Die Grundlchrcn der
mathematischen Wissenschaften in Einzeldarstellungen
Band 183
J. L. Lions • E. Magenes
Non-Homogeneous
Boundary Value Problems
and Applications
III
Springer-Verlag Berlin Heidelberg New York

Die Grundlehren der
mathematischen Wissenschaften
in Einseldarstellungen
mit besonderer Berucksichtigung
der Anwendungsgebiete
Band 183
Herausgegeben von
J. L. Doob • A. Grothendieck . E. Heinz . F. Hirz^bruch
E. Hopf • W. Maak . S. MacLane . W. Magnus -J. K. Moser
M. M. Postnikov . F. K. Schmidt. D. S. Scott. K. Stein
Geschaftsfiihrende Herausgeber
B. Eckmann und B. L. van der Waerden

J. L. Lions • E.Magenes
Non-Homogeneous
Boundary Value Problems
and Applications
Translated from the French by
P. Kenneth
Volume III
Springer-Verlag Berlin Heidelberg New York 1973

J. L. Lions E. Magenes
University of Paris University of Pavia
Title of the French Original Edition:
Probl&nes aux limites non homogfcnes et applications (tome III)
Publisher: S. A. Dunod, Paris 1968
Translator:
P. Kenneth
Paris
Geschaftsfuhrende Herausgeber:
B. Eckmann
Eidgenossische Technische Hochschule Zurich
B. L. van der Waerden
Mathematisches Institut der Universitat Zurich
AMS Subject Classifications (1970)
Primary 35J20, 35J25, 35J30, 35J35, 35J40, 35K20, 35K35, 35L20,
Secondary 46E35, 46F15, 46F99
ISBN 3-540-05.832-X Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-05.832-X Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the
German Copyright Law where copies are made for other than private use, a fee is payable to the
publisher, the amount of the fee to be determined by agreement with the publisher.
(C) by Springer-Verlag, Berlin • Heidelberg 1973. Printed in Germany.
Library of Congress Catalog Card Number 71-151 407

Preface to the English Translation
The present translation follows the French edition without change,
except for some corrections which were suggested to us by the remarks
of C. Baiocchi and M. L. Bernardi, to whom we express our sincerest
thanks. We have added a complementary bibliography. We also wish to
thank P. Kenneth for his excellent work of translation.
Paris/Pavia, March 1972
J. L. Lions E. Magenes

Introduction
1. Our essential objective is the study of the linear, non-homogeneous
problems:
(1) Pu = f in 6, an open set in R^,
/2) \Qiu = Sj on d& (boundary of (9),
\or on a subset of the boundary d& 1 < j < v,
where P is a linear differential operator in & and where the Q/s are linear
differential operators on dO.
In Volumes 1 and 2, we studied, for particular classes of systems
{P, Qj}, problem (1), (2) in classes of Sobolev spaces (in general constructed
starting from L2) of positive integer or (by interpolation) non-integer
order; then, by transposition, in classes of Sobolev spaces of negative
order, until, by passage to the limit on the order, we reached the spaces
of distributions of finite order.
In this volume, we study the analogous problems in spaces of infinitely
differentiable or analytic functions or of Gevrey-type functions and by
duality, in spaces of distributions, of analytic Junctionals or of Gevrey-
type ultra-distributions. In this manner, we obtain a clear vision (at least
we hope so) of the various possible formulations of the boundary value
problems (1), (2) for the systems {P, Qj} considered here.
2. One difficulty in this direction is connected with the (locally
convex) topologies, which are "naturally" tied to the spaces of Gevrey-
type functions and their duals (a difficulty which did not appear in
Volumes 1 and 2, where, for the essential part, all the spaces were Hilbert
or Banach spaces). The indispensable minimum on this subject is given
in Chapter 7 (1). No doubt a number of our results, in particular in
Chapters 10 and 11, could be improved (in the sense of strengthening
certain topologies) by a tighter topological analysis of the situation;
however, the technical difficulties seem incompatible to us with the
interest of the eventual complements; for this reason we have reduced
the topological considerations to their strict minimum.
3. Once having introduced the Gevrey-type spaces and their duals,
as well as their vector-valued analogues, we consider the problems of
type (1), (2) in the following order:
1) elliptic problems (Chapter 8),
(x) First chapter of this Volume; we continue the numbering of the chapters
from the preceding Volumes.

