(Rn) if m(V) = O. This
denseness was obtained for an arbitrary such set V if p  2, and for a certain class of such sets V,
which were called quasibroken in [29], if 1 < p < 2. Later on it was shown that V with m(V) = 0
may be arbitrary in the case 1 < p < 2 also (Samko [36]; see alao the case of mixed nonn spaces

588 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Lp(Rn) in [36]). The denseness of V in Lp(Rn), 1 < P < 00, for V an origin or a union of the
coordinate hyperplanes was earlier established by Lizorkin [5], [8].
The space , cOJTesponding to the case V = {O}, is dense in Lp(R1j Ixp) provided that
1 < P < 00 and "y > -lor P = 1 and"Y > -I, but is not an integer, we refer to Muckenhoupt,
Wheeden and Wo-Sang Young [1, Theorem 6.10], where the space Soo was considered. This
consists of Schwartzian functions with Fourier transfonns identically equal to zero near the origin,
which is somewhat narrower than {o}. We observe that in Samko [29], [36] the denseness of
similar spaces narrower than v was indeed shown.
25.2. du Plessis [2] observed that the Sobolev theorem is reduced to the case n = 1
n n
immediately, since Ixln  II IXjl, or more precisely, Ixln  nn/2 n IXjl. This follows from the
j=l j=l
fact that the geometric mean is dominated by the arithmetic one. So
J Ip(y)dy < J IIp(y)ldy
Ix - yln-a - n
R" R" n IXj - yj 11- a ln
j=1
and then the one-dimensional Hardy-Littlewood theorem 5.3, being applied to each variable,
gives the desired estimate for ilIa Ipllg, t =  - .
We note that the Sobolev inequality ilIa Ilig  ell flip was shown by Lieb [1] to admit a
maximizing function 1 such that equality holds. A similar function does not exist for the Bessel
potential - see Lieb [I, p. 352]. The sharp constant c was explicitly evaluated by Lieb [1] in the
case q = pi or P = 2 or q = 2.
25.3. In the limiting case P = n/a the operator Ia is not defined as an absolutely convergent
integral on the whole space Lp(R n ), but IIIalpllBMO  clllplip on the set which is dense in Lp,
where BMO is the space of functions with bounded mean oscillation. We refer to  4.2 (note 3.3)
and  17.2 (note 13.1) in the one-dimensional case, where the references to the multidimensional
case are also given. Harboure, Macias and Segovia [2] obtained weighted estimates for the Riesz
potential in the case p = n/ a. The paper by Mizuta [2] is also relevant. It concerns the so
called total differentiability of Riesz potentials of functions in Lp, p = n/a with an integer a.
We mention the paper by Strichartz [2] as well, who studied the Riesz (and Bessel) potentials of
functions in BMO. Chanillo [2] considered commutators b(x)(Ia I) (x) - Ia(bI)(x), 1 e Lp, with
be BMO. In this connection see alao Komori [1] for the estimate
IIbI a 1 - 1 I a bllHP  cllbllH" IIIlIr,
1 1 1 a
-=-----
p q r n
In the case n = 1 the conunutator bDa - Da b was shown by Murray [1] to be bounded in L2 if
and only if Da b e BMO. The discussion of such iterated conunutators may be found in Murray
[2].
25.4. The analogue of the Sobolev Theorem 25.2 for the mixed norm space Lji(R n ) as in
the definition in  24.4 and 24.12, was given by Benedek and P&IlZone [1]. fa is bounded from
Lp(Rn) into Lq(RR), p = (Plt.",Pn), q = (ql,...,qn), if 1 < Pi < n/a and l/qi = I/Pi - a/no
A more general assertion in this direction was obtained by Lizorkin [7], who considered the wide
class of convGlution operators, including in particular the anisotropic Rieu potentials
J [p(x - y)]a-n Ip(y)dy
R"
(29.6)
where p(x) is a non-negative a-homogeneous function of degree 1 (i.e. petal Xl,"" ta"Xn) = tp(x)
for t > 0 with aj > 0, L: aj = n), non-vanishing as Ixi = 1. Lizorkin proved the boundedness of

 29. ADDITIONAL INFORMATION TO CHAPTER 5
589
n
the potential (29.6) from Lp(Rn) into Lq(Rn) with 1 < Pi  qi < 00, a = L: ai(l/pi - I/Qi),
i=l
::r :h:n::: ::: :.: :::1(: 3] ( ;0 :;a ) re;/ =:at=
J=l
situation of the weighted mixed nonn spaces and on subspaces of different dimensions was proved
by KokiJashvili [1]. In this connection we observe that the Sobolev type different dimensions
theorems for Riesz potentials were first obtained by D'in [2]. Besov, D'in and Nikol'skii [I, p. 34]
give such a theorem in the general anisotropic case.
We mention alao Adams and Bagby [I], where some cases of estimates for the usual
(isotropic) Riesz potential operators from Lp into Lf are specified.
25.5. The continuity of potential type integrals
19lp = f Ix - tla-nlp(t)dt, Ip e Lp(O),
{}
in the case of a bounded region 0 e R n and l' > n/a was first noted by Sobolev [2, p. 48].
Moreover (Sobolev [3, p. 256]),
{ Ha-nlp (0),
Ig'P e H1,l/p' (0),
H1(0),
if 0 < a - n/p < I,
if a - n/p = 1,
if a - n/p > I, 1/1' + 1/1" = 1.
(29.7)
The definition of the space H>..,k (0) in the one-dimensional case is in  1 (Definition 1.6). The
case of the kemel A(x, 11) Ix -lIl a - n with the first factor Holderian in x, but under the particular
value l' = 00, was treated by Mihlin [2, p. 196]. There is a similar result for 0 = Rn, close to
(29.7). The Riesz potential JOIp with Ip e L1(R n ) n Lp(Rn), 1 < l' < 00, and n/p < a < 1 + n/p
is a Holderian function: lalp e h.nlp (R n ) = {g : g(x + h) - g(x) = o(lhl a - nlp ) unifonnly in x
as Ihl - o} (00 Plessis [1]). This result was extended to certain convolution operators by Cotlar
and panzone [1] and to more general operators by Kilbas [9].
The estimate (29.7) for the integrals 19lp in the case a - n/p > 1 is specified as follows:
19lp e Ha-n/p(o) if a - 7'1./1' ;: 1,2,... and 19lp e Ha-nlp,l/,,' (0) if a - n/p = 1,2,.... This
is a corollary of the generalized Holderness of the potentials more general than 19lp (Kilbas [12]).
25.6. In the case of radial functions the expression
o
(1: Ip)(x) f  n l ( ". ) f Ip(ly I) dy = -2- f pa-1U ( El ) Ip(p)dp, 0 < a  00,
,... Ix - IIl n - a "Yn(a) P
IYI<o 0
holds, where U(,\) = ;'(::;) -#'1 (T' 1- y; y; ,\2) (Rubin (17). Theorem 25.3 has the essential
specification for radial functioD8. Let Ba be the ball in Rn centered at the origin and with radius
a, 0 < a  00. The following theorem holds (Karapetyants and Rubin [1] if n = 2 and Rubin [17]
in the general case).
Theorem 29.1. Let Ip(!xl> e Lp(Bo; Ixl"), 1 < l' < 00, v < n(p - 1), 7'1.  1, 0 < a < n,
o  m  a, m < 1/1', Q = 1'(1 - mp). Theft IIlglpllLp(B.iIZll'o)  cIlIpIlLp(B.ilzll')' with
VO = q(mn - a + v/p) if v > ap - 7'1. aftd vo = q(mn + e - np) if v  ap - 7'1., e> 0, ezcept the
cale v  ap - 7'1. if CI = 00.

590 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
The Fool of this theorem is based on the representation which is of interest itself of the Riesz
potential with a radial density in terms of one-dimensional fractional integration in the radial
variable: namely
(I:'P)(vr> = 2-Otrl-R/2(12.(R-0t)/2-1 t;!'P(VT»(r), 0 < r  a,
where 12 and t;! are the Riemarm-Liouville fractional integration operators given by (2.17)-
(2.18). Rubin [17, 22] is relevant where the inversion of potentials with a radial density and a
characterization of the range were given.
25.7. In the papers by KokiJashvili and Gabidzashvili [1] and Gabidzashvili [4] and also
the book of Kokilashvili [2, p. 36-45] Theorem 25.4 on weighted est1 ( .mates for the ) z potential
was extended to the anisotropic potentials (29.6) with p(x) = t IXj 12/Otj (Otj > 0,
J=l
j = 1,..., n). Applications to imbedding theorems for general weighted Liouville spaces were
alao given. The limiting case (1',)-1 (EOti)-R = n - Ot may be found in Gabidzashvili [5].
Gabidzashvili [1]-[3] extended Theorem 25.4 to certain potential type operators TOt generalizing
(29.6) in the space X of homogeneous type, and proved the theorem on two-weighted estimate for
TOt. He also gave Koosis-type theorems which allow one to have the weight r by a given weight
p, so that TOt is bounded from Lp(Xjp) into Lq(Xjr) and vice versa. A more general situation
of homogeneous measure space can be found in Kokilashvili and Kufner [1] and [2], where the
conditions on the measure were in particular found for the boundedness within the frames of
Lebesgue, Lorentz and Orlicz spaces.
25.8. The weighted Theorems 25.3 and 25.4 were extended to more general cases involving
the two-weighted estimates from Lp(R'I; pd into Lq(RRj P2), including the weak type estimates.
These questions have recently been developed extensively and essentially (far promoted). The
two-weight problem of weak type inequalities for the Riesz potential was first solved in Sawyer [I],
and more effectively by Gabidzashvili [1]-[3]. As for norm inequalities, the two-weight problem
was completely solved by Sawyer [2], and in other terms by Gabidzashvili and Kokilashvili
[10]-[11]. For various other investigations and generalizations we refer to the survey by Dyn'kin
and Osilenker [1], the book by Kokilashvili [2] and his papers [3]-[9] and the papers by Gatto,
Gutierres and Wheeden [1], Harboure, Macias and Segovia [I], [2], Stromberg and Wheeden [I], [2],
Heinig [1], [2], Andersen [2], Chanillo and Wheeden [1], Ruiz and Torrea [I], Gabidzashvili [4]-[6],
Kokilashvili and Gabidzashvili [2], Gabidzashvili and Kokilashvili [1], Gabidzashvili, Genebashvili
and Kokilashvili [1] and Fofana [1]. The reader can also find other references in these papers.
The Muckenhoupt-Wheeden weight condition (25.41) is not always easily checked, so
Abdullaev [1] gave the weighted boundedness conditions in the case when the set of singularities
of the weight functions Pi and P2 satisfied certain assumptions and Pi and P2 were functions of
the distance from this set.
25.9. Mapping properties of the operator lOt within the frames of the Hardy spaces HP (RR),
o < p < +00, which coincide with Lp(RR) in the case 1 < p < 00 up to the equivalence of
norms, were investigated by Stein and Weiss [2]. Their results were developed by Krantz [I], for
operators more general than lOt. The case of certain generalized Hardy spaces can be found in
Taibleson and Weiu [1] and Younpheng Hang [1]. Two-weighted estimates of the Riesz potentials
in the weighted Hardy space HP(R'I; p) were given by Stromberg and Wheeden [I], [2] and Gatto,
Gutierres and Wheeden [2].
25.10. Sobolev's theorem 25.2 was extended to the generalized Riesz potentials
Ot f 0(11)
lo,'P = Iyln-Ot 'P(x - y)dy
R"

 29. ADDITIONAL INFORMATION TO CHAPTER 5
591
with O(y) = O(y/lyl) e Ln/(n_a)(Sn-d by Muckenhoupt [1]. In this paper the limiting cases
l' = n/a and l' = 1 were also considered. In particular, the one-dimensional Zygmund-Flett result
(see  4.2, note 3.2) was extended to the n-dimensional case:
{ } IIp
1 1(I81p)(x)I'dx  e 1[1 + IIp(x)l)pog(1 + IIp(x)l)]l- aln dx,
B B
where B is a ball in Rn. Another proof for 0 == 1 in the cases l' = n/a and l' = 1 was given by
Hedberg [I), who also proved the nmltiplicative inequality
1I[alplir $ elllpll-'lIlnlpll:, Ip(x)  0,
where 0 < a < n, 0 < 0 < I, 1 < l' < 00, l' < q $ 00, l/r = (1 - 0)/1' + O/q.
A more general case of O(y) e Lr(Sn_d with r  n/(n - a) was considered by Muckenhoupt
and Wheeden [I), who proved the estimate IIlxlll(lglp)(x)lI, $ ell Ixlll Ip(x) lip provided that
o < a < n, 1 < p < n < a, l/q = q/p - a/n and a + max(-n/p, -1/1' - (n - 1)/r') < #J. <
-a + min(n/q',I/q' + (n - 1)/r').
The problems of inversion of the operators [8 were treated by Samko [21], [28] in the elliptic
case and [19] in a special non-elliptic case.
25.11. The mapping properties of the operator [01 in the Orlics space LM(R n ) were
investigated by O'Neil [2]. O'Neil's theorem fonnulated in  9.2, note 5.9 for the one-dimensional
case is valid for n  1 also if in the restrictions on M(u) a is replaced by a/no Kokilashvili and
Krbee [1]-[4], and also Kokilashvili [2. p. 78-82], extended the weighted Theorem 25.4 for Riesz
potentials [a to Orlicz spaces, and Kokilashvili and Krbec [3]-[4] gave applications to imbedding
theorems for weighted Sobolev-Orlicz spaces. Theorem 25.4 was also generalized to the case of
multiple fractional integrals in weighted Orlicz spaces - Kokilashvili [4].
O'Neil [1] established the analogue of the Sobolev theorem for the Lorentz spaces Lp".
Flett [9] considered the case of Lorentz spaces with weights which are powers of a function Ip(x)
satisfying the condition m{x : Ip(x)  y} $ cy. The general case of Muckenhoupt-type weights
for Lorentz spaces can be found in the papers by Kokilashvili [2, p. 64-71], [5], [6]. Mapping
properties of the Riesz potential in Morrey spaces were considered by Adams [1].
25.12. The Sobolev-Orlicz spaces L)(Jl'I) and E)(Rn) of fractional order a =
(a 1, . . . , an), a j  0, generalizing the Liouville spaces L a) (R n ) and consisting of functions
f e LM (EM)' which have partial Liouville derivatives of order aj in each variable belonging to
LM (EM respectively), were investigated by Chuvenkov [1].
25.13. Connections between Besov spaces (their definition may be found in Nikol'skii [6] or
Besov, n'in and Nikol'skii [1]) and fractional differentiation was revealed by Lizorkin [4], [9] who
had shown that Bessel fractional integro-differentiation aa - (27.8) - realizes the isomorphism
between Beeov spaces: Ga(B;,') = Btr, 1 < l' < 00, 1 $ 0 $ 00, 1 $ r < 00. Herz [I, p. 315]
showed that a similar statement is true for the Riesz fractional integration: [a (A r , ) = A a+ , r
p, p,
under the appropriate interpretation of the Besov space. That is A;" is defined as the completion
in the Besov nonn of the space of infinitely differentiable functions, the Fourier images of which
have a compact support not containing the origin. The obvious relation to Lizorkin space () is
evidently seen here.
25.14. The Riesz potential arises naturally in the theory of Radon transfonn - Helgason [2,
p. 20, 29 etc].
We also observe that Bredimas [6] used one-dimensional fractional integration in the problem
of inverting the spherical Radon transfonn.
25.15. The modification of the unilateral Riesz potentials (25.71) adapted to the case of a

592 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
half-space, namely
01 Cn 1 :1:.. 1 (Xn - Yn)OI
lo+<p = r(0I) Ix _ yln <p(y)dy,
° R..-l
xeRt.'
and coinciding with the Riemann-Liouville integral (5.1) if n ::: I, was considered by Rubin [20].
In particular, it was clarified whether the Riesz potential/Ol<p, <p e Lp(Rn), coincides with the
unilateral potential I+ 'fjJ in the half-space Xn > 0, where 1/1 e Lp also. This is the case, if
01 < lip, and then one can explicitly construct the operator A bounded in Lp(Rt.) such that
(lOI<p)(X) = (J+A<p)(x), Xn > o. IT 01 > lip, in order that such an equality is satisfied, it is
. 8; f( :I:' , 0 ) _ , _ ) . _ ]
nece888l')' and suffiCIent that (j J - 0, X - (Xl,"" xn , 1 - 0,1,..., [01 - lip under the
:1:..
appropriate interpretation of the trace.
Similar modifications
B OI Cn-I 1
+<p = r(0I)
IYI<I:l:1
(lx12 _ ly12)01
I In <p(y)dy,
x-y
01 Cn-I 1
B_<p = rea)
IYI> 1:1:1
(lx12 _ ly12)01
I In <p(y)dy,
x-y
of unilateral Riesz potentials, connected with a sphere in R n were introduced in Rubin [19], [22],
where the following results were obtained: i) the semi group propertYi ii) inversion fonnulae given
below in the case 0 < 01 < 1 for simplicity, namely
(BOI)-l 1= r(n/2) I(x) + aCn_I 1
+ r(n/2 - a) IxI201 r(l - 01)
1111<1:1:1
I(x) - I(y) d
(l x l 2 - lyl2)0Ilx _ yln y,
(B OI )-l 1= OICn-I 1
- r(1 - 01)
IYI>I:l:1
I(x) - I(y) d
(lyl2 _ Ix12)0I1x _ yln Yi
Hi) the representation of the Riesz potentiallOl<p as a composition
1 01 <p = 2- 01 B/2IYI-0i B/2 <p
d. (25.76), and also Rubin [28].
25.16. The radial limits of the Riesz potential were studied by Mizuta [I], who showed that
if <p e L p (Jr1; Ixl-.6) with {3  0, p> 1 and ap + (3 < n, then there exists a Borel set E C Sn-l
with the Riesz capacity cOIp(E) = 0 such that lim (JOI<p)(rO') = 0,01 > 0, for each 0' e Sn-I \ E.
r-oo
This question in a more general setting is discussed in Kurokawa and Mizuta [1].
25.17. Kurokawa [1] proved that the Riesz (and Bessel) potentials approximate the identity
operator as 01 - +0 both almost everywhere and in the Lp-nonn. Namely, i) let 1  P < 00
and I e Lp(Rn), then 1 01 1 converges to I at each Lebesgue point of Ii ii) let 1 < q < p and
I e Lp n Lq, then 1 01 1 converges to I in the Lp-nonn.
25.18. The periodic analogue of the Riesz potential, i.e. the operator generated by the

 29. ADDITIONAL INFORMATION TO CHAPTER 5
593
multiple Fourier expansion of the form
[0I<p "-# L <Pklkl- Ol e ib
Ikl;tO
where k = (kb"" k n ), a > 0, 0  Xi < 21r, i = 1,... ,n, was considered by Cheng Min-teh and
Chen Yung-ho [1] (1956), [2] (1957) and Wainger [1] (1965). The one-dimensional case may be
seen in  19.3. Certain developments can be also found in Cheng Min-teh and Deng Dong-gao [1]
(1979).
25.19. The Riesz-type generalization
:r0l<p = bn(OI)]-l f <p(x 0 y)IIOI-ndy,
R"
+
i;Y = (lnYb" .,lnYn), Rt. = {y ERn: Yl > 0,..., Yn > O}
of Hadamard's constructions (18.42)-(18.44) was considered by Emgusheva and Nogin [1] together
with the corresponding Riesz-type fractional derivatives in Rt. which are invariant relative to
dilatations.
25.20. Let lex) be locally Lebesgue integrable in R n and let g(x) = [Oil be the Riesz
potential, 0 < a < n. The inequality
f I ( ) ( )I d < h f Il(x)ldx
hn 9 x - 9 Xo x - C Ix _ xoln-OI(h + Ix - xoD
Ix-xol<h R"
1
holds, which is obtained by direct estimations, c depending only on a and n (Samko). This easily
yields the convergence of the left-hand side to zero as h - +0 under the sole assumption that
(IOIllD(xo) is finite. This proves Love's conjecture (see  17.2, note 12.6). Love himself developed
proofs for the case n = 2, but they were lengthy and he noted the fact that this conjecture can
also be proved by means of Theorem 1.11 from Landkof [1].
26.1. In connection with Theorem 26.3 we observe that the truncated Riesz derivative D 1
of the function 1 E [OI(L p ), 1 < p < n/OI, converges as e - 0 not only in the norm of Lp(Rn),
but almost everywhere too. This may be derived from (26.40) by means of an approach already
known, that is, via comparison of the approximation of the identity with the maximal function -
Stein [2, p. 77-78]. There exists, however, the explicit estimate obtained from (26.40) by direct
means (see Samko [20] or [31, p. 78]). Thus
I(D J)(x) - <p(x)1  Cl sup «t,(x) + c2\11e(X) + e'. -0I(c31<p(x)1 + qll<pIDp, 1 = [01 <P ,
0<'<£
which yields the almost everywhere convergence. Here 1* is the same as in (26.39),
«t,(x) = en f l<p(x - y) - <p(x)ldy,
IYI<'
1
\II,(x) = e'. -01 f tOl-'. -l«t,(x)dt,
£
and Ci (i = I, 2, 3, 4) are absolute constant. The almost everywhere convergence of the
hyperaingular integrals Dol with a characteristic O(y) not necessarily homogeneous was
investigated by Nogin [8], [11].

594 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
26.2. The hypersingular integral (Dof)(x) with a homogeneous characteristic O(y/lyl)
of functions sufficiently smooth and vanishing at infinity, admits the following estimates. Let
f(x) e c'(Rn), I> a, and If(x)1  e(l + Ixl)-N 1 , I(vj f)(x)1  e(l + IxD-N, for Iii = 1 with
NI > a, N'J. > n. Then
I(Dof)(x)1  e(l + IxD-min(a+Nl,N,I+a)
if 0(0") e LICSn-d, and
I(Dgf)(x)1  e(l + Ixl)- min(a+NltN,n+a)
if 0(0") is bounded (Samko [26], [31, p. 89)).
26.3. The continuity modulus, of fractional order in general, w-y(J,6) = sup 1I'Yfllx,
I h l<6
X = Lp(Rn), 1  p < 00, or X = BC(Rn), of the hypersingular integral DOf is estimated as
follows:
6
( D a f 6 ) < eb ( S ) w-y(J, S) + e f w>.. (J, t) a ( t ) dt
w-y {}, - sa t 1 + a '
o
00
where aCt) = f O(tO")dq, t > 0, b(S) = f t- 1 - a a(St)dt, "y > 0, 0 < a < .x  I. Here 1 is the
5-- 1 1
order of the difference, defining Dgf. In the case 0 = const both aCt) and b(S) are constants and
.x may be chosen any such that .x > a (Samko and Yakubov [4]; the case of integer "y and .x and
X = C can be found in paper [2] by the same authors).
26.4. In the case of the characteristic O(y') equal to the spherical hannonic Ym(y') the
hypersingular integral Dgf admits the simple expression in terms of D a - m f:
Dy...f = .xYm(V)D a - m f if a  m,
D y ... f = .xy m(V)[a-m f if a  m,
where .x is the constant depending on a, n, m and on the type of the hypersingular integral
(Samko [26], [31, p. 90)). Compare these formula with the representation (26.83) in the case
a = m.
The paper by Horvath, Ortner and Wagner [1] is also in a sense relevant. In it there were
considered distributional convolutions with the kernels en a jxi Ixl- j + a - n , the power xi being
taken in the certain graded algebra associated with the Lale operator, 80 that the components
of xi with respect to the canonical basis of the algebra are hannonic polynomials Yj(x). See also
Ortner [2].
26.5. There exists a characterization of the space [a(L p ), 1 < p < n/a, not in
terms of Da f as in Theorem 26.8 and 26.9, but in terms of Strichartz's constructions
j t- 2a - 1 ( f 1 (:Yf)(X)ldY ) 2 dt. This characterization is given in a more general context of
o IYI<1
the Lorentz spaces Lp,q - Bagby [3].
Added in proofs. The characterization of the range [a(H p ) in these terms, Hp being Hardy
spaces, 0 < p < 1, was given by Strichartz [3].
26.6. IT we discard the condition f e Lq(R n ) in the characterization of the space [a(L p ),
(see Theorems 26.8 and 26.9), we are in a situation when f(x) "contains" a polynomial, because
the function f(x) is detennined by the given semi norm IIDa flip omy. The thorough investigation
of the question how does the function f(x) "turns into" a polynomial at infinity with dependence
of the defining seminorm, was undertaken by LizOI-kin [to], [11]. In this connection we observe

 29. ADDITIONAL INFORMATION TO CHAPTER 5
595
that the space IOI(L,,) with 1 < p < 00, p  nia may be treated as a subspace of the quotient
space S' I pOI-nip where S' is the Schwartz space of temperate distributions and pOI-nip is the
subspace in S' consisting of polynomials of a degree not exceeding a - nip. For p = 2 we refer to
Pryde [1] and to Davtyan [1], [2] in the case 1 < p < 00. Davtyan considered a more general case
of the non-isotropic Riesz potentials. This assertion for isotropic potentials was known earlier,
but the authors cannot find the cOlTesponding reference.
The paper by Kurokava [2] is also relevant. There the author dealt with the case of an
integer Ot and investigated the representation of the function I in the "Beppo Live spaces"
{I e 1)' : 1); I e Lp, Ijl = a} in the form I = P(x) + Iklp, Ip e Lp, where P(x) is a polynomial
and I k r.p is a modification of the Riesz potential. The latter is defined by the kernel obtained
from the usual Riesz kernel kOl(x - y) by subtracting its partial Taylor sum at the point (-y).
Kurokawa [3] gave also some weighted Lp-norm inequalities for the operator I k . In this connection
a similar ErdeIyi modification of the one-dimensional Riemann-Liouville fractional integral must
be mentioned (see  9.2, note 5.6).
26.7. Samko [20] considered hypersingular integrals with the characteristic {1 (y) which , in
contrast to the homogeneous case as in (26.47), stabilizes as y - 00 and y - 0, and cleared up
the influence of such a characteristic on the convergence of the hypersingular integral. This is
question B from  26.4. As in  26, this was done in the Lp,r-setting also, i.e. I e Lr while the
convergence of DOl I is considered in Lp. The same question B with more general characteristics
including both homogeneous and stabilizing ones was considered by Nogin [1], [11]. The latter
author's investigation generalizes the results by Wheeden [5] concerning the case r = p, but
Wheeden's assumptions on the characteristic were weaker.
26.8. Functions I(x) e 100(L p ), 1 < p < nlOl, have all the weak (generalized) derivatives
of integer order IPI  a and 111),6 Ilir  c111DI,61 Ilir  c211DOI I lip , where l/r = lip - (01 - IPD/n
(Samko (17), [31, p. 109]).
26.9. The space Lp,r(R'I) treated in  26.7 coincides with the Sobolev-type space Wr
in the case of an integer 01, the latter consisting of functions I e Lr such that 1); I e Lp (Rn ),
Ijl = a (Samko and Umarkhadzhiev [1], [2] and Samko [31, p. 112]). Hence we can note as a
corollary that
100(Lp) = {I : Ie Lq, 1)j Ie Lp, Iii = a},
1 < p < nlOl, q = npl(n - Otp), Ot = 1,2,...,n - 1.
In the same papers the characterization of the space 100(L p ), ° < a < nip, was given in terms of
the higher derivatives: I(x) e 100(L p ) if and only if
Ie Lq, 1); Ie Lr, Ijl = 01, D OI -[OI]1)j Ie Lp,
where l/q = lip - Olin, l/r = lip - (01 - [OI])/n.
26.10. The following coordinate characterization of the space 100(L p ) involves infonnation
on the behaviour of function I(x) e 100(L p ) in each variable Xi, i = 1,2,..., n (Nogin and Rubin
[3]). Let
(1!XIp)(x) =  f Ip(x - te;) dt,
· ,on (01) It1 1 -0I
Rl
ei = (0,...,0,1,0,...,0),
'--v-'
i-1
be the partial Riesz potential in the variable Xi and let
Ir(L p ) = {I : I = Irlp, Ip e Lp(R n )}, II/lIrr(L p ) = IIlpllp, i = 1,2,... ,n.

596 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
n
Theorem 29.2. Let 1 < p < 00 and 0 < Ot < lip. Then [OI(L p ) = n [f(L p ) and the norm,
i=l
n
IIf1l10(L ) and L IIflll(L ) are equil1alent.
., i=l · P
An extension of this theorem to the case of anisotropic Riesz potentials of the type (29.6)
was given by Emgusheva and Nogin [2]. In fact, they dealt with a more general situation,
considering the anisotropic spaces
L:. r = {n,,) , II/lId II r'  I(i la; (F J)(() II. < 00 }
li = (Otl,"', Otn)  0 and characterizing it in tenus of Riesz derivatives both in the whole space
R n and in each variable separately, or in various unifications of variables.
The space L:,r was in fact considered earlier by Davtyan [2], [3], who used the anisotropic
hypersingular DOl f, introduced by Lizorkin [6], to define this space as L:,r = {f : f e Lr, Da f e
L p }. The latter is an evident generalization of the space (26.91). Davtyan showed that
L:,r = Lr n [ci(L p ), where [ci(L p ) is the range of te anisotropic Riesz potential of the
type (26.9). He also proved the inversion theorem DOl [a f = f, f e Lp, 1  l'  niOl.,
1/ 01 - = n- 1 LOt;l.
26.11. The characterization of the weighted space [OI[L p (Rn j p)) of fractional integrals
with weight p(x) satisfying the Muckenhoupt-Wheeden condition (25.43) was given by Nogin
and Samko [4]. Thus f(x) e [OI[L p (R n j p)], 1 < l' < nlOl, if and only if f(x) e Lq(p q1p ) with
l/q= IIp-Otlnandeitherthereexists lim DlffinthenormofLp(Rnjp)orIlDlffIlL ( R"' p) < c.
£ -0 ., , -
In the case n = 1 this statement was obtained by Andersen [1]. A more general situation for the
spaces L;,r (PI, P2) generalizing the spaces (26.91), PI e Ar and P2 e Ap being Muckenhoupt-type
weights was studied by NOlin [9], [12]. The case of power weights was previously considered by
Nogin [6].
26.12. The following theorem on the simultaneous approximation of functions and their
Riesz derivatives holds. Every J-nction f(x) with the finite norm
IIflir + II DOl flip
(29.8)
can be approximated by Co (Rn)-J-nction, with re,ped to this norm, 1 < l' < 00, 1 < r < 00 and
a > O. We refer to Samko [17,  5] if 1 < l' < nlOl, Nogin and Samko [2] if Ill' - Olin  l/r  Ill'
and Nogin [5, p. 28] in the remaining case.
26.13. A function ,,(x) is called a multiplier in the space X, if "f e X and lI"fllx  cllfllx
for all f eX. In order that a bounded function ,,(x) to be a multiplier in the space [a (L p ) of
fractional integrals, 1 < l' < nlOl, it is necessary and sufficient that the operators
N£1j) = f Ne(x,Y)Ij)(y)dy,
R"
1
Ne(x, y) = L () f Itl-n-OI(r #J)(X)l_II,OI(X - Y - vt, t)dt, I> a,
11=1 Itl>£
where m,OI(X, h) is the kemel (26.32), should be bunded in Lp(Rn) unifonnly in e. The
conditions which follow are sufficient: 1) J Itl-n-OII1:"lInl OIdt < 00, 2) for each i = 1,... ,1- 1
R"
there exists a number OIi such that 0 < OIi < min(OI,I- i) and J Itl-n-OI+OIall"lIn/(n_OIa)dt <
R"

 29. ADDITIONAL INFORMATION TO CHAPTER 5
597
00. In particular, functions ,,(x) e C>"(Rn), .x > a, are multipliers in [a(L p ) (Samko [24]).
{ I, x e 0
The discontinuous function Xo(x) = , where 0 is the half-space or a ball, is a
0, x fl 0
multiplier in [a (L p ) if 1 < p < 1/ a, 0 < a < 1. The case of the region 0 satisfying the se>-called
Strichartz condition may be found in Strichartz [1] for Bessel potentials, and in Nogin and Rubin
[3] for Riesz potentials.
26.14. The operator of multiplication by functions of the fonn e(x)(1 + Ix1 2 ) -11/2, II > 0, is
a compact operator from the space [a (L p ) into the space [a-e (L p ) provided that 0 < e < a and
II > e, c(x) e C>'(Rn), .x > a - e (Umarkhadzhiev [1]).
26.15. The Riesz derivative Da I as the limit (26.26) was treated by Emgusheva and Nogin
[1] in a more general setting with integration in (25.60) over R n \ G e instead of {t : It I > e}, where
G e is an arbitrary neighbourhood of the origin with the only assumption that m(K n G e ) - 0
as e _ 0 for every compact set K in R n . We observe that G e may thereby be unbounded. In
particular, it was shown that existence of D a I does not depend on the choice of G e .
1
26.16. The generalized Marchaud-type differences L: ci/(x - kiY) ( 9.2, note 5.11) may
i=l
be used in multidimensional Riesz differentiation Da I as well. This was done by Kuvshinnikova
[1] together with an application to inversion of some multidimensional potential-type operators.
26.17. In connection with Theorem 26.10 and the assertion (26.97') we mention the
following statement of Samko [34]. Let I(x) e Lr(R n ), 1 < r < 00. Then lim D I exists in
e-O
Lp(Rn), 1 < p < 00, if and only if II(E - pt)a/ll"  eta, t e R, where Pt is the Poiuon
operator (25.47). H the limit exists, then
II(E - Pta)/lIp  AtaliD a I lip
and lIhlllp  elhlaliDalllp, h eRn.
26.18. We note that the apparatus of hypersingular integrals DOl with homogeneous
characteristics and the technique developed in  26 were essentially used in a recent paper by
Kochubei [1] in constructing and investigating the fundamental solution of the Cauchy problem
for the periodic pseudo-differential equation
m
Ut(x, t) + L(Atu)(x, t) = I(x, t), x e R n , t e (0, T]
t=o
where At are pseudo-differential operators with symbols at (x, t,) homogeneous in respect to (.
The fractional spherical parabolic equation tt + (_6)a/2 u = 0 with the Beltrami-Laplace
operator 6 was considered by Liu Gui-Zhong [I], who in fact characterized the subspace of
solutions in L2(sn-l).
The investigation of the initial value problem for the non-linear equation Ut = (la u ) . u
can be found in Ponce [1].
27.1. The following Bernstein-type inequality
II(E - )a/2gllp  c(1 + (l"2)a/2I1gllp
holds for Bessel differentiation (E - )a/2g = F- 1 (1 + 112)a/2Fg in the case of functions
9 e £p(R") with Fowier transfonns supported in the cube Ijl  (1", j = 1,...,n. Also, the
Bohr-Favard-type inequality
II(E - )-a/2gllp  e(1 + (l"2)-a/2I1gllp
is valid, where G a = (E - )-a/2 is the Bessel fractional integration operator and the functioD8

598 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
9 e Lp(Rn) have Fowier transforms vanishing in the same cube. The constant c in both cases
does not depend on g(x) (Lizorkin [5]).
27.2. The .Ri,otropic Bellel potentiGI is defined as
GOIr.p = f GOI(X -1I)Ip(y)dy, 01 = (OIb'" ,an),
R"
where GOI(X) is the kernel with the Fowier transform
GOI(X) = [1 + p2(x)r Ol ./ 2 ,
n
1 1 L 1
-=- -,
a. n aj
j=1
n
p(x) being the positive solution of the equation L: x]p-2IJ j (x) = I, aj = a.faj' The space
j=1
GOI(L p ) of functions representable by the anisotropic Bessel potentials GOIIp, Ip e L,,(R n ),
coincides with the Sobolev space LOI)(Rn). This consists of functions f e Lp(Rn) with Liouville
fractional partial derivatives 8 01 ;ff8xj; also belonging to Lp(Rn). Moreover, the space GOI(L p )
is characterized in terms of the corresponding anisotropic hypersingular integral (Lizorkin [8]).
The case a1 = ... = an was presented in  27.3.
27.3. We indicate here the integral operators which are similar by their nature to Bessel
potentials, but have a unilateral character similar to the potentials (25.71). We denote
G%1p = g _Ip,
% ( X ) = d n -1 (Xn)%K n / 2 (lxl) e L ( R n )
gOi rea) Ixln/2 1,
where d n -1 = 21-n/21r-n/2, a > 0 - the unilateral Bessel pot.entials, g) = ( V I + 112 - a T
in)-OI. Similarly to (27.22) the equality
00
G%1p = r;a) f y::-1 (My..Ip(', Xn T Yn»(x')dYn
o
holds and similarly to (25.76) the representation G OI = G:/ 2 G;/2 is valid. Here
f Kn/2( V ly'12 + ya) , ,
(My"lp(-, Xn - Yn»(X') = dnYn (lY'12 + ya)n/4 Ip(x - Y , Xn - yn)dy.
R..-l
The operators inverse to G% are realized as hypersingular integrals wit.h weighted differences
as at the end of  27.4), thus
(G%)-1 f = x:, I) f I: (yf)(x)dy, I> a,
R+

 29. ADDITIONAL INFORMATION TO CHAPTER 5
599
where
I
(f)(x) = f(x)ey,.IYln/2K n / 2 (IYI) + L () (_I)I:(klyl)n/2 Kn/2(klyl)f(x - ky).
1:=1
Theorem 29.3. Let a > 0 and 1 < p < 00. Then G%(L p ) = L, and the norm, IIG%""IIL; and
1I""lIp are etl'u.i11t.&lent. Let
00
""; = f f Iyl-ny;a(yf)(x)dy, I> a.
e R"
Then f e G%(Lp), 1 5 p 5 00, if and only if lim o ""; e Lp(Rn). If f e G%(L p ), 1 < p 5 00,
e-
(L p )
then lim ",,'t ezisb almo,t everywhere 01.0.
e-O
Besides the operators G%, one may consider unilateral potentials of the fonn
Cn-I f (Yn)%e-IY,,1
rea) Iyln ",,(x - y)dy,
R"
r(n/2)
C n -l = 1r n / 2 .
(29.9)
Their action in Fourier images is reduced to multiplication by (1 + I'I T in)-a (d. (25.75) with
e = 1). In the case n = 1 these potentials coincide with the operators (E  V)-a considered in
 18.4.
Operators G% defined via Fourier images are found in the book by Triebel [1, p. 282], where
the mapping property G : L, - L;+P was proved for them. fu explicit fonn the potentials
G%"" were given in Rubin [20], [24], where in particular, the inversion of these potentiaJa and
Theorem 29.3 were proved. The potentials (29.9) were considered in Eskin [1].
27.4. We note the characterization of the space Ga(L p ) of Bessel potentials in tenus of
"tnmcated" integrals (27.46), (27.54) and (27.55).
Theorem 29.4. Let 1 < p < 00, a > 0, a '# 2,4,6,.... Then f(x) e G a (L p ) iJ and only if
Vi f e £p, 1,;15 lal, and one of the Jollowing conditions is sati,fied:
i) the .equence Dt..e f of integral, (27.46) converges in Lp -norm;
ii) sup IID e fllp < 00.
e>O ,
We observe that hypersingular integrals D J have a certain advantage in comparison with
the Rieaz derivatives Da used in Theorem 27.3. That is, they contain the characteristic .xa(lyl)
exponentially vanishing at infinity.
Theorem 29.5. The 'pace Ga(L p ), a > 0, 1 5 p < 00, con,ist, oj tho,e and only tho,e
J-nction. f(x) e Lp Jor which the ,equence  f converge, in Lp-norm, where  f i, one oj
the integral. (27.54) and (27.55), or the truncation oj the integral (27.56).
The former of these theorems was established by Nogin [9], the latter by Rubin [23], [25].
27.5. The presentation of the Bessel potentials theory from the point of view of fractional
powers of operators can be found in Fisher [3)-[6]. We mention also the paper by Fujiwara [2],
where the domain of the fractional power (E - )a/2 in the half-space (under the third boundary
condition) was characterized in tenus of the interpolation spaces [H2,P(,>, v (Ri'>]. The

600 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
previous paper Fujiwara [1] is also relevant. Purely imaginary fractional powers of an elliptic-type
differential operator of the second order were treated by Fujiwara [3].
27.6. Ginzburg [1] studied trace problems in certain weighted Bessel potential spaces in the
strip {(x, y) e Rn+l : x e R n , 0 < y < b}, defined via Bessel fractional differentiation in x and
nonnal differentiation of the first order in y.
28.1. Functions I{) depending only on the Lorentz distance p() = V -  - ... - , the
Lorentz-invariant functions, admit the expression for the Fourier-Laplace transfonn similar to the
Bodmer result (25.11), namely
00
f . (2 )(n-2)/2 f
e,.zl{)[p2«()]d( = ; (-2)/4 l{)(t)t(n-2)/4 K (n_2)/2( V- tp2 (z» dt.
[_p (z)] n
Ra 0
There the function I{)(P()] is supported in the light cone K+ = { : p2() > O'l > O} and
z = x + iy, x eRn, y e K; = {y : p2(y) > 0, Yl < o} and KII(z) is the McDonald function,
and it is assumed that the integral in the left-hand side converges absolutely (Domingues and
Trione [1]). In this connection the book by Leray [1] is relevant, where the expression for the
Fourier-Laplace transfonn of Lorentz-invariant functions was given in terms of a composition of
the one-dimensional Fourier-Laplace transfonn with fractional integration, this being similar to
(25.14).
28.1'. Kipriyanov and Ivanov [1] introduced the hyperbolic Riesz potential of type
(28.19) within the frames of a general Lorentz space X defined as a smooth manifold in R n
equipped with the Lorentz structure and the metric rzy. This was defined by the equation
ch 2 rzy = [x, y]2[x, x]-l[y,y]-l with [x,y] = XlYl - X2Y2 - ... - XnYn, this also being known as
Lobachevskii space. In fact the authors dealt with the following modification
(101 I)(x) = Hn 1 (0I) f f(y)sh OI-lrzydy
D",
of the Riesz hyperbolic potentials, Dz being the corresponding analogue of the cone (28.18).
They proved Sobolev-type theorem on the boundedness of [01 from Lp(X) into Lq(X) with
1 < p < q < 00, l/q = l/p - OI/n, together with the weak-type estimate in the case p = 1.
28.1". Riesz [7] introduced surface hyperbolic single- and double-layer potentials. The
fonner was given by f g(O')rOl-n(O' - x)ds q with the Lorentz distance, and Sex) a part of a
S(z)
given surface S which is cut off by the cone K;(x) (d. (28.21». Ivanov and Kipriyanov [1]
applied such potentials with 01 = 2 to the Cauchy problem for the wave equation with the initial
data u(x) = hex),  = g(x), xeS.
28.2. The spaces
£;,r = £;,r(R n + 1 ) = {f: IIflir + IIF-l(l12 + i-r)0I/2 F flip < oo},
01 > 0, 1  p < 00, 1  r < 00,
(compare with the spaces L;,r = {f : IIflir + IIF-llIOIFflip < oo}, see (26.91» were introduced
and investigated in Nogin and Rubin [I], [5]. They generalize the spaces HOI(L p ) and 'HOI(L p )
and coincide with the fonner if 1 < p < 00, 0 < 01 < 1 + n/2, r = (n + 2)p/(n + 2 - OIp) and
with the latter if 01 > 0, 1 < p = r < 00. The spaces £,r consist of functions f e Lr(Rn+l)
representable as the potential HOII{) with I{) e Lp(]lR+l):
£;,r = Lr n HOI(L p ), 1  p, r < 00, 01 > O.

 29. ADDITIONAL INFORMATION TO CHAPTER 5
601
They are characterized by the relations
/:';,r = {f : f e Lr, TOt f e Lp}, 1  p, r < 00, Ot > 0,
/:, p Ot r = {f : f e Lr, sup liT: flIP < oo}, 1 < p < 00, 1  r < 00, Ot > 0,
, £>0
where TOt is the operator (28.52) and Tf' f is the truncated integral in (28.52). The spaces
/:',p = 'HOt (L p ) admit a similar characterization in tenus of the hypeninplar integral (28.54).
H 0 < 01 < 2, the characterization of the space 'HOt(L p ) in tenus of hypeningular integrals
with first order differences was also obtained by Chanillo [1] and Nogin [5]. In the case 0 < Ot < 1
Bagby [1] used other means - Strichartz-type constructions - for the same purpose. There is the
characterization of the space 'HOt (L p ) with 01  1 in non-constructive terms - via reduction as in
Gopala Rao [1, 2].
Nogin and Rubin [1], [5] obtained the "separated" characterization of the spaces /:';,r' which
reveals different behaviour of functions f(x, t) e /:';,r in the space variable x and in the time
variable t, thus
Ot { Ot Ot/2 }
/:'p,r = f: f e Lr, Df e Lp, Dc f e Lp, ,
where 1 < p < 00, 1  r < 00, 01 > 0 and
I
D:f= lim - d 1 ( ) f ,",(-I)k( k ')f(x-k1l,t)I1II-n-Otd71
£-+0 n I 01 L.J
(L p (R"+l» , 1711>£ k=O
is the "panial" Rien derivative in x and D/2 f is the partial Riesz derivative in t. Instead of the
space derivative D:f the set of partial coordinate derivatives D:.f, i = 1,2,...,n, may be used
to give the similar "coordinate" characterization of the spaces /:',;. These characterizations were
applied by Nop and Rubin [I], [5] to investigate multipliers in r,,r and establish a connection of
/:'p,r with Sobolev spaces in the case of integer 01. Some auflicient tests for multipliers in tenus of
the spaces C>"(Rn) were, in particular, given in these papers. The wide clue of regions in Rn+l,
containing the clue of so called Strichartz regions was also defined, the characteristic functions
of which are multipliers in /:',r' Multipliers in 'HOt(L p ) were studied by Bagby [1]. Chanillo [1]
showed that functions in Co (Jr1+ 1 ) are multipliers in 'HOt (Lp ).
Nogin and Rubin [1], [5] showed that the space r," m = 1,2,..., consists of functions
f(x, t) e Lp, which have the partial derivatives 8 2m f /8xJm, j = 1,..., n, and 8 m f /8t m in
Lp(Rn+l). In the case of odd 01 = 2m - 1, m = 1,2,..., a similar aseertion was obtained, the
only difference being that smoothness order m - 1/2 with respect to t is fractional. Thus the
partial Rien derivative D';'-1/2 was used. In the e r = p statements similar to above were
proved by Gopala Rao [1], but his results contain superfluous information on mixed derivatives in
the sufficiency part. Nop and Rubin managed to avoid this exCe811 information.
Bagby [1] gave interpolation theorems for the spaces 'HOt (L p ).
We alao observe that the spaces 'HOt(L p ) coincide up to the equivalence of the BOnos with
the anisotropic spaces of functions in Lp which have the generalized Liouville derivatives of order
01 in Xj' j = 1,..., n, and of order 01/2 in t, all beloDling to Lp. This is proved by means of
the Lp-multipliers technique. Such spaces are particular cases of the general anisotropic spaces
investigated by Lizorkin [5], [6], [8].
28.3. As it was shown by Nogin and Rubin [2], the hyperaingular integrals  f and '1'Ot f -
(28.52) and (28.54) - which invert the potentials f = HOtr.p and f = 'HOtr.p, r.p e Lp, respectively,
may be interpreted as the almoet everywhere limits of the corresponding "tnmcated" integrals.

602 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
28.4. Fractional powers of the hyperbolic operator  - J;.s - i 4- were investigated by
Sprinkhuisen-Kuyper [1], [2].
28.5. One more variant of fractional integration is defined by the relation
F(T,a J) = e-!lzIO j(x), cv > 0, t > 0,
in Fourier images. This was investigated by Takano [2], who showed that the operators TOI f
form a semigroup of the clus (CO) in the space L p (Jr1j p), with a Muckenhoupt-type weight p(x)
for every Ot > O. The generator of this semi group is the hypersingular operator DOl. This was
interpreted by Tabno as the closure of the operator F-ilx 1 01 F in the space Lp (Rn j p). The
case p(x) = (1 + 1x12)-m/n, m > n, was considered earlier in Takano [1]. This aemigroup in the
one-dimensional case arOlle earlier in some probability problems (see Kat [1] and Elliott [1] in the
case 0 < Ot < 1 and S. Watanabe [1] in the case 0 < cv < 2).
28.6. In connection with the various fonns of fractional integra-differentiation of functions
of many variables, we mention the case of functions given on the unit sphere in R n . There are
many papers devoted to fractional powers of the Beltrami-Laplace operator on the sphere and
their generalizations and analogues. We refer the reader to the review paper by Sa.mko [30], where
other references can be found. We also cite Samko [22] and Pavlov and Samko [1], where the
direct analogues of Riesz integro-differentiation on the sphere were considered, including spherical
hypersingular integrals. See also Colzani [1] and Vakulov and Samko [1].
28. T . There is a generalization of fractional integro-differentiation to the case of many
variables which involves integration over a cone with the vertex at the point x e Rn instead of
the half-axis (-oo,x) in the one-dimensional case. This is discussed in the book by Vladimirov,
Drozhzhinov and Zav'yalov [1]. The definition is as follows. Below r is a closed convex acute solid
cone in R n ; r. = {( e g& : ( . x  0 "Ix e r} is a cone conjugate to rj C = int r. is the set of inner
points of r.; Sf. is the convolution algebra of temperate generalized functions with support in rj
H(C) is the algebra of functions holomorphic in the tube region TC = {z = x+iy: x e Rn, y e C}
which are Laplace transfonns L(g] = (g(),eiz.) of generalized functions 9 e Sr' The function
K:C(z) = f eiz. d( is called the Cauchy kernel of the tube region TC. The open convex acute
r
cone C is called regular if [K:C(x)]-l e H(C).
The operation of fractional integro-differentiation is introduced as follows. Let C = int r.
be a regular cone. Then K:C(z) = [K:C(Z)]OI e H(C) for any cv e Ri. Let 8r() = L-i[K:C(z)](()
be the Laplace original of the power K:C (z) defined as the characteristic function of the cone r,
if Ot = o. The operator of fractional integro-differentiation is defined as the convolution with the
distribution 8r()' These operators fonn an abelian group with respect to cv.
It can be shown that the tensor product of the one-dimensional Liouville fractional integrals
of the same order, and Riesz potentials with the Lorentz metric are particular cases of the above
operators.
28.8. Another approach to fractional integro-differentiation connected with cones in R n
was earlier developed by Gindikin [I], [2] (see also Vainberg and Gindikin [1]). In contrast to
note 28.7 the order of fractional integration is "multidimensional" here. Let V be a convex cone
in g&, not containing straight lines, such that there exists a group G(V) of linear transfonns
preserving V, with the following property. For all points x and y e V there exists a unique
transfonnation VI e G(V) for which tJI(x) = y. IT we fix a point e e V, then we can transfer the
multiplicative structure of the group G(V) to V: xy = g(x)y, where g(x)e = x, g(x) e G(V).
A function satisfying the condition f(xy) = f(x)f(y) is called a complex power function. Every
such function f, nonnalized by the condition fee) = I, has the fonn f(x) = n [xk(X)]PJl fxp,
k=i
p = (Pb"" PI) e C ' . The number I is called the rank of the cone V, Xk (x) are fractionally
rational functions of coordinates of the point x. The cone V is defined by the inequalities
Xt(x) > 0, k = 1,..., I. Let d/.i be the measure invariant with respect to the group G(V). It is
connected with eudidian one by the relation dJ.l. = xddx, where d = (db"" d/) e R ' . Further, let

 29. ADDITIONAL INFORMATION TO CHAPTER 5
603
ry (p) be the Siegel integral of the second kind, or r-fWlction of the cone V. It may be represented
as a product of usual r-functions. The Riemann-Liouville operator of order P = (PI' .. . ,PI) e C '
is defined as
00
("ylp)(x) = ry1(p) f Ip(Y)(Y - x)p+ddy
:e
where integration is calTied out over the cone (x, 00) consisting of points y such that y - x e V.
In Gindikin [1] a detailed investigation of the operators"y was Wldertaken, and applications
to constructing fundamental solutions of certain differential equations and solving some problems
of integral geometry were given. The Abel equation "ylp = f was considered.
Analysis similar in a sense to the latter was developed earlier by Ga.-ding [1] for the case
of cones of synunetric and Hermitian matrices. Faraut [1] also considered. the case where the
Riesz-type potential was introduced for a symmetric cone in a Jordan algebra and its applications
to the Cauchy problem were given.
The Riemarm-Liouville operators connected with homogeneous cones were also dealt with
in the paper by Watanabe [1].
We observe that the Riemann-Liouville operators "y were used by V.S. Rabinovich [1] in
investigating multidimensional integral equations of convolution type, the symbol of which has
a singularity at infinity of the type of complex power fWlction related to some cone. In this
paper the analogues of the Sobolev-Slobodetskii function spaces were constructed in tenus of
the operators "y. This was applied to the problem of Noether properties for convolution type
equations .
28.9. Let Ac(b) be the pyramid (28.60), and x eRn, k = (k 1 ,.. .,k n ) be a vector-fWlction.
Let us denote k(x) = kJCx).. .kn(xn) and consider the Abel-type equation
f k(A. (x - t»lp(t)dt = f(x), x e Ac(b).
Ac(:e)
(29.10)
which is more general than (28.65). Let the vector-function lex) exist for the given kernel k(x)
such that
f I(A. (x - t»k(A. (t - 1"»dt = 1
U(:e,T)
(29.11 )
where O'(x, 1") is the region (28.64). Then (29.10) is solved by the same method that was applied
to (28.65), and the unique solution of (29.10) is
",(x) = fr ( t oi/ 8:i ) f I(A. (x - t))J(t)dt
1=1 J=1 Ac(:e)
(Kilbas) .
Equation (29.10) is a pyramidal anogue of the SoBine equation (4.1), while (29.11) is such
an analogue of the condition (4.2").
28.10. Beautroux and Burbea [1] considered the "fractional radial differentiation" operator
for holomorphic functions fez) = L::akzk in the unit ball in en defined as DOl f = E(lkl +
l)OI akz k. They introduced the Sobolev-type space A,OI of holomorphic functions with the finite
norm II DOl fll p ,9 where IIIIp,9 denotes Lp-norm with respect to the measure (1 -lIzIl 2 )9- 1 dv(z),
q > o. They proved imbedding theorems with respect to the parameters p, q, 01, and gave

604 CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Lipsbitz-type estimates for functions in A,a' Similar results in the context of more general
Hardy-Bergman-type spaces H,f,f1 can be found in Jevtic [2].
28.11. Ricci and Stein [1] defined fractional integration in the context of nilpotent groupe
by considering convolutions with kernels supported on an analytic homogeneous manifold V in
the group. Typical examples of these convolutions are fractional integration along a homogeneous
curve 11 = z1c, z e R. Ie > 0, or the case when V is the n-dimensional forward light cone in
,R"+l. They investigated the problem of Lp -+ Lq boundedness of such convolutions.

Chapter 6. Applications to Integral
Equations of the First Kind with
Power and Power-Logarithmic Kernels
In this chapter we present applications of fractional integrals and derivatives to
integral equations of the first kind
M<p == f k(x,t)<p(t)dt = I(x), x EO,
o
the kernels k(x,t) of which have a singularity of the type Ix _tl a - 1 , 0 < Q < 1,
or, more generally Ix - tl a - 1 ln m Iztl ' Fractional integrals arise naturally in the
former case, the latter leading to their generalization considered in S 21. The
kernels k(x, t) are assumed to be real-valued for simplicity, while the right-hand
side and the unknown function may be assumed to be complex-valued.
In the case k(x, t) = c(x, t)lx - tl a - 1 the investigation and solution of the
above equation is based on the idea to single out the fractional integration operator
explicitly. Namely, the equation M<p = I generally speaking can be represented in
the form [a N<p = I, N = va M, where [a and va are suitable forms of fractional
integr<rdifferentiation. The equation N <p = g, 9 = va I, obtained after inverting
the Abel equation proves to be an equation of the second kind under rather weak
assumptions on c(x, t). In the case when c(x, t) is continuous at t = x this is a
Fredholm equation of the second kind. If the function c(x, t) is allowed to have a
jump at t = x, this case being of special interest and often occurring in applications,
then the situation proves to be more difficult. The equation of the second kind
which results is of a singular type. The theory of such equations is well known
and we assume the reader to be familiar with the simplest ideas and facts of this
theory. An excellent presentation may be found in the well known books of Gahov
[1] and Muskhelishvili [1]. Nevertheless, we display the main results of this theory
as required in the present chapter in the preliminary subsection S 30.1. In certain

606
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
simple cases for example in S 30.1, we shall manage with minimal knowledge of the
singular integrals considered in S 11.
Some types of singular equations (the "dominant" ones, in particular) are
solvable in closed form, i.e. via quadratures. This will allow us to obtain an
explicit solution - in closed form - of a certain class of the equations of the first
kind considered. As regards the most general case, we shall obtain results which
have a qualitative nature, and characterize the solvability of the equation. Among
these the reader will find the criterion of normal solvability and the formula for the
index (see definitions for these notions in S 31.1).
We call attention to subtlety connected with the representation of the operator
M in the form M = [01 N, which will always be our particular concern in the
investigation of the equations. This representation obviously assumes the identity
M = [01 VOl M to hold, but in general [01 VOl f f; f - see for example (2.60). We
can be sure of the validity of the relation [OIV OI f = f by taking functions f
representable by the fractional integral of order Q as in (2.59). So we must check
the fact that the range of the operator M is embedded into the range of the
fractional integration operator [01. We shall always keep this checking in mind.
As for the equations of the first kind with a kernel involving a power-
logarithmic singularity, the scheme of investigation is the same. Fractional
integration is replaced by the operators [OI, m considered in S 21, and fractional
differentiation by the operators inverse to IOI,m realized as a convolution with the
Volterra function.
 30. The Generalized Abel Integral Equation
We begin with a consideration of the equation which has become known as the
generalized Abel equation. The case of the equation on the whole axis will be
easier in a sense. In the first subsection we give necessary preliminaries concerning
the solu tion of singular integral equations for the reader's convenience.
30.1. The dominant singular integral equation
The equation
00
( ) ( ) a2(x) f cp(t)dt - f( )
al x cP x + - - - X.
11' t-x
-00
(30.1)
is known as the dominant singular integral equation on the whole axis. We assume
the coefficients al(x) and a2(X} to be real-valued for simplicity, while the right-hand
side and the unknown function may be assumed to be complex valued. Let
al(x), a2(x) E H>"(RI), 0 < A $ 1; a(x) + a(x) f; 0, x E R I .
(30.2)

i 30. THE GENERALIZED ABEL INTEGRAL EQUATION
607
We denote
G( ) - al(X) - ia2(x) _ i8(z)
X - - e ,
al(x) + ia2(x)
8(x) = argG(x).
(30.3)
The integer
00
x = 2 1 darg G(x)
-00
(30.4)
is called the index of (30.1). Equation (30.1) is solvable in closed form and its
solvability is characterized by the following theorem.
Theorem 30.1. Let assumptions (30.2) be satisfied. If x  0, then (30.1) is
unconditionally solvable in Lp(R 1 ), whatever the right-hand side f(x) E Lp(R 1 ),
1 < p < 00, is and its general solution is given by
M 00
(x) = ECI: a2(x)Z(X) + al(x) f(x) - a2(X)Z(X) 1 f(t)dt
<p 1:=1 (x + i)1: A(x) A(x)1r -00 Z(t)(t - x) (30.5)
where CJ: are arbitrary constants, which may be complex the sum being omitted if
x = 0, A(x) = a¥(x) + a(x) and
( . ) -_/2
Z(x) = :  ; vi A(x)e 'YCz) ,
_ 1 1 00 In [( +: )- G(t)]
-y(x) - _ 2 . dt.
1r1 t - x
-00
If x < 0, (30.1) is solvable if and only if
00
1 f(x)dx = 0,
« )1: k=I,...,lxl.
Z x) x + i
-00
(30.6)
We remark also that
Z(x),
1 >.. . 1
Z(x) EH (R),
(30.7)
if al(x), a2(x) E H>"(Rl), 0 <  < 1.

608
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
The singular equation of the form
00
a1(X),p(X) + .!. f a2(t),p(t) dt = g(x)
11' t-x
-00
(30.8)
is also solvable by quadratures, unconditionally solvable for x  0 and has the
general solution
N 00
,p(x) = '" CI:Z(x) + a1(x) (x) _ Z(x) f a2(t)g(t)dt
L.J (x + i)1: A(x) 9 11' Z(t)A(t)(t - x)
1:=1 -00
(30.9)
where A(x) and Z(x) are the same as in (30.5). The necessary and sufficient
condition for solvability of (30.8) in the case x < 0 is
00
f a2(x)g(X) d = 0
Z(x)(x + i)1: x ,
-00
k = 1,... ,Ixl.
The characteristic singular equation on an interval
"
a1(x)cp(X) + a2(x) f cp(t)dt = f(x), a < x < b,
11' t-x
a
(30.10)
has a more complicated solvability picture. Let as before
a1(x), a2(x) E H>"([a, b])j a(x) + a(x) -:F 0, x E [a, b).
(30.11)
Solutions of (30.10) may be looked for in the space H. = H.(a, b) of Holderian
functions with integrable singularities at the end-points of the interval (cf.
Definition 13.1). This is caused by the fact that the solutions of (30.10) have in
general singularities at the end-points. Sometimes it is desirable to find solutions
bounded at one end or at both. We denote
H = H([a, b)) = U H#([a, b]) = H. n C([a, b)),
#>0
(30.12)
H: = H. n C([a, b», H; = H. n C«a, b]).
(30.13)

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
609
We shall use one of the spaces H., H:, H; or H as a space in which solutions are
to be found.
In contrast to the case of the whole axis the index of (30.10) is known to
depend on the space of solutions. Let G(x) = e i8 (z) be the function (30.3). Let us
choose the value of arg G( x) so that
o  8(a) < 21r.
(30.14)
Let
na = { 0,
1,
if we look for solutions bounded as x --+ a,
if we admit for solutions unbounded as x --+ a
and nb is defined similarly. By X we denote any of the spaces H., H:, H; or H.
We set
K= KX = [ 8 ) +n. +n6- 1
= [ 8 ) +{ I
for the space H. ,
for the space H: or H; ,
for the space H
(30.15)
and
8(a)
J.la = 1 - na - -,
21r
J.lb = 8(b) _ [ 8(b) ] _ nb.
21r 21r
where [8(b)/(21r)] denotes the entire part of the number. We emphasize that
J.la = J.la(X) and J.lb = J.lb(X) and -1 < J.la < 1, -1 < J.lb < 1.
Theorem 30.2. Let conditions (30.11) and (30.14) be satisfied and
f ( ) f.(x)
x = (x - a)l-II-(b - x)l-III1'
f.(x) E H.
Then (30.10) is unconditionally solvable in the space X if Va > J.la1 Vb > J.lb and
x  0 and its general solution is
ll'} ( "' ) a2(x)Zo(X) ( _ ) IJ- (b - ) lJll p ( ) al(x)f(x)
T "" A(x) x a x N-l x + A(x)
b
_ a2(x)ZO(x) f (  ) IJ- (  ) IJ" f(t)dt
1rA(x) a t-a b-t Zo(t)(t-x)' (30.16)

610
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
where A(x) is the same as in (30.5), P M - 1 (x) is a polynomial of degree x-I with
arbitrary coefficients (PM-l(X) == 0 if x = 0) and
Zo(z) = exp 2 [i (:t +8(a) In(z - a) - 8(b)ln(b - Z)] E H.
(30.16')
If x < 0, (30.10) is solvable if and only if
b
f /?z -)a)-Idz) - 0, k= 1,..., Ixl.
Zo x x - a 1'. b - x IJb -
a
This condition being satisfied, (30.10) has the unique solution given by (30.16) with
P M -l == O.
30.2. The generalized Abel equation on the whole axis
In this subsection, using Liouville fractional integra-differentiation I we solve
following equations of the first kind
:& 00
01 f cp(t)dt f cp(t)dt .
M cp == u(x) (x _ t)1-OI + v(x) (t _ X)1-OI = f(x),
-00 :&
(30.17)
:& 00
MOI1/J == f u(t)1/J(t)dt + f v(t)1/J(t)dt = (x).
(x - t)1-OI (t - x )1-01 g
-00 :&
(30.18)
effectively in closed form.
Equations (30.17) and (30.18) are known as the generalized Abel equations wit4
exterior and interior coefficients, respectively. These equations can also be written
as
00
M Ol = f Cl(X) + c2(x)sign(x - t) ( )d - f( )
cp - I 11 cp t t - x,
x - t -01
-00
(30.17')
00
M - 01 = f Cl (t) + c2(t)sign (x - t) /. ( )d _ ( )
cp- I P 'P t t-gx,
x - t -01
-00
(30.18')
where
( ) u(x) + v(x)
Cl x = 2 '
( ) u(x) - v(x)
C2 X = 2 .
(30.19)

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
611
Equations (30.17) and (30.18) are evidently rewritten in terms of the operators of
fractional integration:
MOI<p == u(x)r(Q)I+<p + v(x)r(a)lcp = I(x),
(30.20)
MOI,p == r(a)I+(u,p) + r(a)I(v1/J) = g(X),
(30.21)
where I are the fractional integration operators (5.2) and (5.3).
We shall seek solutions of (30.17) and (30.18) in the space Lp(R I ), 1 < p < l/a.
A principal question for (30.17) and (30.18) - as well as equation of the first kind
- is the following. How can we characterize admissible right-hand side functions
I(x) and g(x) of (30.17) and (30.18) for a given space of solutions, Lp(R I ), in the
present case? We shall see that the ranges MOI(L p ) and MOI(L p ) coincide with the
space 100(L p ) of fractional integrals, which was studied in detail in S 6, or perhaps
differ from 1 01 (L p ) by a finite-dimensional space.
As for the coefficients u(x) and v(x), we shall assume that
u(x), v(x) E H>"(RI), A > a,
(30.22)
in the case of (30.17) and
u(x), v(x) E H>"(RI), A > 0,
(30.23)
in the case (30.18). Here RI is the axis, completed by the unique infinite point
and H>" CR I ) is the Holder space defined by (1.6). Clarifying condition (30.22), we
remark that it is caused by the requirement that multiplication by u(x) and v(x)
preserves the space of functions representable by fractional integrals of order a (see
Theorem 6.7).
We suppose that the coefficients u(x) and v(x) do not vanish simultaneously:
u 2 (x) + v 2 (x) f; 0, x E R I .
(30.24)
The key point in solving (30.17) and (30.18) is the connection between
the fractional integration operators I and the singular operator S, which was
established in S 11.
A. Solution of the generalized Abel equation with exterior
coefficients.
Applying (11.10) to (30.20), we have
MOI<p == r(a)[(u + v cosa7r)I<p + vsina7rSI+<p] = I,

612
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
i.e. we represent the operator MO/ as a composition:
MO/'P == No[t.'P = I,
(30.25)
where
00
a2(x) f (t)dt
No=al+a2S=al(x)(X)+- -,
11' t-x
-00
(30.26)
al(x) = r(cr)[u(x) + v(x) cos cr1l'], a2(x) = r(cr) sin cr1l'v(x). (30.27)
It is clear now that the solutions of (30.17) will be given in closed form: we are to
solve successively the singular equation
No = I
(30.28)
of the form (30.1) using known results and then invert the Abel equation. While
solving these equations in the way mentioned we have to guarantee the solvability
of the Abel equation [t.'P = . Namely, we seek solutions 'P E L" and suppose that
IE [O/(L p ). But inverting this Abel equation we do not know, generally speaking,
that  E [0/ (L p ). So we have to make sure that any solution (x) of the dominant
singular equation (30.28) is representable by the fractional integral of order cr,
when the right-hand side is. This will be shown in Lemma 30.1 below.
We solve now (30.28) by using the result in (30.5). As was noted above we are
to solve this equation in the space [O/(L p ). We shall find all the solutions of this
equation in the space L" q = p/(I- crp), (taking into account that [O/(L p ) C Lq by
Theorem 5.3) and then show that these solutions belong to [O/(L p ), if IE [O/(L,,).
It follows from (30.24) and (30.22) that the assumptions (30.2) are satisfied. The
index x of (30.28) (see (30.4» in terms of functions defined in (30.19) has the form
00
x =  f d arg [Cl(X) + iC2(X)tg cr 2 11' ] .
-00
(30.29)
The number x is integer-valued: x = 0, :i:l, :i:2,... .
If x  0, the general solution of the singular equation (30.28) IS gIven,
according to (30.5), by the relation
 ( ) =  v(x )Z(x) u(x) + v( x) cos cr1l' /( )
X  CI: (x + i)1: + r(cr)A(x) x
00
_ v(x)Z(x) sincr1l' f I(t)dt
A(x) 1I'r(cr) Z(t)(t - x)'
-00
(30.30)

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
613
where C1c are arbitrary (complex) constants,
A(z) = u 2 (z) + 2u(z)v(z) cas a1r -+: v 2 (z),
(30.31 )
and the function Z(z) is equal to
( . ) -1ft/2
Z(z) = : ; vi A(z)e 'Y(z),
(30.32)
If x < 0, the necessary and sufficient conditions of solvability for (30.26)
appear according to (30.6): the right-hand side is to be orthogonal to a finite
number of linearly independent functions.
Lemma 30.1. Let al(z),a2(z) E H>"CR 1 ), ..\ > a. ]f f(z) E ]OI(L p ), then all the
solutions  ELf' q = p/(I- ap), of the equation al  + a2S = f belong to ]OI(L p )
as well.
Prool. Since f E Lf(Rl), all the solutions in Lq(Rl) are given by (30.5). Then
the second and the third terms in (30.5) belong to ]OI(L p ) by (30.7), Theorem 6.7
and Corollary 1 of Theorem 11.4. Further, by (5.28) we have (z + i)-k = ]t-tplc,
where tplc(Z) = const(z + i)-::-OI E Lp(Rl) for every p  1, k = 1,2,.... So
(z + i)-k E 101(Lp) and then also Z(z)v(z)(z + i)-Ic E ]OI(L p ) by Theorem 6.7. .
In view of Lemma 30.1 solutions <p(z) of the starting equation (30.17) or
(30.25) may by found by the relation
00
( ) 01..... D Ol..... _ _ a f (z) - (z - t) dt
tpz=D+....., +..... ( ) ,
r 1 - a t 1 +01
o
(30.33)
where (z) is the function (30.30). The solution tp E Lp(Rl) of the Abel equation
was found in accordance with Theorem 6.1.
Thus, taking Lemma 30.1 and Theorem 30.1 into account, we see that the
initial equation (30.25) is solvable for any right-hand side f E ]a(L p ) if x  O. This
means that ]OI(L p )  MOI(L p ) if x  O. On the other hand, MOI(Lp)  ]a(L p ),
which follows directly from (30.17) in view of (6.1) and Theorem 6.7. So
MOI(L p ) = ]OI(L p ) if x  O.
(30.34)

614
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
In the case x < 0 (30.34) is valid "up to a finite-dimensional subspace", which
is produced by the finite number of linearly independent solvability conditions
(30.6).
Thus we have proved the following theorem.
Theorem 30.3. Let assumptions (30.22) and (30.24) be satisfied and let x be the
indez (30.29). Equation (30.17) is unconditionally solvable in Lp(Rl), 1 < p < I/Ot,
for any right-hand side f(x) E IOt(L p ), if x  0, and its general solution is given
by (30.33) and (30.30). If x < 0, then (30.17) is solvable for those and only those
right-hand sides f(x) E IOt(L p ), which satisfy the conditions
00
f f(x)dx = 0,
()( )A: k=I,...,lxl,
Zx x+i
-00
(30.35)
If these conditions are satisfied, the equation has the unIque solution given by
(30.33) and (30.24) with CA: = O.
B. Solution of the generalized Abel equation with interior coefficients.
Applying (11.10) to (30.21) and taking the commutation property of (11.12)
into account, we obtain
MOt<p == r(Ot)I.t[( u + v COB Ot1r)rp + sin Ot1rS( vrp)].= 9,
i.e. we represent the operator MOt as a composition
MOt<p == I.tNorp = 9
(30.36)
where
00
- 1 f a2(t)<p(t)dt
No<p = al<P + S(a2<P) = al(x)rp(X) + - ,
1[' t-x
-00
(30.37)
and al (x) and a2(x) are coefficients (30.27). Now we are to solve successively the
Abel integral equation and then the singular integral equation
Norp = D+g
(30.38)

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
615
of the form (30.8). Here
00
D a - Q f g(x) - g(x - t) dt L
+g - r(1 _ Q) t1+a E p
o
provided that 9 E [a(L p ) (we have used Theorem 6.1 here). It remains to solve
(30.38) in the space Lp. In contrast to the previous case of exterior coefficients,
when we solved the singular equation first and the Abel equation afterwards, we
deal with the inverse order for solving now. Hence there arises no need to represent
the solution of the singular equation by fractional integral as before.
Solving (30.38) using (30.9), we have
( ) =  CI:Z(x) u(x) + v(x) cas a1r ( D a )( )
<p x  (x + i)A: + r(a)A(x) +g x
00
_ sin a 11' Z(x) f v(t)(D+g)(t) dt
1I'r(a) Z(t)A(t)(t - x) ,
-00
(30.39)
provided that x  0, with x the same as in (30.29). Here A(x) and Z(x) are
functions (30.31) and (30.32).
In the case x < 0 there arise necessary and sufficient solvability conditions for
(30.38):
00
f V« X» (g):) dz = 0, k= 1,..., Ixl.
Z x x+i
-00
Us.ing (5.17) for fractional integration by parts, we can transform these orthogonality
conditions to the form
00
f g(x)tPl:(x)dx = 0, k = 1,... ,lxi,
-00
(30.40)
where
00
tPlc(X) = .!!.- f v(t)(t + i)-Icdt .
dx (t - x)aZ(t)

To justify the applicability of (5.17) we have to show that v(t)(t + i)-A: /Z(t) E
[a(L,). This in fact was established in the proof of Lemma 30.1.
As a result, we arrive at the following theorem.

616
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Theorem 30.4. Let assumptions (30.23) and (30.24) be satisfied. Equation
(30.18) is unconditionally solvable in Lp(RI), 1 < p < I/Ot, for any right-hand
side g(x) E [Ot(L,), if x  0 and its general solution is given by (30.39). [f
x < 0, (30.18) is solvable for those right-hand sides g(x) E [Ot(L p ) which satisfy
orl.hogonality conditions (30.40), then in this case the unique solution is given by
(30.39) with C1c = O.
30.3. The generalized Abel equation on an interval
We consider now the equations of the above type in the case of an interval:
 b
f <p(t)dt f <p(t)dt
u(x) (x _ t)I-Ot + v(x) (t _ x)l-Ot = f(x),
a 
a < x < b,
(30.41)
 b
f u(t)<p(t)dt f v(t)<p(t)dt = ( x )
(x - t)i-Ot + (t - x )i-Ot 9 ,
a 
a < x < b,
(30.42)
where 0 < Ot < 1. This case is of especial interest because of the various applications
where such equations arise - see references in S 34.2, note 30.8. In contrast to
S 30.2, the behaviour of the coefficients at the end-points of the interval will have
an essential influence on the existence of solutions, their quantity and the nature
of the singularities.
We shall give the complete solution of (30.41) and (30.42) in closed form and
clarify the influence of the end-points on the solutions of the equation.
We shall seek solutions of (30.41) and (30.42) in the space H. of Holderian
functions with integrable singularities at the end-points (see Definition 13.1). We
recall that it consists of functions of the form
<p. (x)
f(x) = (x _ a)1-£l(b - x)l-£2
(30.43)
where €1 > 0, €2 > 0 and <p.(x) is Holderian on [a, b]. As for the right-hand sides
f(x) and g(x), we assume that
f(x),g(x) E H,
(30.44)
where H: is the space defined in (13.53) or (13.59). This is justified by
Theorem 13.14, stating that fractional integration of order Ot maps the space H.
onto H: one-t<rone.
Comparing the setting of the question with that of the case of the whole axis,
we remark that the difference in choice of the space for the solution is caused by

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
617
the fact, that Holder spaces are not convenient for the whole axis. This is because
of specificity connected with the infinite point (although they may be used, too),
while the spaces Lp on the interval [a, b] prove to be less convenient than H. since
it is not easy then to characterize singularities on the right-hand sides.
As regards the coefficients u(x) and v(x), we assume that
u(x), v(x) E H>"([a, b]),  > Q; u 2 (x) + v 2 (x) f; 0, x E [a, b].
(30.45)
As in S 30.2, the method used to solve the considered equation will be based
on the connection between fractional integrals via a singular integral. In this case
(11.16)-(11.19) will be used.
A. Solution of the generalized Abel equation with exterior
coefficients.
We rewrite (30.41) in terms of the fractional integration (2.17)-(2.18):
u(x)I:+Y'+ v(x)Ib_Y' = r(I Q ) f(X).
(30.46)
One can eliminate Ib_Y' or I:+Y' using (11.18) or (11.19), respectively. Both ways
are equivalent and give the solutions which differ only in form. Substituting the
fractional integral Ib_Y' from (11.18) into (30.46), we arrive at the singular equation
b
( ) ..... ( ) a2(x) f (t) d _ f(x)
al x 'OJ! x + t - (b ) '
1r t-x -x Ot
a
(30.47)
where
al(x) = u(z)+v(x)COSQ1r, a2(x) = v(x) sin Q1r
(30.48)
and
:&
 1 f Y'(t)dt
(x) = (b _ x)o (x _ t)I-Ot '
a
(30.49)
Thus we have to solve the singular equation (30.47) and then invert the Abel
equation (30.49). In this process we must take care to find such solutions  of
(30.47), for which (30.49) is solvable.
We use well known results (Gahov [1] and Muskhelishvili [1]) from the theory
of singular equations on an open curve, the finite interval in our case. Let
G(x) = al(x) - ia2(X) = u(x) + e-Otiv(x) = e i8 (:&)
al(x) + ia2(X) u(x) + e Ot1rl v(x)
(30.50)

618
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
in correspondence with S 30.1, where the value of 8(x) is fixed by the condition
o  8(a) < 21r,
(30.51 )
and let x be the integer (30.15).
We solve (30.47) basing ourselves on Theorem 30.2 and taking into account
the fact that the right-hand side of (30.47) is in H., if 1 E H;. In the case x 2: 0
we obtain
:&
j cp(t)dt _ a2(x)Zo(X) ( _ ) IJ G (b _ ) a+ lJ I> p ( ) al(x) /( )
(x _ t)l-a A(x) x a x x-I X + A(x) x
o
_ a2(x)Zo(x) j b ( x _ a ) 1'. ( b _ x ) a+ lJ I> I(t)dt
1rA(x) 0 t - a b - t Zo(t)(t - x) (30.52)
where A(x) and Zo(x) are the functions (30.31) and (30.16'),
8(a)
J.lo = 1 - no - -,
21r
J.lb = 8(b) _ [ 8(b) ] -nb
21r 21r
(30.53)
and Px-l (x) is a polynomial of degree x-I with arbitrary coefficients (Px-I(X) == 0
in the case x = 0).
It remains to invert (30.52) as the Abel equation relative to I;'(t). The principal
fact that we are to investigate is to discover whether the right-hand side of (30.52)
allows the solution of this equation. Since we need to find any solution I;' E H.,
we need to clarify, by Theorem 13.14, whether the right-hand side belongs to the
space H;, if I(x) E H;. We begin with the following lemma.
Lemma 30.2. The weighted singular operator
_ 1 j b ( X _ a ) lJ. ( b - X ) 1'1> I(t)
8 f - - - - -dt
1'.,1'1> 1r t - a b - t t - x
o
maps the space H; onto itself provided that a-I < J.lo  a and a-I < J.lb  a.
Proof. Let H(€I'€2) be the space (13.51). By Theorem 11.1 the operator 81'.,1'1>
maps H(€I'€2) onto itself, if
J.lo < A+€l < l+J.lo, J.lb < A+€2 < 1+J.lb.

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
619
Since H: is a union (13.53) of the spaces H(c},c2), the operator S,...,,... then
preserves this union invariant, if Po ::; a < 1 + Po and Pb ::; a < 1 + Pb which
coincides with the assumptions of the lemma. .
We turn now to the right-hand side of (30.52) and recall that f(t) E H;".
By Theorem 11.1 we have Zo(z) E H([a,b]). Since multiplication by a function
belonging to H>"([a, b])  > a preserves the space H:, the right-hand side of (30.52)
belongs to H: by Lemma 30.2, provided that
a - 1 < Po ::; a, a-I < J.lb ::; a.
(30.54)
By (30.53), in order to satisfy the latter of these conditions we need to choose
nb = 1.
(30.55)
As regards the former condition, we have
no = 1,
no = 0,
if O(a)/(21r) < 1 - a,
if O(a)/(21r) 2: 1 - a,
(30.56)
and then the index (30.15) takes the form
x = [ O(b) ] + { I,
21r 0,
if O(a) < (1 - a)21r,
ifO(a) 2: (l-a)21r
(30.57)
(we recall that 0 ::; O( a) < 21r).
Remark 30.1. The relation O(a) = (1- a)21r corresponds to the case u(a) = O.
Thus the integer no and nb being chosen by (30.55) and (30.56), the right-hand
side of (30.52) belongs to H:, and therefore (30.52) is solvable relative to 'P by
Theorem 13.14. Inverting this Abel equation, we obtain the general solution of the
generalized Abel equation (30.41) in the case x  0:
N Z
( z ) = "" c sin a1r  J u(t) + v(t) cos a 11' f(t)dt
'P  1c'P1c + 1r dz A(t) (z - t)o
Ic-l 0
_ ( sina1r ) 2  J z v(t)Zo(t) dt J b (  ) ,... ( b-Z ) ""+o /(s)ds
11' dz A(t)(z-t)o s-a b-s Zo(s)(s-t) ,
o 0
(30.58)

620
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
where CIc are arbitrary constants and I;'k(X) are solutions of the corresponding
homogeneous equation which are absent if x = 0:
z
d j v(t)Zo(t)(t - a)IJ.+k-l(b - t)IJb+ a
<p1c(X) = dx (x _ t)a dt,
(I
(30.59)
with Zo(t) and A(t) functions (30.16') and (30.31) and
{ -8(a)/(211") , if 0  8(a)/(21r) < 1 - a,
J.la = 1- 8(a)/(211"), if 1- a $ 8(a)/(211") < 1,
J.lb = 8(b)/(211") - [8(b)/(211")] - 1.
(30.60)
By interchanging the order of integration (30.58) is transformed to
N b
"" sin a1l" d j
I;'(x) = L.J Ckl;'k(X) + -;- dx K(x,t)f(t)dt,
k=l a
(30.61)
where
K ( x, t ) u(t) + v(t) cos a 11" ( _ t ) -a
A(t) x +
sin a1l" j z (  _ a ) IJ. ( b _  ) a+IJb Zo()v()
+ 1I" Z o(t) t - a b - t A()( - t)(x - )a'
a
We can transform also (30.58) to another form not containing integrals in the
principal value sense. In view of (11.16) we have
. b a
SID a1l" j ( s - a ) 1;'( s )ds ( ) -na f a
- - -=-cosa1rl;'x+va+b_l;'.
11" t-a s-t
a
Replacing here a by 1 - a, we obtain the representation of the singular integral in

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
621
the form
b t b
sinQ1I' f cp(s)ds - ( ) sinQ1r d f  f ( t -a ) I-a cp(rdr
- --COSQ1I'cpt +-- - .
11' s-t 11' dt (t-)l-a r-a (r-)a
a a 
(30.62)
Applying this representation to the singular integral in the last term in (30.58), we
arrive at
. z
'" sin Q1I' d f u(t)f(t)
cp(x) = L.:' CA:CPA:(x) + ----;- dx A(t)(x _ t)a dt
A:-l a
2 z t b
( sin Q1I' ) d f v(t)Z(t)dt d f dr f cp(s)ds
- ----;- dx A(t)(x - t)a dt (t - r)l-a Z(s)(s - r)a'
a a T
(30.63)
where
Z(t) = (t - a)l-a+ IJ -(b - t)a+ IJ . Zo(t).
As regards the case x < 0, the singular equation (30.47) IS solvable by
Theorem 30.2 if and only if
b
f (t)("')" ar'd) - 0, k = 1,..., IKI.
ZO x x - a IJ_ b - x a+IJ. -
a
(30.64)
If these conditions are satisfied, (30.47) has a unique solution which is given by the
same resulting relation (30.52) with the term containing P._I(X), being omitted.
The final result of this investigation is stated in the following theorem.
Theorem 30.5. Let the assumptions in (30.45) be satisfied, let
u(x) + e- a1ri v(x)
8 ( x ) = arg ( . ( , 0  8 ( a ) < 211',
u x)+e a1rl v x)
and let x be the integer (30.57). The generalized Abel equation (30.41) is
unconditionally solvable in the space H. for each right-hand side f(x) E H:, if
x  0, in particular, if 8(b)  O. Its general solution is given by (30.58) or (30.63).
If x < 0, the equation is solvable if and only if the conditions in (30.64) are

622
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
satisfied and then it has the unique solution given by (30.58) or (30.63) under the
choice C1c = O.
B. Solution of the generalized Abel equation with the interior
coefficients.
As in the case of the whole axis, the equation with interior coefficients is easier
because we are to solve the Abel equation first and afterwards the singular
equation.
Equation (30.42), that is
I:+(Ulp) + I_(v'(J) = r(1,,) 9(Z)
is transformed by means of (11.16) to
I:+[(u + v cos "")'(J + sin "..r;a S(r:v'(J)] = r(1,,) 9(Z).
Inverting the Abel equation relative to the expression in the square brackets, we
obtain the singular equation
b
1 f a2(t)1jJ(t)
al(x),p(X) + - dt = gl(X),
1[' t-x
a
(30.65)
where al (x) and a2(x) are the same as in (30.48) and
1jJ(x) = (x - ay:rcp(x),
(x - a)a a
gl(X) = r(Q) (Va+g)(x)
(30.66)
:&
= sinQ1r (x - a)a f g(t)dt .
1r dx (x - t)a
a
It remains to solve (30.65). Its solvability is determined by the same integer x as
above and the expression for its solutions is known - Gahov [1] or Muskhelishvili
[1]. The derivation of the final result is left to the reader.
30.4. The case of constant coefficients
We consider especially the generalized Abel equations (30.17), (30.18) and (30.41),
(30.42) in the case when the coefficients are constant.