Introduction
VII
2) general evolution problems (Chaptei 9),
3) parabolic problems (Chapter 10),
4) hyperbolic problems, or problems well-posed in the sense of
Petrowski or of Schroedinger (Chapter 11).
We proceed according to the following steps, which are analogous in
principle (but with entirely different "technical details") to those of
Volumes 1 and 2:
(i) study of the regularity of problem (1), (2), i.e.: assuming "regular"
data / and gj (in the sense: C°°-functions, or Gevrey functions, or analytic
functions), we study the corresponding regularity of u;
(ii) by transposition of the isomorphism established from the results
of type (i), we deduce from them, (after obtaining trace theorems) the
solutions of problems (1), (2) in spaces of distributions, of analytic
functional or of Gevrey functional.
Thus, the "basic tools'' are regularity theorems and trace theorems,
which are established for each situation considered.
In particular, for elliptic equations, the starting regularity theorem
states that the solutions of the boundary value problems with analytic
data are analytic (a theorem which, moreover, appears as a particular
case of a much more general result on elliptic iterates; see Chapter 8) and
the main trace theorem characterizes, as functional analytic on the
boundary, the solutions of the elliptic equations whose right-hand terms
are distributions which "do not grow too rapidly at the boundary".
A similar situation for parabolic problems is discussed in Chapter 10,
in which we obtain complete characterizations by Gevrey-type functions
and functionals.
The other problems (hyperbolic, well-posed in the sense of Petrowski,
or of Schroedinger) are studied according to the same principles, but
since, here, "optimal" regularity results do not exist, we do not obtain
the most general results (certain questions remaining open in this
direction).
A very brief Appendix gives some applications to the calculus of
variations and to optimal control theory in Gevrey-type spaces.
4. As for the preceding volumes, each Chapter ends with comments
and, except for Chapter 7, a list of open problems.
5. The applications of the theory of non-homogeneous boundary
value problems given in this volume and in Volumes 1 and 2 are not
exhaustive; various other applications relative to numerical analysis are
given in Aubin [1], [2], [3], Bossavit [1], Lions [10] and applications
to non-linear problems in Lions [9] for example.
The authors warmly thank G. Geymonat for his constructive
criticism.
Paris, June 18, 1969.

Contents
Chapter 7
Scalar and Vector Ultra-Distributions.
1. Scalar-Valued Functions of Class Mk 1
1.1 The Sequences {Mk} 1
1.2 The Space 3>Mk(&) 2
1.3 The Spaces @Mk(3f) and &Mk(&) 5
2. Scalar-Valued Ultra-Distributions of Class Mk ; Generalizations .... 6
2.1 The Space 3>Mk(&) 6
2.2 Non-Symmetric Spaces of Class Mk 7
2.3 Scalar Ultra-Distributions of Beurling-Type 8
3. Spaces of Analytic Functions and of Analytic Functional 9
3.1 The Spaces Jt(jT) and #\X) .' 9
3.2 The Spaces tf\r) and #\r) 10
4. Vector-Valued Functions of Class Mk 11
4.1 The Space 3)Mk(J;F) 11
4.2 The Spaces ®uk(tf\ F) and Sm^\ F) 12
4.3 The Spaces @±,Mk(Sl F) 13
4.4 Remarks on the Topological Properties of the Spaces 2mJ^\ F),
#Mk(S;F),®±iMk(J;F) 15
5. Vector-Valued Ultra-Distributions of Class Mk\ Generalizations .... 16
5.1 Recapitulation on Vector-Valued Distributions 16
5.2 The Space 2>'Mk{J\F) 18
5.3 The Space @'±tMk(S; F) 20
5.4 Vector-Valued Ultra-Distributions of Beurling-Type 21
5.5 The Particular Case: F = Banach Space 22
6. Comments 22
Chapter 8
Elliptic Boundary Value Problems in Spaces of Distributions and
Ultra-Distributions.
1. Regularity of Solutions of Elliptic Boundary Value Problems in Spaces
of Analytic Functions and of Class Mk; Statement of the Problems and
Results 25
1.1 Recapitulation on Elliptic Boundary Value Problems 25
1.2 Statement of the Mk -Regularity Results 26
1.3 Reduction of the Problem to the Case of the Half-Ball 29

Contents
IX
2. The Theorem on "Elliptic Iterates": Proof . . 30
2.1 Some Lemmas 30
2.2 The Preliminary Estimate 32
2.3 Bounds for the Tangential Derivatives 35
2.4 Bounds for the Normal Derivatives 48
2.5 Proof of Theorem 1.3 54
2.6 Complements and Remarks 55
3. Application of Transposition; Existence of Solutions in the Space @'(Q)
of Distributions 58
3.1 Generalities 58
3.2 Choice of the Form L; the Space E(Q) and its Dual 61
3.3 Final Choice of the Form L; the Space Y 64
3.4 Density Theorem 65
3.5 Trace Theorem and Green's Formula in Y 66
3.6 The Existence of Solutions in the Space Y 70
3.7 Continuity of Traces on Surfaces Neighbouring F 70
4. Existence of Solutions in the Space &mAQ) of Ultra-Distributions . . 74
4.1 Generalities 74
4.2 The Space EmAQ) and its Dual 75
4.3 The Space Y^k and the Existence of Solutions in Ym^ 79
4.4 Application to the Regularity in the Interior of Ultra-Distribution
Solutions of the Equation Au = f 82
5. Comments 83
6. Problems 85
Chapter 9
Evolution Equations in Spaces of Distributions and Ultra-Distributions.
1. Regularity Results. Equations of the First Order in t 87
1.1 Orientation and Notation 87
1.2 Regularity in the Spaces ^+ 88
1.3 Regularity in the Spaces @+tM, 91
1.4 Regularity in Beurling Spaces 94
1.5 First Applications 95
2. Equations of the Second Order in / 98
2.1 Statement of the Main Results 98
2.2 Proof of Theorem 2.1 99
2.3 Proof of Theorem 2.2 100
3. Singular Equations of the Second Order in t .107
3.1 Statement of the Main Results .107
3.2 Proof of Theorem 3.1 107
4. Schroedinger-Type Equations Ill
4.1 Statement of the Main Results Ill
4.2 Proof of Theorem 4.1 Ill
4.3 Proof of Theorem 4.2 113
5. Stability Results in M^-Classes 114
5.1 Parabolic Regularization 114

X
Contents
5.2 Approximation by Systems of Cauchy-Kowaleska Type (I) ... 117
5.3 Approximation by Systems of Cauchy-Kowaleska Type (II) ... 121
6. Transposition 125
6.1 Orientation 125
6.2 The Parabolic Case 125
6.3 The Second Order in / Case and the Schroedinger Case 128
7. Semi-Groups 129
7.1 Orientation 129
7.2 The Space of Vectors of Class M% 129
7.3 The Semi-Group G in the Spaces D(A°°; Mk). Applications . . . 135
7.4 The Transposed Settings. Applications 143
7.5 Another Mk -Regularity Result 148
8. Mk -Classes and Laplace Transformation 151
8.1 Orientation-Hypotheses 151
8.2 Mk -Regularity Result 151
8.3 Transposition "... 153
9. General Operator Equations 153
9.1 General Results 153
9.2 Application. Periodic Problems 158
9.3 Transposition 160
10. The Case of a Finite Interval ]0, T[ 161
10.1 Orientation. General Problems 161
10.2 Space Described by v(0) as v Describes X 162
10.3 The Space Smh 164
10.4 Choice of L 167
10.5 The Space Y and Trace Theorems 168
10.6 Non-Homogeneous Problems 170
11. Distribution and Ultra-Distribution Semi-Groups 174
11.1 Distribution Semi-Groups 174
11.2 Ultra-Distribution Semi-Groups 180
12. A General Local Existence Result 181
12.1 Statement of the Result . 181
12.2 Examples 184
13. Comments 186
14. Problems 187
Chapter 10
Parabolic Boundary Value Problems in Spaces of Ultra-Distributions.
1. Regularity in the Interior of Solutions of Parabolic Equations 191
1.1 The Hypoellipticity of Parabolic Equations 191
1.2 The Regularity in the Interior in Gevrey Spaces 195
2. The Regularity at the Boundary of Solutions of Parabolic Boundary
Value Problems 203
2.1 The Regularity in the Space 3>{Q) 203
2.2 The Regularity in Gevrey Spaces 204
3. Application of Transposition: The Finite Cylinder Case 208
3.1 The Existence of Solutions in the Space @'{$>)'. Generalities, the
Spaces X and Y 208