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
623
A. Equation on the whole axis.
If the coefficients u and v are constant, the solution of (30.17) may be obtained
avoiding an application of the general theory of singular integral equations with
variable coefficients (but the connection with the singular integral, as before). Let
us consider these equations given in the form (30.17'):
00
M a -  f CI + c2 si gn(x -t) ( ) dt - f( )
<p - r(Q) Ix-tI1-a <p t - x,
-00
x E R 1 ,
(30.67)
where 0 < Q < 1, Cl and C2 are constants, c + c f; O. The factor l/r( Q) is taken
for symmetry with the inverse operator.
We note that the integral Ma<p is the Feller potential considered in S 12. The
following theorem presents the explicit form for the inverse operator (M a )-l. We
stress that the integrals in (30.68) are interpreted as conventionally convergent in
the norm of the space Lp( Rl). They converge also almost everywhere.
Theorem 30.6. Equation (30.67) is solvable in Lp(Rl), 1 < p < I/Q, for any
right-hand side f(x) E Ia(L p ) and has the unique solution
00
( x) = Q f Cl+c2sign(x-t) [f ( x)_f ( t)]dt
<p Ar(1 - Q) Ix - tl 1 + a
-00
(30.68)
00
= Q f 2Clf(x) - (Cl + c2)f(x - t) - (Cl - c2)f(x + t) dt
Ar(1 - Q) t 1 + a '
o
where A = 4[c cos 2 (Q1f/2) + c sin(Q1f/2)].
Proof. We have
Ma<p = u1t.<P + vI<p, u = CI + C2, v = Cl - C2
(cf. (12.10». So the connection (11.10) between the fractional integration I:t: and
the singular operator yields
Ma<p == It.N<p = f, N<p = al<p + a2 S <p,
(30.69)
with al = u + v cos Q1f and a2 = v sin Q1f. Since S2<p == -<p by (11.2), the singular

624
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
operator N with constant coefficients is invertible and
N-1 _ al _ a2 S
<p - a 2 + a2 <P a 2 + a2 <po
I 2 I 2
Inverting the Abel equation in (30.69) by Theorem 6.1 and then the singular
operator, we obtain
_ 1
<p = N ID+f = 2 2 (aID+f - a2S D +f)
a l + a2
(30.70)
where D+ is the Marchaud fractional derivative (5.57), (5.60). Substituting SD+f
from (11.11') into (30.70), we obtain
<p(x) =  [(CI + c2)D+f + (CI - c2)Df]
(30.71 )
which gives (30.68). .
We single out the important particular cases of (30.67):
00
f <p(t)dt
Ix - tll-a = f(x),
-00
00
f sign (x - t)
Ix _ tll-a 1/J(t)dt = g(x),
-00
(30.72)
where <p(t) , 1/J(t) E Lp(RI), 1 < p < 1/Ot, and f(x),g(x) E [a(L p ). According to
(30.68) solutions of these equations are given by the relations
00
( ) - Ot Ot1r f f(x) - f(t) dt
<p x - 21r tg T Ix _ tl l + a '
-00
(30.73)
00
Ot Ot1r f g(x) - g(t) .
1/J(x) = 21r ctg T Ix _ tl l + a sIgn (x - t)dt.
-00
(30.74)
Thus we have obtained the inversion of the potentials [a and H a see (12.1), (12.2),
(12.1') and (12.2'). In the case of sufficiently good functions f(x) and g(x), e.g.
differentiable ones and those such that If(x)1  c(1 + Ixl}-II, v > 1 - Ot, these
relations may also be written in the form
00
( ) _ 1 Ot 1r d f sign (x - t) f( ) dt
<p x - -tg-- t
21r 2 dx Ix - t la '
-00
(30.75)

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
625
00
1 Q1I' d f g(t)dt
,p(x) = 211' ctg 2" dx Ix _ tl a '
-00
(30.76)
We note that a simple condition on the right-hand side j(x), sufficient for the
solvability of (30.67), is that it may be represented in the form
j.(x) >.. .
/(x) = (1 + Ixl)lI ' j.(x) E H (R 1 ), A > Q, II> Q.
(30.77)
Then /(x) E [a(L p ), 1 < p < I/Q, by Theorem 6.6 and so (30.67) is solvable in
Lp. For functions /(x) of the form (30.77), the inversion given in (30.68) may be
also written in the form
00
<p(x) = 1 d f C2 + clsign(x - t) j(t)dt.
Ar(l - Q) dx Ix - tl a
-00
(30.78)
B. Equation on the interval.
As regards the equation
z b
f <p(t)dt f <p(t)dt
u (x _ t)l-a + v (t _ x)l-a = j(x),
a z
a < x < b,
(30.79)
with 0 < Q < 1 and constant u and v, we shall obtain its solution from the general
relation (30.58). The function 8(x) from (30.50) is now constant:
u + e- a 1l'i v
8(x) == 8 = arg . , 0 < 8 < 211'
u + e a 1l'I v
(30.80)
(it is clear that 8 = 0 if and only if v = 0).
Lemma 30.3. The inequalities 0 < 8 < (1 - Q)211' and (1 - Q)211' < 8 < 211' are
equivalent to the inequalities uv < 0 and uv > 0, respectively.
Prool. From (30.80) we have
sin(z/2), we obtain
u l_e(.+"O)' -a1l'i
V = ea . -1 e
Since 1 - e iz = _2ie iz / 2 x
! _ sin([8 - (1 - Q)211']/2)
v - sin(8/2)
which easily yields what is required. .

626
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
We derive from Lemma 30.3 that the index x, defined in (30.57), of (30.79)
equals 1 if the coefficients u and v are of the same signs and equals 0, if they have
different signs.
We conclude then from (30.58) that the generalized Abel equation (30.79)
with the coefficients u and v of different signs has a non-trivial solution in the case
f(x) == 0:
:&
cpo(x) = .!!.-. f (x - t)-a(t - a)-8/(211")(b - t)a-l+8/(211")dt
dx
a
1
d f ( b_X ) a-l+8/(211")
= - s-a(1- s)-8/(211") S + - ds
h x-a
o
( ( /( 1 ( ) a-2+8/(211")
_ b - a) 1 - Q - 8 21r» f -a (1 _ ) -8/(211") b - x d
- ( ) 2 S S S + s.
x-a x-a
o
Applying (13.18), we obtain
const
cpo(x) = (x _ a)a+8/(211")(b _ x)1-8/(211") ' 0 < 8 < (1 - Q)21r.
(30.81 )
Taking (30.63) and (30.81) into account we state the resulting conclusion in
the following theorem.
Theorem 30.7. Equation (30.79) is solvable in the space H. whatever the
right-hand side f(x) E H: was. The solution is unique, if uv > 0, and contains
one arbitrary constant, if uv < O. All the solutions are given by the formula
:&
C u sin Q1r d f f(t)dt
cp(x) - (x _ a)a+8/(211")(b _ x)1-8/(211") + A  dx (x - t)a
a
2:& t b
v ( sin Q1r ) d f Z(t)dt d f dr f f(s)ds
- A ----;- dx a (x - t)a dt a (t - r)1-a T Z(s)(s - r)(30.82)
where c = 0, if uv > 0, and c is arbitrary, if uv < 0, and A = u 2 + 2uv COS Q1r + v 2 ;
Z(t) = (t - a)2-a-8/(211")(b - t)a-l+8/(211"), if uv > 0, and Z(t) = [(t - a)/(b -
t)p-a-8/(211"), if uv < O.
Let us consider the following important particular cases. The Carleman

 30. THE GENERALIZED ABEL INTEGRAL EQUATION
627
equation
b
f cp(t)dt = f(x)
Ix - tp-a
a
(30.83)
which attracted the attention of many authors (see references in S 34.1) is contained
in (30.79) under the choice u = v = 1. The result in (30.82) gives its unique
solution if we choose c = 0 and
a1r ( b _ t ) a/2
A = 4cos 2 -, Z(t) = -
2 t-a
By simple calculations it may be shown that (30.83) with the right-hand sides
f(x) = (x - a)n, n = 0,1,2,..., and f(x) = [(x - a)(h - x)](a-l)/2
has the solution
t ( n - a/2 ) ( b_z ) 1c
n! n . a1r r(1 - a) 1c=O n - k a-b
cp(x) = -;(b - a) SID 2"" r(n + 1 _ a) [(x _ a)(b _ x)]a/2 '
( b - a ) a
cp(x) = 2B (a, 1y-) [(x - a)(b - x)]-(1+a)/2,
respectively.
There exists another inversion formula for the Carleman equation (30.83):
b
tg(a1r/2).!!.-. f sign (x - t) f( t ) dt
cp(x) 21r dx Ix - tl a
a
2 b
+ sin :: /2) d f M(x, t)[(t - a)(b - t)]-a/2 f(t)dt,
a
(30.84)
where
b
f V (t - a)(b - t) - V (y - a)(b - y)
M(x, t) =
t - Y
a
dy
X
Ix - yla[(y - a)(b _ y)](1-a)/2

628
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Compare this with (30.75). Let us prove this result. Equation (30.52) in application
to the considered particular case (30.83) has the form
1 :1: cp(t)dt =! I(x) _ t Q1r I b [ (x - a)(b - x) ] 01/2 I(t)dt .
(x - t)l-a 2 21r g 2 (t - a)(b - t) t - x
a a
(30.85)
On the other hand, since cp(t) is the solution of (30.83), subtracting (30.85) from
(30.83), we have
b b /2
1 cp(t)dt =! I(x) + t Q1I" 1 [ (x - a)(b - x) ] a I(t)dt .
(t - x)l-a 2 21r g 2 (t - a)(b - t) t - X
:I: a
(30.86)
Inverting (30.85) and (30.86) as Abel equations and summing up the result, we
obtain the relation
( ) 1 d ( 2 Q1r I-OI l . 2 Q 1r A 1 - a I)
2cp x = r(Q) cos(Q1r/2) dx cos T B - SID T 8-01/2
where (12.44), (12.45) and (12.48) are used for brevity. By simple transformations
we represent the second term as
1
AI-a 8-01/21 = AI-a 8(1-01)/21 + 2 r(1 )' 01"-
1r - Q SID T
b b
X 1 dt 1 ([ r(t) ] a/2 _ [ r(r) ] (1-01)/2 ) I(r)dr
It - xl a r(r) r(t) r - t '
a a
where r(t) = (t - a)(b - t). We apply (12.47) to the first term here, after which
(30.84) is obtained by simple calculations.
As regards another particular case
b
1 sign (x - t) 1/J(t)dt = g(x)
Ix - tll-a
a
(30.87)
this differs from (30.83) by the fact that the corresponding homogeneous equation
has a non-trivial solution. The result in (30.82) yields the general solution of
(30.87):

 31. EQUATIONS WITH POWER-TYPE KERNELS
629
:&
C 1 Q1(' d j g(t)dt
1jJ(x) - (x _ a)(1+a)/2(b _ x)(1+a)/2 + 21(' ctg T dx (x _ t)a
a
+ cos 2 T d j :& ( t - a ) (1-a)/2 dt
1('2 dx b - t (x - t)a
a
t b ( 1 ) /2
d j dr j ( b - S ) -a g(S)d8
X dt (t _ r)1-a;::-; (8 _ r)a'
a T
(30.88)
The relation
b
C 1 Q1(' d j g(t)dt
1jJ(x) = (x _ a)(1+a)/2(b _ x )(1+a)/2 + 21(' ctg T dx Ix - tl a
a
b
cos 2 !!.!:. d j
- 21('2 2 dx N(x, tHt - a)(1-a)/2(b - t)(1-a)/2 g (t)dt,
a (30.89)
is also valid, where
b
j V (t - a)(b - t) - V (y - a)(b - y)
N(x, t) =
t-y
a
sign (x - y)dy
X Ix _ yla[(y _ a)(b _ y)](1-a)/2'
Its derivation is similar to that of (30.84).
 31. The Noether Nature of Equations
of the First Kind with Power-Type Kernels
In this section we shall investigate the problem of normal solvability and the index
(N oether nature), properly interpreted, for equations of the first kind:
j c(x, t)
M <p == Ix _ tp-a <p(t)dt = f(x),
{}
x E 0, n = [a, b], -00  a < b  00.
(31.1 )

630
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
The generalized Abel equations, considered in S30, are their particular cases. We
shall reduce (31.1) to an equivalent integral equation of the second kind. The
latter will be in general singular. We give first the necessary preliminaries from
the theory of N oether operators and from the theory of singular integral equations.
Then we shall consider the case of the whole axis n = R 1 , and in conclusion deal
with the case of a finite interval n = [a, b], -00 < a < b < 00.
31.1. Preliminaries on Noether operators
We give here the bare minimum concerning Noether operators, which is necessary
for the presentation of questions on the solvability of (31.1). These questions
are developed in SS 31.2 and 31.3. The theorems about the Noether properties
of operators in abstract Banach spaces given below are known as the Nikol'skii-
Atkinson-Gohberg theorems. More details on the theory of N oether operators can
be found in the books of Gohberg and Krupnik [4] and S.G. Krein [1]; see also
references in S 34.1.
Let X and Y be Banach spaces. The ring of all linear operators bounded from
X into Y will be denoted by [X --+ Y].
Definition 31.1. The set
Zx(A) = {cp : Acp = 0, cp E X}
of all zeros of the operator A in the space X is called the kernel of the operator
A E [X --+ Y].
For the adjoint operator A. E [Y. --+ X.] we denote similarly
Zy. (A.) = {1/J : A.1/J = 0, 1/J E Y.}.
This subspace in Y. is called the co-kernel of the operator A.
Definition 31.2. The operator A E [X --+ Y] is called normally solvable in X if
its range A(X) consists of elements orthogonal to the co-kernel:
f = Acp, cp EX, {::} (f, 1/J) = 0, 1/J E Zy. (A .).
Theorem 31.1. The operator A E [X --+ Y] is normally solvable in X if and only
if its range A(X) is closed in the space Y.
We introduce the following notation for the dimensions of the kernel and
co-kernel:
n = nA = dimZx(A), m = mA = dimZy.(A.).

 31. EQUATIONS WITH POWER-TYPE KERNELS
631
The number mA is called the deficiency number of the operator A. Sometimes both
numbers mA and nA are called the deficiency numbers, the former of them being
also known as the nullity of the operator.
Definition 31.3. The operator A E [X -+ Y) is called the Noether or Noetherian
operator, if it is normally solvable in X and its kernel and co-kernel are
finite-dimensional: n < 00, m < 00.
The ordered pair (n, m) is called the dimensional characteristic or d-
characteristic of the operator A. The difference x = n - m is called the
index of the operator. We shall use the designation x = xx_y(A) and write
xx(A) in the case X = Y. It is known that the index of the operator is stable
relative to small perturbations.
For Noether operators the following statements hold.
Theorem 31.2. If A E [X -+ Y) and B E [Y -+ Z) are Noether operators, then
the operator BA E [X -+ Z) is also Noetherian, and
xx_z(BA) = xx_y(A) + Xy_z(B).
Theorem 31.3. If the operator A E [X -+ Y) is Noetherian and T is a completely
continuous operator from X into Y, then the operator A + T is also Noetherian,
and xx_y(A + T) = xx_y(A).
Theorem 31.4. The operator A E [X -+ Y) is Noetherian if and only if it admits
the left and right regularizers R, E [Y -+ X) and R,. E [X -+ Y): R,A = E + Tl
and All,. = E + T2, where T1 and T2 are operators completely continuous in X and
Y, respectively.
The usual examples of N oether operators are the singular integral operators
(30.1), (30.8) and (30.10). We recall known results concerning the Noether
properties of these operators which will be needed below.
Theorem 31.5. The singular integral operator
00
No<p = a1(x)<p(X) + a2(x) f <p(t)dt = f(x)
11' t-x
-00
(31.2)
where a1 (x), a2(x) E C( it 1 ) and are real-valued, is Noetherian in the complex-
valued space Lp(R1), 1 < p < 00, if and only if a(x) + a(x) f; 0, x E it 1 . The
operator No has the d-characteristic (x,O) if x  0 and (0, Ixl) if x  0, where x
is the integer (30.4).

632
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
A similar operator on a finite interval will be considered with a "regular" term:
b b
N<p == al(x)<p(X) + a2(x) f <p(t)dt + f (x, t)<p(t)dt = f(x).
11' t-x
a a
(31.3)
In the case of an interval the conditions of normal solvability and the relation
giving the index are already dependent on the space of the solutions. We shall give
two known assertions. The first will concern the "classical" setting of the question
of normal solvability, when one considers the orthogonality to solutions not of the
adjoint equation N*,p = 0, but of the homogeneous transposed equation:
b b
al(x),p(X) - .!. f a2(t),p(t) dt + f (t, x),p(t)dt = O.
11' t-x
(31.4)
a
a
and solutions of (31.3) are sought in the space of functions, Holderian in (a, b) and
having integrable singularities at the end-points with non-fixed orders. The second
assertion will deal with the case when the Noether nature is interpreted in the
sense of Definition 31.3 and the operator in (31.3) is considered in the space Lp(p)
with the weight function p(x) = (x - a)#J(b - X)II.
We recall that the coefficients al(x) and a2(x) in (31.3) and (31.4) are assumed
to be real-valued. As regards the kernel (x, t) we assume that
 ( x t ) = A(x, t) + B(x, t)sign(x - t) 0 < g _ < 1 ,
, Ix - tl 1 - e '
where A(x, t) and B(x, t) are functions, Holderian in both variables.
Let H*, H;, H; and H be the function spaces considered in S 30.1. As
previously in S 30.1, G(x) denotes
G(x) = al(x) - ia 2 (x) .
al(x) + ia2(X)
Since al(x) and a2(x) are real-valued, we have G(x) = e i8 (z). We consider the
value of 8(x) = argG(x) to be chosen by the condition 0  8(a) < 211' with 8(x)
extended by continuity to other points x E (a, b).
Theorem 31.6. Let al(x), a2(x) E H, a(x) + a(x) f; 0, x E [a, b), and 8(a) f; 0,
8(b) f; 211'k, k = 0,::1:1, :1:2,.... Then (31.3) with the right-hand side f(x) satisfying
the assumptions of Theorem 30.2, is solvable in the spaces H*, H:, H; or H if

 31. EQUATIONS WITH POWER-TYPE KERNELS
633
and only if
b
f f(X),pj(x)dx = 0,
a
(31.5 )
where {,pj} is the complete system of solutions of (31.4) in the spaces H, H;, H:
or H*, respectively. The difference of the numbers of linearly independent solutions
of the homogeneous equation (31.3) and of the solvability conditions given in (31.5)
is equal to the index x given by (30.15).
Theorem 31.7. Let a1(x), a2(x) E C([a,b]). The operator (31.3) is Noetherian
in the space Lp(p), 1 < p < 00, p(x) = (x - a)#J(b - X)II, -1 < p. < p - 1,
-1 < v < p - 1, if and only if
1) aHx) + a(x) -:F 0, a ::;; x ::;; b;
II) 8(a) f; 21f' 1P , 8(b) f; -21f'( mod 21f').
These conditions being satisfied, the index of the operator (91.9) is equal to
x= [ 8(b) + I+V ] + [ 1+1£ _ 8(a) ]
21f' p p 21f'
(cf. (90.15)). In the case (x, t) == 0 the operator given by (91.9) has the
d-characteristic (x,O) if x  0 and (0, Ixl) if x::;; O.
In the following subsections we consider the question of the Noether nature
of the operator in (31.1). Equation (31.1) as an equation of the first kind is
not normally solvable generally speaking, so the question of its "Noetherness"
requires the proper setting. The operator M will not be in general Noetherian
as an operator acting from certain space X into the same space X. In the case
X = Lp(Rl) it is not even bounded from X into X - see the Hardy-Littlewood
Theorem 5.3. If mesO < 00, the operator M is bounded in Lp(O), but M cannot be
Noetherian as an operator from Lp(O) into Lp(O). Indeed, if M E [Lp(O) -+ Lp(O)]
is Noetherian, then by Theorem 31.4 there exists the bounded operator R such
that RM = E + T, where T is completely continuous in X = Lp(O), mesO < 00,
(under rather weak assumptions on c(x,t», we see then that the identity operator
E = RM - T is completely continuous in X, which is impossible.
A natural question arises: may one construct, for the operator M non-
Noetherian from X into X, such a space Y that the operator M is Noetherian
as an operator from X into Y? The construction of such a space for achieving
N oether properties is sometimes called the nonnalization of the operator.
By the Hardy-Littlewood Theorem 5.3 the operator M with a bounded function
c(x, t) is bounded from Lp(O), 1 < p < I/Ot, into L,(O), q = p/(1 - Otp). It is not,
however, a Noetherian operator from Lp into Lq. Let us show this for c(x, t) == 1.
Correspondingly with Theorem 31.1 we have to show that the range M(L p ) is not
closed in L,. For c(x, t) == 1 by (30.34) we have M(L p ) = IOt(L p ) C Lq, where
IOt(L p ) is the space of fractional integrals. We recall that IOt(L p ) f; Lq - see (6.5).

634
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Since infinitely differentiable finite functions belong to [Ot(L p ) as for example, by
Theorem 6.5, the space [Ot(L p ) is dense in Lq and consequently is not closed in Lq.
In the general case under rather weak assumptions on c(x, t) we shall show
that M(L p )  [Ot(L p ) and that the range M(Lp) is closed in [Ot(L p ). In other
words, we shall treat the operator M as Noetherian from Lp into [Ot(L p ), the latter
being considered as a Banach space relative to the norm (6.17).
31.2. The equation on the axis
The equations to be considered are
00
f c(x, t)
(M cp )(x) == Ix _ tl l - Ot cp(t)dt = f(x), 0 < a < 1.
-00
(31.6)
We allow the function c(x, t) to be discontinuous on the diagonal t = x:
( ) { u( x, t),
c x,t =
v(x,t),
t < x,
t > x,
(31. 7)
so that
:e 00
(M )(x) == f u(x, t)cp(t)dt + f v(x, t)cp(t)dt = f(x).
cp (x - t)1-Ot (t - x )i-Ot
-00 :e
(31.6')
In the case of continuity at the diagonal (u(x, x - 0) = v(x, x + 0» the operator
M will always be of the Fredholm type i.e. with the index equal to zero, in the
corresponding setting. It will be of the Fredholm type also, if, e.g. v(x, t) == o.
We define now the class of admissible functions u(x, t) and v(x, t) in (31.6).
Let R be the half-plane {(t, x) : t < x} and R: = {(t, x) : t > x}.
Definition 31.4. A function u(x, t) defined in the closed half-plane R belongs to
the class H;(R) if i) u(x, x) E C(RI); ii) u(x, t) is Holderian in x of order A,
o < A < 1, uniformly with respect to t:
IXI - x21>" -2
IU(XI' t) - U(X2' t)1  A (1 + IXII)>"(1 + I X 21)>'" (Xl, t), (X2, t) E R+.
(31.8)
The class H;(R:) is defined similarly. Writing c(x, t) E H;(Ri) will denote
>.. -2 >.. -2
that u(x,t) E H:e(R+), v(x,t) E,H:e(R_).
We observe that functions in H;(Ri) are bounded, but are not necessarily
continuous in t. For example u(x, t) = x(1 + x 2 )-lsignt E H;(R), A = 1.

 31. EQUATIONS WITH POWER-TYPE KERNELS
635
A. Imbedding M(L p ) C IOt(L p ), 1 < p < l/Q. We represent the operator M
as
:& 00
M = f u(t, t)cp(t) dt f v(t, t)cp(t) dt K+ K-
cp (x - t)l-Ot + (t _ x)l-Ot + u cp + v cp,
00 :&
(31.9)
where
:&
K+ = f u(x, t) - u(t, t) ( t ) dt
u cp (x - t)l-Ot cp ,
-00
(31.10)
00
K- = f v(x, t) - v(t, t) ( t ) dt
tJ cp (t - x)l-Ot cp .
:&
By means of (11.10) we obtain
Mcp = r(a)I+Nocp + Kcp + K;cp,
(31.11 )
where
00
sin Q1I' f v(t t)cp(t)dt
Nocp = [u(x, x) + v(x, x) cos a1l']cp(x) + - ' .
11' t-x
-00
(31.12)
Theorem 31.8. If c(x, t) E H:(RJ with  > a, then M(Lp)  IOt(L p ) and
II MOt cpllrO(Lp)  cllcpllp, 1 < p < II a.
(31.13)
Prool. By (31.11) and the boundedness of the singular operator in Lp(Rl) (see
Theorem 11.2) it is sufficient to prove the statement of the theorem for Kj cp
and K;cp. We apply Theorem 6.2. Since u(x,t) is bounded, we conclude that
Kjcp E L, by the Hardy-Littlewood Theorem 6.3. To verify the conditions (6.19)
we use the relation
00
(K;t <P)(z)-(K;t <P)(z-t)= f u(z, z-t-S)z-t. z-t-s) <P(z-t-s)ds
o
00
+t Ot f [u(x, x-ts)-u(x-ts, x-ts)]k(s)cp(x-ts)ds,
o (31.14)

636 CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
where k(s) is the function (6.10). Let us prove (31.14). For J(x) = Kj<p we have
00
J(x) = f[u(x, x - s) - u(x - s, x - s»)sa-l<p(x - s)ds and so
o
00
J(x)- J(x-t)=t a f [u(x, x-t-st)-u(x-t-st, x-t-st»)(s+l)a-l<p(x_t-st)ds
-1
00
_t a f [u(x-t, x-t-st)-u(x-t-st, x-t-st»)sa-l<p(x-t-st)ds
o
00
= f [u(x, x-t-s)-u(x-t, x-t-s»)sa-l<p(x-t-s)ds
o
00
+t a f k(s+l)[u(x, x-t-ts)-u(x-t-ts, x-t-ts»)<p(x-t-ts )ds,
-1
which gives (31.14). By this result
00 00
n a K+ - Q f  f u(x, x-t-s)-u(x-I, x-t-s) ( x-t-s ) ds
+,e u <p r(l- Q) t1+a sl-a <p
e 0
00 00
+ r(1a) f  f [u(", , ",-Is) -U("'- ta, ",-Is )]k( a)",(",-ta )da
e 0
=Ae(x)+Be(x).
(31.15)
According to (31.8)
00
c f g(x - t)dt
IAe(x)1  (1 + Ix I)>' t 1 + a ->'(1 + Ix - 11)>"
o
00
( ) = f l<p(x - s)lds
9 x 1 .
S -a
o
(31.16)

 31. EQUATIONS WITH POWER-TYPE KERNELS
637
Here g(z) E Lq(R l ), q = p/(I- ap), by Theorem 5.3, so
00 ( 00 ) IIp
IIA.II. c / t-a-ldt \.-£ (1 + Izl)-(1 + Iz - tD-'lg(z)I'dz
clIgllq 1 0 00 t>'-Ot-ldt ( 1 00 00 (1 + Izl)->'IOt(1 + Iz _ tl)->.IOtdZ ) Ot
(31.17)
According to (6.32) the inner integral is dominated by c(1 + t)->'IOt, so IIA£ II" 
00
cll<plI" f(1 + t)->'t>'-Ot-Idt = cllI<pllp.
o
For B£(z) we have
00 00
B.(z) = r(1  0) 1 k(s)ds 1 u(z, z - t) - ;(z - t, z - t) I"(z - t)dI
o £II
or by (6.11)
00 £8
B.(z) = - r(1  0) 1 k(s)ds 1 u(z, z - t) - ;(z - t, z - t) I"(z - t)dt
o 0
£ 00
a l dt l
=- ( ) - k(s)[u(z,z-ts)-u(z-ts,z-ts)]<p(z-ts)ds.
rl-a t
o 0
Hence
£ 00
1 dt 1 s>'lk(s)<p(z - ts)lds
IB£(z)1  c tl->' (1 + Izl)>'(1 + Iz - tsl)>"
o 0
and applying the Minkowsky inequality, we obtain
£ 00 { 00 } IIp
dt >. 1<p(z)IPdz
IIB.II.  c / t'- /s Ik(s)lds _£ (1 + Izl)'(1 + Iz + tsl)'
(31.18)
The inequality
00
1 11/J(z)ldz < 1 + 2°
(1 + Izl)°(1 + Iz - hl)a - (1 + Ihl)a ll1/Jlh,
-00
a> 0,
(31.19)

638 CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
is valid. Indeed let h > 0 since the case h < 0 is reduced to the former one by the
change x = -yo Denoting the left-hand side of (31.19) by J(h), we have
00 ( h/2 00 )
J h 11/J(-x)ldx 11/J(x)ldx
( )  J (1 + .,)"(1+., + h)" + J + J (1 + .,)"(1 + I., - hI)"
o 0 h/2
hl2 00
$(1 + h)-ol11/Jlh + J (1 + h - x)-ol1/J(x)ldx + J (1 + x)-ol1/J(x)ldx
o hl2
( h ) -0
$(1 + h)-01l1/J1l1 + 1 + "2 111/Jlh,
which yields (31.19). By (31.19) we obtain
e 00
IIBelip $ cllcpllp J t>'-ldt J s>'lk(s)I(1 + ts)->'ds
o 0
from (31.18). The inner'integral is estimated as follows:
00
j(t) = J s>'lk(s)I(1 + ts)->'ds  ct- Ot as t  O.
o
(31.20)
In fact since Ik(s)1 $ cs Ot - 2 for s  1, we have
00
j(t) c + c J s>'+Ot-2(1 + ts)->'ds
1
1
=c + c J -Ot( + t)->'
o
lIt
=c + ct 1 ->'-Ot J -Ot(1 + )->'d $ ct- Ot .
o
By (31.20)
IIBelip  cc>'-Otllcpllp
(31.21 )

 31. EQUATIONS WITH POWER-TYPE KERNELS
639
and then, according to (31.15), IID+.eKjrplip  cllrpllp as well.
Since (K;rp)(x) = (Kt.rp-)(-x) with v-(x,t) = v(-x,-t) and rp-(t) = rp(-t),
the same estimate is valid for D+,eK; as well. In correspondence with Theorem
6.2 the statement of the theorem is proved. .
B. Representation of the operator M of the potential type by the
composition M = It. N.
Theorem 6.1 allows one to invert the fractional integration to the right
by fractional differentiation, provided that we deal with the range [01 (L p ) , i.e.
It.D+I == I, I E [OI(L p ). Hence the imbedding M(L p )  [OI(L p ) obtained in
Theorem 31.8 allows to write
Mrp == [t.Nrp, N = D+M, rp E L", 1 < p < 1/0:.
(31.22)
The following lemma presents an explicit expression for the operator N.
Lemma 31.1. Under the assumptions 01 Theorem 31.8 the operator N = D+M,
with M the operator given by (31.6), has the lorm
N = r(o:)N o + T,
(31.23)
where No is the singular operator given by (31.12) and
Trp = T: rp + cos O:1f'T v - rp + sin O:1f'ST v - rp
(31.24)
where
00 00
rj- = 0: f  f u(x, x-t-s)-u(x-t, x-t-s) ( -t- ) d
u rp r(1-o:) t l +OI SI-OI rp x S s,
o 0 (31.25)
00 00
r - 0: f dt f v(x, x+t+s)-v(x+t, x+t+s) ( )d
v rp= r(1-o:) t1+01 SI-OI rp x+t+s s.
o 0 (31.26)
The operators T: and Tv- can be also represented as
:&
T;i", = f(l  0) f T;i(z, T)",(T)dT,
-00
(31.27)
00
T.- '" = f(l  0) f Tv- (z, T)'P( T)dT,
:&

640
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
where
:&
+ ( x r ) = f u(x, r) - u(t, r) dt
u' (x - t)l+a(t _ r)l-a
T
1
_  f u(x,r) - u[r+ s(x - r),r] d
- ( s,
x - r sl-a 1 - s)l+a
o
T
- ( ) f v(x,r)-v(x,t) d
Tv x,r = (t-x)l+a(r-t)l-a t
:&
(31.28)
1
=  f v(x,r)-v[x,x+s(r-x)] ds.
r - x sl+a(1 - s)l-a
o
Proof. From (31.11) we have D+M<p = r(cr)No<p + D+Kj<p + D+K;<p, so
T = D+Kj + D+K;. From (11.10) we conclude that
D+f = cos cr1rDf + sin cr1rSDf
(31.29)
for f E [a(L p ). Since K;<p E [a(L p ) by Theorem 31.8, we obtain (31.24) by
simple transformation using (31.29).
Further, passing to the limit in (31.15) as c -+ 0 and taking (31.21) into
account, we obtain
D a K+ l/'J
+ UT
00 00
cr f dt f u( x, x - t - s) - u( x - t, x - t - s)
r(1 _ cr) tl+a sl-a <p(x - t - s)ds
o 0
00 00
+ cr f k(s)ds f u(x, x - t) - u(x - t, x - t) <p(x - t)dt.
r(1 - cr) 0 0 t (31.30)
The passage to the limit here is easily justified if it is in the norm of Lp. It is
not difficult to show that the integrals in (31.23) exist not only in the sense of
convergence in the Lp-norm, but in the usual sense also for almost all x.
In (31.30) the second term is identically zero in view of (6.11). Expression
(31.26) can be deduced by obvious symmetry arguments. The representation
(31.27) and (31.29) are obtained by easy transformations. .
Remark 31.1. It follows from Theorem 31.8 that the operator T in (31.23) is
bounded in Lp(Rl) at least, if 1 < p < l/cr. It will be shown below that the

 31. EQUATIONS WITH POWER-TYPE KERNELS
641
operator is completely continuous in Lp(RI). Now we observe that the kernels
(31.28) admit the estimates
c(x - r)>'-l
IT.;(x, r)1  (1 + Ixl)>' '
c(r - x)>'-l
IT; (x, r)1  (1 + Ixl)>'
which follow from (31.28) by (31.8), c being a constant.
C. The N oether nature of the operator M.
The previous consideration paved the way for the result on the Noether nature
of (31.6). Let c(x, t) be of the form (31.7).
Theorem 31.9. Let c(x, t) E H;(R'i) with  > n. The operator M is Noetherian
from the space Lp(R 1 ), 1 < p < l/n, into the space IOt(L p ) if and only if
u 2 (x,x) +v 2 (x,x) f; 0, x E R 1 .
(31.31)
Under this condition the index of the operator M is equal to
00
XLp_lOl(Lp)(M) =  f darg{c1(x) + iC 2 (x)tg n 2 1r' }
-00
(31.32)
where Cl(X) = u(x, x) + v(x,x), C2(X) = u(x, x) - v(x,x).
Prool. Since  > n the imbedding M(Lp)  IOt(L p ) and the representation in
(31.22) are valid by Theorem 31.8. Elementary arguments show that the property
of the operator M to be Noetherian from Lp into lOt (L p ) is equivalent to that of
the operator N from Lp into Lp. The condition (31.31) is necessary and sufficient
by Theorem 31.5 for the operator No to be Noetherian and under this condition
the index of the operator No is equal to the integer (31.25). So in correspondence
with Theorem 31.3, it is sufficient to show that the operator T is completely
continuous in Lp(RI). Since a composition of a bounded and continuous operator
is a completely continuous operator again, we observe by (31.24) and the similarity
of the operators Tj and Tv- that it remains to prove the complete continuity of
the operator r: only. We give this statement separately in the following lemma.
Lemma 31.2. If u(x,t) E H:(.R),  > n, then the operator Tj is completely
continuous in Lp(RI), 1 < p < l/n.
Prool. We check the Riesz criterion
II(T.;<p)(x + 6) - (T.;<p)(x)lIp  (6)1I<pllp,
(6)  0,
6-0
(31.33)

642
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
( f I (Tj<P)(X)IPdX ) 1/.  '1(N)III"II., '1(N) N-:::oo 0,
zl>N
(31.34)
which guarantees (see Dunford and Schwartz [1, p.324]) the complete continuity of
the operator Tj. We observe that the representation of the operator Tj given in
(31.27) and (31.28) ''works well" for checking (31.33), while (31.25) is well-matched
to (31.34).
We denote F(x) = (Tj<p)(x). By (31.27) we have
z+6 z
F(x + 6) - f(x) = f Tj(x + 6, T)<p(T)dT + f [T.;(x + 6, T) - T.;(x, T)]<p(T)dT
z -00
= Al +A2.
The estimate for Al is simple in view of (31.8):
6 { 00 } IIp
lid dl,.  ! dT _£ [T,; (., - T + 6, ., )1"(") I' d.,
6
 c f (6 - T )>'-1 dT\I<pllp = X 6 >'1I<pllp.
o
(31.35)
Further,
00 { 00 } l/P
IId211.  ! dT _£ IT,;(" + T + 6,,,) - r,; (., + T,.,)IP 11"(" )1' d.,
We represent Tj(x + T + 6, x) - Tj(x + T, x) as Al + A 2 + A3 where
Z+T
Al = f u(x + T + 6, x) - u(x + T, x) dt,
(x + T + 6 - t)l+a(t - x)l-a
Z
z+r+6
A - f u(x+T+6,x)-u(t,x) dt
2 - (x + T + 6 - t)l+a(t - x)l-a '
z+r

 31. EQUATIONS WITH POWER-TYPE KERNELS
643
Z+T
A. = J [(Z + T + 6 - W'-o - (Z + T - t)-'-O] U(Z /Z;)_:(t, Z) dt.
Z
Let us estimate Al first:
I
c6>'r>' J ds
IAII  (1 + Ix + rl)>' sl-a[6 + r(1 - s)]1+a
o
6>.-a r a
-c
- 1(1 + Ix + rl)>'(6 + r)
where (13.18) is used. Thus
6>.-a
IAt!  CI (1 + Ix + rl)>'rl-a '
(31.36)
The estimate for A 2 is obtained similarly and coincides with (31.36). Further,
applying the mean value theorem to A 3 , we have
1
A c6 J (1 - s - )ads
I 31  (1 + Ix + rl)>'r 2 ->' sl-a(1- s)l+a->'(I- s + 6/r)l+a '
o
where 0 <  < 6/r, so
I
c6 J sa-l(1 - s)>'-l-ads
IA31  (1 + Ix + rl)>'r 2 ->' (1- s + 6/r)l-£(1- s + 6/r)£
o
C 6£
< 1
- (1 + Ix + rl)>'r l ->'+£ '
where CI = Cl(€) and 0 < € < A-a. By the estimates for A}, A 2 and A3 we have
00 00
II A II 6£ J a(r)dr 6>.-a J a(r)dr
2 p  C r l ->'+£ + C rl-a '
o 0
(31.37)
{ } l
where a( r) = -L (1 + Ix + rl}->'P l<p(x )IPdx . It is clear that 0  a( r)  1I<pllp
and it is easily shown that lIall,  KII<pllp for q > max(p, I/A), where K = K(q). The

644
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
application of the Holder inequality in the case r > 1 shows that the integrals in
00
(31.37) are finite and admit the estimate f r- ll a( r)dr  cll<pllp with v = 1 -  + c
o
or v = 1 - Q and c = c(,c,Q,q), provided that the order q of integrability
for the function a(r) is chosen within the limits max(p, 1/) < q < 1/( - c).
The latter leads to the following choice of c which is always possible, namely
max(O,  - IIp) < c <  - Q. Therefore (31.35) and (31.37) yield the inequality
IIF(x + 6) - F(x)lIp  c6£1I<pllp, so (31.33) is satisfied.
Let us turn to verifying (31.34). Using (31.25), we obtain
00 00
rj- c f dt f l<p( x - t - s) I d
I( u <p)(x)1  (1 + Ix I)>' t l + a ->'(1 + Ix _ tl)>' sl-a S.
o 0
Let g(x) be the same as in (31.16). Following the same line as in estimating in
(31.16) and (31.17), we have
( fiT: <pIP dX ) II.  011911. j tl+- ( f [(1 + 1"1)(1  I., _ tl)]/Q ) a
zl>N 0 zl>N
( ) a
00
1 dt 
 01111"11. (N + 1)< f tl+Q- f (1 + 1.,1)(-')/Q(l + I., - tl)/Q
o zl>N
Here c > 0; we choose c < Q and c <  - Q; then the inner integral is estimated
according to (6.32), so that
( ) lip
00
+ p cIII<pllp t>.-a-ldt _ C2
f ITu 1"1 d.,  (N + 1)< f (1 + t)-. - (N + 1)< 111"11.
>N 0
which completes the proof of the lemma and, thereby, that of Theorem 31.9. .
Thus, in view of (31.22), equation (31.6) is reduced to the equation of the
second kind
00
sin Q1I' f v(t, t)
N<p == r(Q)[u(x, x) + v(x,x) COSQ1I"]<p(x) + - -<p(t)dt + T<p = D+/,
11' t-x
-00
(31.38)

 31. EQUATIONS WITH POWER-TYPE KERNELS
645
where T is the completely continuous operator. Theorem 31.6 yields the following
corollary.
Corollary. Let u 2 (x, x)+v 2 (x, x) f; 0, x E R I . Equation (31.6) with the right-hand
side f(x) in [Ot(Lp) is solvable in Lp, 1 < p < I/Ot, if and only if
00
f ,pj(x)(Df)(x)dx = 0, j = 1,..., m,
-00
where ,pj(x) is a complete system of linearly independent solutions of the
homogeneous singular equation N.,p = 0, adjoint to (31.38).
D. Regularization of operators of potential type.
In the conclusion of this subsection we consider the problem of a regularization
of (31.6), i.e. reducing it to a Fredholm equation, not a singular one. Our goal
will be an effective construction - in the explicit form - of the regularizer for the
operator M, treated in correspondence with Theorem 31.4. This regularizer will
be the operator
00
Rf = Ot sin Ot1r f co(x, t) [f(x) - f(t)]dt,
1rA(x) Ix-tl l + Ot
-00
(31.39)
where
A(x) = u 2 (x, x) + 2u(x, x )v(x, x) cas Ot1r + v 2 (x, x),
( ) { u(x,X), t<x,
Co x, t =
v(x, x), t > x.
Theorem 31.10. Let c(x,t) E H:(1i) and let u(x,x), v(x,x) E H>'(RI),  > Ot.
[fu 2 (x,x) + v 2 (x,x) f; 0, x E it l , then the operator (31.39) is a regularizer of the
operator (31.6):
RM<p = <p + '1"<p, MRf = f +'1"'f,
(31.40)
where T' and T" are operators, completely continuous an Lp and [Ot (Lp),
respectively.
Prool. First of all we remark that the operator R is bounded from [Ot (Lp) into

646
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Lp. To see this, we rewrite the operator R in terms of fractional differentiation:
R f u(x, x) ( 01 f)( ) v(x, x) ( 01 f)( )
= r(Q)A(x) D+ x + r(Q)A(x) D_ x,
after which the estimate IIRflip  cllfllro(Lp) follows immediately from Theorem
6.1 and (11.11").
It follows from Lemma 31.2 that the operator
:&
( U[OI _ [OI U ) =  f u(x, x) - u(t, t) ( t ) dt
+ + cp r( Q) (x - 1)1-01 cp
-00
is completely continuous from L" into [OI(L p ), if u(x) E H>'(R}), A > Q. SO, while
considering the composition RM and M R, we may interchange the operation of
multiplication by a Holderian function with the operators It. and [. According to
(31.9) we have M = Mo+Tl, where Mocp = r(Q)tt(ucp)+r(Q)[(vcp) and T 1 is an
operator completely continuous from Lp into [OI(Lp) (see Lemma 31.2). Therefore,
1
RMcp = A (uD+ + vD)(u[+ + v[)cp + T 2 CP
1
= A [(u 2 + v 2 )cp + uD+tvcp + vD[+ucp] + T3CP,
where T2 and T3 are operators completely continuous in Lp(R1). Since D+[cp =
cos Q1rcp + sin Q1rScp and D[+cp = cos Q1rcp - sin Q1rScp by (11.10) and (11.11), we
have
RM cp =  [(u 2 + v 2 + 2uv COS Q1r)CP + sin Q7r(uSv - vSu)cp] + T 3 cp.
The operator uSv - vSu is completely continuous in Lp(R1), 1 < p < 00, since
Su - uS is - see for example Gohberg and Krupnik [4, p. 33]. So RMcp = cp + T'cp,
where T' is completely continuous in Lp.
The validity of the second part (31.40) is verified similarly. .
31.3. Equations on a finite interval
Let us consider the operators of a potential type in the form
:& b b
- f cp(t)dt f cp(t)dt f
M cp = u(x) (x _ t)l-OI + v(x) (t _ x )1-01 + T(x, t)cp(t)dt
a :& a
(31.41 )
= f(x)

 31. EQUATIONS WITH POWER-TYPE KERNELS
647
and investigate them from the point of view of the proper setting of the
problem of normal solvability. In the end of this subsection we shall obtain the
statement on the Noether nature of the operator M as an operator from Lp into
[Ot(L p ) = [Ot[Lp(a, b)]. In such a setting the adjoint operator M* is interpreted as
an operator from the space lOt (L,,)* = D+ (L p ) of generalized functions into Lpl.
In view of wide applications of such equations on a finite interval, we shall be also
interested in the "classical" setting of the problem, when instead of the adjoint
operator M* one deals with the transposed operator
z b b
MT,p = 1 v(t),p(t)dt + 1 u(t),p(t)dt + 1 T ( t X ) ,p ( t)dt = 9 ( X )
(x - t)I-Ot (t - x)l-Ot '
G Z a
(31.42)
and the spaces of solutions cp and ,p are characterized in simple terms, being in
general topological spaces and not Banach ones.
A. Reduction to the equation with the Cauchy kernel.
Applying (11.18) and (11.16) to (31.41) and (31.42), respectively, we reduce
them after elementary transformations to the equations with the Cauchy kernel:
b
( ) ..... ( ) a2(x)(b - x)Ot 1 (t)dt K ..... = f( )
al x 'OJ! x + (b ) ( ) + 'OJ! x ,
11' -tOtt-x
a
(31.43)
b
( ) .1. ( ) _ 1 1 a2(t)(b - t)Ot,p(t)dt K *.I. - ( )
al X Y" x (b ) + Y" - 91 X ,
11' -x Ot t-x
a
(31.44)
where al(x) and a2(x) are functions (30.48) and
z
(x) = 1 cp(t)(x - t)Ot-ldt,
a
b
9,(20) = .  ): 1 9 (t)(t - x)Ot-ldt,
SID Q1I' x
z
(31.45)
1 Ot f
K = r(Q) TD a + '
K',p = r(lOt) D_r,p,
(31.46)
where T is the integral operator with the kernel T(x, t) and r is its transposed
operator. It is necessary to remark that the "right-hand" representation T" =
[b_Db_T" used in the passage to (31.44) is valid for functions ,p which are
considered below, since we shall show that the function T",p is representable by
the fractional integral of order Q.

648
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
B. The class of admissible perturbations T.
It is natural to consider the perturbing operator T to have the kernel T(x, t)
with a singularity of order weaker than 1 - a, in order that the operator K in
(31.4) should have a weak singularity. By means of the expression for fractional
integration by parts we arrive at the requirement that the fractional derivative of
T( x, t) with respect to t of order a
b
71°)(x, t) = - r(I,,) :, J T(x, s)(s - t)-Ods
t
(31.47)
must have a weak singularity. The following lemma presents a simple sufficient
condition for the admissible kernel T(x, t).
Lemma 31.3. Let
( ) { Cl(X, t)(x - t),61- 1 , t < x,
Tx,t= ,6 1
C2(X, t)(t - x) -, t > x,
(31.48)
where a < Pi  1 and cs(x, t) are functions, bounded in [a, b] x [a, b], differentiable
in respect to t as t f; x and such that loci/at I  Ailx - tl- 1 , i = 1,2. Then the
kernel 71 01 )(x, t) is representable as the sum of a Volterra degenerate kernel and
the kernel with a weak singularity:
{ c((z,b) ( b ) ,6-I ( b t ) -OI
71 01 )(x, t) = To(x, t) + r 1-01 - X - ,
0,
t > x,
t < x,
(31.49)
/To(x, t)/  clx - tlmin(,6lt,6)-OI-l.
(31.50)
Proof. Let t < x at first. We have
{ z b }
Tf.0I) 1 d Cl(X, s)ds C2(X, s)ds
· (x, t) = - r(1 _ ,,) dt ! (s - t)o(x - s)1-I" + ! (s - t)o(s - x)'-p,
In the first integral we change the variable: s = t + w(x - t) and then differentiate
under the integral sign, while in the latter integral we differentiate first and then

 31. EQUATIONS WITH POWER-TYPE KERNELS
649
change the variable: s = x + w(x - t). We obtain:
1
-r(1 - a)r!OI)(x t) = PI - a f Cl(X, t + w(x - t» dw
t , (x - t)1+01-,61 w Ol (1 - W)I-Pl
o
1
( ) R -01 f aCl(X,S) (1 - w),61dw
+ x - t 1"1
as WOl
o
(b-t)/(z-a)
a f C2(X, x + w(x - t»dw
+ (x - t)1+01-,6 wl-P(1 + w)1+01 .
o
The integrals in the first and in the third terms are bounded since Ci(X, I) are
bounded. For the second term we have
1 1
f aCl(X, s) (1 - W),61 dw <AI f dw
as WOl - (x - t)w Ol (1- w)l-Pl
o 0
=(x - t)-1 AIB(1 - a,Pl)'
Let t > x. Integrating by parts in (31.47) and then differentiating under the
integral sign with respect to t, we obtain
( "..,( 01 ) ( _ C2 ( x, b)
r l-a)l;' x,t) (b-x)I-,6(b-I)OI
b
f aC2(X,S) ds
- as (s - x)l-P(s - 1)01
t
b
f c2(x,s)ds
+ (1 - (2) (s _ x)2-P(8 _ 1)01 '
t
The first term here is a degenerate kernel, others admit the estimates
b b
f aC2(X, s) ds <A2 f ds
as (s-x)l-,6(S-t)OI - (s-x)2-P(8-t)0I
t t
C
< .
- (I - x)I+OI-P'

650
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
6
J C2 (X, s )ds c
(s - X)2-,6(S - t)a (t - X)l+a-,6'
t
Gathering all the estimates we obtain (31.49) and (31.50).
C. Noether theorems in the "classical" version.
Since equations (31.41) and (31.42). are reduced to the singular equations
(31.43) and (31.44), we can use the Noether theory of the latter equations. The
main work here is the proper choice of the space for the solutions cp and 1/J in the
situation of the "classical" approach, based on Theorem 31.6. The problem of such
a choice did not arise when we considered the operator M in (31.41) as Noetherian
from L,(a, b) into [a[L,(a, b)] in the setting prescribed by Definition 31.3.
In the theorems below, the value
u(a) + v(a)e-a"'i
8(a) = arg .
u(a) + v(a)e a ,...
is chosen within the bounds 0  8( a) < 211'.
Theorem 31.11. Let u(x), v(x) E H>'([a, b]), A > Q, and f(x) E H:, let the
kernel T(x,t) satisfy the assumptions of Lemma 31.3 and let u 2 (x) + v 2 (x) f; 0,
8(a) f; 0, 8(a) f; (1- Q)211', 8(b) f; 211'k, k = 0, :1:1,.... Equation (31.41) is solvable
6
in the space H. if and only if f f(x )1/Jj (x )dx = 0, where {1/Jj} is a complete system
a
of solutions of the homogeneous equation (31.42) (with g(x) = 0), which have the
form
1/J(x) = (x - a)-a(b - x)-a1/J.(x)
(31.51)
with a function 1/J.(x), Holderian on [a,b]. The difference between numbers of
linearly independent solutions of (31.41) and (31.42) is evaluated by (30.57).
The proof of this theorem is actually set up by the reduction to (31.43) and
(31.44), and follows as we shall show from the known facts for such equations. Let
us rewrite (31.43) and the homogeneous equation corresponding (31.44) in the form
6
al(x)6(X) + a2(x) J 6(t) dt + K66 = /6(x),
11' t-x
a
(31.52)
6
al(x)1/J6(X) - !. J a2(t)1/J6(t) dt + K;1/J6 = O.
11' t-x
a
(31.53)

 31. EQUATIONS WITH POWER-TYPE KERNELS
651
Here b(X) = (b - x)-a(x), /b(X) = (b - x)-af(x), ,pb(X) = (b - x)a,p(x),
Kb = (b - x)-aK(b - t)a and Kb = (Kb).. Equations (31.52) and (31.53) are
singular integral equations with the N oether nature presented in Theorems 31.6
and 31.7.
The problem to be solved is the following. If we seek solutions <p(x) in the
space H., what space is to be chosen for solutions ,p(x) in order that the spaces for
solutions of (31.52) and (31.53) are in agreement with each other as prescribed by
Theorem 31.6? Since solutions of (31.41) and (31.42) are connected with solutions
:&
of (31.52) and (31.53) by the relations b(X) = (b - x )-a J(x - t)a-l<p(t)dt,
a
-rPb(X) = (b - x)a,p(x), the required choice is determined by Theorems 31.6 and
13.14. By Theorem 13.14 (x) E H:, so b(X) E H.. Theorems 30.2 and 31.6
prompt us to seek solutions b(X) bounded or non-bounded at the point x = a
in the cases 0 < 8(a) < 211'(1 - Q) and 211'(1 - Q) < 8(a) < 211' respectively. Let
for preciseness 8(a) > 211'(1- Q). Then solutions ,pb(X) of (31.52) are sought, by
Theorem 31.6, in the space of functions bounded at both end-points, i.e. in the
space H. Therefore ,p(x) is to be taken in the class of functions of the form
,p(x) = ,pb(x)/(b - x)a, ,pb(X) E H.
(31.54)
The functions ,p(x) may be sought, indeed, in a wider space (31.51). The simplest
way to see this is the following. We might reduce (31.41) and (31.42) to the
singular equations of the type (31.43) and (31.44) or (31.52) and (31.53) with the
weight function (x - a)a instead of (b - x)a. To obtain this we have to apply
(11.17) and (11.19) instead of (11.18) and (11.16). By this method we would arrive
at the equation of the type (31.52) with the right-hand side (x - a)-a f(x) and the
equation of the type (31.53) with the solution ,pa(x) = (x - a)a,p(x). Arguments
repeating those given above show that the choice of the class of the same solutions
,p(x) must be the following: ,p(x) = ,pa(x)/(x - a)a with ,pa(x) E H. Comparing
this with (31.54) we attain (31.51).
Applying Theorem 31.6 to (31.52) we see that (31.43) is solvable in H. if and
b
only if J /(x),pj(x)dx = 0, where {,pj} is a complete system of solutions of the
a
homogeneous equation corresponding to (31.44) in the space (31.51). If (31.43)
is solvable in the space H. for the right-hand side /(x) E H:(C H.), then all
its solutions (x) belong to the subspace H too. We omit the proof of this
assertion, but give some elucidations. For a singular equation, as in (31.52), with
the kernel K:(x, t) which is Holderian in both variables, it is known that if its
right-hand side has integrable singularities at the end-points of the power type
with orders not exceeding a given number, and the Holder property of the kernel
is of sufficiently large order, then the solution, which is a priori considered in H.,
has similar singularities. It is known that this can be proved by the method of
Carleman- Vekua-regularization - (see Gabov [1]) which is based on the inversion

652
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
of the dominant part of the equation. After this the required information about
solutions is obtained from the theory of Fredholm equations. In our case, that
is under our assumptions on the kernel T(x, t), the difference is only in the fact
that we arrive at the Fredholm equation with a kernel having a weak singularity
(see Remark 31.3 below), and the statement is true as well. This requires detailed
estimates which occupy too much space and therefore are omitted.
Since (31.43) is solvable and all its solutions belong to the space H:, then
(31.41) is solvable in H.. It remains to observe that the last statement of the
theorem follows from its validity for (31.52) and (31.53). .
Remark 31.2. The choice (31.51) is that of the space, "adjoint" to the space H.
for solutions of (31.41) and (31.42). This choice of adjoint spaces of (weighted)
Holderian functions for (31.41) which is an equation of the first kind enables
us to obtain the statement of the Noether theorems-type, i.e. Theorem 31.11.
We would like to stress, that the main idea underlying the choice of the adjoint
b
spaces proves to be the requirement that I(x - a)a(b - x)al<p(x)1/J(x)lds < 00. As
a
regards the theory of singular equations, the rule for the choice of adjoint spaces is
b
I 1<p(x),p(x)ldx < 00.
a
Remark 31.3. One can investigate the regularizer (interpreted in the sense of
Theorem 31.4), construct it explicitly and show that the kernel of the regularized
equation, i.e. the Fredholm-type equation of the second kind, has a weak
singularity. Namely, if u(x), v(x) E H>', A > a, and the kernel T(x, t) satisfies
the assumptions of Lemma 31.3, then the kernel of the regularized equation is
dominated by Ix - tI IJ - I with J.l = min(A,PI' P2) - a.
D. The N oether nature of potential type operators from Lp into
[a(L p ).
Let us consider the equation of type (31.41) in the form
z b
M = f u(x, t)<p(t)dt f v(x, t)<p(t)dt - f( )
<p - (x - t)l-a + (t _ x)l-a - X.
a z
(31.55)
We suppose that the functions u(x, t) and v(x, t) satisfy the assumptions
1) they are Holderian of order A > a in x uniformly in t:
IU(XI' t) - U(X2' t)1 :5 Alxl - x21\ A > a,
- >.
Iv(xl, t) - V(X2, t)1 :5 Alxl - x21, A > a,
(31.56)
where Xl  t, X2  t and Xl :5 t, X2 :5 t, respectively, and A and A do not depend
on t;

 31. EQUATIONS WITH POWER-TYPE KERNELS
653
2) u(z,z) = u(z,z - 0) E C([a,b]), v(z,z) = v(z,z + 0) E C([a,b]).
Following the same lines as in S31.1 (see (31.9)-(31.12» and applying (11.16),
we reduce (31.55) to the form
M'{J = r(o:)I:+NOI'{J + Tl'{J + T 2 '{J = f(z),
(31.57)
where
b
NOI'{J = al (z )cp(z) + .!. f ( y - a ) 01 a2(y) '(J(y)dy,
11' z-a y-z
a
(31.58)
and al(z) = u(z, z) + v(z, z) cas 0:11', a2(z) = v(z, z) sin 0:11', and
:&
T l/'J = f u(z,y) - u(y,y) ( ) d
IT (z - y)l-OI '(J Y y,
a
(31.59)
b
1'- - f v(z,y) - v(y,y) ( ) d
2'{J - ( ) 1-01 '(J Y y.
y-z
:&
(31.60)
We shall prove that the operators Tl and T2 are completely continuous from
Lp into 100(L p ), 1 < p < I/o:. Firstly we show that the operators of the form
(31.55) are bounded from Lp into 100(L p ). We denote
:&
f u(z, t)
M 1 '{J = (z _ t)l-OI '(J(t)dt,
a
b
f v(z,t)
M 2 '{J = (t _ Z)1-OI '(J(t)dt.
:&
Lemma 31.4. Let -00 < a < b < 00, 0 < a < 1, 1 $ p < 00. The
operator M 1 with the function u(z, t) satisfJing the assumptions 1) and e) is
bounded from Lp(a, b) into 1\[Lp(a, b)]. If u(z, z) f; 0, a $ z $ b, then
M 1 [L p (a, b)] = 1:+[Lp(a, b)].
Proof. Let p > 1. We have
%
M f u(t, t)'{J(t) d 1',
1'{J = (z _ t)I-OI t + 1'{J
a

654
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
where T 1 is the operator (31.59). Let us show that Tl is bounded from
L,,(a,h) into I\[Lp(a,h»). First of all, using Theorem 13.3, we shall prove that
Tl(Lp) C 1:+ [Lp(a, b)]. In accordance with this theorem let
1:-£
.7. ( x ) = f f(x)-f(t) dt a<x<b ,
Y"£ (x - t)l+a '
-00
where f(t) = (Tl<P)(t), -00 < t < h, and it is assumed that <p(s) == 0 as s < a.
Simple transformations lead to
1:-£ 1:-£
( ) f ( ) d f u(x, s) - u(t, s) dt
£ X = cP s s (t _ s)l-a(x _ t)l+a
a .
00 1:-£'
+ r(a) J k(s)ds J u(x,t) - u(t,t) <p(t)dt,
x-t
o a
(31.61 )
where k(s) is the function (6.10). Hence elementary manipulations which are left
to the reader, give the uniform estimate II£ lip  c with c not depending on c. So
T 1 <P E Ia(L p ), if <p E Lp, by Theorem 13.3. Therefore, by the same Theorem 13.3
T 1 <P = 1:+ V 1 cp,
v; n a T I . Qt/J£(x)
lCP = a+ 1<P = 1m r(l - ) .
£-0 - a
By (31.61) and (6.11) we arrive at
1: 1:
a J J u(x, s) - u(t, s)
Vl<P = r(1 _ a) cp(s)ds (t _ s)l-a(x _ t)l+a dt.
a 8
(31.62)
The kernel of this integral operator is dominated by const (x - s )-1 in view of
(31.56) and so the operator VI is bounded in Lp. This is equivalent to boundedness
of the operator T 1 from Lp into 1:+ [L" (a, b»).
Further, we have M 1 CP = I\[r(a)u(t,t)<p(t) + Vl<P)' It remains for us to
observe that VI is a Volterra-type operator with a weak singularity, so the operator
in the square brackets maps Lp onto itself one-to-one, provided that u(t, t) f; O.
It remains then to consider the case p = 1. Since the above representation for
M 1 <P holds for <P E Lp, p > 1, it evidently holds on a set dense in Ll' Since the
operators Ml, I:+and VI are bounded in Ll, this result is extended by continuity
to the whole space L 1 . .

 31. EQUATIONS WITH POWER-TYPE KERNELS
655
Corollary 1. Let u(x, t) and v(x, t) satisfy assumptions 1) and 2). The operators
M 1 and M2 are bounded from L,(a, b) into p;w[L,(a, b)], -00 < a < b < 00,
o < er < 1, 1 < p < l/er.
Indeed, by symmetry we state that the assertion of Lemma 31.4 holds for
the operator M2 as an operator from L,(a,b) into If_[L,(a, b)]. It remains then
to recall the coincidence of the ranges: 1+[L,(a, b)] = If_ [L,(a, b)] in the case
1 < p < l/er.
Corollary 2. Let u(x,t) and v(x,t) satisfy assumption 1). The operators Tl and
T2 are completely continuous from L,(a,b) into p;w[L,(a,b)], -00 < a < b < 00,
o < er < 1, 1 < p < l/er.
Indeed, as in Corollary 1 it is sufficient to consider the operator Tl only. Its
complete continuity from L, into IOt(L p ) is equivalent to that of (31.62) in the
space L,(a, b). As it was observed in the proof of Lemma 31.4, the kernel of the
operator V 1 is dominated by the kernel (x - s)-l with a weak singularity. Then
the operator V 1 is completely continuous in L, - see for example Krasnosel'skii,
Zabreiko, Pustyl'nik and Sobolevskii [1, p. 97]).
We are ready now to give the result on the Noether nature of the operator M
in (31.55) in the following theorem in which
G(x) = u(x,x) + v(x,x)eiOt1r = e i8 (z).
u(x, x) + v(x, x )e IOt1r
Theorem 31.12. Let the functions u(x, t) and v(x, t) satisfy assumptions 1) and
2). Then the operator M in (31.55) is Noetherian from the space L", 1 < p < l/er,
into the space lOt (Lp) if and only if
i) u 2 (x,x) + v 2 (x,x) f; 0,
a  x  b;
(31.63)
1- er p
ii) 8(a) f; 211'- (mod 211'),
p
211'
8(b) f; Ii (mod 211').
(31.64)
These conditions being satisfied, the index of the operator M is given by
x = XL _I01 ( L ) = [  _ er _ 8(6) ] + [ 8(b) +  ]
p P P 211' 211' P
(31.65)
Proo/. Representation (31.57) and Theorems 31.2 and 31.3 show that the property
of the operator M to be Noetherian from L, into lOt (L,) is equivalent to that of

656
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
the operator N a from L, into L,. The latter is equivalent to the property of the
operator
b
1 f a2(Y)
al(x),p(X) + - -.,p(y)dy
1r y-x
a
(31.66)
to be Noetherian in the space L, (p) with the weight p( x) = (x - a) - 01' . The
singular operator and the operator of multiplication by a continuous function are
permutable up to an operator completely continuous in the space L,(p) ...... see
Gohberg and Krupnik [3], [4], so the criterion for (31.66) to be Noetherian is
contained in Theorem 31.12.
Remark 31.4. Theorem 31.12 may be extended to the case of the space L,(p)
with the weight p(x) = (x - a)#J(b - X)II, -1 < p. < p - 1, -1 < II < p - 1. In
correspondence with Theorem 31.7 the conditions in (31.64) are to be replaced by
the conditions 8(a) :F 21r (0/ - a) and 8(b) f; -21r (mod 21r), the conditions
in (31.65) requiring similar changes. The literary citations in S34.1 should also be
referred to.
Remark 31.5. A theorem about the Noether nature - in the sense of Definition
31.3 - of the operator M holds in the case of the spaces H6(p) similarly to Theorem
31.12.
Remark 31.6. In the case IIp < a < 1 the statement of a type such as Theorem
31.12 holds. This gives the Noether nature of the operator M from L, into a
special space (see S34.2, note 31.1).
E. The case v(x, t) = O.
We now specially consider an important, particular case of (31.55), namely
:&
f u(x, t)
(x _ t)l-a cp(t)dt = f(x), 0 < a < 1.
a
(31.67)
It is reducible to a Volterra integral equation of the second kind if u(x, x) f; O.
Indeed (31.57) takes the form r(a)I:+[u(t, t)cp(t)] + T1CP = f. Inverting here the
fractional operator 1:+, we arrive at the equation'
:&
a1r f
u(x, x)cp(x) + -;-- K:(x, s)cp(s)ds = n:+f,
SID a1r
a
(31.68)

 31. EQUATIONS WITH POWER-TYPE KERNELS
657
where
:&
K- ( X S ) = f U(X, s) - U(t, S) dt
, (t - S )l-a(X - t)l+a
IJ
1
=  f U(X,S) - U(S +(X - S),S) ds.
x - S l-a(1- )l+a
o
according to (31.62). Here IK-(x, s)1 $ c(x - s)>'-I if u(x, s) satisfies (31.56).
Thereby, in view of Lemma 31.4 the following result is proved.
Theorem 31.13. Let u(x, t) satisfy (31.56) and let u(x, x) be continuous on [a, b]
and u(x, x) =I 0, a  x  b. For every f(x) E I:+(L,,), p  1, (31.67) is equivalent
to the following Volterra equation of the second kind:
:&
<p(x) + f A(x, s)<p(s)ds = g(x),
a
(31.69)
where g(x ) =  ( Da a+ f)(x) and the kernel A ( x,s) = .a7rK'. :&, , has a weak
U,,:&,:&J 81na7rU :&,:&
singularity. Therefore, (31.67) is unconditionally and uniquely solvable in L,(a, b)
for any f(x) E I:+(L,).
The case Q  1 may be similarly considered, provided that u(x, t) is
differentiable up to order [Q] and (8[a]u(x, t»/8x[a] satisfies (31.56) and the
requirement of continuity at t = x.
31.4. On the stability of solutions
Solutions of integral equations of the first kind are not stable in general and the
problem of inversion of such equations is known as an ill-posed problem. The
simplest equation of the first kind - the Abel equation:
:&
f <p(t)dt
(x _ t)l-a = f(x),x > at
(I
(31.70)
is not stable for example in the 'space C([atb]). In fact let us choose fn(x) =
n-a[(x __ a)/(6 -- a)]n so that IIfnlle = n- a -+ 0 as n -+ 00. The solution
( ) n-a
<pn(x) = rla) V:+fn of (31.70) is equal to <pn(x) = :: n!(=i:) by
(2 26) S II II (b-arr(n+l) (b-tr " f (1 66) Th .
. . 0 <Pn e = nor ( a r ( n+l-a ) -+ r a In vIew 0 ',' ere IS no

658
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
stability in the case of the space Lp(a, b), 1 5 p ::; 00 also, which can be justified
by the same example modified in the following way, namely
/n(X) = n- a + 1 / p [(x - a)/(b - a)]n.
(31. 71)
Then
(b - a)l/p
II/nllp = n a - 1 / P (np + l)1/p -+ 0,
(b - a)l/p-a
IICPnilp -+ r(a)pl/P f; 0
(31. 72)
as n -+ 00.
The instability of solution of the equations of the first kind is a reflection of
the simple fact that the operator generated by the left-hand side of the equation,
maps the considered space of solutions onto its proper subspace, the space itself
being essentially wider than this subspace. Therefore the inverse operator is not
bounded. This means that for equations of the first kind it is natural to estimate
the proximity of solutions and that of the right-hand sides in different metrics.
Namely, the proximity of right-hand sides is to be considered in a stronger metric,
so that when dealing for example with (31.70) in a certain space X we have to
take another space Y for the right-hand sides providing the boundedness of the
inverse operator from Y into X. Such an approach can naturally give stability of
the solutions. The spaces X = Lp and Y = Lq are not suited for this goal whatever
p E [1,00] and q E [1,00] were, since the inverse operator is never bounded from
Lq into Lp. We can achieve stability by taking Y = I:+(X), i.e. requiring that
the class of right-hand sides consisted of functions representable by the fractional
integral of a function in the given space. (Compare this with similar arguments
in connection with the normal solvability of integral equations of the first kind in
the end of S31.1.) In other words, it is natural to define stability for example for
(31.67) in such a way that not the condition 1I/lIx -+ 0, but IIV:+/llx -+ 0 would
imply IIcplix -+ O. This means that under such a definition of the stability we are
interested in the a priori estimate
IIcplix 5 cIlV:+/lix.
(31.73)
Of course, the idea formulated above is trivial for the simplest equation such as
(31.70), but it proves to be substantial and useful in a more general situation of
(31.67) and of (31.55) or (31.41) all the more so. This idea is effective under the
choice, for example, of X = Lp or X = H>., X = H>'(p), since the spaces I:+(X)
are well studied and characterized in the case of such X - see U3. We recall also
that the space I:+(L p ) was shown to be coincident with the Sobolev-type space
Ha,p - see S 18.4.
We confine ourselves to the illustration of the afore mentioned idea by the
example of (31.67).

 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
659
Theorem 31.14. Let u(x, t) satisfy the assumptions of Theorem 31.13. The
solution of (31.67) exists, is unique in any of the spaces L,(a, b), 1  p < 00, and
admits the estimate
IIcpll"  cIlV:+fll,
(31.74)
whatever the right-hand side f was, the constant c depending only on n and the
kernel function u(x, t).
Proof. By Theorem 31.13 it is sufficient to prove the estimate IIcpll,  cllgll, for
the Volterra integral equation (31.69) of the second kind with a weak singularity in
the kernel. The latter estimate is a consequence of the boundedness of the operator
inverting such equations in the space L" we refer for example, to the book of
Mihlin [1], the estimate (6) on p. 44 and the Corollary on p. 94. .
Theorem 31.14 with Theorem 13.5 yield the following corollary.
Corollary. If u(x, t) satisfies the assumptions of Theorem 31.13, then the solution
cp(x) of (31.67) admits the estimate
1I<p1I. :S c (11/11. + tt- 1 - a W. U ,t)dt) , 1 < p < 1/0,
where w,(f,t) = IIf(x+t)-f(x)II,. Here, evidently, a simple estimate IIcpll, 
cllfllHA, 1 < p < l/n, 0 < n < A  1, follows.
We might obtain similar it. priori estimates for more general (31.55) and
(31.44). For this goal we must use the reduction of these equations to singular
integral equations, presented in S31.3 and the stability of solutions of the latter
equations. We do not elaborate on these considerations.
We also do not concern ourselves here with the problems connected with
the regularization in the sense of Tihonov of the ill-posed problem of solving the
integral equation of the first kind. There are many investigations devoted to this
goal. We refer, for example, to the paper by Zheludev [1], directly concerning the
Abel equation (see also [2]), where numerical methods of solutions were suggested
based on the method of Tihonov's regularization. The papers by Gerlach and
Wolfersdorf [1], Hai and Ang [1], Gorenflo [6] and Ang, Gorenflo and Hai [1] are
also relevant. We also note the paper by Savelova [1] which deals with some
questions of stability of fractional differentiation, including the multi-dimensional
case.
 32. Equations with Power-Logarithmic Kernels
The present section deals with the solution of integral equations of the first kind
with power-logarithmic kernels and a variable limit of integration

660
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
:&
_ ( 1 ) f c(x - t)(x - t)a- 1 In,6 -Lcp(t)dt = j(x),
r a x-t
a
(32.1 )
a < x < b,
where [a, b) is a finite interval of the real axis, a > 0, -00 < {3 < 00, "y > h - a. In
particular, the inversion formula for convolution operators of the form
:&
(I:f ",)( z) == rl",) f (z - tr1ln fJ z  t ",(t)dt = f( z),
a
(32.2)
a < x < h,
is given. The solution of the latter equation unlike the case {3 = 0 considered
in S 2 requires the application of special functions of the Volterra type. This is
necessary for the following reason. In the case of a natural power of a logarithm
({3 = m = 1,2,...) the function x a - 1 ln m X (which is naturally considered instead
of xa-llnm(1/X» is obtained by ordinary differentiation of a power function x a - 1
with respect to a m times: x a - 1 ln m X = (/TIx a - 1 /da m . Therefore it is natural to
expect that the power-logarithmic function x a - 1 In.8 (1/ x) with an arbitrary real
exponent {3 can be obtained by fractional differentiation (1/ x )a-l with respect
to a. However after doing this we obta.in instead of the function xa-1In.8(1/x)
itself a certain special function known as a Volterra function, whose main term of
asymptotic expansion is equal to x a - 1 In,6(1/x) - see (31.11) and (32.37) below.
For this reason we first give some properties of the special Volterra functions
and prove some identities for them. We then solve in closed form the integral
equations of the form
:& m
r(l",) f (z - W- I L:Amk lnk(z - t)",(t)dt = f(z),
a k=O
(32.3)
a < x < h, a > 0,
with constant coefficients Amk and integer nonnegative powers of logarithms. In
conclusion we obtain the criterion for solvability of (32.1) and, in particular, of
(32.2). We Dote that unlike (32.3) the unknown function cp(t) is given via the
solution of certain Volterra integral equations. We also observe that the results
below can be extended to integral equations of the type (32.1)-(32.3) but with
variable lower limit.

fi 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
661
32.1. Special Volterra functions and some of their properties
We consider the function
00
1 xa+TrUdr
p(x, iT, Q) = r(a + r + l)r(iT + 1)
o
(32.4)
called the Volterra function as given by Erdelyi, Magnus, Oberhettinger and
Tricomi [3, 18.3] and M.M. Dzherbashyan [2, Ch. 5, S 11, p. 261]. We recall some
its properties. The function p(x, iT, a) is defined for ReiT > -1 and is an analytic
function of x with branch points x = 0 and x = 00 and has no other singularities.
It is an entire function with respect to a. It can be extended to an entire function
with respect to iT by the relation
(_I)n 1 00 dR [ xa+T ]
p(x, iT, a) = r(iT + n _ 1) r u + n drn r(a + r + 1) dr,
o
(32.5)
ReiT > -n - 1,
and hence, in particular,
dR-I [ a ]
p(x, -n, a) = (-It- 1 da n - I r(: + 1) ,
n = 1,2,...
(32.6)
From (32.4) the differentiation formula
dR
- d p(x, iT, a) = p(x,iT, a - n)
x n
(32.7)
follows. The asymptotic expansion of the Volterra function at zero is given by the
relation
( 1 ) -U-I [ N-I( l)n( 1) ( 1 ) -n
p(z,u,a)=zo In; ?;. - ;+ n p(I,-n-l,a) In;;
+0 ((lnrN)].
Reu> -I, largln G) I 0, z -+ 0,
(32.8)

662
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
where (0' + l)n is the Pochhammer symbol given in (1.45).
In Lemma 32.1 below we prove two integral identities which are used in the
sequel. They involve certain special functions. We denote for simplicity
(Ig)(a) = { (Ig)(a),
rD:/J g) (a)
if {3 > 0,
if {3 < 0,
(32.9)
where Ig and V: fJ 9 are the right-sided fractional integral (5.3) and fractional
derivative (5.8) considered on the half-axis (0, +00). We denote by 1[g(T, x)](t)
an application of the operator I to a function g( T, x) with respect to T at the
point t. We introduce the functions
lI(x) = JJ(x, 0, 0), IIh(X) = (d/dx)lI(xe h ),
(32.10)
[ 1 ( x ) T-l ]
JJt,/J(x) = I: r( T) :Y (t),
-00 < {3 < 00,
(32.11)
and we shall call these Volterra functions also. The following lemma is true.
Lemma 32.1. The following integral identities
:&
f ta-1[lnt + h -1jJ(a)]lIh(x - t)dt = _xa-l, a> 0,
o
(32.12)
:&
f JJa,/J(t)JJ1-a,-/J(x - t)dt = 'Y, ° < a < 1, -00 < (3 < 00,
o
(32.13)
are valid for the Volterra functions (32.10) and (32.11).
Proof. Let h E R 1 . We consider the relation
:&
f ta-l ( t ) U-l a+u-l
x - eh(a+u-l)dt = x eh(a+u-l),
r(a)r(O') r(a+O')
o
a > 0, 0' > 0,
(32.14)
which is directly checked by (1.68) and (1.69). We first apply the integration
operator I with respect to 0' to (32.14) and then the differentiation operator 1;1

 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
663
with respect to a. Then in view of Stirling's formula (1.63) we obtain the relation
z 00
J ta-l [ f / ( a )] J ( X - t ) T-1eh(T-l) xa+u-l
- ha I t + h - - dt d - h(a+u-l)
f(a)e n f(a) f(r) r--f(a+O")e .
o 0
Setting here 0" = 1 and using (1.67), (32.4) and (32.6), we have
z
J ta-1Pnt + h -1jJ(a)]II«x - t)eh)dt = x a /a, a> O.
o
(32.15)
Differentiating this result with respect to x and applying (32.7) and the estimate
II(X) = O(ln(l/x)-I) as x -+ +0, following from (32.8), we derive as a result the
first identity (32.12) in view of (32.10).
Let now -00 < {J < 00. Similarly to the previous case {J = 1, where ordinary
operators of the integration and differentiation of the integer order were applied to
(32.14), we transform this relation first applying the operator I with respect to
a to (32.14) and then the operator I;/J with respect to 0" - see (32.9). Setting
0" = 1 - a (0 < a < 1) and h = -In 1', l' > b - a, in the relation thus obtained
and taking (32.11) into account we obtain another identity (32.13). The lemma is
proved. .
Remark 32.1. By (32.8) the function defined by (32.11) has the asymptotic
behavior
!Ja,I/(Z) = (r-I (In;t [ r(la) + P d: ( r(la» ) In-I; + 0 (In- 2 ;) ],
x -+ +0, (32.16)
at zero. Therefore (32.13) can be continuously extended to the cases a = 0, {J > 0
and a = 1, {J < O. If we set a = 1, (J = 1 in (32.16), then according to (32.7) it is
not difficult to obtain the following asymptotic result at zero:
IIh(X) = O(x- 1 1n- 2 (I/x» as x -+ +0
(32.17)
for the function IIh(X) defined by (32.10).

664
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
32.2. The solution of equations with integer non-negative
powers of logarithms
First we consider the simplest equation of the form (32.3) in the case m = 1:
:&
r(I",) J(z - It- 1 [In(z - I) + A](I)dt = I(z). '" > O.
a
(32.18)
Without loss of generality we suppose that All = 1 and A 10 = A. We apply the
operator
:&
(J:+g)(x) = J IIh(X - t)g(t)dt
(32.19)
a
where IIh has the form (32.10) to both sides of this equation and set h -1/J(a:) = A.
Then from (32.12) it follows that (32.18) is solvable together with the equation
:& :&
r(I",) J (z - l)a-l(I)dt = - J "h(Z - 1)/(1).
a a
(32.20)
According to S 2, the unique solution of the latter equation, and therefore of
(32.18) is given by the formula
cp(x) = -('D:+J:+/)(x)
(32.21)
where V:+ is the Riemann-Liouville fractional derivative given in (2.30).
Now we consider (32.3) with m = 2:
:&
rt",) J (z - It- 1 [In2 (z - I) + A 21 In( z - I) + A20]( I)dl = I( z). '" > O. (32.22)
a
Again without loss of generality we suppose that A2 2 = 1. Setting h = h2 in
(32.12) and differentiating it with respect to a: and adding (32.12) multiplied by
h l - 1jJ( cr) to the relation thus obtained we find
:&
J t a - l {ln2 t + [h2 + h l - 21/J(a:))lnt + [h 2 -1/J(a:)][h l -1/J(a:») -1/J'(a:)}lIh(X - t)dt
o
= -xa-1Pnx + h l - 1jJ(cr»), a: > O.
(32.23)

S 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
665
Let us suppose now that the constants h l and h 2 are connected with the constants
A 20 and A 2l by the relations
h l + h 2 - 2,p(a) = A2l' [hl - ,p(a)][h2 - ,p(a)] - ,p'(a) = A 20 ,
(32.24)
and J:.+ and J: are the operators (32.19) with h = h l and h = h 2 respectively.
Applying first the operator J: and then the operator J:.+ to the both sides of
(32.22) and taking (32.23) and (32.24) into account we find
:&
r(I,,) l(a: - tt- 1 cp(t)dt = (-1)2(J.j.J:/(f»(a:).
a
(32.25)
Hence it follows that (32.22) is solvable together with the latter equation and its
unique solution has the form
<p(z) = (-1)2(V:+J:.+J:+/)(z).
(32.26)
Analyzing the solution of (32.18) and (32.22) we see that the relations in
(32.12) and (32.23) played the main role in constructing the solutions given in
(32.21) and (32.26) respectively. Similarly, for solving (32.3) with an arbitrary
m the leading role is to be played by the identity which generalizes (32.12) and
(32.23) and is given by the following statement.
Lemma 32.2. Let m = 1,2,... and the constants Amj = Amj(m, a) be defined by
the recumnt relations
Amm(m, a) = 1,
m> l'
- ,
Alo(l, a) = h l - ,p(a), ..., AlO(m, a) = h m - ,p(a);
Am,m-l(m, a) = Am-l,m-2(m, a) + Al.o(l, a),
m>2'
- ,
(32.27)
d
Amj(m, a) = Am-l,j(m,a)AlO(l,a) + da Am-lJ(m,a) + Am-l.j-l(m,a),
j = 1,..., m - 2, if m  3;
d
Amo(m, a) = Am-l,O(m, a)AlO(l, a) + da Am-l.o(m, a), m  2.

666
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Then the identity
Z m m-l
f t a - 1  Amj(m, Q) ln j t IIhn& (x - t)dt = _x a - 1  Am-l,j(m - 1, Q) ln j x
o J=O J=O
(32.28)
is valid.
Prool. If m = 1, then in view of (32.27) we have A ll (I,Q) = 1, AlO(I,Q) =
h 1 - ,p(Q). If m = 2, then A 22 (2, Q) = 1, A 21 (2, Q) and A 20 (2, Q) are given by
(32.24). Therefore (32.28) coincides with (32.12) for m = 1 and with (32.23) for
m = 2. Changing h 1 to h 2 and h 2 to h3 in (32.23), differentiating this relation
with respect to Q and summing it with (32.23) itself (with h 1 = h 2 and h 2 = h 3 )
multiplied by AlO(l, Q) = h 1 -1/J(Q), we obtain (32.28) with m = 3. Continuing
this process and using the method of mathematical induction we prove that (32.28)
is valid for an arbitrary natural m, and this completes the proof. .
Let now the constants h 1 , . . . , h m be connected with the constants Am,m-l, . . . ,
Amo(Amm = 1) by means of (32.27) and let J:+,..., J:+ be the operators given
in (32.19) with h = hi,..., h m respectively. Applying the operators J:+,..., J:;
successively to both sides of (32.3), using Lemma 32.2 and the assertions of S 2 we
obtain the following theorem.
Theorem 32.1. Let the constants hl"'" h m be connected with the constants
Amm = 1, Am,m-l"'" Amo by (32.27) and let J:+, . . . , J:+ be the operators given
in (32.19). Then (32.3) is solvable together with the equation
P;+cp = (_I)m J:+,..., J:+ f
(32.29)
and its unique solution has the form
cp = (-I)m1':+J:+,..., J:+ f
(32.30)
where 1:+ is the fractional integral (2.17) and 1':+ is the fractional derivative
(2.30).
In particular, if Q = 1, then the unique solution of (32.3) with the pure
logarithmic kernel has the form
cp(x) = (_I)m :x J:+,...,J:.;f
(32.31)

 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
667
Now we investigate (32.27) which reflects connections between the constants
Amm, Am,m-l, . . . ,Amo involved in (32.3) and the constants hI,. . . ,h m contained
in its solution (32.30). If m = 2, then (32.27) coincides with (32.24) which by the
Vieta theorem is the same as the numbers hI - .,p( a) and h 2 - .,p( a) which are roots
of the quadratic equation
Z2 - A 21 Z + A20 + .,p'(a) = O.
(32.32)
If m = 3, then (32.27) has the form
A33(3, a) = 1,
A 3 2(3, a) = hI + h 2 + h3 - 3.,p(a),
A 31 (3, a) = [hI - ,p(a)][h2 - ,p(a)] + [hI -1/J(a)][h3 - ,p(a)]
+ [h 2 - .,p(a)][h3 -1/J(a)] - 3,p'(a),
A30(3, a) = [hI -1/J(a)][h2 - ,p(a)][h3 - ,p(a)] - A3,2(3, a).,p'(a) - .,p"(a).
By the Vieta theorem this is equivalent to the fact that the numbers hI - ,p( a),
h 2 - ,p( a) and h3 - ,p( a) are roots of the cubic equation
Z3 - A 32 Z 2 + [A 31 + 3,p'(a)]z - [A 30 + A32,p'(a) + ,p"(a)] = O.
(32.33)
Similarly, in the general case of an arbitrary natural m the numbers h 1 -
1/J( a), . . . ,h m - 1/J( a) are roots of a certain algebraic equation of power m, namely
zm - am_IZ m - I + ... + (_I)m- 1 alz + (-I)m ao = 0,
(32.34)
with the coefficients am-b", ,al,aO expressed via the constants Am,m-l"" ,Amo
(Amm = 1) involved in (32.2), for example, am-l = Am,m-l' etc.
32.3. The solution of equations with real powers of logarithms
We pass on to solving (32.1). We note at once that (32.13) leadsto the following
statement.
Lemma 32.3. The solvability of the equation
:&
f JJQ,{j(x - )<p(t)dt = f(x)
a
(32.35)

668
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
in L( a, b) is equivalent to that of the equation
:& :&
f cp(t)dt =  f !ll-OI,-fJ(x - t)f(t)dt
a a
(32.36)
in L(a, b) where !lOl,{3(X) and !ll-OI,-fJ(x) are the functions (32.11).
We introduce the notation
{ ( ) 01-1
1 !: In-{31.
r(OI) .., :& '
U Ol ,{3 = U OI ,{3(X) = -1
P () In-{3-1 ;,
a > 0, -00 < P < 00,
(32.37)
a = 0, 0 < P < 00,
and observe that u OI ,{3(x) is the main term of asymptotic expansion of the Volterra
function (32.11):
!lOl,{3(X) = uOl,{3(x)[1 + 0(1)] as x -+ +0.
In the following the ensuing lemma plays a fundamental role.
Lemma 32.4. Let c(x) be an absolutely continuous function on (0, b - a] and the
relations
Co = lim c(x),
.2:-0
( ) _ c( x) U OI ,fJ (x)
r x - ( ) - Co
!l0l ,fJ Z
(32.38)
be valid. Then for any function cp(x) E L(a, b) the relation
:& :&
f c(x - t)UOI,fJ(X - t)cp(t)dt = Co f JJOI,{3(X - t)cp(t)dt
o a
:& t
+ f JJOI,{3(X - t)dt f 1/J(t - r)cp( r)dr
(32.39)
a
a
is true where
:& 11
1/J(x) = - f r'(y)dy f !lOl,{3(t) :x !ll-OI,-{3(x - t)dt E L(O, b - a).
o 0
(32.40)

 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
669
Proof. First we show that ,p(x) E L(O, b - a). Let Q > 0, p > ° and [,8] be the
entire part of p. In view of (32.16) we have
b-a b-a 11 dt b-a ln -,6 1.l n ,6 J.-.
I 1,p(z)ldz:5c I Ir'(y)ldy I t ' - a I (z  t)1+:-' dz
o 0 0 11
b-a 11 b-a ]
=c I I r'(y)ldy l _ I dt I [ In  + In r ]
t -01 X - t t
0011
x [( In,6-]  -In,6-] r ) + In,6-] r ] In-,6 r(x - t)-I-OIdx.
x-t t t t
Applying the binomial formula and carrying out the change x = tr we obtain
b-a ] b-a 11
I 1,p(z)ldz:5c L (]) I Ir'(y)ldy I t'!!.a
o 1=0 0 0
. (I 1 ,6-[,6] )
b-a IIn :I::t r In :I::t + In,6-] t
x I In P -[.8JH 1. (z _ t)'+a dz
11 t
] b-a 11 (b-a)/t 1 1 I 1 ,8-]+i I I I I i
"" ( [,8] ) I I ' ( )Id I dt I n T=T + n T=T d
c i ry y t (r-l)1+01 r
1=0 0 0 1I/t
] b-a (b-a)/1I 00
:5c 1; (]) I 1r'(y)ld Y ( I In T + I In b  a )
1-0 0 I (b-a)/1I
I I 1 ,6-]+i I I I i
In T-I + In T-I
x (r _ 1)1+01 dr
(I 1 ,6-]+i I I i )
] ( [,8] ) I b-a 1 00 In T':l + In r':l Inr
c  i Ir'(y)ldy (r _ 1)1+01 dr
1=0 0 1
< +00,
The cases Q = 0, p > 0 and Q > 0, p < ° are considered similarly.

670 CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
:&
We use Lemma 32.2 now. Setting j(x) = f c(x - t)u a ,,6(x - t)<p(t)dt and
a
taking (32.38) into account we have
:&
 f fll-a,-,6(x - t)j(t)dt
a
:& t
=  f I-'l-a,-p(z - t)dt f[cOl-'a,p(t - T) + r(t - T)I-'a,p(t - T)]I"(T)dT
a a
:& :& t
= Co f <p(t)dt +  f fll-a,-,6(x - t)dt f r(t - T)fla,,6(t - T)<p(T)dT.
a a a (32.41)
One may check directly that
:& t
 f fll-a,-,6(x - t)dt f r(t - T)fla,,6(t - T)<p( T)dT
a a
:& t
= f dt f 1jJ(t - T)<p(T)dT.
a a
Then from (32.41) it follows that
:& :&:& t
 f I-'l-a,-p(z - t)f(t)dt = Co f I"(t)dt + f dt f .p(t - T)I"(T)dT.
a a a a
But according to Lemma 32.3 the relation
:& t
j(x) = f fla,,6(x - t)(co<p(t) + f 1jJ(t - T)<p(T)dT]dt
a a
is valid. Hence (32.39) follows. The lemma is thus proved. .
Further we consider the operator
:&
(T1/J<p)(x) = f 1/J(x - t)<p(t)dt.
a
(32.42)

 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
671
It is completely continuous in Lp(a, b), 1  p  00, and has zero spectral radius,
Zabreiko [1]. Therefore the operator caE + Tt/J is invertible in L,(a, b) and the
following statement giving the criterion of solvability of (32.1) in L,(a, b) follows
from Lemma 32.4.
Theorem 32.2. Let a function c(x), c(O) ¥ 0, satisfy conditions of Lemma 32.4
and let u a ,,8(x) have the form (32.37). The equation
:&
f c(x - t)u a ,,8(x - t)cp(t)dt = f(x)
a
(32.43)
is solvable in L,(a, b), 1  p  00 if and only if the free term f(x) is representable
in the form
:&
f(x) = f lla,,8(x - t)X(t)dt, X(t) E L,(a, b).
a
(32.44)
This condition being satisfied the equation has the unique solution given by the
expression
'{J = (coE + T",)-1 :'" (  i 1'1-a,-p('" - t)f(t)dt )
where (caE + Tt/J)-l is the operator inverse to the operator
(32.45)
:&
(caE + Tt/J )cp(x) = cacp(x) + f 1/J(x - t)cp(t)dt
a
(32.46)
and the function 1/J(x) is defined by (32.40), (32.11) and (32.38).
Corollary. Equation (32.2) is solvable in L,(a, b), 1  p  00, if and only if the
free term f(x) is representable in the form (32.44). This condition being satisfied
the equation has the unique solution cp given by the expression
'{J = (E + T",) -1 :'" (  i 1'1-a,-p( '" - t)f( t)dt )
(32.47)
A characterization of the space of the right-hand side f of (32.43) via a
convolution with the Volterra function JJl-a,-,8 is contained in Lemma 32.3 which
is a generalization of Theorem 2.1 for equations of the form (32.44).

672
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Remark 32.2. It is easily shown that the relation 1:/ <p = c, <p E L1" is impossible
in general if c ¥ 0 in the case p  1/0:, 0 < 0:  1. Namely, if p < 1/0:, {3 E Rl or
p = 1/0:, (3  -1 + 0: then <p == c == o.
 33. The Noether Nature of Equations of the
First Kind with Power-Logarithmic Kernels
The present section deals with the investigation of the N oether nature of integral
equations of the first kind with power-logarithmic kernels
b
- f c( x, t) P 1 _
K<p = Ix _ tll-a In Ix _ tl <p(t)dt - f(x),
a
a < x < b,
(33.1 )
where [a, b] is a finite interval of the real axis, -00 < {3 < 00, 0  0: < 1 ({3 < -1
if 0: = 0) , "( > b - a, and the function c(x, t) has a jump of the first kind at the
diagonal x = t:
{ u( x, t), t < x,
c(x, t) =
v(x, t), t > x.
Equation (31.1) considered in S 31 is a particular case of (33.1) for 0 < 0: < 1,
(3 = O. We shall study the Noether nature of the operator K in the corresponding
proper setting, namely from the space Lp == L1'(a, b), 1 < p < 00, into a special
Banach space X. We shall show that the appearance of the factor In P Iztl
weakening or strengthening a singularity of the kernels at the diagonal x = t
(depending on the sign of (3) and in the kernel of the potential type operator (33.1)
changes only the range of the operator K. As for the Noether nature of (33.1), it
remains the same in the sense that the Noether properties of K from Lp into X
are equivalent to that of a certain singular integral operator in Lp which does not
depend on {3.
The scheme of investigation of the Noether properties of (33.1) is the same
as for (31.1) with power kernels in S 31. We note only that the presence of
the logarithmic factor with power unity considerably complicates the proof of
the imbedding theorems for the ranges of the operators 1:/ and I!, with
power-logarithmic kernels (see (21.1» playing an important role, and the identities
giving a connection between these operators with singular integral operators. To
derive the former we use properties of the Volterra function (32.11) obtained in
S 32, and to obtain the latter we use the method of differentiation of arbitrary
order with respect to a parameter in the relation connecting fractional integrals
with singular operator. The results obtained are used in the investigation of the
property of the operator (33.1) as to whether it is Noetherian.
(33.2)

 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
673
33.1. Imbedding theorems for the ranges of the operators I:f
an d pJ/ ,fj
b-
Let 0  a < 1, -00 < {3 < 00. We recall the notations:
x fj -::L.
OI,fj _ 1 f 10 x-t ( )
10+ cp - r(a) (x _ t)l-OI CP t dt,
o
b fj -::L.
OI,fJ _ 1 f In I-x
I b _ cp - r(a) (t _ X)l_OI CP(t)dt.
x
(33.3)
If (3 = 0, 0 < a < 1, we shall use the symbols 1:+ cp and I b _ cp since in this case the
integrals (33.3) coincide with the fractional integrals (2.17) and (2.18). In the case
a = 0 we shall omit the factor l/r(a) in (33.3) and suppose that (3 < -1.
We introduce the Banach spaces of functions representable by (33.3) with
power-logarithmic kernels with the density cp E L,(a, b), 1 < p < 00:
I:/(L,) = {f: f = I:fcp, cp E L,(a,b); II flllcr' ( L ) = IIcpIlL p }'
-+ p
(33.4 )
I:!(L,) = {f: f = I:!cp, cp E L,(a, b); IIfIlI:':(Lp) = IIcpIlL p }'
(33.5)
We obtain the imbedding theorems for such spaces as a corollary of a more general
assertion. We introduce the Banach space
x
I:+(L.) = {I : I(z) = f g(z - t)1"(t)dt, I" E L.(a, b); II/III:+(L.) = III"IIL. },
o
g(x) E L(O,b - a), 1  p  00.
(33.6)
more general than (33.4) and (33.5). The following theorem is derived from the
result of S 32.
Theorem 33.1. Let 1'00,fj(X) be the Volterra function (32.11) and uOI,fJ(x) be its
main part (32.37) near the origin and c(x) be an absolutely continuous function on
(0, b - a] and Co = lim c(x). Then the following statements are trne:
x-o
1) If Co f; 0, then
l:cr'(L,) = l:+'(L,).
(33.7)

674
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
2) If Co = 0, then the imbedding
[ CUOI./I ( L ) -+ [IJOI./I ( L ) _ IUOI./I ( L )
0+ , 0+ 'I' - 0+ ,
(33.8)
is valid and the operator of this imbedding is completely continuous.
Prool. The assertions of theorem follow from Lemma 32.4 of the previous section:
the former in view of the boundedness of (32.46) in L,(a,b) and the latter in view
of the complete continuity of the Volterra operator (32.46) in L,(a, b). .
Corollary. Let
.1  p < 00, 0  Q < 1, 0 < 6 < Q if Q > 0, -00 < Pl < P2 < 00.
Then the imbedding
1:(1 (L'P) -+ [:f'J(L,) if Q  0 (P2 < -1 for Q = 0),
1:(1(L'P) -+ l:f'J(L,) -+ 1:;6(L'P) if Q> 0, Pl > 0
(33.9)
(33.10)
are valid and the operators of these imbeddings are completeiy continuous.
33.2. Connection between the operators with
power-logarithmic kernels and the singular operator
Earlier in S 11, the relations in (11.16) and (11.17) connecting the Riemann-
Liouville fractional integration operators 1:+ and I b _ with the singular operator
S were found. They played an important role in the investigation of the N oether
nature of the potential type operators (31.55) with the power-type kernel on a
finite interval [a, b). It is natural to expect that a connection between the operators
I::t and I! with power-logarithmic kernels and the singular operator S are to
play similar role for the potential type operators (33.1) with the power-logarithmic
kernel. Proceeding from (11.16) and (11.17) we shall find such connections by
applying the method of differentiation of an arbitrary order with respect to the
parameter Q to these equations. It will allow as to pick out a natural power of
logarithm under differentiation of an integer order, while a modification of this
method will help to cover the case of an arbitrary order.

 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
675
We introduce singular integrals with power-logarithmic kernels
1 I " ( t - ) 0/ lnm t-a
(Sa'O/lmCP)(X) = -  t z-a cp(t)dt,
1r x-a -x
a
"
1 I ( b - t ) 0/ ln m .!=.!.
(S",O/lmCP)(X) = ;: b _ x b _za cp(t)dt,
a
(33.11 )
dq dq
Sa,O/CP = Sa,O/,OCP, S",O/CP = S",O/,OCP'
Lemma 33.1. If m = 0,1,2,. .. and
1 < p < 00, -1 + IIp < Q < IIp,
(33.12)
then the operators Sa,O/, m and Sb,O/,m are bounded in the space L,(a,6).
Proof. The assertion of the lemma follows from Theorem 11.3 if m = 0 and from
Theorem 1.5 if m = I, 2, . . . .
Remark 33.1. It is known that the conditions given by (33.12) are not only
sufficient but also necessary for boundedness of the operators Sa,O/ and S",O/ in the
space L,(a,6), Gohberg and Krupnik [5].
Remark 33.2. Let II Sa 0/ m II = max II Sa 0/ m Il L be the norm of the operator
, I II/PULp=l I I P
So,O/, m ' Then Theorem 1.5 yields the estimate
00
liS II < .!. I to/-l/, Iln m tl dt 2
a,O/, m - 1r It _ 11 ' m = I, ,...
o
( 33.13)
The similar estimate for the operator S",O/, m is also valid.
We rewrite (11.16) and (11.17) in the form
I b _ = 1:+ [cos(Q1r)cp + sin(Q1r)Sa , O/cp], cP E L,(a, b), 0  Q < IIp,
(33.14)
1:+ = Ib_[cos(Q1r)cp - sin(Q1r)S",O/cp], cP E L,(a, b), 0  Q < IIp,
(33.15)

676
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
interpreting them in the limiting case a = 0 as <P = <po Similar identities are also
true for the operators I::t and I:! with power-logarithmic kernels. Namely, the
following theorem holds.
Theorem 33.2. Let <p E L,(a, b), 1 < p < 00, 0  a < IIp, -00 < (3 < 00. Then
the relations
I:.:..fJ <p = I:f[cos(a1l')<p + sin(a1l')So,a<P + Tl<P],
I::t <P = I:! [cos( a 11') <P - sin(a1l')Sb,a<P + T2<P],
(33.16)
(33.17)
are valid where Tl and T2 are operators completely continuous in L,(a, b).
Prool. We have to prove (33.16). If {3 = 0 it coincides with (33.14). If {3 f; 0 then,
as was said above, to obtain the required result we apply differentiation of order
{3 with respect to a to (33.14). We begin with the case of natural (3 = 1,2,....
Multiplying (33.14) by r(a) and differentiating the relation thus obtained (3 times
with respect to a and dividing by r( a) we have
I:!'P= t (  ) I::t-jNt ({3= 1,2,...),
J=O
(33.18)
where
N  - dj(cosa1l') E E j ( j ) {ji-isin(a1l') S .
J - d' + . {)" (I a I'
aJ I aJ -I ' ,
i=O
Hence we obtain (33.16) for natural {3 by the Corollary of Theorem 33.1 and
Lemma 33.1.
Let now {3 be an arbitrary real number. It would natural to think that we have
now to apply differentiation (or integration) of an arbitrary order (3 to (33.14).
However we can not use the operators of Liouville fractional integra-differentiation
since the operator So,a of the form (33.11) does not preserve the space L,(a, b) if
IIp < a < 1, and (33.14) is not true if a  1. It turns out that in this situation
the truncated operator
-fJ a ( r(t) + )
(I+f)(a) = ca'Y 1_ ---:ytPa,ef (a),
c a = { l/r(a),
1,
a> 0,
a= 0,
(33.19)
is well-suited for our purpose, where P:' e is the projection
(p+ f)(t) = { f(t), t: [a, a + c] 0  a < a + c < IIp.
a,e 0, tE[a, a + c] ,
(33.20)

 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS 677
We replace Q by t in (33.14) and then apply (33.19) with respect to t.
A. First we transform the left-hand side of the relation obtained. For {3 > 0
(if Q = 0 we take (3 > 1) by (33.19) we have
] £ 6
-fj t _ 01  r(t)dt 1 tp(y)dy
(1+1 6 _",)(",) - Co'Y r(,8) 'Y'(t - ..)1-11 r(t) J (y - ",)'-'
01 :&
6 £ T
J cp(y)dy J rfJ-l ( y - X )
= C OI (y _ X )1-01 r({3) 1 dr.
:& 0
U sing the result
£ [ 00 ]
r fJ - 1 X T 'Y -fj 1
J r(,8) () dT = (In;) 1 - r(,8) J '-p-'e- T dT
o £ In ;.
({3 > 0),
(33.21 )
we find
6
( lfJ It l/'J )( X ) = ( IOI,-fj l/'J )( X ) - C J >'fJ(t - x) l/'J ( t ) dt.
+ 6-T 6- T 01 (t_X)l-OI-£T
:&
(33.22)
Here
00
>'fJ(X) = X-£ In- fJ 2 J e-TrfJ-ldr = 0 ( In- l 2 ) as x -+ 0
r({3) x x
£In;'
(33.23)
according to the relation
j e-'tO-'dt = uO-'e- u [1 + 0 CI )] as lul-+ 00
u
(33.24)
as given by Erdelyi, Magnus, Oberhettinger and Tricomi [1, 6.9.2(21), 6.13.1(1)].
In view of Theorem 33.1
6
J >'fj(t - X) ( ) dt 01+£
C OI (t_x)l-OI-£ tpt =1 6 _ Vltp,
:&

678
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
where VI is a completely continuous Volterra operator in L,(a, b). Everywhere in
the following we shall denote by V 2 , V 3 ,... Volterra operators completely continuous
in L,(a, b). Hence according to (33.14) and the Corollary to Theorem 33.1 we have
b
f A{3(t - x) ( )d a+e N TT I a -{3" V;
C a (t _ x)l-a-e cP t t = I a + a+e VICP = a.t V2 N a+e ICP,
:&
where Ntcp = cos(t1r)CP + sin(t1r)Sa"CP. Substituting this expression into (33.22) we
obtain the relation
I -{3 I t I a -{3 I a -{3"
+ b- cP = b": cP - a.t V3CP
(33.25)
where V3 = V2Na+e VI.
Let now (3 < 0, cr > O. Then, using (33.18)-(33.20) and (33.23) and (33.21)
and (33.23) we have
d[-{3]+1 r ( t )
j{3 It cP -c ",a ( _I ) [-{3]+1 11+{3+[-{3]_p+ It cP
+ b- - a, dcr[-{3]+1 t "It a,e b-
b
d[-{3]+1 f ( ) a
=c a 'Y a (-I)[-{3]+1 r - x
dcr[ - {3]+ I "I
:&
[I -1-{3-[-{3] -1- \ ( )]
X n T-:& _ A I+{3+[-{3] r - X cp(r)dr
r-x (r-x)l-e
f b ( ) a In[-{3]+l -2-
=I:..:-{3cp-C a r-x ( )IT_:& AI+{3+[-{3](r-x)cp(r)dr.
:& "I r - X (33.26)
Taking (33.23) into account, by Theorem 33.1 and (33.14) we find
b ( ) a In[-{3]+l -.:L
C a f TZ (T-Z)t:: .x1+P+[-PJ(T-Z)(T)dT= I_V4
:&
= I:+N a V 4 cp = I:.t-{3V s N a V 4 cp.
Then from (33.26) we obtain (33.25) with V 3 = VsN a V 4 .
B. Now we transform the expression j cos(t1r)I:+cp. If {3 > 0, then by

 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
679
(33.18)-(33.20) we have
z a+£ t
-P t _ a j cp(y)dy 1 j cos(t1r) ( X - Y )
1+ cos(t1r)Ia+cp - Ca'Y X _ Y r(p) (t _ a)l-P ---:;- dt.
a a
(33.27)
Integrating the inner integral by part and taking (33.21) into account we obtain
a+£ t
1 j cos(t1r) ( X - Y ) ( X - Y ) a _P 'Y -
r(p) (t - a)l-P ---:;- dt = ---:;- In X _ y [cos(a1r) - A(X - y)],
a
where
00
(u) = j
£In;'
e- T ; dT + j sines + n)rds j
o £In;.
TP-l
e- r r(p) dT -+ 0
as u -+ O.
Hence according to Theorem 33.1 (33.27) takes the form
i cos(t1r)I:+cp = I:+-P[cos( a1r)cp + V 6 CP].
(33.28)
For p < 0 a similar result is found in the same way as in case A.
C. Finally we consider the expression i+ sin(t1r)Sa,tCP. Let p > O. Carrying
out integration by parts as in case B, then according to (33.11) we have
z ( £ )
d ' R-l
-P t . y. X - Y SfJ
1+1.+ sm(t1r)S...<p=c a ! (z_y)'-a I sm(a+s)1r (,) riP) ds (S..a<p)(y)dy
£ [ Z In- P -L. ]
+ I ! (z_y)la A(z-y, T)(ST+a.l<p)dy dT,
(33.29)
where
£ In .1.
A(U,T) = rp) j'e-'s fJ - 1 Sin(n+sln- 1 'fJrds.....o as U.....O.
r In ;.
The expression similar to the first term on the right-hand side of (33.29) (with
sine replaced by cosine) was considered in case B. Therefore using Theorem 33.1

680
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
we reduce (33.29) to the form
j + sine tr )8.,0'1' = I.j.-II [sine "..)8.,0'1' + Vd i V. (8,+0,1'1' )dr]. (33.30)
Here the operator-function Vi is completely continuous in L,(a, b) for T > O. If
T = 0, then complete continuity does not hold in view of the first statement of
Theorem 33.1.
If we return to the proof of Lemma 32.4 of the previous section we can check
that the operator-function V T is continuous with respect to T in the operator
topology on (0, c) and uniformly bounded with respect to T on [0, c). Also in view
of (33.13) we have
00
IIST+a,ll!  c =  f t a + T - l /'lt - ll- 1 IIntldt.
o
£
Therefore the operator f VTST+a,ldT is completely continuous in L,(a, b), and so
o
we obtain the result
jI:+ sin(t1l')Sa,a<P = 1:+-.8 [sin ( Q1I')Sa,a<P + Vs<p).
(33.31 )
from (33.30). In the case (3 < 0 an analogous result is found by arguments similar
to those above.
Collecting (33.25), (33.28) and (33.31) obtained in the cases A, Band C we
arrive at (33.16). Relation (33.17) is derived from (33.16) by applying the operator
A : f(x) -+ f(a + b - x) to the left and to the right. The theorem is proved. .
From (33.16) and (33.17) the coincidence of the space given in (33.4) and
(33.5) follows. Namely, the following assertion is true.
Corollary. If 0  Q < IIp, then the Banach spaces I:/(L,) and I!(L,)
coincide up to the equivalence of norms:
1:/ (L,) = I!(L,)':YIa,{j(L,).
(33.32)

 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
681
33.3. The Noether nature of equations (33.1)
We consider (33.1) which according to the supposition (33.2) has the form
z b
- f u(x,t) fj 'Y ( f v(x,t) fj 'Y
Kcp = ( )1- In -cp t)dt + ( )1- In -cp(t)dt.
x-t 01 x-t t-x 01 t-x
a z
(33.33)
where -00 < {3 < 00, 0  a < 1 ({3 < -1 if a = 0).
We assume the functions u(x, t) and v(x, t) to satisfy the following conditions
similar to conditions 1) and 2) for (31.55) in the case 0 < a < 1: thus
1) u(x,x) = u(x,x - 0) E C([a,b]), v(x,x) = v(x,x + 0) E C([a,b]);
2) in the case a < 1 the functions u(x, t) and v(x, t) are Holderian of order
 > Q with respect to x and uniformly in t;
3) in the case a = 1 the functions u(x, y) and v(x, t) are differentiable with
respect to x and the inequalities
l au l Cl
ax  (x - t)l-£l '
I av I < C2 0
ax - (t - X)1-£l' €l > ,
lu(x, t) - u(t, t)1  C3(X - tY, Iv(x, t) - v(t, t)1 < C4(t - xY, €2 > 0,
hold where Cl, C2, C3 and C4 are some positive constants.
Let us consider the case 0  a < IIp ({3 < -1 if a = 0). By the Corollary of
Theorem 33.2 we shall study the mapping properties of the operator K from the
space L,(a, b) == L" 1 < p < 00, into the space [OI,fj(L,) defined by (33.32).
We represent the operator given in (33.33) as
K = K OI ,,8 + r:,fj + T;',8
(33.34)
where
K 01 ,fj cp - [01 ,fj Ul" + [01 ,fj Vi"
- a+ T b- T'
(33.35)
z
( 1?,fj )( x ) =  f u(x, t) - u(t, t) In,8....:L (t)dt
1 cp r(a) (x - t)l-OI X - tCP ,
a
b
( ,fj )( x ) =  f V(x, t) - v(t, t) In,8....:!...- ( t ) dt.
2 cp r(a) (t-X)1-OI t-xCP
z
(33.36)

682
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Lemma 33.2. [f 1 < p < 00, 0  Q < l/p and u(x, t) and v(x, t) are Holder'ian
functions of order A > Q with respect to x uniformly in t, then T:,/J and T;,/J are
completely continuous operators from L,(a,b) into [OI,/J(L p ) and
mOl ,/J _ [ 01 ,/J V; - ,..,,01 ,/J _ [ 01 ,/J iT
.11 - a+ 1, .L2 - b- Y2,
(33.37)
where the operators VI and V2 are completely continuous in L,(a, b).
Prool. If 0 < Q < 1, {3 = 0, then the assertion of thus lemma coincides with
LeJ"ma 31.4 and
mOl ,0 [ 01 V; - ,..,,01,0 [ 01 iT
.11 = a+ 1, .L2 = b- Y2,
(33.38)
where the operators
:& :&
V; = QSinQ1r j ( s ) ds j u(x,s)-u(t,s) dt
1<P 1r <P (t - S)1-OI(X - t)1+01 '
a 8
(33.39)
b 8
y; = QSinQ1r j ( s ) ds j v(x,s)-v(t,s) dt
2<P 1r' <P (s _ t)I-OI(t _ X)I+OI
:& :&
(33.40)
are completely continuous in L,(a, b).
We transform (33.38) by using the operator given in (33.19), following the
same lines as for (33.14) and (33.15) in the proof of Theorem 33.2. After some
transformations similar to those above with Theorems 33.1 and 33.2 being taking
into account, we have
T:,/J = [:f[sin(Q1r)Vl + T 3 ], T;,/J = [..:/J[sin( Q1r)V2 + T 4 ] (0  Q < l/p),
where the operators T3 and T4 are completely continuous in L,(a, b). Hence the
truth of the theorem follows in view of the complete continuity of the operators VI
and V2 in L,(a, b). .
From (33.27) it follows that the Noether nature of the operator given in (33.3)
is equivalent to that of the model operator given in (33.35). The latter in view of
Theorem 33.2 is representable in the form
KOI,/J<p = [:f[NOI<P + Ts<p]
(33.41 )
where N 01 is the singular integral operator given by (31.58) and Ts is a completely
continuous operator in L,(a, b).

 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
683
We conclude from (33.41) that the Noether nature of the operator K from
Lp(a, b) into IOI,IJ(L,) is equivalent to that of the singular integral operator
& 01
sin(a1l') f ( t - a ) v(t, t)cp(t)dt
NOICP = [u(x, x)+v(x, x) cas(a1l')]cp(x)+ - (33.42)
11' x-a t-x
a
in L,(a, b). Using here Theorem 31.7 about the Noether nature of the operator N OI
we obtain the following statement similar to Theorem 31.12.
Theorem 33.3. Let 1 < p < 00, 0  a < IIp and -00 < {3 < 00 if a > 0 and
(3 < -1 if a = 0 and let the functions u(x, t) and v(x, t) satisfy conditions 1) and
e) indicated above. The operator given by (33.33) is Noetherian from L,(a, b) into
IOI,IJ (L p ) if and only if
1) u 2 (x, x) + 2u(x, x)v(x, x) cas a1l' + v 2 (x, x) f; 0, a  x  b; (33.43)
2)
8(a) f; 211'( -a + IIp), 8(b) f; 211'1 p ' (mod 211'),
(33.44)
where
i9(z) _ u(x, x) + v(x, x)e- i0l 1l'
e - u(x, x) + v(x, x)e i0l 1l' , 0  8( a) < 211'.
These conditions being satisfied the index of the operator K is equal to
(33.45)
{ k,
'" = "'L _I01'/'(L ) =
., ., k - 1,
if 0  8(a) < 211'( -Q + IIp),
if211'( -Q + IIp) < 8(a) < 211',
(33.46)
where k is the integer number defined by the condition
8(b) - 211'k E (-211' Ip, 211' Ip/).
(33.47)
In particular, if a = 0 and (3 < -1, then the operator given by (33.33) is
Noetherian if and only if
u(x,x) + v(x,x) f; 0, a  x  b,
(33.48)
and the index is equal to '" = "'L.,_I0,II(L.,) = O.
Comparing Theorem 31.12 with Theorem 33.3 we obtain an interesting fact.

684
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Theorem 33.4. The appearance of the factor In,6 Iztl in the kernel of the
potential type operator
b
- 1 f c( x, t)
M<p= r(Q) Ix_tI1-a <P(t)dt
a
which weakens or strengthens (depending on the sign of (3) a singularity of the
kernel on the diagonal x = t, changes the range of the operator but does not
influence its Noether properties in the sense that if 0 < Q < IIp, then the Noether
nature of the operator K from L,(a,b) into [a,,6(L,) is equivalent to that of the
singular integral operator given by (33.42) in L,.
Remark 33.3. Theorem 33.3 may be extended to the case of the space L,([a, b], p)
n
with the general power weight p( x) = n Ix - XI: l#Jk . The theorem about the
1:=1
Noether nature of the operator K from the Holder weighted space H&(p) into the
generalized Holder weighted space H;+a,,6(p) is also valid - S 34.2, notes 33.1 and
33.2.
Remark 33.4. Theorem 33.3 presents the N oether nature of the operator given
by (33.33) in the case 0  Q < IIp. A similar assertion is also true in the case
IIp < Q  1 (see S 34.2, note 33.3). In particular, if Q = 1 and {3 > 0 the Noether
nature of the operators given in (33.1) with pure logarithmic kernels is valid. More
general investigations of the N oether nature of operators with such kernels have
also been carried out (see S 34.2, note 33.4).
 34. Bibliographical Remarks and
Additional Information to Chapter 6
34.1. Historical notes
Notes to  30.1. Here we confined ourselves to the case of equations on the axis and on an
interval of the axis. The reader may meet the theory of singular integral equations in more detail
in the books by Gabov [1] and Muskhelishvili [I), and also Gohberg and Krupnik [4], where these
equations are considered on curves in the complex plane.
Notes to  30.2 and 30.3. The generalized Abel equation, both in the case of inner and
external coefficients, on a finite interval and on a half-axis first &rose in Zeilon [1] (1924). In
connection with this paper, which was unknown to researchers who dealt with generalized Abel
equations between 1960 and 1970, one should refer also to  17.1, Notes to  11.2 and 11.3. In
this paper by Zeilon a certain fonnalism was proposed of reducing the generalized Abel equation
to the characteristic singular equation, or to the Hilbert boundary value problem for analytic
functions connected with the latter. This paper also contained the first attempt to consider
systems of generalized Abel integral equations with constant coefficients. Zeilon suggested a
method of reduction to a singular equation, which is surprisingly original for the year 1924. He

 34. ADDITIONAL INFORMATION TO CHAPTER 6
685
stated that the solution of the generalized Abel equation could be reduced to the successive
solutions of a singular equation and the usual Abel equation. However, he did not give either
a correct solution of the singular equation or an investigation of its solvability at all. Such a
solution was known to be obtained much later by Muskhelishvili [1] and by Gabov [1]. The
solution of the generalized Abel equation with a sufficiently complete investigation of solvability
was first given in this way by Sakalyuk [1] (1960) and [4] (1965) in the case of a finite interval.
Sakalyuk used Carleman's method of analytic continuation. The restricted assumptions made in
these papers were essentially weakened by Wolfersdorf [2] (1965) and Samko [1] (1967) and [5]
(1968). In the papers by Samko the solution was based on another method presented in  30. It is
based on the direct connection with the singular operator which allows one not only to clarify the
connection between admissible solutions and right-hand sides of the generalized Abel equation,
but to investigate general equations of the first kind with a power-type kernel as well.
The generalized Abel equations (30.17) and (30.18) on the whole axis were solved by Samko
[4] (1968), (7) (1969) and [9] (1970). The presentation in  30.2 and 30.3 follows Samko [27,  3
and 9] (1978), see also a survey by Samko [32].
Notes to  30.4. The explicit solution (30.67) was given by Samko [10] (1970), [14] (1973).
Refer also [13] (1971) where the solution of this equation was obtained in the form (30.63). The
cases in (30.72) were noted by Samk.o [5] (1968). These cases were in general known earlier.
We refer to (30.73) in Stein and ZygImmd [2] (1965) for functions <p e L2(Rl) with a compact
support and (30.74) in Heywood (1967). Heywood [2] (1971) showed that the integrals in (30.73)
and (30.74) are conditionally convergent for almost all x if I(x) e 1 01 (L,,), 1 < p < 1/01. The
solution of (30.72) in certain spaces of generalized functions was investigated by Jones [2] (1970).
The generalized Abel equation (30.79) with constant coefficients on a finite interval was
specially considered by Wolfersdorf [3] (1969), who gave its solution in terms of integrals with
the hypergeometric kernel. The solution in the form (30.82) was given Samko [27,  9] (1978)
and it was presented in the book by Gabov [1] (1977). The assertion of Lemma 30.3 was noted
by Wolfersdorf [3] (1969) and repeated by Ganeev [1] (1979), [2] (1982). In the latter paper the
solution of the generalized Abel equation with constant coefficients on a finite interval was also
given via the hypergeometric function, but it differed from the result of Wolfersdorf [3].
The solution of (30.83) was first given by Carleman [1] (1922). His solution may be seen in
 34.2 below. Another form of this solution was obtained by Ahiezer and Shcherbina [1] (1957)
(see also  34.2). Williams [1) (1963), who considered (30.67) in connection with some problem of
electrostatics, anived at the same result. Krein [1] (1955) arrived at a certain method of solution
of integral equations in his investigations connected with the inverse Stunn-Liouville problem. In
particular, he obtained the solution of the Carleman equation (30.83). In this connection we refer
also to Mhitaryan [1] (1968).
The fonn (30.84) for the solution of the Carleman equation was indicated by Samko [3]
(1968) who also obtained (30.89).
Notes to  31.1. Theorems 31.1-31.4 are well-known theorems in the theory of Noetherian
operators. In abstract Banach spaces they were first obtained by Nikol'skii [3] (1943) in the
FredhoJm case when the index is equal to zero, and they were extended by Atkinson [1] and
Gohberg [1] (1951) to the case of non-zero index. One may use Gohberg and Krein [1] (1957) and
Kato [1] (1958) together with the books cited in the beginning of  31.1 to make an acquaintance
with the theory of Noetherian operators in Banach spaces in more detail.
Theorems of the type 31.5-31.7 are well-known in a more general case of an arbitrary smooth
curve in the complex pJane. We gave formulations only for the case of the axis or any interval
required. This case was considered especially by Widom [1] (1960). Theorem 31.5 is well-known,
as for example in Gohberg and Krupnik [4]. Theorem 31.6 is a classical version of statement on
the validity of Noether theorems for singular integral equations on an open contour - see the
books by Gahov [1] and Muskhelishvili [1]. However, the assumpt,ions on the kernel K:(x, t) in
these books were less general. Theorem 31.7 in a more general fonn was obtained by Gohberg
and Krupnik [1-3] (1968-1971). We used the expression given by Karapetyants and Samko [2]
(1975) for the recalculation of the index when writing the index in an explicit form.
Notes to  31.2. The proper setting of the Noether nature for equations of the first kind
with a power-type kernel developed here was suggested by Samko [6] (1968). He investigated the

686
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Noether properties of (31.6) in such a setting in the papers [12] (1971) and [14] (1978). The
presentation in  31.2 follows the latter paper in the main. Theorem 31.10 was proved by Rubin
[2] (1972).
Notes to  31.3. The results of this subsection were obtained by Samko [1] (1967), [6]
(1968) and [27] (1978) except for Theorem 31.12 concerning the Noether nature in L" on a finite
interval, which was given by Rubin [3] (1973). The result for L,(p) noted in Remark 31.4 was
obtained by Rubin [1] (1972) including the case a = -00 or b = 00. The Noether nature in L,(p)
in a more general weighted case was investigated by Rubin [2] (1972), [8] (1975).
The criterion of the Noether nature for the operator (31.55) from H6(p) into H;+Ot(p)
mentioned in Remark 31.5 was obtained by Rubin [7] (1974). We remark that the decisive
point here is the application of Theorem 13.13 on the isomorphism between the spaces H6(p)
and H+Ot(p), realized by the Riemann-Liouville operators (2.17) and (2.18). It allows one to
characterize the class of the right-hand sides f of (31.55) in the same simple tenns as the solution
c.p itseH.
The case vex, t) :: 0 in (31.55), i.e. the case of (31.67), was known long ago. It was already
investigated by Volterra [1] (1896) - see also Volterra [3, p. 100-101] - under an assumption on
the function u(x, t) which was stricter than in  31. We managed to weaken the assumptions On
the function u(x, t) in  31 by using the Marchaud form of fractional differentiation.
Notes to  31.4. The arguments on stability presented here should be considered as
well known. The use of the space lOt (Lp)(:: HOt,,) as the space of right-hand sides for integral
equations of the first kind with a weak singularity was suggested by Samko [5] (1968), [9] (1970).
Theorem 13.13 may be considered as an assertion following from Volterra's investigations [1]
(1896), if we use a subsequently obtained result concerning the boundedness in L, of the resolvent
operator of the second-order Volterra integral equation with a weak singularity.
Notes to  32.1. The idea of an application of differentiation and integration with respect
to the parameter Ot to (32.14) which leads to the relation in (32.12) was suggested by Volterra
[2] (1916). The analogous application of Liouville differentiation with respect to the parameter Ot
leading to (32.13) was demonstrated by Rubin [4] (1973), [9] (1976) and [to] (1977).
Notes to  32.2. The problems of the solution in closed form of the integral equations
(32.3) with power-logarithmic kernels and variable upper limit were first considered by Volterra
[2] (1916). The presentation of these results also contained in Volterra and Peres [1] (1924) and
[2, Ch. 7, O. 16181] (1936). Volterra found the solution of the simplest equation (32.18), and
indicated the method of solution of a more general equation in (32.3). This method is based on
the identity, generalizing the equality in (32.15), namely
:t: m m-I
J t Ot - I L Bmk Ink t II[(X - t)eh]dt = - x: L Bm-I,k Ink x, Ot > 0,
o k=O k=O
(34.1 )
where II(X) has the form (32.10), d. the relation in (32.28). We also observe that the solution of
the simplest equation (32.18) is contained in the books by M.M. Dzherbashyan [2, Ch. 5,  1.1,
p. 264] and Volterra [3, p. to2].
Lenuna 32.2 and Theorem 32.1 obtained by Kilbas were not mentioned earlier, as well as
equivalence of (32.27) in Lemma 32.2 to the fact that the numbers hI' h2, . " , h m in tenns of
which the solutions (32.20) of (32.3) is expressed, are connected with the roots of the algebraic
equation (32.34).
Notes to  32.3. The results of this subsection were obtained by Rubin [10] (1977).
Notes to  33.1. Theorem 33.1 was proved by Rubin [10] (1977). Earlier he obtained the
imbeddings in (33.10) in the cases of natural and nonnegative real powers of logarithm in the
space L,,(a, b) - see [4] (1973) and [9] (1976), respectively.
Notes to  33.2. The results of this subsection were given by Rubin [4] (1973) and [10]
(1977).
Notes to  33.3. The Noether nature of the potential type operators (33.1) with the
power-logarithmic kernel from the space L,(a, b), 1 < p < 00, into the spaces (33.32) was

 34. ADDITIONAL INFORMATION TO CHAPTER 6
687
investigated by Rubin for natural, nonnegative real and arbitrary powers of logarithm in the
papers [4] (1973), [9] (1977), respectively.
34.2. Survey of other results (relating to SS 30-33)
30.1. Carleman [1] obtained the solution of the equation
1
f Ip(t)dt = f(x) 0 < x < 1,
Ix - tll-a '
o
(34.2)
in terms of the contour integral
1
i sin ¥ d f dt f [ t(t - 1) ] a/2 f(lJ)dlJ
Ip(x)=-- - - -
21r 2 dx (t - x)a IJ(IJ -1) IJ - t
rr 0
(34.3)
where r  is an arbitrary closed contour intersecting the positive real half axis R only in the
point x.
30.2. The solution of the Carleman type integral equation
a
f Ip(y)dy
Ix 2 _ y211-a = f(x), 0 < x < a,
o
which is reducible to (34.2) by changes of variable is known in the form
a t
2r(1 - a) sin ¥ 1 d f t2adt d f f(IJ)lJl-adlJ
Ip(x) = - 1r [r (¥)] 2 x a dx  (t 2 - x 2 )a/2 dt 0 (t 2 _ 1J2)a/2
This differs from that derived from (30.82) or (30.84) - see Ahiezer and Shcherbina [1]. Later a
similar result for (34.2) was obtained by Williams [1].
30.3. The integral equation of the first kind
00
f c(x - t)
Ix _ tll-a Ip(t)dt = f(x),
-00 < x < 00,
(34.4)
-00
with a difference kernel, where
{ u(x),
c(x) =
vex),
x > 0, >. . 1
, u(x),v(-x)eH (R+), '\>01,
x < 0,
is reduced to an integral equation of the second kind with difference singular kernel
00 00
sinOl1r f v( -It - xl) f
Nip == al Ip(x) + - Ip(t)dt + T(x - t)Ip(t)dt = g(x)
1r t-x
-00
-00

688
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
where al = u(O) + v(O) C06 CV1r, 9(X) = rTaJD..f. The kernel T(x), which is evaluated via
u(x) and v(x) explicitly is a continuous function everywhere except for the origin, and has the
estimates IT(x)1  clxl cv - 1 as x - 0 and IT(x)  c(I + Ixl)-cv-l as x - 00. The operator
N has the structure N<p = >"<p + p.S<p + hI * <p + S(h2 * <p) where >.. and p. are constants, and
hI (x), h2(x) e Ll (Rl). IT the operator N is invertible, then the operator N-l has the same form
- Samko [16], [27,  5].
30.4. The generalized Abel equation (30.79) with constant coefficients on a finite interval
was solved by Wolfersdorf [3] in the form (a = 0, b = 1):
<p(x)
c
(x - a)cv+8/(2"')(I _ x)I-8/(2,..)
{ Z l } 1
+ sin CV1r ..!!...- u f f(t)dt - v f f(t)dt - >.. f P(x, t)f(t)dt.
A1r dx (x - t)CV (t - x)CV
o Z 0
Here c = 0 if uv > 0 and c is arbitrary if uv < 0, A = u 2 + 2uv cos CV1r + v 2 ,
>.. _ _ vsin 2 CV1r r(2 - O/(21r»r(I - cv)
- A1r 2 (cv + I)r(I - cv - O/(21r»
. f > 0 d ' - vsincv,.. I'(I-8/(2,..YI'(-CV) . f < 0
I uv an" - A ,..  I'( 1-cv-8 ( 2,.. » I uv ,
1 ( (J x-t )
P(x, t) = [x(I - x)r -cv2Fl 2 - -,I + CVi 2 + CVi - ( )
21r X 1 - t
if uv > 0,
and P(x, t) = 1z {[X(1 - x)]-CV 2 Fl (1 - -/;r, CVi 1 + CVi Z(I--:t» ) } if uv < 0 where (J is given in
(30.80). Compare this with (30.84) and (30.89) in the cases u = v and u = -v. Another form of
solution was given by Ganeev [2] also in terms of hypergeometric fWlctions.
30.5. The generalized Abel equation (30.41) is also solved in closed form in the case when it
is considered on a smooth curve in the complex plane. This case was investigated by Sakalyuk [3]
and Peters [2]. The latter author also solved the generalized Abel equation (30.42) on a smooth
curve and the equation
f <p(t)t + k f <p(t)t = f(z), z e c,
(z - t) -cv (t - z) -cv
C.. C..
(34.5)
in the case of a closed contour C, C az and C za being two its constituent arcs. Here k ':# 1 and
k ':# e 2 ,..icv. The cases k =.1 and k = e 2 ,..icv which imply the degenerate cases of the corresponding
Riemann bOWldary value problem were considered by Chumakov and Vasil'ev [1]. We note that
by reduction to (34.5) on a closed contour Peters [2] solved, for example, the equation
,..
f sign (x - t)
Isin(x _ t)II-CV <p(t)dt = f(x), 0 < x < 1r.
o
A modification of (30.41) which corresponds to the case of system of intervals was considered by
Chumakov [2].
30.6. The "exceptional" cases of the generalized Abel equation (30.41), when u(x) and
v(x) vanish simultaneously at a finite numbers of points of the interval [a, b], was investigated by

 34. ADDITIONAL INFORMATION TO CHAPTER 6
689
Vasil'ev [2, 5].
The solution of (30.41) in the case when u(x) and vex) may be infinite at the points a and b
was given by Orton [1] in some spaces of generalized functions.
30.7. The algorithm for solving the equation
z b
f P(x - t) f P(x - t)
u(x) (x _ t)l-a <p(t)dt + vex) (t _ x)l-a <p(t)dt = l(x)
a z
more general than (30.41), where P(x) is a polynomial, was indicated by Sakalyuk [2]. Peters
[2] considered such an equation in the case of an open smooth curve. We refer also to the case
vex) == 0, l(x) == 1 in Neunzert and Wick [1].
We also observe that some generalizations of (30.41) with the Gauss hypergeometric function
(1.72) in the kernel on a finite interval and a closed contour was considered by Chumakov [1], and
Peschanskii [1], respectively.
30.8. The generalized Abel equation (30.41) on a finite interval has many applications. We
note some of them. First of all there are boundary value problems for differential equations of
mixed type as for example, Wolfersdorf [2]. We also note the paper by Bzhikhatlov [1] where
an equation similar to the generalized Abel equation (30.41) was obtained while solving the
boundary value problem for equations of degenerate type. Wolfersdorf [4] gave applications of
the generalized Abel equation to problems of game theory. Applications of the particular cases,
when u == v == 1 or u == -v == 1, to problems of magnetohydrodynamics have been indicated by
Lundgren and Chiang [1]. Applications of (30.41) with constant coefficients to contact problems
of the theory of creep and plasticity were given by AlUtyunyan [1, 2] and AlUtyunyan and
Manukyan [1]. Similar applications in a more general case of a variable coefficient of friction were
given by Rubin [15].
We alao observe that the special case cosa1r1+<p - 1f_<p = 1 of the generalized Abel
equation arose in an application to a non-linear Hilbert boundary value problem - Hatcher [1].
30.9. Although the first attempt to solve systems of generalized Abel integral equations
was undertaken by Zeilon [1], their real investigation was begun comparatively recently. More
serious approaches were made by Lowengrub and Walton [1] and Walton [1]. In the former
paper a system of two equations of the type (30.41) - considered in a non-general form - was
reduced to the Riemarm boundary problem for two pairs of functions by the method of analytic
continuation. In the latter paper the system of two equations more general than those of the form
(30.41) was reduced to a complete singular equation. Investigations in the papers were influenced
by applications to mixed problems of the theory of elasticity. We refer also to Lowengrub [1] in
this conntion.
Interesting results for system of generalized Abel equation were obtained by Vasil'ev [1, 3,
4]. After constlUcting some Hermitian forms by the given system he obtained the criterion of
uniqueness of the solution and the criterion of absolute solvability of the system in terms of sign
definiteness of these forms. In certain cases the numbers of solutions and of solvability conditions
were expressed via the rank and the signature of these forms.
In particular, Vasil'ev [4] reduced systems of the generalized Abel equation U 1:+ <p+ V 1 b _ <p =
1, where <p and 1 are vector-functions and U and V are matrix-functions, to a system of singular
integral equations. He gave a similar investigation for systems of the form 1:+ (U <p) + 1 b _ (V <p) = 1.
Complete results on the solvability of systems in tenns of the method of Hermitian forms were
given in the case of constant coefficients, for example, when U and V are number matrices. In
the case of systems of the form
00
f
01 + 02 sign (x - t) ( )d - 1( )
11 <p t t - x,
Ix - t -a
x eR 1 ,
-00

690
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
where Cl and 02 are number matrices Vasil'ev [I, 3] obtained explicit relations for inversion of
the type (30.68).
Vasil'ev [6] considered systems of generalized Abel equations on the whole axis containing a
vector-function Ip( -x) together with a vector-function Ip(x).
We alao note the papers by Penzel [I, 2] where systems of generalized Abel equations were
studied on the half-axis R in the spaces Lp(Rj p) with weight p = x'y.
30.10. The integra-differentiation equation
1
,X1p(x) = f Sir (x It) Ip'(t)dt, 0 < a < I,
x-t a
o
appeal'S in problems of approximation of function, under the application of the variational method
to the approximation of functions by positive polynomial operators - Kogan [I, 2]. In these
papers, by inversion of the right-hand side, this equation was reduced to a homogeneous Fredhohn
equation of the second kind with a positive defined operator.
30.11. Some results on the solvability of the non-linear integra-differential equation
:&
f F(x, t, Ip(t), Ip'(t»(x - t)a-1dt = l(x), x> 0, 0 < a < I,
o
are contained in the paper by Sadowska [1].
31.1. The generalized Abel equation (30.41), i.e. the equation MIp = u(x) 1:+ lp+v(x)If_ Ip =
I, was solved in  30 in the case 0 < a < 1. It may be considered in the case a  (0,1) as
well. H a < 0, then this relation can be interpreted as a differential equation of fractional order.
A comprehensive investigation of the equation MIp = 1 with an arbitrary a e Rl was given by
Rubin [11]. In particular, for every a, he constructed a pair of Banach spaces X and Y connected
with the spaces L,(a,b) such that the operator M is Noetherian from X into Y.
31.2. The Noether nature of integral equation of the first kind
MIp == f e(x, y)lx - yla-llp(y)dy = l(x), x eO,
{}
where the function e(x, y) is discontinuous at the diagonal x = y was investigated by Rubin
[2, 5] in the case when 0 is a union of intervals of the whole axis and M is considered as an
operator from L,(Oj p) into Ia[L,(Oj p)]. Rubin [6] also investigated the Noether properties of
the operator M in the case when 0 is a sufficiently smooth curve in the complex plane.
31.3. Rubin [8], [13] investigated the Noether nature of the operator M of potential type
(31.1) in a more general case, when e(x, t) may have jumps not only for t = x but for t = ak'
k = 1,..., m, x = bj' j = 1,..., n as well, the case ak = bj being especially important. This
generalization is interesting, particularly for the reason that it allows one to consider the equations
n °k+l
L f ek(x, t)lx - tla-11p(t)dt = l(x)
k=l Ok
"with a finite number of potential type kernels".
31.4. The Noether nature of potential type operators stated in Theorem 31.9, can be
demonstrated under weaker assumptions about the behavior of u(x, t) and vex, t) at infinity.

 34. ADDITIONAL INFORMATION TO CHAPTER 6 691
n
That is we restrict ourselves to the case of degenerate functions u(x, t) = L: ak(x)bk(t),
k=l
m
v(x,t) = L: Ck(x)dk(t), and if we consider the Noether property not from L, into IOt(L p ) but
k=l
from L into L+Ot. Here L = L(Rl) is the Sobolev space of fractional smoothness, the
potential type operator being considered as closed one - Skorikov [1].
31.5. The N oether nature of the equation
00
f (:'12 <p(t)dt = f(x)
-00
more general than (34.3), with an "almost difference" kernel was considered by Skorikov [4]. He
obtained the criterion of the Noether property and the index formula in the case when m(x, y)
has a jump of the first kind with respect to y if y = o. The investigation was based on the
reduction to a certain convolution-type singular equation.
31.6. Samko and Vasil'ev [1] singled out some classes of equations more general than the
generalized Abel equation which are reduced to complete singular equations with a meromorphic
kemel, and therefore can be solved in closed form.
:&
31.7. Problems of similarity of the operator 1 0 +<p+ fk(x,t)<p(t)dt to the fractional
o
integration operator 10+ in L,(o, a) were considered by Kalisch [2] and Malamud [1], [2]. We
recall that the operators A and B are named similar in the space X if there exists a linear
operator e in X invertable in X such that AC = CB.
31.8. Atkinson [1] considered the equation
:&
f u(x, t)
( ) <p(t)dt = f(x),
xp - tp Ot
o
o < x < b,
0< Ot < 1,
which coincides with (31.67) if p = 1. He showed that the unique solution <p of this equation
has a smoothness of the form <p(x) = x,Ot-l+.6tb(x), VI e en [0, b], if the additional smoothness
assumptions on the kernel u(x, t) e e n + 2 {(x, t) e R2: 0  t  x  b} and the free term
f(x) = x,6g(x), g(x) e en+1[0,b] hold and the inequality pOt + (3 > 0 is satisfied. The method of
investigation is similar to the consideration of (31.67) on reduction to a VoltelTa integral equation
of the second kind, and on applying the method of successive approximation to the latter in a
special fractional space.
31.9. In connection with the stability problems for the VoltelTa equation (31.67) of the first
kind, Vessela [1] proved the estimate
"<pU,  c(ll<p'II;/Cl+Ot)lIfll;!(1+Ot) + IIfll,)
provided that u(x, t) and Ut(x, t) are continuous. The estimates of such a kind are inspired by
the fact that some information on the derivative of the unknown solution is known in certain
applications. A more general norm of the fractional Sobolev space lIP,' was also dealt with. A
generalization of such an estimate presented in the context of the ideas of regularization was given
by Ang, Gorenflo and Hai [1].

692
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
32.1. The equation of the fonn

1 c(x - t)k(x - t)<p(t)dt = f(x), x e (a,b),
o
which is more general than (32.1) was investigated by Rubin and Volodarskaya [1] and Rubin [13],
[14]. Here the function c(x) satisfies certain conditions connected with the absolute continuity on
[0, b - a], and the kernel k(x) has the fonn
(34.6)
k(x)=x a - 1g (In;) , aO, "Y>b-a.
(34.7)
Here the factor 9 (In;') introduces such a singularity (or zero) into the kernel which is weaker
than that of the power fWlction. For example g(x) may have the form
n
g(x) = II In" ;,
k=1
Ink = In.. .In,
"'-.-'
k times
-00 < (3k < 00.
In the cited papers the weakening or strengthening of the singularity of the kernel is obtained
by the application of a convolution operator with respect to the variable a, the kernel of this
convolution being, in general, a distribution.
32.2. Rubin [12] obtained the characterization and inversion relations for the convolutions
1:+ 1 <p, <p e Lp(a, b) ( 32.2) in terms of difference constructions of the Marchaud derivative type.
We denote
1 00 x'-a ( r l(a» )
(x) = - r(t _ a + 1) exp t rea) dt.
o
Then the following statement is true.
Theorem 34.1. A" eiste"ce of the limit
-£
(Bf)(x) = lim 1 [f(x) - f(Y)]'(x - y)dy
£-0
a
where we take f(x) == 0 6e,0"d the intenJal [a, b), in L,(a, b) u the necessary condition for the
representa6ilit, of a J-nction f e Lp(a, b) in the form f = 1:+ 1 <p with 0 < a < 1, <p e Lp (a, b) if
1 < p < 00 o."d it is sufficient if 1  p < 00. Moreover, the relation
<p(x) = f(x)(x - a) - (BJ)(x)
holds. Compo.re thi. with the similar Theorem 19.! for fractional integrals in  19.
32.3. Linker and Rubin [1] investigated question of restriction, continuation and "sewing"
of the spaces of fWlctions representable by the convolutions (32.2) with a density <p e L,(a, b).
Similar results for fractional integrals were given in  13.3.
32.4. Knowledge of asymptotic expansions of integrals with power-Iogaritlunic kernels
allows us to find an asymptotic solution (see  16.5) of the corresponding equations. In this
context we refer to Kilbas [8] and  23.1 (Notes to  21.5).

 34. ADDITIONAL INFORMATION TO CHAPTER 6
693
32.5. Volterra and Peres [1] considered the equation
j ex - 1)0-1 [ t AmI: lnl:(x - t) + (x - t)aa(x, t) ] <p(t)dt = l(x), a > 0,
o 1:=0
(34.8)
a(x, t) being a certain given function, which is more general than (32.3). By means of the relation
in (34.1) this equation was reduced to a simpler Volterra equation of the first kind in which
logarithmic factors were absent. In particular, if a = 1 and m = 1 (34.8) was reduced to a
Volterra integral equation of the second kind under some smoothness assumptions on a(x, t) with
respect to x.
32.6. Veber [4] obtained the inversion formula for the simplest integral equation (32.2) on
the axis:
:&
r;a) / (x - t)a- 1 ln(x - t)<p(t)dt = lex)
-00
in a certain space of generalized functions which is invariant with respect to Liouville fractional
integra-differentiation. It is interesting that here, unlike the case of a finite interval, the
homogeneous equation (J == 0) has the non-trivial solution <pet) = cexp(e"'Ca)t) in the considered
space of generalized functions.
32.7. The behavior of the convolution operator (32.19) with the special Volterra function
(32.10) and with h = ,pea) was investigated by Kilbas [4] in the ordinary H6(p) and generalized
H,I: (p) weighted Holder spaces. The boundedness of this operator in H,I: (p) (k = 1,2,...) and
complete continuity in H6(p) were proved.
32.8. The solution of the equation
b
1 / I X+Y I
- In - <p(y)dy = l(x),
1r x-y
a
o  a < x < b < +00,
(34.8')
with a = 0 in the form
, F::\ ../X 1/2 1/2
<p(vxJ = 7(Z>b_Z>0+ l(v'Y»(x)
where z>2 and z>t are the Riemann-Liouville fractional differentiation operators (2.22) and
(2.23), was first obtained by Ahiezer and Shcherbina [1] (1957) as a particuJar case of solution of
more general than (34.8') integral equation (39.10'), with Gauss hypergeometric function in the
kemel. A closed-form solution and the solvability conditions of this equation in the case of any
a  0 were investigated by Vasilets [1] in the weighted spaces L,([a,b]ip) with the power weight
p(x) = (x - «I)"(b - X)II.
33.1. Theorem 33.3 in the case of the natural powers oflogarithm was extended by Rubin
[4] to the weighted spaces L,([a, b) : p) ,!here
n
p(x) = II Ix - x1c I"k, a = Xl < . . . < Xn = b.
1:=1
(34.9)
Ru [9] indicated that this result was tlUe for real nonnegative powers of logarithm.
33.2. Kilbas [6] investigated the Noether properties of the operator (33.1) with power-
logarithmic kernels with natural powers oflogarithm from the weighted Holder space H6(p) into

694
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
the generalized weighted Holder space H+Ol,,, (p), where 0 <  < 1,0 < + Ot < 1, P = 1,2,..., p
is the weight in (34.9). We note that the main role here is played by the theorem on the
isomorphism between the spaces HS(p) and H+Ol,,,(p) realized by the operators I:f and r::!
with power-logarithmic kernel if P is natural. Its proof uses the properties of the convolution
operator (32.19) with the special Volterra function (see  32.7), together with the results of  21.
33.3. H <p e Lp == L,(a, b), lip < Ol < 1, then the relations similar to (33.16) and (33.17)
are true
r::! <p = 1:![C06(Ol1r) + sin(Ol1r)Sa,Ol_1<P + T3<P] + c!C<P),
(34.10)
r:f <P = 1:![C06(Ol1r) - sin(Ol1r)S&,Ol_1 <P + T 4 <P] + C2(<P),
(34.11)
where T3 and T4 are completely continuous operators in L, and c!C<P) and C2(<P) are some
functionals. These relations were proved by Rubin [10] who applied them to the investigation
of the Noether properties of the operator (33.1) from L, into r:f(L,) ED R if lip < Ol < 1 -
remark 32.2. He also proved statement similar to Theorems 33.3-33.4.
This method of investigation does not work in the exceptional case Ot = lip, since the
operators Sa,Ol and S&,Ol involved in (32.16) and (32.17) are bounded if lip - 1 < Ol < lip, and
the operators Sa,Ol-1 and S&,Ol-1 involved in (34.1) and (34.11) are bOWlded if lip < Ot < 1 + lip.
Therefore the problem of the investigation of the Noether properties of the operator (33.1) in the
case Ot = lip remains open.
33.4. The Noether nature of the potential type operators (33.1) with pure logarithmic
kernels (Ot = 1, P > 0) acting from the space L" was investigated by Rubin [4] in the case of
the first power of the logarithm, and by Kilbas [2], [5] in the case of natural and real powers of
logarithm.
KUbas [3], [5], [7] investigated the Noether properties of the operator of the fonn
m &
X<p ==  J Kj(x, t)ln,6j Ix  tl <P(t)dt = lex), -00 < a < b < 00.
J=la
more general than the operator (33.1) with pure logarithmic kernel (Ot = 1, P > 0). He considered
this operator as acting from the space L,([a, b]i p), 1 < p < 00, where p is the weight (34.9) into
a special space. Here Pj  0 (j = 1,2,..., m), "y > b - a and among the functions Kj (x, t) there
may be both continuous and piecewise continuous ones with jumps at the diagonal x = t. Other
results connected with such equations may be seen in the papers by Kilbas [10], [11].
33.5. Rubin and Volodarskaya [1] and Rubin [13] investigated imbedding of the ranges of the
convolution operators (34.6) which are more general than the imbedding (33.10). The introduction
of the scalled generalized Volterra functions and the investigation of their asymptotics by means
of properties of the Laplace transfonn lie at the base of these papers.
33.6. The Noether nature of potential type operators with a more general kind (34.7) of
singularity was considered by Rubin and Volodankaya [1] and Rubin [14].

Chapter 7. Integral Equations of the First
Kind with Special Functions as Kernels
Abstract. In this chapter we shall be concerned with applications of fractional
integra-differentiation to the investigation of one-dimensional integral equations of
the first kind
b
f K(x, t)f(t)dt = g(x),
a
-00 $ a < x < b $ +00,
(1)
with the kernels K(x, t) containing special functions. Such equations are closely
connected with integral transforms - see S 1.4. Many problems from other fields of
mathematics such as differential equations discussed in Chapter 8, function theory
and others, also problems in physics, mechanics and other natural sciences are
reduced to them. The so-called dual and triple integral equations - see typical
examples in the last section of this chapter - can be written in the form (1).
Equations of the form (1) are equations of the first kind and therefore the
problem of their inversion is an ill-posed problem. Many methods of solution are
known for such equations, which depend on the type of the kernel. The convolution
equations, i.e. the equations of the form (1) with K(x, t) = k(x - t), have been
most studied - Gabov and Cherskii [1], Titchmarsh [1]. Hirshman and Widder
[1], H.M. Srivastava and Buschman [3]. We observe that the latter book contains
many equations with special functions as kernels and their solutions, as well as a
large bibliography of the papers devoted to solution in closed form of convolution
equations with variable upper or lower limits of integration.
Solutions of equations such as (1), which are generalizations or modifications
of the Abel integral equation (2.1), and also solutions of the composition type
equations connected with the Abel equation are constructed in this chapter. It
turns out that the most effective method is the factorization method, also called
the composition expansion method, which gives a representation of some classes of

696 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
the operator (1) as the compositions of fractional integra-differentiation operators
together with other operators which have known inversion formulae. Many
mathematicians throughout the world have made considerable contributions to the
development of this method, which was started for the equations considered in the
early 1960s - see S 39.1. However, we note that long before this the implicit idea of
the factorization method was used by the Russian mathematician Sonine [4, 5] at
the end of the 19th century and the method itself was in fact applied by Lebedev
[1] in 1948 - see S 39.1 and S 39.2 (notes 37.3 and 35.7).
 35. Some Equations with Homogeneous Kernels
Involving Gauss and Legendre Functions
In this section we consider the Mellin convolution integral equations with the
Gauss hypergeometric function or the Legendre function as kernels which are
important in applications. We shall show that these equations can be inverted in
terms of compositions of two fractional integra-differentiation operators with power
weights, or in terms of operators involving Gauss or Legendre functions and usual
differentiation in a quite symmetrical way. We shall use the results of S 10.1 in
these investigations.
35.1. Equations with the Gauss function
We deal with the following four integral equations involving the Gauss hypergeomet-
ric function as a kernel
j z (x _ r)c-l ( .. X ) _
r(c) 2 F l a, b, c, 1 - -;;: cp( r)dr - g(x),
e
(35.1)
j z (x _ r)C-l ( r )
r(c) 2 F l a, b; c; 1 -; cp( r)dr = g(x),
e
(35.2)
d
j (r_x)C-l ( X )
r(c) 2 F l a, b; c; 1 -;: cp( r)dr = g(x),
Z
(35.3)
d
j (r x y-l ( r )
(c) 2 F l a,b;c;I-; cp(r)dr=g(x),
Z
(35.4)
considered on an interval 0  e < x < d  00. In accordance with Remark 10.3
we denote the operators on the left-hand sides of (35.1)-(35.4) by 11+(a, b)cp,
21+(a, b)cp and 31:1_ (a, b)cp, 41:1_ (a, b)cp if d < 00 or 31: (a, b)cp, 41: (a, b)cp if d = 00.

 35. KERNELS INVOLVING GAUSS AND LEGENDRE FUNCTIONS 697
By the relations given in (10.22)-(10.29) the operators jIC(a, b) are the
compositions of two one-sided fractional integrals or derivatives with power
weights. The conditions for such a represent ability are given by Theorem 10.4
and Remark 10.3, which show that if rp(x) belongs to the space Lp or some of
its subspace, then under corresponding conditions the operators jIC(a, b) map a
certain subspace of L" and some or other of the relations given in (10.22)-(10.29)
are valid. This means that if the right-hand side, i.e. a function g(x), is taken from
the space Dj indicated in Table 10.2 which is mapped by the operator jIC(a, b),
then the corresponding equation jIC(a,b)rp = 9 is uniquely solved by successive
inversion of two integra-differentiation operators composing jIC(a, b). Realizing
these inversions we obtain from (10.22)-(10.29) the following representations of
solutions of (35.1)-(35.4):
rp(x) = x-a I;xa I;Cg(x),
rp(x) = x- b I;cxc-a I;';xa+b-cg(x),
rp(x) = I:;cxa I;';x-ag(x),
rp(x) = x a + b - c I;':x c - a I:;cx-bg(x),
rp(x) = x- b I:cxc-a Iixa+b-cg(x),
rp(x) = x-a Iixa I=Cg(x),
rp(x) = I:cxa Iix-ag(x),
rp(x) = x a + b - e Iixc-a I=cx-bg(x).
(35.5)
(35.6)
(35.7)
(35.8)
(35.9)
(35.10)
(35.11 )
(35.12)
To formulate the corresponding theorem we denote by Ej a column of numbers
which is obtained after replacing (10.22)-(10.29) by (35.5)-(35.12) respectively in
column Ej of Table 10.2. Then the following statement is true.
Theorem 35.1. We consider (35.1) and (35.2) with e = 0 on the interval (0, d)
and (35.3) and (35.4) with d = 00 on the interval (e, 00) provided that Rec > O. If
the conditions Aj from Table 10.e are satisfied for the operators Bj and a given
function g(x) E Dj, 1  p < 00, then the corresponding equation Bjrp = 9 of the
forms (35.1)-(35.4) has a unique solution given by the expression Ej where e = 0
and d = 00. The statement of the theorem remains tnJe when e > 0 and d < 00
correspondingly, and in these cases the assumptions in the conditions Aj involving
p are omitted.
Prool. In fact, all the conclusions of the theorem were obtained in the proof of
Theorems 10.2 and lOA and Remark 10.3. We additionally note that, for example,
if e > 0 then the singular point T = 0 of the kernel of (35.1) lies beyond the interval

698 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
of integration. This fact allows us to remove the conditions containing p from
the assumptions Al and A 2 , since singularities of the function 1;'( r) and those of
the kernel at the point r = 0 are not added. Similar assertions are true for other
operators in (35.2)-(35.4). .
It follows from (10.30)-(10.32) that (35.1)-(35.4) can be considered as four
different forms of writing of the same equation, for example, the first equation.
Other equations differ from this only by changes of variables, functions and
parameters.
It should be noted that except for the relations given in (35.5)-(35.12) we can
use other forms of representations for the solutions of (35.1)-(35.4), in particular,
via integral operators involving the hypergeometric function as a kernel. Such
representations can be obtained from (35.5)-(35.12) by using the relations given
in (10.4) and (10.5) or similar to them. Taking (35.5) as an example we shall
construct two solutions of (35.1). It is clear that the solution given in (35.5) can
be written in the form
<p(z) = z-o ( d ) m zOz-°G--bzoI-c)-(m-b)g(z)
or
<p(z) = z-o r;;zo I::;c+m ( d ) m g(z), m = 1,2,... ,
It should be emphasized that the latter relation is not equivalent to (35.5) since
the function g( x) here is assumed to belong to the subspace of functions in L, (e, d)
representable by the form 9 = I:.tX, X E L,(e, d), i.e. having the derivative g(m)(x)
and g(e) = g'(e) = ... = g(m-l)(e) = O. Comparing two last expressions with
(10.24) and taking (10.19) into account we arrive at the following representations
for the solution of (35.1):
{ z }
r x - r m-c-l r
<p(z) = z-o dz m zo! ( r(J _ c) 2F, (-0, m - b; m - c; 1-;) g(T)dT ,
o < Rec < m, 9 E I+(L".( e, d»;
(35.13)
j z (x _ r)m-c-I ( r )
I;'(x) = r(m _ c) 2 F I -a, -b; m - c; 1 -;- g(m)(r)dr,
e
(35.14)
0< Rec < m, 9 E Cm([e,d]) g(e) = g'(e) = ... = g(m-l)(e).
It is clear that L".(e, d) = L,(e, d) if e > 0 as is seen in (10.2). Similarly we can
transform other relations given in (35.6)-(35.12).

 35. KERNELS INVOLVING GAUSS AND LEGENDRE FUNCTIONS 699
35.2. Equations with the Legendre function
Different particular cases of equations (35.1)-(35.4) with the kernel involving
Chebyshev, Legendre, Gegenbauer, Jacobi, etc. polynomials arise in applications
to differential equations as for example in S 40.2. Here we consider equations that
will be most useful below, i.e. equations with the Legendre function given in (1.79),
(1.80) as a kernel:
:&
1 (x 2 - t 2 )-14 2 P: () f(t)dt = g(x),
e
(35.15)
:&
l(z2 _t 2 )-"/2p: (;) f(t)dt = g(z),
e
(35.16)
d
1 (t 2 - x 2 )-/J/ 2 P: (T) f(t)dt = g(x),
:&
(35.17)
d
l(t 2 - z2)-"/2p: () f(t)dt = g(z),
:&
(35.18)
provided that 0  e < x < d  00 and Re p < 1 in all cases and that the integrals
indicated above converge at the variable end point. We shall obtain solutions of
these equations by the Mellin transform. We consider the first equation in more
detail.
In (35.15) we change the variables and the functions by the substitutions
t1-/J f(t)H(t - e) = 2<p(t 2 ), t 2 = r,
x 2 = y, y / r = ",
(,,- 1)-/J/ 2 H(,,- I)P:(VTi) = h(,,),
g(.jY) = gl(Y),
(35.19)
where H() is the step function, namely H() = 1 if  > 0 and H() = 0 if  < O.
Then this equation has the form
00
1 h(y/r)<p(r)r-1dr = gl(Y)'
o

700 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Applying the Mellin transform defined in (1.112) to the latter and 11.14 (1) from
the book by Marichev [10] and taking the convolution theorem given in (1.115)
into account we obtain the relation
_ _ -/J r(1 - s)r(I/2 - s) _
<p (s) - 2 r«l + JJ + v)/2 - s)r«JJ _ v)/2 _ s) 9d s ),
ReJJ < 1, Re(2s - JJ - v) < 1, Re(2s + v - JJ) < O.
(35.20)
The inverse path from (35.20) to its original <p(t), and hence f(t), call be carried
out by various methods. We shall indicate three of them here, which lead to
different forms of solutions.
A. Using the representation given in (10.35) and grouping the gamma-
functions by different ways (vertically and crosswise) we obtain the following two
representations for the function <p(y):
<p(y) = 2-/J(y(l-/J-II)/21+1I-1)/2)(y1+(I1-/J)/21-11-1)/2y-l/2)91 (y), (35.21)
<p(y) = 2-/J (y(l-/J-II)/21+II)/2y-1/2)(y1+(I1-/J)/21-I1)/2-1)91 (y). (35.22)
These representations coincide for sufficiently good functions gl (y) (for which
the operators in the parenthesize are commutative) since they differ from each
other by replacing v by -v - 1 which is not essential according to the property
Pt(z) = PII_1(z). Making the inverse change in (35.21) by the expressions given
in (35.19) we arrive at the following representation for the solution of (35.15):
f( ) - 2 1 -/J -1I 1 (/J+1I-1)/2 2+1I-/J l (/J-1I-1)/2 -1 ( )
x - X e+jz X e+jz X 9 X
(35.23)
where l:+;z is the operator defined in S 18.2, see (18.41).
B. Another representation for the solution can be obtained if we multiply the
numerator and denominator of (35.20) by r(1 + (JJ + v)/2 - s) and then apply the
duplication formula given in (1.61) for the gamma-function together with (10.35)
and its analogue in the form
-y+ioo
fJ/ 2 / 0I -fJ -0I/2 f( ) =  f r(1 - a - 2s) f - ( ) -' d
x 0+ " r=X x . r( IJ 2 ) s x s,
,yZ 21rz 1 - fJ - S
-y-ioo
(35.24)
21' < 1 - Rea.

 35. KERNELS INVOLVING GAUSS AND LEGENDRE FUNCTIONS 701
Then
. ( ) -2 11 r(1 - 2s)r(1 + (p. + 11)/2 - s) . ( )
<p s - r(1 + p. + 11- 2s)r({p. _ 11)/2 _ s)91 s ,
<p(y) = 2 11 (y-(IJ+II)/2 1:;)(yl+(II-IJ)/2 IO.:-l y (IJ+II)/2)gl (y),
and finally the desired solution of the equation (35.15) can be written in the form
f(x) = (2x )11+1 I;';j;: 1:1 11 g(x).
(35.25)
C. In many cases the third form of representation for the solution of (35.15)
via an expression almost symmetric with (35.16) is used. To obtain it firstly we
note that the difference between parameters of gamma-functions in the numerator
and the denominator in (10.35) is equal to (3 - Q and if Re({3 - Q) < 0, then the
integral on the right-hand side of (10.35) corresponds to the fractional integral
x P I+P x- OI , and if Re ({3 - Q) > 0 then it corresponds to such a fractional
derivative. Analogously, the difference of parameters in (35.20) is equal to 1 - P
with Re(I- p.) > 0, i.e. the right-hand side of (35.20) corresponds to the fractional
derivative of a function gl (y). So in terms of originals it is convenient to write
this fractional derivative as a composition of the operators including the usual
differentiation operators.
To return to originals, we use the fact that according to (10.35) multiplication
by (1 - {3 - 8)n in terms of the Mellin transform corresponds to the operator
x fJ ( d )n xn-fJ when Q-{3 = -n and that the correspondence x fJ / 2 ( d ) n x(n-p)/2
+-+ (1 - {3 - 2s)n follows from (35.24). Multiplying and dividing the right-hand
side of (35.20) by (1 - {3 - s)n or (1 - {3 - 2s)n with the corresponding numbers
{3 = 1 + (II - p)/2 or (3 = n we write the right-hand side of (35.20) in the form
2-IJr(l-s)r(I/2-s) ( P-II ) .
- - S 91(8),
r({1 + p. + 11)/2 - s)r({p. -11)/2 + n - 8) 2 n
2- IJ - n r({1 - n)/2 - 8)r(1 - n/2 - s) .
r({1 + P + 11)/2 _ s)r({p _ 11)/2 _ s) (1 - n - 28)n91 (8).
If 1 - Re p - n < 0, then the first gamma-functions correspond to the kernel of
''fractional integral" of the form (,,- 1)-IJl/2 H(,,- I)Pt'1 1 (,,-1/2) with specially
selected parameters PI and 111; see Marichev [10, 11.13(4)]. The second multipliers
correspond to the above operators of usual differentiation with power multipliers
which can lie both outside and inside of the integral. In the latter case additional
assumptions on the function 91 (y) are needed. Passing to the originals in the given

702 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
expressions after the changes given in (35.19) we finally obtain the following four
representations for the solution of (35.15):
f(x) =X- II (  ) n Xn+1I
xdx
:&
X f (X 2 - r 2 )(n+ IJ )/2-1 p+:-IJ (;) g( r)dr,
e
(35.26)
:&
f(x) =x- n f (X 2 - r 2 )(n+ IJ )/2-1 p+:-IJ (;) r 1 - 1J - 1I
e
X (  ) n (r2n+IJ+II-1g(r»dr,
rdr
(35.27)
f(x) =X IJ + n - 1 cr { X1-1J
dx n
:&
f (,,2 - T 2 )(n+ pJ /2-1 T -n p;-n- p (;) 9(T)dT},
e
(35.28)
:&
f(x) = f (X 2 - r 2 )(n+ IJ )/2-1 p;-n-IJ (;) g<n)( r)dr.
e
(35.29)
The latter relation contains an operator in the form (35.16). Thus, replacing
here g(n)( r) by f( r), f(x) by g(x) and J.l by 2 - n - J.l and taking the mutual
invertibility of (35.15) and (35.29) into account we obtain the representations for
the solution of (35.16) in the form
f( ) - I IJ-I-2 I "+1 (2 ) -11-1 ( )
x - e+ e+;:& X 9 X ,
(35.30)
n :&
f(") = (:,, ) f (,,2 - t 2 )<n+pJ/2-1 p;-n- p (i) g(t)dt,
e
(35.31 )
:&
f(x) =x- n f (x 2 - t 2 )(n+ IJ )/2-1 p;-n-IJ Ci) t lJ + n - 1
e
x ( :, r {t'-Pg(t»dt,
(35.32)
cf. (35.25) with (35.30) and (35.28) with (35.32).

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 703
Replacing in (35.15) and (35.16) t 2 -1J I(t) by l(t- 1 ), xlJg(x) by g(x- 1 ), x by
x-I and e by d- 1 we arrive at (35.18) and (35.17). This property allows us to use
the solutions obtained for finding corresponding solutions of (35.17) and (35.18).
In particular, the solutions
I(x) = (x/2)"+l 1:::"-2xl-IJ-1 1;:ixlJ-II-3g(x),
I( x ) = 2"+1xlJ+I+1 l-lI-lxl+II-1J I IJ + II x- II - 1 g( x )
d- iZ d-
(35.33)
(35.34)
follow from (35.30) and (35.25). We shall not write analogues of (35.26)-(35.29)
which can be obtained in the indicated way. We only note that solutions of (35.17)
and (35.18) are representable by expressions different from (35.28)-(35.29) and
(35.31 )-(35.32) respectively, only by changing the intervals of integration of (e, x)
to (x, d) and (x 2 - t 2 )(n+IJ)/2-1 to (_I)n(t 2 - x2)(n+IJ)/2-1.
Using the method which led us to the solution given in (35.25) one can see by
direct evaluation that besides (35.33) and (35.34) other forms for the solution of
(35.17) and (35.18) which are admissible are
I(x) = (2x)"+1 lii;I::" g(x),
I(x) = 1:::"- 2 1;:i(2x)-1I-1g(Z)
(35.35)
(35.36)
respectively.
On the basis of results from S 10.1 and using Theorem 18.1, Lemma 31.4,
Remark 10.3 and the symmetry property of the Legendre function Pt(z) =
PII_l(z), by which we can assume Rev  -1/2 without loss of generality, the
results obtained can be stated as the following theorem.
Theorem 35.2. Let Rep < 1, Rev  -1/2 and 0 < e < d < 00. Equations
(35.15), (35.16) and (35.17), (35.18) are solvable in L,(e, d), 1  p < 00, if and
only il g E 1:+ IJ (L,( e, d» and 9 E Il: IJ (L p (e, d» respectively. These conditions
being satisfied, each of the equations has the unique solution given by (35.25),
(35.30) and (35.35), (35.36).
The cases e = 0 and d = 0 in (35.15), (35.16) and (35.17), (35.18) respectively
are more complicated and therefore they require further investigations (see S 39.2,
note 35.3).
fi 36. Fractional Integrals and Derivatives as
Integral Transforms
When introducing the fractional integro-differentiation operators 1:+ and 1_ in
S 2, we in fact considered separately three cases namely integration if Rea > 0,

704 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
differentiation if Rea < 0 and integro-differentiation of imaginary order if Rea = 0,
a f; O. However, there is a way of defining these operators for all values of a at the
same time if we use Fourier or Mellin transforms. Thus we may write down the
equations
1'+ioo
01 f( ) 1 f r(1 - a - s) f - ( ) -' d
10+ x= 21ri r(l-s) s+ax s,
-y-ioo
(36.1)
Re(s + a) < 1,
-y+ioo
If(") = 2i f r() o) r(B + O).,-'dB,
-y-ioo
(36.2)
'Y = Res > 0,
see (10.35) and (10.36), which follows from (7.17) and (7.18) after applying
the inverse Mellin transform defined in (1.113). Here f-(s) denotes the Mellin
transform of a function f(x) given in (1.112).
The integrands on the right-hand sides of (36.1) and (36.2) contain the
ratio of two gamma-functions. This stimulates us to consider more complicated
constructions of such a kind involving the ratio of arbitrary products of gamma-
functions as for example constructions similar to the integrand in (1.95) defining
the Meijer G-function. Realization of the latter approach in terms of originals
will lead to an integral transform of type (1.44) with the Meijer G-function as a
kernel where k(." t) = n ( if I :: ). Its particular cases were indicated in S 104.
However the language of originals proves to be inconvenient for constructing the
theory of integral convolution transforms. There are two reasons for this. First of
all, as in the case of 19+ and 1, a definition of the transform can prove to depend
essentially on parameters of the G-function though the G-function itself depends
on its parameters analytically. Secondly, the G-function exists not for all values
of m,n,p,q. For example, it does not exist if p = q = 0 or m = n = 0, p,q  O.
Therefore it is more convenient to use the language of the Mellin transforms in
order to define integral convolution-type transforms by using the Parseval relation
given in (1.116).
36.1. Definition of the G-transform. The spaces 911';-:;(L) and
Lc,-y) and their characterization
Taking h-(s) in (1.116) to be a ratio of products of gamma-functions as in (1.95)
we arrive at the following concept.