Contents
XI
3.2 Space Described by Vv as v Describes X 212
3.3 Trace and Existence Theorems in the Space Y 216
3.4 The Existence of Solutions in the Spaces @'S)r(Q) of Gevrey Ultra-
Distributions, with y > 1, s > 2m 221
4. Application of Transposition: The Infinite Cylinder Case 224
4.1 The Existence of Solutions in the Space @'+{R; 3>'(Q)): The Space
X__ 224
4.2 The Existence of Solutions in the Space ^+(R; 3>\Q)) • The Space
y+ and the Trace and Existence Theorems 229
4.3 The Existence of Solutions in the Spaces ^'.^(R; ^'r{Q)), with
r>l, s>2m ' 236
4.4 Remarks on the Existence of Solutions and the Trace Theorems in
other Spaces of Ultra-Distributions 239
5. Comments 243
6. Problems 244
Chapter 11
Evolution Equations of the Second Order in t and of Schroedinger Type.
1. Equations of the Second Order in /; Regularity of the Solutions of
Boundary Value Problems 246
1.1 The Regularity in the Space @(Q) 246
1.2 The Regularity in Gevrey Spaces 248
2. Equations of the Second Order in /; Application of Transposition and
Existence of Solutions in Spaces of Distributions 251
2.1 Generalities 251
2.2 The Space ^_,y([0, T]; @V{Q)) and its Dual 253
2.3 The Spaces X and Y 254
2.4 Study of the Operator V 258
2.5 Trace and Existence Theorems in the Space Y 260
2.6 Complements on the Trace Theorems 263
2.7 The Infinite Cylinder Case 265
3. Equations of the Second Order in t; Application of Transposition and
Existence of Solutions in Spaces of Ultra-Distributions 268
3.1 The Difficulties in the Finite Cylinder Case 268
3.2 The Infinite Cylinder Case for m > 1 269
4. Schroedinger Equations; Complements for Parabolic Equations .... 273
4.1 Regularity Results for the Schroedinger Equation 273
4.2 The Non-Homogeneous Boundary Value Problems for the
Schroedinger Equation 274
4.3 Remarks on Parabolic Equations 276
5. Comments 277
6. Problems 277
Appendix. Calculus of Variations in Gevrey-Type Spaces 279
Bibliography 290

Contents of Volume I
(Published 1972)
Chapter 1 Hilbert Theory of Trace and Interpolation Spaces
Chapter 2 Elliptic Operators. Hilbert Theory
Chapter 3 Variational Evolution Equations
Contents of Volume II
(Published 1972)
Chapter 4 Parabolic Evolution Operators. Hilbert Theory
Chapter 5 Hyperbolic Evolution Operators, of Petrowski and of Schroedinger.
Hilbert Theory
Chapter 6 Applications to Optimal Control Problems
Appendix Boundary Value Problems and Operator Extensions

Chapter 7
Scalar and Vector Ultra-Distributions
In this chapter we introduce certain spaces of scalar or vector-
valued, infinitely differentiable functions and the spaces of
ultra-distributions derived from them by duality. These notions are essential to
the remaining part of the text. However, in order not to burden the
presentation, particularly with techniques from the theory of topological
vector spaces which are in a certain sense marginal to the theory of
partial differential equations, we state only the definitions and properties
of these spaces and refer the reader to the original texts for the proofs.
1. Scalar-Valued Functions of Class Mk
1.1 The Sequences {Mk}
Our aim is to generalize the notion of distribution of L. Schwartz [1]
(see also the recapitulation given in Chapter 1, Section 1) on an open set
Q in Rw, by taking as fundamental space, instead of 3}{Q), a "smaller"
space of infinitely differentiable functions in Q whose derivatives are
bounded by suitable sequences of positive numbers. Let us therefore
introduce, once and for all, the fundamental hypotheses on the sequences
to be considered.
Let {Mk}, k = 0, 1, 2, ..., be a sequence of positive real numbers
such that
(1.1) M*<M%_xMh+1 VA>1
[logarithmic convexity condition),
(1-2) £^<+~
k=l Mk
(non-quasi-analyticity condition),
I ^ere exists a constant H such that
{'' \Mk+1<HhMk Vk