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 705
Definition 36.1. The G-transform of a function f(x) is defined by the integral
(Gf)(z) '" (c;.n I :: 1/(1») (z)
=  f r [ (bm)+s,
- 21ri (a;+I) + s,
q
1 - (an) - s ] f - ( ) -, d
1 _ (b+1 ) _ s s x s,
(36.3)
where
[ (b m ) + s, 1 - (an) - 8 ]
r (a;+I)+8, 1-(b+I)-s
_ r [ bl + S, . . . , b m + s,
- an+l+S, ..., a,+8,
1 - al - s,
1 - b m + 1 - 8,
. . . , 1 - an - S ]
. . . , 1 - b q - s
m n
n r(b j + s) n r(1 - aj - s)
j=1 j=1
=
, q
n r(aj +s) n r(l-bj -S)
j=n+l j=m+l
(36.4)
f-(s) is the Mellin transform - see (1.112) - of a function f(x) on the line
u = {s,Res = 1/2} = {1/2 - ioo,I/2 + ioo}; (an) = al,a2,...,a n ; (a;+I) =
a n +l,a n +2,...,a,; (b m ) = bl,...,b m ; (b+I) = bm+t,...,b,; and the components
of p- and q-dimensional vectors (a,) and (b q ) are complex numbers which satisfy
the conditions
Re aj f; 1/2 + I, j = 1, .. . , n;
(36.5)
Reb j f; -1/2 -I, j = 1, .. . , m.
It is clear that if m = n = p = q = 0, then
(egg I : If(t»(x) = f(x).
(36.6)
It is natural to care about the convergence of the integral in (36.3). Since in
accordance with (1.65) the gamma-function has a power-exponential asymptotic
expansion as 11m s I --+ 00 then the space of functions is to be characterized by f- ( s)
with power-exponential weight at infinity. This justifies the necessity to introduce
the fullowing three definitions.

706 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Definition 36.2. The ordered pair (c. , 'Y.) where
c' = m + n - p  q . r' = R£ (Oj - t b j ) ,
(36.7)
is called a characteristic of the G-transform (36.3), the value
17 = 2sign c. + sign 'Y.
(36.8)
is called an index of the G-transform and the function
( ) [ (bm)+s, 1-(a n )-s ]
H s = r (a;+l) + s, 1 _ (b+l) - s
(36.9)
and the number p + q are called an image of the kernel and an index of the
complexity of the G-transform, respectively.
Definition 36.3. Let c,'Y E Rl and 2signc + sign'Y  O. We denote by rot(L)
the space of functions f(x), 0 < x < 00, representable in the form
1 f -
f(x) = 211'i f.(s)x 'ds,
q
(36.10)
f.(s) = s-'Ye-,..cIIm"IF(s),
(36.11)
where F(s) E L(O'). We denote roto.(L) as rot-1(L) for brevity.
Definition 36.4. We denote by Lcl'Y) the space of functions f(x), 0 < x < 00,
satisfying (36.10) and (36.11) where F(s) E L 2 (0'), 2signc + sign'Y  0, and the
integral along 0' is assumed to be mean square convergent.
The pair (c,'Y) is called a characteristic of the space rot;:(L) and Lcl'Y).
It follows from (1.65) that the asymptotic relation
H(s) f'OoJ Isl-'Y. e-c."'IIm"l, IImsl-+ 00,
(36.12)
holds and hence the integral
h(s) = 2i f H(s)x-"ds
q
(36.13)

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 707
is convergent everywhere in the case c.> 0 except at the point x = 1 if '1 = 0,
o < 'Y.  1, p = q, and it is divergent everywhere if 17 = O. However, if
17 = 0 and p f; q, then the function h(x) may be defined via the convergent
integral of the form given in (36.13) with the contour u coinciding with the line
Res = 1/2 + €sign(q - p), € > O. Therefore the case 17 > 0 is connected with the
usual direct G-transform realized by the Meijer G-function. The cases 17 < 0 are
singular and, in particular, they correspond to the inverse transform of Laplace,
Stieltjes and Meijer-type transforms if c. < 0 and of a fractional differentiation if
c. = 0, 'Y. < O. As for the cases 17 = 0 they correspond to Watson type transforms,
i.e., in particular, to the Narain transform - see Marichev [10, Sec. 8.3], the Hankel
transform, and ¥- and H-transforms considered below in S 36.7 when p f; q, or
to transforms of the type of integrals of imaginary order when p = q. The given
definitions of the spaces rot;'(L) and Le,'Y) take the behavior of H(s) at infinity
into account. Namely, if c + c. > 0 or c + c. = 0 and 'Y + 'Y.  0 (which can be
briefly written as the inequality 2sign (c + c.) + sign ('Y + 'Y.)  0) then the integral
in (36.3) is absolutely convergent in the case of rot(L) or in square mean in the
case of Le,'Y), respectively.
We give some properties of these spaces.
1) The relation LO,O) = L 2 (0, 00) is true.
2) x- 1 /(x- 1 ) E rot;'(L) or Le,'Y) if and only if I(x) E rot;'(L) or Le.'Y),
respectively.
3) The sets of spaces rot;'(L) and Le,'Y) are well-ordered in the sense that
rot;;'Y,(L) C rot;'(L),
L(e','Y') C L(e,'Y)
2 2
(36.14)
if 2sign(c' - c) + sign('Y' - 'Y) > O.
4) The spaces rot;:(L) and Le,'Y) with the norms
Il/lIroz- 1 = f IF(s)dsl, II/IIL(c,r) = IIFIIL(q)
(c.,.) 
q
(36.15)
and the usual operations of addition and multiplication by scalar are Banach spaces
which are isometric to L( -00,00) and L 2 ( -00,00), respectively.
5) The following theorem gives a characterization of spaces rot;'(L) and Le,'Y)
in terms of spaces rot- 1 (L) and L2(0, 00), respectively.
Theorem 36.1. a) The spaces roto.(L) and Lo,'Y) consist of functions I(x)
representable in the lorm I(x) = x-'YlJ+rp(x) where rp(x) E rot- 1 (L) and rp(x) E
L2(0, 00), respectively.
b) The spaces rot(L) and Le,'Y) with c > 0 consist of functions I(x) for

708 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
which there exist the constants Mj depending only on j(z) such that
II [(z d: f g (1- H m-: r _1/2 z :z )] !(n 2e Z)11 < MJ,
m> c+1', n= 1,2,3,...,
(36.16)
and the norm an (36.16) is evaluated in the space rot- 1 (L) and L 2 (0,00),
respectively.
The proof will be given only for the space LcJ"(), since it is similar for the case of
rot(L ).
a) Let c = O. Then by definition j(z) E LoJ-Y) if and only if
j(z) = 2i f s--YF(s)z-'ds, F(s) E L2(U),
q
(36.17)
where the integral converges in square mean. We transform (36.17) to the form
1 f r(1 - s) -.
j(z) = 21ri r(I+1'_ s ) F 1 (s)z ds,
q
(36.18)
_-yr(I+1'-s)
F1(Z) = s r(1 _ s) F(s).
According to (1.66) F 1 (s) belongs to L2(U) if and only if F(s) E L2(U), Since l'  0
then r(1 - s)/r(2 + l' - s) E L2(U) and in accordance with the Parseval relation
(1.116) in the space L 2 - see, for example, the book by Titchmarsh [I, p. 127,
Theorem 73 and p. 71, the formula 2.1.17] - we have
1
1 f r(1 - s) -8 f (1 - t)'Y
21ri r(2 + l' - s) F 1 (s)z ds = r(1' + 1) <p(zt)dt
q 0
where <p(z) is the inverse Mellin transform of a function F 1 (s). Then from (36.18)
we obtain the relation
-y -  f r(1 - s) 1+-Y-8 _  T'Y+1
Z j(z) - 21ri dz r(2 + l' _ s) F1(S)Z ds - dz 10+ <p(z).
q

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 709
Since F 1 (s) E L 2 (0') then <p(x) E L2(0,00) - see, for example, the books by
Titchmarsh [1, p. 126, Theorem 71] and M.M. Dzherbashyan [2, p. 58, Theorem
1.16] - and therefore <p(x) E L(O, E) for any E > O. Hence it follows that
d lJ':l<p(x) = lJ+<p(x) and therefore f(x) = x--r lJ+<P(x).
b) Let c> O. A function f(x) belongs to Lc,-r) if and only if
f(x) = 2i f s--re-1rcllm8IF(s)x-8ds, F(x) E L 2 (0').
q
It follows from (1.65) that the functions F( s) and
Fl(S) = sm--r e-1rcllm81 F(s)r- 1 (1/2 - c - 'Y + m + 2cs),
m > 'Y + c,
belong or do not belong to the space L 2 (0') simultaneously. The function
smr-l(1/2 - C - 'Y + m + 2cs) satisfies the conditions of the theorem in the book
by M.M Dzherbashyan [2, Subsection 2.3.2]. Therefore Le,-r) coincides with the
space Lf of Dzherbashyan [2, p. 90] where
(s) = sm /r(I/2 - c - 'Y + m + 2cs),
m > 'Y + c.
Hence the product
sme-2c.lnn rr n ( 1 + 2cs ) s E 0',
k + m - c - 'Y - 1 / 2 '
k=l
converges boundedly to the above function (s) as n -+ 00. The latter implies that
sm e -2c8lnn IT ( 1 + k+m--r-1/2 ) /(s) is uniformly bounded with respect to s
k=l
and n. Then by a statement due to M.M. Dzherbashyan [2, p. 90] f(x) belongs
to L = Lc,-r) if and only if the estimate in (36.16) is true. The theorem is thus
proved. We just note that in the case of rot;'(L) the corresponding result by Vu
Kim Than [4] for the space rot. 1 (L) instead of the latter cited statement should be
used. .
36.2. Existence, mapping properties and representations of
the G-transform
We now consider the question of the existence and mafcpin g properties of the
operator of the G-transform in the spaces rot(L) and L 2 c ,-r), and also obtain its

710 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
representations in the usual form via the integrals containing Meijer G-functions
in the kernel along the ray (0,00). The following statement answers the questions
concerning existence and mapping properties.
Theorem 36.2. The G-transform defined in (36.3) with the characteristic (c., 1'.)
exists in the spaces rot;'(L) and LcJ'Y) if and only if
2sign(c + c.) + sign (1' + 1'.)  O.
(36.19)
Under this condition the G-transform isomorphically maps the spaces rot(L) and
L (c,'Y) h ntJ-l (L) d L (c+c. ,'Y+'Y.) . I
2 onto t e spaces ;'!J'-c+c. ,'Y+'Y. an 2 , respect'we y.
Prool. Let f(x) E rot(L) or LcJ'Y). Then it follows from (36.10) and (36.11)
that f.(s) = s-'Ye-1rcllm'IF(s), F(s) E L(O') or F(s) E L 2 (0') respectively. Since
(c.,1'.) is the characteristic of the G-transform, then the asymptotic relation given
in (36.12) holds with the values defined in (36.7). Hence taking s EO' into account
we obtain the representation
H(s )f. (s) = s-'Y-'Y. e- 1r (c+c.)IIm 81 F I (s)
(36.20)
where Fl(S) E L(O') or Fl(S) E L2(0')' Hence the integral in (36.3) defining the
G r . f ' . . f t . . th ntJ-I (L) L (c+c.'Y+'Y.)
-translorm, 1 It eXISts, IS a unc Ion 10 e space ;'!J'-c+c. ,'Y+'Y. or 2 I .
We find now the conditions for the existence of this integral. Since Ix-'I = x- 1 / 2
for s E 0' and Fl(S) E L(O') or L 2 (0'), then only the power-exponential weight
s-'Y-"Y. e- 1r (c+c.)I Im8 1 influences on its existence. If c + c. > 0, then this weight
decreases exponentially as IImsl --+ 00 for any value of l' + 1'.. If c + c. = 0, then
this weight decreases only for l' + 1'. > 0, or it is bounded for l' + 1'. = O. These
can be unified into the same condition of the form (36.19). If these conditions are
not satisfied, then the weight increases at infinity and therefore the integral (36.3)
is divergent. Isomorphism of the map is clear from the above arguments. .
Theorem 36.3. Let the inequality
4signc. + 2sign 1'. + sign Ip - ql > 0
(36.21 )
and the conditions
Rebj > -1/2, j = 1,2,...,m; Reaj < 1/2, j = 1,2,...,n,
(36.22)
hold. Then the G-transform defined in (36.3) exists in the space rot- 1 (L) and may

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 711
be represented as the following Mellin convolution integral
00
(Gf)(x) = f n (  I (ap) ) f(y) dy
,q y (b q ) y
o
(36.23)
containing the Meijer G-function.
P1'OOJ. If the condition in (36.21) holds, then the G-function n (z I :: )
exists and is integrable on any interval [c, E], 0 < c < E < 00, everywhere including
the singular point x = 1 provided that c. = 0, p = q. For 0 < 'Y. < 1 this function
has a singularity of order 0«1- x)'1.-1) as observed from 8.2.1.48 in Prudnikov,
Brychkov and Marichev [3]. Consider firstly p  q. Then this G-function has
asymptotic estimates near the singular points x = 0 and x = 00 which may be
written in the form (see Marichev [10, Sec. 8.3] or [12]):
O(lxl b ),
X -+ 0, b < min Reb k ,
- 1km
G mn ( X I (a,) ) -
,q (b q )-
0(lxI 0 - 1 ) + clxl P cos[( q - p)x 1 /(q-p) + 6], x -+ 00,
a > max Reak and c = 0, 6 = const or (36.24)
- 1kn
c f; 0, q  p + 2, c. = 0,
(q - p)p = (1 + p - q)/2 - 'Y..
Hence it follows that
00 1 00
f Gn (z I ::nz'-ldZ = f O(z6)z'-ldz + f O(z.-l)z.-ldz
o 0 1
(36.25)
for c. > 0 or c. = 0 and 'Y. > 0, q = p. Therefore according to (36.22) the
left-hand integral converges boundedly if s E u. The latter means that there exists
a constant c > 0 such that for any c > 0, E> 0 and t E R1 the estimate
E
1 f K(x)x it - 1 / 2 dxl  c,
£
K ( x ) - n ( X I (a,) ) x'-1
-,q (b q ) ,
holds - see Vu Kim Than [4]. Further, the inequality q  p + 2 following from the
one q > p for c. = 0 is to be taken into account. Then from (36.21) and (36.24) it

712 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
follows that the additional term of the form
00
g 1 X'-IX[2(q-p»)-1_1/2 COS[(q - p)X 1 /(q-p) + 6]dx
1
00
= g(q - p) 1 t(q-p)8i-l/2 COS[(q - p)t + 6]dt,
1
(36.26)
8 = Ims, s E (1,
arises on the right-hand side of (36.25) in the remaining case c. = 'Y. = 0, q  p+ 2.
The latter integral in (36.26) converges boundedly also. So the conditions given
in (36.21) and (36.22) ensure a bounded convergence of the integral in (36.25) for
s E (1. Hence by the Parse val relation - see (1.116) - in the space rot- 1 (L) the
integral in (36.23) is convergent and is equal to (36.3) - see Vu Kim Tuan [1].
Now let p  q. Applying the reflection relation (1.36) and the translation
relation (1.97) for the G-transform to the right-hand side of (36.23) and making
the change of variables x = l/x' and y = l/y' we have
(Gf) (  ) = 1 00 G':.m ( X' I -(bq» ) f (  ) dy' .
x' x' qp y' -(ap) y' y' y'
o
(36.27)
Using now property 2) of the space rot- 1 (L) given in S 36.1, we easily arrive at the
case p  q. .
Remark 36.1. If c' = 'Y' = 0 and p = q, then the G-function a;.n (x I :: ) has
a non-integrable singularity of order 0«1- x)it/l- 1 ), Im1/J = 0, at the point x = 1
and hence the integral in (36.23) is convergent in such case only for f(x) = 0. The
condition in (36.21) excludes this case.
Theorem 36.4. Let 2signc. + sign 'Y.  ° and let the conditions in (36.22) be
satisfied. Then the G-transform exists in the space L 2 (0, 00) and can be represented
in the form
00
( G f)( ) - d I Jn+l ( X I 1, (a p ) + 1 ) ( ) d
x - dx p+l,q+l y (b q ) + 1,0 f Y Y
o
(36.28)
where (a p ) + 1 = al + 1, a2 + 1, .. . , a p + 1. If additionally 2sign c. +sign ('Y. -1/2) > ° I
then the G-transform defined in (36.3) exists in the space L2(0,00) and may be
represented as in (36.23).

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 713
Proof. If2signc. + sign 1'.  0 then the function given in (36.9) is bounded on the
line u in view of the asymptotic estimate in (36.12). Therefore H(s)/(I-s) E L2(U)
and (36.3) may be written in the form
(Gf)(x) = _ 2 1 . d d f H I (8) f.(s)x l - 8 ds.
1n x -8
q
(36.29)
Applying now Parseval's relation, given in (1.116), in the space L2(0,00) to the
right-hand side of (36.29), and the translation relation (1.97) for the G-function, in
particular, leading to the relation
Gm,n+l ( I 0, (a p » ) ,n+l ( 1 1, (ap) + 1 )
x p+l,,+1 x (6,),-1 = p+l,,+1 x (6,)+1,0
(36.30)
we arrive at the representation in (36.28) without difficulty.
If 2signc. + sign (1'. - 1/2) > 0, then not only H(s)/(I- s) E L2(U) but also
H(8) E L2(U) since H(s) = O(lsl-'Y.). Therefore Parseval's relation can be applied
to (36.3) immediately which yields (36.23). .
Remark 36.2. Since, for example, (GA I  I f(t») (z) = L {t f m ;z} and
( G8 I  I f(t») (z) = L -1 {t f m ; z} - see (1.119)-(1.121) - and the Meijer
G-function of the form (36.23) or (36.30) does not exist in the case m = n = 0
of the G-transform, then not every G-transform may be represented by (36.23)
or (36.28). The cases 2signc. + sign1'. > 0 and 2signc. + sign1'. < 0 of the
G-transform correspond to the direct and inverse classical integral convolution
transforms and the cases 2signc. + sign1'. = 0, i.e., c. = 1'. = 0, correspond to
the Watson transform for q f; p ( see the books by Titchmarsh [1, Chapter 8] and
M.M. Dzherbashyan [2, Chapter 2, Section 1]) or to the fractional-type integrals of
imaginary order I and I for q = p. This can be seen in more detail in S 36.1.
Thus the general approach to the theory of integral convolution transforms with
homogeneous kernels is realized via the representation in the form (36.3).
36.3. Factorization of the G-transform
We introduce the following definition.
Definition 36.5. We call any representation of the operator in (36.3) by a
composition of other G-transforms with less indices of complexity - see Definition
36.2 - a factorization of the G-transform.
The most simple G-transforms except the trivial one in (36.3) are those which
have indices of complexity equal to 1. They are connected with the Laplace
transforms given in (1.119)-(1.121) and are defined in the following way.

714 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Definition 36.6. The following transforms of a function f(x)
xOlA%x-OIf(x) = XOlL{t%OI-If(fFI);X%I}
00
= f (i) 01 e-(Zlt)%l f(t) t ,
o
(36.31 )
x Oi Alx-OI f(x) = x OiTI L -1(f FOI f(t%l); X TI }
'Y+ ioo
1 f f-(s)x-8
= 211'i r(:i:a :i: s) ds,
'Y- ioo
(36.32)
Re(s + a)  0,
with L{<p(t);p} and L-l{g(p);x} given in (1.119) and (1.120) or (1.121) are called
the direct and inverse modified Laplace transforms with power multipliers.
It is also expedient to consider G-transforms where indices of complexity are
equal to 2 as simple transforms. These are fractional integrals and derivatives Ig+
and I, the Stieltjes, Hankel and Meijer transforms, the sine- and cosine transforms
and their inverse transforms. In accordance with the convolution theorem in (1.115)
or Parseval's relation in (1.116) compositions of several G-transforms correspond to
the multiplication of the transforms of their kernels in the form (36.9), which leads
to an increase in the numbers of the gamma-functions in the transform of a kernel.
This leads again to a G-transform with greater index of complexity. Therefore it is
natural to hope that any G-transform under some conditions can be factorized in
terms of compositions of other G-transforms, and in particular, in terms of simple
G-transforms enumerated above. In fact, the following assertion is true.
Theorem 36.5. Let G I ,..., G, be G-transforms such that transforms of their
kernels HI (s), . . . , H,( s) satisfy the condition
HI(S)H2(S)... H,(s) = H(s)
(36.33)
where H(s) has the form (36.9). Let also the transforms Gl,..., G, have indices
of complexity less then G and let them have characteristics (ci, 1'i), . . . ,( c, 1';),
respectively. Then if the condition in (36.19) holds the G-transform - see (36.3)
- in the space rot(L) and Lc,'Y) can be factorized via the transforms G I ,. .. ,G,
arranged in some order
(Gf)(x) = (Gi,.. . Gi'JGi1f)(x)
(36.34)

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
715
if and only if (ii, .. . ,i,) is a reaJTangement of the numbers (1,2, . .. ,I) such that
the inequalities
2sign ( c + t Ci; ) + sign ( r + t r;; )  0,
3=1 3=1
(36.35)
k = 1,2,. . . ,I,
hold. For any group of transformations G 1 ,..., G, of the above form there always
exists at least one reamlngement satisfying (36.35).
Proof. The existence of G-transforms satisfying the condition in (36.3) is clear. For
example, this condition holds if we set 1 = p+ q and take r(b i + s), i = 1,2,..., m,
r(l- ai - s), i = 1,2,...,n, r- 1 (ai + s), i = n + 1,...,p, r- 1 (1- bi - s),
i = m + 1,... ,q as the transforms of the kernels H}(s),..., H,+q(s) respectively,
which yields the factorization via the direct and inverse modified Laplace transforms
defined in (36.1) and (36.2) - see Remark 36.2. It is clear from (36.33) that the
relations
,
"" c"f = c.
L.J · ,
i=l
,
L -y: = ,.
i=l
(36.36)
are true. Applying Theorem 36.3 1 times in succession we easily obtain necessary
and sufficient conditions given in (36.35) which guarantee the existence of the whole
compositions (Gi le ... Gi'JGhf)(x), k = 1,2,..., I. These compositions belong to
(' ') 1c 1c
the spaces rot;'\,(L) or L 2 c .'Y where c' = c + L: c;', -y' = -y + L: -yr. Hence
· j=l J j=l J
( 1 c+c. 'Y+'Y.
by 36.36) we have that (Gf)(x) E rot;-+c..'Y+'Y.(L) or (Gf)(x) E L 2 · ,
respectively for k = I.
We show now the existence of the rearrangement (;1,"" i,) for which the
conditions, given in (36.35), hold. For this purpose we choose the indices ;},. .. ,i,
in the following way. Let i} be the index of the largest number among ci. If there
are several such c;, then we take ;1 as the index of the largest number among the
numbers ,: in the pairs (c;, -y:) where ci are the largest numbers and i 1 is an
arbitrary index ; from this set if there are several such numbers -y:. Withdrawing
the pair (ci, -y;) with such indices we consider the rest of the pairs and choose the
next index i2 and the corresponding pair among them in a similar way. As a result
after such a choice we have the inequalities
2sign (ci; - Ci;+l) + sign (-y:; - -Y:;+I)  0,
(36.37)
j = 1,2, . .. ,1- 1.

716 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
We show that (36.35) follows from (36.37). We assume on the contrary that (36.35)
is not true, for example, for k = 1- 1:
( '-1 ) ( I-I )
2sign c + ?: ci; + sign "I + ?: "Ii;
1=1 1=1
< o.
(36.38)
That is, the inequality
c + c +... + c < 0
'I " - I
(36.39)
or
c + c! +... + c! = 0
'I "-1'
(36.40)
"I + "Iii + .. . + 'YZ_I < 0
is true. According to (36.36) from (36.19) we have that
2si g n ( c + c! +... + c! ) + si gn ( 'V + 'V +... + 'V ) > 0
'I "-I I "1 "'-1 - ,
i.e.
c + c! +... + c! + c! > 0
'I "-I "
(36.41 )
or
c + ci l + . . . + ci'_1 + ci, = 0,
(36.42)
"I + "Iii + .. . + 'YZ_I + "Ii,  0
hold.
If (36.39) and (36.41) or (36.39) and (36.42) or (36.40) and (36.41) hold
simultaneously, then ci, > 0 and since ci l  ci  .,.  ci, by construction then
c + ci 1 + . . . + ci,_1 > 0 which contradicts (36.39). Now we suppose that the
system of conditions in (36.40) and (36.42) holds. Then ci, = 0 and 'YZ > O. So
it follows from (36.19) and (36.40) and the inequalities ci l  ci  ...  ci, = 0
that ci l = ci = ... = ci, = 0, c = 0, from (36.37) that ,i l  ...  ,i, > 0 and
from the condition 2signc + sign'Y  0 that "I  O. The latter contradicts (36.40)
and therefore (36.38) is not true. Similarly one may prove that the inequalities in
(36.35) hold for the values k = 1- 2,... ,1. .
36.4. Inversion of the G-transform
Theorem 36.6. Let g(x) E rot;c.,'Y+'Y.(L) or g(x) E L+c.,'Y+'Y.. Then the
G-transform
(G-1g)(z) = (G::;n..-m I i;::U:? I g(y)) (z) = I(z)
(36.43)

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 717
is the inverse of the G-transform defined in (36.3) with the notation (Gf)(x) = g(x).
If additionally the conditions of Theorem 36.5 are satisfied, then the inverse operator
in (36.43) is factorized by the relation
f(x) = (G- 1 g)(x) = (Gi;.lGi./ ... Glg)(x).
(36.44)
Proof. Since 9 E rot;-c..'Y+'Y.(Lc+c. .'1+'1.», then by Theorem 36.2 the G-transform
in (36.43) exists and maps a function 9 into a function f in the space rot;'(Le.'Y».
Applying now the G-transform in (36.3) to (36.43), evaluating the left composition
directly, reducing all gamma-functions and using (36.3) we obtain the function g(x)
on the left-hand side.
The inverse G-transforms Gi- 1 involved in (36.44) have the characteristics
J
(-ci j , -1';), j = 1,2,..., I. Therefore the conditions in (36.35) with respect to
(36.44) have the form
, ,
2sign (c + c. - L ciJ + sign (1' + 1'. - L 1'iJ  0,
j=k j=k
(36.45)
k = 1,2,..., I.
It is clear that (36.45) and (36.35) coincide and hence Theorem 36.5 enables us
to factorize the inverse operator (36.43) by (36.44) for the above rearrangement
(i 1 ,...,i,). .
Remark 36.3. If the characteristic (c, 1') of the space rot;, (L) or Le.'Y) satisfies
the condition
, ,
2sign (c - L Icn + sign (1' - L hn  0
i=l i=l
(36.46)
then the restrictions in (36.35) are true for all rearrangements and therefore the
order of application of th-e operators Gij in (36.34) and Gl in (36.44) may be
arbitrary, i.e. these G-transforms are commutative in such a case.
We consider the important case when all the Gi, i = 1,2,..., I, are modified
Laplace transforms as defined in (36.31) and (36.32). Applying the Mellin transform
- see (1.112) - to these relations we see that the transforms of the kernels in (36.9)
containing only one gamma-function in the numerator or in the denominator
r(:i:a:i: s) +-+ x a A:i:x-a, Re(a + s)  0,
r-. 1 (:i:a:i: s) +-+ x a Alx-a, Re(a + s)  0,
(36.47)
(36.48)

718 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
correspond to such operators. Hence each of the above Laplace transforms
corresponds to one of the gamma-function in the G-transform (36.3). That is, this
general G-transform can be factorized via the compositions of the operators
x bj A x- bj
+ ,
j = 1,2,..., m;
X Oj - l A_xl-Oj, j = 1,2,. . . , n;
,..Oj A -l,..-Oj J n + 1 P .
"" + "" , . = ,..."
(36.49)
,..bj-1 A - _ l,..1-bj , J m + 1 q
"" "" '= ,..., .
under certain conditions. By the symbols
AI",' ,Am, Am+l,'" ,A m + n , Am+n+l"" ,Am+p, Am+,+l,'" ,A,+q
we denote these operators respectively taken in indicated order. Their
characteristics being evaluated by (36.7) are equal to (1/2, -Reb j ), j = 1,2,... , m,
(1/2,Reaj), j = 1,2,...,n, (-1/2,Reaj), j = n+ 1,...,p, (-1/2,-Reb j ), j =
m + 1,..., q respectively. We denote them by (8 1 ,6),..., (8,+q,p+q) in the
indicated order. Theorems 36.5 and 36.6 can be formulated with respect to such a
factorization in the following way.
Theorem 36.7. Let the conditions in (36.19) and (36.22) and the inequalities
Re aj > -1/2, j = n + 1, . . . , p;
(36.50)
Rebj < 1/2, j = m + 1,..., q,
hold. Then the G-transform defined in (36.3) can be factorized in the spaces
rot(L) and Lc,..,) via a composition of modified Laplace transforms with power
multipliers (36.49) applied in same order
(Gf)(x) = (Ai p +" ... Ai'JAi1f)(x)
(36.51)
if and only if (i l ,..., i,+q) is a rearrangement of the numbers (1,2,...,p + q) such
that the inequalities
k k
2sign ( c + L () i j) + sign ( "y + L i j)  0,
j=l j=l
(36.52)
k = 1,2,...,p+q,
hold.

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 719
Theorem 36.8. Let g(x) E Jt:;;.c. ,'Y+'Y. (L) or L+c. ,'1+'1. and the conditions
in (36.19), (36.22), (36.50) and (36.52) be satisfied for a certain reamlngement
(iI, .. . ,i,+q) of the numbers (1,2,... ,p + q). Then the G-transform defined in
(36.43), inverse to (36.3), can be factorized via the operators AI, . . . ,A,+, with the
characteristics (8 1 ,6), . . . , (8,+" ,,+q) by the relation
f(x) = (G- 1 g)(x) = (A . :- 1 A . :-l .. . A . :- 1 g)(x ) .
1  p+v
(36.53)
Remark 36.4. In the case of operators in (36.49) the transforms of the kernels
Hl(S),..., H,,+q(s) are single gamma-functions of the form r(b j + s), r(l- aj - s),
r-1(aj + s) or r- 1 (1- b j - s) respectively. We can combine these gamma-functions
into various groups by different methods, for example in pairs. As a result we
can obtain many variants of composition expansions for the G-transform by using
(36.34), for example, via fractional integrals and derivatives, Hankel, and Stieltjes
and Meijer transforms and their inversions.
Remark 36.5. If the conditions in (36.22) or (36.50) are violated for some indices
j provided that (36.5) hold, then instead of the corresponding operators in (36.49)
there arise operators such that in the kernels of the direct Laplace transforms
r ( _ .. ,II
the function e- z will be changed to e- Z - L:. For such indices j the
k=O
gamma-function corresponding to the image of the kernel can be changed by three
gamma-functions by the formula
r(a + s) = (_I)k r(a + k + s)r(l- a - k - s) ,
r(1 - a - s)
-k < Re(a + s) < 1- k.
(36.54)
Under a suitable choice of k the condition violated for the latter will be now
satisfied. But such an operation leads to an increase of the index of complexity of
the G-transform.
From what has been said above we conclude that the following statement
holds which is important in the theory of integral transforms.
Theorem 36.9. The G-transfonn and, in particular, the classic direct and
inverse integral transforms of the convolution type is the composition of a certain
number of direct and inverse Laplace transfonns of the form (36.31) and (36.32),
whose admissible order of application depends on the space of functions f and the
parameters of the transfonns. For sufficiently good function spaces of the type
rot;:(L) or Lc,'Y), with the condition, for example, 2c - p - q > 0, the operators

720 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
AI,... ,A'+f are commutative and their composition forms the G-transform defined
in (36.3).
36.5. The mapping properties, factorization and inversion
of fractional integrals in the spaces 91r;:;(L) and Lc,..,)
Theorem 36.10. The fractional integro-differentiation operators x-a(I+f)(x)
and x-O(If)(x) of an arbitrary complex order a are both defined in the spaces
rotO:,(L) and LoJ"") with "I' = max(O, -Rea). These operators map the spaces
rot (L) and L c,..,) with 2sign c + sign ("I - "I')  0 isomorphically onto the spaces
rot;''+Rea(L) and Lc,""+Rea), respectively and they can be factorized via the
composition of the operators given in (36.31) and (36.32) by the following relations
under the cOfTesponding conditions indicated below C '
(Ig+f)(x) = x-I A:1x a A_xf(x)
(36.55)
if f(x) E rotO,(L) or LoJ"') and Rea > -1/2;
(Ig+f)(x) = x a - 1 A_x-a A:1x a + 1 f(x)
(36.56)
if f(x) E rot/;J_Rea(L) or L1/2,-Rea) and Rea > -1/2;
(1;:+I)(z) = (  r z-IA:1za+'A_z/(z)
(36.57)
if f(x) E rotO,;(L) or LoJ"') and k > -Rea - 1/2;
(If)(x) = x a A+1x- a A+x a f(x)
(36.58)
if f(x) E rotO,;(L) or Lo,..,) and Rea < 1/2;
(If)(x) = A+x a A:;l f(x)
(36.59)
if f(x) E rotl2J..,(L) or L1/2,..,) and Rea < 1/2;
(If)(x) = x a A:;lx- a I+A+xa-1c f(x)
(36.60)

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 721
if f(x) E rotO,(L) or Lol") and k -1/2 < Rea < k + 1/2.
Proof. Let n be an integer such that Rea + n > O. Then we have
x-Ot(Ig f)(x) = x- Ot  ( I Ot + n  f f-(S)X-'dS )
+ dx n 0+ 21r1
q
= x- Ot cr  f f-(s)lOt+nx-'ds
dx n 21ri 0+
(1
cr 1 f [ I-S ] +
= x- Ot __ r f-(s)x Ot n-'ds
dx n 21ri 1 + a + n - s
q
1 f [ I-S ] cr +
= x- Ot  r f-(s)-x Ot n-'ds
21r1 1 + a + n - s dx n
q
= 2i f r [11:  s] /*(s).,-'ds.
q
(36.61)
All interchanges of the order of integration and differentiation that have been made
in (36.61) are valid because of the absolute convergence of the integrals given
above. The latter is true by Isl-" F(s) E L(O') - see (36.11) - or by Parseval's
relation in the space L 2 (0') and Isl-" F(s) E L 2 (0'). If now the transform of the
kernel is written as the product r- I (1 + a - s)r(1 - s), then the factorizations in
(36.55) and (36.56) with the corresponding conditions on the spaces and Rea are
easily obtained as particular cases of Theorem 36.7.
Let now a be an arbitrary complex number and k be a positive integer such
that Rea + k + 1/2 > O. Then by (1.47) the product r- l (1 + a - s)r(1 - s) can
also be written in the form
r [ 1 1 - s ] = (1 + a - s)l:r- 1 (1 + a + k - s)r(l - s).
+a-s
Now by the arguments given after Remark 36.3 and the relations 8, 14 and 12
in Brychkov, Glaeske and Marichev [1, p. 24], and also by the choice of k, the
images (1 + a - s)l:, r- I (1 + a + k - s) and r(1 - s) correspond to the operators
x- Ot ( t: )1: xOt+I:, x-l-Ot-I:A:lxl+Ot+1: and x-lA_xl the characteristics of which are
(0, -k), (-1/2, Rea + k) and (1/2,0), respectively. Hence the factorization of the
form (36.57) follows. The remaining relations are proved in the same way. The
first statements of this theorem are the direct corollaries of Theorem 36.2. .
Corollary 1. Any function f(x) in the space rot;,(L) or Lc.,,) can be represented

722 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
in the form f(x) = x--Y Ici+rp(x) where rp(x) E rot;'(L) orrp(x) E LCIO), respectively.
The proof follows from the fact that x--y Ici+ is the G-transform with the
characteristic (0,1') and therefore it is an isomorphic mapping of the space
rot;,( L) or Lc,O), respectively onto the space rot (L) or Lc,-y) provided that
2sign c + sign l'  o.
Corollary 2. If a = i8 is a pure imaginary number then the operator I as an
automorphism in the spaces rot(L) and Lc,-y).
The proof follows directly from the fact that the operator x-iS I& IS the
G-transform with the characteristic (0,0).
36.6. Other examples of factorizations
This subsection deals with examples of factorizations in terms of fractional integrals
of Hankel and Bessel transforms, generalized Laplace transforms and potential type
integrals. For convenience we shall denote by {h(x)}rp the Mellin convolution in
00
(1.114) and set {h(l/x)}rp = f h(t/x)rp(t)t- 1 dt.
o
The following statements are valid.
Theorem 36.11. The Hankel transform of modified form
00
{J.(2.fi)}f = J J. (2/f) f(t) t ,
o
Rev> -1,
(36.62)
maps the spaces rot(L) and Lc,-y) isomorphically onto itself and it can be.
factorized as the composition
{J II (2yX}} f = x- 1 - 1I / 2 A:IX 1 + 11 A+x- II / 2 f(x).
(36.63)
If additionally f(x) E rot/,RA!1I/2(L) or f(x) E L/2,RA!1I/2(L), then the composition
in another order
{J II (2vx)} f = xll/2 A+X- l - II A:lx 1 + 11 / 2 f(x)
(36.64)
can be also admitted.
The proof follows from the fact that if Rev > -1, then the operator in (36.62)

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 723
admits the representation
{J II (2vx)}f = 2i I r(S + 1I/2)r- 1 (1 + 11/2 - S)f.(s)x-'ds
q
(36.65)
according to the sixth relation given in (1.118) and the Parseval relation - see
(1.116). So it is factorized via two operators xll/2 A+X-II/2 and X-l-II/2 A:lx 1 + 11 / 2
as seen in the arguments after Remark 36.3. Their characteristics are equal to
(1/2, Rell/2) and (-1/2, -ReIl/2), respectively. So applying Theorem 36.5 we
obtain the conditions guaranteeing the existence of the compositions of these
operators in one or another order as well as (36.63) and (36.64) themselves. .
Theorem 36.12. The Bessel ¥ -transform of the modified form
00
{Y.(2v1z)}f = I Y. (2/f) f(y) ; ,
o
IReIlI < 1,
(36.66)
where ¥II(Z) is the Bessel function of the second kind (Erdelyi, Magnus,
Oberhettinger and Tricomi fe, 7.e.l]) maps the spaces rot;:(L) and LCt7)
isomorphically onto itself and it can be factorized via the composition of the Hankel
operators J II = {J II (2..jX)} and the fractional integrals or derivatives of the form
K = x- II / 2 I: 1 / 2 x(II+1)/2 and I = x-(1I+1)/2 I2 xll/2. If c = 0 and II < 1/2, then
the operator J II can be in any position and the operator K must be applied after I:
¥II = KIJ II = KJIII = JIIKI.
(36.67)
If 2signc + sign ( l' - 1/2)  0, then the order of application of the operators is
arbitrary.
Proof. In accordance with (1.116) and 9.4(1) in the book by Marichev [10] we
form the G-transform as
(G f)( ) 1 Ir [ 8+11/2,8-11/2 ] f . ( ) -' d
y x = 21ri s _ (II + 1)/2, (3 + 11)/2 _ s 8 X 8.
q
(36.68)
Its characteristic is equal to (0,0) and so by Theorem 36.2 it exists in any spaces
rot;'(L) and LCt"') and is an isomorphic mapping of this space onto itself. If
addiionally the conditions in (36.22) which have the form -1 < Re II < 1 are
satisfied, then according to Theorem 36.3 the integral in (36.68) is transformed

724 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
into the integral in (36.6). Now we write the transform of the kernel in the form
r [ s+II/2 ] r [ s-II/2 ] r [ 11/2+1-8 ]
II /2 + 1 - 8 8 - (II + 1 )/2 (3 + 11)/2 - s .
U sing the second relation in (1.117) it is not difficult to establish the following
Mellin correspondences
r [ a + S ] +-+ xOl:-ox-c,
c+s
Re(a + 8) > 0,
(36.69)
r [ b - S ] I-d Id-b b-l
d _ 8 +-+ X 0+ x ,
Re(b- 8) > 0,
(36.70)
r [ :] .... .,(o-d+l)/2{Jo+d_l(2v'Z)}.,(d-o-l)/2,
(36.71)
Re(a + d), Re(a + s) > 0,
from (36.1), (36.2) and (36.65). Applying these relations to the above image of the
kernel 1 it is easy to obtain the whole factorizations indicated in this theorem. The
conditions on the characteristics of the spaces are deduced from Theorem 36.5. .
Theorem 36.13. The Bessel H-transform of the modified form
00
{H. (2v'Z)}/ = J H. (2/f) /(1/)  , -2 < Rev < 0,
o
(36.72)
where HII(z) is the Struve function (Erdelyi, Magnus, Oberhettinger and Thcomi
[2,7.5.4]) maps the spaces rot(L) and Le,,,,} isomorphically onto itself and it can
be factorized as the following compositions
{H II (2v'Z)}f = (J II )(x(II+1)/2 1:1/2 1!2x-(1I+1}/2)f(x),
{HII (2VX}} f = x(II+1}/2 1: 1 / 2 x-II/2( J II )xll/2 1!/2 x-(1I+1}/2 f(x),
(36.73)
(36.74)
{H II (2VX}}f = (x(II+1)/2 1:1/2x- II / 2 )(x ll / 2 1!2x-(1I+1}/2)(JII)f(x),
(36.75)
where f(x) E rot- 1 (L) or f(x) E L 2 (0,00) in the first two cases, and f(x) E
rot;'/2( L) or L,1/2 in the third case. Moreover, the operators on the right-hand

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 725
sides of (36.73) and (36.74) are commutative and the operator (J II ) in (36.75) is
commutative with the operators x(II+1)/21: 1 / 2 x- Il / 2 and x"/2 11 2 x-(1I+1)/2.
The proof is similar to that of Theorems 36.11 and 36.12 and is based on the
representation of the transform {H II (2yX}}f in the form
{H (2yX}}f = f r [ s + 11/2 ] r [ s + (II + 1)/2 ]
II 21r1 1+1I/2-s s+II/2
(1
(36.76)
x r [( -= :2_-sS ] f"(s)z-'ds
provided that -2 < Rell < 0, which follows from 9.5(1) in the book of Marichev
[10]. .
Theorem 36.14. The generalized Laplace transform
00
D.{f(y); z} = 2-./ 2 f e-%(2o)-' (D. m !(y) d: ,
o
(36.77)
Rell < 1,
where DII(z) is the parabolic cylinder function (ErrUlyi, Magnus, Oberhettinger and
Tricomi [2,8.2]) maps the spaces rot;'(L) and Le..y) isomorphically onto the spaces
rot;1/21'Y-ReIl/2(L) and L+1/2''Y-ReIl/2, respectively and it admits the following
composition expansions
DII{f(y); x} = xl/2 1:"/2 x(II-1)/2 A+f(x),
DII{f(y); x} = A+xl/2 1:"/ 2 X(II-1)/2 f(x),
(36.78)
(36.79)
where Rell < 0 and 2signc + sign(-y + (1 - ReIl)/2)  0 in the latter case.
Proof. In accordance with 8.30( 1) in the book by Marichev [10] we consider the
G-transform
(Gnf)(z) = 2i f r [s  tl  02] f"(s)z-'ds.
(1
(36.80)
It has the characteristic (1/2, -Rell/2) and therefore exists in any space rot(L)

726 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
or L(e,y) and ma p s it into rot-I ( L ) or L(e+l/2,..,-ReIl/2) b y Theorem 36 2
2 e+l/2,..,-ReIl/2 2 . .
The reducibility of (36.80) to (36.77) follows from Theorems 36.3 and 36.4 and the
factorization relations (36.78) and (36.79) follow from Theorem 36.4. It is easy to
formally write these relations on the basis of the correspondence
[ s, s + 1/2 ] [ S + 1/2 ]
DII +-+ r S + (1 _ v)/2 = r[s]r s + (1 - v)/2
(36.81)
+-+ (A+)(x 1 / 2 r:,"/ 2 x(I-1)/2).
The theorem is proved. .
In the conclusion of this subsection we illustrate another proof, although
formal, of the decomposition (12.39) for the potential type integral in (12.34)
00
1 01 - 1 f cp(t)dt
o cp - 2r(a) cos(a1r/2) Ix - tp-OI'
o
x> O.
(36.82)
For any function I E rot(L) (Le,..,» we have the representation
x-O(I!: f)(x) = 2i f r [s _ :/;, a i  :j2 _ s] f"(s)x-'ds,
q
(36.83)
see 2.5(1) in the book by Marichev [10]. Using now (36.69) and (36.70) we easily
obtain the following factorization
x-OI(Ig I)(x) = (X-OI I/2xOl/2)(x-0i/2 I;2)/(x)
(36.84)
where an arbitrary order can be admitted. The transform of the kernel may be
decomposed into two fractions by another way which leads to the factorization via
Hankel transform.
36.7. Mapping properties of the G-transform on
fractional integrals and derivatives
Theorem 36.15. Let the conditions
2sign(c + c.) + sign (1' + 1'. + Rea)  0,
2sign c + sign ( l' + Re a)  0
(36.85)

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 727
hold. Then there exists the G-transform defined in (36.3) of the operator
x-a(Ig+f)(x) in the spaces rot(L) and Lc,..,) and it is evaluated from the relation
(Gz-a(I+f))(z) = (G;+11 (:),  l/(t») (z).
(36.86)
The proof follows from Theorem 36.5 where we have to set I = 2,
Gi l f = x-a Ig+f and Gill = Gf in (36.3) and take the G-transform
(G;+11 (:),  I/(t») (z) instead of that in (36.3).
Theorem 36.16. Let the conditions of Theorem 36.15 be satisfied. Then there
exists the G-transform defined in (36.3) of the function x-a(I/)(x) in the spaces
rot(L) and Lc,..,) and it is evaluated from the relation
(Gz-a(If)(z) = (c;:'t'11 : () l/(t») (z).
(36.87)
The proof is similar to that of Theorem 36.15.
36.8. Index laws for fractional integrals and derivatives
For fractional integrals and derivatives there are known the so-called index laws
given by (10.4), (10.5), (10.42) and (10.43), see also Theorem 10.7. In this
subsection we prove the theorem characterizing the conditions under which index
law as well as the analogue of Theorem 10.7 for the operators Ig+ and I in the
spaces rotO,(L) and Lo,..,) hold.
Theorem 36.17. Let (nl"" ,nn) and (PI,... ,Pn) be two sets of arbitrary
complex numbers. Then the operators x P . I-P. x-a. are commutative in the space
{f(x) : I(x) = x 6 g(x), g(x) E roto,;(L) or g(x) E Lo,..,)} with
i
"y = max{O, max Re L(Pij - nij ), {i l ,..., il:} C {I, 2, . .. ,n}}, (36.88)
j=l
6 > max Reni - 1/2. (36.89)
lSiSn
Moreover, if in addition (Pit..., Pn) is a certain reafTtJngement of the set

728 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
(Qt,..., Qn), then the composition of all these operators is an identity operator
(x fJi ,. I" -fJi,. x- ai ,. ) ., . (XfJil I;l -fJi l x-ail )f(x) = f(x).
(36.90)
If the conditions in (36.88) or (36.89) are not satisfied, then the above operators
are not commutative.
Proof. Let the representation f(x) = x 6 g(x) where g(x) E rotO,(L) or LoJ"') be
true. Then by (36.10) we have
R a. -{J. R a. -(J. 1 f 6
x,...illo.t l 'lx-ailf(x) = X,...illo.+.l 'I 21ri g-(8)X -a i l - 8 ds.
(1
Since Re(6 - Qi l - s) > -1 according (36.89), then for m > Re(Pi l - QiJ we find
XfJillail -(Jil x-ail f(x) = X{Jill- m lail -{Ji l +m  f g- ( 8 ) x6-ail -8 ds
0+ 0+ 0+ 21ri
(1
=X{Jill- m  f r [ 1 - Qi l - S + 6 ] g-(8)x6-{Jil +m- 8 d8
0+ 21ri 1 - Pi l - 8 + 6 + m
(1
_ (Ji 1 f r [ 1 - Qi l - S + 6 ] - ( ) cJ"I 6-{Ji +m-' d
-x 1 - 9 8 -X 1 s
21ri 1 - Pi 1 - S + 6 + m dx m
(1
- 1 f r [ 1 - Qi l - 8 + 6 ] - ( ) 6-8 d _ ( )
- 21r; 1 - Ph - 8 + 6 9 8 X 8 - cp :t .
(1
All transpositions of operators that have been carried out are valid since all integrals
we have used converge absolutely or (according to Parseval's relation) in L2, in view
of the condition g- (8 )8'" E L( 0') or g- (8 )8'" E L 2 ( 0') respectively. The constructed
function cp(x) belongs to the space {f(x) : f(x) = x 6 g(x), g(x) E rotO,,(L) or
Lo,..,'), where 'Y' = 'Y + Re(Qi l - Pi l )}. The conditions in (36.88) and (36.89)
rewritten in the form
1c
'Y- = max{O,maxRe L(,Bij - Qij)' {i l ,... ,i rc } C {1,2,... ,n} \ id,
j=l
6> max ReQi - 1/2,
l<i<n
f;ti;

 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS 729
with respect to the parameter "I' are also satisfied. So we can apply
operators of the type xfhJ 1;.+'J -{3i'J x-ai'J with i2 E {I,..., n} \ il to rp(x) again.
Continuing this procedure we obtain by Theorem 36.5 that the composition
(x{3i.. 1;" -(3i.. x-a i ..). .. (x{3i l 1;I-{3il x-ail )f(x) exists for any rearrangement
(il,..., in) and moreover it is representable in the form of the G-transform
 f r [ 1 - ai.. + 6 - s,..., 1 - ail + 6 - S ] g.(S)X6-8ds
21r1 1 - {3i.. + 6 - s, . . . , 1 - {3i l + 6 - s
q
= f r [ 1 - al + 6 - s,..., 1- an + 6 - S ] g.(S)X6-8ds,
21r1 1 - {31 + 6 - s, . . . , 1 - {3n + 6 - s
q
(36.91 )
see (10.48), and therefore it does not depend on other applications of composing
operators. The change of this order leads to the change of the order of the
gamma-multipliers in (36.91). If in addition {{31,'" ,{3n} = {al,"', an}, then all
gamma-functions in (36.91) are cancelled and the right-hand side has the value
x 6 g(x) which yields (36.90).
Let now the conditions in (36.89) be broken, Le. let there exists a number
j such that 6 =:; Rea; - 1/2. At first let 6 < Reaj - 1/2. Then the operator
x{3jl;-{3jx-ajf(x) does not exist for the function f(x) = x- l / 2 +£e- z E roto.(L)
or Lo"y) if € < Reaj - 1/2 - 6. Similarly, in the case 6 = Reaj - 1/2 the function
f (x) can be found in the space indicated in the formulation of this theorem such
that the above operator does not exist on this function.
Finally we suppose that the condition in (36.88) is not satisfied for a certain
set (il,..., il:)' Then by Theorem 36.5 the composition n (X{3ij 1;;-{3i j x-aij)f(x)
j=l
does not exist in the space rotO,(L) or Lo,..,). .
Theorem 36.18. Let the conditions of Theorem 36.17 be satisfied except (36.89)
instead of which we assume the opposite inequality
6 < max Reai - 1/2
lSjSn
(36.92)
to be hold. Then the statements of Theorem 36.17 with 10+ replaced by 1_ are tme.
The proof is similar to that of Theorem 36.17.
Corollary 1. Let "I = max(O, -Rea, -Re{3, -Re(a + (3» and 6 > max( -Rea,
-Re(3) - 1/2, or "I = max(O, -Rea, -Re(a + (3» and 6 > -Rea - 1/2. Then the
first and the second relations in (10.4) hold in the spaces {f(x) : f(x) = x 6 g(x),
g(x) E roto.(L)} and {f(x) : f(x) = x 6 g(x), g(x) E Lo,..,)}, respectively.

730 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
The proof is obtained on the basis of the representation
-a-P I a I p f( ) 1 f r [ p + 1 - s . ] r [ 1 - s ] f . ( ) -8 d
X 0+ 0+ X = 21ri a + P + 1 _ s p + 1 _ s s X s.
q
Corollary 2. Let'Y = max(O, Rea, Rep, Re(a+p» and 6 > max(Rea, ReP)-1/2.
Then the relation
101 x-alP x-P f( x ) - IP x-PIa x-OI f( x )
0+ 0+ - 0+ 0+
(36.93)
holds in the spaces {f(x) : f(x) = x 6 g(x), g(x) E roto,;(L)} and {f(x) : f(x) =
x 6 g(x), g(x) E Lo,'Y)}.
The proof follows from Theorem 36.17 and Corollary 1.
Corollary 3. Let'Y = max{O, -Rea, Re(a + P)} and 6 > max(O, -Rep, -Re(a +
P»-1/2. Then (10.42) holds in the spaces {f(x) : f(x) = x 6 g(x), g(x) E rotO:(L)},
and {f(x) : f(x) = x 6 g(x), g(x) E Lo,'Y)}.
Proof. Taking the result a + P + / = 0 into account we rewrite the left-hand
side of (10.42) in the form l:-P xOl+Px- P Ig+x- OI - P Ig+x P f(x). This composition
exists and has the value
 f r [ l+a+p-s ] r [ l-s ] r [ I+P-s ] f.(s)x- 8 dS=f(X)
21r% 1 - s 1 + p - s 1 + a + P - s
q
provided that the conditions of the Corollary are satisfied. This completes the
proof, cr. the proof of Theorem 10.6. .
 37. Equations with Non-Homogeneous Kernels
In this section we consider some classes of integral equations of the first kind
solvable in closed form, the kernels of which are not directly representable in the
form K(x/t). These are (1) equations in which the kernel is of the type K(x - t),
and which takes the form K(In 8), 8 = y/r, after the substitutions x = In y and
t = In r, (2) certain non-convolution equations with Bessel functions, (3) equations
of composition type decomposed into more simple invertible equations, and (4)
also equations connected with the integration of special functions with respect to
a parameter - Kontorovich-Lebedev and Mehler-Fock type transforms.

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
731
37.1. Equations with difference kernels
We investigate the following seven left-sided Abel-type integral operators
:&
j ( t)a-l
(1:/'>' I)(x) = x (Q) 1 F 1 (I; Q; (X - t»/(t)dt,
a
(37.1)
:&
a>. j (x - t)a-1 -
(Aa.t I)(x) = r(Q) J(a-1)/2((X - t»/(t)dt,
a
(37.2)
:&
a>. j (x - t)a-1 -
(Ba-t I)(x) = r(Q) J a / 2 - 1 (A(X - t»/(t)dt,
a
(37.3)
:&
a,>' _ j (x - t)a-1 - . r::----;
(C(I+ I)(x) - r(Q) Ja-1(AVX - t)/(t)dt,
a
(37.4)
:&
( a,>' ( _ j (x - t)a-1 -
D a + I) x) - r(Q) Ja(A(X - t»/(t)dt,
(I
(37.5)
:&
(E';;' I)(z) = j (z (;-' Ja-,(>. V (z - I)(z - 1+ "'()/(I)dt,
(I
I arg 1'1 < 11',
(37.6)
j :& ( X - t ) a-1 I(t)
(8:+/)(x) = 2sh r(Q) dt
(I
(37.7)
and the corres p ondin g ri g ht-sided O p erators la,fJ,->' A a ,>. B a ,>' C a ,>. D a ,>.
&- , &-, &-, &-, &-,
E:">"'" and 8f_ obtained from (37.1)-(37.7) by replacing x - t by t - x and [a, x]
by [x, b), b  00. In the case when b = +00 we shall use the index - instead
of the symbol 00- for the above operators - see, for example, (10.56). In the
expressions given above Q (ReQ > 0), (3, A and l' are some complex parameters,
1Fl(a;c;z) is the confluent hypergeometric function defined in (1.81) and 1,,(z) is
the Bessel-Clifford function defined by
_ _. ( Z ) -" _ 00 (_z2/4)1c
J,,(z) = I,,(.z) = r(v + 1) 2 J,,(z) = ?; (v + 1)l:k! '
(37.8)
The function 1,,(z) will occur below as well.

732 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
According to the obvious relation 1,,(0) = 1,,(0) = 1 we obtain
[ OI,fj,O _ A OI,O _ B OI,O _ C Ol,O _ D OI,O _ E OI,O,-y _ [ 01
a+ - a+ - a+ - a+ - a+ - a+ - a+'
(37.9)
This allows us to consider the operators given in (37.1)-(37.6) and (37.7) -
taking the asymptotic estimate 2sh (T /2) f'OoJ T as T --+ 0 into account - as certain
generalizations of the fractional integration operators [:+.
Proceeding from the representations of kernels of the above operators via
series, and using Legendre's duplication formula given in (1.61) it is not difficult
to write the following relations characterizing the structure of the operators in
(37.1 )-(37.4):
[01 ,fj ,>. _  ((3) I: )..1: [01+1:
a+ - L...J k! a+
1:=0
= [:+(E - )..[+)-fj,
(37.10)
AOI,>. _  (Ot/2h ( _)..2 ) 1: [01+21:
a+ - L...J k! a+
1:=0
- [01 ( E + )..2[2 ) -01/2
- a+ a+'
(37.11)
B OI ,>. =  ( Ot + 1 ) (_)..2)1: [01+21:
a+ L...J 2 k! a+
1:=0 I:
- [01 ( E + )..2 [2 ) -(01+1)/2
- a+ a+ '
(37.12)
00 ( \2 ) 1:
,>. _ 1 A 01+1:
c:+ - L k! -"4 [a+
1:=0
= I+ exp ( _ 2 I+ ) ,
(37.13)
where E is the identity operator. The sums in the middle parts of these relations
are called the Neumann generalized series. Since the operators [;+ are bounded
in L,(a, b), p  1, b < 00 - see the proof of Theorem 2.6 - then these series
may be summed for 1)..1 < 1I[+IIL(a,b)' After evaluating the sums one may omit
this restriction on ).. and on the sums in (37.10)-(37.13), since the operators in
(37.1)-(37.4) are analytic functions with respect to)... A similar approach can
be applied to the operators in (37.5)-(37.7) and to the corresponding right-sided
operators, but in the case when b = +00 we must be careful while using Neumann
generalized series to take into account the asymptotic estimates of the special
functions in the kernels at infinity.

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
733
We also note that from (37.10) and (10.58) the interesting operator relation
1 01 - P e>'z 1 P e->'z - [01 ( E _ >..[1 ) -p
a+ a+ - a+ a+
(37.14)
follows where e::l:>'z denotes the operator of multiplication by the function e::l:>.z -
see the relation after (18.77).
On the basis of (37.10)-(37.13) and the semigroup property in (2.65) it is easy
to write the relations
[OI,P,>' r!,6,>. _ [0I+'Y,P+6,>.
a+ a+ - a+ '
(37.15)
A 01,>' _ [0I/2,OI/2,i>'101/2,0I/2,-i>'
a+ - a+ a+ '
A OI ,>. A'Y,>' - AOI+'Y,>' ( 37.16 )
a+ a+ - a+ '
BOI,>' _ 1 01 / 2 ,(0I+1)/2,i>'1 01 / 2 ,(01+1)/2,-;>.
a+ - a+ a+ '
A OI ,>. B'Y,>' - BOI+'Y,>' ( 37.17 )
a+ a+ - a+ ,
[I B OI ,>' _ A(0I+1)/2,>.
a+ a+ - a+ '
C Ol ,>'c:.,6 - COI+'Y,&W ( 37.18 )
a+ a+ - a+ '
rtOI,>'c:.,i>. _ [OI+'Y
va+ a+ - a+ '
C OI ,>. [p - COI+P,>.
a+ a+ - a+
and similar expressions for the right-sided operators.
From the connection of the operators in (37.1)-(37.7) with fractional integrals
noted in (37.9) we can conclude that the operators in (37.1)-(37.7) have the same
range in L,(a, b) as [:+, i.e. the following statement is true.
Theorem 37.1. Let -00 < a < b < 00, 0 < Q < 1, 1 $ p < 00. Then the
operators in (37.1)-{37.7) are bounded from L,(a, b) onto [:+[L,(a, b)] C Lp(a, b).
The proof follows directly from the properties IFI (I; Qj 0) = 1.,(0) = 1, 2sh( T /2) f'OoJ
T as T -+ 0 and Lemma 31.4.
Comparing (37.1)-(37.7) with (1.122) it is easy to see that, if a = 0 or a > 0
and a function f(t) is defined to be zero on the interval 0 < t < a, then the
operators in (37.1)-(37.7) can be written as the Laplace convolutions in (1.122).
Then after evaluating the Laplace transforms of the kernels by the corresponding
relation 6.10(5) in Erdelyi, Magnus, Oberhettinger and Tricomi [1] and 2.2.12.8
(cases 1:+ 1 , [:+2, [), 2.12.9.3 if n = 0, 2.12.11.5 and 2.4.10.4 in Prudnikov,
Brychkov and Marichev [2] and [1] respectively, it is not difficult to obtain the
following Laplace transforms of (37.1)-(37.7) by the convolution statement in
(1.123):
(L1:'>' f)(p) = p-OI(1- )..p-l )-P(Lf)(p),
(LA;+>' f)(p) = (p2 + )..2)-0I/2(L/)(p),
Re).. > 0, Rep> 0; (37.19)
Rep> IIm>"l;
(37.20)

(LB/ f)(p) = p(p2 + A 2 )-(Ot+l)/2(Lf)(p), Rep> IImAlj
(LC+>' f)(p) = p-Ot exp[-A 2 /(4p)](Lf)(p) , Rep> OJ
(LD+ f)(P) = C+ V;2 + A2 ) a (Lf)(p), Re p> IIm AI;
(LE+>',-r f)(p) = ( 2 ) Ot-l exp[(p- Vp2 + A2)')'/2] (Lf)(p),
p + Vp 2 + A 2 Vp 2 + A 2
734 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Rep> IImAI;
(LSOt f)(p) = r(p + (1 - a)/2) (Lf)(p)
0+ r(p + (1 + a)/2) ,
(37.21)
(37.22)
(37.23)
(37.24)
2Rep> Rea - 1. (37.25)
Comparing the right-hand sides here with those of (37.10)-(37.13) we observe that
the first four relations are obtained from (37.10)-(37.13) by replacing 1+ by p-l -
see (7.14).
As before, from (37.19)-(37.25) we easily obtain the operator relations in the
form
EOt,>',o - DOt-I,>. AI,>. DOt,>. D/J,>' - DOt+/J,>'
a+ - a+ a+ , a+ a+ - a+ ,
EOt,>.,-r E/J,>.,6 - AI,>' E Ot +/J-l,>',-r+ 6
a+ a+ - a+ a+ '
1 Ot +/J s:!+1 _ 1 Ot + 1 ,1,--r/ 2 s:!-II/J+l,1,-r/ 2
a+ a+ - a+ a+ a+ '
E Ot,>',-r D /J,>' - E Ot+/J,-r
a+ a+ - a+ .
(37.26)
(37.27)
(37.28)
(37.29)
in addition to (37.15)-(37.18).
We consider now the inversion problem for the operators in (37.1)-(37.7).
(37.19)-(37.25) show that the Laplace transforms (Lh)(p) of the kernels h(x) - see
(1.119) - of (37.1)-(37.7) and the corresponding inverse values [(Lh)(p)t 1 have
the same form and differ from each other only by values of the parameters. In the
simplest case of the fractional integration operator 1+, to which (Lh)(p) = p-Ot
with the condition Re a > 0 corresponds, this condition for the Laplace transform
pOt of the kernel of the inverse operator Dg+ obliges us to represent pOt in the form
pOt = pnp-(n-Ot), where Re(n-a) > 0, with the value pn corresponding to the
operator ( t: )n = D o + - see (1.124). Similar operations are to be carried out when
inverting the operators in (37.1)-(37.7).
Taking the above arguments into account we shall construct the solution of
the equation
(1:/'>' f)(x) = g(x), a < x < b  00.
(37.30)

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
735
Using (37.10), (10.58), (10.59) and 6.3(7) in Erdelyi, Magnus, Oberhettinger and
Tricomi [1] we formally arrive at the following representations for the solution of
(37.30):
f(x) = ((I:/,>')-l g }(x) = I;':(E - AI+)P g(x)
= 1-1 I'+m-a ( E _ All ) /J I-m g( x )
a+ a+ a+ a+
= (  ) ' I'+m-a,-p,>. (  ) m ( x )
dx a+ dx 9
( d ) ' / :1: ( t)'+m-a-1
= dx  + m _ cr) 1 F 1 (-P; 1+ m - cr; A(X - t»gCm)(t)dt
a
= (:J H(z),
(37.31)
f(x) = e>':I: I;! e->':I: I+ag(x),
f(x) = I:+ae>.:l:I;.fe->.:l:g(x),
(37.32)
(37.33)
f(x) = e>':I: (  ) ' I'+m-a,/J-a,->. (  ) m (e->':I: g(x».
dx a+ dx
(37.34)
Proceeding analogously from (37.20)-(37.25) we can formally write the
following representations for the operators inverse to (37.2)-(37.7):
f(x) ={(A:)-lg}(x) = (A;.;,>'g)(x)
=(1;; + A2)' A2m-a'>'(I;; + A2)mg(x)
( d2 2 ) ' / :1: (x - t)21+2m-a-1 - (37.35)
= dx2 + A r(21 + 2m _ cr) J'+m-(a+1)/2(A(X - t»
a
x ( :'2 + 2 r g(t)dt,
((B+>')-lg}(x) = I+(A;';-l,>'g)(x),
(37.36)
( d ) ' / :1: ( t)'+m-a-l
((C:+>.)-lg}(x) = - d  ) il+ m - a - 1 (Avx=t)gCm)(t)dt ,
X a + m - cr (37.37)

736 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
((V:.;')-lg}(Z) = (V;;'>g}(Z) = 2' [v::;a.>  G) A;i" l:;' g] (z),
k - 1  Rea < k, k = 1,2,...,
(37.38)
{(E:+>','Y)-1 g }(X) = (E+al>"-'Y A;;,>'g)(x)
= (E:+ a + 1 ,>.,-'Y D;'>' A;;,>'g)(x),
k-lRea<k+l, k=I,2,...,
(37.39)
{(S:+)-1 g }(X) = (S;;g)(x)
[ 2k-a ( -1 a + 1 k) ( -1 1 - a ) ] ( )
= So+ 10+ +  - k 10+ +  k 9 x,
2k- - 2  Re a < 2k, k = 1,2,...
(37.40)
To justify these inversion relations we denote by R( x) the integral operator
next after the signs ( d )' in (37.31), (37.34) and (37.37) and after the signs
( + A2)' in (37.35) and (37.36). Then conditions for the invertibility of the
operators in (37.1)-(37.7) can be collected in the following general theorem.
Theorem 37.2. Let 0  a < x < b < 00 in the equation [h*f](x) - see (1.122), let
h be one of the operators defined in (37.1)-{37.7), Rea > 0, letg be a given function
on [a, b] and let f be an unknown function. We assume that f(x) = g(x) = 0
for 0 < x < a in the case a > O. Then the solution f = h- 1 g of this equation
exists and is unique in the space f E L,(a, b), b < 00, if g(x) E 1:+(L,(a, b»,
p  1. In the case when p = 1 this solution can be represented by the corresponding
relations given in (37.31)-{37.40) provided that the following additional conditions
are satisfied:
1) 0 < Rea < I+m, I,m = 1,2,..., 9 E Acm([a,b]), g(a) = g'(a) =... =
g(m-1)(a) = 0, R E AC'([a,b]), and R(a) = R'(a) = ... = R(l-1)(a) = 0 for
(37.31), only if 9 E Acm or R E AC', then 1=0 or m = 0, for (37.31);
2) 1  p < 00 and Re{3 > 0 or Re{3 < 0 but g(x) is representable in the form
g(x) = (1:;f3,p)(x), ,p(x) E L,(a,b), for (37.32);
3) 1  p < 00 and Re(a - (3) > 0 or Re(a - (3) < 0 but g(x) = (1:+1/J)(x),
,p(x) E L,(a, b), for (37.33);
4) the conditions similar to those of 1) for (37.34)-{37.37);
5) 0 < Rea < k, 9 E AC k + 2 ([a, b]) and g(a) = g'(a) = ... = g<k+1)(a) = 0 for
(37.38);
6) 0 < Rea < k + 1, 9 E AC k + 5 ([a, b]) and g(a) = g'(a) = . .. = g(k+4)(a) = 0
for (37.39);
7) 0 < Rea < 2k, 9 E AC 2k ([a, b]) and g(a) = g'(a) = ... = g<2k-1)(a) = 0 for
(37.40).

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
737
The proof follows from the existence, uniqueness and coincidence of the
corresponding Laplace transforms of the equations and their inversions, and
also from the existence of all given operators in the indicated spaces of functions
9 and f. The condition 9 E 1:+(L,(a, b» is obtained on the basis of Theorem
31.13 and the conditions 2) and 3) follow from Theorem 10.9 if we take into
account the fact that for the existence of the corresponding fractional derivatives
the condition 9 E 1:+(L,(a, b» in the last subcases in 2) and 1) must be changed
by the condition 9 E 1:';.6 [L,,(a, b)] or 9 E 1:+ [L,(a, b)]. The conditions of the
form g(a) = g'(a) = ... = gCm-l>(a) = 0 ensure the vanishing the terms outside of
integrals when using (1.124). .
Remark 37.1. According to (37.8) the replacement A by iA in all relations of this
subsection for the operators defined in (37.2)-(37.6) is equivalent to replacing the
functions I" and ill by each other.
Remark 37.2. Making the reflection operation in all relations of this subsection,
i.e. replacing x by a + b - x, and the corresponding changes in the functions f and
g, it is not difficult to write the corresponding relations and results concerning the
. ht . d d t 1 01,.6,->' A OI,>. B OI,>. C Ol,>. D Ol,>' E OI,>.,,. d S Ol . th
rIg -SI e opera ors b- 'b-' b-' b-' b-' b- ,an b- 10 e case
when b < 00. If b = 00, then such results are in general still valid under stronger
conditions.
37.2. Generalized operators of Hankel and ErdtHyi-Kober
transforms
The generalized operators of Hankel and Erdelyi-Kober transforms considered here
will be used later in S 38 when considering certain types of dual integral equations.
Generalized Erdelyi- Kober operators differ from the operators
:&
l(x 2 -t2)"/2J,,(A v x2 -t 2 )tp(t)dt = 1/J(x),
o
Rev > -1,
(37.41 )
00
1 (t 2 - x 2 )"/2 J,,(A V t2 - x 2 )tp(t)dt = 1/J(x),
:&
Rev> -1,
(37.42)
only by the weight factors. The operators in (37.41)-(37.42), in their turn are
obtained from the operators C+>' and C'>' - see (37.4) and the text below - by
the changes Q = V + 1 and 2"+ltf(t 2 ) = A"tp(t) and replacing x by x 2 . After these
changes we observe that from Theorem 37.2, (37.37) and Remark 37.2 the inversion

738 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
relations for (37.41) and (37.42) follow in the form
:&
cp(x) = A  f t(x2 - t 2 )-(1I+1)/2 1_ 11 - 1 (A V X 2 - t 2 )1/J(t)dt
dx
o
(37.43)
and
00
cp(x) = -A d f t(t 2 - x 2 )-(1I+1)/21_ 1I _ 1 (A Vt 2 - x 2 )1/J(t)dt
:&
(37.44)
respectively provided that A > 0, -1 < Rev < 0 and 1/J E AC([O, b]), and
1/J(x) = O(e->':&x ll - 1 / 2 -£) as x -+ 00 and € > 0 for (37.42). If we replace the
symbols III and J II with each other, which is equivalent to replacing A by iA in
(37.41)-(37.44), then (37.41) and (37.43) are valid under the same conditions.
However the conditions for the validity of (37.42) and (37.44) are changed since
the integral in (37.42) must converge and the function III(z) in the kernel has an
exponential growth at infinity.
On the basis of (37.41) and (37.42) we introduce the following generalized
Erdelyi-Kober operators
:&
J>.(1], n)f(x) = 2 a A1-ax-2a-2'1 f t 2 '1+1(x 2 - t 2 )(a-1}/2 J a - 1 (A V x 2 - t 2 )f(t)dt,
o (37.45)
00
R>.(1], n )f(x) = 2 a A 1-a x 2'1 f t1-2a-2'1(t2 - x 2 )(a-1}/2 J a - 1 (A vt 2 - x 2 )f(t)dt,
:& (37.46)
where n > 0,1] > -1/2. We also define the operators Ji>.(1],n) and Ra>.(1],n) via
the right-hand sides of (37.45) and (37.46) by replacing J a - 1 by la-l'
The generalized operator of Hankel transform is defined by the relation
8 ( a, b, Y ) f(x) = 2ax2u-a(x2 _ a 2 )-U
1], n, U
00
x f r 1 - a - 2u (r 2 - y 2 t J2'1+a( V (x2 - a 2 )( r 2 - b 2 »f( r)dr.
y
(37.47)
It is obvious that this operator is connected with 8'1,a,2 given in (18.19) by means
of the relation
8 ( 0, 0, 0 ) = 8 ( 0, 0, 0 ) = 8'1 a 2.
1], n, U 1], n, 0 I ,
(37.48)

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
739
We give formally, without indications of the space of functions some properties
of the operators introduced above. As usual these properties can be checked by
direct calculations for sufficiently good functions and then can be extended to
functions in Lp.
1. It is obvious that if A --+ 0, then the operators in (37.45) and (37.46)
coincide with the Erdelyi-Kober operators (18.8):
Jo(TJ, a) = 1",00 Ro(TJ, a) = K",OI'
(37.49)
2. On the basis of (18.11) we have
Jo(TJ,O) = E, RO(TJ,O) = E.
(37.50)
3. Translation relations
J>.(TJ, a)x 2 ,6 J(x) = X 2,6 J>.(TJ + (3, a)J(x),
R>.(TJ, a)x 2 ,6 J(x) = X 2,6 R>.(TJ - (3, a)J(x),
(37.51)
(37.52)
follow from (37.45) and (37.46).
4. Evaluating the corresponding repeated integrals, using 2.15.35.2 with
b 2 + c 2 = 1 in Prudnikov, Brychkov and Marichev [2], it is not difficult to prove
the results
h>.(TJ + a, (3)J>.(TJ, a) = J>.(TJ+a,{3)h>.(TJ,a) = 1",01+,6'
Ri>.( TJ, a )R>. (TJ + a, (3) = R>. (TJ, a )Ri>. (TJ + a, (3) = K",OI+fJ
(37.53)
(37.54)
provided that a > 0, {3 > 0 - see (37.18).
5. Taking the last relations into account and using (37.50) we can define the
operators J>. (TJ, a) and R>. (TJ, a) for a < 0 via solving the corresponding integral
equations (see S 18.1) that is .
h ('I. Q )f(") = ., - 2a-2. ( 2.,dd., ) n .,2n+2 a - 2. h ('I, Q + n )f("),
(37.55)
R ('I, Q )f(.,) =.,20 (- 2"" r .,2n-2. R ('I - n, Q + n)f (.,).
(37.56)

740 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
where -n < a < 0, n = 1,2,.... Hence we deduce the relations
J i >.I(TJ, a) = J>.(TJ + a, -a), J;I(TJ, a) = Ji>.(TJ + a, -a).
Ri>.l(TJ, a) = R>.(TJ + a, -a), RI(TJ, a) = Ri>.(TJ + a, -a).
(37.57)
(37.58)
6. For the operators in (37.45) and (37.46) the following analogue of the
formula for integration by parts
00 00
f xf(x)J>.(TJ,a)g(x)dx = f xg(x)R>.(TJ,a)f(x)dx
o 0
(37.59)
is valid.
7. By calculating the compositions of the operators in terms of the relations
2.12.35.2 and 2.12.35.6 in Prudnikov, Brychkov and Marichev [2] the following
operator relations
J;,('1 + ", ,8)8 (0, 0, A ) = 8 ( 0, A, <r - '1- t" + ,8)/2) (37.60)
TJ, a, U TJ, a +p,
R,('1, ,,)8 ( l' 0, A ) = 8 (0, A, <r + '1 + t" + ,8)/2) '(37.61)
TJ a, p, U TJ, a + p,
8 ( 0, A, A ) 8 e, 0, <r _ '10_ "/2) = 1>('1," + ,8), (37.62)
TJ + a, p, U TJ, a,
8(°' 0, ° ) 8 ( A, 0, '1 +,,0+ ,8/2) = R;,('1, " + ,8). (37.63)
TJ, a, ° TJ + a, p,
can be proved, which connect the operators in (37.45)-(37.47) with each other.
8. Using (37.62) with a + P = ° we can deduce that the equation
S ( O, y, Y ) f(X)=9(X)
TJ, a, U
(37.64)
has the solution
f(X)=S ( y,
TJ+ a,
0,
-a,
 ) g( z )
(37.65)

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
741
from whence the operator relation
S-1 ( 0, y, Y ) = S ( y,
TJ, a, U TJ + a,
0,
-a,
)
(37.66)
follows.
37.3. Non-convolution operators with Bessel functions
in kernels
In this subsection we consider two following non-convolution operators with Bessel
functions in kernels:
:&
-+ f (x - t)a-1 -
(Ja,>.f)(x) = r{a) J a - 1 (A Vt {X - t»f{t)dt,
o
(37.67)
:&
-- f (x - t)a-l -
(Ia,>.f)(x) = r(a) Ia-1(A V x(x - t»f(t)dt,
o
(37.68)
where J,,(z) and I,,(z) are the Bessel-Clifford functions defined in (37.8). We
shall also deal later with the operators differing from (37.67) and (37.68) by the
replacement of the symbols J and j by I and J, respectively. We shall denote
these operators by I: >. and J; >.. It is obvious that
I ,
-+ - - +
Ia , >. - J a,i >.'
J; >. = I; i>.'
I I
(37.69)
All the above operators coincide with the fractional integral
-+ -+ -- -- a
J a 0 = Ia 0 = J a 0 = Ia 0 = 10 +
.. , .. ..
(37.70)
in the case >.. = 0, while for other values of >.. they are also closely connected
with the fractional integral, being compositions of the operators Jt>. or I: >. with
10+ 1 in one or another order. These properties are consequences of the foilowing
statement.
Theorem 37.3. Let Rea> 0 and f E L,(O,b), b> 00, p  1. Then the operators
introduced in (37.67) and (37.68) are defined and are bounded from L,(O, b) onto

742 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
the space I<f+[L,(O, b»). If also Re{3 > 0, then the following composition relations
/J -+ -+
Io+(JOI,>.f){x) = (JOI+/J,>.f)(x),
- /J -
(1;,>.I o +f){x) = (1;+/J,>.f)(x)
(37.71)
(37.72)
hold.
Proof. The first statement follows immediately from Lemma 31.4. (37.71) and
(37.72) are proved by direct evaluation, which will be shown, for example, for
(37.72):
z t
f (x - t)0I-1 - f (t - r)/J- 1
r(Q) IOI_1(A Y X(X - t»dt r({3) j(r)dr
o 0
z z
= f rf)J (z - tt-l(t - T)fJ-lla_I(> "j z(z - t)}dt.
o T
Changing the variable t = x + (r - x)(J2 and taking (37.8) and the relation 2.15.2.6
in Prudnikov, Brychkov and Marichev [2] into account we easily obtain the value
of the inner integral and thereby the right-hand side of (37.72). .
It is important to note that (37.71) and (37.72) enable us to define the
operators J:,>. and i;,>. for negative values of Q by the method which has already
been applied in S 18.1 and in the definitions in (37.55) and (37.56). Namely, we set
(J.t,!)(z) = (  ) n (Jt+n,,!)(z),
(l;,>.j)(x) = (l;+n,>.j<n»{x),
(37.73)
-n<ReQ$I-n, n=I,2,...
(37.74)
We also note that (37.71) and (37.72) yield the representations
(J:'>.f)(x) = Ig;l(J(>.j)(x),
(l;,>.j){x) = (il,>'Ig.; 1 j)(x),
(37.75)
(37.76)
-+
which characterize an important role of the operators J 1 ,>. and l>. in the

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
743
investigation of (37.67) and (37.68) with an arbitrary a. The following statements
contain the conditions for the validity of (37.73)-(37.76).
Theorem 37.4. Let f E Ac(n-1)([0, b)), n = 1 + [-Rea] and Rea < O. Then the
constructions in (37.73) and (37.74) exist.
Theorem 37.5. Let f E L,(O, b), b < 00, and p  1. Then (37.75) IS
true for Rea > 0 while (37.76) is valid for Rea > 1 or for 0 < Rea  1 if
f E 1J+a[L,(0, b)].
The proof of these theorems are obvious and follow from the connection of the
operators in (37.67) and (37.68) with the fractional integrals mentioned above in
(37.71) and (37.72).
Now we consider the inversion problem for the operators in (37.67) and (37.68).
As it follows from (37.75) and (37.76) this problem is reduced to the inversion
problem for the operators it>. and j>.. To solve the latter we prove the following
auxiliary statement.
Lemma 37.1. The identity
:&
j a a
r ar Jo(>. vt (r - t» ar 1 0 (>' v x(x - r»dr
t
(37.77)
a a
= x ax Jo(>' vt (x - t» - t lJt 1 0 (>' V x(x - t»
is valid.
Proof. We denote by A the left-hand side of (37.77) and expand the Bessel
functions in the integrand in series with summation over k and I, respectively.
Differentiating these series with respect to r and replacing I by k - m, 0  m 5 k,
we obtain
A - -   (-I)mm(>.2t/4)m(k - m)(>.2x/4)I:-m j :& ( _ ) m-1 ( _ ) 1:-m- 1 d
-   (m!)2«k-m)!)2 r r t x r r.
1:_0 m_1 t
We evaluate the inner integral by the relation 2.2.6.11 in the hand-book by
Prudnikov, Brychkov and Marichev [1] which yields the value
00 ( ) 1:-1 ( >. ) 21: 1:-1 ( )
A = - L x: 2 2" L  (-I)mt m xl:-m(tk + (x - t)m).
1:=0 () m=1

744 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
It follows from 4.2.3.1 and 4.2.3.18 in the cited hand-book that the finite sum is
equal to -tkx 1c - kx( _t)1c. Hence we derive the right-hand side of (37.77). .
Theorem 37.6. Let F(x) E AC([O,b]). Then the equation
:&
 (Jt>.f)(x) = f(x) + f f(t) :X Jo(>. y t(x - t»dt = F(x)
o
(37.78)
has a unique solution of the form
:&
f(x) = x-I j>.(xF(x»' = F(x) - x-I f F(r)r :r 10(>' Yx(x - r» dr. (37.79)
o
Proof. Replacing x by r in (37.78), multiplying both sides of these equation by
a
r ar 10 (>. y x(x - r»
and then integrating with respect to r from 0 to x we have
:& :&
f f(r)r  lo(>' Yx (x - r»dr+ f r :r 10(>' Y x(x - r»dr
o 0
T :&
X f f( t) :T J o (> .v t ( T - t»dt = f F( T) T :T lo( >. "; z ( z - T })dT.
o 0
To evaluate the inner integral we interchange the order of integration in the second
term on the left-hand side and use Lemma 37.1. As a result we arrive at the
relation
:& :&
Z f f(t) :z Jo(>. ";t (z - t})dt = f F( T)T :T 10 (>. ";z (z - T})dT.
o 0
(37.80)
Now it is easy to obtain (37.79) by subtracting (37.80) from (37.78). .
Theorem 37.7. Let 0 < Rea < 1, 9 E 1<f+[L 1 (0, b)] and g(O) = O. The operator
inverse to (37.67) exists on such functions g(x) and can be represented in both of

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
745
the following forms:
:&
(J: , >.)-l g (X) = x-I f lo(>, .j x(X - t»d[t(lo+g)(t»),
o
(37.81)
(J:>.)-l g (X) = x- 1 (11_a,>.<P)(X),
<p(X) = xg'(X) + (1 - cr)g(X).
(37.82)
Proof. (37.81) follows directly from the relations (37.75), (37.78) and (37.79)
applied to the equation (J: >.f)(x) = g(x). To obtain the representation in (37.82)
we introduce the notation I
d
dt [t(Io+ag)(t») = (Io+a<p)(x)
(37.83)
and apply (37.76) to (37.81). Then the right-hand side of (37.81) has the
form x- 1 (11_a >.<p)(x). We prove that <p(x) = xg'(x) + (1 - cr)g(x) whence the
representation in (37.82) follows. For this purpose we rewrite (37.83) in the form
<p(x) = 1+lxl-axa Io+g(x) and apply (10.12)
<p(x) = x a 10+ 1+lxl-ag(x)
d
= x Q _(x 1 - a g(x»
dx
(37.84)
= (1 - cr)g(x) + xg'(x)
provided that g( x) is sufficiently smooth. The latter relation can be obviously
extended to the function g( x) E IC+ (L 1 (0, b». .
Theorem 37.8. Let 0 < Recr < 1, x- 1 h(x) E L 1 (0, b) and h(x) E IC+(Ll(O, b».
Then the operator inverse to (37.68) exists on such functions h(x) and can be
represented in the form
(l- )-lh(x) = xl-a[xaJ+_ x- 1 h(x»).
a , >' dx 1 a , >'
(37.85)
The proof follows directly from (37.67), (37.68), (37.81) and Theorem 37.7 after
replacing x a (x 1 - a g(x»)' by f(x), xf(x) by h(x) and cr by I-cr. The conditions for

746 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
h(x) ensure the existence of the operator inverse to (37.68) and its representation
by the form (37.85). .
37.4. Equations of compositional type
In a series of problems of mathematical physics one sometimes meets integral
operators, generating integral equations of the first kind with Volterra kernels, such
that they may be represented as compositions of more simple invertible operators.
We call these equations of a compositional type. Some typical examples of such
equations are considered in this subsection.
A. If we set x = y and denote u(y, y) = <p(2y) and change the variables
(1 + 8)y = t, 2y = x, r(t)t IJ + P - 1 = f(t),
1 ( 2 ) l-IJ-2,
1 3 ;- <p(x) = g(x)r(p)
(37.86)
in the solution (40.28) of the hyperbolic equation in (40.19), then we arrive at the
integral relation
:t:
j (X-t),-l..... ( x-t A 2 t(X-t» )
r(p) =2 p,1 - fljPj, 4 f(t)dt = g(x)
o
(37.87)
involving the Humbert function defined in (40.25) in the kernel.
The inversion formula for this relation can be obtained by two methods. One is
by considering the solution of other boundary value problem for the corresponding
hyperbolic equation (40.19). The other is by studying the structure of the operator
in (37.87). We shall illustrate both ways. To use the first we consider the second
Cauchy-Goursat boundary value problem with the conditions
u(x, x) = <p(2x), uy(x,O) = 0
(37.88)
for (40.19). We stress that the function in (40.28) satisfies the condition uy(x, 0) = 0
provided that a constant is given by (40.31). According to the papers by Kapilevich
[lj (5.1), (5.3)] and Gordeev [1] the solution of the second Cauchy-Goursat problem
has the form

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
:&-y
u(x, y) = f [<p'(t) + (p. + p)t- 1 <p(t)]H(x, y; t/2, t/2)dt
o
747
(37.89)
:&+11
+ f [<p'(t) + 2- 1 (p. + p)<p(t)]R(x, y, t/2, t/2)dt,
:&-y
where Ii and R are the Green-Hadamard and Riemann functions respectively for
(40.19). Their explicit representations via triple series were given by Kapilevich
[3, p. 1481]. Setting y = 0 in (37.89) and taking the condition u(x,O) = r(x) into
account we arrive at the relation
:&
r( x) = f [<p'(t) + (p. + p)t- 1 <p(t)]H(x, 0; t/2, t/2)dt.
o
(37.90)
The kernel of this equation can be easily obtained from (5.31a) in Kapilevich [1]:
H - ( 0 ) - _ 2 2, -I' 1J+ 2 P R - 2 '..... ( 1 . 1 . R A 2 R )
x, , TJ, TJ - x x TJ 1'::'2 p., - p., - p, - 2xTJ ' -  '
2 1 - 2 , -Ii
it = r(1 _ p)r(p + 1/2) ' R = x(x - 2TJ), P < 1.
(37.91 )
Substituting (37.91) into (37.90) and using the changes in (37.86) we obtain the
relation
:&
f(x) =22,-1/ 3 itr(p)X- 1 f[(g(t)tl-2'-IJ) + (p. + p)t- 1 g(t)t 1 - 1J - 2 ,]
o
2 ( t - X A 2 X )
X (x - t)-'t lJ + 'S2 p.,1 - p.; 1 - p; ---U-' 4(t - x) dt.
Hence taking the values it and 13 from (40.31) into account we find finally the
inversion formula for (37.87) in the form
:&
1 f (x - t)-'..... ( t - X A 2 X ) 1-
f(x) =;- t' r(1 _ p) '::'2 p.,1 - p.; 1 - Pi 2t' 4(t - x) d(t 'g(t»,
o
O<p<1.
(37.92)
The functions 9 and f in these relations must naturally be such that the integrals

748 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
here exist. For this the conditions t#J+ 1 g (t) --+ 0, t 2 -#J g (t) --+ 0 and tf(t) --+ 0 as
t --+ 0 and the condition 9 E AC 1 ([0, b]) are sufficient.
The fact that (37.92) really inverts (37.87) follows from the uniqueness
theorems for the solution of the initial Cauchy problem and the second Cauchy-
Goursat problem for (40.19) with A = ib, b > O.
The result obtained coincides with the known inversion relations in special
cases, below.
1. If A = 0, then (37.87) and (37.92) according to (40.25) and (1.80) have the
form
:&
1 (z2 - t 2 )('-1)/2 p;' (;) f(t)dt = g(z),
o
(37.93)
:&
f(x) =  1 t'(x 2 - t 2 )-'/2 p#J (-) d(t 1 -, g(t».
o
(37.94)
The latter relation can be also rewritten by taking out the derivative of the integral
1
f(x) =  1 "(1 - "1 2 )-,/2 p ( ! ) :'(g(XTJ)xl-')TJ1-' dTJ
x #J "1 "1 ax
o
1
=  1 g(XTJ)x1-'(1 - "1 2 )-1'/2 p ( ! ) dTJ
 #J "1
o
(37.95)
:&
=  1(x 2 - t2)-'/2p#J (i) g(t)dt.
o
(37.93) and (37.95) are the special cases of (35.16) and (35.31) for e = 0 and n = 1,
respectively.
2. If A = p. = 0, then (37.87) and (37.92) become the reciprocal relations
(I+f)(x) = g(x) and f(x) = d IJ+'g(x).
3. If p. = 0, then in accordance with (40.25) and (37.8), (37.87) and
(37.92) can be rewritten in the form of the relations (l:>.f)(x) = g(x) and
f(x) = x- 1 (J 1 _,,>.<pd(x) where CPl(X) = xg'(x) + (1- p)g(x), cf. (37.67), (37.82)
and (37.69).
We consider the structure of the operator in (37.87). For this purpose taking

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
749
(18.41) into account we evaluate the composition
:&
, -1 -+ )( I (X2 - y2),-1 2y
1 0 +;:&2 x (lOl,>.f X) = r(p) y dy
o
11
I (y - t)0I-1 -
x r(a) 10l-1(>' V t (y - t»f(t)dt
o
[ 00 ( A2 ) 1c 2 1 :& f(t)t1c dt ]
= ?;"4 (a).k! 0 r(p)r(a)
:&
X I (x 2 - y2),-I(y - t)1c+OI-ldy
t
=[E...](x - t),,+1c+0I-1(x + t),-1
I I ( t ) ,,-1
x (1 - r)'-l r 1c+0I-1 1 - t  : r dr
o
=[E...](x - t),+1c+0I-1(x + t)p-l r(p)r(k + a)
r(p + a + k)
( t - X )
X 2Fl 1- p,a + k;p+ a + k;-
t+x
=  ( A2 ) 1: 2(2x)p-l 1 :& f(t)t1c(x _ t)p+1:+0I-1
 4 k!r(p + a + k)
-o 0
( X - t )
X 2Fl 1- p,p;p + a + k; 2;"" dt
:&
=2(2x),,-1 I (x - t)p+0I-1
r(p + a)
o
( X - t A 2 )
x =:2 p,1 - p;p + a; , 4t(x - t) f(t)dt
(37.96)
provided that Rep > O. Thus we arrive at the operator which differs from (37.87)

750 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
by a change of variable only. Hence (37.87) in the form
j z (x _ t)p-l..... ( X _ t A 2 )
r(p) '::'2 fl,1 - fljP;, ""4 t (X - t) j(t)dt
o
= (2X)I- IJ I+jz'J(2x)-I(I:_IJ,>.f)(x), Rep> Refl > 0
(37.97)
follows. In particular, in the case A = 0 we obtain from (37.97) the following
compositional representation for the operator in (37.93)
z
j (",2 _ t 2 )('-' )/2 p;' G-) f( t)dt = (2",)'- p I+;%, (2",) -, Ig+ P f( ",). (37.98)
o
This relation conforms to (35.16) and (35.30) if we take the operator equality
r;;;z'J1J+ = (2x)-1 into account.
The following statement is a consequence of the above arguments.
Theorem 37.9. Let Rep > 0, g(x) E AC 1 ([O, b]), b < 00, and t IJ + 1 g(t) -+ 0 and
t 2 - IJ g(t) -+ 0 as t -+ O. Then (37.87) is invertible by (37.92) in the space of
functions f(x) E AC(O, b», b < 00, such that tf(t) -+ 0 as t -+ O.
We also note that under the appropriate conditions by using (37.82) the
solution of (37.87) can be represented in the compositional form
f(x) = x- 1 (J 1 + IJ _ p ,>.CP)(x),
d
cp(x) = 2 IJ x P - IJ dx [x 2 + IJ - p I;':z'J xIJ - 1 g(x)].
(37.99)
B. Now we consider the integral equation
j z (x _ tp-l ':' ( , .. t ) _
r(-y) .....1 a,a,p,-y,I-;,A(x-t) f(t)dt-g(x),
o
Re-y > 0,
(37.100)
where
....., )  (a)j(a')rc(p)j xi rc
'::'1 (a, a ,p;-y;x,y = L...J (). "k' Y,
. L_ O -y 3+ rc ]. .
3,"'-
Ixl < I,
(37.101)

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
751
is one of the Humbert functions (see Erdelyi, Magnus, Oberhettinger and Tricomi
[1; 5.7(25)]). We shall investigate this equation in the space of functions
Q, = {I: f(x)x' E L1(0,b), b < oo}. The following statement is true.
Theorem 37.10. The integral equation in (37.100) has the solution 1 E Q"
q < min(O, Re(1' - a - (3», and q $ 0 when Re(1' - a - (3) > 0, il and only il
g(x) E Iri+(Q,). This condition being satisfied the equation has a unique solution.
If also min(Rea ' , Re(1' - a), Re(1' -,8» > 0, then the solution of (37.100) can be
represented in the form
I(x) = e>.:t: 1 0 ':' e->.:t: Ig:OI'- x fJ 10':x- fJ g(x).
(37.102)
Proof. It follows from (37.101) and 2.10(1) and 2.10(12) in Erdelyi, Magnus,
Oberhettinger and Tricomi [1] that the function defined in (37.101) has the
following asymptotic expansions
{ 0«1 - xp-OI-fJ), Re(1' - a - (3) < 0,
31(a, ai, (3; 1'; x, y) = O(ln(1 - x», l' - a - (3 = 0,
0(1), Re(1' - a - (3) > 0,
(37.103)
as x -+ 1. Hence we obtain that the integrand in (37.100) can be rewritten as
t'/(t)[O(t-') + O(P- OI -fJ-,)] for t -+ 0 which guarantees the existence of this
integral. Applying now Lemma 31.4 in this case when p = 1 relative to the space
Q, we obtain that 9 E IJ+ (Q,) if 1 E Q,. The inverse assertion follows from
Lemma 31.4 as well.
Let now 9 E Iri+ (Q,) and let the conditions of the second part of theorem
be satisfied. Then by Theorem 10.9 the operator IJ+ OI ,OI',>. defined in (10.55)
transforms the function 1 E Q, into (Iri+ OI ,OI',>. f)(x) E Iri+ OI (Q,). This means that
R - 01 01' >. +
x-fJ(Iri+ ' , I)(x) E L1(0, b), and so the operator Ig+ n, Rea + n > 0, can be
applied to the latter function. Considering Ig: n x-fJ (!J+ OI ,OI',>. I)(x) as a repeated
integral, changing the order of integration and using the integral representation
1
';:' ( I (.l.. ) - r(1') f 01-1 (1 ) -0I-1 (1 ) -fJ
....1 a,a'fJ'1',x,y -r(a)r(1'- a ) u -u -ux
o
x 1F1(a ' ; l' - a; y(1 - u»du,
Rea> 0, Re(1' - a) > O,x ft [1,00),

752 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
for the function 31 we obtain
:&
, j ( t ) 'Y+n-l
101+nx-p ( fJ-0I,0I ,>. f )( x ) = x-P x -
0+ 0+ r(+n)
o
X 2 1 (cr + n, cr', {3;  + n; 1 - tlx, A(X - t»f(t)dt.
(37.104)
On the basis of (37.103) and (37.104) we can conclude that x fJ I:nx-fJ
(!J';OI,OI',>. f)(x) E !J:n(Qq) and hence I:nx-p(!J';OI,OI',>, f)(x) E fJ:n-p(Ld.
Therefore both sides in (37.104) can be differentiated n times which yields the
relation
:&
1 01 -P ( Tr -OI,OI',>' f)( ) = -P j (x - t)'1-1
o+x 10+ x x r()
o
X 2, (a,a',p;"Y; 1- ;,(z - t») f(t)dt.
(37.105)
Finally, taking (10.58) into account we arrive at the following compositional
representation for (37.100):
j :& (x - t)'Y-l ':' ( , .. t )
r( ) ....1 cr, cr ,(3, , 1 - ;' A(X - t) f(t)dt
o
P I OI -fJ Tr-OI-OI' >':& 1 01 ' ->':& f( ) ( )
= x o+x 10+ e o+e x = g x .
(37.106)
Inverting now each operator in this composition we obtain (37.102). .
37.5. The W-transform and its inversion
In this subsection we consider some properties of the so-called W -transform, which
is introduced by analogy with the G-transform from S 36, and is a generalization
of the so-called integral transform with respect to an index. Examples of the latter
transform are the known Kontorovich-Lebedev and Mehler-Fock transforms - see
S 1.4 and (37.109), (37.110) and (37.141) below.
Definition 37.1. The W -transform of a function f(x) is defined by the integral
(Wf)(z) '" (w,:;n I v'(J:r) l/{t») (z)

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
= 2i ! r[V- ix - s, v + ix - S]
q
753
[ (Pm) + s, 1 - (an) - s ] . )
X r (a+l) + S, 1 _ (P;"+1) _ s f {I - s ds,
(37.107)
where v (Re v > 1/2) and components of the vectors (a p ) and (Pq) - see explanation
to (36.3) - are complex parameters satisfying the conditions in (36.5), f.(s) is the
Mellin transform of a function f(x) and (f is the contour (f = {s, Res = 1/2}.
It is obvious that the following relation
(W I)(z) = (G;+211- v + iz, ,8r - iz, (a.) I f1(Y») (1)
(37.108)
with It (y) = y-l f(y-l) which connects G- and W-transforms is valid - see (36.3).
Definition 37.2. The transforms
00
Kiz{f(t)} = ! Kiz{t)f(t)dt,
o
(37.109)
00
K i -;l{g(t)} = + ! tsh1rtKit(x)g(t)dt,
1rX
o
(37.110)
where Kiz(t) is the Mcdonald function - see (1.85) - with an imaginary index
v = ix are called direct and inverse Konlorovich-Lebedev transforms, respectively.
The formulations and proofs of the following theorems use the terminology of
S 36.
Theorem 37.11. The W-transform defined in (37.107) with the characteristic
(c. + 1, 'Y. - 2Rev + 2) where Rev> 1/2 exists on functions in the space rot(L)
if and only if
2sign(c. + c + 1) + sign('Y + 'Y. - 2Rev + 2)  O.
(37.111)
The prool follows from Theorem 36.2 since according to (37.108) the W-transform
can be considering as the G-transform evaluated at the point 1 and having the
characteristic (c. + 1, 'Y. - 2Rev + 2). .

754 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Theorem 37.12. Let the conditions in (36.22) and the inequality
4sign(c. + 1) + 2signb'. - 2Rell + 2) + sign 12 + p - ql > 0
(37.112)
hold. Then the W -transform defined in (37.107) exists on functions in the space
rot-1(L) and can be represented in the form
00
(W f)(x) = J c;'+2 (111 - v + ix, P)v - ix, (a:.) )/(I)dl.
o
(37.113)
If we take the arguments in the proof of Theorem 37.11 into account, we see
that the proof follows from Theorem 36.3.
Theorem 37.13. Let the conditions in (36.19) and in (37.111) hold. Then the W-
transform can be represented on functions in the space rot(L) as compositions of
the G-transform, defined in (36.3) with the direct K ontorovich-Lebedev transform,
defined in (37.109) by the relation
(W f)(x) = 2 2 - 2 . K% {12.-1 (Gn I P: I h(Y») (  ) },
(37.114)
- see h(y) in (37.108).
Proof. It follows from Theorem 37.11 that the W-transform exists on functions
in the space rot(L) under the conditions of this theorem. We take 9.3(1) in
Marichev [10] into account and rewrite it in the form
00
r[1I - ix - S, II + ix - s] = J K 2iz (2..jY)y,,-,-ldy.
o
(37.115)
This enables us to transform (37.107) to
( ) 1 J [ (Pm) + s, 1 - (an) - s ] . )
(Wf) x = 21ri r (a;+l) + S, 1- (p;n+l) _ s f (1- s ds
q
00
X 2 J K2iz(2y'Y)y,,-,-ldy.
o
(37.116)

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
755
The condition in (36.19) allows us to interchange the order of integration in (37.116)
by Fubini's theorem - see Theorem 1.1. This easily yields the representation
(37.114) after changing the variable 2.JY = t and using (36.3). .
Theorem 37.14. Let f(x) E rot-1(L). Then the Kontorovich-Lebedev transform,
defined in (37.109) has the following compositional representation in terms of two
direct modified Laplace transforms and the inverse Laplace transforms, defined in
(36.31) and (36.32) and (1.119), respectively
Ki.;i{f(t)} = -/ix-l/2 A:lxl/2 A+x-l/2 L{f(t)j ch(2/../X)}.
(37.117)
Proof. The integral representation for the Mcdonald function
00
Kiz(t) = 1 e- tchu cosuxdu
o
(37.118)
- see Prudnikov, Brychkov and Marichev [1; 2.4.18.4] - enables us to write (37.109)
in the form
00 00
Kiz{f(t)} = 1 f(t)dt 1 e- tchu cos uxdu.
o 0
(37.119)
Since f(t) E rot-I (L), then the estimate
00 00
IKiz{f(t)}1 = 211f' 1 dt 1 F(s)t- 8 ds 1 e- tchu cosuxdu
o q 0
00 00
 2 1 IF(s)lds 1 1 e-tchUt-l/2dtdu
q 0 0
1 1 1 00 .;:idu
= - IF(s)lds rT"":": < +00
21f' vchu
q 0
follows from (37.119). This allows us to change the order of integration in (37.119):
00 00
Kiz{f(t)} = 1 cos uxdu 1 e- tchu f(t)dt.
o 0

756 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Making the change of variable u = 2/# and replacing x by ..;x and using
the factorization of the cosine Fourier transform which is obtained from (36.62)
and (36.63) when v = -1/2 we arrive at (37.117) if we take the relation
J -1/2( z) = -/2/( 1f' z) cos z into account. .
Theorem 37.15. Let 1 E rot;'(L), 1/2 < Rev < 3/4 and the inequality
2sign(c + c.) + sign("Y + "Y. - 1/4)  0
(37.120)
hold. Then the inverse W-transform operator «WI)(x) = g(x» defined in (37.107)
has the form
I(x) =  ( cq-m,p-n 1 -(er;+l), -(ern) I -3/2+11 K- 1 { ( ::. ) }) (x ) . (37.121 )
2 p,q _({3';+1), -((3m) y 2i0JY 9 2
If also q - m - n > m+ n - p, Re{3j < 1/4, j = m+ 1,..., q, xe 21rz g(x) E L 1 (0, (0),
and the conditions in (36.22) hold, then (37.121) can be represented in the lorm
00
I(x)=+ j tsh21f't
1f' x
o
c q - m ,p-n+2 ( 1 1+V+it, l+v-it, 1-(er l n + 1 ), 1-(er n » ) ( t ) dt
x 1'+2,q x 1_ ( am+1 ) 1- IJ ) g.
fJq , fJm (37.122)
Prool. Applying the operator K;i{g(T/2)} to the W-transform (WI)(x/2) =
g(x/2) we obtain the relations
00
K;i {g () } = 1f'X j Tsh 1f'TK iT (2yX)(W I) () dT
o
00
= 7r2 j Tsh 7rTKi. (2v'Z)dT
o
1 j ( iT ) ( iT )
x 21f'i r v - s -"2 r v - s + 2"
q
r [ ((3m) + S, 1 - (ern) - S ] I . ( 1 )d
x ( n+1 ) + 1 _ ( am+1 ) _ - S s.
er p s, fJq S (37.123)

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
757
The condition in (37.120) enables us to use the results by Vu Kim Tuan and
Yakubovich [1] which justify the interchange of the order of integration in (37.123)
provided that 1/2 < Rell < 3/4. Evaluating now the inner integral using (19) from
the cited paper
00
f rsh1rrKiT(2yX)r(lI- S - ir/2)r(lI- S + ir/2)dr = 21r2XIl-"
o
(37.124)
we can rewrite (37.123) in the form
K2;P {g (i)} = 2z"-1/2 (G;.n I d. 1;/ G)) (z).
(37.125)
To obtain now the representation in (37.121) it is sufficient to use Theorem 36.8
relative to the G-transform given in (37.125) and the reflection and translation
relations (1.96) and (1.97) for the G-function.
Now we shall prove that the solution can be represented in the form (37.122).
Since q - m - n > m + n - p and the conditions in (36.22) hold, then by Theorem
36.3 (37.121) can be rewritten as
f( x ) = f oo Gg-m,p-n ( X yl -(a+1), -(an) )y - ll d y
21r 2 p,g _(P+1), -(Pm)
o
00
X f rsh 1rr KiT (2..jY)(W f) (r /2) dr.
o
(37.126)
To obtain now (37.122) it is sufficient to interchange the order of integration in
(37.126) and then apply the relation
00
f Y -II K. ( 2 /;;\ y Gg-m,p-n ( x y 1 _(a+1), -(an) ) d y
ar V:I/ p,g _([31;'+1), -(Pm)
o
_ !G,-m,p-n+2 ( 1 11- ir, II + ir, _(Q+1), -(an) )
- 2 p+2,g x _(P+1), -(Pm)
and the property of G-function given in (1.97). The interchange of the order
is possible according to the conditions rsh1rrg(r/2) E £1(0,00) and Pj < 1/4,

758 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
j = m + 1,... ,q, and the estimate
00
f G g-m,p-n ( ... y 1 -(cr;+I), -(cr n » ) -il K (2 h:\ d
p,g .., _(P;"+I), -(Pm) Y iT vYJ Y
o
00
< const f y-ReIl(xy)-1/ 4K O(2PJdy < +00.
o
The theorem is thus proved. .
37.6. Application of fractional integrals to the inversion of the
W-transform
As was shown in Theorem 37.13 the W-transform is representable as a composition
involving the G-transform, defined in (36.3). The latter can be factorized via the
operators in (36.31) and (36.32) which may lead to fractional integra-differentiation
operators after pairing according to Theorem 36.10. To give details we investigate
an important special case of the W-transform defined in (37.107) which in its turn
generalizes the known Mehler-Fock transform. Namely, we consider the transform
defined by
(F./}(z) = (wr.: I \,td I /(t») (z)
(37.127)
=  f r(v - ix - s)r(v + ix - s)r(1 - crl - s) f.(1 _ s)ds
- 211"i r(l- PI - s)r(1 - P2 - s)r(1 - P3 - s)
q
where Re v > 1/2 and the parameters crt, PI, P2 and P3 satisfy the conditions in
(36.5). We shall call it the F 3 -transform since the kernel of the integral in (37.127)
has the form (10.47) and is connected with the Gorn function F3 given in (10.45).
Picking out the functions r(v - ix - s) and r(v + ix - s) from the kernel we
calculate the characteristic (36.7) for the four remaining gamma-functions:
c. = -1, "I. = Re( cr l - (PI + P2 + (3»'
(37.128)
N ow it is easy to formulate the four following theorems which are special cases
of Theorems 37.11 - 37.15.
Theorem 37.16. The F 3 -transform defined in (37.127) with the characteristic
(0,"1. - 2Rev + 2) where Rev > 1/2 exists on functions in the space rot;,i(L) if

 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
759
and only if
2signc + sign (1' + 1'. - 2Rev + 2)  O.
(37.129)
Theorem 37.17. Let the conditions in (36.22) with respect to the parameters
Ot1, {31, {32 and (33 be satisfied and
1'. - 2Re v + 2 > O.
(37.130)
Then the F 3 -transform defined in (37.127) exists on functions in the space rot- 1 (L)
and can be represented in the form
00
(F 3 f)(x) 1 f (t - 1)1-211-,8
r(2 - 2v - (3)
1
X F. ( 1 - " - /1 - iz, a' , 1 - " - /1 + iz, b ' ;
2 - 2" - /1; 1- t, 1- D/(t)dt,
(37.131)
where
Ot1 = {31 + {32, {31 = a', {32 = b', {33 = (3.
(37.132)
Theorem 37.18. Let f(x) E rot;:(L), 1/2 < Rev < 3/4 and
2sign(c - 1) + sign (1' - Re{3 - 1/4)  O.
(37.133)
If the conditions
Re{3 < 1/4, Rea' < 1/4, Reb'.< 1/4 (37.134)
hold and xe 21rz g(x) E L(O, (0), then we have the following inversion relation for
de!
the F 3 -transform (g( -x) = g( x»:
00
(-1 )(x) = xll-1 f r(1 - v - a' + it)r(1 - v - b' + it)r(1 - v - (3 + it)
3 9 211' r(1 - v - a' - b' + it)r(2it)
-00
x 3F2(1 - v - a' + it, 1- v - b' + it, 1 - v - {3 + it;
1 - v - a' - b' + it, 2it + 1; l/x)x i 'g(t)dt.
(37.135)

760 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Theorem 37.19. Let Rev > 1/2 and the parameters crl = PI + P2, PI, f32 and
P3 = P satisfy the conditions in (36.22) for m = 0, n = p = 1, q = 3 and let the
inequality
2sign ( c - 1) + sign (1' - Re P)  0
(37.136)
hold. Then the following factorization of the F 3 -transform in the space rot(L)
is valid via the Kontorovich-Lebedev transform (37.109), the modified Laplace
transforms (36.1) and (36.2) and the fractional integro-differentiation operators:
(F3/)( z) = 2 2 - 2. K 2 ;$ ([t 2 .-'t' A: 't' A: 't d Ig+t 7 ' G / G) )] (  ) }, (37.137)
(F3/}(z) = 22-2. K 2 ;$ {[t 2 .-'t' A:'t' Ig+t d A:'z1' G / G))] (  ) }, (37.138)
(F 3 /)(z) = 2 2 - 2 . K2;$ { [t 2 .-'t' 19+t' A:'t d A:'t " (i / ( i) )] ( t: ) }. (37.139)
The values of the parameters b, €, d, 6 and 1'1 with the condition Re6 > 0 for
each of these relations are given in Tables 37.1 - 37.3, respectively.
Table 37.1
b e d 6 "Yl
a' -1 b' - a' 1 - b' + {3 a' + b' - {3 - a' - b'
a' - 1 {3 - a' 1 + b' - {3 a' -a' - b'
{3-1 b' - a' 1 + a' - b' b' -a' - b'
Table 37.2
b e d S ")'1
a' - 1 1 + b' - a' {3 - a' - b'-1 a' 1-{3
{3-1 1 + b' - {3 b' - 1 a' 1 -a'
a' - 1 1 + {3 - a' a' - 1 a' + b' - {3 1 - b'
Table 37.3
b e d S ")'1
b' {3 - a' - b' - 1 a' - {3 a' 1 - a'
b' {3 - a' - b' - 1 {3 - a' a' 1 - b'
{3 -a' - 1 a' - b' a' + b' - {3 1 - a'

 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
761
The proof follows from Theorem 36.5 if we take the arguments in the proof of
Theorem 37.11 into account.
As has already been noted, the F 3 -transform includes the Olevskii and
Mehler-Fock transforms which are obtained from (37.107) when a' = 0 and a' = 0,
(3 = 1/2 - v respectively, by using (1.79), namely:
00
(2 F d)(z) - r(2 _ v _ p) J (t _1)1-2_-P
1
(37.140)
X 2F1(1- V - {3 - ix, 1 - v - {3 + ix; 2 - 2v - (3; 1 - t)f(t)dt,
00
g(x) = J t 1 / 4 - 1I / 2 (t - 1)1/4-11/2 P/fi:J:(2t - l)f(t)dt.
1
(37.141)
These expressions differ from the representations (8.55) and (8.42) in Marichev
[10] by some changes of variables and functions. The conditions in (37.130)
for the validity of (37.140) and (37.141) have the form 2 - 2Rell - Re{3 > 0
and 3/2 - Rell > 0 respectively, and it is satisfied owing to the assumption
1/2 < Rell < 3/4 of Theorem 37.15.
As regards the Mehler-Fock transform of the form (37.141), the following
statement which is derived from Theorem 37.19 immediately is valid.
Theorem 37.20. Let 1/2 < Rev < 3/4 and let the inequality
2sign(c - 1) + sign(-y - Rell- 1/2)  0
(37.142)
hold. Then the transform defined in (37.141) can be factorized in the space rot(L)
by the relation
g(z) = 2 2 - 2 - K2i% { (t 2 _- 1 t 6 A: 1 t< A:1t d G f G))) (  ) },
(37.143)
where the parameters have the values: b = -1, c = 1/2 - v and d = 1/2 + v or
b = -1/2 - v, c = v - 1/2 and d = 1.
 38. Applications of Fractional Integro-
differentiation to the Investigation
of Dual Integral Equations
Solutions of many applied problems in mechanics with mixed conditions are reduced
to the so-called dual and triple integral equations. It is typical of such equations

762 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
that an unknown function satisfies different integral relations on different intervals.
One may find the fundamentals of the theory of such equations in detail, and
including surveys of various methods of their solution, for example, in the books
by Sneddon [3, 7 or 6], Uflyand [1] and Virchenko [3] - see also the survey by
Popov [1]. We consider here some typical examples of the applications of fractional
integro-differentiation to reducing dual and triple equations to a Fredholm integral
equation of the second order, the theory of which is well known.
38.1. Dual equations
Example 38.1. In investigations of mixed boundary value problems in
mathematical physics using the Hankel transform, one often meets dual integral
equations with the Bessel function given in (1.83) in kernels of the form
00
f t- 2 °[1 + R(t)]w(t)J(xt)dt = F(x), 0 < x < 1,
o
00
f t- 213 W(t)J II (xt)dt = G(x), 1 < x < 00.
o
(38.1)
Here R(t), F(x) and G(x) are the given functions and w(t) is an unknown one. To
represent (38.1) in the form which is suitable for using the Kober operators defined
in (18.5) and (18.6) we make the following changes of variables
,p(y) = y-l/2W(2..Jii), I(x) = 2 2o x-o F( ..;i),
g(x) = 2 213 x- 13 G( ..;i), k(y) = R(2..Jii).
(38.2)
Then by using (18.19) these equations are rewritten in the form
SIJ/2-o,20,1 (1 + k),p = I,
SIl/2-13,213,1,p = g,
(38.3)
where the functions I and 9 are given on the intervals It = (0,1) and 1 2 = (1,00),
respectively.
We first consider the case k(y) == O. Let
A = (p. + 11)/2 + {3 - Qt.
(38.4)
Applying the operators in (18.5) and (18.6) respectively to (38.3) and using (18.21)

 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
763
we arrive at the relations
1:/ 2 + 01 ,>.-,.,1 = 1t/2+0I,>._,.,S,.,/2-OI,201,1.,p = S,.,/2-0I,>.-,.,+201,1.,p,
(38.5)
K;/2_0I,1I_>.9 = K;/2_OI,II_>' SII/2-/J,2/J,l.,p = S,.,/2-OI,>.-,.,+201,l.,p.
Let a function h be defined by the expression
h = { 1:/2+01,>.-,.,1,
K;/2_OI,II_>.9,
xEh,
x E 12.
(38.6)
Then (38.5) can be rewritten as a single equation
S,.,/2-0I,>.-,.,+201,1,p = h,
(38.7)
for which the inversion relation has the form
.,p = SIl/2+/J,,.,->.-201,l h
(38.8)
according to (18.20). Substituting here the value from (38.6) and making the
inverse changes given in (38.2) we obtain the explicit solution w( t) of the dual
equation (38.1) for R(t) == O. All the operations given above are valid under
appropriate conditions on parameters and functions.
The solution of (38.1) can be obtained by another method. To obtain it we
represent the function 1 as 1 = h + h where
ft={
on h,
on 12,
12= {
on h,
on 12,
(38.9)
and similarly to this we write 9 = 91 + 92. Then by (38.5) we have
1+ 1 - K-
,.,/2+01,>.-,., - ,.,/2-01,11->.9.
(38.10)
Since the functions h and 92 are known, then (38.10) can be considered as
the equation relative to unknowns 12 and 91 in the form
1+  - K- - K- 1+ 1
,.,/2+OI,>.-,.,J2 ,.,/2-01,11->.91 - ,.,/2-01,11->.92 - ,.,/2+01,>.-,., 1.
(38.11)

764 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
The first term on the left-hand side vanishes on the interval Ii and we obtain the
relation from which the function 91 can be found by the second result in (18.17).
Hence the function 9 = 91 + 92 will be obtained. Finding now a function ,p via 9
from the second result in (38.3) and using the inversion relation given in (18.20)
we finally arrive at the representation of the solution in the form
,p = SII/2+fJ,-2fJ,19.
(38.12)
(38.11) can be considered on the interval 1 2 . Then the second term on
the left-hand side vanishes and after similar arguments we may arrive at the
representation of the solution in another form
,p = SIJ/2+OI,-201,1/.
(38.13)
One can show that the constructed solutions in (38.8), (38.12) and (38.13) are
equivalent to each other.
Now we consider the case of (38.1) with an arbitrary function R(t). We
shall use the above solution given in (38.8) in the special case R(t) == 0 as a
"trivial" solution containing some unknown function h = hI + h 2 of the form (38.9).
Substituting it into (38.3) we arrive at the system
1 = SIJ/2-OI,201,I S II/2+fJ,IJ->"-201,l h + SIJ/2-OI,201,lkS II / 2 + fJ ,IJ->"-201,l h ,
9 = SII/2-fJ,2fJ,I S II/2+fJ,IJ->"-201,l h = K;/2_fJ,>"_lI h .
(38.14)
Inverting the second of these equations by the second relation in (18.17) we find
the function h on the interval 1 2 : h 2 = K;/2_OI,II_>..92' Taking the expression
h = hI + h2 and the first relation in (18.22) into account we can write the first of
these equations on the interval II in the form
I'02+fJ,IJ->" hI + SIJ/2-0I,201,lkS II / 2 + fJ ,IJ->"-201,1 hI
= 1 - SIJ/2-OI,201,lkS II / 2 + fJ ,IJ->"-201,lh2'
(38.15)
Inverting now the operator It/2+fJ,IJ->" by the first equation in (18.17) and using
the first result in (18.21) we finally arrive at the relation
hI + SIJ/2-OI,>"-1J+201,lkS II / 2 + fJ ,IJ->..-201,lh l = H,
H = 1:/ 2 + 01 ,>"-1"1 - SIJ/2-OI,>"-1J+201,lkS II / 2 + fJ ,IJ->..-201,lh 2

 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
765
on the interval 1 1 . Under appropriate conditions on the given functions this will
be a Fredholm integral equation of the second kind relative to the function h 1 . Its
kernel contains the function k and the right-hand side is a known function since the
function h 2 has already been obtained. Constructing the solution of this equation
we can find the function h and then the solution of (38.1) by (38.8) and (38.2).
Example 38.2. Here we consider the dual integral equation of the form
00
f t-/J- II (t 2 - k 2 )13 J/J(xt)w(t)dt = F(x), 0 < x < 1,
1c
00
f JII(xt)'I(t)dt = G(x), 1 < x < 00,
1c
(38.16)
where k  O. As in the previous example we use notations It = (0,1) and
1 2 = (1, (0), After the substitutions
w(t) = t"+ 1 ,p(t), F(x) = (x/2)/J- 2 fJ f(x), G(x) = (2/X)" g(x)
(38.17)
and the use of the notation in (37.47), (38.16) has the form
s (: /l 2fJ,
s ( 0, 0,
v, -v,
 ) ,p{z) = f{z),
) ,p{z) = g{z),
x E 1 1 ;
x E 12.
(38.18)
Applying the operators in (37.45) and (37.46) respectively to (38.18) and using the
properties in (37.60) and (37.61) we arrive at the equation
s (', p, fJ2) ,p{z) = h{z), z E I, U h
(38.19)
where h = h 1 + h 2 - see (38.9) - and
h 1 (x) = Ji1c(P - {3, (3 - p)f(x),
h2(x) = R Ic ({3, v - (3)g(x).
(38.20)

766 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Using now the inversion relation given in (37.65) we find
I/J(z) = S (: : P2) h(z),
(38.21 )
whence it is not difficult to obtain an explicit solution of (38.16) after the inverse
change (38.17).
Example 38.3. Now we consider the dual integral equation of another type
PIJ{H(r)1jJ(r);x} = f(x), 0  x  a;
PIJ{1jJ(r);x} = g(x), a < x < 00, I/JI < 1/2,
(38.22)
where
00
PIJ{1jJ(r);x} = f P::i/2+iT(chx)1jJ(r)dr, 0  x < 00,
o
(38.23)
is the inverse generalized Mehler-Fock integral transform with the associated
Legendre function P:=i/2+ir(z) defined in (1.79) in the kernel. We note here
that if we replace 1jJ(r) by 1r- 1 rshT1rT(1 - 11+ ir)r(1 - II - ir)g(r) and g(x) by
211-3/2sh 1/2-11 xf(ch 2(x/2» in the second relation in (38.22) and put /J = 1/2 - II,
then by (8.42) and (8.43) in Marichev [10] we obtain the inversion of the direct
modified Mehler-Fock transform defined in (37.141). The function H(r) in (38.22)
is assumed to be known.
We introduce analogues of the Erdelyi-Kober operators defined by the following
fractional integrals of one function by another one namely chx (see  18.2)
I%IJ{cp(t);x} =(1r/2)%1/2r- 1 (1/2 =F /J)(shx)-(IJ=fIJ)/2
:&
X f (sht)(l+IJ%1%IJ)/2(chx - cht)-1/2=fIJcp(t)dt,
o
I/JI < 1/2,
K%IJ {cp(t); x} =( 1r /2)=f1/2r-1(1/2 =F /J)(shx )(IJ%IJ)/2
(38.24)
00
x f (sht)(1-IJ=f 1 =fIJ)/2( cht - chx)-1/2=fIJcp(t)dt,
:&
I/JI < 1/2.
(38.25)

 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
767
Without writing down the operators obtained by inverting (38.24) and (38.25), we
denote them as I; and 1<; respectively. Then by direct evaluation one may
prove two following compositional relations:
I:{P#J{x(r); t};} = (.1'eX)(X), X E L 1 (0, 00),
(38.26)
K;' {P# {xC T); t}; z} = F, L(:)tt ; z},
X
- E L 1 (0,00),
W
(38.27)
where IJJI < 1/2, w(t) = r(I/2 + JJ + it)r(I/2 + JJ - it)[r(I/2 + it)r(I/2 - it)]-l,
and .1'e and .1'8 are the sine and cosine Fourier transforms defined in (1.108) and
( 1.109).
We introduce the notation
G(x) = { G1(X),
G 2 (x),
o :5 x :5 a,
a < x < 00,
(38.28)
where G2(X) = v'2/1r 1<;1{g(t);x}, a < x < 00, and G 1 (x) is a certain as yet
unknown function. Then applying the operators I: and 1<;1 respectively to
(38.22) and taking (38.26) and (38.27) into account we obtain the relation
.1'e{H(r)1jJ(r);s} = F(x), 0  x :5 a;
{ 1jJ(r) }
.1'8 w(r)thr1r ;x = G(x),
a < x < 00,
(38.29)
where F(x) = v'2/1r I:{f(t); x}, 0 :5 x  a. Inverting the second of them and
using (1.111) we find the value
./>(T) = w( T)th 1fT [( ) (F,G,}(T) + (':' ) (F,G 2 )( T)]
(38.30)
where the symbol (:) (F,G)( T) here and below means that the integration in the
corresponding integral in (1.109) is carried out only on the interval (a, b) instead
of (0,00). To find the unknown function G 1 we substitute (38.30) into the first
equation in (38.29) and as a result we obtain the Fredholm integral equation of the
second kind. After solving it and finding the function G, defined in (38.28), the
required function 1jJ(r) can be obtained by (38.30).

768 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
38.2. Triple equations
Example 38.4. We consider triple integral equations of the form
00
f t- 2 {3 JII(xt)W(t)dt = Fl(X), 0 < X < a,
o
00
f t- 2Ot JIJ(xt)W(t)dt = G2(X), a < x < b,
o
(38.31)
00
f t- 2 {3 JII(xt)w(t)dt = F3(X), b < x < 00,
o
where Fit G 2 and F3 are given functions and W is an unknown one. We make the
substitutions
1/J(t) = t-lw(t), /(x) = (2/x)2{3 F(x),
9(x) = (2/x) 2Ot G(x),
(38.32)
setting
 { F( x), x e Ij,
F(x) = L..J Fj(x); Fj(x) = _
j=l 0 xelj,
(38.33)
It = (0,a),I 2 = (a, b), 13 = (b,oo)
and defining similarly the function G(x), the functions G l , G 3 and F 2 being still
unknown. Then by (18.19) the system of equations in (38.31) has the form
SIl/2-{3,2{3,2,p(X) = /(x),
SIJ/2-0t,20t,2,p(X) = 9(X).
(38.34)
From the latter equation we obtain
,p(x) = SIJ/2+Ot,-20t,29(X),
(38.35)
by using (18.20) where 9(X) = 9l(X) + 92(X) + 93(X), i.e. 9 contains the unknown

 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
769
functions 91 and g3. To find them we introduce the notations
K = K II / 2 - fJ ,(/J-II)/2-0I+fJ'
(38.36)
I = 1/J/2+0I,(II-/J)/2+fJ- 0I
and apply the operator I-I inverse to I to the first relation in (38.24). Then
we apply the operator K to the second result in (38.34) and use the relations in
(18.17) and (18.21). This leads us to the result 1- 1 /= Kg:
1- 1 /= I-I St/J = 111/2+fJ,(/J-II)/2+0I-fJSII/2-fJ,2fJ,2t/J(X)
= SII/2-fJ,(/J-II)/2+0I+fJ,2t/J( x),
Kg = KII/2-fJ,(/J-II)/2+fJ-OIS/J/2-0I,201,2t/J(X)
= SII/2-fJ,(/J-II)/2+0I+fJ,2t/J( x).
(38.37)
In the same way after analogous applications of the operators K- 1 and I it is not
difficult to obtain a second relation in the same form, which together with the first
one gives the system of equations
1- 1 /(x) = Kg(x), K- 1 /(x) = Ig(x).
(38.38)
Equation (38.38) can be rewritten in the form
() r' It (20) = (:) Kg,(2O) + (: )KI/3(2O) + (') Kg.(2O) , (38.39)
x E I},
(': ) r' f.(2O) = () Ig,(2O) + (:) Ig2(2O) + ( :) 11/3(20), (38.40)
x E 13.
on the intervals h and 1 3 , respectively. Inverting these relative to gl and g3
respectively we arrive at the following results for these unknown functions:
g, (20) = - (:) rt ) K g.(2O) + (: ) r' (  ) r' It (20)
- (:)K- ' (:)K9 2 (2O), (38.41)

770 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
1/3(20) = - (: )rl ()1 91 (2O) + (: )rl (': )K- 1 h(2O)
- (:)rl()192(2O).
(38.42)
Excluding 93 from here we finally arrive at the relation
91 (20) = ( : ) r 1 ( ': ) K ( : ) r 1 (  ) 191 (20)
+ (: )K- 1 { ()rlf1(2O) - ()K92(2O)}
- (: )rl (': )K(: )rl { (': )r1fa(2O) - ()192(2O)}'
(38.43)
Evaluating the compositions of operators given in (38.43) one can see that this
relation, under appropriate assumptions on the functions, is a Fredholm integral
equation of the second kind relative to 91. Considering this equation to be solvable
and using (38.42), (38.35) and (38.32) we can find the solution of the system of
equations (38.31).
At the end of this section we shall prove that (38.31) in the special case
F 1 = F3 = 0 is equivalent to the system of equations from the following example.
Example 38.5. We consider the triple integral equations
- rot-I { r( + s/6) l"- ( S ) ' X } = 0 X E I I U 1 3 ' ,
r( + fJ + s/6) T' ,
-1 { r(1 + '7 - s/O') _ }
rot r(1 + '7 + a _ s/O') <p (s); X = 92(X),
(38.44)
X E 12,
where Ij (j = 1,2,3) are the intervals in (38.33), while a, fJ,, '7, 6> 0,0' > 0
are real parameters and the unknown function <p-(s) = rot{<p(x); s} is the Mellin
transform - see (1.112) - of a certain function <p(x).
Using the notation of (38.33) and (23.1) and (23.2) we reduce the system of
equations in (38.44) to the form
Ig+ iU ,'1<P(X) = 9(x), li6,(<P)(X) = /(x)
(38.45)
or

 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
(:) I;..{'Pl(Z)+ (:) I;..{'P2(z) + (  ) I;..{'P3(z) = 0, Z E I"
( ) I.f+;u..'Pl(Z) + ( :) I.f+;u..'P2(z) = g(z),
771
x E 1 21 (38.46)
( ';' ) I;..{'P3(Z) = 0,
x E 1 31
on the corresponding intervals. Using (18.17) we invert the third, the second and
the first of the equations (38.46) relative to the functions 'P3, 'P2 and 'Pi and obtain
the relations
'P3(X) = 0,
'P2( z) = - ( : ) IO.t;u..+a ( n I.f+;u..'Pl (z) + (:) I O .t;u..+ag 2 ( z),
(38.47)
'Pl(Z) = - (: ) I:.HII () I;..{'P2(z).
Substituting the value of 'Pi from the third relation into the second one we finally
arrive at the Fredholm equation of the second kind
'P2(Z) =(: ) IO.t;uMa ()I.f+;u.. (: )I:m ()I;'.{'P2(Z)
+ (: ) I o .t;u..+ag 2 (z).
(38.48)
relative to 'P2' After inverting this equation it is not difficult to restore the solution
of the system of equations in (38.44).
In conclusion we note that if we set
Fi = F3 = 0,
G2(X) = (x/2)2a+i g2 (x),
p. = ao + Po +  + 1],
v =  + 1],
a = (ao - Po + 1] -  - 1)/2,
(38.49)
P = (1] -  - 1)/2,
w(t) = S'1I{3o+-'112'P(t)
in (38.31), then according to (18.19), (18.22), (23.1) and (23.2) this system of
equations has the form (38.44) with a being replaced by ao and P by Po and

772 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
6 = u = 2. Obtaining the solution of this system of equations we find the solution
of the system of equations given in (38.41), (38.49):
w(t) = S",{30+-"cp(t).
 39. Bibliographical Remarks and
Additional Information to Chapter 7
39.1. Historical notes
Notes to  35.1. The equation of the fonn (35.4) with d = 1 and variable lower limit was first
considered by Higgins [3] (1964). He obtained its solutions in the form. (35.11) and (35.14) in the
space of sufficiently smooth functions. We note that earlier special cases of this equation with
Chebyshevand Legendre polynomials in the kernel were studied by Ta Li [1] (1960), [2] (1961)
and Buschman [1] (1962) respectively (see  39.2, note 35.1 below).
Methods of fractional integra-differentiation were first applied by Love [2,3] (1967) to
studying (35.1) - (35.4) in the spaces Qq and Rr given in  17.1, Notes to  10.1 (see  39.2, note
35.2 below).
Theorem. 35.1 was not published previously, but the case p = 1 is contained in the papers by
Love mentioned above.
Notes to  35.2. Equation (35.16) was first obtained and inverted by Copson [5, p. 353]
(1958) when solving the Dirichlet problem for the hyperbolic equation (40.19) with .x = 0 in the
first quadrant. However, this result was not specially noted in the above paper. Therefore it
is generally accepted that the paper by Buschman [3] (1963) was the first where the inversion
relations for (35.16) and (35.18) were obtained. Using methods of fractional integra-differentiation
and the theory of generalized functions Erdelyi [9] (1964), [12] (1967) found the solution of (35.15)
and (35.17) in the form. (35.25) and (35.35) (see  39.2, note 35.3 below).
Notes to  36.1 and 36.2. The idea of defining fractional integrals and derivatives via
the inverse Mellin transform. operators given in (36.1) and (36.2) was suggested in the papers by
Kober [1] (1940) and Erdelyi [4] (1940), though previously it was suggested by Zeilon [1, p. 4]
(1924).
The definition of the G-transfonn via the Mellin-Barnes integral in (36.3) was first introduced
by Vu Kim Than, Marichev and Yakubovich [1] (1986) in the fonn which slightly differs from
(36.3). Earlier the special case of such a transfonn in the fonn (36.23) was considered by Narain
Roop [1] (1959), and this case was investigated more carefully in Narain Roop [2-4] (1962-1963)
and Fox [2] (1961). The space roto(L) was introduced by Vu Kim Tuan [1] (1985) who denoted
it by L- 1 . The more general se rotl(L) was introduced by Vu Kim Tuan [4] (1986). The
space Lc.-y) is a special case of the space Lf which was introduced by Akopyan [1, p. 6] (1960) -
see also M.M. Dzherbashyan [2]. Theorems 36.1-36.4 were proved in the papers by Vu Kim Tuan
[5] (1986) and Vu Kim Tuan, Marichev and Yakubovich [1] (1986).
Notes to  36.3. The idea of the factorization of integral transfonns, i.e. their
representation as compositions of other "more simple" integral transforms, probably first appeared
in the expansion expression of the Stieltjes transform. represented as the composition of two direct
Laplace transforms - Widder [1] (1938), [2] (1946) and also Erdelyi, Magnus, Oberhettinger
and Tricomi [4, Ch. 14]. Tricomi [2] (1935) pointed out the expansion fonnula of the Hankel
transform. via the composition of direct and inverse Laplace transforms. Compositions of such a
kind were systematically applied by Hirschman and Widder [1] (1958) to studying convolution

 39. ADDITIONAL INFORMATION TO CHAPTER 7
773
type transforms. As far as the G-transform of the form (36.23) is concerned, the idea of its
factorization was developed in Fox [3] (1963), [6] (1971) and Rooney [5] (1983) (see  39.2, notes
36.1-36.4 below).
The technique of factorization of the G-transform of the form (36.3) via simpler G-transforms
of such a kind, together with special tables and notations were first developed formally, i.e. without
characterization of all conditions and spaces of functions, by Brychkov, Glaeske and Marichev [1]
(1983), and also [2] (1986). There were prior investigations by Higgins [4,5] (1965) and Marichev
[3] (1973), [6] (1976). The factorization problem in the space rot(L) was solved in the paper
by Vu Kim Than, Marichev and Yakubovich [1] (1986) where Theorem 36.5 was proved.
Notes to  36.4-36.8. Theorems 36.6-36.8 and 36.10-36.13 were proved by Vu Kim Tuan,
Marichev and Yakubovich [1] (1986) and by Marichev and Vu Kim Tuan [2] (1985), [3] (1986)
respectively. The other results in these subsections were obtained by Vu Kim Than and have not
been published previously.
Notes to  37.1. The solution of (37.1) with .x = 1 was obtained by K.N. Srivastava [5]
(1964) in a more cumbrous form that given in (37.31)-(37.34). The solution of (37.1) in the form
(37.31) with I = 0 and a = 0 was given by Wimp [2] (1965) in terms of the Laplace transform,
we refer to Rusia [4,6]. The most perl'ect investigation of the equation (37.1) was C&lTied out
by Prabhabr [1] (1969) (the case .x = 0, a = 0), [2] (1970) who used methods of fractional
integra-differentiation with Rea > 0 and found the solutions (37.32) and (37.33) when ReJ3 > 0
and Re J3 < Re a, respectively.
Equation (37.2) with a = 0, a = 2n + 1 and .x = i, i.e. the equation

f (x - t)n In (x - t)j(t)dt = 9(X)
o
where In(z) is the modified Bessel function given in (1.84), was first considered by Tedone [1]
(1914) for integer non-negative n. In the case of natural n he reduced this equation to the
simplest fonn with n = 0 and obtained its fonnalsolution. The solution of (37.2) with a = 0 and
a = 1 was fOWld by Elrod [1] (1958) by using (37.20).
A formal solution of (37.41) which differs from (37.4) only by change of variables was
obtained by Burlak [1] (1962) and Sneddon [2] (1962). Sufficient conditions for its solvability were
indicated by Srivastav [1] (1966). We note only that (37.4) in another form waS already solved by
Sonine in 1884. For this see  39.2, note 37.3 below and also the Brief Historical Outline at the
start of this book.
The solution of (37.5)-(37.7) was given by Marichev [8] (1978). The presentation in this
subsection fonows this paper.
Notes to  37.2. The solutions (37.43) and (37.44) of (37.41) and (37.42) were fonnalIy
obtained by Burlak [1] (1962) and Sneddon [2] (1962); see also the papers by Srivastav [1] and
Rusia [3]. The necessary conditions for the existence of the solution of (37.41) in the space of
continuous functions were given by Bha.ratiya [1] (1965), and the criterion of solvability for this
equation in the space L2(0,00) was given by Soni [1] (1971).
It is generally considered that the solution of (37.41) and hence (37.4) was first given by
BurJak and Sneddon. However, in actuality the equation which was solved by Sonine [4] (1884),
see alao [6] (1954), is reduced in the special case to (37.41) by quadratic change of variables
(see  39.2, note 37.3 below). We also note that Sneddon, who did not know of this paper by
Sonine, called the operator in (37.41) the Sonine operator on the basis of the fact that Sonine had
calculated two special integrals of such a kind of Bessel functions, which are known in literature
as Sonme integrals. For these we refer to (2.54) and the handbook by Prudnikov, Brychkovand
Marichev [2, equations 2.12.4.6 and 2.12.35.12]. We also observe that the inversion formula for
(37.41) with v = 0 in the space L2(O,OO) was indicated in the paper by Rozet [1] (1947) (see
 39.2, note 37.4 below).

774 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
The generalized Erdelyi-Kober operators defined in (37.45) and (37.46) and generalized
operator of the Hankel transform given in (37.47) were introduced by Lowndes [1] (1970), who
had proved (37.48)-(37.66) and other results.
Notes to  37.3. The operator inverse to the operator (37.67) in the form (37.81) was
first constructed by Bakievich [1] (1963) for 0 < cv < 1. However, in the special case cv = 1 the
inversion of this operator, in the form of the relations in Theorem 37.6, was obtained earlier in the
paper by Velma [1] (1945), and also his book [3, p. 69, 70]. In this paper the expressions of the
form (37.78) and (37.79) and more general relations proved in 1942 and connecting the solutions
of the Laplace equation u = 0 and the Helmholtz equation u + .x 2 u = 0 were indicated. In
this connection,  40.2 and  43.2, note 40.1 should be seen.
A special case of (37.71) written in another form was indicated in the paper by Henrici [I,
p. 256] (1957). Other results of this subsection are the development of investigations given in the
paper by Lowndes [9] (1985) where a modified form of the operator (37.68) was indicated and
studied. Theorem 37.7 was mentioned in the paper by Marichev [1] (1972) as a special case of a
more general statement.
Notes to  37.4. The results of this subsection were given by Marichev [1] (1972).
Notes to  37.5. Special cases of the scalled integral transforms with respect to an
index, or with respect to a parameter of special functions were discovered and investigated
comparatively recently. The book by Lebedev [4] presents the fundamentals of the theory of
Kontorovich-Lebedev transforms defined in (37.109), (37.110), and Mehler-Fode transforms given
in (37.141), (38.23). In the paper by Wimp [1] published in 1964 the considerably more general
transform (37.113) with the Meijer G-function in the kernel was introduced. Its inversion formula
was obtained, and five special cases were considered. For this the book by Marichev [10, Section
8.4] should be seen. A more compact form for the inversion relation given in (37.122) was found
by Yakubovich [1] (1985), who also investigated the compositional structure of this transform.
The presentation in this subsection follows the paper by Vu Kim Than, Marichev and Yakubovich
[1] (1986).
We also note that different representatives of the above transforms with G-functions and
H-functions in the kernel were considered in the papers by Kalia [1] (1970) and Shah [1] (1972).
Notes to  37.6. The F3-transfonn in the form (37.131) was introduced in the paper by
Brychkov, Marichev and Yakubovich [1] (1986). Theorems 37.16-37.20 were proved by Yakubovich
and have not been published previously.
Notes to  38.1 and 38.2. The solution of the dual equations (38.1) in the case
R = G = J.l. = v = 0, cv - (3 = 1/2, and F == 1 was first obtained by Beltrami [1] (1881). We
also note that earlier, Weber [1] (1872) had formulated problems for partial differential equations
reducible to dual equations. Different special cases of (38.1) with R = G = J.l. = v = 0 in the
main were studied much later in the papers by King [1] (1935), Busbridge [1] (1938), Tranter
[1-3] (1951954), Gordon [1] (1954) and Noble [1] (1955), and also Titchmarsh [1] (1937, 1948).
Various methods which mainly differ from methods of fractional integro-difl'erentiation were used
in the above papers.
Fractional integration of order 1/2 was first applied by Copson [3] (1947) to solve dual
equations of the fonn (38.1). In this context see also Noble [2] (1958) and Sneddon [1] (1960).
This method was developed in the papers by Copson [6] (1961), Peters [1] (1961) and Lowengrub
and Sneddon [1] (1962), [2] (1963) in order to solve (38.1) in the case J.l. = v, R == o.
Fractional integration with Sonine operators (see  39.2, note 37.3) was first applied by
Ahiezer [1] (1954) to solving dual equations of the form (38.16) with F = J.l. = v = 0 and modified
order of conditions on x. In his paper [2] (1957) this result was generalized to the case v = -J.l..
The papers by Peters [1] (1961), Burlak [1] (1962), Lowndes [1] (1970), etc. were devoted to
the investigation of (38.16) with arbitrary J.l. and v. However, all these articles contained an
improper setting of the problem. Instead of integration over (k, 00) in the first equation in (38.16),
integration over (0, 00) was taken. But this is not COITect for arbitrary {3 if the values of the
function (t 2 - k 2 )/J on 0 < t < k have not been specified.
It should be noted that in the papers by ErdeJ.yi and Sneddon [1] (1962) and Sneddon [2]
(1962), the systematical application of the Erdelyi-Kober type operators given in (18.1)-(18.6) to

 39. ADDITIONAL INFORMATION TO CHAPTER 7
775
the theory of dual, triple, etc. equations was begun. At the same time, the investigations in
these and many following papers were canied out formally without studying the spaces of the
functions, and the conditions on the parameters.
Subsections  38.1 and 38.2 were written using material from the papers by Erdelyi and
Sneddon [1] (1962) and Sneddon [2] (1962) - Example 38.1, Lowndes [1] (1970) - Example 38.2,
Virchenko and Ponomarenko [1] (1979) - Example 38.3, Virchenko and Makarenko [2] (1975) -
Example 38.4, Lowndes [2] (1971) and Sneddon [5] (1975), [6] (1979) - Example 38.5, with some
modifications. We alao note that the solution in (38.8) was obtained by Titchma.rsh [I, p. 334]
(1937), when #J. = v, and by Peters [1] (1961), while the solutions in (38.12) and (38.13) were
fOWld by Noble [1] (1965) and the method of "trial" solution (Example 38.1) was used by Gordon
[1] (1954) and Copson [6] (1961).
39.2. Survey of other results (relating to  35-38)
35.1. The integral equation
I
I (t x)c-l ( t )
-r( c) -#'1 a, Pi Ci 1 -; I(t)dt = g(x),
:&
o < xo $ x  I,
(39.1 )
was considered by Higgins [3] who obtained two forms for its solution
I(x) =Irexp rlx-P g(x),
1
I(x) = 1 I (t - x)a-e+m- 1 t p (I-m-a g})(t)
r(a - c + m) 1-
:&
X I ( m, -Pi a - c + mi 1 - ;) dt,
91(T) = T-fJg(T), m = 1,2,...,
under the appropriate assumptions for a sufficiently smooth right hand side g(x). Here If_ is the
fractional integro-differentiation operator given in (2.18) or (2.34). We note that the above paper
by Higgins was the first one in which the tedmique of factorization of the 1-transfonn by using
fractional integra-differentiation operators was applied. The solution of the equation
1
I(x) = rImC) I (t - x)m-c-I X 1 (-a, -Pi m - Ci 1 - ;) g(m)(t)dt,
:&
m = 1,2,...,m > Rec> 0,
which differs from (39.1) by replacing 1 - tlx by 1 - x/t was obtained by Wimp [2] by using the
Laplace transform. Earlier special cases of this equation with the Chebyshev polynomial of the
first kind Tn (x) in the kernel
1
I  J(I)dl = g(,,), 0 < "0  "  I,
t 2 - x 2
:&
(39.2)
and with the Legendre polynomial Pn(X) in the kernel

776 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
1
f Pn(t/x)f(t)dt = g(x), 0 < xO  x  I,
:&
(39.3)
were considered in the papers by Ta Li [I, 2] and Buschman [I], respectively. The classical scheme
for the solution of the Abel equation in (2.1) (see  2.1) was used in these papers. In the paper
by Widder [4] (39.2) and (39.3) were solved by the method based on the Laplace transform. We
also observe that in the paper by Erdelyi [7] the equation of the form (39.3) was solved by using
Rodrigues formula. The solutions of the equations of the form (39.3) in which Pn (x) is replaced by
the generalized Legendre polynomials Qmk(X) and by the generalized Rice polynomials HlOt,fJ)(x)
and by the hypergeometric fWlction ¥"3(-n, a + (3 + n + I, ,'iP, P' ,I + ai x), were obtained by
R.P. Singh [1] and Dixit [I, 2], respectively.
The solution of the integral equation
1
f (t - x)Ot pOtlfJ) ( : - 1) f(t)dt = g(x), 0 < Xo  x  I,
:&
(39.4)
with the Jacobi polynomials
pOtlfJ)(X) = (a + l)n[n!]-l:.aFl (-n,n + a + {3 + Ii a + Ii (1 - x)/2),
which is a special case of (39.1), was first obtained by Higgins [2] - see also K.N. Srivastava [4]
and Rusia [5]. In the papers by Bhonsle [I, 2], Rusia [2, 7], C. Singh [3] and K.N. Srivastava [3,
6-8] solutions of the equations of homogeneous type containing Jacobi polynomials in the kernel
were obtained.
35.2. We denote by Qq(a, b), 0  a < b  00, the space of functions <p(x) given on (a, b)
almost everywhere and such that the fWlction xq<p(x) is locally integrable on (a, b). Love [2]
obtained necessary and also sufficient conditions for the existence and uniqueness of the solution
of (35.1) and (35.2) in Qq(O, d), 0 < d < 00, and their explicit solutions in the form (35.5)-(35.8).
These investigations were continued in the paper by Love [3] where (35.9)-(35.12) for the solutions
of (35.3)-(35.4) in the space Qq(d,oo), 0 < d < 00, were proved.
We note that the representations for the solution of the equation of the form (35.2) as the
compositions of two Erdelyi-Kober-type operators given in (18.1) and (18.2) were formally done
by Buschman [5]. Miiller and Richberg [1] obtained the inversion formulae in the special case of
(35.3) with Chebyshev polynomials of the first kind in the kernel
00
f p !(')d' = g(x), x> 0,
t 2 - x 2
:&
which is an analogy of (39.2).
R.K. Saxena and Kumbhat [3] deduced the inversion relations for the operators
:&
x-,,-6 f ( t )
reS) t"(x-t)6-11 a,{3i S i 1 -; f(t)dt,
o
(39.5)
00
r:) f t-"-'Y(t - X)'Y-1:fl (a, {3i "Yi 1 - ) f(t)dt
:&

 39. ADDITIONAL INFORMATION TO CHAPTER 7
777
for J(x) e L,(O,oo), 1  p  2j under the appropriate assumptions on the parameters. Goyal
and Jain [1] found the inversion relations for the operators more general than (39.5) with Wright
hypergeometric function in the kernel.
Braaksma and Schuitman [1] obtained the inversion relations for the operator of the form
(35.4)
00
1 f( X ) C-l ( t ) dt
(AJ)(x)= r(c) 1-; i"l a,b j c j l-; J(t)t:
:&
in the space of test functions T('x, JL) and for the operator A' conjugate to A in the space of
generalized functions T'('x,I-') (see  23.2, note 18.3).
U sing methods by Love [2, 3], Prabhakar [4] constructed the solution of the equation
b
f (tm - xm)c-l ( X m )
r(c) l a, bj Cj 1 - t m mt m - 1 J(t)dt = g(x),
:&
x> 0, Rec > O.
(39.6)
McBride [1] obtained invetsion relations for the equation HJ = 9 (see  23.2, note 18.5)
more general than (35.1), and also for similar generalizations of (35.2)-(35.4). The investigations
were given in the spaces of test F,I-' and generalized F;'I-' function considered in  8.4.
Saigo [I, 2] found the inversion relations for the operators t:/,,, and I:!'''' -00  a <
b  00 (see  23.2, note 18.6).
In the paper by Smirnov [5] the inversion relation for the equation of the first kind with
Gauss hypergeometric function in the kernel, which arose in solving the Trikomi problem for
equations of mixed type with two degenerate lines, was obtained by duction to an equation of
the form (35.1).
35.3. Using methods of fractional integro-differentiation Erdelyi [9] obtained necessary and
also sufficient conditions for the existence of the solution, integrable on a finite interval, of (35.15)
for ReJL < 1 and Rell  -1/2, and he constructed the explicit solution of this equation in the
form (35.25). ErdeIyi also indicated the form (35.35) for the solution of (35.17) with d = 00, which
was investigated earlier by K.N. Srivastava [1]. There is a mistake in his inversion expression -
see also Habibullah [2].
In the paper by Erdelyi [12], (35.15) with arbitrary real JL and II was studied in the space
of generalized functions with support on (0,00) and conditions for the solution to be an usual
function were found.
Smirnov [4,6] indicated sufficient conditions for the invertibility of two equation of the form
(35.15) and (35.16) written in another way and gave their inversion relations.
Din Khoang Anh [1] obtained the conditions for the existence and uniqueness of the solution
of (35.15)-(35.16) with e = 0 and of (35.17)-(35.18) with d = 00 in the spaces L,«O,a),x'Y) and
L,«a,oo),x'Y), 0 < a < 00, respectively, and constructed the inversion relations for them.
We note that a series of pages by Buschman [2], Higgins [1] and K.N. Srivastava [2] were
devoted to solving equations of the form (35.17) with Gegenbauer ultraspherical polynomials
C(x) in the kernel instead of p,Nx).
35.4. In the paper by Sneddon [4] the general method based on the Mellin transform defined
in (1.112) and the convolution theorem in (1.115) was suggested for solving the equation
a
f k (7) J(t) t = g(x), 0  x  a, a  00.
:&
(39.7)
As examples, solutions of (35.4), (35.15), (35.17) with d = 00, (39.1 )-(39.3), etc. were given.

778 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
35.5. Kalla and Saxena [2] obtained the inversion relations for the operators
:&
JJx-'1- 1 f ( at'" )
reI-a) -#'1 a,p+mi'"Yi;iJ t'1f(t)dt,
o
(39.8)
00
re: a) f -#'1 (a,p + mi'"Yi :t: ) t- 6 - 1 f(t)dt
:&
(39.9)
more general than (23.5) and (23.6) for f(x) e L,(o, 00), 1  p  2, or f(x) e rotp(O,oo), p> 2,
provided that the conditions
'"Y,:#O,-I,-2,..., Re('"Y-a-{3»m-l, m=I,2,...,JJ>O,
Ret1 > max(l/p,l/p'), lip + lip' = I, I arg(1 - a)1 < 11"
hold. Here rotp(O, 00) means the subspace of L,(O,oo), p> 2, consisting offWlctions which are
the inverses of the Mellin transfonu of fWlctions in Lpl(-ioo, ioo) - see also  23.2, note 18.1.
In the paper by Parashar [1] equations more general than (23.5) and (23.6) with the Meijer
G-function in the kernel were considered, and the uniqueness of their solutions in the space
L l (0, 00) was proved provided that certain conditions hold. The inversion relations for such
equations in Lp(O, 00), 1  p  2, or in rotp(O, 00), p> 2, were obtained in Kalla [3] by using the
Mellin transfonn. These results were extended by Kalla [12], Kalla and Kiryakova [1] and Kalla
[11] to more general equations with the Fox H-fWlction and an arbitrary fWlction, respectively in
the kernels. One may find other generalizations of (23.5) and (23.6) in  23.2, note 18.4.
35.6. Love [6] obtained necessary and also sufficient conditions for the solvability of the
integral equation
00
f Ha, bi Ci -xlt)t b f(t)dt = g(x), 0 < x < 00.
o
He constructed the solution of this equation on the basis of the representation of the left-hand
side as the composition of the Riemarm-Liouville fractional integration operator in (5.1) and the
Stieltjes transfonn (see  9.2, note 7.3). Another fonu for the solution of this equation via the
inverse Mellin transfonn was given earlier by Sw&roop [1].
These investigations were continued in the papers by Prabhakar and Kashyap [1] and Love,
Prabhakar and Kashyap [1] where the integral equations
00
f tc-l ( t )
rec) :fl a,bici-; f(t)dt = g(x),
o
00
f tc-l
r(c) I F I (ai Ci -xt)f(t)dt = g(x)
o
were solved on the basic of the representation for the left sides as the compositions of
the Riemann-Liouville fractional integration operator given in (5.3) and Stieltjes and Laplace
transfonns, respectively.
35.7. Lebedev [1] proved that the integral equation
a
(Tf)(x) =  f K ( 2../iX ) f(t)dt = g(x), 0  x  a,
11" t+x t+x
o
(39.10)

 39. ADDITIONAL INFORMATION TO CHAPTER 7
779
where K(u) = (1I"/2)l (1/2,1/2; 1; u 2 ) is the elliptic integral of the first kind has the solution
a t
I(x) = - f tdt d f Tg(T)dT .
11" dx vt2 _ x 2 dt 
z 0
This result is based on the representation for the operator T as the composition of the left-sided
and right-sided fractional integration operators by the function x 2 :
z a
(TI)(x) =  f dt f I(T)dT .
11" v x2 - t2 
o t
This method was simultaneously applied by Copson [3] to solving a two-dimensional equation
in electrostatics. It should be noted that the papers by Lebedev and Copson were the first
publications where integral equation of the first kind with special functions in the kernels were
solved by means of their representations as the simpler equations of Abel type, where solutions
were already known. In other words, the method of factorization for integral operators with
special functions in the kernels was used in these papers for the first time.
We also note that Lebedev [1] applied his result to the equation
f a (xt)m ( 1 1 4xt )
(x + t)2m+l -Fl m + 2' m + 2 i 2m + Ii (x + t)2 I(t)dt = g(x),
o
and in the paper by Kalla [10] a similar equation with other parameters
f a -2a ( 1 4xt )
(x + t) l a, 2i Ii (x + t)2 I(t)dt = g(x),
o
05 x 5 a, 0 < a < 1,
was solved by method of Lebedev. In the paper by Ahiezer and Shcherbina [1] a fonnal solution
of the equation
a
f 1 ( f p + I .!. 4x 2 t 2 ) t dt - x
(x 2 + t2)p 7Fl 2' 2 '2' (x2 + t 2 )2 I() - g( ),
o
o < x < a,
(39.10')
was given where 0 < a 5 +00, 0 < 2p < q < 2p + 2. IT p = 1/2, q = 2, then (39.10) is a special
case of (39.10')i see also Williams [1].
35.8. By using Erdelyi-Kober type operators given in (18.1)-(18.4) Lowndes [4, 6] obtained
a solution of the integral equation of the first kind
b
f K(x, t)<p(t)dt = I(x), a < x < b,
a
(39.11)

780 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
in the following cases:
00
(T6x u ,,-lt 6 1-& f
K(x, t) = r(1 _ a)r(1 - (3)
max(z,t)
TU( a-" )+6(,6 -1-& )-1
?/J(T)dT,
(T U _x u )a(T6 -t 6 ),6
a = 0, b = +00, 0 < a, {3 < I, (T > 0, 6> 0, -00 < J.l.," < 00;
and
min(z,t)
(T8x- u ( a+" )t 6 (1-I-&-,6)-1 f
K;t(x, t) = r(a)r({3)
T u (,,+l )+I-&6-1?/J( T )dT
(XU - T u )1-a(t6 - T6)1-,6 ,
a
K;2(X, t) = x 1 - u (a+,,+1-") C;  x 1 - U )" (x u (a+,,-l)-l K 1 (x, t»,
b<+oo, (T>O, 8>0, a>O, (3>O, -00 <J.l.,V < 00,
where ?/J(t) is a certain fWlction. The paper by Williams [1] is also relevant.
As examples the equation
a
f <p(t)dt
K>..(plx - tl) Ix _ tl>" = f(x), 0 < x < a, Rep > 0, l.xl < 1/2,
o
with the modified Bessel function K>..(z), and the equation of the form (39.10') were considered.
Ahner and Lowndess [1] fOWld the solution of (39.11) with the kernel K(x, t) more general than
the above equations, and applied their results to solving dual and triple integral equations of the
form (38.16), (38.44).
36.1. Fox [3, 5, 6] developed a method of solving the integral equation
00
Kf(x) == f k(xt)f(t)dt = g(x), x> 0,
o
(39.12)
provided that the Mellin transform k.(s) = rot{k(x)j s}, - see (1.112) - of the kernel k satisfied
certain conditions.
On the basis of the convolution theorem in (1.116) for the Mellin transform the method of
solving (39.12) in L2(O, 00) was developed in the paper by Fox [3] provided that the fWlction
k. (s) satisfied the functional equation
n
. II r(a; + ("j + 1 - s)/m; )r(a; + ("; + s)/m;)
k (s)k.(1 - s) = .
j=l r«,,; + 1 - s)/mj )r«"j + s)/mj)
(39.13)
The simplest case when k.(s)k.(1 - s) = 1 is well-known, as for example, in the book by
Titchma.rsh [I, p. 401]. In the paper by Fox [3] the solution of (39.12) wit.h the kernel k(x)
satisfying (39.13) was expressed via the Erdelyi-Kober type operators defined in (18.1)-(18.4). For
example, if (39.13) holds with a1 = a, "1 = " and m1 = m, then the fonnal solution of (39.12)

 39. ADDITIONAL INFORMATION TO CHAPTER 7
781
has the fonn
00
f(x) = f (Ig+im;'1/mk)(xt)(Ig+;m;'1/mg)(t)dt
o
where the operator Ig+;U,'1 is given in (18.1). As an example, the solution of (39.12) with
k(x) = #xO/ C06(X - 0/1r/2) was given also.
In the paper by Fox [5, 6] on the basis of direct and inverse Laplace transforms defined in
(1.119) and (1.120) a method was developed by which a fonnal solution of (39.12) with the kemel
k( x) such that
n [ n ] -1
k.(s) = !! reO/jS + {3J) g rhl:s + 151:)
(39.14)
can be obtained. As examples, the solutions of (39.12) were obtained with k(x) = sin x,
k(x) = J.,(x) and k(x) = x" K.,(x) where J.,(x) and K.,(x) are the Bessel function of the first
kind defined in (1.83) and the Macdonald function given in (1.85), respectively. This method
was applied by Fox [7] and by R.U. Venna [8] to finding the inversion relations for the Vanna
transform defined in (9.6) and for the transform of the form (39.12) with the Meijer G-function
G:(x) - (1.95) - in the kernel, respectively.
36.2. It is said that k(x) and hex) fonn a pair of Fourier kernels if they satisfy reciprocal
inverse relations
00 00
f k(xt)f(t)dt = g(x), f h(xt)g(t)dt = f(x),
o 0
(39.15)
and each of them can be considered as the inversion expression of one another. IT h(t) '# k(t) and
h(t) = k(t) then the Fourier kemels are called non-symmetrical and symmetrical Fourier kernels,
respectively. Often more general relations than (39.15)
00 :&
f k1(xt)f(t) t = f g(t)dt,
o 0
00 :&
f h1 (xt)g(t) t = f f(t)dt,
o 0
(39.16)
:& :&
where kdx) = fk(t)dt, h1(X) = fh(t)dt are considered. They are reduced to (39.15) provided
o 0
that differentiation of the integrand is possible.
Investigation of the Meijer G-function as a symmetric Fourier kernel was first given by
Narain Roop [1]. These investigations were developed in the paper by Fox [2] where the new
general function of hypergeometric type called later the Fox H-function was first introduced and
studied. The theory of Fox H-function may be found in the paper by Braaksma [1] and in the
book by Mathai and Saxena [1] and in the handbook by Prudnikov, Brychkov and Marichev [3].
Problems connected with symmetrical Fourier kernels were also considered by Marichev
[10] and Verma R.U. [4]. Using a method based on the direct and inverse Laplace transforms
R.U. Verma [5, 6] reduced equations of the fonn (39.12) with G- and H-functions, respectively in
the kemels, to the equations with symmetrical Fourier kernels, and constructed the solutions of
such equations in closed fonn.
In the paper by Kesarvani [2-5] G- and H -functions were studied as non-symmetrical Fourier
kernels. We also note that Kesarvani [7] found necessary and sufficient conditions for functions

782 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
f(x), g(x) e L2(O, 00) to satisfy the dual equations (39.15) with
k(x) = hex) = "YJ1.1'/2x(1'-1)/2G'P 2 ( J1.X)1' I al,"', ap, -al,"', -ap )
p, q bl, . . . , b q , -bit. . . , -b q
where G2q is the Meijer G-function defined in (1.95).
In the case when h(t) = k(t) in (39.15) or in (39.16), functions f(x) and g(x) are called
k-transforms of each other. Mainra [1] and B. Singh [1] studied the classes of k-transforms
(39.15) in which Fourier kernels are respectively the generalized Watson functions w:::..:': (x)
expressed via multiple integrals of products of Bessel functions, and via the function formed
by the application the Erdelyi-Kober-type operators given in (18.1) and (18.3) to the Watson
function. Soni [2] proved that if f(x), g(x) e L2(O, 00), Rea > 0 and Re'1 > -1/2, then f(x) and
g(x) are k-transforms if and only if the functions It,af(x) and Kq,af(x) where It,a, Kq,a are
the Kober operators defined in (18.5), (18.6) are the same k-transforms.
36.3. A series of papers by Fox [2], Kesarwani [5], R.K. Saxena [2, 4], K.C. Gupta and
Mittal [I], Rattan Singh [I], Kalla [9], Buschman and Srivastava [I], Kumbhat [I], Nasim [1] are
concerned with finding inversion relations in Ll(O,OO) or in L2(0,00) of equation of the form
(39.12) with the Fox H-function or its special cases in the kernel. Solutions include the fractional
integration operators of Erdelyi-Kober-type given in (18.1) and (18.3) in some special cases -
R.K.Saxena [2], Kalla [9], Buschman and Srivastava [1]. As examples the inversion relations were
obtained for the Vanna transform given in (9.6), and for the Hankel and the Meijer transforms
(see  1.4).
In the papers by V.P. Saxena [I], Bhise and Madhavi Dinge [1] and Madhavi Dinge [1]
integral operators of the form (39.12) with the Fox H-function in the kernel were represented
as compositions of the EnIelyi-Kober-type operators defined in (18.1) and (18.3), and operators
of the form (39.12) with the Fox H-function of less order. R.U. Venna [7] formally constructed
the solution of a two-dimensional integral equation with a kernel which is a product of two Fox
H-functions of one variable.
We note that Raina and Koul [1,2] and Raina [1] proved that the fractional integrals defined
in (5.1) and (5.3) of Fox H-functions are H-functions also but of greater order.
36.4. Rooney [5] investigated the mapping properties on the integral transform
00
(KJ)(x) = j Gn ( xt I al,"', ap ) f(t)dt, x> 0,
bit . .. , b q
o
from the weighted space £"" = {I : [I"" 1(")1' "-I,/,, < 00,1 $ , < 00 } - see Rooney [3] -
onto £1-1-'," By using the Mellin transforms he obtained a characterization of the range of
the operator Kf in terms of the ranges of the Erdelyi-Kober-type operators given in (18.1) and
(18.3), and of the modified Hankel and Laplace transforms:
(39.17)
00
(H1c,'1 f )(X) = j (xt)1/1c-l/2 J'1(1kl(xt)l/1c)f(t)dt, Re'1 > -I,
o
00
(L1c,af)(x) = j(xt)-a e - I 1c I (zt)l / k f(t)dt, k = :1:1,:%2,..., x> 0,
o
under the appropriate assumptions on the parameters of Meijer G-function in the kernel. In

 39. ADDITIONAL INFORMATION TO CHAPTER 7
783
the case l' + q = 2m + 2n, the conditions and the expression for inversion of the equation
(Kf)(x) = g(x) were given.
Let
'H-, f(x) = x--,-1/2 Hl,-,[t-,+1/2 f( t)](x) ,
'Df is a fractional derivative in (5.8). Gasper and Trebels [4] proved the bOWldedness of the
operator 'D'H-, from £01+-,+1/2+1/", into £-,+1/2+1/"" and deduced necessary conditions
for 'H-,-multipliers of type (1',1').
36.5. A series of papers is concerned with the application of the Erdelyi-Kober-type
operators (18.2) and (18.4) to finding inversion relations for the integral transforms (39.12) with
special Macdonald fWlction KII(x) and Whittaker fWlction Wk,IJ(x) in the kernels.
Saksena [I], Fox [5], Okikioulu [3,6] and Manandhar [1] used the Erdelyi-Kober-type
operators defined in (18.1)-(18.3) and the Mellin transform given in (1.112) to finding inversion
relations in L,(O, 00) for the transform
00
x-, f (xt)0I-1/2 KII_I/2(xt)f(t)dt = g(x), x> 0,
o
(39.18)
in the cases l' = 2, "y = 0, 01 = V > 1/2; l' > I, v > 1/1', "y  1 - 2/1' > "y - v, 01 + 1 - 1/1' >
v > -01 + 1/1' and 1  l' < 00, veRI, 01 > Iv - 1/21 + 1/2 + 1/1" (1/1' + 1/1" = 1) if v > 0 and
01 > Ivl - 1/1" if v < 0, respectively.
Okikioulu [3] showed that the operator in (39.18) can be represented as the composition
of the modified Laplace transform and the ErdeIyi-Kober-type transform defined in (18.3), and
it is bOWlded from L,(O,oo) into Lq(O,oo), l/q = 1 - "y - 1/1' > O. He also proved that
integral transforms of the form (39.18) with replacing yOI-I/2 K II _ 1 / 2 (y) by yl/2-11 J II _ 1 / 2 (y),
yl/2-II Y 1 / 2 _ II (y) and yl/2-IIH1/2_II(y) where JIJ(Y), YIJ(Y), HIJ(Y) are Bessel and Struve
fWlctions - as given in (1.83), and Erdelyi, Magnus, Oberhettinger and Tricomi [2, 7.2.1, 7.5.4] -
are representable as compositions of the Erdelyi-Kober-type fractional integrals defined in (18.1),
(18.3) and the cosine- and sine- Fourier transforms given in (1.108), (1.109).
The Kober operators in (18.6) were applied by Swena [1] and R.K. Saxena [1] to obtaining
inversion relations for the Vanna transform given in (9.6). We also note that the inversion of the
Varma transform in terms of the direct L and inverse L -1 Laplace transform operators was fOWld
by Fox [7].
H .M. Srivastava [1] obtained the inversion relation for the transform
00
f (xt)II-1/2 e -zt/2W k + 1 / 2 ,IJ(xt)f(t)dt = g(x), x> 0,
o
(39.19)
which coincides with the Vanna transform in the special case J.l. = v. H.M. Srivastava proved that
if f(x) e L2(O, 00) is the solution of (39.19) and -v  k < v+ 1/2, then f(x) = L-l1jlI_kg(x)
where L-l is the inverse Laplace transform operator and ljlI_k is the ErdeIyi-Kober operator
given in (18.2). Other representations for the solution of (39.19) were given by Pathak [1].
We also note that McKellar, Box and Love [1] and Love [8] applied fractional integra-
differentiation to finding inversion relations for the integral transform of the form (39.12) with
the Struve function HII(z) in the kernel- Erdelyi, Magnus, Oberhettinger and Tricomi [2, 7.5.4].
36.6. K.J. Srivastava [1-3] applied the Kober operators defined in (18.5) and (18.6) to the
investigation of the generalized Mellin transform given in (9.7) and acting from Lp (0, 00) into
00
L,'(O,oo) (1  p  2,1/1'+1/1" = I), and to studying in L2(O, 00) the transform f wIJ,II(xt)f(t)dt
o

784 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
00
with the Watson function W#J,II(X) = x l / 2 f t- l J II (t)J#J(x/t)dt in the kernel, where JII(x) is the
o
Bessel function given in (1.83) - K.J. Srivastava [1] and [2], [3], respectively.
36.7. Marichev [1,2] considered the equation
:&
f (x - t)c-l ( , , x t )
J(t) r(c) F3 cv,a,{3,{3jcjI-7,I-; dt=g(x),
o
o  a < x < b  00, Rec > 0,
involving the Appel function F3 of two variables defined in (10.45) in the kernel. He obtained the
solution of this equation via the composition of three fractional integro-dift'erentiation operators
or via the operator with the function F3 in the kernel (see  10.3). Similar equations with the
Appel function Fl in the kernel were studied by Marichev and Vu Kim Than [1]. Vu Kim Tuan
[6] proved necessary and sufficient conditions for the existence and uniqueness of the solution of
the Abel-type equation
:&
f (x t)c-l ( t ) a
(c) ; #L(X, t)J(t)dt = g(x),
o
#L(X, t) being an analytic function, #L(X, x) '# 0, in the space of function J(x) e Qq (see  17.2,
notes to  10.1). He extended these results to equations containing Kummer, Gauss and Appel
hypergeometric functions: #L(X,t) = IFtCajbj,X(x - t», i''t(a,bjcjI- x/t), F 3 (a,a'jb,b'jcjI-
x/t.I - t/x).
36.8. H.M. Srivastava and Buschman [1] considered operators of the fonn
:&
x-'Y- a ! (x + at)a- l t'Y J(t)dt,
o
00
x 6 ! (t + bx)f3- 1 t - 6 -f3 J(t)dt,
:&
from which the Kober operators in (18.5) and (18.6) are derived if a = -1 and b = -1. They
proved that the composition of these operators is the integral operator with a homogeneous kernel
involving the Appel function Fl and, in particular, the Gauss hypergeometric function #'1'
36.9. Habibullah [1] investigated the integral operators
00
AJ(x) = x>' ! (xt)b-l-#Ha, bj Cj -xt)J(t)dt,
o
00
BJ(x) = x>' !(xt)O-IIFl(a,Cj-xt)J(t)dt,
o
00
C J(x) = x>' f (xt)o-I1fJ(a, Cj xt)J(t)dt,
o
where -#'1 is the Gauss hypergeometric function defined in (1.72), IF} is the confluent
hypergeometric function given in (1.81) and 1fJ is the Tricomi function - ErdeIyi, Magnus,

 39. ADDITIONAL INFORMATION TO CHAPTER 7
785
Oberhettinger and Tricomi [1,6.5]. He proved that the operators A and B can be represented
as compositions of the Erdelyi-Kober-type operators given in (18.1) and the generalized Stieltjes
and Laplace transforms, respectively, and the operator C can be represented as a composition
of the generalized Stieltjes and Laplace transforms (see  1.4 and  9.2, notes 7.3, 7.8). On the
basis of these results Habibullah proved the boundedness of the operators A, B and C from
L,(O, 00), p  I, into Lq(O, 00), l/q = 1 - l/p - .x  0, and found their inversion relations under
the appropriate assumption on parameters. We note that this type of result in another space of
functions was previously obtained by SWaI'OOp [1]. We refer also to Marichev [10, Sect. 8.2] with
respect to the operators A and B, and to Love [6] with respect to the operator A.
36.10. Let Ij;OI be the Kober operator defined in (18.5), let -#'1 be the Gauss
hypergeometric function given in (1.72) and let Tn be the integral operator (Tnl)(x) =
:&
(-I)n x l(x) + Ik(t/x)l(t)dt. EnJelyi [5] proved that if lex) e L2(O,OO), k(x) = [B(n, II +
o
1)]-lxll/2Hl - n, v + n - Ii f,C + Ii x), v> -I, then Tn can be represented in the form
(Tnl)(x- 1 ) = (1t/200l)-1 R1t/2,OIl(x), Rg(x) = x- 1 g(x- 1 ).
36.11. By using the fractional integrals x P 1+x'Y and x P lx'Y Marichev [6] proved relations
of compositional type which are special cases of (36.34). Analogues of (11.27)-(11.30) which
realize the connection between certain pairs of the integral operators with special functions in
the kernel and the singular operator are more interesting. We indicate two pairs of such a type,
namely
{J_ II (2VX)}1p = (COSV1I" + sin V1I"X- II / 2 Sx ll / 2 ){JII(2VX)}Ip,
(39.20)
{J II (2VX)}1p = {J- II (2VX)}(cosv1I" - sinV1I"X Il / 2 SX- II / 2 )Ip,
00
where IRevl < 1/2, {J II (2VX)}1p = I J II (2#)Ip(t)C 1 dt, and
o
(a, b)lp(x) = [C08C1r + ")'x c - a - b sx a + b - c + .xSh10+(a, b)lp(x),
(39.21 )
210+ (a, b)lp(x) = <t1(a, b)[eo8C1r - ")'x a + b Sx- a - b - .xxcSx-C]Ip(x),
where jl 0 +(a, b) (j = 1,2), Il(a,b) (k = 3,4) are the operators in (10.18)-(10.21) and
")'=
sin a 11" sin b1l"
sin(e - a - b) 11" ,
.x = sin(e - a)1I"sin(e - b) 11" ,
sin(c - a - b)1I"
e '# a + b,
00
(Slp)(x) = .!.. f Ip(t)dt .
11" t-x
o
These relations were previously obtained in Marichev [3]. All expressions of this kind hold for
sufficiently good functions, and can be proved by applying the Mellin transform defined in (1.112)
to both sides of the equation.
We note in addition that by using relations of such a type and the Mellin transform,
Marichev [7] indicated the class of complete singuJa.r equations with power-logarithmic kernels
solved by quadratures.
36.12. H.M. Srivastava [7] employed the Riemann-Liouville fractional integro-differentiation
in identities involving infinite series. He showed how such exercises culminate in linear and
bilinear generating functions for a wide variety of special functions.