2 1. Scalar-Valued Functions of Class Mk
(sufficient condition for stability with respect to differentiation),
(1.4)
there exists a constant c1 such that
k) Mk_jMj < c^Mk Wk and V/, 0 < / < k
(sufficient condition for stability with respect to multiplication and
composition). D
Essential examples of sequences {Mk} are the Gevrey sequences:
(1.5) Mk = (k\)s, Mk = kks, Mk = r(sk + 1),
where s is real and > 1 and r is the Euler function.
1.2 The Space ^Mk(ii)
For all definitions and notations of the theory of topological vector
spaces to be used in the sequel, we refer the reader to Bourbaki [1], [2],
Grothendieck [4], Garnir-de Wilde-Schmets [1], Horvath [1], Treves [1].
Let Q be an arbitrary, non-empty, open set in Rw.
Definition 1*1. £#Mk (Q) denotes the space of infinitely differentiable
functions x -> cp{x) with compact support in Qy such that there exist two
positive numbers c and L [dependent on cp) with
(1.6) sup |D>(*) | < cLhMk-, \oc\=k, k = 0, 1, 2, ... Q
The following are some properties of <2)Mk (Q) (see for instance Rou-
mieu [1], [2]):
a) @Mk (@) does n°t reduce to {0}: for every ball Se of radius s
contained in Q, there exists a Qe £ <2>Mk (Q) satisfying the conditions:
Qe(x) > ®y support of Qe(x) contained in Se, j qe{x) dx = 1;
b) there exists a partition of unity by functions of <2)Mk (Q), for every
open covering of Q;
c) @Mk {&) is stable with respect to differentiation, that is if
<P € @mAQ) then D> € ®Mk (fi), V*;
d) <2)Mk (Q) is an algebra;
e) @Mk (Q) is dense in 2{Q).
Properties a), b) and e) follow from (1.1) and (1.2), c) from (1.3) and d)
from (1.4). D
Remark 1.1. We shall not always use all the hypotheses (1.1), ...,
(1.4); we shall specify whenever necessary. D
A "natural'' topology is introduced in @Mlc (Q) as follows.

1.2 The Space @Mk{&) 3
For jf a compact set contained in Q and L a positive number,
consider the subspace <2)Mk (Q; jf, L) of @Mk (Q) made up of the
functions y with support in X and such that (1.6) holds for this fixed L, c
still depending on cp. We can easily see that @Mk (Q; Jf, L) is a Banach
space for the norm
D>(*)
(1-7) Ml= sup .-_
|aJ=ft,fe=0,l,...| ^ ^Jfc
From the algebraic point of view, we evidently have
as (jf, L) varies over the set of couples such that Jf is an arbitrary
compact set contained in Q and L an arbitrary positive number; this
set is filtering for the relation
pf, L) < {X\ V) if X C tf'> L < L'.
We also have
®Mk(Q'yXyL)C®Mk(Q\X\L') if (X,L)<(X',L')
with continuous injection.
Thus it seems natural to provide the space @Mk (Q) with the inductive
limit topology of the topologies of the spaces <3Mk (Q; X, L) (that is,
the finest locally convex topology for which the injection of @Mk (Q; X, L)
into @Mk (@) is continuous). This amounts to defining
(1.8) ®Mk (Q) = ind Hm 9Mk (Q; X, L),
where X increases monotonically to Q and L increases monotonically
to + oo; and it suffices for X and L to vary over a sequence Xn and Ln
with the same properties. D
Remark 1.2. Since X is bounded, we can easily see that the norm
(1.7) for QfMh (Q; X, L) is equivalent to any one of the following norms:
||a?II = sup 7 --- fixed j) with 1 < p < oo.
LrMh
|p>IIl»(Q)
\a\=k,k=0,l,... L Mk
For p = 2,we may also consider the norm
M [&£> {L*Mk?j
for which 9Mk (Q; X, L) is a Hilbert space. D