786 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Saigo and Raina [1] and Saigo [8] evaluated the generalized fractional integrals and derivatives
given in  23.2, note 18.6 of a general class of polynomials with essentially arbitrary coefficients of
several elementary functions, and of the Bessel and the modified Bessel functions given in (1.83)
and (1.84). A similar problem for the two fractional calculus operator introduced by R.K. Saxena
and Kumbhat [2] and involving the Fox H-function was studied by Saigo, Kant and Koul [1].
37.1. A series of papers by Widder [3], Buschman [4], Khandeka.r [I], K.N. Srivastava
[9,10], Rusia [1,5], C. Singh [1,2,4], B.K. Joshi [I], H.L. Gupta and Rusia [1] were concerned with
the solution of special cases of (37.1) with the generalized Laguerre polynomials Lg(x) or with
the Whittaker functions M1c II(X) in the kernel. Methods based on the Laplace transform or on
reducing such equations to the Abel integral equation (2.1) were used.
By using the Laplace transform S.D. Gupta [1] solved the equation
:e
f eo(:e-t)(x - t)a L,II[(X - t)IJ]f(t)dt = 9(x), 0 < x < a,
o
involving the generalized Laguene polynomial L,IJ(z) in the kernel which is defined via the
Laplace transform by
r(mJL+a+1)
L{xaeo:e La (x), p} = (p _ a ) -a[l _ ( p _ o)-IJ]-m
m,IJ' r(mJL + 1) ,
a> 0, Re(p - a) > 0,
d. (37.19).
By using the Laplace transform H.M. Srivastava [3] obtained the inversion relation for the
integral equation more general than (37.1) with the confluent hypergeometric function of several
variables
00
«I2(at...., arj bjzt...., zr) = L
ml,...,mr=O
(at)ml ... (ar )mr z;nl
(b)ml+...+mr mi!
m r

mr!
He also indicated that this method can be applied to solving the general equation (4.2)
in the case when the Laplace transform of the kernel Lk(p) is representable in the form
Lk(p) = [(p - a)n(Lkt)(p)]-l where kdx) is a certain function.
37.2. By using the Laplace transform Kalla [2] and T.N. Srivastava and Y.P. Singh [1]
obtained the solution of the equations - see (37.2) - namely
:e
f (x 2 - t 2 )11 J II (.X(x 2 - t2»)e-b(x-t) f(t)dt = 9(x),
o
:e
f (x - t)1I Jp.(x - t)IJ)f(t)dt = 9(X),
o
where JII(x) is the Bessel function defined in (1.83) and J(x) is the Bessel-Maintland function
(see  23.2, note 18.2).
The method of fractional integro-differentiation was applied by Prabhakar [3] to solving the

 39. ADDITIONAL INFORMATION TO CHAPTER 7
787
integral equation
:&
1 (x - t),6-1 E:,,6[.x(x - t)]f(t)dt = 9(X), Re{3 > 0,
a
and a similar equation with variable lower limit where E:,,6(z) = f: k!f(:,6) ' Rea> 0, is
k=O
the generalized Mittag-Leffler function.
A method based on the Laplace transfonn was used by H.M. Srivastava and Buschman [2]
to find the solution of the most general convolution equation
:&
I (X - t)a-l H;J:t [ X - t I (ab Ad,..., (ap,A,) ] f(t)dt = 9(X)
(b1, 19 1),...,(b q ,19 q )
o
(39.22)
with the Fox H-function in the kernel. As examples, special cases of this equation with the
generalized Wright function , q -1, a special case of the Meijer G-function of the fonn G;: (x)
and the generalized hypergeometric function pF,-1 were considered. In this connection we also
point out the paper by Nair [I], who considered convolution equations with the generalized
Wright function p'iflq and the generalized Bessel-Mailtand function JC(x). In the paper by
H.M. Srivastava [2] a solution of the equation similar to (39.22) with variable lower and infinite
upper limits was obtain by using the Mellin transfonn. Also, as special cases, the equations with
the G-function of the fonn Gt(x), with the generalized hypergeometric function of the fonn
pFq(x), with the Whittaker function W>',#J(x) and with Bessel functions J.,(x), K.,(x) and y.,(x)
were considered.
37.3. Sonine [4,5] (1884) and also [6, p. 151], proved that the operators in (4.2) and (4.2')
with the kernels
k(x) = (2x)-' J_,(2i..ftY) ,
(2i..ftY)-'
(2x)p-lJ,_1(2JXY)
l e x ) 0 < P < I '
= (2y'XY)P-l '
(39.23)
and their special cases with p = 1/2 of the fonn
k ( ) _ cos(2iJiY)
x- v:;x ,
and also the operators with the kernels
lex) = cos(2JiY)
.;;rx
(39.24)
x-p e- lI :&
k(x) = r(1 _ p) ,
:&
lex) = 1 1 T,-2e-T:&dT.
r(p - 1)
-00
have the property given in (4.2") together with the fractional integro-differentiation operators
k(x) = xa-1/r(a), lex) = x-a 1r(1 - a), 0 < a < 1.
After the quadratic changes of variables and functions
x = (2, t = T 2 , a = c 2 , 2..[Y =.x, 2 1 -'Tf(T 2 ) = F(T), g«(2) = G«()i
x = a + b 2 - 2, t = a + b 2 - T 2 , 2i..[Y =.x, 2f(t) = .J;F(T), 9(X) = G()

788 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
the operators in (4.2) and (4.2') with the kernels (39.23) have the form
 ( ) -'I'
J  J_.(i). ,,/{2 - T 2 ) F(T)dT = G«),
c
T (  ) '_l
F(-r) = dd-r J -r 2 .x - 2 J,_1(.x V-r 2 - 2) G()d(,
c
o < p < 1,
which coincide with (37.41), (37.43) and the operators in (4.2') with the kernels (39.24) have the
fonn
b
J COS(.x)
 -rF(-r)d-r = G(),
V-r2 - 2

b
F(-r) = _2.. J m) (G«)d(.
1r-r d-r 2 _ -r2
T
The authorship of the latter expression is attributed to D.S. Jones [1] in some papers.
37.4. Rozet [1] obtained the inversion relation for the integral equation (39.12) in L2(0, 00)
under the appropriate assumptions on the kernel K(x, t). As an example he found relations for
the solutions of the equations
x
J Jl(2) J(t)dt = g("),
x - t
o
x
J In( vX(X + t»
.;x+i" f(t)dt = g(x),
( x + t)n
o
the first of which is the limiting case of (37.68) with a = 0, .x = 2i.
The inversion relation for the non-convolution equation (J+ ,J)(x) = g(x) with the operator
a , '"
given in (37.67) in the case a = 1 was obtained by Mackie [1]. The paper by Soni [4] is an
example of more general result (see  39.2, note 37.11 below).
By using the non-convolution operators given in (37.67) and the fractional integra-
differentiation operators defined in (5.1) and (5.8), Soni [1] solved the problem of the
00
characterization ofthe space of functions f(x) E L2(O, 00) such that xll/2 J J II (2..;;i)t- Il / 2 f(t)dt E
o
L2(O, 00) when v > O.
Lowndes [9,10] investigated properties of the operators
x
J ( X t ) (a-l)/2
l>'(fl,a)f(x) = 2a-l.xl-ax-('1+a) t'1 : Ja_l(.x VX(X- t» f(t)dt,
o
a  0, .x  0,
l>'(fl,a)f(x) = x m -'1 1>.(O,a + m)(d/dx)mx'1 f(x), a  0, .x  0,
where m is the positive integer for which 0 < a + m  I, and of the operators li>.(fl, a) defined

 39. ADDITIONAL INFORMATION TO CHAPTER 7
789
by the above relations with JOI-l being replaced by 101-17 where JII(z) and I I1 (z) are the Bessel
functions given in (1.83) and (1.84), d. (37.68).
37.5. In the paper by Pollard and Widder [1] the method of operational calculus was
applied to finding the solution of the equation
:&
/ k(x - t)/(t)dt = 9(x), x> 0,
o
the kernel k(u) = uAru exp (- iu) (u > 0) of which is connected with the solution of the heat
equation. The solution of this equation is expressed in terms of the Riemann-Liouville fractional
integra-differentiation operators of order 1/2.
37.6. Let J>.('1,OI) and R>.('1,OI) be the operators in (37.45) and (37.46) and let
Lp,1I == Lp([O,oo);x IlP - 1 ) be the weighted space. IT 01  1/2,1 < p < 00, then the operators
J>.('1,OI) and R>.('1,OI) are bounded in Lp,lI when v < 2 + 2'1 and v > -2'1, respectively. IT
o < 01 < 1/2, 2/(1 + 201) < p < 2/(1 - 201), then J>.('1, 01) and R>.('1, 01) are bounded in Lp,1I if
v < min(I, 2/p) + 201 + 2'1 and v > max(I,2/p) - 201 - 2'1, respectively - Heywood and Rooney
[1] and Heywood [3].
37.7. Following the papers by Prabhakar [1,2] and using q-fractional integrals (see  23.2,
note 18.15) Prabhakar and Chakrabarty [1] obtained the inversion relations for the q-integral
analogue of (37.1) with IF} being replaced by the basic confluent hypergeometric function l()l in
the kernel (see Slater [1]).
37.8. Pinney [1] considered the integral equation
00
/ p(t)Ip( '; x 2 + t 2 )dt = I(x), x> 0,
o
(39.25)
and found its solution
00
Ip(x):::r. _.!. / 1( ';X 2 + t 2 )q(t)dt
xdx
o
1(/2
provided that for a function p(x) there exists the function q(x) such that f p(rcos Ip)q(r sin Ip)dlp= 1
o
for any r > O. He also obtained sufficient conditions for the truth of the latter relation, which is
an analogue of (4.2") for the Sonine equation (4.2). He also considered the following special case:
pet) = t l - 2IJ L;IJ(kt 2 ), 0 < Re < I,
where
. 00 k
L-IJ ( z ) = -r ( vI ) '" rck - v) z
II 11"  + + L..J r(k +  + 1) k!
k=O
is the generalized Legendre function - the Legendre polynomial LIJ(z) for the natural v = n. In
particular, pet) _ t l - 2 IJ as v _ 0, and
pet) = t 1 - IJ J_IJ(OIt), 0 < Re < I,
where J- IJ (z) is the Bessel function of the first kind given in (1.83); in particular, pet) = cos OIt
if  = 1/2. In the case pet) = t l - 2IJ , 0 < Re < I, the connection of (39.25) with the Abel

790 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
equation (2.1) was shown and when pet) = cosat (39.25) was reduced to the equation
00
1 c06 j(t)dt = g(x), x> O.
t - X
:&
37.9. Sani [3] found the integral representation
:&
1 (x - t)"/2JI [2 Vk(x - t)] j(t)dt
o
:& [ t ]
k"/2 d
= rev + 1) ! (x - t)" di! JO [2 v' kIt - T )] J( T )dT d'
for the operators of the form (37.4), and proved the necessary and sufficient conditions of
solvability in L2 (-00,00) for the equation
:&
d 1 Jo[2 V k (x - t)] j(t)dt = g(x), -00 < x < 00,
-00
and obtained its solution in the form
00
j(x) = _.:!... 1 Jo[2 Vk(t - x)] g(t)dt.
dx
:&
Soni [3] also investigated the conditions for the uniqueness of the solution of the equation
:&
.:!... 1 Jo[2 Vk(x - t)] j(t)dt = g(x), a> -00,
dx
a
in L2(a, +00), because the corresponding homogeneous equation (g == 0) can have a non-trivial
solution in L2(a, +00). Similar investigations were done for the operators with variable lower
limit - see also Bouwkamp [1]. By using the method of representation via the composition of two
equations of the form (37.4), Williams [1] solved (39.11) with the kernel
00
K(x,t) = 1(-r 2 - b 2 )r-r-m-n-1Jm (x-r)J n (t-r)d-r.
b
We note here that other integral equations with the Bessel function J II (x) in the kernel
considered on the whole line arise in applications. Two such equations can be obtained from
(40.26) and (40.27) if we carry out differentiation with respect to y for y = 0 and take (40.45) and
(40.44) into account in the former relation and if we set y = 0 and take (40.32) and (40.47) into
account in the latter relation. Then for J.l. = 0 after the substitution -rex) = 129(X) with Ipl < 1/2,
p ::/= 0, we obtain the relations

 39. ADDITIONAL INFORMATION TO CHAPTER 7
791
00
f It (2' J_,(.xlt - xDdt = 9(X), -00 < x < 00,
-00
00
ctgp1r f signet - x) d -
vex) = -- I 1 1 2 [g(t)J p _1(.xlt - xl)]dt,
211" t - x -, dt
-00
where JII(z) is the Bessel-Klifford function _given in (37.8). One may check that the second
relation inverts the first under appropriate assumptions on the function 9(X).
37.10. Let KII(z) be the Macdonald function defined in (1.85) and let YII(z) be the Bessel
function of the second kind - Erdelyi, Magnus, Oberhettinger and Tricomi [2, 7.2.1]. By using
the Mellin transform and results 21 and 20 from Table 9.1 V.K. Vanna [1] found the solution
of equations similar to (37.4) with variable lower and an infinite upper limits, in which J II is
replaced by K II and by Y II:
00
f (t - x)(11-1)/2 K II (2..;t=X)f(t)dt = 9(X),
:&
00
f (t - x)(1I-1)/2YII(2..;t=X)f(t)dt = 9(X).
:&
37.11. By using the method of reducing to a boundary value problem for the partial
differential equations
8 2 <p
8x8y + c(x,y)<p = 0, c(x, y) = c(y,x), <p(x,O) = f(x), x> 0,
<p(O,y) = -fey), y > 0,
(39.26)
in the first quadrant x > 0, y > 0 Mackie [1] obtained a solution of the integral equation
:&
f K(x, t)9(t)dt = f(x), 9 = !. ( 8<P _ 8<P ) I
2 8x 8y :&=11
o
where K(x, t) = R(x, Xi t,O) and R(x, Xoi t, to) is the Riemann function of (39.26) - Definition
40.2. As examples (37.67) with cv = 1 and the equations
:&
f Pn(x/t)<p(t)dt = 9(X),
o
:&
f Pn(t/x)<p(t)dt = 9(X)
o
with the Legendre polynomials Pn(z) in the kernels were solved, d. the equation (39.3).
37.12. Prabhakar [5] considered the equations
:&
f (x t)c-1
(c) (11 (a, hi Ci 1 - x/t, .x(x - t»f(t)dt = 9(X), cv < x < {3,
cv
(39.27)

792 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
:&
f (x - t)c-l
r(c) «11 (a, bj Cj 1 - tlx, ,x(x - t»f(t)dt = g(x), cv < x < (3,
cv
(39.28)
where «IlCa,bjcjx,y) is the Humbert function given in (10.64), cv > 0, (3 < 00. In particular,
if ,X = 0, then (39.27) and (39.28) coincide with the Love equations (35.1) and (35.2), and if
b = 0 they then coincide with the equations of the form (37.1). Prabhakar fOWld the criterion of
solvability for (39.27) and (39.28) in LlCcv, (3) and obtained their solutions in the form
f(x) = e>':&x- b I;;e->':&xb I:;Cg(x) , f(x) = I:;ce>':&x b I;e->':&x-bg(x)
respectively. Here 1:+ is the Riemann-Liouville fractional integro-differentiation operator defined
in (2.17). Similar results were proved by Prabhakar [6] for the equation of the form (39.27) and
(39.28) with variable lower and finite upper limits and for more general equations. For example,
for the equation
(j
f [bet) - h(x)]c-l ( hex) )
r(c) «11 a, bj Cj 1 - h(t) , 'x(h(x) - h(t» f(t)dh(t) = g(x),
:&
cv < x < (3,
where h(t) e COO[cv, (3], h'(t) > o. In particular we note that if ,X = 0 and h(t) = t m , then the
latter equation is reduced to (39.6) and if b = 0 and h(t) = t, then it is reduced to (37.1).
37.13. Tanno [1] investigated the inversion problem for the convolution integral transforms
00 ioo
J k(x - t)f(t)dt = g(x) where k(x) = k J [l(t)]-le:&tdt and 'et) is a meromorphic function
-00 -ioo
with real zeroes and poles. He also considered such a problem for more general transforms. The
book by Hirschman and Widder [1] should be consulted in this connection. As examples, solutions
of the Abel equation (2.1), the Higgins equation (39.1) and the equation of the form (39.4) were
obtained.
37.14. R.K. Saxena and Sethi [1] fOWld the operational relations similar to (37.60) and
(37.61), and connected the generalized Hankel transform operator given in (37.47) with the
modifications of the operators in (23.5) and (23.6).
37.15. Tedone [1] considered the integral equation
:&
f (x - t)n [m(X - t)f(t)dt = g(x), m  0, m + n  0,
a
(39.29)
with the modified Bessel function given in (1.84) in the kernel. On the basis of properties of the
function Im(z) he proved some difference-recurrence relations for the operator in (39.29). This
was used to reduce the solution of (39.29) with m, n = 0,1,2, . " to the successive solutions of
simpler equations with m = n = 0 -  39.1, Note to 36.1 - and the differential equation. Also
(39.29) was reduced to the solution of the VoltelTa integral equation with a difference kernel if
m = 0,1,2,..., n = -1,-2,..., m + n > O.
37.16. Yakubovich, Vu Kim Tuan, Marichev and Kalla [1] proved a theorem similar to
Theorem 37.15 in the space L2(R) and considered the special cases of the W-transform given
in (37.107).
The Kontorovich-Lebedev transform defined in (37.109) and the Mehler-Fode transform
given in (37.141) of generalized functions were investigated by Glaeske [1,2] and Glaeske and Hess
[1-3].

 39. ADDITIONAL INFORMATION TO CHAPTER 7
793
38.1. When investigating the special case of (38.1) Lebedev [2] obtained the Shlemil'ch
equation
r /2
f  ;: - f <p(rc068)d8 = g(r)
r 2 - t 2
o 0
containing the fractional integral of order 1/2 by the fWlction x 2 -  18.2. The inversion relation
of this equation was used by Lebedev and Uflyand [1].
38.2. In the case G = 0, J.I. = v Tranter [3] represented the solution of (38.1) as a series
00
L: ak J II+2k+n (x) of Bessel fWlctions given in (1.83), and as a result he reduced this problem to
k=O
solving the infinite system of linear equations with respect to ak'
Cook [1] used the integral analogue of such a method and reduced this system to Fredhohn's
integral equation of the second kind.
Nasim and Aggarwala [1] suggested a method for solving (38.1) based on its decomposition
into two systems of more simple dual equations with F(x) = 0 and G(x) = O.
38.3. Walton [1,2] studied the problem of the uniqueness for the solution of the dual
equation (38.1) in the space of generalized functions.
38.4. Buschman [6] and Kesarvani [6] indicated that by using the Erdelyi-Kober-type
fractional integration operators given in (18.1) and (18.7) and the Mellin transform defined in
(1.112) the dual integral equation (38.1) with R(t) = 0 and the dual integral equation with Meijer
G-functions in the kernel, respectively can be reduced to a single equations of the same form
given on the half-line [0,00).
By using the Mellin transform Nasim and Sneddon [1] investigated dual equations with
kemels of general form. As an example, dual equations of the form (38.1) with different Bessel
fWlctions and trigonometric functions in the kernels were considered.
38.5. By using two-dimensional analogues of Erdelyi-Kober operators -  29.2 (note 24.2)
- Makarenko [1] studied dual and triple integral equations with Bessel fWlctions in the kemels.
Using the method in Example 38.2 Virchenko and Gamaleya [1] considered the system of dual
equations of the form (38.16).
38.6. By using fractional integro-differentiation Sethi and Banerji [1] reduced equations
more general than (38.16) to the Fredhohn equation of the second kind.
38.7. The system of dual equations associated with the Mehler-Fock transform and reduced
to (38.22) when 9 = J.I. = 0 and H(T) = th'T1r/T was first investigated by Grinchenko and Ulitko
[1]. Using fractional integrals of order 1/2 by the function chx, they reduced such a system to
the Fredholm integral equation of the second kind.
Two systems with the same kemels and two systems with trigonometric kemels were solved
by Babloyan [1] by using a similar method. A system of the form (38.22) with the integer J.I. was
first investigated by Ruhovets and Uflyand [1]. These results were developed by Virchenko and
Makarenko [1].
We also note the paper by Lebedev and Skal'skaya [1] who studied the system (38.22) with
J.I. = 0 and H(T) of a special form and obtained the equation with the Gauss hypergeometric
fWlction
:&
f ;( t l (a,-a j jl -7) dt = g(x)
1
instead of fractional integrals. They used the result by Love [2] ( 35.1) for the inversion of the
last equation.
Investigating three dual equations with the generalized associated Legendre fWlction in the
kernel Virchenko [1] used the inversion relation for the more general equation with the Gauss
hypergeometric fWlction

794 CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
x
f 1 ( chx - ch t )
<p(t)(chx-cht)C- l a,bici dt=9(X),
chx+d
a
C > O.
S.P. Ponomarenko [2] considered the system. of dual equation ohnatrix type pIJ [A j ( 'T)" ( 'T)] =
f(;)(x), j = 1,2, and using (38.24)-(38.27) she reduced this system to a system of Fredhohn
equations of the second kind.
Together with the equations (38.22) similar equations associated with the inverse Kontoro-
vich-Lebedev transform were considered in some papers. Dual and triple integral equations of
such a type were investigated by Lowndes [3] and the former equation was studied as the limiting
case of the latter equation. One case of dual equations was solved in closed form, and other ones
were reduced to the Fredhohn integral equation of the second kind.
38.8. Equation (38.31) generalize the systems from the papers by Cook [3] and Borodachev
[1] and intersect with the system from the paper by S.P. Ponomarenko [1]. All these systems are
reduced to Fredhohn equations of the second kind by using Erdelyi-Kober-type operators.
38.9. Virchenko and Makarenko [3] and Makarenko [1] considered dual and triple integral
equations with kernels more general than JII(z) namely the Watson function WIJ,II(X) -  39.2
(note 36.6.) above. These equations were reduced to Fredhohn equations of the second kind.
38.10. The method given in  38.1 (Example 38.1) was used by V.irchenko [2] to find the
inversion relations for two dual equations with the functions l and #'2 in the kemels,"by their
reduction to the equation Af = 9 where A is the operator in  39.2 (note 36.9).
38.11. K.N. Srivastava [11] and Virchenko and Romashchenko [1] considered triple integral
equations of the form (38.22) with #A. = 0 and arbitrary #A. respectively. The result from the paper
by Pathak [2] was generalized in the latter paper.
By using the Erdelyi-Kober operators given in (18.5) and (18.6), Lowengrub and Walton
[1] and Walton [3] reduced the system of triple equations of the form (38.31) to a system of
generalized Abel integral equations -  34.2 (note 30.9).
Lowndes and Srivastava [1] reduced a certain class of triple series equations involving the
generalized Laguerre polynomials. to the triple integral equations of the form (38.31) and obtained
an exact solution of these triple series equations.
38.12. Quadruple and n-tuple integral equations were considered in some papers. Using the
method of Cooke [2], Ahmad [1] investigated the 4-tuple equations with the functions J II (z) in
the kernels. The method used in fi 38.1 when solving the system (38.16) was applied by Dwivedi
and Trivedi [1] to similar triple and quadruple equations.
A further generalization was made by Vu Kim Tuan [5] who considered n-tuple equations
with the Meijer G-function in the kernels of the form
00
d f Gm,li+l ( I o ,(a.>,(cp» ) ) d ( )
dx x P+1i+I,Qi+m+1 xy (b m ), (i)' -1 fey y = 9i x ,
o
ai-l < x < ai,
00
d f mi,l+l ( 1 1, (al), (4J ) - . (
x dx G pi +1+I,Q+mi+ 1 xy (b;"J, (dq),O f(y)dy - 9. x),
o
where 0 < ao < al < ... < an-l < an = 00 is the disjunction of the half-line [0,00) into
intervals, 9i(x) are the given functions and the remaining parameters satisfy the appropriate
assumptions. Vu Kim Tuan proved that this system is solvable in the space f(x) e L2(O, 00) if
and only if 9i e L2(ai_lt ai), i = 1,2,..., n. In this case a unique solution was found in closed
form. Vu Kim Tuan used the methods from the papers by Fox [4] and R.K. Saxena [3,4] where
solutions of dual equations with functions more general than the Fox H-function in the kernels
were formally constructed, and the Erdelyi-Kober operators were applied.
ai-l < x < ai,

Chapter 8. Applications to
Differential Equations
In this chapter we shall be concerned with the applications of fractional integro-
differentiation to the investigation of partial differential equations of the Euler-
Poisson-Darboux-type, and of ordinary differential equations of fractional and
integer order. In particular, the Cauchy, Dirichlet and Neumann singular initial
value problems are solved in closed form, and the existence and uniqueness
theorems for certain types of differential equations of fractional order are proved.
The resulting solutions have wide applications to Tricomi-type boundary value
problems for partial differential equations of the mixed type.
fi 40. Integral Representations for the Solution
of Partial Differential Equations of Second Order
via Analytic Functions and Their Applications
to Boundary Value Problems
The present section deals with certain aspects of the general theory of elliptic
equations with anaJytic coefficients as developed in the book by Vekua [3] (1948),
and their applications to the so-called generalized Helmholtz two-axially symmetric
equation. We prove that if the Erdelyi-Kober operator 1",01 defined in (18.8),
and the generalized Erdelyi-Kober operator J>.('1, Q) defined in (37.45) are applied
to analytic functions in each variable, then we obtain the solutions of the above
Helmholtz equation. These results are applied to the solution of certain boundary
value problems.

796 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
40.1. Preliminaries
Let there be given the following general homogeneous partial differential equations
of the second order with two independent variables and with analytic coefficients
Hu == u('1 + a(, 7])U( + b(, 7])U'1 + c(, 7])u = 0,
Hl U == U zz - U yy + Al(X, y)u z + Bl(X, y)u y + CI(x, y)u = 0,
(40.1 )
( 40.2)
( 40.3)
Ku == U zz + U yy + A(x,y)u z + B(x,y)u y + C(x,y)u = 0,
and equations of the form
H.v == v('1 - (a(,7])v)( - (b(,7])v)'1 + c(,7])v = 0, (40.4)
K.v == Vzz + V yy - (A(x,y)v)z - (B(x,y)v)y + C(x,y)v = 0, (40.5)
conjugate to (40.1) and (40.3). Generally speaking we assume that the complex
variables x, y,  and 7] are connected with each other by the relations
 = x + iy, 7] = x - iy,
( 40.6)
x = ( + 7])/2, y = ( - 7])/2i.
We denote by t the number conjugate to  and note that t = 7] holds only in the
case when x and y are real.
We introduce the differential operators
a l ( a .a ) a l ( a .a )
a = 2" ax - " ay ,. a7] = 2" ax +" ay
(40.7)
In particular, the following formal relation is obvious namely
a 2 a 2 a 2
4 aa7] = ax2 + ay2 = A2
( 40.8)
where A2 is two-dimensional Laplacian.

 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
797
If the coefficients of (40.1 )-( 40.3) are connected with each other by the relations
4a(, 7]) } = A (  + 7]  - 7] ) :!: 'B (  + 7]  - 7] )
4b( , 7]) 2 '2i I 2' 2i '
(  + 7]  - 7] )
4c(,7]) = c '2i '
( 40.9)
A 1 (x,y) = A(x,-iy), B 1 (x,y) = iB(x,-iy),
C 1 (x, y) = C(x, -iy),
then (40.1)-(40.3) and (40.4)-(40.5), respectively can be transformed into each
other by the changes given in (40.6) and replacing y by iy, while (40.2) becomes
(40.3). Nevertheless, for real , 7], x and y the above equations are classified in
the following way. Equations (40.1), (40.4) and (40.2) are called hyperbolic-type
equations in the first and second canonical forms, respectively, and (40.3) and
(40.5) are called elliptic-type equations.
Let  = x + iy be a point lying in a certain simply connected domain  in
the complex plane C, x and y themselves may be complex. Then we denote by i>
a domain containing the point 7] connected with  by (40.6). Further we denote
by (, i» a (Cartesian) product of  and i> called a cylindrical domain. We
introduce the following definition:
( 40.10)
Definition 40.1. A simply connected domain  in the complez plane C is called
a basic domain of (40.1) and (40.9) if their coefficients a, b, c are analytic with
respect to variables (, 7]) E (, i», the functions A, B, C are analytic in (x, y) E 
and the relations in (40.9) hold.
We indicate the most important representation relations via analytic functions
for the solution of (40.3), which will widely used below in special cases. First such
a relation uses the concept of a Riemann function.
Definition 40.2. A solution v = R(, 7]) = R(, 7];0, 7]0) of the Goursat boundary
value problem for the conjugate equation (40.4) with the conditions
"
R 1(=(0= exp f a(o, t)dt,
"0
(
R 1"="0= exp f b( r, '1o)dr
(0
(40.11)

798 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
is called the Riemann function of the operator H given in (40.1) corresponding to
the point (o, 7]0) E .
We note the following properties of the Riemann function:
a) if a, b E Cl(), C E C(), then the Riemann function R of the operator H
in the domain  exists and it is an unique;
b) the relations R,,(o, 7]) = a(o, 7])R(o, 7]) and Rd, 7]0) = b(, 7]o)R(, 7]0)
hold on the characteristics  = o and 7] = 7]0;
c) the normalization condition R(o, 7]0;0, 7]0) = 1 holds;
d) the reciprocity condition R.(,7];o,7]o) = R(o,7]oj,7]) is valid, i.e. the
function R( , 7]; o, 7]0) with respect to the last variables o and 7]0 is the Riemann
function R. of the conjugate equation (40.4) corresponding to the point ,7].
More detailed information about the Riemann function and its tables in
different special cases of equations of the form (40.1) may be found in the books
by Babich, Kapilevich, Mihlin, Natanson and others [1], Kosljakov, Gliner and
Smirnov [1] and in the paper by Andreev, Volkodavov and Shevchenko [1]. Other
references are also cited there.
The following statement is true - Vekua [3, p. 31].
Theorem 40.1. Let  be a certain basic domain of (40.9). Then all solutions of
(40.3) regular in  - that is, those expanded in the Taylor's series in  - can be
represented in the form
 (
u(x,y) = aoR(o,7]o;,t) + f (t)R(t,f/O;,t)dt + f .(T)R(o,T;,t)dT,
o qo
(o, 7]0) E (, i».
(40.12)
Here ao is a certain constant and () and .(7]) are certain functions analytic in
 and  respectively, and uniquely defined by means of u(x, y) provided that the
relations in (40.6) hold.
The following statement gives several similar integral representations for the
solution of (40.3) by means of functions satisfying conditions simpler than those
for the Riemann function.
Theorem 40.2. Let "Y(, 7], 8) be a function analytic in the cylinder (,,)
where  is a certain basic domain of (40.1), and let this function satisfy (40.1)
with respect to  and 7]. Let in addition rp() be an arbitrary function analytic in
, and let all integrals given below exist. Then the integral
u(x, y) = f "Y(, t, 8)rp(8)d8
L
( 40.13)

 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
799
along any closed contour L lying in  satisfies (40.3) provided that the conditions
in (./0.6) and (./0.9) hold. If in addition the relation
lim [ a a "Y(,7J,8) +a('7J)"Y(''1,8) ] = 0,
8- 71
J [ :( "Y({, '1,8) +b(0/J'r({, 'I, 8)] = 0,
( 40.14)
( 40.15)
respectively hold, then the integrals of the form

u(x, y) = f "Y(,,8)cp(8)d8,
o
(40.16)
t
u(x,y) = f "Y(,,8)cp(8)d8,
to
( 40.17)
respectively, where o lies on the boundary of, are regular solutions of (40.3) in
.
Remark 40.1. Theorem 40.2 with the conditions  = x + y,  = x - y, and the
conditions in (40.10) instead of those in (40.6) and (40.9) is also true with respect
to (40.2). In this case the conditions for the functions "Y(,, 8) and cp() to be
analytic may by weakened up to the existence of their continuous derivatives of the
second order.
Remark 40.2. The following functions

R(8, '10;, '1), '10 E ; f R(t, '10;, 7J)(t - 8YJt- 1 dt, Rea > 0,
8
are two examples of a function "Y(, '1, 8) satisfying the conditions in Theorem 40.2.
Theorem 40.3. If A, Band C are analytic in , then any solution of (40.3)
regular in  is also analytic in  with respect to x and y.

800 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
40.2. The representation of solutions of generalized Helmholtz
two-axially symmetric equation
In this subsection we obtain different integral representations for the solutions
of two mutually connected cases of (40.3) and (40.2), namely the generalized
Helmholtz two-axially symmetric equation
>. _ 2p 2p 2
H p . 1J U = UJ:z + U yy + -;-U z + yU y + A U = 0, p, p., A - const,
( 40.18)
and the corresponding hyperbolic equation
>. _ 2p 2p. 2_
h p . lJ u = U zz - U yy - yU y + -;-U z + A U - 0,
( 40.19)
obtained from (40.18) after replacing y by -iy. We also consider some special
inversion relations for above equations.
In the special case p = A = 0 (40.18) is the so-called equation of axially
symmetric potential theory. This is discussed in more detail in S 41.1 and S 43.1,
Notes to U 41.1 and 41.2. Here we only note that (40.18) itself arises, with the
special values x = X, y = Y, 2p. = m - 1 and 2p = n - 1, if we need to find the
monochromatic solution of the form
U = u(X, Y, t) = U1(X, Y)e:i:i>'t,
X 2 = x + x + . . . + x;', y 2 = y + y + . . . + y
for the D'Alambert wave equation
m {Pu n {J2u 1 {J2u
L ox 2 + L oy2 - A2 ot 2 = 0
k=l k k=l k
( 40.19')
in the space with co-ordinates (Xl, X2,"', X m , Yb Y2,... Yn) and time t.
The possibility of an effective construction of the solution of (40.18) is enabled
by two remarkable properties of the operator H :.1" There are "the relations of
conformity"
H>' ( x l - 2IJ u ) = x l - 21J H>' U
P.IJ p.1-1J '
( 40.20)
H>' ( yl- 2p U ) = yl-2p H>' U
'1'.1' 1-'1'.1' '
and a compositional representation for the solution of (40.18) by means of the
Erdelyi-Kober operators, defined in (18.8) and (37.45), of harmonic functions even

 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
801
in x and y. This representation follows from the relation
I (Z) [ (y) J (z) (O 0) A I H >. [ (Z) I (y) J (Z) (O 0)1
-1/2,1-' -1/2,'1' >. , u = p,1-' -1/2,1-' -1/2,p >. , ,
( 40.21)
where the indices x and y in the above operators imply a variable to which these
operators are applied. According to (40.20) each solution u of (40.18) corresponds
to another solution of this equation obtained from u by replacing fl by 1 - fl, or
p by 1 - p, and by multiplying by x 1 - 2 1-' ,or y1-2p, respectively. It follows from
(40.21) that if I is harmonic, i.e. satisfies the Laplace equation AI = 0, then
the function u = H;'I-'I{/2'I-'IY{/2,pJiZ)(0, 0)1 is the solution of (40.18). Similar
properties are also true for (40.19).
Equations (40.20) is easily verified by direct evaluation. Equation (40.21) can
be proved as a direct corollary of Lemmas 40.1-40.2 given below. These lemmas
characterize the application of the Erdelyi-Kober-type operators to the differential
operator
L(Z) = x-2'1-1 x2'1+1  =  + 2TJ + 1 ,
'1 dx dx dx 2 X dx
(Z) d 2
L_1/2 = dx 2 '
( 40.22)
These lemmas as well as Lemma 40.3 below were proved in the papers by Lowndes
[7,5,9] - see also Erdelyi [8, 10] and S 43.1, Notes to SS 40.2 and 40.3 and S 43.2,
note 40.1.
Lemma 40.1. Let I(x) E C 2 (0, b), b> 0, and 1(0) = 1"(0) = O. Then
J1 S )(0, O)/I/(z) = (::2 + 2) J1 s )(0,0)/(z).
Lemma 40.2. Let cr > 0, I E C2(0, b), b > 0, x 2 '1+1 I(x) is integrable at zero, and
x 2 '1+1 I'(x) --+ 0 as x --+ O. Then
JiZ)(TJ, cr)Lz) I(x) = (L1.OI + .x 2 )Ji z )(TJ, cr)/(x)
and, in particular, if A = 0, then I'1,OILz) I(x) = L101I'1,0I1(x).
Lemma 40.3. Let cr > 0, I(x) E C 2 (0, b), b> 0, and x'1+1: 1(1:)(x) with k = 0,1,2
are integrable at zero. Then
(x-'1- OI la,>.x'1)x 2 L<:'1I(x) = x2(L<:'1 + .x 2 )(x-'1- 0I 101,>.x'1)/(x),

802 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
where J a ,>. is the operator obtained from the operator in (37.68) by replacing j by
J.
Lowndes [9] proved that by using Lemma 40.3 we can find the full system of
solutions of (40.18) in a neighbourhood of the origin in the form
Un(x, y) = Anr- IJ - P J2n+IJ+p('xr)PP-1/2,IJ-1/2)( cos 28),
x = rcos8, y = rsin8, An - const, n = 0, 1,2,...,
( 40.23)
where JII(z) is the Bessel function given in (1.83), and pOJb)(z) is the Jacobi
polynomial - Erdelyi, Magnus, Oberhettinger and Tricomi [2, 10.8]. Lowndes used
the known result for (40.18) with ,X = 0 obtained by the method of division of
variables in polar coordinates. In particular, we also note that in the case of (40.18)
the system in (40.23) has the form Anr-PJn+p('xr)C(cos8) and AnrnC(cos8),
where C(z) are Gegenbauer polynomials if p = 0 and p = ,X = 0, respectively.
However, to find integral representations for the solutions of (40.18) it is more
suitable to use another approach based on Theorem 40.2 and (40.21). We denote
by r(x, y, t) a special solution of (40.18) connected with the solution 1'(, 1], t) from
Theorem 40.2 by means of (40.6). Using the relation in (40.21) and its special cases
when pp'x = 0, we can conclude that this solution should be found in the form
r(x,y,t) = x O y b p C 8(x,fj)
( 40.24)
where x = -p/( 4xt), fj = _,X2 p/4, p = (x - t)2 + y2, and a, b, c must be defined.
Substituting (40.24) into (40.18) and carrying out some operations we arrive at
an equation in partial derivatives of second order with respect to 8(x, fj). This
equation is a linear combination of two equations of the form 5.9(29) from Erdelyi,
Magnus, Oberhettinger and Tricomi [1], with 8 = =2 and the coefficients x- 2 and
,X2, in two cases only connected by means of (40.20): namely
1) a = -p, b = 1 - 2p, c = p - 1, Ot{3 = p(1 - p), Ot + (3 = 1, l' = Pi
2) a = -p, b = 0, c = -p, Ot{3 = p(1 - p), Ot + (3 = 1, l' = 1 - p.
So we conclude that 8(x, fj) = =2(Ot, (3; 1'i X, fj) where
...  (Oth ({3)1: xl: y'
=-2(Ot,{3;1'i x ,y) =  () kill '
1:,1=0 l' 1:+1 .,
Ixl < 1,
( 40.25)
is one of the Humbert confluent hypergeometric functions with the parameters
Ot = P and (3 = 1 - p. Integral representations for such a function in terms of the
Legendre function P:(z) = PII(z) given in (1.79), and the Bessel-Clifford function

 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
803
given in (37.8) were obtained by Kapilevich [1, p. 1243] in the form
1
=2(11,1 - 11; "'(; x, y) =( "'( - 1) 1(1 - tp-2 P _#(1 - 2xt)
o
x I-Y-2(2 v' (1 - t)y)dt,
( 40.25')
Re"'( > 1.
It should be noted that if xt < 0, 11 f; 0 then the variable x = -p/(4xt) can
be on a cut - a singular line - of the Humbert function. To exclude this we
multiply the special solution in (40.24) with the parameters indicated in 1) by
the piecewise constant multiplier It r(t)lt 1# (sign xt + 1)lsi g n#l, where OOI, It is a
constant. Then we integrate the expression obtained respect to the parameter t in
accordance with (40.13). As a result we find the function
( ) _ I 1 1 _# 1-2, 1 00 r(t)ltl#(signxt + 1)lsign#1
u x, y - 1 X Y [(x _ t)2 + y2]1-'
-00
..... ( P >.2p )
x .::. 2 H 1- H' p ' -- -- dt
,-, ,-" 4xt' 4 .
( 40.26)
Similarly on the basis of 2) we obtain the second function
00
-# 1 lI(t)ltl#(signxt + 1 )Isign #1
u(x, y) =hlxl [(x _ t)2 + y2],
-00
..... ( P >.2p )
X'::'2 JJ,1 - 11; 1 - p; - 4xt ' - 4 dt,
( 40.27)
which is obviously the solution of (40.18) as well as the first one. We note that if
p = 0, -1, -2,... and p = 1,2,... respectively, then the functions in (40.26) and
(40.27) are not defined, and a special approach is required to constructing solutions
in these cases.
One may satisfy oneself by direct verification that the constructed solutions
of the form (40.24) satisfy the corresponding condition in (40.14). Therefore on
the basis of (40.16) and Remark 40.1 after replacing y by -iy and the interval
-00 < t < 00 by 0 < x - y < t < x + y, t = x + By in (40.26) we obtain the function

804 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
1
( ) _ I -IJ f r(x + 8y) ( 8 ) IJ
ux,y -3 x (1-8 2 )1-, x+ Y
-1
... ( y2(1 _ 8 2 ) 2 )
X =-2 J.l,1 - J.l;P; 4x(x + 8y) ' "4 y2 (1 - 8 2 ) d8,
( 40.28)
which satisfy the hyperbolic equation (40.19) when p > O.
Making the inverse replacement y by iy in (40.28) we arrive at the relation
u(x,y) = - i(2i)1-2'/ 3 x- 1J f r(O')O'IJ( - -1)2,-1
L
... ( y2( _ -1)2 2y2 _ ) 
x =-2 J.l,1 - J.l;P; 16xO' ' 16( -  1)2 T'
0' = x + iycoscp,
( 40.29)
L = {= ei'P, 0  cp  11'}, p> 0,
which after the change 8 = y cos cp can be also rewritten in the form
y
u(x, y) =13x-lJylyl-2, f r(x + i8)(x + i8)IJ(y2 - 8 2 ),-1
-y
( 82 _ y2 2 )
X 32 J.l,1 - J.l;P; 4x(x + i8) ' 4(8 2 - y2) d8, p> O.
( 40.30)
In order that the above integrals in (40.26)-(40.30) may exist, the functions
r and v must satisfy certain conditions given below and in the theorems of the
next subsection. It should be noted that the integral operators in these expressions
are compositions of the form I{/2JIJI<.!{/2.,Jiz)(O, O)f - (40.21) - which will be
observed below while investigating their boundary values.
From the above arguments we arrive at the following statement.
Theorem 40.4. Let P > 0 and J.l  0 and let r(z) be an analytic function of
z = x + iy in the neighbourhood of z = 0 and let it be an even function of x and y,
i.e. r(z) = r(z) , r(-z) = r(z) , Rer is an even function of x and y, Imr is an odd
function of x and y, and 1m r = 0 when xy = O. Then (40.29) and (40.30) give all
the classic solutions u E C 2 of (40.18) in the neighbourhood of (0,0) even of x and
y for which u(x, :1:0) is bounded. When 0 < p < 1/2 and 0 < J.l < 1/2 respectively,
then the relations uy(x, 0) = 0, or more exactly uy(x, y) = O(y), and uz(O, y) = 0,

 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
805
or more exactly uz(x, y) = O(x), respectively, are true. If in addition
13 = [B(p, 1/2)]-1 = r(p + 1/2)[y'1fr(p)]-I,
(40.31 )
then the functions given in (40.28)-{ 40.30) satisfy the Dirichlet condition
u(x,O) = r(x)
( 40.32)
and u(x, y) = r(x) + O(y2) as y -+ O. If r(z) has singularities at the points z = ZI
and z = -zl, then :i:ZI, :i:zl are singular points for the solution u(x, y) given in
(40.29), (40.30) too.
The proof of the basic conclusions of this theorem can be found in the book by
Gilbert [2]. The condition in (40.32) can be verified directly by the passage to the
limit as y -+ 0 in (40.28) and (40.29).
Theorem 40.4 shows that the operator in (40.29) maps analytic functions into
solutions of (40.18). In the general case its inverse operator was constructed as a
cumbersome series by Gilbert [2, p. 205-206]. As far as the special case JJ = A = 0
is concerned, such an operator can be found explicitly in a relatively simple way -
Gilbert [1]. Next we prove that other special cases of the inverse operator - when
JJA = 0 - can be constructed in closed form.
Case A. Let p. = 0, p > O. We assume that r(z) = r(z) . Then (40.30) with
the condition in (40.31) has the form
1- 2, 7/
u(." y) = B(P, 1/2) J T(" + ;y)(y2 - 8 2 )"-1 J'- l (> .v y 2 - 8 2 )d8
-7/
7/
2 y l-2, J -
= B(p,I/2) Rer(x + iy)(y2 - 8 2 ),-1 J,-1 ( A Vy 2 - 8 2 )d8
o
= 1r- 1 / 2 r(p + 1/2)J1 Y )( -1/2,p)Rer(x + iy)
= 1r- 1 / 2 r(p + 1/2)yl-2p(:: f)(w),
f(t) = Rer(x + i../i)t- 1 / 2 , W = .;,
( 40.33)
where 1 11 (z) is the Bessel-Clifford function given in (37.8), J17/\", Q) and C:: are
the operators in (37.45) and (37.4) applied with respect to the second variable.
Using the inversion relations of these operators given in (37.57) and (37.37) we

806 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
obtain the following representation for the operator inverse to (40.33):
Rer(x + iy) =y'1rr- 1 (p + 1/2)Ji)(p - 1/2, -p)
y
_ 2 1 - m y'1ry f (y2 - t 2 )m-,-1
r(p + 1/2) r(m - p)
o
X i m _._ 1 (>. Jy2 - t 2 ) G :, ) m (t 2 '- l u(z, t»tdt.
( 40.34)
Now the corresponding statement follows from Theorem 37.2.
Theorem 40.5. Let g(y) = u(x, V'Y)y,-1/2 E ACm([O, b», b < 00 for all x and
g(O) = g'(O) = ... = g(m-1)(0) = 0 and 0 < p < m. Then each solution u(x, y)
of (./0.18) with fl = 0 cOf7'esponds by (-10.3-1) to a certain harmonic function
Rer(x + iy). If u(x, y) is an even function of x and y, then Rer(x + iy) is also an
even function of x and y and the relation r(i) = r(z) holds for the corresponding
analytic function r(z).
Case B. Let A = 0, p > O.We assume that an even function u is given on the
boundary r of the unit disk Izi  1:
u(x,y) Ir= f(cp), r = {x = coscp, y = sincp, Icpl  1r},
f( cp) = f( -cp) = f( 1r - cp), cp 1= :i:1r /2.
( 40.35)
By substituting z = x + iy = e irp , x + i8 = t in (40.30) and using (40.35) we arrive
at the following integral equation:
z
-"' I 1 2-2, f
 y r(t)t"'[(z - t)(i - t)],-l
zyB(p, 1/2)
z
( (Z-t)(t-i» )
x 2 F 1 fl,1 - fl;P; 2t(z + i) dt = f(cp),
( 40.36)
Icpl < 1r, cp 1= :i:1r /2.
Integration in (40.36) is taken along the vertical interval joining the points i and
z. This interval can be replaced by a segment of a circle r of the form t = e iOl ,
lad  cp, since r(t) is analytic in the unit disk and other integrands are analytic
everywhere except the points t = z, i, 0, 00 lying on the boundary or beyond the
corresponding segment of a circle. According to the Cauchy theorem, the value

 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
807
of the integral is unchanged for such a replacement of the contour of integration.
In order to choose a single-valued branch of (40.36) with respect to the variable
z we make a cut along the half-line y = 0, x  0 and set Irpl < 1r. Then
[(z - t)(z - t)],,-1 = [2t(cOSQ - cOSrp)],-1 becomes a single-valued function with a
jump at the point t = -1, and (40.36) has the form
I V' r( eia)eia('+IJ) ( 1 ( COS Q ) )
2 F 2 fl, 1 - fl; P; - 1 - - dQ
(cos Q - COS <p)I-, 2 COS rp
-V'
= 2 1 -'B(p, 1/2) sin rpl sin rpI2,-2 cos lJ <pf(<p).
( 40.37)
Since the argument of the hypergeometric function must be lie beyond the cut
[1,00), then the condition Irpl < 1r/2 must be assumed, and therefore a passage
across the singular line x = 0 of the equation is excluded.
Splitting the integral in (40.37) into two integrals over [-rp,O] and [0, rp],
o $ <p < 1r /2, and making the substitution Q = -Ql in the first integral and taking
into account even of r(t) and (1.79) we arrive at an integral equation of a Mellin
convolution type with a Legendre function in the kernel given in (35.18), where
f(cosQ)lsinQI = Re(r(t)t'+IJ), 1] = COSQ, X = cosrp, d = 1,
( 40.38)
0$Q<1r/2,
2-' -Ii
o $ <p < 1r/2, g(x) = r(p + 1/2) (1 - x 2 )'-1/2xIJ f(arccos x).
U sing Theorem 35.2 concerning the inversion of this equation we can formulate the
following statement.
Theorem 40.6. Let p > 0, fl < 1/2, let f(<p) be such a function that the
corresponding function, g(x) from (40.38) is representable in the form g(x) =
(If_h)(x) where h E L 1 (£, 1), £ > 0, and let (If_h)(x) be the fractional integral
defined in (2.18). Then the solution of (40.36) on the circle Izl = 1, xy f; 0, in
the space of even functions r(i) = r( -i) = r(t) is reduced to the solution of the
following Hilbert boundary value problem - Gahov [1].
The Hilbert problem 40.1. It is required to find a function r( z) =
(x, y) + ;1](x, y) analytic in a quarter of the disk Izi < 1, x > 0, y> 0, continuous
right up to its boundary I and such that limit values of the real and imaginary
parts of r(z) satisfy on I the linear relation
a(t)(t) + b(t)1](t) = c(t), tEl,
( 40.39)
where a = c = 0, b(t) = 1 on the axes x = 0 and y = 0 and a(t) = COSQ(p + p),
b( t) = - sin Q(p + p), c( t) = f( cos Q) sin Q, 0  Q < 1r /2, on a quarter of the circle.

808 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Under the circumstances f is expressed via the given function j(t;) in (40.35)
by the relations (35.34), (40.38). The solution should be found in the space of
functions r( z) which are bounded near z = 1 and have an integrable singularity at
the point z = i.
Remark 40.3. If JJ > 1/2, then 1 - p. < 1/2 and the analogue of Theorem 40.6
obtained by using the property in (40.20) is true. Then t1-/J must be substituted
instead of t/J and JJ must be replaced by 1 - JJ in (35.4).
Proof of Theorem 40.6. After inversion of (35.18) we arrive at the problem of
finding the limit value r(t) of the function r(z) which is even and analytic in the right
half-disk, if its boundary value in (40.38) of the form Re[r(t)t'+/J] = j(cosQ)lsinad
is known. As was noted in Theorem 40.4 the function r(z) must have the property
Imr(z) = 0 on the coordinate axes, i.e. when xy = O. These three conditions may
by joined to the one given in (40.39) with the indicated coefficients.
We evaluate the index x and the number of solutions of the Hilbert
problem 40.1. Following the book by Gahov [1, S 30] we rewrite the relation
Re [r(t)t'+/J] = c(t) in the form
2c(t) = t'+/Jr+(t) + t,+/Jr+(t) = t'+/Jr+(t) + t'+/J r-(t).
Hence we deduce the boundary condition of the corresponding Riemann problem
r+(t) = _e- 2ia ('+/J)r-(t) + 2e- ia (,+/J)c(t), t E r.
( 40.40)
Here r+(t) is the limit value on the circle Izl = 1 of a function r(z) analytic in
Izl < 1 as z -+ t = e ia , 0  Q < 1(/2, and r-(t) is the limit value on Izl = 1 of
a certain function r-(z), Izi > 1, which is not necessarily analytic. By a similar
method the relation r+ = r- reflecting the analyticity of r on the axes is reduced.
Since r(t) is even, then the homogeneous condition r+(t) = _t- 2p - 2 /Jr- (t)
can be continued to the whole circle except the singular point t = -1 lying on the
cut. After a circuit around -1 arg t- 2 ,-2/J has a jump which equal to -41(p + p.)
- Gahov [1], S 43.2]. So the index x is equal to [-2p - p] + 1 in the space of
solutions integrable at the point t = -1, and the value Ix\ characterizes the number
of solutions of the homogeneous Hilbert problem when x  0, or the number of
solvability conditions of the corresponding inhomogeneous problem when x < O.
The solution of this problem itself can be constructed from the expressions given
in the book by Gahov [1, S 46]. Substituting the above function r(z) into (40.30)
with A = 0 we obtain the solution u(x, y) of the following Dirichlet problem.
The Dirichlet problem 40.2. It is required to find a real-valued function
u(x, y) even in x and y and continuous in the disk Izi  1 everywhere except
perhaps the line x = O. This function must satisfy (40.18) with A = 0, 0 < p < 1/2,

 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
809
in the disk when xy f; 0 and with initial data indicated in (40.35) on the circle
Izl = 1 and it must be continuous and bounded on the line y = 0 and uy(x, 0) = 0;
this function is bounded on the axis x = 0 if only J.l < 1/2.
40.3. Boundary value problems for the generalized Helmholtz
two-axially symmetric equation
In this subsection on the basis of (40.26) - (40.28) we construct solutions of
the Dirichlet and Neumann problems in the half-plane for (40.18) and of the
half-homogeneous Cauchy problem in the characteristic triangle for (40.19). We
also indicate the character of the behavior of the solutions near singular lines, and
the conditions for functions which guarantee the existence of integrals.
At first we clarify the behavior of (40.26) as x -+ O. It follows from (40.25)
that the representations
_ 00 11
'::'2(cr,P;1';x,y) = L - ( ) 1I2Fl(cr,P;1'+ I;x)
'=0 l' 1 .
 (cr)I:(P)A:xl: - .
= L...J () k! J+I:-l (2z..jY)
1:=0 l' I:
(40.41)
are true. Substituting an expansion of the Gauss hypergeometric function of the
form (10.13) into (40.41) we obtain the following leading term of (40.41) as x -+ 00,
I arg( -x)1 < 11':
82(0,,8; r; z, ,I)  r [;:  := :] of. (r - 0; 11)( -z )-0
+f [;: :=:] of.(r -,8; 11)( -z)-II, 0 '1,8;
( 40.42)
B(cr,cr;1';z,y)-r [ l' ] (-x)-aln(-x), P=cr.
cr,1'- cr
The latter relation follows from 2.10(7) in ErdtHyi, Magnus, Oberhettinger and
Tricomi [1]. Applying (40.42) to (40.26) we find that if J.l < 1/2, then the value
u(O, y) exists; if J.l > 1/2, then the limit lim IxI2#J-lu(x, y) exists; and if J.l = 1/2,
z-o
then the limit lim In-1Ixlu(x, y) exists. These properties show that the solution
z-o
given in (40.26) has a power singularity of order O(x l - 2 #J) when J.l > 1/2 and a
logarithmic singularity of order O(ln Ixl) when J.l = 1/2 on the line x = O.
Now we use the asptotic expansion OF1(V; -y) = O[y(1-211)/4 cos(2..jY +

810 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
1r(1 - 2v)/4)] as y -+ 00 - Marichev [10, (6.21)]. Then from (40.42) we find that
S2(a,p;-y; x, y) = O(y1/4+(0I--Y)/2x-0I) + O(y1/4+(P--Y)/2x- P ),
af;p, x,y,-+oo.
( 40.43)
Therefore the integrand in (40.26) has the following estimate as t -+ 00, namely
O[r(t)ltI IJ + p - 3 / 2 ] if  f; 0 and O[r(t)ltI2p-2(1 + ItI 2IJ - 1 )] if  = O. Hence the
improper integral in (40.26) is convergent if the functions r(t)ltI IJ + P - 3 / 2 and
r(t)ltI 2P - 2 (1 + ItI 2IJ - 1 ), are integrable at infinity.
The above investigations lead to the following two statements.
Theorem 40.7. Let r(t) be a continuous function bounded on the axis (-00,00) and
satisfying the following conditions at infinity: Ir(t)1 < CltI1/2-IJ-p-e when p. > 0,
 f; 0 or Ir(t)1 < CltI2-2IJ-2p-e when p.  1/2,  = 0 and Ir(t)1 < CltI 1 - 2p - e when
o < p. < 1/2,  = 0 where C is a constant and c > O. Then the Dirichlet problem
in the half-plane y > 0 which consists of finding the solution u E C 2 (y > 0, X f; 0)
of (40.18) with the initial data in (40.32) at the points x, -00 < x < 00, x f; 0, is
solvable for p < 1/2, p f; 0, -1, -2, . " and its solution is expressed by (40.26) with
the coefficient
1 1 = [B(l/2, 1/2 - p)t 1 = r(l - p)[y'1rr(I/2 _ p)]-1.
( 40.44)
The behavior of the derivative u y as y -+ 0 is characterized in Table 41.1 when
q = O. The behavior of u and U z as x -+ 0 is also given in Table 41.1 but with
q = 0 and replacing p by p. and y by x. The solution given in (40.26) has the
following behavior as y -+ 00:
u(x, y) = O[y-P-IJ-1/2(1 + y2IJ-1)] when  1= 0,
u(x, y) = O[y-2 IJ -1/2(1 + y4 IJ -2)] when  = o.
Theorem 40.8. Let v(t) be a continuous function on the axis (-00,00) and let
1I( t) satisfy the following conditions at infinity: 11I(t)1 < CltIP-IJ-1/2-e when  f; 0
and 11I(t)1 < CltI 2p - 1 - e (1 + ItI 1 - 2IJ ) when  = 0 where C is a constant and c > O.
Then the weighted Neumann problem in the half-plane y > 0 which consists of
finding the solution u E C 2 (y > 0, X f; 0) of (40.18) with initial data
lim y 2p u y (x, y) = lI(x), -00 < X < 00, x f; 0,
y_O+
( 40.45)

 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
811
is solvable for p > -1/2, p f; 0,1,2,. .., and its solution is expressed by the relation
u(x, y) = U2(X, y) + C 1 U01(X, y) + C2U02(X, y), C 1 , C2 - const.
( 40.46)
Here U2(X, y) has the form (40.27) with the coefficient
1 2 =
B(p, 1/2)
21r
( 40.47)
and the functions
UOj = r- P - IJ J e -l)j(p+IJ)(Ar), r 2 = x 2 + y2, j = 1,2,
are singular solutions of (40.18) satisfying the homogeneous condition of the form
(40.45). The behavior of the solution U and its derivatives as x --+ 0 and y --+ 0
are characterized in Table 41.1 in the same way as was indicated in Theorem 40.7,
and the solution's behavior as y --+ 00 is obtained from Theorem 40.7 on the basis
of (40.20).
Remark 40.4. (40.46) shows that the solutions of problems with data on singular
lines can not be unique without additional restrictions on the space. In the case
p = 1/2 one more solution involving the function In(p/y) in the kernel can be
added to (40.46). For more details we refer to S 41.4.
Remark 40.5. Expanding the function in (40.27) with the constant (40.47) in the
neighbourhood of the point p: U2 = p- 1 U20 + U2 + O(p) we write the second term
of this expansion
00 00
- ( ) _ lxi-I' f (t)II IJ ( ' 1) lsignlJl '" (J.l)A:(1 - p)1e
U2 x, Y - 21r II t slgnxt + L.J (k + l)!k!l!
-00 1e,I=O
Ie ( A 2 ) I [ Ie+l 1 ]
x (_-E-) _-1:.. lop + C - L -:- dt,
4xt 4 j=1 J
p = (x - t)2 + y2,
( 40.48)
where C is the Euler-Mascheroni constant. This function is the solution of the
problem given in Theorem 40.8 for the singular case p = O. If A = P = 0 and the

812 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
condition of the form
00
f v(t)dt = 0
-00
( 40.49)
necessary for solving the Neumann problem in the half-plane holds, then (40.48)
leads to the known Dini relation for the solution of the Neumann problem in the
half-plane y > 0 for the Laplace equation
00
u(z,y) =  f v(t)ln[(z - t)2 + y2]dt.
211'
-00
( 40.50)
Remark 40.6. If p  1/2, then (40.27) represents also the solutions of the
following weighted Dirichlet problems
lim y2 P -l u (z,y)=- 1 1 2 v(z), p>I/2, p¥I,2,3,...,
1/-+0 - P
(40.51 )
lim In- 1 yu(z, y) = v(z), p = 1/2,
y-+o
( 40.52)
and (40.48) is the solution of the problem in (40.52) also.
In conclusion we note that on the basis (40.28) we can similarly obtain the
solution of the following half-homogeneous Cauchy problem for the hyperbolic
equation (40.19)
u(z,O) = r(z), lim y2 P Uy (z, y) = 0, 0 < z < 1,
y-o+
( 40.53)
and make the corresponding statement about its solvability.
Theorem 40.9. If r E C 2 ([0, 1]), 0 < p < 1/2, and '3 be given in (40.31), then
the Cauchy problem in (40.53) in the triangle D = {O < z - y < z + y < I} has the
classic solution u E C 2 (D) of the form (40.28).
 41. The Euler-Poisson-Darboux Equation
The present section deals with integral representation for the solutions of the
Euler-Poisson-Darboux equation in the elliptic and hyperbolic cases, and their
applications to the construction of the solution of the Dirichlet, Neumann and
Cauchy boundary value problems. As in S 40 we show by using the Erdelyi-Kober

 41. THE EULER-POISSON-DARBOUX EQUATION
813
operators defined in (18.8) that solutions of the simplest equations with constant
coefficients are reduced to solving the Euler-Poisson-Darboux equation.
41.1. Representations for solutions of the
Euler-Poisson-Darboux equation
The generalized Euler-Poisson-Darboux equation
_ P-u - pu"
E(P, P )u = u" -  = 0, P" P - const,
-17
( 41.1)
is of especial importance while solving problems of axially symmetric potential
theory. This equation corresponds to (40.1) with a(, 17) = _pC-) /( - 17), b(, 17) =
P/( - 17) and C(, 17) = O. After the substitutions given in (40.6) it has the form
+ _ 2q _
E u - U zz + u,y + -U z + -u, - 0,
y y
P = p + iq, P- = p - iq.
( 41.2)
( 41.3)
We note that the Laplace equation in R"
U Z1Z1 + UZ'Jz'J + .. . + U z .. z .. = 0,
( 41.3')
the solution of which is sought in the form
U(Xl,...,Xn)=u(x,r), x=x n , r= v x+"'+X-l'
with the axis of symmetry r = 0, is reduced to (41.2) with q = 0, 2p = n - 2 and
y = r. Such a solution is called a 2p + 2-dimensional axially symmetric potential
and satisfies the equation
2p
U zz + U rr + -U r = O.
r
( 41.3")
Since the operator in (41.3") is even with respect to r, then each solution of (41.3")
is an even function of r, and if p = 0 then the 2-dimensional axially symmetric
potential is a harmonic function.
A further two cases will be different. We shall assume p and P- to be real or
complex and conjugate numbers with ReP = ReP- = p while investigating (41.1)
or (41.2), respectively.
By direct evaluation we prove the following statements.

814 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Lemma 41.1. A function u is the solution of the equation E({3, (3*)u = 0 if and
only if the function
v(,7]) = (7]-),6+,6.-1u(,7])
( 41.4)
satisfies the equation E(1 - (3*, 1 - (3)v = O.
Lemma 41.2. Iful(, 7]) satisfies (41.1), then the function
,6 ,6- (  + d c7] + d )
u(, 7]) = (+ b)- (a7] + b)- Ul a + b ' a7] + b
( 41.5)
where a, b, c and d are arbitrary constants such that ad - bc f; 0 is also a solution
of (41.1).
Lemma 41.3. Let {3 > 0, (3* > 0, and let r(8) be an arbitrary function continuously
differentiable twice. Then the integral
1
J 1 ({3,{3*) = f r[ + (7] - )t]t,6.-1(1- t),6-1dt
o
( 41.6)
satisfies (41.1) and it also satisfies (41.2) in the case p > 0 provided that the
conditions in (40.6) hold. -
We note that Lemma 41.3 follows from Theorem 40.2 if we set
"Y(,7],8)=C(-8)-,6(7]-8)-,6., C - const. (41.7)
Then the integral of the form (40.17), i.e.
"
J 2 ({3, (3*) = f "Y(, 7], 8)1I( 8)d8, TJo = ,
"0
( 41.8)
where 11(8) is an arbitrary function continuously differentiable twice and {3 < 1,
(3* < 1, will satisfy (41.1) and by using Lemma 41.1 ( 41.8) transforms into
(41.6). It is obvious that (41.6) and (41.8) are linearly independent solutions if
{3 f; 1/2, {3* f; 1/2, p f; 1/2.
Theorem 41.1. If 0 < {3 < 1, 0 < {3* < 1 and either {3 f; 1/2 or {3* f; 1/2, then

 41. THE EULER-POISSON-DARBOUX EQUATION
the general solution of (41.1) is given by
815
1
U(, 1]) = f r(8)t,6.-1(1 - t),6-1dt
o
1
+ (1] - )1-,6-,6. f 1I(8)t-,6(1 - t)-,6. dt
o
=J1(P,p.) + J2(P,P.) = J(P,p.),
8 =  + (1] - )t,
( 41.9)
where r(8) and 11(8) are arbitrary junctions continuously differentiable twice. If
o < p < 1, p f; 1/2 then the general solution of (41.2) is given by (41.9) where r(8)
and 11(8) are arbitrary continuous functions.
In the case P = p. = p = 1/2 such a solution has the fonn
1
u(, 1]) = f r(8)(t - t 2 )-1/2dt
o
1
+ f 1I(8)(t - t 2 )-1/2In[(t - t 2 )(1] - )]dt.
o
(41.10)
Remark 41.1. In particular, if r(8) and 11(8) are analytic functions in a certain
domain 'D, i.e. for (, 1]) E 'D, then (41.9) and (41.10) present all solutions of (41.1)
analytic in 'D.
We characterize briefly the process for finding the general solution of (41.1)
for other values of the parameters P and p.. Let - k < p. < 1 - k, -I < P < 1 - I,
k, I = 1,2,3,...,p + p. f; -1, -2, -3,.... Then according to the inequalities
o < 1 - k - p. < 1 and 0 < 1 - I - P < 1 the function h (1 - k - p. , 1 - I - P)
given in (41.6) is the solution of the equation E(I - k - rr, 1 -1- P)u = O. We
evaluate the derivative
1c I 1
/)';'11 1,(1 - k - {1', 1 -1- {1) = f T(HI)(8)r P (1 - t)-P'dt,
o
(41.11)
provided that r( 8) is sufficiently smooth. It is obvious that the integral on the
right-hand side of (41.11) satisfies the equation E(1 - P" 1 - P)u = O. Therefore

816 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
multiplying (41.11) by (1]_)1-f3-f3. and using Lemma 41.1 we obtain the function
1
(1] - )1-f3-f3. f rCI:+l)(8)t-f3(1 - t)-f3. dt, 8 =  + (1] - )t,
o
which is the solution of (41.1). After similar transforms with the second solution
J2 we arrive at the general solution of (41.1) in the form
81:+1
u(, 1]) = (1] - )1-f3-f3. 81:81]' J(I- k - p., 1 - 1- P),
( 41.12)
where J(P, p.) is given in (41.9).
Using the inequalities p. + k > 0 and (3 + I> 0 we can obtain another form of
the general solution
_ _ 1-f3-f3. al:+l J(P + I, p. + k)
u(, 1]) - (1]) al:a1]' (1] _ )l-f3-f3.-I:-l '
(41.13)
If, for example, P = p. = 1/2 - k, k = 0,1,2, . . . , then J 1 and J2 will become linear
dependent solutions and it should be substitute the right-hand side in (41.10)
instead of J into (41.12) and (41.13). If one of P and p. is an integer, then
(41.1) can be integrated in quadratures by the cascade Laplace method - Babich,
Kapilevich, Mihlin, Natanson and others [1,p.43].
It should be noted that we can arrive at the representation in (41.9) by using
the Riemann method and the Riemann function of (41.1):
RE(, 1]; o, 1/0) = (1] - )f3+f3. (1] - 0)-f3 (1]0 - )-f3. 2F1(P, p.; 1; 0'),
0' = (- 0)(1] - 1]0) , (41.14)
(- 1]0)(1] - o)
where 2F1 is the Gauss hypergeometric function glVen 10 (1.73) - Babich,
Kapilevich, Mihlin, Natanson and others [1, p. 48].
It is not difficult to transfer the above results to the case of the elliptic
equation (41.2). Thus from (41.7), (41.3), (41.6) with Imt = 0 we obtain the
special solution of (41.2):
'Y(, 1], t) =C1 (- t)-f3 (t -()-p
=C1 exp[-(p+iq) In( -t)- (p- iq) In(t-()]

 41. THE EULER-POISSON-DARBOUX EQUATION
817
=Cl exp[-2p In I -tl-(p+iq)i arg( -t)-(p-iq)i arg(t-)]
( 41.15)
=cll _tl- 2p exp[-(p+iq)i,p- (p-iq)i( 1r-,p)]
= c2r2P e 2Q t/J,
where
rl = I - t I = v' (x - t)2 + y2,
,p = arg( - t) = arccos[(x - t)/rl], Yl  O.
( 41.16)
( 41.17)
Setting C2 = 11 r(t) or C2 = 1 2 v(t) in (41.15), and integrating along the axis
-00 < t < 00 and using Lemma 41.1 with respect to (41.2) (see also (40.20», we
arrive at the following two solutions of (41.2):
00
1-2p I r(t)e2Qt/J
u(x, y) = ' 1 Y [(x _ t)2 + y2p-p dt,
-00
( 41.18)
00
I v(t)e2ft/J
u(x, y) = 1 2 [(x _ t)2 + y2]P dt.
-00
( 41.19)
These representations are analogues of the solutions given in (40.26) and (40.27)
for (40.18) coinciding with (41.12) when q = 0 for p = A = O. (41.18) and (41.19)
are also analogues of J 1 (f3,p.) and J 2 (P,p.). The analogy of the solution given
in (41.10) and containing the logarithmic function is the solution of (41.2) with
p = 1/2 of the form:
1 00 r(t)e2Qt/J [ Y ]
u(x,y) = 13 C 1 +C2 1n ( )2 2 dt,
v' (x - t)2 + y2 x - t + y
-00
( 41.20)
where C 1 and C2 are arbitrary constants.
Carrying out the changes given in (40.6) and the substitution (1 - 2t)y = 8
and taking (41.3) into account we obtain the following analogue of (40.30):
1/ ( ) iQ
u(x, y) = ' 3y l-2 p I r(x + i8)(y2 - 8 2 )P-l : : d8.
-1/,
(41.21 )

818 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
In particular, if q = 0 and r(i) = r(z) , then from (41.21) we obtain the special
case of (40.33), with  = 0:
Y
u(x, y) = 2l 3 yl-2 p f Rer(x + i8)(y2 - 8 2 )p-ld8
o
= l3r(p)I1/2,,,Rer(x + iy),
( 41.22)
where Il is the Erdelyi-Kober operator defined in (18.8) and applied with respect
to y. The following analogue of Theorem 40.4 is valid for (41.21) and (41.22).
Theorem 41.2. Let p > 0 and let r(z) be analytic function in the neighbourhood
of z = O. Then (41.21) presents all classical solutions u E c 2 of (41.2) in the
neighbourhood of (0,0). If in addition r(z) is an even function of y and satisfies
the condition r(i) = r(z ) , then (41.22) presents all classical solutions u E c 2 of
(41.2) with q = 0 even with respect to y in the neighbourhood of (0,0). Under the
circumstances the value u(x, :1:0) is bounded and if 0 < p < 1/2, then uy(x,O) = 0
and uy(x,y) = O(y) as y --+ O. If in addition (40.31) holds, then the functions
u(x,y) given in (41.21) and (41.22) satisfy (40.32).
The rel ation in verse to (41.22) is obtained from (40.34) when  = 0, and hence
jm-p-l( V y2 - t 2 ) = 1, provided that the conditions in Theorem 40.5 hold.
In conclusion of this subsection we note that on the basis of the relations in
S 17.2, note 16.2 we can deduce the connection between the asymptotics of r(z)
and u(x, y) as y --+ +00. Indeed, if
00
Rer(x + iy)  L dl:(x)y-21: as y --+ +00,
1:=0
( 41.23)
then from (41.22) and (17.17) we obtain the following asymptotic expansion of the
solution u(x,y) of the boundary value problem defined in (40.32) and (40.31) for
(41.2) with q = 0, namely
( ) l r ( ) [  (-I)l:rot(Y){9(X+iY);2k+l}
u x, y 3 P L.J k!r ( _ k ) 21c+l
1:=0 P Y
( 41.24)
00 r(1/2 - k) ]
+ {; dl:(x) r(1/2 + p _ k)y21:
as y --+ +00,
where rot(Y){g(x + iy);2k - I} is the Mellin transform defined in (1.112) of the
function g(x + iy) with respect to y at the point 2k + 1.

 41. THE EULER-POISSON-DARBOUX EQUATION
819
41.2. Classical and generalized solutions of the Cauchy
problem
After replacing y by iy the elliptic equation (41.2) is transformed into a hyperbolic
equation of the form
_ 2q 2p
E U = U zz - U yy - -U z - -u y = O.
y y
( 41.25)
The last equation is characterized by the Cauchy problem considered in the
characteristic triangle D- = {O < -y < x < 1 + y} with initial conditions
U(x,O) = r(x), 0  x  1, lim (-y) 2p u y (x,y) = v(x), 0 < x < 1. (41.26)
y--O
The solution of this problem (with exactness to coefficients) can be obtained from
(41.9) and other relations for the general solution in Subsection 41.1 if we set
 = x + y, 17 = x - y, p. = p + q, P = p - q
( 41.27)
in the above expression. Thus the following statement is true.
Theorem 41.3. If r E C2([O, 1]), v E C2«O, 1», 0 < p. < 1, 0 < P < 1 and
P + rr < 1, then the Cauchy problem given in (41.26) for (41.25) in the domain
D- is properly set and its solution U E C2(D-) is given by
1
u(x,y) = - Al(-y)1-2p f v(x + y(l- 2t»(I- t)-p. t-Pdt
o
( 41.28)
1
+ B(P fJ") f T(Z + 11(1 - 2t»(1- t)P-1t P . -lclt,
o
where Al = [(1 - 2p)B(1 - p., 1 - P)]-l.
Corollary. If the condition
U(x, -x) = 1jJ(x), 0  x  1/2,
( 41.29)
holds on the characteristic y = -x, then the following representation of v(x) via

820 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
r( x) and 1/J( x) is valid:
_ _ 2 1 - 2p r(1 - P) fj  fj. (  )
II(X) - r(l- 2p) x dx 10+1/J 2
2 1 - 2p r(2p)r(1 - P) d 2p
+ r(1 - 2p)r(p.) dx 10+ r(x),
( 41.30)
where IC+ is the fractional integration operator defined in (2.17).
Proof. Substituting y = -x into (41.28) and using (41.29) after replacing 2xt by
17 and 2x by x we obtain
1/J () = - A I 2 2p - 1 r(l- p.)IJ+.fj. X-fjll(X)
+ r(2p) x l - 2P lfj x fj -- 1 r ( x )
r(p.) 0+ .
Applying the operator IC -1 to this relation we find
2 1 - 2P Xfj . ( X )
II(X) = - A 1 r(1 _ p.) IC+ -11/J 2
2 1 - 2p r(2p) fl fj.-l 1-2p fj ( ) fj.-l
+ Alr(p.)r(1 _ p.) r- 10+ x 10+ r x x .
Using now (10.12) we evaluate the composition in the last term:
I fj.-l 1-2P l fj fj.-l ( ) _ I -fj (1 2p-1 1-2P l fj -fj ) 2p-l ( )
0+ x o+x r x - 0+ 0+ x o+x x r x
- I -fj (l fj -fj I 2p-l 1-2p ) 2p-l ( )
- 0+ o+x 0+ x x r x
_fj d 2p ( )
= x dx 10+ r x ,
which yields (41.30). .
Remark 41.2. The relation in (41.30) is also true provided the conditions
o < p + p. < 1, p. > 0, p < 1 less restrictive than those in Theorem 41.3 hold.
Remark 41.3. (41.30) is widely used while investigating equations of mixed type
- see, for example, the books by Tricomi [1], Bitsadze [1-3] and Smirnov [2, 7].
However, (41.30) is usually proved by more a complicated method which does not
use (10.12).

 41. THE EULER-POISSON-DARBOUX EQUATION
821
In many cases, especially while solving boundary value problems for equations
of mixed type, the conditions T, II, U E C 2 from Theorem 41.3 lead to restrictions
on the given functions, which are verified with difficulty. This problem is solved by
considering formal solutions of the type (41.28) which are not sufficiently smooth
and can be approximate by solutions in the space C 2 . Now we consider a class Rl
of such a generalized solutions of the Cauchy problem introduced by K.I. Babenko
[1, 2].
Definition 41.1. The function in (41.28) with {3 = /3- = p is called a generalized
solution of (41.25) with q = 0 in a class Rl in the domain D- = {O < -y < x <
1 + y} if 0 < p < 1 and
T E H£W 1 ([0, 1», Ql > 1- Pi II E H£W([O, 1», Q2 > P,
(41.31 )
where H>'([O, 1» is the space of Holderian functions defined in S 1.1.
The following statement is true.
Theorem 41.4. Let generalized solution given in (41.28) with q = 0 of the Cauchy
problem defined in (41.25), (41.26) be in the class Rl' Then U z , u y E C(D-), the
function U y satisfies the second condition in (41.26) and there exists a sequence
{Un}l' Un E C 2 (D-), of classical solutions of (41.25) such that lim Un = U in
n-oo
any closed triangle D; = {€ < -y < x < 1 + y}.
Proof. Taking the coordinates (41.27) in (41.28) and using Lemma 13.2 according
to (41.31) we arrive at the representations
8
T(8) = T(O) + 1(8 - s)-P+£cp(s)ds,
o
( 41.32)
8
11(8) = 11(0) + 1(8 - s)P-l+£1jJ(s)ds,
o
where € > 0 is sufficiently small and cp,,p E C([O,l]). Substituting these values
into (41.28) for rr = {3 = p and changing the order of integration in the domain
o < s < 8,  < 8 < '1, we obtain

822 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
u(,,,) = - A I 2 2p - 1 (,,- )1-2PB(1 - p, 1 - p)lI(O) + r(O)
   
+ (JdsJdO+ JdsJdO)
o  ,
X [-AI22p-l,p(s)(7J - 8)-P(8 - )-P(8 - 8)p-l+£
( 41.33)
+ (B(p,p»-1(7J - )1-2p<p(8)("_ 8)P-l(8 - )P-l(8 - 8)-P+£].
Using now the substitutions 8 = (,,- )t +  or 8 = (s - 7J)t + 71 we evaluate the
inner integrals by (1.73). Then we have
u(, 71) = - A 1 2 2p - 1 B(1 - p, 1 - p)(7J - )1-2Pll(0) + r(O)
- 2 2p - 1 AIB(1 - p,p + c)(7J - )-P

x J 1/I(s)('1- s)'2F, (p, 1 - p; 1 + t;  =; ) ds

+ B(p,1 - p + c) (71 _ )-P
B(p,p)

x J l"(s)('1- s)'2F, (p, 1 - p; 1 + t;  =; ) ds


+ J 1"(-)('1- s)-'+<2F, (p,p - t; 2p;  = : )dS
o

_ AIB(I- P, 1- p) J ,p(s)( _ 8)p-l+£
2 1 - 2p (,,_ )2p-l
o
(  - 71 )
X 2Fl 1 - p, 1 - p - c; 2 - 2p;  _ s ds.
( 41.34)
All the above integrals are proper integrals and have continuous derivatives with
respect to  and 71 in the domain 1). The derivative of the last integral in (41.34)
with respect to  also exists since by (10.13) (- 8Y-l+£ 2Fl () = 0(1) as
8 -+. Derivatives of all terms in (41.34) except the first and the last ones have
order 0[(7J-)-P] as 71 -+. Applying the operator (7J-)2p ( : - : ) to (41.34)

 41. THE EULER-POISSON-DARBOUX EQUATION
823
and passing to the limit as TJ --+  we find
lim(TJ - )2p(U'i - ud = - A I 2 2P B(1 - p, 1 - p)(1 - 2p)
'1-

x ["'(0) + J .p(s)({ - S)'-1+<dS]
o
= - 22PII()
and this passage is carried out uniformly for 0    8 < 1. Hence after the
substitutions given in (41.27) we obtain the second relation in (41.26).
Since cP,1/J E C([O, 1»), then by Weierstrass's theorem there exist sequences of
functions {CPn( S )}=l' {,pn(s)}=l E C 2 ([0, 1» tending to cP and 1/J respectively,
uniformly on any interval [0,8 0 ], 8 0 < 1. By (41.32) they correspond to the
functions Tn, II n , n = 1,2,..., which belong to C2([0,1» also. But then the
corresponding solutions in (41.28) denoted by Un belong to C 2 (D-) and Un --+ U
too. .
41.3. The half-homogeneous Cauchy problem
in multidimensional half-space
We consider the Cauchy problem in the half-space Rt.+ l = {(x, y) x =
(Xl,' .. , X n ) E R n , y> O} for the hyperbolic equation
 8 2 u _ 8 2 u _ 2p 8u = 0
L.J 8x2 8y2 Y 8y
k=l k
( 41.35)
with the half-homogeneous initial conditions
u(X,O) = T(X), uy(x,O) = 0,
( 41.36)
where T(X) E C 2 (R n ) is the given function.
It is known, (Courant and Hilbert [1, p. 466]) that if 2p = n - 1, then the
solution of the problem given in (41.35) and (41.36) is unique, and it can be
represented as the spherical mean Mn(x, y; T) of the function T in the space R n by
u(Z, II) = Mn(z, II; T) = IS:-.I J T(Z + yt)du.
5"'_1
( 41.37)
Here t = (t 1 ,. . . , tn) lies on the surface of the unit sphere Sn-l Le. It I = 1, du is
an element of the unit sphere surface area and ISn-11 = 21f'n/2/r(n/2) is its surface

824 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
area. Its value is evaluated by (25.9) when f == 1. In particular, the relations
Mn(x,O;r) = r(x),
M1(x, y; r) = 2- 1 [r(x - y) + r(x + y)]
( 41.38)
( 41.39)
follow from (41.37).
We apply the Erdelyi-Kober operator 1",00 defined in (18.8) and denoted by
1l below, with respect to y to the solution in (41.37). Then using Lemma 40.2
with  = 0 we select the parameters of 1l so that the operator in (41.35) with
2p = n - 1 will be transformed into the operator in (41.35) with arbitrary 2p. In
this way we construct the function
r(p + 1/2) (y)
u(x, y) = r(n/2) In/2_1,p+(1_n)/2Mn(x, y; r).
( 41.40)
According to Lemma 40.2 this function is the solution of (41.35) and the constant
coefficient r1::) is chosen so that the conditions in (41.36) hold in view of
( 41.38).
We note that if we use the definition of the Erdelyi-Kober operator 1",01 for
Q < 0 considered in subsection 18.1, then (41.40) remains the solution of the
Cauchy problem given in (41.35) and (41.36) in the case p < (n - 1)/2. However,
Bresters [2] proved that the solution given in (41.40) is not a unique in the case
p < O.
By using (23.1) (41.40) can be represented in the form
1'+ioo
r(p + 1/2) f r«n - s)/2) (y) -,
u(x,y)= r(n/2) r«(I-s)/2+p) rot {Mn(x,y;r);s}y ds,
1'-ioo
o < 'Y < 1,
(41.41 )
where rot(Y) is the Mellin transform operator defined in (1.112) and applied with
respect to y. Such a form of the solution of the Cauchy problem is admissible for
2p f; -1, -3,.. ., and it is useful while investigating the asymptotic behavior of
this solution.
As an example of such an investigation we find the asymptotic expansion of
the solution u(x, y) as y -+ 00 in the case when n = 1, p > 0 and r(t) has the
simplest asymptotic expansion of the form
00
r(t) -- L a1c t - 1c , t -+ 00.
1c=O
( 41.42)

 41. THE EULER-POISSON-DARBOUX EQUATION
825
Then, as was done by Berger and Handelsman [1],
00
M 1 (x, yj r) f"oJ L Ct(x)y-21:,
1:=0
( 41.43)
where co(x) = ao,
21:-1 ( 2k 1 )
CI:(x) = E  a21:-j(-XY,
j=O J
k = 1,2,. . .
is a polynomial of degree 2k -1. Hence by using (17.18) we obtain the asymptotic
expansion of the solution of (41.25) with q = 0:
( ) f"oJ r(p+ 1/2) [  2(-I)I:!J]t<Y){M}(x,y;r);2k+ I}
u x, y -Ii f;;, k!r(p _ k)y2l:+1
 r(I/2 - k) ] 2t
+  r(p + 1/2 - k) CA:(x)y- .
( 41.44)
In conclusion of this subsection we note that Lemma 40.2 in the case of
arbitrary , enables us to construct the solutions of the Cauchy problem for
arbitrary homogeneous linear hyperbolic equations of second order with constant
coefficients, on the basis of the solution given in (41.37). Indeed, the above
equations by changes of variables can be reduced to the equation
n
L V ZIcZIc - V yy - 2v = O.
t=1
( 41.45)
According to Lemma 40.2 the solution v(x, y) of this equation can be transformed
to the solution u(x, y) of (41.35) with 2p = n - 1 by the relation
r(n/2) (y) ( In -1 )
u(x, y) = -Ii J i >. -2'  v(x, y),
( 41.46)
where J1 Y ) (71, Q) is the generalized Erdelyi-Kober operator J>.(71, Q) defined in
(37.45) and applied with respect to y. Then (41.36) is true for the solution v(x,y)

826 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
also, and the inverse transform is given by the expression
_ -Ii (y) ( n 1 - n ) .
u(x, y) - r(n/2) J>. 2" - 1,  Mn(x, y, r).
( 41.47)
If on the basis of the solution v(x, y) we construct the function
y
v(x, y) = f v(x, t)dt,
o
( 41.48)
then it is also the solution of (41.45) satisfying the half-homogeneous initial
conditions
v(x,O) = 0, vy(x,O) = r(x)
( 41.49)
symmetric with (41.36). Replacing r(x) by II(X) in (41.48) and adding it with the
function given in (41.47) we arrive at the solution of the Cauchy problem of the
general form (41.26) with p = 0 for (41.45).
41.4. The weighted Dirichlet and Neumann problems in a
half-plane
In this subsection we apply the results given in (41.18)-(41.20) to solving the
Dirichlet and Neumann problems in the half-plane y > 0 for (41.2) with any
parameters p and q. The following analogues of Theorem 40.7 and 40.8 are true.
Theorem 41.5. Let r be a continuous function bounded on the real line (-00,00).
Then the Dirichlet problem in the half-plane y > 0 which consists of finding the
solution u(x, y) E C 2 (y > 0) of (41.2) in the space of functions bounded at infinity
under initial conditions in (40.32) is properly set for any real q only in the case
p < 1/2. The solution is given by (41.18) with
I _ (1 - 2p)B(1 - p, 1 - p)
1 - 2 2P 1re Q 1l' .
( 41.50)
Theorem 41.6. Let II be a continuous function absolutely integrable on the real
line (-00,00) and if p  0, then in addition III(t)1 < CltI 2p - 1 - e , c > 0, It I --+
00, C - const. Let also the necessary condition in (40.49) be satisfied. Then the
weighted Neumann problem in the half-plane y > 0 which consists of finding the
solution u E C2(y > 0) of (41.2) vanishing at infinity and satisfying the boundary
condition in (40.45) on the axis -00 < x < 00 is properly set only in three cases:

 41. THE EULER-POISSON-DARBOUX EQUATION
827
1) p> q, q is an arbitrary real number; 2) -1/2 < p < 0, q = 0 and 3) p = q = O.
In the first two cases the solution of the above problem is given by (41.19) with
12 = _22p-211'-le-q1rB(p, P),
( 41.51)
and in the third case the solution is given by (40.50) and is continuously connected
with (41.19) and (41.51) only for q = 0, p -+ O. These solutions contain integrals
which converge absolutely and uniformly in the set {y  0, Ixl < R} and these
solutio n vanis h at infinity of order u = O(p-2 p -l) and U z , u y = O(p-2 p -2) as
p = V x 2 + y2 -+ 00. ,
Theorem 41.7. Let p = 1/2 and r be a function continuous on the real line
and Ir(t)1 < Cltl- e , € > 0, It I -+ 00, C - const. Then the weighted Dirichlet and
Neumann problems in the half-plane y > 0 which consists of finding the solution
u(x, y) E C 2 (y > 0) of (41.2) vanishing at the infinity and satisfying the conditions
lim In- l yu(x, y) = lim yuy(x, y) = r(x), -00 < x < 00,
y-+O y-+O
( 41.52)
respectively, are solvable and their solutions are given by (41.20) where 1/J has the
form (41.17) and
1;1 = -(1+e2q1r)(CI-AC2), A = In4+2C+1/J(1/2-iq)+1/J(1/2+iq), (41.53)
or by the equivalent relation
00
e- q1r f r(t)e2q.p eAC-IyC
u x,y = - 10 dt,
( ) 2chq1l' V(t - x)2 + y2 «x - t)2 + y2)C
-00
( 41.54)
where C, C I and C 2 are arbitrary constants and 1/J(z) is defined in (1.67).
Remark 41.4. Differentiating (41.18) with respect to y, and taking the limit as
y -+ +0 in the equation obtained and in (41.19), provided that p < 1/2 and (40.32)
and (40.45) hold, and further, making the changes of variables
CI = chq1l', C2 = -shq1l', cr = 1 - 2p,
f(x) = r(x), <p(t) = r(cr)l2eq1rll(t),
we obtain the relations in (12.11) and (30.78) which reflect connection between the
direct and inverse Feller transforms.

828 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
The following boundary value problems are solvable for other parameters p
and q.
A. If p > 1/2 and q is any real number, then the weighted Dirichlet problem
(40.51), -00 < x < 00, is solvable and its solution is given by (41.19), (41.51).
B. If p = 0, q f; 0, then the weighted Neumann problem
lim In- 1 YUy(x, y) = II(X)
y-+O
(41.55 )
is solvable and its solution is given by (41.19) with p = 0, 1 2 = [4qe q ,.. shq1r]-l.
C. If p < 0, q f; 0, then the Neumann problem
uy(X,O) = II(X), -00 < X < 00,
( 41.56)
is solvable in the space of functions bounded at infinity and its solution is given by
(41.18) with
t
r(t) = _pq-l f 1I(8)d8 + C, C - const,
-00
where C = 0 if u vanishes at infinity.
D. If q = 0, p = -1/2, then the weighted Neumann problem
lim (y In y)-lu y (x, y) = II(X), -00 < x < 00,
y-+O
( 41.57)
is solvable and its solution is given by (41.19) with q = 0, 1 2 = -1/2.
E. If q = O,p < -1/2, then the weighted Neumann problem
lim y- 1 u y (x, y) = II(X), -00 < X < 00,
y_+O
( 41.58)
under the conditions (40.49) and III(t) I < Cltl- 2 - e , c > 0, It I --+ 00, is uniquely
solvable in the space of functions vanishing at infinity together with their derivatives
of the first order and its solution is given by (41.18) with q = 0 and
t
r(t) = f (t - 8)1I(8)d8,
-00
I _ 2f(1 - p)
1 - y'1rr( -1/2 _ p)'
In order for the above problems to become uniquely solvable we must assume
additional conditions on the solutions, and remove the singular solutions of (41.2) of
the form yl-2Prp-2e2qt/J, yl-2p, r2p e 2Q t/J, 1 and In y, rllln(yr12)e2ft/J if p = 1/2.

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 829
Table 41.1
p < -! -! -!<p<O 0 <0 O<p<! ! >!
q 0 0 0 0 O V V V
u=O 1 1 1 1 1 1 my 1I1-2p
Uy = 0 y ylny y-2p 1 1 1I-2p 11- 1 1I-2p
Table 41.1 reflects the dependence of the orders of the solution u of (41.2) and
its derivative u y as y --+ 0 on the parameters p and q.
This table also show which weights correspond to properly setting Neumann
and Dirichlet problems with conditions on the singular line y = o.
 42. Ordinary Differential Equations
of Fractional Order
Equations in which an unknown function y(x) is contained under the sign of a
derivative of fractional order, i.e. equations of the form
F(x, y(x), V:;Wl (x)y(x), V::W2(X)Y(X),..., V::wn(x)y(x» = g(x),
( 42.1 )
where V:j = V::+ or V::_ are called ordinary differential equations of
fractional order. By analogy with the classical theory of differential equations,
differential equations of fractional order are divided into linear, homogeneous
and inhomogeneous equations with constant and variable coefficients. Differential
equations of fractional order are studied both in the space of regular functions,
i.e. functions summable to a certain power and continuous and differentiable up to
certain order in a classical sense, and in various spaces of generalized functions.
Analogues of the Cauchy and Dirichlet problems for differential equations of
fractional order often arise in applications. Thus if we would like to find the
solution y(x) of (42.1) with initial conditions
Vy(x)\Z=zo = b}, Vy(x)\z=zo = b 2 , ...,
VfJ b+ ",y(x) \ _ = b m , xo,b,b 1 ,...,b m ,{31,...,{3m - const,
Z-Zo
then we say that we are dealing with the solution of the Cauchy-type problem for
(42.1). If the values of the unknown function or of its derivatives of integer or
fractional order are given at the end points of a certain interval [XO, Xl] then we
say that we are dealing with the Dirichlet-type problem for (42.1).
In this section we shall be concerned with the proper setting of certain
problems for differential equations of fractional order. Thus we shall study
questions concerning the solvability of these equations in certain spaces of functions.

830 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Further, the applications of the theory of fractional integra-differentiation and the
theory of differential equations of fractional order to integration of certain classes
of differential equations of integer order will be considered.
42.1. The Cauchy-type problem for differential equations and
systems of differential equations of fractional order of general
form
We need to find a function y( x) satisfying the equation
dOl
dxOl Y(x) = f(x,y), n-l<an, n=I,2,...,
( 42.2)
n = -[-a), where here and below dOlOI = 'Dg+, with initial conditions
d Ol - 1c I
dOl-I: y(x) = bl:,
x z=+o
k = 1,2,..., n,
( 42.3)
where f(x, y) is the given function and a, b 1 ,..., b n are given constants.
We shall consider a series of theorems on the existence and uniqueness of the
solution of the above problem.
We denote by Rn the following set of points (x, y) in a domain D lying in
Rx R:
Rn = {(." y) ED: 0 < .,  h, I.,n-a y (.,) - r(" : + 1) I  a} ,
n-l hn-I:bl:
a > {;. r(a _ k + 1)'
( 4204)
where a, h and b o are certain constants.
Theorem 42.1. Let f(x, y) be a real-valued function continuous an D and
Lipschitzian with respect to y:
If(x, yt) - f(x, Y2)1  AIYl - Y21
( 42 A')
and let the condition SUP(z,y)ED If(x, y)1 = b o < 00 hold.
Then there exists a unique continuous solution of the Cauchy-type problem
given in (42.2), (42.3) for n = 1 in the domain Rl C D. For the case n = 1 we
refer to (4204).

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 831
Theorem 42.2. Let f(x, y) satisfy the conditions in Theorem 42.1. Then there
exists a unique continuous solution of the Cauchy-type problem given in (42.2),
(42.3) for n = 1,2,... in the domain Rn C D.
Theorem 42.3. Let h:(x, Yl,.. ., Ym), k = 1,2,..., m, be real-valued functions
continuous in a domain Dm C R x R!" and satisfying the conditions
m
Ih:(x, Y1,..., Ym) - h:(x, Z1,..., zm)1 :5 A L IYi - zil, k = 1,2,..., m,
i=1
and sup Ifl:(x,Yl,...,Ym)1 = M < 00. Then there exists a unzque
(z .Yl.. ...y", )ED",
l:=l,2.....m
continuous solution of the Cauchy-type problem
dOl
- d YI: (x) = fl: (x, Yl , . . . , Ym),
x Oi
d Ol - 1 I
d 01-1 YI:(X) = bl:,
x z=+o
k=I,2,...,m,O<Q:51,
in the domain
R, = {(z,V"... ,Vm) E Dm : 0 < Z  h,
1 1-01 ( ) bl: I
X YI: X - r( Q) :5 a,
k= 1,2,...,m},
where a > Mh/r(Q + 1).
Theorem 42.4. Let f(x, Y1,.. ., Ym) satisfy the conditions in theorem 42.3. Then
there exists a unique continuous solution of the Cauchy-type problem
dOl
- d YI:(x) = fl:(x, Yl,..., Ym),
x Oi
d Ol - j I
dx Ol - j YI:(x) = bl:. j ,
z=+o
k = 1,2,..., m, j = 1,2,..., n, n - 1 < Q :5 n,
in the domain

832 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
R.. = {e z, Vb . . . , Vm) E Dm : 0 < z $ h,
Izn-aV.e Z ) - reo< : + 1) I $ a, Ie = 1,2,..., m},
where
n-l hn-jb.
> '" Ie"
a  r ( Q- '+ 1 ) '
,=0 J
bi,o = M, k = 1,2,. . . , m.
Theorem 42.5. Let Pie (x), k = 0,1,..., m, and J(x) be functions continuous on
an interval (0, h). Then the Cauchy-type problem
m d(m-le)Ol
Ple(x) dx(m-le)Ol Yi(x) = J(x), 0< Q $ 1,
d leOl - 1 I
d leOl-l y(x) = ble,
x z=+o
k = 1,2,..., m,
has a unique solution continuous on (0, h).
Theorem 42.6. Under the conditions in Theorem 4e.5 the Cauchy-type problem
m cI<m-le)Ol
LPIe(x) dx(m-1c)Ol Y1c(x) = J(x), n -1 < Q $ n,
1e=0
d leOl - j I
d leOl-j y(x) = bleJ,
x Z=+O
k= 1,2,...,m, j= 1,2,...,n,
has a unique solution continuous on (0, h).
The prool of Theorems 42.1-42.6 differs little from the proofs of the corresponding
theorems for differential equations of integer order. Therefore we prove Theorem
42.1.
Integrating (42.2) where d--QQ = Ig+ we have
d- a dOl d- a
dx-Ol dxOl Y(x) = dx-Ol J(x,y),

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 833
and hence according to the property in (2.61) and the condition in (42.3) we obtain
:e
X a - 1 f (x - t)a-1
y(X) = 6 1 r(Q) + r(Q) J(t, y)dt.
a
( 42.5 )
So, the problem in (42.2) and (42.3) is reduced to (42.5). We show now that if
the continuous function J(x, y) satisfies (42.5), then it satisfies (42.2) and (42.3).
Indeed, applying the operator dOl to (42.5) we have
d a 6 1 iJO d a d- a
dxa Y(x) = r(Q) dx a xa - 1 + dx a dx-a J(x,y),
dOl ( _
and, hence, d:e Ol y x) - J(x, y).
The condition in (42.3) for n = k = 1 can be obtained without difficulty if we
apply the operator dOl--\ to (42.5):
d a - 1 b iJO-1 d- 1
dx a - 1 y(x) = r() dx a - 1 x a - 1 + dx- 1 J(x, y)
:e
=6, + r;l) f 1(1, y(l»dl,
a
and then set x = o.
It follows from our arguments that (42.5) is equivalent in the above sense to
(42.2) with initial conditions given in (42.3).
We accomplish the rest of the proof by the method of successive approximations,
although the method of compressed mappings - Kolmogorov and Fomin [1, p. 73]
- may be also used. Let
x a - 1
Yo(x) = b 1 r(Q) '
:e
x a - 1 f (x - t)a-1
Yn(x) = b 1 r(Q) + r(Q) J(t,Yn-1(t»dt,
o
n = 1,2,....
( 42.6)
First of all we require the points (x, Yn (x» to be lie in R1 for 0 < x  h. The

834 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
estimate
:&
IZ1-OYn(Z) - r) I = ;:; J (z - t)O-l f(t, Yn-l(t»dt
o
b o x b o h
< <
- f(a + 1) - f(a + 1)
( 42.7)
follows from the condition sup I/(x, y)1 = boo If the condition bohlf(a + 1) < a
(:&,y)eD
holds, then (x, Yn(x» E R1 for 0 < x  h.
Now we estimate the difference Yn(x) - Yn-1(X). By (42.7) we have
IY1(X) - yo(x)1  boxOt If(a + 1)
 bohOt If(a + 1).
By using the Lipschitz condition given in (42.4') and the last estimate from (42.6)
with n = 1 we find that
:&
IY2(Z) - Yl (z)1 = r(i Ot ) J (z - t)o-l(f(t, Y1 (t» - f(t, Yo (t»dt
o
:&
 rtOt) J(z - t)O-lIYl(t) - Yo(t)ldt
o
:&
A J 01-1 bot Ot Ab o h20t
 f(a) (x - t) f(a + 1) dt  f(2a + 1)'
o
Repeating such an estimate many times we finally arrive at the inequality
IYn(x) - Yn-1(x)1  An-1bohnOt If(na + 1).
It follows from here that the sequence Yn (x) tends to a certain limiting function
y(x) uniformly with respect to x (0 < x  h). This limiting function is continuous
in (0 < x  h) and satisfies the inequality
Ix1-Oty(x) - b1/f(a)1  a,
which is obtained if we pass to the limit in (42.7) as n -+ 00. Now carrying out

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 835
the passage to the limit in (42.6) as n -+ 00 we obtain (42.5) according to the
continuity of j(x, y).
We prove that the solution y(x) is unique for sufficiently small h. Let
Ah Oi /r(Ot + 1) < 1 and we suppose that there exist two solutions y(x) and Y(x) of
the problem considered. Substituting them into (42.5) and subtracting one from
the other we obtain
:&
f ( t)0I-1
Iy(x) - Y(x)1 = x (Ot) [j(t, y(t» - j(t, Y(t»]dt
o
:&
 r) f (z - t)O-'III(t) - ¥(t)ldt.
o
We suppose that the difference Iy(x) - Y(x)1 admits a maximal value 6 at a certain
point x =  lying on the interval 0 < x  h. Then for x =  from the last inequality
we have that 6  Ar- 1 (Ot)6h Ol Ot- 1 or 1  Ah Oi /r(Ot + 1) which contradicts the
assumption. This completes the proof of Theorem 42.1.
Theorem 42.2 is proved similarly to Theorem 42.1, only in this case we set
n b 1c x Ol - 1c
yo(x) = L r(Ot - k + 1)'
Ie=!
Theorems 42.3 and 42.4 are extensions of Theorems 42.1 and 42.2 to the case
of systems of differential equations of fractional order. Theorems 42.5 and 42.6 are
special cases of Theorems 42.1 and 42.2. .
In conclusion of this subsection we consider two examples in which Theorems
42.1 and 42.2 are used.
Example 42.1. We solve the following Cauchy-type problem
dOl
- d y(x) = y(x),
x Oi
n - 1 < Ot  n,
d Ol - 1c I
dx Ol - 1c y(x) = b le ,
:&=+0
k=I,2,...,n.
Using the proof of Theorem 42.1 we have

836 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
n a-k
1/0(:1:) =  b. r(o:l:_ A: + 1)'
:e
I/m(:I:) = 110(:1:) + rto) J (:I: - tj«-1 I/m-1 (t)dt.
o
Hence for m = 1,2,... we find
n 2a-k
1/1(:1:) = 110(:1:) +   b. r(2: _ A: + 1) '
n 3a-k
1/2(:1:) = 1/1(:1:) + 2  b. r(3: _ A: + 1) '
. . . ,
which in the general case yields
n m+l aj-k
'" ""1 X
Ym(x) = L.J b k L.J >..'- r(a' _ k + 1) '
k:1 j=1 J
m = 1,2,...
After we pass to a limit as m --+ 00 we obtain the following representation for the
sol u tion:
n 00 aj-k
'" ""1 X
y(x) = L.J b k L.i >..'- r(a' _ k + 1)
k=1 j=1 J
00
= L b/cx a - k Ea,l+a_k(>"X a ),
k=1
where E a ,{3(z) is the Mittag-Leffler function defined in (1.91). In particular, if
a = n = 1, then
00 j-l
y( x ) = b '" >..j -1  = b e>.:e
1 L.J r ( . ) 1
j=1 J
and we have "a joining" of the known solution of the Cauchy problem for the
equation of first order and of the above problem for the equation of order a.
Example 42.2. Now we construct the solution of the Cauchy-type problem for

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 837
the inhomogeneous differential equation
dOl
- d y(x)-Ay(x)=h(x), n-l<crn,
x Oi
with initial conditions
dOl-I: I
d Ol-I: y(x) = bl:,
X z=o
k = 1,2,...,n.
Similar to the previous example we have
z
Ym(z) =VO(z) + rto) J (z - t)a-'Ym_,(t)dt
o
z
+ r(l cr ) J (x - t)0I-1h(t)dt,
o
and hence
n m+l OIj-1:
'" ""1 X
Ym(X) =  bl: f;: A J - r(crj - k + 1)
m Aj-l J z
+  1 r(crj) (x - t)OIj- 1 h(t)dt.
J- 0
Passing to a limit as m --+ 00 we find the solution of the Cauchy-type problem
n
y(X) = L bl:xOl-1: EOI.l+OI_t(AX Ol )
1:=1
z
+ J (x - t)0I-1 EOI,OI[A(X - t)OI]h(t)dt.
o
42.2. The Cauchy-type problem for linear differential equation
of fractional order
We consider the linear differential equation of fractional order
n-l
q..y(X) + LPI:(X)q"-k-ly(X) + Pn(X)Y(X) = J(x), (42.8)
1:=0

838 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
where
I7"\U" _ -n0',,-1-n0'''-1 -nO'o
.IJ - £.'0+ £.'0+ ... £.'0+'
k = 1, 2, . . . , n,
I7"\UO _ -nO'o-1
.IJ - £.'0+ '
rc
Urc = L aj - 1, k = 0,1, . . . ,n, 0 < aj  1, j = 0,1, . . . , n
j=O
(42.9)
- obviously, arc = Urc - Urc-1, k = 1,2, . . . ,n, ao = Uo + 1 - and prc(x) and f(x) are
given functions. We require to find the solution y(x) of this equation satisfying the
initial conditions
U"y(x)lz=o = h, k = 0, 1,..., n - 1.
We begin the investigation of this Cauchy-type problem for the case
( 42.10)
Prc(x)=O, k=O,I,...,n,
i.e. we consider the equation
U"y(x) = f(x)
(42.11)
with initial conditions given in (42.10).
The following statement is true.
Theorem 42.7. Let a function f(x) E L 1 (0,a) be represented in the form
dO'..-l _
f(x) = dxO'..-1 f(x), an < 1,
( 42.12)
where j(x) E L 1 (0, a). Then there exists a unique solution of the Cauchy-type
problem defined in (42.11), (42.10) represented in the form
z
n-1 xu" f (x - tt..- 1
y(x) = L brc r(1 + urc) + r(u n ) f(t)dt.
rc=o 0
Proof. It is obviously follows from (42.9) that
dO',,-1 d
u" = U"-l.
dxO',,-1 dx
(42.13)

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 839
Hence, (42.11) may be rewritten as
dOl.. -1 d
_q"-ly(X) = f(x)
dx a ..- 1 dx
or
d- 01..
q..-ly(X) = - d f(x) + b n -1. (42.14)
x-a..
Thus the problem defined in (42.11) and (42.10) is reduced to the problem given in
(42.14) with initial conditions in (42.10) where k = 0,1,..., n - 2. Applying again
(42.13) to (42.14) similar to the above arguments we obtain
d-a..-1 d- a .. d- a ..- 1
q..-y(x) = d d f(x) + d b n -l + b n -2.
x-a..-1 x-a.. x-a..-1
( 42.15)
Now the problem defined in (42.11) and (42.10) is reduced to the problem given in
(42.15) with initial conditions (42.10) where k = 0,1,..., n - 3. Continuing this
process we reduce (42.11) to the equivalent equation
y(x) = -q.. f(x) + -q"-lbn_1 + ... + -q.. f(x)
which has the form
n-l qlc
y(x) = L b lc r(t ) + -q.. f(x)
k=O + Uk
( 42.16)
if we take into account the relation
x qlc
-qlc b k = b k r(1 + Uk) '
We prove that the above function y( x) satisfies the initial conditions in (42.10).
For this we apply the operator qo given by
n-l qlc-qo
-qOy(x) = L bk x + qO-q.. f(x)
1:=0 r(1 + UTe - uo)
to (42.16). Setting here x = 0 we obtain the condition in (42.10) for k = O.
Applying the operator ql to (42.16) and setting then x = 0 we arrive at the
condition in (42.10) for k = 1. Continuing this process we verify that all conditions

840 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
in (42.10) are satisfied. The uniqueness of the solution of the above Cauchy-type
problem follows from (42.16) too. .
Now we formulate the main theorem of this subsection.
Theorem 42.8. Let the functions PI:(x), k = 0,1,..., n, be Lipschitzian, i.e. they
satisfy the Holder condition of order A = 1 - Subsection 1.1 - on an interval [0, a],
and let f(x) be continuous function in [0, a] which can be represented in the form
d Ota - 1 _
f(x) = dx Ota - 1 f(x), ° < ern < 1,
( 42.17)
where J(x) E L 1 (0, a). If ero > 1 - ern, then the Cauchy-type problem defined in
(42.8) and (42.10) has an unique solution continuous on [0, a].
Proof. We set Uay(x) = (x). Then (42.8) has the form of the Volterra integral
equation of the second order
:&
(x) = w(x) + f W(x, t)(t)dt,
o
( 42.18)
where
(x - t)u..-l
W(x,t) = - Pn(x) r(O"n)
n-l (x _ t)Ua-Uk- 1
- LPn-l:-l(X) r( ) '
1:=0 O"n - 0"1:
( 42.19)
n-l X Uk
w(x) = f(x) - Pn(x) L bl: r(1 )
1:=0 + 0"1:
n-l n-l U..-Uk
- LPn-l:-l(X) L b m r(1 x 0" _ 0" r
1:=0 m=1: + m I:
( 42.20)
( 42.19) shows that the kernel W (x, t) has a weak singularity at t = x. A pp lying
the method of successive approximation to (42.18) we find that this equation
admits not more than one solution (x) E L1(0,a) continuous on [O,a]. Hence,
by Theorem 42.7 we deduce the uniqueness of the solution of the problem defined
in (42.8) and (42.10). The proof of the existence of the solution of this problem

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 841
according to Theorem 42.7 is also reduced to the proof of the representation
dOt ,. -1 _
(x) = dx Ot ,.-l (x), i(x) E L 1 (0, a).
(42.21 )
Indeed, since (42.18) admits not more than one solution (x) = o"y(x), then by
Theorem 42.7 it is sufficient to prove that the function
n-l X 0lc 1 / :1:
y(z) =  b. f(l + IT.) + f{IT n ) 0 (z - tt.-l(t)dt
is the solution of the problem given in (42.8) and (42.10). Thus we must to
verify the condition given in (42.21) and in Theorem 42.7. We shall now do this
verifi cation.
Taking into account the conditions ao > 1 - an, an > 0 and the inequalities
Ulc + an  Uo + an = ao - 1 + an > 0, Urn - Ulc + an > an > 0,
in accordance with (2.44) we write the relations
X0lc dOt,.-1 x° Ic + Ot ,.-1
r(1 + UIc) - dx Ot ,. -1 r(ulc + an)'
k = O,I,...,n- 1,
X0".-Olc dOt,.-1 x°".-olc+ Ot ,.-1
- ,
r(1 + Urn - UIc) dx Ot ,.-1 r(u rn - Ulc + an)
m=k,k+l,...,n-l,
which together with (42.17) allow us to write (42.20) in the form
w(x)
dOt,.-1 _ d Ot ,.-1 n-l x 0Ic + Ot ,.-1
dx Ot ,.-l f(x) - Pn(x) dx Ot ..- 1  b lc r(ulc + an)
n-l d Ot ,.-1 n-l x 0 ".-olc+ Ot ,.-1
- L Pn-Ic-l(X) d -1 L r( r
x Ot ,. Urn - Ulc + an
1c=0 rn=1c
( 42.22)
Now we apply the result by Dzherbashyan and Nersesyan [6, p. 17], who proved
that for any functions g(x) E L 1 (0,a) and plc(X) E C([O,a]) there exists a unique
function G(x) E L 1 (0,a) satisfying the relation
d- Ot d- Ot
Pic (x) dx-Ot g(x) = dx- Ot G(x), 0  a < 1.
( 42.23)

842 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Similar statements may be found in Lemmas 3.2 and 10.1.
On the basis of the above result, (42.22) can be rewritten as
d Ol .- 1
w(x) = d 1 W(X),
X Ol .-
( 42.24)
where w(x) is a certain function in L 1 (0,a). Further from (42.18) and (42.19) we
have
d-U" n-l eFt -u..
(x) = w(x) - Pn(X)- d (x) - "Pn-l:-l(X) d (x),
x-U..  xUt- U ..
1:=0
and hence according to (42.23) we obtain
d- N
(x) = w(x) + dx- N v(x), v(x) E L 1 (0, a),
( 42.25)
where
x = min{O"n, O"n - 0"0,..., O"n - O"n-l} = min{O"n, an}.
Hence it follows from (42.24) and (42.25) that
d Ol .- 1 d- N
 ( x) = d 1 w(x ) + - d V ( x).
x Ol .- x- N
If now x  I-an, then the representation in (42.21) has been proved. If x < I-an,
then according to (42.23) there exists PEN such that (p - l)x < 1 - an < px and
d Ol .- 1 d- PN
(x) = dx Ol ..- 1 w(x) + dx- PN v(x), v(x) E L}(O,a).
This completes the proof. .
In conclusion of this subsection we indicate that the simplest Cauchy problems
for differential equations of fractional order
Vo+Y - AY = 0, x > 0, IJ';OIylz=o = 1, ° < a < 1,
( 42.8')
and
Vy - AY = 0, x> 0, y(O) = 1, 0< a < 1, (42.8")
have the solutions y = x Ol - 1 EOI,OI(AX Ol ) and y = exp( _A 1 / OI X), respectively. Here
the fractional derivatives V o + and V are defined in (5.9) and the Mittag-Leffler
function EOI,{J(z) in (1.91).

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 843
42.3. The Dirichlet-type problem for differential equation of
fractional order
We consider the differential equation of second order with fractional derivatives in
the form
Ly =y"(X) + ao(x)y'(X)
m
+ L al:(x)Vg+(WI:(X)Y(X» + a m +l(X)Y(X) = f(x)
1:=1
( 42.26)
on the internal [0,1], where 0 < QI: < 1 and functions ao(x), a m +l(x), at(x), wI:(x),
k = 1,2,..., m, f(x) are continuous on [0,1].
In the theory of boundary value problems for the equations of the form (42.26)
the following two theorems are of importance. The first of them is an analogue of
the Hopf principle - Bitsadze [3, p. 25, 26].
Theorem 42.9. Let Wt(x), k = 1,2,..., m, be non-decreasing positive functions
on [0,1] satisfying the Holder condition with exponents XI: > QI:, 0 < QI: < 1,
k = 1,2,..., m, and ak(x) E C[O, 1], ak $ 0, 0 < x < 1, k = 1,2,..., m + 1. If
y E C 2 (0, 1) is the solution of (42.26) which differs from a constant, then the
positive maximum and negative minimum of y(x) can be only at the end points
x = 0 or x = 1.
Proof. We suppose the opposite, i.e. there exists Xo, 0 < Xo < 1, such that
max y(x) = y(xo) > O. We note that if <p(t) is continuous on [0, x] and satisfies
0<J:<1
the -Holder condition of order X > Q at the point t = x and has a maximum at
this point then (13.1) yields (Do+<p)(x) > O. Therefore, since functions Wk(X) are
positive and non-decreasing on [0,1], then WtY has a positive maximum at the
point xo. Hence there exists 6 > 0 such that the inequalities
o < y(x) < y(xo), al:(x)(Dg+WtY)(x) $ 0, k = 1,2,..., m,
hold for any x E [xo - 6, xo]. From the equation Ly = 0 we have
m
y" + ao(x)y' = - L al:(x)Dg+wl:(x)y - a m +l(x)y.
k=l
Hence
y" + ao(x)y'  0, x E [xo - 6, xo].
( 42.27)

844 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
On the interval [xo - 6, xo] we introduce the preliminary function
z(x) = y(x) + eg(x),
where g(x) = exp( -J.lx) - exp( -J.lxo), J.l > 0, 0 < e < [y(xo) - y(xo - 6)]/g(xo - 6).
Substituting y(x) = z(x) - eg(x) into (42.27) we have
z" + ao(x)z'  eJ.lexp( -J.lx)[P - ao(x)].
If we choose J.l such that J.l > ao (x) for x E [xo - 6, xo], then we obtain
z" + ao(x )z' > O.
( 42.28)
If only z(x) has a maximum on the open interval (xo - 6,xo), then the conditions
z' = 0 and z"  0 hold at the corresponding point. This contradicts (42.28).
Therefore z( x) can have maximum only at the point Xo since
y(xo) - y(xo - 6)
z(xo - 6) <y(xo - 6) + ( 6) g(xo - 6)
9 xo-
= y(xo) = z(xo), z'(xo)  O.
But 0  z'(xo) = y'(xo) + eg'(xo) = y'(xo) - eJ.l exp( -J.lxo) and hence y'(xo) 
eJ.l exp( -J.lxo) > 0 which contradicts the necessary condition of the extremum
y'(xo) = O. Thus the extremum of y(x) can not be at the inner point of the interval
[0,1]. .
The second theorem is an analogue of the Zaremba-Giraud principle - Bitsadze
[3, p. 26].
Theorem 42.10. Let the conditions in Theorem 42.9 be satisfied. If y E
C[O, 1] U C 1 (0, 1] U C 2 (0, 1) is the solution of the equation Ly = 0 given in (42.26)
and
y. = max y(x) = y(l) > 0
Ozl
(y. = min y(x) = y(l) < 0),
Ozl
then y'(l) > 0 (y'(l) < 0). If y. = y(O) > 0 (y. = y(O) < 0), then y'(0) < 0
(y'(O) > 0) provided that the additional conditions hold, namely y(x) E C 1 [0, 1),
WI:(x) E C 1 [0, co], WI:(O) f; 0, k = 1,2,..., m, where co is a small positive number.

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 845
The proof follows directly from Theorem 42.9.
Definition 42.1. We say the problem of finding the solution of (42.26) with the
boundary condition
y(o) = y(l) = °
( 42.29)
is the Dirichlet-type problem for this equation.
Theorem 42.11. Let the coefficients of (42.26) satisfy the conditions in Theorem
42.9 and ao(x) == 0, at(x), wI:(x) E C 1 [0, 1], k = 1,2,..., m. Then the Dirichlet-
type problem defined in (42.26) and (42.29) is solvable unconditionally and uniquely
in the space of functions C[O, 1] n C 2 (0, 1).
Proof. It is not difficult to verify the following equalities
<p(x) = x<p(x) - (x - 1)<p(x)
:& 1
= d [1 t<p(t)dt + 1 (t - 1)<p(t)dt]
o :&
:& 1
= dd x [1 t<p(t)dt + x(x -1)<p(x) + 1 (t - 1)<p(t)dt - x(x - 1)<p(x)]
o :&
 :& 1
= dx'J [1 (x - t)t<p(t)dt + 1 x(t - 1)<p(t)dt]
o :&
, 1
= dx'J 1 G(x, t)<p(t)dt,
o
where
G(x,t) = { t(x -1), t  x,
x(t - 1), t > x.
( 42.30)
Therefore in the case ao(x) == ° we can rewrite (42.26) as

dx'J (x) = 0,
( 42.31)

846 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
where
1
(x) =y(x) + f G(x, t)a m +1(t)y(t)dt
o
m 1 1
+ L f G(x, t)al:(t)(Vgtwl:y)(t)dt - f G(x, t)j(t)dt.
1:=10 0
(42.32)
From the immediate verification of this by using (42.29) we can prove that
(O) = (1) = O. Hence, from (42.31) we derive the relation
(x) = O.
( 42.33)
Hence we have proved that the Dirichlet-type problem given in (42.26) and
(42.29) is equivalent to the integral equation (42.32) and (42.33). The immediate
verification shows that this equation is a Fredholm equation of the second kind.
In view of Theorems 42.9 and 42.10 and the boundary conditions in (42.29) the
homogeneous integral equation (42.32) and (42.33) with j(x) = 0 is equivalent
to the Dirichlet-type homogeneous problem, and therefore it has only the trivial
solution y = O. From here we deduce that the Fredholm inhomogeneous equation
is solvable unconditionally and uniquely, and hence the Dirichlet-type problem is
also solvable unconditionally and uniquely. .
42.4. Solution of the linear differential equation of fractional
order with constant coefficients in the space of generalized
functions
We consider the linear differential equation of fractional order
n
L ajlajy(x) = j(x),
j=1
( 42.34)
where here and below [aj = I+ = V;';j, and with constant complex non-vanishing
coefficients a1, a2, . . . , an and different real exponents a1, a2, . . . , an. This equation
generalizes the Abel integral equations of the first and second kind, and the usual
linear differential equations of integer order with constant coefficients.
We shall find the solution y(x) of (42.34) in the space S+ of tempered
distributions with support in [0,00). One may obtain more detailed information
about this and other terms and notation of this subsection in the book by
Vladimirov [2].

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 847
Let f(x) E S+. We write (42.34) as the convolution
k(x) * y(x) + f(x)
( 42.35)
where
n
k(x) = Eajfa;(x),
j=1
( 42.36)
and a generalized function fa(x) E S+ is
{ x+- 1 /r(n), n > 0,
fa(x) = (N)
fa+N(x), n  0, n + N > 0, N = -[-n] + 1.
( 42.37)
We apply the Fourier-Laplace integral transform to both sides of (42.35). For
f(x) E S+ this transform is defined by
F(z) = L[f(x)](z) = L[J)(x + iy) = V(f()e-(YJO](x),
( 42.38)
where V[g()](x) is the Fourier transform of a generalized function g() E S+
defined in the usual way - Vladimirov [2, p. 105]. We introduce the notation
Y(z) = L[y(x)](z),
n a . ei7ra ;/2
K(z) = L[k(x)](z) = '" J .
L.J za,
j=1
( 42.39)
where the branches of the power functions are given by the condition za; > 0 for
z = x > 0, j = 1,2,..., n. Since f(x), y(x), k(x) E S+, then F(z), Y(z), K(z) is
analytic in the upper half-plane C+ = {z : Imz > O} in the complex plane C.
We denote by H the set of functions G(z) of complex variable z = x + iy,
analytic in C+ and satisfying the estimate IG(z)1  M(1 + IzI2)'/2(1 + y-'),
z E C+, for certain real non-negative constants M, p and q which do not depend on
z. The set H called the Vladimirov algebra, is a multiplicative algebra relative to
addition and multiplication of analytic functions, and multiplication of a function
by a complex number.
The following assertion is true - Vladimirov [2, p. 173].
Theorem 42.12. The algebras S+ and H are isomorphic algebraically and
topologically. The Fourier-Laplace transform is an isomorphism of S+ onto H
and the relation L[k(x) * y(x)](z) = K(z)Y(z) holds for any generalized functions
k(x), y(x) E S+.

848 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Thus (42.34) has a solution in the space S,+ if and only if the solution Y(z) of
the algebraic equation K(z)Y(z) = F(z) is in the Vladimirov algebra H. Further
if Y E H, then a unique solution of (42.34) is given by the relation
y(X) = L- 1 [F(z)IK(z»).
( 42.40)
Here L -1 is the inverse Fourier-Laplace transform which maps H onto S,+.
If a function K(z) of the form (42.39) does not have zeros in C+, then
II K(z) E H and hence Y(z) = F(z) I K(z) E H. In this case (42.34) is solvable in
S+ for each f(x) E S,+.
Now we assume that the function K(z) has I zeros z = Zj E C+, j = 1,2,..., I.
This set can not be infinite since then K(z) == O. Therefore the function
Y(z) = F(z)1 K(z) is in H provided that the conditions F(zj) = L[f(x)](zj) = 0,
j = 1, 2, . . . , I, hold.
We denote by N and No the sums of orders of zeros with positive and zero
imaginary parts respectively, of the function K(z). Also if K(z) vanishes at origin
of coordinates, then we shall not take this zero into account. Then the following
expression is true
1 1
N = -[arg K(z)]-y - -(No - m Otj).
2 2 1Jn
(42.41 )
This follows from the generalized principle of argument - Gabov [1, p. 100]. Here
[w h means a change of w after a circuit in a positive direction along a closed
contour 'Y consisting of an upper half-circle surrounding all zeros of K (z) and of an
interval of the real axis joining this half-circle.
Thus we obtain the following statement.
Theorem 42.13. Let N  0 defined in (42.41) be the sum of orders of zeros with
positive imaginary parts of a function K(z) of the form (42.39) analytic in C+
and being the Fourier-Laplace of the generalized function k(x) given in (42.36). If
N = 0, then (42.34) is solvable in the space S+ for each f(x) E S,+. If N > 0 and
Z1, Z2,..., z, are all zeros of K(z) with orders r}, r2,..., r" r1 + r2 + ... + r, = N,
such that Imzj > 0, j = 1,2,..., I, then (42.34) is solvable in S+ if and only if
the function F(z) = L[f(x)](z) has zeros Z1, Z2,..., z, with orders more or equal to
rl , r2, . . . , r" respectively. If the solution exists, then it is unique and is given by
(42.40). In the case N = 0 the solution of (42.34) can be also represented in the
form y(x) = f(x) * 90(x) where 90(Z) = L -1[11 K(z)] is the fundamental solution
of the operator k( x )*, i. e. a generalized function with support in [0, +00) satisfying
the equation k(x) * 90(X) = 6(x) where 6(x) is the Dirac delta-function.
Let now the right-hand side f(x) in (42.34) be in the space IY+ consisting of
generalized functions with support in [0,00). This is obviously wider than S,+. We
construct the solution y(x) which also is in D'+.

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 849
1/ K (z + ic) E H and the generalized function
Let c be a real non-negative constant such that c > max IZj I. Then
1 j  1
g(x) = e Cz L-l[l/ K(z + ic)]
( 42.42)
is, in general, in the space IY+. We note that g(x) does not depend on c and the
relation g(x) = go(x) holds in the case N == O.
We consider the following convolution
k(x) * g(x) == L -1[K(z)] * e Cz L -1 [1/ K(z + ic)]
= e CZ (L- 1 [K(z + ic)] * L-l[I/K(z + ic)])
= e CZ 6(x) == 6(x),
where 6(x) is the delta-function. From this we obtain that the function g(x) is the
fundamental solution of the operator k(x)* in the space D and
y(x) == f(x) * g(x).
( 42.43)
Hence we arrive at the following statement.
Theorem 42.14. If f(x) E D, then a unique solution of(42.34) in the space D+
is given by (42.43) where g(x) is the fundamental solution of the operator k( x)* in
the form (42.42).
Remark 42.1. A generalized function g(x) in the space D+ can be in a narrower
space. So if g(x) and f(x) are continuous on [0,(0), then y(x) in (42.43) is
continuous on [0,00) and (42.43) has the form
Z
y(x) = f f(l)g(x - t)dt, x > o.
o
This representation is also true in the case when g( x), f( x) E LoC = {<p : <p E
L2(a, b)Va, bE R 1 }.
42.5. The application of fractional differentiation to
differential equations of integer order
We consider two examples in which the theory of fractional differentiation will be
applied to the integration of ordinary differential equations of the second and nth
order.

850 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Example 42.3. We consider the following equation of second order
2 J1-y dy
(a2 + b 2 x + C2 X ) dx2 + (al + bIX) dx + aoy = O.
( 42.44)
We shall find its solution as the fractional derivative y = 1)oz(x) with 1)o = 1)o+
order p of which must be defined. Using the Leibniz relation given in (15.11) for
fractional derivative in the form
x1)tl z(x) =1)tl(xZ(x» - (p + 1)1)oz(x),
x1)t2z(x) =1)t2(x2z(x» - 2(p + 2)1)tl(xZ(x»
+ (p + l)(p + 2)1)oz(x),
from (42.44) we obtain
1)t2([a2 + b2X + C2x2]Z(X»
+ 1)tl([al + bix - 2C2(P + 2)x - b 2 (p + 2)]z(x»
+ 1)o ([ao - bl(p + 1) + (p + 1)(p + 2)C2]Z(X» = O.
We shall find the solution of this equation in the space of functions z(x)
integrable on any finite interval and satisfying the conditions z(xo) = z'(xo) = 0

which we need in order that the operator relations 1)o 1z = 1z1)o and 1)o b =

1)o be valid when p <.0. Then the previous equation can be written in the
form
v. { d [ d (02 + b2 z + C2 z2 ) + 01 + biz - 2C2(P + 2)z - b,,(p + 2)]
+00 - b l (p + 1) + C2(p + l)(p + 2) }Z(Z) = o.
( 42.45)
We define a parameter P as one of solutions of the quadratic equation
ao - b 1 (p+ 1) + C2(P+ l)(p+ 2) = O.
( 42.46)
Then from (42.45) we obtain the simple differential equation
dz 2 [
dx (a2 + b 2 x + C2 X ) = -z al + bix - (2 + p)(b2 + 2C2X)]

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 851
the solution of which is easily given by the method of separation of variables,
namely
z(x) = (a2 + b2X + C2X 2 )P+1 exp { - f a + b 1 x 2 dX } .
a2 + 2 X + C2 X
( 42.47)
Here the parameters a, band c must satisfy appropriate conditions which ensure
that the relations z(xo) = z'(xo) = 0 hold.
Thus the solution of (42.44) has the form
y(x) = Voz(x)
( 42.48)
with the parameter P being defined in (42.46).
The integral in (42.47) can be evaluated in different forms in accordance with
the relations between its parameters. The detailed investigation of all its values
and the corresponding representations of the function z(x) was accomplished by
Holmgren [2] (1967). Here we give only the results concerning one of these cases.
Let C2 f; 0, b - 4a2c2 f; O. Then a2 + b2X + C2 x2 = C2(X - a) (x - (3),
and after simple evaluation, the function z(x) from (42.47) has the form
Z ( x ) = +1 ( x - a ) p-q+1 ( x - lJ ) p-r+1 where q =  r = - and
2 fJ c'J(a-{j) ' c'J(a....{j)
the conditions Re(p - q) > 0 or Re(p - r) > 0 when Xo = a or Xo = (3 respectively
must be satisfied.
The special case of (42.44) is the hypergeometric equation - Erdelyi, Magnus,
Oberhettinger and Tricomi [1, 2.1(1)] - for which a2 = 0, b 2 = 1, C2 = -1, al = C,
b l = -(a + b + 1), ao = -ab. Then the corresponding equation (42.46) yields the
values Pl = a-I and P2 = b - 1. From (42.47) and (42.48), for the first of these
parameters PI = a - I, we obtain the following representation for the solution of
the hypergeometric equation:
y(x) = vg+ 1 X o - c (1- xy-b-1.
( 42.49)
This is a modification of the Euler integral representation given in (1.73), and also
the expression 3 in Table 9.1. Evaluating the integral in (42.49) we arrive at the
representation of the solution via the hypergeometric function in the form
_ r(a - C + 1) l-e . .
y(x)- r(2-c) x 2Fl(1+a-c,l+b-c,2-c,x).
Example 42.4. We consider the following ordinary differential equation of order

852 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
n with binomial coefficients
n dle y
L(ale + ble x ) dx le = O.
le=O
( 42.50)
Using the integra-differentiation operators of arbitrary order we construct one of
the solutions of (42.50) in the form of (n - I)-dimensional integral. For this we
first introduce the polynomials
n
<p(x) = L ale xle ,
le=O
n n
1/J(x) = Lblex le = b n II(x - Ale),
le=O le=l
(42.51 )
where we denote by Al, A2,"', An the roots of 1/J(x) such that Aj f; Ale, j f; k,
j, k = 1,2, . . . ,n. Making the substitution
y = exp(Alx)Y(X), Al f; 0,
( 42.52)
and using the identity
r (e>'lZY(x» = e>'lZ ( AI + .!!.- ) Y(x)
dx m dx
we rewrite (42.50) in the form
I" (\1 +  ) Y(z) + zt/> (\1 +  ) Y(z) = O.
( 42.53)
We shall find the solution of this equation as a derivative of order p, thus
dP
Y(x) = - d YI(X)
x P
where the function YI(X) is integrable on any interval (0, a) and satisfies the
conditions
YI (0) = Y (0) = .. . = yp-I)(O) = O.
( 42.54)

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 853
Then (42.53) is reduced to the following equation
dP+ I { d-l [ ( d ) d-l ( d ) ]
dxp+l dx-l <P Al + dx - (p + 1) dx-l ,p Al + dx
d- l ( d ) }
+x dx-l ,p Al + dx YI(X) = O.
( 42.55)
We introduce the polynomials
,pI (X) = X-I,p(AI + x), <PI (X) = X-I [<p(AI + x) - (p + 1),pI(X»),
( 42.56)
Since ,p(AI) = 0 then ,pI (x) has order n - 1. By setting
<p(AI) <p(AI)
p + 1 = ,p1(0) = ,pH AI) = crl,
( 42.57)
we obtain that X<PI (x) vanishes at x = 0 and hence <PI (x) is a polynomial of order
n - 1. We assume that crl < O. Then the relation
, (:20 ) 1/'(20) + 2O.p, ( :., ) 1/,(20) = 0
( 42.58)
follows from the homogeneous Abel equation of order -p - 1 given in (42.55) and
written in terms of the notation in (42.56). Hence under the conditions in (42.54)
and crl < 0 (42.50) is reduced to the equation (42.58) of order n - 1.
Continuing by analogy the above procedure of lowering the order, we arrive at
the following system of differential equations of orders crj - 1:
tfX 1 - I
y(x) = e>'lZ YI(X)
dx Ot1 - 1 '
dOt-l
( ) _ >'U Z ( )
YI X - e dxOt-l 112 x ,...,
( 42.59)
d Ot ..- 1 - 1
Yn-2(X) = e>'l."-Z d I Yn-l(X).
x Ot ..- 1 -
Here Yn-l(X) = exp(AI,n-Ix)(an + bnx)-Ot.. is the solution of the last simple
equation of first order
n-' (:20 ) !/n-'(2O)+ 2O.pn-' ( :., ) I/n-' (20) = 0,
( 42.60)

854 CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
and x = l.m is the root of the equation ,pm (x) = 0 and
,pm (x) = X- 1 ,pm_l (l.m-l + x), m = 1,2,.. ., n - 1,
,po(x) = ,p(x), 1.0 = 1,
(42.61 )
<Pm(x) = x-l[<Pm_l(l,m_l + x) - Qm,pm(x»), m = 1,2,..., n - 1,
<PO(x) = ,p(x), (42.62)
Qm = <Pm-l (l.m-l)/,p:n-l (l.m-l), m = 1,2, ..., n.
( 42.63)
We prove that 1,m = m+l - m' m = 1,2,... ,n - 1. From (42.56), (42.51)
and (42.61) we have the relations
n
,p2(X) = x-l,pl(l.l + x) = x- 1 b n II (x + 1 + 1,1 - I:).
1:=2
Since x = 1.1 is a root of the equation ,pl(X) = 0, then we can set 1.1 = 2 - 1'
Continuing such arguments for 1,m' m = 2,3,..., n - 1 we obtain
n
1.m = m+l - m, ,pm(x) = b n II (x + m - I:),
I:=m+l
m = 1,2,..., n - 1.
( 42.64)
The conditions in (42.54) and Ql < 0 are concerned with the function Yl(X).
Similar conditions for other functions YI: (x) have the form
YI:(O) = yHO) = ... = yn-I:)(O) = 0, k = 1,2,..., n - 1,
QA: < 0, k = 1, 2, . . . ,n, an = 0,
( 42.65)
the last condition an = 0 follows from Yn-l(O) = O. Under these assumptions after
convoluting the system in (42.59) we obtain the following representation for one of
solutions of (42.50):
da1-1 da-1
Y ( x ) =b-a"e>'lZ e(>'->'l)z ...
n dxa1-1 dxa-1
d a ..- 1 - 1
x e(>...->...-l)Zx- a ...
dx a ..- 1 - 1
( 42.66)

 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 855
One may prove that (42.65) yields the following conditions for QA::
Q1 < 0, Qn < -1, Qn-1 < -1,
( 42.67)
Qn-2 < -2,. . . ,Q2 < -n + 2.
They can be extended if we use analytic continuation of the Abel integrals in
( 42.66).
The special case of (42.50) for n = 2, a2 = 6 0 = 0, 6 2 = -6 1 = 1 is the
Kummer degenerate hypergeometric equation
xy" + (e- x)1/ - ay = 0.
( 42.68)
For this equation
cp(x) = ex - a, ,,(x) = x 2 - x,
1 = 1, 2 = 0, Q1 = e - a, Q2 = a,
and (42.67) has the form Q1 < 0, Q2 < -1. The solution given in (42.66) for
(42.68) has the form
dC- a-1
Y ( x ) =e:& e-:& x-a
dx c - a - 1
:&
f ( t ) a-c
=e:& X - e-'t-adt
r(1 + a - e)
o
r(1 - a) 1-c:& ( . 2 . )
r(2-e) x e 1 F 1 1 - a , -e,-x,
( 42.69)
where 1F1 is the Kummer degenerate hyper geometric function defined in (1.81),
and give also by formula 9 in Table 9.1. Here the conditions e - a < 0, a < -1
can be extended to the ones e - a < 1, a < 1 which ensure the convergence of the
integral.

856
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
fi 43. Bibliographical Remarks and
Additional Information to Chapter 8
43.1. Historical notes
Note to  40.1. The presentation in this subsection follows the book by Vekua [3;  6, 10]
(1948).
Notes to  40.2 and 40.3. The method of fractional integro-differentiation for the
differential equations in generalized axially symmetric potential theory was first considered by
Weinstein [2-5] (1952-1955). In the paper [5] (1955) and also [7] (1960), he proved relations
connecting the solutions of (40.19) with J.l. = -\ = 0 with eaCh other for different values of the
parameter p by means of the fractional integral (Lemma 40.2 with -\ = 0). This idea was
developed by Erdelyi [8] (1963), [10, 11] (1965) and [14] (1970) who investigated the properties
of the differential operator Lz) given in (40.22). In particular, in the papers [8] (1963) and
[10] (1965) he proved Lemma 40.2 in the case -\ = 0 and its analogue for the ErdByi-Kober
operator K",OI defined in (18.8) -  43.2, note 40.1. On the basis of these statements Erdelyi
found relations connecting the solutions of (40.18) and (40.19) with J.l. = -\ = 0 with each other
for different values of p by means of the ErdByi-Kober operators. The results of Erdelyi were
generalized by Lowndes [5] (1979), [7] (1981), [9] (1985) who proved Lemmas 40.1-40.3 and the
relations in (40.23). We also note that such an idea embryo was in fact suggested in Poisson [1]
(1823). "Expressions of confonnity" of the fonn (40.20) were indicated by Weinstein [3] (1953)
though such results had already appeared in Darboux [1] (1915).
The presentation in 4O.2-40.3 follows the papers by Marichev [4] (1976), [9] (1978) with
certain modifications in (40.21)-(40.23) and with inserting the regulator (signxt + 1)lsign IJI in
(40.26), (40.27) and (40.48).
Notes to  41.1 and 41.2. Equation (41.1) as a special case of a more general equation
was first obtained by Euler [I, p. 177, 426-432] (1772) in connection with his investigations
concerned with the motion of air in pipes of different section, and with the vibration of strings of
variable thickness. Euler found the solution of (41.1) with 0 < {3 = (3. < 1/2. Such an equation
of the fonn (41.25) with q = 0 was solved by Poisson [1] (1823), who obtained the hyperbolic
analogue of the representation for the solution given in (41.22) called the Poisson representation.
The general solution of (41.1) for {3. = {3 was given by Riema.nn [I, p. 40, 381-395] (1860).
He constructed the solution of the Cauchy problem by using a certain auxiliary function, and the
method was called by his name later - (41.14). The fundamental solution of (41.2) with q = 0,
i.e. the solution of (41.3"), was first indicated by Beltrami [1] (1881) in the case 2p = 1. This
result was extended to p > 0 only in 1948 by Weinstein [1], who found two representations for
such a solution.
We note that in 1915, much later than Euler and Poisson, (41.25) with q = 0, 0 < p < 1
occurred in the book by Darboux [1] who called it the Euler-Poisson equation in connection with
investigations of problems of the curvature of surfaces. Thus many authors call equations of
the fonn (41.1), (41.25) and (41.26) Euler-Poisson-Darboux equations, although Euler-Poisson
equations are their more precise names.
Great attention to such equations arose after the first edition of the book by Tricomi [1]
in 1923. In this book equations of the fonn (41.1), (41.2) and (41.25) with q = 0, p = 1/6
played the main role in the investigation of a boundary value problem for equations of "mixed"
elliptic-hyperbolic type YUzz + Uyy = 0, and later called the Tricomi equation. For more details
on this subject we refer to Bitsadze [1-3] and Smirnov [1-3,7].
One may find more detailed historical infonnation in the papers by Weistein [3] (1953), [8]
(1965). We note only that [3] contains probably the first reducibility of the general equations
(41.3') to (41.3"), and the representation in (41.22) relative to (41.2) with q = 0, 0 < p < 1/2.
The presentation in the beginning of Subsection 41.1 follows the book by Tricomi [1] (1957).
The relations in (41.18) and (41.19) were obtained by Marichev [5] (1976). Theorem 41.2 for

 43. ADDITIONAL INFORMATION TO CHAPTER 8
857
q '# 0 was proved by Marichev. The relations in (41.23) and (41.24) were found by Berger and
Handelsman [1] (1975).
Theorem 41.3 was proved by Gordeev [1] (1968). In the case {3. = {3 this result in other
notation was known earlier - Bitsadze [1-3], Smimov [1-3] and Gilbert [2]. The space Rl was
introduced by K.1. Babenko [1] (1951) who proved Theorem 41.4 - see also [2] (1985).
Notes to  41.3. Equation (41.35) was first considered by Poisson [1] (1823) in the case
n = 3, p == 1. The Cauchy problem given in (41.35) and (41.36) was investigated by Weinstein
[2] (1953) for different values of the parameter p. He obtained the solution of this problem in
the form (41.40). Weinstein indicated and used the relations u = yu p + 1 , uP = yl- 2p u 1 - p ,
connecting the solutions u = uP of the equation of the form (41.35) with each other for different
values of p - (40.20).
The presentation in subsection 41.3 follows the paper by Berger and Handelsman [1] (1975)
except for (41.46)-(41.49) obtained by Lowndes [8] (1983).
Notes to  41.4. The Dirichlet problem for (41.2) with q = 0 was first considered by Velma
[2] (1947). In particular, he proved Theorem 41.5 in this case. Theorem 41.5 for any real q as
well as other results in subsection 41.4 were obtained by Marichev [5] (1976) with certain more
accurate definitions in the case p = 1/2.
Notes to  42.1. The paper by O'Shaughnessy [1] (1918) was probably the first where
the methods for solving the equation nl/2y = y/x were discussed. Two solutions of such an
equation were suggested by O'Shaughnessy and discussed by Post [1] (1919). They were essential
different because really they were the solutions of two different equations, namely V2y = y/x
and n1.}2y = y/x, but this was not taken into account by O'Shaughnessy and Post. Later
Mandelbrojt [1] (1925), see also Volterra [3, p. 99] (1982), arrived at a differential equation of
1
fractional order in investigating the extremum problem for the functional J F[V:+y(x)j x]dx.
o
Mandelbrojt had assumed that the cOlTesponding variations are equal to zero and obtained the
differential equation of fractional order Fa [V:+y(x)jx] = 0 with Cauchy conditions. The paper
by M. Fujiwara [1] (1933) where, in particular, the equation (+y)(x) = (ax- 1 )ay(x), a > 0,
containing the Hadamard fractional differentiation operator V+ defined in (18.54) was considered,
may also be regarded as a prehistory of the theory of differential equations of fractional order.
The first serious step in this theory was made by Pitcher and Sewell [1] (1938) who proved
Theorems 42.1 and 42.2 on the existence and uniqueness of the solution of the Cauchy-type
problem for the equation (V:+y)(x) = l(x, y) under other conditions than those in subsection
42.1. Ban-ett [1] (1954) obtained the solution of (42.11) provided that the conditions in (42.10)
hold. These results were generalized later in the papers by AI-Bassam [4] (1965), [8] (1982),
AI-Abedeen [1] (1976) and Al-Abedeen and Arora [1] (1978). They proved certain theorems
similar to the corresponding theorems from the theory of linear ordinary differential equations.
The presentation in  42.1 is based on the simplified results from the above papers.
Notes to  42.2. The main presentation in this subsection follows the paper
by M.M. Dzherbashyan and Nersesyan [6] (1968). The solution of (42.8') is given in
M.M. Dzherbashyan (7), and the solution of (42.8") is well known, for example, Titchmarsh [1].
Notes to  42.3. The Dirichlet problem for (42.26) was investigated by M.M. Dzherbashyan
[8] (1970), Nahushev [4] (1976), [5] (1977) and Aleroev [1, 2] (1982). The presentation in  42.3
follows the above papers by Nahushev.
Note to  42.4. The presentation in this subsection follows the papers by Didenlro [1..".s]
(1984), Kochura and Didenko [1] (1985).
Notes to  42.5. The idea of applying fractional integra-differentiation to constructing
solutions of ordinary differential equations was first suggested by Liouville [3] (1832), who
considered (42.44). Using the idea of Liouville this equation was studied by Holmgren [2] (1867),
Sohncke [1] (1867) and Letnikov [4, part III] (1874). A more complete and detail investigation
was given by Letnikov.

858
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Equation (42.50) was also investigated by many authors. The methods of fractional calculus
were used by Letnikov [9] (1888), Nekrasov [2] (1888), [4] (1891) and Karasev [1] (1957), and also
Alekseevskii [1] (1884).
The presentation in  42.5 follows the papers by Letnikov [4] (1874), [9] (1888) and Holmgren
[2] (1867).
43.2. Survey of other results (relating SS 40-42)
40.1 Erdelyi [8, 10, 14] proved the statement for the second Erdelyi-Kober operator in (18.8)
similar to Lemma 40.2 in the case .x = o.
Lemma 43.1. Let cv > 0, f e 0 2 (0,00) and x 2 '1-1 f(x) and x 2 '1 f'(x) be integrable at infinity.
Then
LZ) K-'1,Otf(x) = K_'1,OtLzJ.Otf(x)
where Lz) and K'I,Ot are given in (40.22) and (18.8), re'pectively.
It was noted that by using the relation Lz)(x-2'1 f(x» = x- 2 '1 Lc.:J f(x) the relations
lOt L(z) f L(z) lOt f lOt L(z) f L(z) lOt f
O+;z '1 = '1-Ot O+;z' -;z '1 = '1-Ot -;z
follow from Lemmas 40.2 and 43.1 Wider the appropriate assumptions indicated by Erdelyi [8].
These relations were generalized by Lowndes [5], thus
JOt L(z) f - ( L(z) + .x 2 ) JOt f ROt L(z) f - ( L(z) _ .x 2 ) ROt f
>. '1 - '1- Ot >., >. '1 - '1-Ot >.
where the operators Jf and Rf are connected with the generalized Erdelyi-Kober operators
J>.(fl, cv), R>.(fl, a) defined in (37.45) and (37.46) by the expressions
Jf = x 2a + 2 '1J>.(fl,CV)X- 2 '1 f, Rff = x- 2 '1 R>'(fl,CV)x 2Ot + 2 '1 f.
In particular (Lowndes [7]), the following representations hold
Z
J>.f(x) = Jf(x) = JIf'(x) = f Jo(.x ...jx 2 - t 2 ) f'(t)dt
o
Z
= f(x) - .x f p J1(.x ...jx2 - t 2 ) f(t)dt,
x 2 - t 2
o
f(O) = OJ
00
R>.f(x) = Rf(x) = -RIf'(X) = f Jo(.x ...jt 2 - x 2 )f'(t)dt
Z
00 .
= f(x) - .x f  J1(.x ...jt2 - x 2 ) f(t)dt,
t 2 - x 2
Z
provided that t- 1 / 2 f(t) - 0 as t - 00 and the operators inverse to J>. and R>. are evaluated from

 43. ADDITIONAL INFORMATION TO CHAPTER 8
859
the results (J>.)-1 f(x) = Ji>.f(x), (R>.)-l f(x) = >.f(x) - (37.57) and (37.58). We mention
that another form of the first relation was indicated by Vekua [3, p. 69].
Lowndes [7] also proved two statements similar to Lenuna 40.1.
Lemma 43.2. Let f e C 2 (b, 00), b > 0, and x- 1 / 2 f(x) _ 0, x 1 / 2 f'ex) _ 0, x- 1 / 2 f"(x) - 0,
as x - 00. Then R>.f"(x) = (-&s - ,X2) R>.f(x), ,X  o.
Lemma 43.3. Let f e C2(b, 00), b > 0, and f(k)(x) = O(e- 6z ) as x - 00, S > ,X  0,
k = 0, 1,2. Then Ri>.f"(x) = (-&s + ,X2) R;>.f(x).
All these results were applied to solving certain boundary value problems for the Laplace
equation with mixed boundary conditions. It was noted that on the basis of Lemma 43.2 we can
obtain the fundamental801utions v = Ko('xr) and v = r- 1 e->.r of the generalized Helmholtz-type
equations (2 - ,X2)v = 0 and ( 3 - ,X2)v = 0 from the fundamental solutions u = -Inr,
r = V x 2 + y2 , and u = r- 1 , r = V x 2 + y2 + z2, ofthe Laplace equations 2u = Uzz + Uyy = 0
and 3U = Uzz + Uyy + Uzz = 0 using the result v = R>. u.
Let Ip('l, cv) be the operator given in  39.2 (note 37.4),
( z ) d d d 2 d
M = x 1 --y _x 1 +-y - = x 2 _ + (1 + ")')x-.
-y   d 
Lowndes [10] proved that if cv > 0, f e C 2 (O, b), b > 0, x'1+m f(m) (x), m = 0,1,2, is integrable
at 0, and x'1+1 f(x) - 0 as x - 0, then
I p ('l, cv)M(l+'l)f(X) = [M(l+'l) + (1'x)2]I p ('l, cv)f(x)
where l' = ,X or l' = i,X,'x > O. He obtained a similar statement in the case cv < 0, and applied
these results to finding the complete set of solution of (40.18) and the equation
m
L a2u a 2 u 21' au 2
-+-+--+,X u=O,
ax 2 ay2 y ay
k=l k
l' > -1/2,
from the corresponding complete set of solutions of the above equations with ,X = o.
The results similar to Lenunas 40.2, 40.3 and 43.1 and connecting the operator Lz) with
the operators I:' and t.'fj, defined in  23.2, note 18.6 were obtained by Kilbas, Saigo and
Zhuk [1]. For example, they proved that jf cv > 0,  and (3 are real numbers, v = mineO,  - (3),
S = 0 if  '# {3 and S = 1 if  = (3, f e C2(O, b), b > 0, X Il + k (Inx)6 f(k)(x), k = 0,1,2, is integrable
at 0, then
IOI,fj'x2L(z) f( x ) - x 2 L(z) IOI,fj' f( x )
0+ -fj/2 - fj/2 0+ .
40.2. The method used in  40.2-40.3 and based on Lemmas 40.1-40.3, 43.1-43.3 was
applied by Erdelyi [11] to the generalized Stokes-Beltrami system
y 2p uz = Vy, y 2p uy = -vz,
(43.1 )
the solutions (u, v) of which called (21' + 2)-dimensional conjugate symmetric potentials. Such
an idea is conditioned by the fact, follows from (43.1), that u is a solution of (41.3") with
r = y, called a (21' + 2)-dimeftsional ,ymmetric potential. Extending the investigations by
Pahareva and Virchenko [1] and also Polozhii [1-3], Erdelyi [11, p. 221] proved that if (u,v)

860
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
is a (21' + 2)-dimensional potential, then (f Y ) 1/2 u, y2a I o (Y)v) is a (21' + 2a + 2)-dimensional
'1'- ,a ,a
potential provided that l' > -1/2, l' + a > -1/2. Here Il is the Erdelyi-Kober operator 1",a
given in (18.8) and applied with respect to y.
ErdByi [14] used the technique of fractional integra-differentiation to extend the results of
Friedlander and Heins [1] who considered (40.19) with IJ. = .x = 0, i.e. (41.25) with q = o. Erdelyi
applied his idea to derive representations for the solutions of the fonn (41.6) from the solution of
the wave equation. Earlier this idea was used by Copson and ErdByi [1] to investigating solutions
of a certain boundary value problem for (40.19) with .x = O.
40.3. Representations more general than (41.22) of the solution via analytic functions of
the fonn (41.22) for (40.18) with IJ. = .x = 0 and of the fonn (40.33) for (40.18) with .x = 0 were
obtained by Krivenkov [1] and Henrici [I], respectively.
40.4. Copson [5] considered the Dirichlet problem for the hyperbolic equation (40.19) with
.x = 0 in the quadrant x > 0, y > O. By using the Riemann method he constructed the solution
of this problem for x < y and x > y. He then showed that this solution and its derivatives
are continuous for sufficiently large IJ. + l' when they pass across the line y = x, provided that
the given boundary values u(x, 0) and u(O,y) satisfy the equation u(x,O) = u(O, x) if IJ. = 1', and
the equation u(O, x) = :t x-I IIJ-l,p-IJU(x, 0) if l' > IJ.. Here 1",a is the ErdByi-Kober
operator defined in (18.8).
40.5. Weinstein [7] studied certain properties of the operator Lz) defined in (40.22) and
applied them to investigating the solutions of the equation
n ( )
a 2 u PI: au
-+-- =0
?; ax XI: aXI: '
PI: = const.
( 43.2)
In particular, he proved that if u( xl, x2, . . . ,Xn) is any solution of this equation, then
2-n-Pl-p'J-".-'P.. ( Xl x2 Xn )
r u r 2 ' r 2 ,..., r 2 '
n
r 2 = LX'
1:=1
is another solution. In the case 1'1 = ... = pn = 0 this property is sometimes called Kelvin's
theorem.
40.6. Radzhabov [1-4] and Radzhabov, Sattarovand Dzhabirov [1-2], and also the papers
cited there, investigated in detail the properties of solutions, including fundamental solutions, for
equations of elliptic type of the fonn (40.18) and (43.2) with singularities in coefficients, and for
certain their analogues and iterative generalizations. They obtained integral representations for
the solutions of these equations and the solutions of the certain boundary value problem such
as Dirichlet, Neumann, "mixed", etc. They used and developed the ideas connected with the
relations given in (40.20) and Lemma 40.2, and the methods by Gilbert [2] and Weinstein [6].
They found the structure of solutions for the iterative equations of high order decomposed as
compositions of equations of the fonn (43.2).
Such equations of mainly hyperbolic type, and the Cauchy and Cauchy-Goursat boundary
value problems for them were investigated by Kapilevich [1-4], etc. He used the technique of
hypergeometric functions of several variables. In particular, in the papers [1-3] he first constructed
the Riemann and Green-Hadamard functions for (40.19), and found the solutions of the Cauchy
and Cauchy-Goursat problems. In [4] Kapilevich obtained the solution of an analogue of a
half-homogeneous problem given in (41.36) for the equation
( a2 a 8 2 8 2 ) m 2m
-+--+b -- u-c u=O
88 2 8 a8 8x 2

 43. ADDITIONAL INFORMATION TO CHAPTER 8
861
in terms of the integral operator involving the function oFm-l(at.....am-ltz) in the kemel-
ErdByi. Magnys. Oberhettinger and Tricomi [1].
40.7. Chen [1. 2] investigated properties of tne solutions of the equation mu:u + U = 0
near the singular line  = 0 with dependence on properties of the functions U(x,O) = Uo(x)
and U(x,O) = utCx). in particular. on their Holder nature - Chen [1]. Making the changes
2p = m(m + 2)-1. r = (1 - 2p)I/(I-2p) and U(x.) = u(x.r) he reduced this equation to the
one given in (41.3"). 0 < p < 1/2. and applied the method of analytic continuation in a complex
domain in two variables to the latter equation. This method is presented in the papers by Lewy
[1. 2] and it is adjoined to the known method given in the book by Vekua [3]. According to
such an approach each solution U(X.71) of the above equation corresponds to a certain analytic
function. the real and imaginary parts of which are expressed for ( = 0 via fractional integrals of
the functions Uo(x) and Ul (x). In this paper Chen [2] indicated such integer solutions U(X.71).
fractional derivatives of which have the norms satisfying special estimates in terms of the norms
of utCx) or l-l(Uo(x) - Uo(a» in the space Lq.
Usanetashvili [1] proved the existence and uniqueness of the regular solution of the mixed
boundary value problem for the equation m Uzz + u = O. m = const > O. in which the values
of the conormal derivative of the desired solution are specified in the elliptic part of thedmundary,
and the conditions on the line of degeneracy  = 0 contain the Riemann-Liouville fractional
integrals and derivatives defined in subsection 2.3.
40.8. The uniqueness and existence of the solution of certain boundary value problems with
boundary conditions involving the Riemann-Liouville fractional integra-differentiation operators
were proved for equations of "mixed" type by Hasanov [1]. lsamukhamedov and Oramov [1].
Ivashkina and Nevostruev [1]. Ktunykova [1]. Nahushev [6]. Salahitdinov and Mengziyaev [1]. For
the generate hyperbolic equations they were proved by Kumykova [2. 3]. Orazov [2], Salahitdinov
and Mirsubarov [1. 2].
Nahushev and Borisov [1] and Zhemukhov [1] investigated the first. second and mixed
boundary value problems for the loaded parabolic equation. and the Darboux problem for the
second-order degenerate loaded hyperbolic equations. respectively. The loaded parts contained
the Riemann-Liouville fractional integrals and derivatives -  43.2. note 42.5.
40.9. Kochubei [2] considered the Cauchy problem
(DCa)x)(t) = Ax(t). 0 < t < T. x(O) = Xo.
for the equation with the closed linear operator A in a Banach space. and the "regularized"
fractional derivative
(DCa)x)(t) = (vg+x)(t) - [r(1 - a)]-l t -a x (O). 0 < a < 1.
where vg+ is the fractional differentiation operator defined in (2.22). He found conditions on the
resolvent (A - 'xE)-1 of the operator A which yield the unique solvability of the Cauchy problem.
The results were applied to the "mixed" problem for differential equations in partial derivatives
with fractional differentiation in the "time" variable.
The case when A = L is the second order elliptic differential operator in n variables was
especially treated in Kochubei [3]. The corresponding Cauchy problem:
(a) )
(D, x (y. t) = Lx(y. t),
yeR n .
o < t < T.
u(y.O) = Ip(Y).
known as the fractional diffusion problem. was proved to have a unique solution under the
appropriate assumption on Ip(Y) and the coefficients of the operator L. In the case when L is
Laplacian. the fundamental solution was explicitly found in term of the Fox H-function, and its
integrability in y over R n was proved. In this connection we refer to the papers by Wyss [1] and
Schneider and Wyss [1].

862
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
40.10. Chanillo and Wheeden [1] applied two-weighted estimates for the fractional
integration operators -  29.2, note 25.8 - to obtain information about the number of negative
eigenvalues of generate elliptic operators in divergence fonn. Kennan and Sawyer [1] used their
characterization of the so-called trace inequality for potential operators to study the domain and
essential spectrum of the SchrOdinger operator H = - - V, where  is Laplacian and V is a
nonnegative measurable function on R n .
41.1. Following Vekua [2] (Theorem 41.5), Olevskii [1] solved the Dirichlet problem for
(41.2) with q = 0 in the cases of a half-sphere and a half space in Jr1, under certain conditions on
p. In the case of a half-sphere this result was extended by Huber [1] to arbitrary p. The analogues
statement for the equation of the form (40.18) with .x = 0 in the case of a quarter sphere in Rn
was obtained by Hall, Quinn and Weinacht [1] for any p and J.l..
Volkodavov [1] and Evsin [I, 2] constructed the fundamental solutions of (41.2), and solved
the Dirichlet problem for the semi-disk {x 2 + y2 < I, y> O} and the Neumann-Dirichlet problem
for the domain for which the boundary contains the interval [-1,1] of the axis Ox, respectively.
41.2. The Cauchy problem given in (41.35) and (41.36) was studied by Diaz and Weinberger
[I], Blum [1] considered all values of p including the singuJar ones, 2p = -I, -3, -5,.... We also
note that one may find reviews of investigations concerning the singular Cauchy problem given
in (41.36), and the regular Cauchy problem with the conditions of the fonn (41.36) on the line
y = e > 0, in the papers by E.C. Young [1] and Asral [I], respectively.
41.3. Saigo [2, 4, 5], see also [6-8], studied the three boundary value problem -for the
Euler-Poisson-Darboux equation (41.1) in the domain 0 <  < f1 < 1 with boundary conditions
involving the integral operators If'" and I!'" given in  23.2, note 18.6. The first
Goursat-type problem has the following boundary conditions
I a,b,p* -a-l (0 ) ( )
0+ U , f1 = </'1 f1 ,
If,P-c-lU(, 1) = </'2()'
Two other problems are the so-called problems with shift, and their investigation was begun by
Nabushev [1]. In this context we refer also to Bzhikhatlov, Karasev, Leskovskii and Nahushev [1].
The boundary conditions have the form
U(,) = </'1 (),
AI:'P. -a-lu(O,) + BltP-P* ,c,P* -a-lu(, 1) = </'2()
in the second problem and
u(,) = </'1(),
A(b+P+p. -1 I:'P* -a-1u(O,)
+ B(1 - )c+P+P*-1 Ir: p - p * .c.p*-a-lu(, 1) = </'2()
in the third one. Here A, B, a, b, c, d are given constants and </'1 and </'2 are given functions in
all three problems. The second problem coincides with that in the paper by Nabushev [1] in the
case when A and B are given functions and {3. = {3, a = -b = -c = {3 - 1. All these problems
were reduced to singulal' integral equations with a Cauchy kernel, and methods from the book by
Gabov [1] were applied to their solution.
Srivastava and Saigo [1] expressed solutions of the above problems in terms of Kampe de
Feriet functions of two variables. They also considered some of the special circwnstance in which
the Kampe de Feriet function can be reduced to the relatively simpler hypergeometric functions.
We also note that Orazov [1] generalized the results of Nabushev [1]. Volkodavov and Repin
[1] solved one more problem of such a type with the operators I:'" and I!'''' Repin [1]

 43. ADDITIONAL INFORMATION TO CHAPTER 8
863
investigated the above second boundary value problem for the equation y 2 UZZ - usfsf + duz = 0
in the special domain.
41.4. By using a Fourier transform in the space of the generalized functions Bresters [3]
constructed the solution of the Cauchy problem given in (41.36) for the equation
n
"" 8 2 1£ _ 8 2 1£ _ 2p 81£ _ 2u = o.
L.J 8x 2 8y2 y 8'1/
k=l k
A relation for the solution has the form
1
2 f C08(Y)
u(x,'I/) = _ I 1 -r(x+ yt) c-:? dt
Sn V 1 - t 2
-1
- (41.37) - and generalizes the classical solution for such a problem from the paper by E.C. Young
[1].
41.5. Bureau [1-3] investigated the Cauchy problem for the equation in partial derivatives
of the hyperbolic type. In particular, the wave equation (40.19) and the Euler-Poi88OD.-Darboux
equations (41.1) and (41.35) were considered. He used the concepts of a finite and logarithmic
parts of di vergent integrals denoted by p J and pi respectively, and related to the Hadamard
definition -  5.5. In particu1ar, by using these ideas some results from the papers by Weinstein
[1-3] were extended. We give some of his definitions.
Let A(t) e em be represented in the form
m-l
A(t) = L AX) (t - x)k + Bm(t), Ak(X) = A(k)(x),
i=O
in the neighbourhood of a point x and let
z
I.(x) = f A(t)(x - t)'dt,
a
,
P(Yi3,U+ 1) = L(-I)k k:) yk-U.
k=O
Then we set
z-£
pJI-m-lJ(x) = lim [ f A(t)(x - t)-m-lJdt - P(ei m -I,m + JL)]
£-0
a
z
=P(a - Xi m -I,m + JL) + f Bm(t)(x -1)-m- lJ dt,
a
z
m 1 Am_l(X) f Bm(t)
pJI_m(x)=P(a-xim-2,m)+(-I) - ( In(x-a)+ ( ) dt,
m -I)! x - t m
a

864
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
( ) ( ) m Am-l (x)
p lI_m x = -1 ( ) i
m-l !
plI.(x) = 0,
8 ':# -m,
where 0 < #J. < I, m = 1,2,3,....
Bureau proved basic properties of p j 18 and p 118 such as various estimates, conunutation
with the operator of differentiation and connection with the Cauchy principal value of an
integral. He extended all of these results to multidimensional integrals, and made a survey of the
applications of these ideas to partial differential equations.
41.6. Cheng [1, 2] investigated some "mixed" boundary value problems with a condition on
the singular line for (41.25) with q = 0 and its generalization.
41.7. Wood [1] investigated the problem of finding the fundamental solution of the equation
Uzz + Uyy + Uu + w 2 c- 2 (z)u = 0 provided that the fundamental solution for the simpler equation
Uzz + Uzz + w 2 c- 2 (z)u = 0 is known. In the case c(z) = z this problem was solved by using
fractional integration.
41.8. Clements and Love [1] considered two "mixed" Heumann-Dirichlet problems
concerning finding a function V( r, z), r = V x 2 + y2 , hannonic in the half-space z > 0 provided
that the values V z and V are given in different parts of the plane z = 0 with a jump for r = a
and r = b. According to Copson [3] the solution of these problems has the form
00 11'
1 f f d
VCr, z) = - 0'(p)p8p
211" (z2 + r 2 - 2rpc06)1/2
o -11'
and therefore can be reduced to solving in succession, the Abel-type equations
z
f (X1(p)dp = hex),
V x 2 - p2
a
b
f pO'(p)dp = hex),
Vp2 - x 2
z
and the equation
c
j() 2(-1)6 f (xt)6 Jet) d - ( )
x + 11" 1 _ x 2 t 2 t - Y X ,
o
o<x<c=,
where S = 0 or S = 1.
41.9. Belonosov [I, 2] investigated certain boundary value problems for the bihannonic
equation 2u = 0 in the case of plane two-connected domains. To solve these problems he used
fractional differentiation j( 01) in a complex plane defined in (22.4) with £, = (-ioo, ioo) and the
expressions connecting j(OI) with the Laplace transform -  7.2 and  9.2, notes 7.2 and 7.4.
41.10. Shinbrot [1] indicated certain sufficient conditions for the existence of a weak solution
u with the fractional derivative V[:+ u, 0 < 01 < 1/2, in the "time" variable for the Navier-Stokes
equation. He obtained the norm estimate in L2 for such a derivative. This problem was first
considered by Lions [1] who found such an estimate in the case 0 < 01 < 1/4 under certain
assumptions on the dimension of the space.
41.11. Senator [1] obtained Schauder and Lp estimates for the solution u of elliptic
boundary value problems with the boundary conditions containing pseudodifferential operators
with non-smooth symbols and normal derivatives of fractional order.

 43. ADDITIONAL INFORMATION TO CHAPTER 8
865
41.12. Berens and Westphal [1] considered the Cauchy problem
d
dx w(x,t) + 1'6+ w (x,t) = 0, x> 0, t > 0, 0 < "y < I,
lim IIw(x, t) - fo(t)IIL = 0, 1 $ p < 00,
J:-O P
where 1'6+ is the Riemann-Liouville fractional differentiation operator defined in (5.6) applied
with respect to t. They constructed the solution of this problem in the form w = WJ:) fo where
W'Y is a semi group of a class 00 in Lp (0, 00) for any x > O.
42.1. By using the Picard method and the Schauder fixed-point principle, Tazali [1] proved
two theorems which yield conditions for the existence of the solution y(x) of the Cauchy-type
problem
(1':+y)(x) = f(x,y(x», a < x  a + h, h > OJ
(1':.;l y )(x)IJ:=a = b, 0 < cv  1.
These statements generalize the known results by Caratheodory in the case cv = 1.
Grin'ko [1] proved the existence and uniqueness theorem for the solution of a nonlinear
differential equation with a generalized fractional derivative inverse to (10.19) in the Holder
weighted space H6([a, b], p), p(x) = (x - a)#J(b - X)II, on a finite interval [a, b] of the real line. He
also constructed the approximation solution of this equation, and obtained the estimate for such
a solution.
Arora and Alshama.ni [1] investigated the stability properties for the Cauchy-type problem
given in (42.2) and (42.3). Hadid and Alshamani [1] proved the estimate at infinity for the
solution of this problem under the appropriate assumptions on f(x,y).
42.2. A series of papel'S is concerned with the investigation of systems of linear differential
equations of fractional order mainly in the space V' of generalized fWlctions -  9.1, notes to
 8.1 and 8.2.
Veber [3] fOWld the solution of the Cauchy problem for the system of equations y(a)(x) =
Ay(x), 0 < cv < 2, with a constant matrix A. The asymptotic behavior as x - 00 of different
solutions of this system including the fWldamental matrix was studied by Imanaliev and Veber
[1] and Veber [5, 6]. The Cauchy problem for the system
y(a)(x) = A(x)y(x) + f(x), n - 1 < cv  n, n = 1,2,...,
( 43.3)
with the continuous matrix-function A(x) for x  0 was investigation by Veber [7], and the
fundamental solution of such a system with a constant matrix A(x) = A = const was considered
by Veber [8]. The case of a single equation was considered in the paper by Veber [1].
Veber [6] proved the criterion of "passivity" for the systems
Ay'(x) + By(a)(x) + Oy(x) = g(x), 0 < cv < I,
( 43.4)
more general than (43.3) with constant matrices A, B, C, and fOWld quasi-asymptotic expansions
of their fWldamental solutions as x - 00. The definition of the above concepts can be fOWld,
for example, in the books by Vladimirov (2j p. 86, 278] and Vladimirov, Drozhzhinov and
Zav'yalov [lj p. 34, 58, 209]. We also note that the solution of the equation of the form (43.4)
with 0 < Ot < 1 was constmcted by Seitkazieva [1], who used the method based on the Laplace
transform.

866
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Bykov and Botashev [1] reduced one problem in studying (watering the fWTOws) to an
equation of the form (43.4):
:e
y'(x) = q - .x 1[1 + b(x - t)-a]y'(t)dt, 0 < a < 1.
a
One may find other examples of applied problems reduced to differential equations and systems
of differential equations with fractional derivatives in the paper by Veber [8].
42.3. Srivastava, Owa and Nishimoto [1] proved the following statement in terms of the
fractional differentiation operator fll(z) == (1'+,,,,f)(z) - (22.17), (22.18) and (22.21) - in the case
when 0 = £",(z) or 0 = £0(%)'
Theorem 43.1. Let 1p(1I; z) '# 0 be analytic in a domain D in the complete plane z. If
( . ) = { 1 Ip( II + 1; z) d } / ( { 1 Ip( II + 1 i z) d } )
Ip II, Z exp "' ( II' Z ) Z exp "' ( II' Z ) Z
.... , .... , -II
and fll(z) e:rist., then the equation fll(z) = 1p(1I; z)f(z), zeD, ha, a solution of the form
f(z) = k (exp {j <p(:;: z) dz } t.. k ;£0 - cons', zED.
In the paper by Nishimoto, Owa and Srivastava [1] such a result was obtained for more
general equation fll(z) = 1p(II;z)f(z) + 1p(II;Z)9(Z). .
42.4. In the papers by Wiener [I, 2] and other papers cited there the theory of fractional
integra-differentiation (in sense of a finite part) by Hadamard considered in  5.5 was applied to
investigating various differential equations of fractional order. In particular, the equation in the
theory of polarography
(1'2y)(X) - czay(x) = x- 1 / 2 , x> 0, -1/2 < a  0,
was considered. Special solutions for such equations with power-type coefficients were fOWld by
Wiener [3], Hadid and Grzaslewicz [1] and Campos [9].
42.5. Nahushev [2] considered the problem of the proper solution for the equation
m
Kalp == 1'g+xplp(x) + E aj(x)1';,iIp(x) + b(x)lp(x) = c(x), 0 < x < I,
j=1
where
0< a < I, (3 > 0, a> al > ... > 0 > ... > am,
b(x), c(x) e C([O, 1]).
{ Cl ([0, I]),
aj (x) e C([O, 1]),
if aj > 0
if aj < O.
Let 0'1«0,1» be a Banach space of functions Ip(x) e C«O,I» with the nonn IIlplill =
max Ix'Ylp(x)l, and let "Y be a constant. Nahushev proved that if {3 < a - (signaj + l)aj/2,
:ee[O,I]
then for each c(x) e 00«0,1» there exists a unique solution of the above equation in the space
Cp«O,I».

 43. ADDITIONAL INFORMATION TO CHAPTER 8
867
In another paper Nahushev [3] studied the "mixed" boundary value problem for the equation
ym uzz + Uyy + a(x"y)uz + b(x,y)uy + c(x,y)u = 0
in a bounded domain D e {y > O} the boundary of which contains the interval 0 < x < 1 of the
axis Ox. The boundary condition on this interval has the fonn lim K 1 u = \I1(x), 0 < x < 1, where
y-O
K 1 is the value of the above operator K 01 when 01 = 1. On the basis of the extremum principle
obtained in the paper the stability and uniqueness of the solution of this problem were proved,
and a question of the existence of such a solution was considered.
42.6. Aleroev [1, 2] investigated the spectrwn of the Dirichlet problem for the differential
equation
u"(x) + a(x)Vg+u(x) = lex), 0  x  1,
(43.5)
when 0 < 01 < 1. He proved that the problem with the conditions u(O) + pu'eO) = u(l) = 0
and lex) = 0, a(x) = ,X for p  0 does not have negative eigenvalues in the space of functions
e[O, 1] n e 2 (0, 1], and the problem with the conditions au(O) + pu'eO) = 'Y, Qu(l) + pu ' (I) = 'Y,
a(x) = ,X has eigenvalues but not more than a continuum set. The inequality
00
L 1'x1:1- 2  11 6 r 2 (2 - 01)
1:=1
for the eigenvalues ,X = 'xl: of the problem a(x) ='x, u(O) = 0, u(l) = 0 was also proved. We
also note that earlier Nahushev [5] showed that ,x = 'xl: is an eigenvalue of the latter problem if
and only if 'xl: is a zero of the Mittag-Lemer function E2-0I ,2( -,X) defined in (1.91). All these
zeros, the moduli of which are sufficiently large, are simple zeros and they have the estimates
'xl: = O(k 2 - 0I ) as k - 00 - Dzherbashyan [2, p. 142].
The asymptotic behaviors of the eigenvalues, and problems of the unifonn convergence
for the differential operator yen) - 'xy given on a finite interval of the real axis with boundary
conditions concentrated at the end points of this interval and containing the Riemarm-Liouville
fractional derivatives given in  2.3, were considered by Bogatyrev [1] and Amvrosova [1].
42.7. Using the relations of realization of operators, for example, see Gel'fand and Shilov
[2. p. 151], Malakhovskaya and Shilananter [1] suggested a method for constructing a generalized
solution of the Cauchy problem for the integra-differential equation
z
Qn (  ) y(x) - a j - ( yet) dt = lex), x> 0,
dx x - t)OI
o
with the conditions yj (0) = aj' j = 0,1, . . . , n - 1, where 0 < 01 < 1 and 01 is a rational number
and Qn(X) is a polynomial of order n.
42.8. Leskovskii [1] constructed the linear independent solutions of the homogeneous
equation (42.34) with specific distinct exponents OIj' -1  OIj < 0, in the form of a Mittag-LefBer
function defined in (1.91). The special solution of the inhomogeneous equation (42.34) with
OIj = (j - 1)/n was first found by Davis [2] - (4.6) and  4.2, note 2.5 and  30 and 34.
An equation of the same type with constant coefficients
n ,x. j c
'xol(x) + '" - ( J (11- x)OI;-l l(y)dy = 0,
L.JrOl')
j=l J z
ReOlj > 0,

868
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
was studied by Alonso [1]. He proved that the solution of this equation has the form J(x) = e-"
if and only of the parameter fl satisfies the conditions
n
Refl > 0, O + L jfl-OIj = 0, C = 00.
j=1
Alonso alao investigated properties of solutions of the cOlTe8pOnding inhomogeneous equation in
the case n = 1.
By analogy with lineal' ordinary differential equations with constant coefficients, Campos [9]
presented a method for solving simple equations of the form (42.34) with fractional derivatives
in a complex plane given in  22.1, in terms of the roots of the characteristic pseudopolynomial
n
E ajrOlj = O.
j=1
42.9. Equations of the form (42.34) with piecewise constant coefficients were investigated
by Kachura and Didenko [1].
42.10. The abstract Cauchy problem for the equation of hypergeometric type
d'd P ( d )
t dt II (t dt + Pj - 1) u(t) - At II t dt + Otj u(t) = 0,
j=1 ;=1
t> 0,
with the closed. lineal' operator A and constant coefficients OIj and I3j was considered by Bragg
[I, 2]. By using fractional integrals he obtained the expressions connecting the solutions of this
equation for different values of parameten and applied these expressions to investigating the
Cauchy problem for the degenerate equation of hyperbolic type
Utt - tmu + vt m / 2 - 1 u = 0, t > 0, m  2, u(x, 0) = ",,(x), Ut(x,O) = ",,(x).
42.11. Yu. Rabinovich and Nesterov [1] considered the differential operaton Knu:
n dR-leu
Knu = "" Pk(Z)---,:- = 0
L.J d%n-..
i=O
( 43.6)
where Pn (%) are polynomials of certain degrees. They found the conditions tmder which the order
n of the operator in (43.6) can be lowered by using fractional derivatives defined in (22.30) and
(22.33'), and in the relation similar to the one in (22.33') in the case zo = 00.
In the paper by Nesterov [1] such derivatives were applied to constructing solutions of
differential equations of the Fuchs type with. singular points
, n ,
II (z - ak)nU(n) + L Q('_I)j(Z) II (% - Clk)n-jU(n- j ) = 0
k=1 j=1 1e=1
where Qm(z) is a polynomial of degree m.
42.12. A series of papers is concerned with the application of methods of fractional
integre>-differentiation to solving linear differential equations of second order of the form (43.6)
with polynomial coefficients.
AI-Bassam [2, 3, 5, 7, 9-12, 14-15], AI-Bassam and Kalla [1] considered differential and
integra-differential equations reduced by the Leibniz rule given in (17.11), and other properties of

 43. ADDITIONAL INFORMATION TO CHAPTER 8
869
fractional integrals and derivatives to operator equations of compositional type
I;':p(x)I;;.;.q(x)I:: n - 1 y(x) = 0, n = 1,2,...,
( 43.7)
m
where p(x) and q(x) are products of the form n (ak + bkx)O/kelJ:I:. Necessary and sufficient
k=l
conditions under which certain classes of differential equations of second order are equivalent to
(43.7) were proved, and their solutions were constructed. Examples of the equations for Gauss,
Hermite, Kummer, LagueJTe, Legendre and Jacobi functions, for the generalized hypergeometric
functions 2F2 and 3F2' and for orthogonal polynomials were considered. On the basis of
( 43. 7) solutions of these equations are represented via the cOlTesponding fractional integrals and
their compositions -  9.3 and  10. In this connection see also the paper by AI-Bassam [13]
where equations with solutions including generalized power series and, in particu1ar, analogues of
exponential, trigonometric and hyperbolic functions were considered.
Higgins [6] suggested an original method for obtaining general solutions of the inhomoge-
neous hypergeometric equations. This method is based on the application of direct and inverse
Laplace transforms given in (1.119) and (1.120), and Erdelyi-Kober transforms defined in (18.1),
(18.2) with respect to parameters.
Nishimoto [7]-[13], and also his papers published in J. ColI. Engng. Nihon Univ. 1988.
B-29, 1989. B-30, applied the fractional integro-differentiation operatol'S III given in  43.2, note
42.3 to studying special solutions of ordinary and partial differential equations of the Fuchs type
connected with Gauss, Kummer, LagueITe, Legendre, etc. special functions. In this connection
we refer also to Nishimoto and Kalla [I, 2] and Nishimoto and Tu Shih-Tong [I, 2].
42.13. Fedosov and Yanenko [1] proved that the equation in partial derivatives of half-integer
order
n
'" k/2 (n-k)/2
LJakV+,:l:V+,y u(x,y) = I(x,y),
k=O
ak - const,
-  24.2 - can be investigated by means of representating their operators as compositions of n
invertible operators of the fonn V + O/jV where
:I:
V + l/u(x,y) =   f (x - t)-1.2 u (t,y)dt
J v 1r dx
-00
and Ot j are the roots of a certain characteristic polynomial. In particular, in the case n = I,
aO = 0, al = 1 the relation for the general solution of this equation
:I: ,
u(x,y) =1r-l/2 f f (a - )1/2 : [/(,y + 0/2(x - a)) - O//(a, 11 + 0/2 (x - a) +  - a)]dda
-00 -00
+ { q(y + 0/2 x ),
0,
0/  0,
0/ > 0,
where q(t) is an arbitrary function was indicated. The case n = 2 was also considered in detail.
42.14. Yaroslavtseva [1] constructed the operators Fj'; = 1,2, for transfonnation of the
operator Pinto D2, i.e. the operators F with the property PFv = FD 2 v on the elements v of a

870
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
certain function space where
( d2 d 2 )
P = - - + 2l1ctglp- - II ,
dlp2 dip
D = -i.!!....
dip
In particular, the operator Fl has the fonn
rp
Fl v = 211%+ 1/2) (sinlp)1-211 f v(t)(cost - C06Ip)II- 1 dt
11" (II)
o
for real II > 0, and the fonn of a convolution of a test function v with the functional sin211(1p/2)
for other values II.
These results were applied to reducing the Cauchy problem for the differential equation
n
L al:pn-I: u = 0, al: - const, 0 < Ip < 11",
1:=0
to the Cauchy problem for the ordinary differential equation
n
L(-I)n-l: atW 2n-21: = o.
1:=0
42.15. Biacino and Miserendino [2] considered properties of the operator
Lu =  /Ja(x)Dau, lal = al + a2,
lal:S4
a = (ab (2), x = (Xb X2),
with fractional derivatives Da u defined in the papers by Biacino and Miserendino [I, 3] - also
 29.2, note 24.10. On the basis of the representation
Lu = L aa(x)Dau + L aa(x)Dau
lale{2,3,4} lal<2
the conditions for the index Lu to be equal to zero were indicated, and the mapping properties of
the operator L is Sobolev spaces were studied.
42.16. Sprinkhuizen-Kuyper [1] proved a series of theorems about the solution of the
Cauchy problem for the equation
1 d I ( d2 II d ) I:
( --- ) - + -- I(x) = g(x),
x dx dx 2 x dx
0< x  1,
with conditions of the form 1(j)(I) = 0, j = 0,1,...,1 + 2k -1. For g(x) e C([O, I)] the above
problem has the solution lex) = ItJl g(x) where

 43. ADDITIONAL INFORMATION TO CHAPTER 8
871
1 1 ( 2 2 ) >'+IJ-1
IIJ,>'/(x) _ 1  y1-IJ
II q'x+JL) 2
:&
( JL + v-I JL x 2 )
X 2 F 1 ,X + 2 ' '2i ,X + JLi 1 - y2 I(y)dy,
in the space I(x) e C2k+l ([0,1]). Some properties of this operator with a Gauss hypergeometric
function in the kernel were also studied in this paper.
These results were developed by McBride [7, 9] and Dimovski and Kiryakova [2] who
considered operators more general than (9.5). For such operators their integral representations
in the form of compositions of te ErdByi-Kober type fractional integra-differentiation operators
given in (18.1) and (18.2) and in terms of the Meyer G - function of the fonn (10.48), were
found. Special cases of these representations which lead to operators with a Gauss hypergeometric
function of the form (10.18) were earlier considered by Sprinkhuizen-Kuyper [1].
42.1 T. Tremblay [2] and Tremblay and Fugere [1] investigated properties of the operators
Da,r = D(zaD)r,
Df3,6,al,...,a m = { D(l-6),f3 II m ( zD + a . ) r; z6f3 } n
n,rl,.",r m Z J'
;=1
where D = t, DC:, a eO, is the fractional derivative given in (22.4)i (3,a; e C(j = 1,2,... ,m),
S = 0 or 6 = Ii r, n, rj (j = 1,2,..., m) are nonnegative integers. In particular, the operator
relations
Da,r Df3,r =D a +f3,r,
Df3,6,al,...,a m _D(l-6)8f3 n z6(1-8)f3n
n,rl,.",r m - Z
n-1 m
X II II (zD + aj - {3(Jn + {3i)r; z68f3n DP-6)(1-8)f3 n ,
i=l ;=1
where (J = 0 or (J = 1 were obtained together with the representation for D:::.;:;;.,am in
terms of the operators z(l-2')')w+(1-'Y)k D k z(2'Y- 1 )w+'Y k where "y = 0 or "y = 1, wee,
k = 0,1,..., n(T! + ... + rm).
As examples the new operator relations for the usual differential operators D and integral
operators D;l were given.

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AUTHOR INDEX
A
Abdullaev, S.K. 590
Abel, M.H. xxi, xxiii, xxiv, xxv, 82
Adamchik, V.S. xxii, 174
Adams, D.R. 589
Adams, R. 583
Agarwal, R.P. 87, 317, 442, 443
Aggarwala, B.D. 793
Ahern, P. 451
Ahiezer, N.I. 302,685,693,774,779
Ahmad, M.I. 794
Ahner, J.F. 780
Ahuja, G. 172
Ahuja, a.p. 454
Akopyan, S.A. 440, 772
AI-Abedeen, A.Z. 857
AI-Amiri, H.S. 454
AI-Bass am, M.A. xxii, 316, 454,
857, 868, 869
Alekseevskii, V.P. 858
Aleroev, T .S. 857, 867
Alexitz, G. 445
Aliniov, S.A. 583
Alonso, J. 868
AI-Salam, W.A. 317,442,443
Alshamani, J.G. 865
Amvrosova, 0.1. 867
Andersen, K.F. 166, 168, 590
Anderson, T.P. 85
Anderssen, R.S. 85
Andreev, A.A. 798
Ang, D.D. 659
Arestov, V.V. 313
Aronszajn, N. 432,583
Arora, A.K. 317
Arora, H.L. 857, 865
Arutyunyan, M.H. 689
Askey, R. 314
Asral, B. 862
Atiyah, M.F. 584
Atkinson, F.V. 685
Atkinson, K.E. 85
B
Babenko, K.I. 821
Babenko V.F. 444
Babenko, Yu.l. xxii
Babich, V.M. 798
Babloyan, A.A. 793
Badalyan, A.A. 89
Bagby R.J. 584, 589, 594, 601
Bagley, R.L. xvii, xxii
Bakaev, N.Yu. 161
Baker, B.B 583
Bakievich, M.I. 774
Balakrishnan, A.V. 120, 161
Balasubramanian R. 85
Banerji, P.K. 793
Bang, T. 432, 433
Bari, N .K. 249, 443
Barrett, J .H. 857
Bavinck, H. 445
Beatroux, F. 603

954
AUTHOR INDEX
Beekmann, W. 315
Belinskii, E.S. 436
Belonosov, S.M. 864
Beltrami, E. 774, 856
Belward, J .A. 302
Belyi, V.1. 436, 449
Benedek, A. 580, 588
Berens, H. 161, 170, 865
Berger, N. 305,318, 319, 857
Bernstein, I.N. 584
Besov,O.V. 580,589,591
Bessonov, Yu.L. 580
Betilgiriev, M.A. 320
Bharatiya, P.L. 773
Bhatt, S.M. 315
Bhise, V.M. 170, 438, 782
Bhonsle, B.R. 776
Biacino, L. 432, 587, 870
Bitsadze, A.V. 820, 843, 844, 856,
857
Bleistein, N. 294, 305
Blum, E.K. 862
Blumenthal, L.M. 435
Bacher, M. 82
Bochner, S. 581
Bogatyrev, S.V. 867
Boman, J. 448
Bora, S.L. 170
Borisov, V.N. 861
Borodachev, N.M. 794
Bosanquet, L.S. xxxii, 83, 84, 160,
163, 164, 277, 312, 313, 315, 434
Botashev, A.I. 866
Bouwkamp, C.J. 790
Box, M.A. 783
Braaksma, B.L.J. 438, 777
Bragg, L.R. 868
Brakhage, H. 86
Bredimas, A. 162, 172, 371, 434,
591
Brenke, W.C. xxii
Bresters, D.W. 584, 863
Brodskii, A.L. 587
Brunner, H. 85
Brychkov, Yu.A. 15, 18, 2.0, 22, 27,
172, 174, 198, 201, 206, 282, 284,
325, 355, 372, 551, 584, 586, 711,
733, 739, 740, 742, 743, 755, 773,
774, 781
Bugrov, Ya.S. 371, 434
Bukhgeim, A.L. 90
Burbea, J. 603
Bureau, F.J. 863
Burenkov, V.1. 371,434
Burkill, J .C. 312
Burlak, J. 773,774
Busbridge,I,W. 774
Buschma.n, R.G. xxii, 23, 431, 695,
772, 776, 777, 784, 786, 787, 793
Butzer, P.L. xix, xxxii, 161, 303,
371,433,434,444,446-448
Bykov, Ya.V. xxii, 866
Bzhikhatlov, H.G. 689,862
c
Calderon, A.P. 583
Campos, L.M.B.C. 452, 866, 868
Carbery, A. 163
Carleman, T. 449, 685, 687
Cartwright, D.1. 90
Cayley, A. 83
Center, R.W. xxv
Chakrabarty, M. 789
Chan, C.K. 85
Chandrasekharan, K. 277
Chanillo, S. 584, 588, 590, 601, 862
Chen Yung-ho 593
Chen, Y.W. 82,84,301,302
Cheng, H. 864
Cheng Min-teh 593
Cherskii, Yu.I. 695
Chiang, D. 689
Cho N ak Eun 454
Choudhary, B. 163
Chrysovergis, A. 432
Chumakov, F.V., 688,689
Chuvenkov, A.F. 168, 591
Civin, P. xxxi, 433
Clements, D.L., 864
Cohn, W.S. 451
Colzani, L. 602
Conlan, J., 171
Cooke, J .C. 793
Copson, E.T. 583, 772, 774, 775,
779, 860, 864
Cossar, J. xxxii, 163, 315
Cotlar, M., 589

AUTHOR INDEX
D
Darboux, G. 856
Davis, H.T. xix, xx, xxx, 87, 302,
434, 867
Davtyan, A.A. 595, 596
Delerue, P. 580
Deng Dong-gao 593
Di Giorgio, M. 587
Diaz, J .B. 453, 862
Didenko, A.V. xxii, 857, 868
Dimovski, I.H. xxii, 302, 436, 454,
871
Din Khoang An 777
Dinghas, A. 87
Ditkin, V.A. 24, 28
Dixit, L.A. 776
Doetsch, G. 162
Doktorskii, R.Ya. 85
Domingues, A.G. 600
Drianov, D.P. 447
Drozhzhinov, YU.N. 602, 865
Duduchava, R. V. 302
Dunford, M. 642
Dveirin, M.Z. 449
Dwivedi, A.P., 794
Dyckhoff, H. 434, 444, 447, 448
Dyn'kin, E.M. 590
Dzhabirov, D.K. 860
Dzherbashyan, A.M. 453
Dzherbashyan, M.M. xix, xxvii, 22,
24, 83, 84, 87, 88, 89, 303, 345,
432, 435, 453, 661, 686, 709, 713,
772, 841, 857, 867
Dzyadyk, V.K. . 432, 444
E
Edels, H. 85
Efimov, A.V. 433,444
Eggermont, P.P.B. 85
Elliott, J. 602
Elrod, H.G., Jr. 773
Emgusheva, G.P. 593,596,597
Erdelyi, A. xxxi, xxxii, 14, 18, 22,
88, 91, 160, 162, 163, 165, 172,
180, 193, 196, 199, 201, 302, 319,
431, 432, 436, 437, 441, 455, 539,
551, 661, 677, 724, 733, 735, 751,
772, 774-777, 783, 851, 856, 858,
859, 860
955
Eskin, G.I. 582, 599
Esmaganbetov, M.G. 444,447,568
Estrada, R., 162
Euler, L. 856
Evsin, V.I. 862
Exton, H. 443
F
Faber, V. 86
Fabian, W. 89, 448, 449
Faraut, J. 603
Fattorini, H.O. 161
Favard, J. 444
Fedoryuk, M.V. 285,584
Fedosov, V.P. xxii, 584, 869
Feller, W. 216,303
Fenyo, S. xix
Ferrar, W.L. 434
Fettis, H.E. 85
Fihtengol'tz, G.M. 485
Fisher, B. xxii
Fisher, M.J. 83,582,599
Flett, T.M. xxxii, 84, 91, 160, 315,
450,452,583,591
Fofana, I. 590
Foht, A.S., 587
Fomin, S.V. 3,24,328
Fourier, J. xxiii, 32
Fox, Ch. xxxii, 437, 442, 772, 773,
781, 783, 794
Fremberg, M.E. 583
Frie, W. 85
Friedlander, F.G. 860
Frostman, O. 581
Fugere, B.J. 871
Fujiwara, M. 857
Fukui, S. 454
G
Gabidzashvili, M.A. 581,590
Gaer, M.C. 317,453
Gahov, F.D. 199, 201, 202, 605,
617,622,651,684,685,695,848
Gaimnazarov, G. 447
Gamaleya, R.V. 793
Ganeev, R.M. 685,688
Garabedian, H.L. 315
Girding, L. 581, 603

956
AUTHOR INDEX
Garnett, J .B. 92
Gasper, G. 163, 164, 783
Gatto, A.E. 590
Gearhart, L. 168
Geisberg, S,P. 167,433
Gel'fand, I.M. 162, 487, 548, 556,
576, 583, 584, 867
Gel'fand, S.1. 584
Gel'fond, A.a. 436
Gel'man, LV. 167
Genebashvili, I.Z. 590
Gerlach, W. 85, 659
Gilbert, R.P. 857, 860
Gindikin, S.O. xxiv, 602, 603
Ginzburg, A.S. 600
Glaeske, H.-J. xxii, 440, 773, 792
Gliner, E.B. 798
Godunova, E.K. 314
Gohberg, I.Ts. 199, 302, 630, 646,
656, 675, 684, 685
Gomes, M.1. xxii
Gopala Rao, V.R. 584, 601
Gordeev, A.M. 746,857
Gordon, A.N. 774,775
Gorenflo, R. xvii, xviii, xxii, 85, 86,
87, 659, 691
Gorlich, E. 434,444,447,448
Goyal, S.P. 439, 777
Graev, M.1. 576, 584
Greatheed, S.S. xxv
Grinchenko, V.T. 793
Grin'ko, A.P. 305, 440, 865
Grunwald, A.K. xxiv, xxv, xxx,
304, 371, 434, 435, 479
Grzaslewicz, R. 866
Gupta, H.1., 786
Gupta, K.C. 171, 782
Gupta, S.B. 786
Gupta Sulaxana K. 315
Guseinov, A.1. 251, 303, 304
Gutierrez, C.E. 590
G willi am, A.E. 451
H
Habibullah, G.M. 777, 784, 785
Hadamard, J. xxvii, 112, 345, 435
Hadid, S.B. 865, 866
Hai, D.D. 659, 691
Hall, M.S. 862
Handelsman, R.A. 288, 305, 318-
320, 857
Harboure, E. 590
Hardy, G.H. xxvii, xxviii, xxix, 84,
86, 88, 91, 160, 161, 277, 304, 314,
315,432,433,435,450,451
Hasanov, A. 861
Hatcher, J .B. 689
Havin, V.P. 582
Hearne, K. 85
Hedberg, L.1. 581
Heinig, H.P. 166, 590
Heins, A.E. 860
Helgason, S. 581, 591
Henrici, P. 774, 860
Herson, D,L. 161
Herz, e.s. 591
Hess, A. 792
Heywood, P. 161, 685, 789
Higgins, T.P. 172, 302, 439, 772,
773, 775-777, 869
Hilbert, R.P. 805
Hille, E. 83, 85, 120, 161, 301, 315,
445
Hirsch, F. 161
Hirshman, 1.1. 452, 695, 772, 792
Holmgren, Hj. xxiv, xxv, xxvi, 82,
83,316,431,435,580,851,857
Hotta, G.e. 315
Horvath, J. 582, 594
Hovel, H.W. 161
Hromov, A.P. 86
Huber, A. 862
Hughes, R.J. 161, 313
Humbert, P. 87
Hvedelidze, B.V. 302
I
Ibragimov, I.I. 315
Il'in, V.P. 580, 589
Imanaliev, M.I. 865
Isaacs, G.L. 83, 312
Isamukhamedov, S.S. 861
Ivanov, L.A. 600
Ivashkina, G.A. 861
Izumi, S. 444

J
AUTHOR INDEX
957
Jackson, F.H. 442
Jain, N.C. 580
Jevtic, M. 451, 604
J oachimsthal, F. xxv
Johnson, R. xxii, 583
Jones, B.F., Jr. 584
Jones, D.S. 685
Joshi, B.K. 786
Joshi, C.M. 88
Joshi, J .M.C. 437
J uberg, R.K. 303
K
Kabanov, S.N. 86
Kae, M. 602
Kalia, R.M. 774
Kalisch, G.K. 162
Kalla, S.L. xxii, xxxii, 170, 439,
778, 782, 786, 792, 868, 869
Kant, S. 786
Kanwal, R.P. 162
Kapilevich, M.B. 746, 747, 798, 860
Karapetyants, N.K. 84,91,92, 169,
308, 311, 432, 685
Karasev, I.M. 858, 862
Kashin, B.S. 92, 310
Kashyap, N .K. 778
Kato, T. 161,'685
Katsaras, A. 443
Kaul, C.L. (see also Koul, C.L.) 317,
318, 584, 782. 786
Kelland, P. xxv
Kerman, R. 862
Kesarwani, R.N. (N arain Roop) 793
Khan, A.H. 443
Khan, M.A. 443
Khandekar, P.R. 786
Kilbas, A.A. xvii, xxii, 84, 303, 305,
319, 434, 435, 440, 448, 589, 603,
859
Kim Hong Oh 451
Kim Yong Chan 451
King, L.V. 774
Kipriyanov, I.A. xxxii, 580, 585, 600
Kiryakova, V.S. 302, 439, 454, 580,
778, 871
Kishore, N. 315
Klyuchantsev, M.1. 441
Kober, H. xxxii, 83, 160, 162, 301,
302,303,431,436,453,772
Kochubei, A.M. 597,861
Kochura, A.I. 857, 868
Koeller, R.C. xxii
Kofanov, V.A. 315
Kogan, H.M. 690
Koh, E.L. 171
Koizumi, S. 452
Kokilashvili, V.M. 308, 581, 589,
590, 591
Kolmogorov, A.N. 3, 24, 328
Komatsu, H. 161
Komatu, Y. 454
Komori, Y. 588
Korobeinik, Yu.F. 436
Kosarev, E.L. 85
Koschmieder, L. 584
Koshljakov, N.S. 796
Kostitzin, V. 86
Kostometov, G.P. 312
Koul, C.L. (see also Kaul, C.L.) 317,
318, 584, 782, 786
Kovetz, Y. 85
Kolbig K.S. xxii
Kralik, D. 444,445
Krantz, S.G. 590
Krasnosel'skii, M.A. 120, 161, 167,
168, 169, 655
Krasnov, V.A. 432,587
Krbec, M. 591
Krein, M.G. 685
Krein, S.G. 5, 630
Krepkogorskii, V.L. 466
Krishna,S. 585
Krivenkov, Yu.P. 860
Krug, A. 435
Krupnik, N .Ya. 199, 302, 630, 646,
656, 675, 684, 685
Kudryavtsev, L.D. 444
Kufner, A. 590
Kumbhat, R.K. 438, 439, 776, 782,
786
Kumykova, S.K. 861
Kurokawa, T. 592, 595
Kuttner, B. 302, 315
Kuvshinnikova,I.L. 597

L
AUTHOR INDEX
958
Lacroix, S.F. XXlll
Lamb, W. 171
Lambe, e.G. 162, 171
Landau, E. 313
Landkof, M.S. 581, 593
Lanford, O.E. 161
Laplace, P.S. xiii
Laurent, H. 415, 435
Lavoie, J .L. 172, 452
Lebedev, N.A. 429,436
Lebedev, N.N. 23, 24, 774, 778, 793,
794
Lee Sang Keun 454
Leibniz, G.W. xxiii
Leont 'ev, A.F. 426, 436
Leray, J. 581, 600
Leskovskii, I.P. 862
Letnikov, A.V. xix, xxi, xxiv, xxv,
xxvi, 83, 160, 174, 302, 304, 371,
433,435,452,479
Levin, V.I. 314
Levitan, B.M. 431
Levy, M .E. 85
Lew, J .S. 288, 318
Lewy, H. 861
Lieb, B.H. 588
Linchuk, N .E. 454
Linfoot, E.H. 315
Linker, A.I. 312, 692
Lions, J .L. 161, 583
Liouville, J. xxi, xxiii, xxiv, xxv,
xxvi, xxviii, 82, 83
Littlewood, J.E. xxix, 84, 91, 160,
304,433,435,450,451
Liu, D. 443
Liu Gui-Zhong 597
Liverman, T.P.G. 432
Lizorkin, P.I. xxxii, 147, 162, 171,
433, 467, 475, 477, 489, 580, 581-
583, 586, 588, 591, 594, 598, 601
Loo, C.- T. 315
Love, E.R. xxii, xxx, xxxi, xxxii, 23,
83,89,92,301,302,432,593,772,
777, 783
Lowengrub, M. 689, 774, 794
Lowndes, J.S. 431, 774, 775, 780,
794
Lu, P. 85
Lubich, Ch. 85, 371, 448
Luke, Y.L. 17,22
Lundgren, T. 689
Liitzen, J. xxiv
M
Macias, R.A. 588, 590
Mackie, A.G. 788, 791
Madhavi Dinge 782
Magaril-Il'yaev, G.G. 580,586
Magnus, W. 14, 18, 22, 88, 91, 160,
172, 180, 193, 196, 199, 201, 441,
455, 539, 551, 861
Mainra, V.P. 782
Makarenko, L.G. 775,793, 794
Malakhovskaya, R.M. 867
Malamud, M.M. 691
Malinovski, H. 85
Malovichko, V.A. 439
Malozemov, V.M. 443
Mamedov, R.G. 442
Manandhar, R.P. 783
Mandelbrojt, S. 857
Manocha, H.L. 88,318
Manukyan, M.M. 689
Marchaud, A. xxix, 109, 110, 111,
116,168,303,469,480,580
Marichev,O.l. xxii, 27, 28,172,174,
190, 191, 198, 201, 206, 282, 284,
301, 325, 355, 372, 437, 551, 584,
700, 701, 707, 711, 725, 726, 733,
739, 740, 742, 743, 755, 772-774,
781, 784, 785, 792, 856, 857
Marke, P.W. xix
Martic, B. 170
Mart in- Reyes, F.J. 167
Martirosyan, V.M. 453
Mathai A.M. 781
Mathur, B.L. 170
Mathur, S.L. 170, 171, 585
Matsnev, L.B. 86
Matsuyama, N. 445
Maz'ya, V.G. 582
McBride, A.C. xix, xxii, xxxii, 157,
162,172,301,431,439,777,871
McClure, J.P. 288, 305, 319
McKellar, B.H.J. 783

AUTHOR INDEX
McMullen, J.R. 90
Meda, S. 581
Medvedev, N.V, 85
Mengziyaev, B. 861
Mhitaryan, S.M. 685
Mihailov, L.G. 87
Mihlin, S.G. 518, 521, 570, 584, 589,
661, 724, 733, 735, 798
Mikolas, M. xx, xxii, xxxii, 162,
315,432-434,452,581
Miller, J .B. 160
Minakshisundaram, S. 277, 315
Minerbo, G.M. 85
Miserendino, D. 587, 870
Misra, A.P. 91
Mittal, P.K. 171
Mizuta, Y. 588, 592
Modi, G .C. 585
Mohapatra, R.M. 315
Montel, P. xxviii, 315, 433, 580, 586
Moppert, K.F. 434
Most, R. 433
Mourya, D.P. 584
Muckenhoupt, B. xxii, 199, 202,
311, 495, 581, 588, 590, 596, 605,
617,622,684,685
Muhtarov, H.S. 251, 303, 304
M ulla, F. 583
Muller, C. 776
Murdaev, H.M. 303, 304, 433
Murray, M.A. 588
Muskhelishvili, N.I. 5, 6, 302
N
N agnibida, N.1. 436
Nagy, B.S. xxxi, 432, 433, 444
Nahushev, A.M. 442,857,861,862,
866, 867
Nair, V.C. 787
Narain Roop (see Kesarwani Roop
N &rain) 23, 772, 781
Nasibov, F.G. 315
N asim, C. 782, 793
Natan80n, I.P. 798
Nauryzbaev, K.Zh. 444
Nekrasov, P.A. 302, 415, 435, 858
Nersesyan, A.B. 83, 88, 89, 303,
440, 453, 841, 857
959
Nessel, R.J. xix, 444
Nesterov, S.V. 868
Neugebauer, C.J. 371
N eunzert, H. 689
Nevostruev, L.M. 861
Nickel, K. 86
Nieva del Pino, M.E. 438
Nikolaev, V.P. 581
Nikol'skaya, N .S. 580, 586
Nikol'skii, S.M. xxxii, 24, 368, 433,
444, 477, 528, 580, 583, 586, 589,
593,595,596,597,600,601,685
Nishimoto, K. xvii, xxii, xxxii, 172,
435,454,866,869
Noble, B. 774
Nogin, V.A. xxii, 552, 582-584
Norrie, D.H. 85
Nozaki, Y. 583
Nunokawa, M. 454
o
Oberhettinger, F. 14, 18, 22, 88, 91,
160, 172, 180, 193, 196, 199, 201,
441, 455, 661, 677, 724, 733, 735,
751,772,783,851,861
Obradovic, M. 454
Ogievetskii, 1.1. xxxii, 433, 444
Okikiolu, G.O. xix, 169, 303, 307,
581, 783
Oldham, K.B. xix, xxii, 172
Olevskii, M.M. 862
Olmstead, W.E. 320
Olver, F. 285
O'Neil, R.O. xxxii, 167, 168, 591
Onneweer, C.W. 443
Oramov, Zh. 861
Orazov, I. 861, 862
Ortner, M. 582, 594
Orton, M. 689
Orudzhaev, G.N. 442
O'Shaughnessy, L. 857
Osilenker, B.P. 590
Osipov, A.V. 85
Osler, T.J. xxxii, 89, 90, 304, 305,
316,317,432,452,453
Ovesyan, K.R. 453
Owa, S. xxii, xxxii, 435, 436, 454,
866

p
AUTHOR INDEX
960
Pacchiarotti, N. 88
Padmanabhan, K.S. 454
Pahareva, N .A. 859
Palamodov, V.P. 584
Paley, R.E.A.C. 315
Panzone, R. 580, 588, 589
Parashar, B.P. 439,778
Pathak, R.S. 783,794
Pavlov, P.M. 602
Peacock, G. xxxii
Peetre, J. 92, 161
Pekarskii, A.A. 451
Pennell, W.O. 90
Penzel, F. 303, 690
Peres, J. 686, 693
Peschanskii, A.I. 449, 689
Pestana, D.D. xxii
Peters, A.S. 688, 689, 774, 775
Petunin, Yu.I. 5
Phillips, R.S. 83, 84, 86, 120, 161
Pichorides, S.K. 200
Pilidi, V.S. 580
Pinkevich, V.T. 444
Pinney, E. 789
Pitcher, E. 857
Plessis, N .du 581, 588, 589
Poisson, S.D. 856, 857
Pokalo, A.K. 444
Polking, J .C. 317
Pollard, H. 789
Polozhii, G.N. 82, 859
P6lya, G. 160, 308
Ponce, G. 597
Ponomarenko, S.P. 794
Ponomanenko, V.G. 447,775,794
Popov, G.¥a. 762
Popoviciu, T. 168
Post, E.L. 316,375,434,447,857
Prabhakar, T .R. 302, 773, 776, 786,
789, 791
Preobrazhenskii, N.G. xxii, 584
Privalov, 1.1 302
Prudnikov, A.P. 15, 18, 20, 22, 24,
27, 28, 172, 174, 198, 201, 206,
282, 284, 325, 355, 372, 551, 584,
711, 733, 739, 740, 742, 743, 755,
773, 781
Pryde, A.J. 595
Pustyl'nik, E.I. 161, 655
Q
Quinn, D. W. 862
R
Rabinovich, V.S. 603
Rabinovich, ¥U.L. 434,868
Rabotnov, ¥u.M. xxii
Radzhabov, E.L. 582
Radzhabov, N. 860
Rafal'son, S.Z. 442
Raina, R.K. 584, 580, 782, 786
Rakesh, S.L. 171
Reddy, G.L. 454
Reimann, H.M. 92
Ren, F. 454
Repin, Q.A. 862
Ricci, F. 604
Richberg, R. 776
Rieder, P. 86
Riekstyn'sh, E.Ya. 288, 305, 318,
319,448
Riemann, B. xxiv, xxv, xxvi, 82, 83,
88,92,856
Riesz, M. xxvii, xxx, 84, 277, 301,
302, 314, 481, 483, 498, 502, 581,
583, 600
Roach, G.F. xxxii
Roberts, K.L. 168
Robinson, D.M. 161
Rogosinskj, W.W. 315
Romashchenko, V.A. 794
Rooney, P.G, 441, 773, 782, 789
Ross, B. xix, xx, xxii, xxiii, xxxii
Rothe, R. xxii
Rozanova, G.1. 314
Rozet, T.A. 773, 788
Rubel, L.A. 317, 453
Rubin, B.S. xxii, 83, 84, 89, 91, 92,
160, 161, 303, 308, 311, 432, 554,
582-584, 589, 592, 595, 597, 599,
600, 601, 686, 687, 690, 692-694
Ruhovets, A.M. 793
Ruiz, F.J. 590
Rusak, V.N. 444
Rusev, P.K. 455

AUTHOR INDEX
Rusia, K.C. 773, 776, 786
Rutitskii, Ya.B. 167, 168, 169
Rychener, T. 92
s
Saakyan, A.A. 92, 310
Saakyan, B.A. 89
Sadikova, R.H. 314
Sadowska, D. 690
Saigo, M. xxii, xxxii, 440, 777, 786,
859, 862
Saitoh H. 454
Sakalyuk, K.D. 84,685,688,689
Saksena, K.M. 170, 438, 439
Salahitdinov, M.S. 442, 861
Samarskii, A.A. 318
Samko, S.G. xvii, xix, xxi, xxii, 83,
161, 164, 168, 171, 302, 303, 304,
306, 311, 432, 433, 434, 447, 448,
485, 507, 528, 529, 530, 537, 544,
546-548, 582, 583, 587, 588, 591,
593-597, 602, 685, 686, 691
Sampson, C.H. 584
Sargent, W.L.G. 312
Sato, M. 444
Sato, T. 82
Sattarov, A.S. 860
Savelova, T.I. 659
Sawyer, E. 167, 590, 862
Saxena, R.K. 782,786,794
Saxena, V.P. 782,783
Schneider, W.R. 861
Schuitman, A. 438, 777
Schwartz, J .T. 467,642
Schwartz, L. 154, 581, 583
Segovia, C. 588,590
Seitkazieva, A. 865
Sekine, T. 454
Semenov, E.M. 5
Semyanistyi, V.I. 162, 581
Senator, K. 864
Sethi, P.L. 793, 794
Sewell, W.E. xxviii, 432, 435, 448,
857
Shabunin, M.L 285
Shah, M. 774
Shanmugam, T.N. 454
Shapiro, H.S. 448
Sharma, B.L. 318
961
Sharma, S. 442
Sharpley, R. 167
Shcherbina, V.A. 302, 685, 693, 779
Shelkovnikov, F .A. 432
Shen, C.Y. 454
Shermergor T.D. xxii
Shevchenko, G.N. 798
Shikhmanter, E.D. 867
Shilov, G.E. 162,487,548,556,583,
867
Shinbrot, M. 864
Sidorov, Yu.V. 285
Simak, L.A. xvii
Singh, B. 782
Singh, C. 776, 786
Singh, Rattan 782
Singh, R.P. 776
Singh, Y.P. 786
Sintsov, D.M. 435
Sirola, R.O. 85
Skal'skaya,LP. 793
Skorikov, A.V. 442, 580,587, 691
Skornik, K. 171
Slater, L.J. 789
Sludskii, F. xxv, 433
Smailov, E.S. 444
Smarzewski, R. 85
Smirnov, M.M. 777, 798, 820, 856
Smirnov, V.I. 429,436
Smith, C.V.L. 171
Smith, K.T. 583
Sneddon, LN. xix, xxxii, 431, 762,
773-775, 777, 793
Sobnak, Sh.D. 371, 434
Sobolev, S.L. 494, 495, 528, 581,
589
Sobolevskii, P.E. 161, 655
Sohi, N .S. 454
Sohncke, L. 857
Solonnikov, V.A. 160
Soni, K. 773,788, 790
Sonine, N.Ya. xxvi, xxvii, 83, 85,
415,431,435,696,773,787,789
Spain, B. 90
Spanier, J. xix, xxii, 172
Sprinkhuizen-Kuyper,LG. 602,870,
871
Srivastav, R.P. 773
Srivastava, H.M. xxxii, 440, 454,
584, 695, 783, 785-787

962
AUTHOR INDEX
Srivastava, K.J. 784
Srivastava, K.N. 776, 777, 786, 794
Srivastava, R. 584
Srivastava, T.N. 786
Starovoitov, A.P. 316
Stechkin, S.B. 249, 444
Stein, E.M. 92, 160, 498, 528, 546,
550,566,581,590,593,604,685
Steinig, J. 312
Stens, R.L. 434, 444, 44448
Stepanets, A.I. 444, 445
Stepanov, V.D. 166
Stolle, H.W. xix
Strichartz, R.S. 588, 594, 597
Stromberg, J .-0. 167, 590
Stuloff, M. 434
Sun' Yung-Sheng 444
Sunouchi, G. 452
Swaroop, R. 785
Szego, G. 308
Szeptycki, P. 432, 583
T
Ta Li 772, 776
Taberski, R. 444, 4448
Taibleson, M. 590
Takahashi, S. 432
Takano, K. 602
Talenti, G. 432
Tamarkin, J.D. xxx, 82, 83, 85, 303,
445
Tanno, Y. 792
Tarasov, R.P. 161
Tardy, P. xxv
Tazali, A.Z.-A.M. 865
Tedone, O. 773, 792
Telyakovskii, S.A. 433,444
Thielman, H.P. 90
Thorin, G.O. 581
Tihonov, A.N. 318
Tikhomirov, V.M. 586
Timan, A.F. xxxii, 315, 444
Titchmarsh, E.C. 24, 86, 169, 695,
708, 709, 713, 774, 775, 780, 857
Tonelli, L. xxix
Torrea, T .L. ,590
Torvik, P.J. xxii
Tranter, C.J. 774, 793
Trebels, W. xix, 163, 164, 303, 313,
444, 783
Tremblay, R. xx, 172,301,452,871
Tricomi, F.G. 14, 18, 22, 88, 91,
160, 172, 180, 193, 196, 199, 201,
307, 441, 455, 661, 677, 724, 733,
735, 751, 772, 783, 791, 820, 851,
856, 861
Triebel, H. 599
Trimeche, K. 441
Trione, E.S. 583, 600
Tripathy, N. 315
Trivedi, T.N. 794
Tseitlin, A.I. xxii
Tu Shih-Tong 869
Turke, H. 312
u
Uflyand, ¥a.S. 762, 793
Ugniewski, S. 85
Ulitko, A.F. 793
Umarkhadzhiev, S.M. xxi, 595, 597
Upadhyay, M. 442
Urdoletova, A.B. 162
Usanetashvili, M.A. 861
v
Vainberg, B.R. 602
Vakulov, B.G. xxii, 497, 581,602
Varma, R.S. 170
Varma, V.K. 791
Vashchenko-Zakharchenko, M. xxv
Vasilache, S. 580
Vasilets, S.I. 693
Vasil'ev,I.L. 688,689-691
Veber, V.K. 162,693,865,866
Vekua,I.N. 774,796,798,856,857,
859, 861, 862
Verblunsky, S. 312, 314
Verma, A. 317,442
Verma, R.U. 584, 781, 782
Vessell a, S. xvii, xxii, 87, 691
Vilenkin, N.Ya. 576
Virchenko, N .A. xxii, 762, 775, 793,
859
Vladimirov, V.S. 172,583,602,846,
847, 865
Volkodavov, V.F. 798, 862

AUTHOR INDEX
Volkov, Yu.I. 449
Volodarskaya, G.F. 692,694
Volterra, V. 83, 85, 89, 686, 693
Voskoboinikov, Yu.E. 85
Vries, G .de 85
Vu Kim Than xxii, 162, 584, 709,
711, 712, 757, 773, 774, 784, 794
w
Wagner, P. 594
Wainger, S. 593
Wall, D.S. 315
Walton, J .R. 689, 793, 794
Wang, F .T. 315
Watanabe, J. 161
Watanabe, S. 602
Watanabe, T. 603
Watanabe, Y. 88, 304, 316
Watson, G.N. 21
Weber, H. 774
Weinacht, R.J. 862
Weinberger, H.F. 862
Weinstein, A. 856, 857, 860, 863
Weiss, G. 581, 590
Weiss, R. 85
Welland, G.V. 581
Westphal, U. xxxii, 161, 170, 312,
371, 434, 446, 865
Weyl, H. xxviii, xxix, xxx, 84, 91,
161, 432, 433
Wheeden, R.L. xxii, 167, 495, 581,
582, 588, 590, 591, 595, 862
Whittaker, E.T. 21, 84
Wick, J. 689
Widder, D.V. 162, 169, 170, 302,
695, 772, 786, 789, 792
Widom, H. 685
Wiener, K. 866
Williams, W.E. 685, 687, 779, 780,
790
Wilmes, G. 371, 447, 433, 582, 587
Wimp, J. 24, 773, 774-776
963
Wing, G.M. 86
Wolfersdorf, L.von 85, 302, 303,
659,685,688,689
Wong, R. 288, 294, 305, 319
Wood, D.D. 864
Wyss, W. 861
x
Xie Ting-fan 444
y
Yakubov, A.Ya. 303, 304,447,448,
594
Yakubovich, S.B. xxii, 757, 772-774,
792
Yanenko, N.N. 869
Yaroslavtseva, V.Ya. 869
Yasakov, A.I. 167
Yoshikawa, A. 161,581
Yoshinaga, K. 161, 162, 580
Yosida, K. 120, 161
Young, A. 85
Young, E.C. 862, 863
Young, L.C. xxxi, 83, 301
z
Zabreiko, P.P. 120, 161, 328, 655,
671
Zaginescu, M. XXll
Zanelli, V. 90
Zav'yalov, B.I. 602, 865
Zeilon, N. xxvii, 302, 684, 689
Zeller, K. 312
Zheludev, V.A. 85,371
Zhemukhov, Kh.Kh. 861
Zhuk, V.A. 859
Zhuk, V.V. 444
Zhong Zhu Zo 454
Zygmund, A. xix, xxx, xxxii, 51, 91,
92, 350, 354, 367, 432, 433, 445,
581,685

SUBJECT INDEX
A
Abel integral equation 29, 30
asymptotic solution 299
generalized 85, 610
multidimensional 458
pyramidal analogue 571
Abel-type integral operator 731
Absolutely continuous function 2
space (see space of absolutely
continuous functions)
Associated Legendre function pt(z)
19
Asymptotic expansion (series) 285
of fractional integral 288, 289,
292, 294, 297, 318, 319
of integral with power-logarithmic
kernel 411
power 286
Asymptotic sequence 285
Asymptotic solution of Abel equation
29
Axially-symmetric potential (see
potential)
B
Banach theorem 13
Bernstein inequality 368, 370
analogue 368, 370
Bernstein type inequality 597
Bessel-Clifford function JII(z) 731
Bessel fractional derivative (E::t: V)a f,
(E::t: D)a f 335
Bessel fractional integral Gal() 333,
482, 540, 598
Bessel fractional integration operator
(see operator of Bessel fractional
integration)
Bessel function of the first kind Jv(z)
19, 20
modified lv(z) 20
Bessel kernel 539
Bessel-Maitland function Jt(z) 437
Bessel potential Gal() 333, 540
anisotropic 598
modification G%I(), esal() 334,541
space L p (Rl) = H a ,P(Rl) = Ga(L p )
336, 541
unilateral G%I() 334, 482, 598
Bessel-type potential 482
Beta-function B(z,w) 17
Binomial coefficient (6) 14
Bisingular integral operator 467
Bochner formula 484
Boundary value problem (see Cauchy,
Dirichlet, Hilbert and Neumann
boundary value problem)

c
SUBJECT INDEX
966
Carleman equation 626,627
Cauchy boundary value problem 812
819, 823, 830 '
Cauchy-type boundary value problem
829, 832, 837
Cauchy formula xxvi, 421
Cauchy integral formula 415
Chen fractional derivative vOt j D Ot j
339, 340 c , c
Chen fractional integral ]g-I{) 338
Cone characteristic 556
Cone light 556
Confluent hypergeometric (Kummer)
function IFda; Cj z) 19
Continuity modulus (integral) wp(J t)
131, 233 '
of fractional order 447
Convolution 12, 25, 28, 154 465
484, 722 ' ,
Cossar fractional derivative 163
(C,a)-method (see method of
summation)
D
D'Alambert wave equation 800
Difference finite (j)(x) 116
of a fractional order (Ot j)(x) 371,
385,453 h
of the vector order (l j)(x ) 469 ,
479 t
weighted Lf, L,,,j 553, 567
with a vector step centered (Ot j)(x)
499 h
with a vector step non-centered
(hj)(x) 499
Differential equation of an integer
order, ordinary 849
Differential equation of fractional
order 829
Cauchy-type problem 829
Dirichlet-type problem 829
linear 837
linear with constant
coefficients 846
ordinary 850
Dirichlet formula 9
pyramidal analogue 571
Dirichlet series 421
Dirichlet-type boundary value
problem 829
Dirichlet weighted boundary value
problem 826, 843, 845
Distribution function 496
Dual equations 762
Dzherbashyan's generalized fractional
integral L(w)1{) 345
E
Erdelyi-Kober operator ]",Ot, K",Ot
332
analogue 766
generalized J>.(71,a), R>.(71, a) 738
Erdelyi-Kober-type operator ]Ot
]Ot . 322 a+;u,,,' .
b-,u,,,
Erdelyi operator 1 0 Ot + o ,]Ot 322
,u,,, b-jU,,,
Euler-Poisson-Darboux generalized
operator 813
elliptic 813
hyperbolic 819
Euler t/1(z)-function 17
F
Factorization
of G-transform 713
of w-transform 754
Favard inequality 370
Feller potential M:f I{) MOtI{) 216 ,
623 ,v ,
analogue ] Ot) I{) 356
Finite part of integral by Hadamard
p.j., pj 112
Formula (see also under Bochner
Cauchy and Funk-Hecke) ,

SUBJECT INDEX
Formula of conformity 800
of fractional integraltion by parts
34
Fourier convolution theorem 25
Fourier kernels 781
Fourier- Laplace series 528
Fourier-multiplier theorem 13
Fourier series 347, 476
Fourier transform :Fcp = :F{cp(t)iX} =
<p(x) 24
convolution 25
multidimensional 473
of fractional integral and
derivative 137, 473, 474
of generalized function 487
of singular integral 200
sine- and cosine- (see sine- and
cosine- Fourier transforms) 25
Fox H-function 781, 787
Fractional derivative (see also Bessel,
Chen, Cossar, Griinwald.:.Letnikov,
Griin w ald- Letnikov-Riesz,
Hadamard, Liouville, Marchaud,
Riemann- Liou ville, Ruscheweyh,
Weyl, Weyl-Liouville, Weyl-
Marchaud fractional derivatives)
in a given direction fOI), X) f
585
in the complex plane Vof, vt8 f ,
Vf 35, 415, 419
of a function by another function
V+;gf 326
of absolutely continuous functions
267
of analytic functions 414-416
of complex order 38
of periodic functions (see Weyl
fractional derivative)
of purely imaginary order Vi8 f
38 a+
Fractional q-derivative 443
Fractional integral (see also under
Bessel, Chen, Griinwald':Letnikov,
Hadamard, Liouville, Riemann-
Liouville, Weyl) 33
in a given direction lf 216,470
967
in the complex plane locp, It8cp,
lcp 94,415,416,418,419,502
of a function by another function
1+jgcp 35, 326
of complex order 38
of generalized functions 146, 151,
154, 157, 158
of purely imaginary order f,
'8 i{J
16_f,l+ 38,89,97
Fractional q-integral ql:v 443
Fractional integro-differentiation of
analytic functions 419, 421
generalization 422
Fractional power of operator 120,
555
Fubini theorem 9
Function (see also Bessel, Bessel-
Clifford, Bessel-Maitland, Beta,
Fox, Gauss, Gorn, Humbert,
Kummer, McDonald, Meijer,
Mittag-Leffler, Riemann, Riemann-
Gurwitz functions)
of bounded mean oscillation 92
Funk-Hecke formula 551
G
Gamma-function f(z) 15
Gauss hypergeometric function
2Fl(a,biciz) 18
Gauss- Weierstrass integral Wtcp 497
kernel W(x, t) 497
Gel'fond- Leon t'ev generalized
integration and differentiation
I"(aiJ), Vn(aif) 426,428
operator of generalized integration
and differentiation I"(aiJ), Vn(aiJ)
426,428
Generalized function (some notions)
Generator of semigroup (see
infinitesimal operator)
Gorn hypergeometric function
F3(a,a',b,b'jCix,y) 193
Griinwald-Letnikov differentiation in
a region 587

968
SUBJECT INDEX
Griinwald-Letnikov fractional
derivative f;) 373
multidimensional f*":J: 479
on a finite interval f:!, f:
386
Griinwald-Letnikov fractional integral
J+ rp, J b _ rp 387
Griin wald-Letnikov-Riesz fractional
derivative r 1t ) 374
H
Hadamard fractional derivative X>%f,
X>+f, X>b_ f 332
Hadamard fractional integral J%rp,
J+ rp, J b _ rp 330, 331
Hadamard property 112
Hankel contour £8 424
Hankel transform
g eneralized S ( a,b,!I' ) f 738
'I,OI,U
modified S'I,OIjU f 325
of modified form {J" (2.jXn f 722
truncated 437
Hardy inequality 104
Hardy space Hp 424
Hardy-Littlewood theorem 66, 102,
103
analogue 91, 466
Heat operator 554, 564
Helmholtz generalized two-axially
symmetric equation 800
Hilbert boundary value problem 807
Holder condition 2, 3
generalized 7
Holder inequality 2, 8
analogue 11
generalized 8
Holder space H>' = H>'(O) 2, 3
gener alized H W , H ([0, 211']) 249,
364
weighted H>'(p)=H>'(O; p), (p)=
H(O; p) 1, 4, 5
Homogeneous function 12
Hyperbolic Riesz potential Ip:J:iOf,
le f 558, 559
Hypergeometric function (see also
under Gauss, Gorn, Humbert,
Kummer)
generalized pF9«ocp); (,89); z) 91
Hypersingular integral DOl f, TOI f 498,
499
annihilation 510
characteristic 518
parabolic TOI f, 'r 0l f 500
symbol Vo(x) 521
truncated D f 500
with homogeneous characteristic
518
with weighted differences T,o f
553
Humbert hypergeometric
1C.. .), 31 (oo .), 32("')
802, 809
function
199, 750,
I
Identity approximation theorem 11
Index laws 177, 301, 307, 727
Index
of complexity of G-transform p + q
706
of G-transform TJ 706
of operator x 607, 631
of singular integral equation x 607
Inequality (see under Berstein, Favard,
Holder, Kolmogorov, Minkowsky,
Opial)
Infinitesimal operator of semigroup
120
Integral equation (see also under
Abel, Carleman, Volterra)
of compositional type 746
of the first kind 606, 696
singular 606
characteristic 608
N oether theory 631, 632
with Bessel function 722, 723

SUBJECT INDEX
with Gauss hypergeometric
function 696
with Legendre function 699
with logarithmic kernel 694
with power kernel 605, 610, 629
Noether nature 630
with power-logarithmic kernel
550, 672
Noether nature 672
Integral transform (see also Fourier,
Hankel, Kontorovich-Lebedev,
Laplace, Mehler-Fock, Mellin,
Meijer, Stieltjes, Varma transform)
23 convolution type 23, 704
with respect to index (parameter)
23, 752
F3-transform 758
G-transform 705, 711, 713
characteristic (c* ,y*) 706
index of complexity (see index of
complexity)
H- and Y -transform 723, 724
w-transform 752
K
Kernel of operator Zx(A) 630
Kober (Kober-Erdelyi) operator 1:;0I<A
K;'OIr.p 322
Kolmogorov inequality 275, 317
Kontorovich-Lebedev transform
K/{J(t)} 753
Kummer function (see confluent
hypergeometric function)
L
Laplace convolution theorem 28
Laplace equation 813
Laplace transform Lr.p = L{ r.p(t)j p} 27,
140
convolution 28
generalized 725
modified A.z.J, A;l J 714
multidimensional 474
969
of fractional integral and
derivative 140, 141, 474
Lebesgue dominated convergence
theorem 10
point 51
Leibniz rule 277
analogue 317
generalized 280
integral analogue 283
Liouville fractional derivative
on axis 'DJ 332
partial and mixed 'D" J 332
Marchaud form 468
truncated D+...+,£J 468
Liouville fractional integral lr.p,
l....z.r.p 94, 419, 462, 502
Liouville space of fractional
smoothness GOI(L p ) 543
Lipschitz space Hl(O), H;, h;, iI;
2, 254, 255
Lizorkin's space  148
of generalized functions ', q,'
150
of test functions ,q, 147,475
Lorentz distance 556
M
McDonald function KII(z) 20
Marchaud fractional derivative 110,
119,468
analogue 225
generalized 168
on half-axis 111, 112
on interval D+J, Db_J 225
truncated D+,£J 226,469
on real line DJ 109, 110, 119
truncated D%,£J 111, 118
Mehler-Fock integral transform 23,
753
Meijer transform 23
generalized 23
Meijer G-function n (zl :) 22

970
SUBJECT INDEX
Mellin convolution theorem 26
Mellin transform rp*(s) = rot{rp(t)iS}
25, 142
convolution 26
multidimensional 474
of fractional integral and derivative
143, 144, 165, 475
Method of summation of integrals or
(0, oc)-method 277
Minkowsky inequality 8
generalized 9
Mittag-Leffler function Eo(z), Eo R(Z)
21 ,
Mixed norm space Lp(Rn) 465
Modulus (see continuity modulus)
Multiplier 598
Muckenhoupt condition 167
M uckenhoupt- Whee den condition 495
M uskhelishvili space H* 246
N
Neumann weighted boundary value
problem 510, 826-828
Neutral periodic function 477
Noether operator 631
o
Operator (see also Abel, Bessel,
Erdelyi-Kober, Erdelyi, Gel'fond-
Leont'ev, Kober, Noether,
Riemann- Liou ville operator)
d-charac teristic (n, m) 631
cokernel Zy*(A*) 630
dificiency numbers 631
heat (see heat operator)
non-convolution with Bessel
functions in the kernel J% , 1%
0,1\ 0,>'
741
normally solvable 630
polysingular 482
of Bessel fractional integration GO
123, 333
of truncation Pal" P+, P_ 211
singular (see singular operator)
tensor product (see tensor product
of operators)
transposed 647
with homogeneous kernel 12
with power-logarithmic kernel!:: '
!:! 388
Opial's inequality 313
O-symbolism 1= O(g), 1= o(g), 1- 9
16
p
Parabolic potentials HOrp, 1{orp 563,
565
space (see space of parabolic
potentials)
Parseval relation 26
Pochhanunerloop 424
symbol (a)n 14
Poincare-Bertrand formula for
interchanging 201
Poisson integral Ptrp 497
Poisson kernel P(x, t) 497
Post generalized differentiation a(V)/,
a[V]f 375, 376
Potential (see also Bessel, Feller,
Riesz potential)
axially symmetric p-dimensional 813
R
Radial function 484
Regularization of divergent integral
529
Riemann function R(, '1i o, '10) 797,
816
Riemann-Gurwitz function (s, a) 20
Riemann-Liou ville frac tional
derivative
left-sided and right-sided
V:+/, vb_I, 35,37, 225,460
mixed and partial 459, 460
pyramidal analogue V Ac !'
VEl I 576, 577

SUBJECT INDEX
of a function by another
function 'D+igf 326
M archaud form
327
Riemann-Liouville fractional integral
left-sided and right-sided
l+r.p, lb_r.p 28,33,232
mixed and partial l::+r.p, 1+r.p
459, 462
pyramidal analogue lAc r.p,
lEI r.p 576, 577
Riemann- Liou ville operators l+, lb_'
'D+,'Db_ 28,33,35,37,225,232
Riemann zeta-function generalized
(see Riemann-Gurwitz function)
20
Riesz differentiation 498
Riesz fractional derivative DOt f 499,
500
Riesz fractional integro-differentiation
483
Riesz kernel k Ot (x) 492
Riesz means (see Riesz normal means)
Riesz mean value theorem 270
Riesz normal means COt(x) 276
Riesz potential IOtr.p 214
analogue [(Ot)r.p 355
anisotropic 588
generalized lnr.p 589
hyperbolic (see hyperbolic Riesz
potential)
modified HOtr.p 481, 565
on half-axis 221
on interval 223
space (see space of Riesz potential)
unilateral 502
modified l+ 'P, Br.p 592
with Lorentz distance (see
hyperbolic Riesz potential)
with radial density 589
Riesz-type polypotential operator !COt
480
Riesz-type potential operator I(Ot)r.p
( ) ,
I+Ot r.p 94,355,419,552
971
Ruscheweyh fractional derivative 430
s
Semigroup property 35,48
Sine- and cosine- Fourier transforms
:F,r.p, :Fer.p 25
Singular integral
with Hilbert kernel H r.p 354
with power-logarithmic kernel
Sa,Ot,mr.p, Sb,Ot,mr.p 675
with weight 618
Singular operator S 199
Sobolev fractional space GOt (L p ) 544
Sobolev limiting exponent 495
Sobolev theorem 494
weighted case 495
Spherical harmonic Ym(O") 529
Spherical mean Mn(x,y;'T) 823
Space (see also Hardy, Holder,
Liouville, Lipschitz, Lizorkin,
Sobolev, Zygmund and mixed
norm space)
of absolutely continuous functions
AC(O) 2
of bounded mean oscillation BMO(a,b)
92
of parabolic potentials HOt (L p ) 564
generalized 600, 601
of Riesz potentials lOt (Lp) 43, 122
of summable functions Lp = Lp(O)
7
with exponential weight Lpw
108
Stieltjes transform 23
T
Table of fractional integrals and
deri vati ves 172
Tay lor formula analogue 46
generalization 88
Tensor product of operators 463
Triple equations 768
Truncated power function y% 22, 94

972
SUBJECT INDEX
v
Volterra functions IL(x,O',a), ILt,{j(x),
v(x) 661, 662
Volterra integral equation 656
w
Weyl fractional derivative vo)J, Do)J
109, 332, 360, 505
multiple and mixed 477
truncated 357
Weyl fractional integral 4 0 )1() 348,
477
multiple and mixed 477
Weyl-Liouville fractional derivative
vo) J 348,477
Weyl-Marchaud fractional derivative
Do) J 353, 357
truncated D( + o) J 357
,f:
y
Young's theorem 12
z
Zygmund generalized space Ao ([0,211"])
364
Zygmund type estimate 249

INDEX OF SYMBOLS
Latin and Gothic
DO) I, DI 352, 357
D...:i:/, D...:i:,el 468, 469
D/, DI 340,470
Dol, Do,el 518,524,552
D+/, Db_I, D+,el 225, 226, 470
V = C:l '"'' a:n ) 458
Vi = alii 458
al...an
Vn(a; I), V{b; I} 426, 428
V(o) I 477
Vol 428
V/, V,e/, V..:i:1 332
Vo) I, V?+I 348,477
V...:i:/, V+..+,el 462, 468
V1:, 9 1 419
V/, VI 339
Vol 415
vg(x) 521
VAc/, VEil 576, 577
V:+/, V b _, V:+I 35,37,225,460
V:+;g/, Vb_;"/, Vt,,1 326
D+>' I, D:",>' I 731
V(OI,02) I V(o,O) 461 471
(al,a2)+' +,€ '
dn,' (0:) 520
dCW V o aCWIe 2
dcw = 0'  46
U1e
A°<p 223
(_A)O I 446
A,(o:) 116,506
A+>' I, A:"'>' I 731
AC(O), Acn(o) 2
( an) 14
(a)n, (a;+l) 705
a(V)/, a[V]1 375, 376
(a - dh/)(x) 375
B°<p 223
B<p 592
B+>' I, B:!" I 731
BMO(a, b) 92
(b m ), (b+l) 705
C w = Cw(R 1 ) 108
C(O), Cm(O) 2
Co(O) 10
CO(x) 276
C:+>. I, C.:..>' I 731
C([a, b]), Cr([a, b]) 159, 160
c. 706
DO I, D I 499, 500
D/, Dtel 109-111,360, 505

974
INDEX OF SYMBOLS
/, +/, b-I 330,331
E 464
(E:f: nYJI I, (E:f: V)Q I 335
EQ(z), EQ,{j(z) 21
E(/3, /3*)u 813
EQ,>"v I EQ,>.,v I 731
a+ ' b-
IFl(a; c; z) 19
2Fl(a;b;c;z) 18
F3(a, a', b, b', c; x, y) 193
Fp, ,IJ = FplJ 158
F. [ ab...,a p ; X ] 91
p 9 /31, . . . , /3 9
:Fep = <p, :F- 1 I = j 24, 473, 484,
486
:Fe ep , :F,ep 25
fn-Q(X) 30
f Q ), I;), I;?.:i: 373, 374, 474
I:, I:l 386, 587
IQ) 585
j (Q) j (Q) 386
a+, b-
GQ ep 333, 482, 540, 598
Gep 334, 482, 598
GQ(L p ), G%(L p ) 123, 334, 483, 543
GQ(x) 539
Gn (zi P:) 22
(Gn I P:? I/(t») (x) 705
(/jQep, (/jQ(x) 541
H, HIJ, H*, H:, H; 246, 608
H ep 354
H>' = H>'(O), H(O) 2, 3, 6
H W , H, H(p) 249
H p 424
HQep, Hep 214
(HQep)(x, t), HQ(L p ) 564
H>'(p) = H>'(O; p), H(p) = H(O; p)
4, 5
H>'Ck n ), H>'Ck n , p) 496
H, H Q , H Q 246,247
H([0,211"]) 364
>. ->.
Hp, Hp 254, 255
H;(.Rl) 634
H;([0,211"]) 367
H Q ,P(R 1 ), H',P([a, b]), H,P([a, b])
232, 336
H>"1c(O), H;'1c (p) 7
H ;,IJ U 800
1{Qep 481, 565
h>' = h>'(O), h(p) 2
>. ->. >.
h p ' h p ' hp ([0,211"]) 254, 367
h>"1c(O), h'1c(p) 7
h;,lJu 800
{h(x)}ep 722
jI+(a,b), 1cI:l_(a,b) 186,187
In(a;/),I{b;/} 426,427
IQep 214,483,492,600
Iep 222
I(OI)ep, IQ)ep 355, 356, 477
IQ(L p ), I(Lp) 43, 122
IQ(L p ), I:i:(Lp) 472
Iep 94, 419, 502
I...:i:ep 462
I (Q) I (Q) I 4
:i: ep, +..;+ 348, 77
Iv(z), iv(z) 20
Iep 338
Iep 216, 470
IRep, IiQep, Ii(L p ) 589, 595
1:01 415, 416
I,t ep, lEI ep 574
Ieep 558
Ip:i:iOep, Ip:i: ep 556
1:+ ep , I6_ep 33,38,459
r::+ep 459, 462
I:+(L p ), I6_(Lp), IQ(L p ) 33, 38,
232
It 8 1 418
I:+jgep, I6_;"ep, It.,ep 325, 329

INDEX OF SYMBOLS
I:f cp. I:! cP 388
I:f(L p ), I:!(L p ), P}/,{j(L p ) 388
I:f'" I, I:!'" I, I.rj,,, I 439
Ic,a,>. Ic,a,>. 7 3 1
a+ ' -
Ia). (0), I(a). (0) 176
I+,,,_ cP 430
I:+;u,,,CP, Ib-;u,,,CP, If:;u,,,cp 322
I",a, It,a 322
-:i:
Ia,>.1 741
qI:v I 443
J:+ cP 387
Jv(z), Jv(z), Jv,m(z), Jf(z) 20
J>.(1],o:)1 738
-:i:
J a,>.1 741
:1 p , :1:, :1 p ,l, :1 0 , 157, 158
:1al 427
cP, +cP, b-CP 330, 331
K:i:, Kt, Kt. 557,558, 559
K",a, K;'a 322
Kv(z) 20
Ki{/(t)}, Ki'/{g(t)} 753
/Cacp 480
/C"a(X) 125, 513
k"a(x) 513
L cP, L -1 9 27
L(w)cp 345
Lp = Lp(O) 7
Lp(p) = Lp(O; p), Lp(Rn; p) 11, 495
Lp(Rn) 457
Lp(Rn) 464
Lp,.(a,b) 176
L) 801
L;(R 1 ) 336
Lp,w 108
Lc,'Y) 706
La ( R n ) La ( Rn ) 505
p,r 'p
£;,r 600
975
Macp, Macp 610
M,tJCP 215
m(O:I"" ,O:n) 176
rotcp, rot { cp(t); s} = cp. (s) 25
rot-I { cp. ( s); x} 25
rot;'(L), rot- 1 (L) 706
Pa(x) 376
P£, (P:f: iO)>' 555
P(x,t), PtCP 497
P:i:CP, Pa"cP 210
Pt(z) 19
p.f., pi, pi 112
Pv cP 603
Qq 301
R n , k n , Rt., Rt...+ 458
R>.( 1],0:) 738
ra(x), r,,(x) 206
S, S', S+ 46, 155
S cP 199
SaCP, S"cP, SaCP 212, 221, 234
Sn-l,ISn-ll 457
S+/, Sb_1 731
Sa,,,1 618
Sa,a,mCP, S",a,mCP 675
S",a;u I 325
S ( a''''Y ) 1 738
",a,u
6:i:, 6(R) 147, 155
T a I, Tea I 500, 547, 549
,!a I, 'I I 565, 567, 569
W(x, t), Wtcp 563, 564
Wr 595
(wnlv(:»I/(Y»)(x) 752
wl',v(x) 437
wp(/, t) 131, 136, 233, 447
X(R 1 ), X 27r 371
Ym(/, x), Ym(u) 528, 529
Y 22, 94
(z+, ZO+, Z-, ZO-) 424

Greek
INDEX OF SYMBOLS
976
[0:], {o:}, () 14,36
B(z,w) 17
f(z) 15
,* 706
,n(O:) 490
-d + :t 562
(dhf)(x), (df)(x,p) 116, 371,
469, 479, 499, 553
d"ar(X, h) 513
6(x-xo) 145
(s,a) 20
x 607, 631
x(o:, I) 119, 469
A;b A;il 714
A([O, 211"]) 364
A 367
AJ(t) 496
Jt(x, u, 0:), Jtt,l3(x) 661
II(X), IIh(X) 662
Dp, (-Dp)<p 554, 555
, ',  147, 150, 151,475,487
+, +, +, (+)' 156, 157, 162
 253
1(,8,6;,;x,y) 199
<p* ( s ), <i'( x) 24, 25
Xar(x; h) 376
\}I, \}I', \}I+ 147, 150, 475, 487
1/;(z) 17
Oar(Y) 523, 530
w(<p, h) 249, 361
wp(f, t) 131, 233
31(0:,0:',,8;,;x,y) 750
3 2 (0:,,8;,;x,y) 802,809