4 1. Scalar-Valued Functions of Class Mk
We have
Proposition 1.1. The injection of <2)Mk (£?; Jf, L) into Q)Mk (Q; Jf', L')
is compact if pf, L) < pf', L').
Proof. Let {cpn} be a bounded sequence in @Mk (Q; Jf, L). By
applying the Ascoli-Arzela theorem we see that there exists a subsequence
{^nj oi {yn} and a function y in @Mk(Q; X*, L), and therefore in
0Mfc (&; <^"> L')>such that
(1.9) lim sup |D>W. (*) - D>(*) | = 0, Vol.
Therefore, it is sufficient to show that
J™ II^ -pI^^l') =0-
Now, thanks to (1.7) and the fact that yn and 99 have their support in Jf,
ipyn,(*)-p>(*)l
II ?>»< - <P UMk w,u) = sup rrWM
< sup sup ttt^^t- (|D>n, (*) -D>(*) |) +
+ sup(y^r sup t*ttIdV»«(*)-d>(*)I^
\D*q)Jx) -Daq)(x)\ /L\NI
^ SUP SUP rr^r +\JT) \\<Pn4 - VhMkiO.JW
0<k<N xW,\*\=k \L ) Mk \^ / fc
|D>n,(*)-D>(*)| , /L\"
SUP SUP m»Af + T7") ' "
<
<
0
with constant c. For fixed s > 0, we can therefore take Are such that
c(L[L')Ne < e[2 and then it follows from (1.9) that there exists an ne
such that for n > we we have
II 9^ -<PhMk{Q;X>,L>)<£- D
Thus we see that <2)Mk (Q) is an inductive limit of a sequence of
Banach spaces En such that
(1.10) En C En+i, with continuous and compact injection.
Following S. Silva [1], if E — indlim En with (1.10), we shall call E
an inductive limit of a regular sequence of Banach spaces. This type of

1.3 The Spaces 9Mlc(^) an^ &Mk(®)
5
space has in particular the following topological properties (see Silva
[1], Yoshinaga [1], Raikov [1], Matagne [1], Komatsu [5]).
(oc) it is separated;
(/?) every bounded set in E is contained in an En, for suitable n, and
is bounded in En;
(y) it is complete, separable, a Montel space (and therefore reflexive),
the strong dual of a Frechet and Schwartz space and therefore also
a (^J5')-space (for the definitions of Schwartz spaces and (^J5')-
spaces see, for example, Horvath [1], Garnir-de Wilde-Schmets
Therefore @Mk (Q) satisfies conditions oc), /?) and y); furthermore it is a
nuclear space (see Mityagin [1]). D
We shall call @Mk (Q) a space of functions of class Mk; when Mk is
given by one of the sequences (1.5) we shall call @Mk {&) ^e Gevrey space
of order s (it is easy to see that the three sequences (1.5) yield the same
space) and we shall often denote it by @S(Q). D
Remark 1.3. We have pointed out properties oc) and /?) of inductive
limit spaces of a regular sequence of Banach spaces. We note that,
given a space E, inductive limit of an increasing sequence of Banach
spaces (i.e. En d En+-[, with continuous injection), E also satisfies
properties oc) and /?) in the following two cases:
1) E is a strict inductive limit (see Dieudonne-Schwartz [1],
Proposition 4);
2) the spaces En are reflexive (see D. G. Schaeffer [1], Appendix 2).
An immediate consequence of properties oc) and /?) then is that E is
reflexive if the En's are reflexive.
1.3 The Spaces 3fMh{X) and ^Mk(0)
Let Q be an open set in Rn and jf a compact set contained in Q.
Definition 1.2. Q)Mh (Jf) denotes the space of restrictions to X of the
infinitely differentiable functions on Q, #-> (p{x), such that there exist two
positive numbers c and L [dependent on cp) such that
(1.11) sup |D>(*) I < cLkMk, \oc I = k, k = 0, 1, 2, ...
xtsr
@Mk ffl) is provided with the following topology. For fixed L,
consider the subspace @Mk (jf; L) of @Mje (jf) made up of the elements
cp such that (1.11) holds for this fixed L, provided with the norm
m\= saP
x€*r,\a\=k
ft=G,],2,...
T>"q>{x)

6 2. Scalar-Valued Ultra-Distributions of Class Mk; Generalizations
(or with the norms of Remark 1.2). It is a Banach space and the injection
of@Mk {yC> L) into$)Mh pf, L') is compact if L < V (see Proposition 1.1).
It is then natural to provide <2)Mlc (jf) with the inductive limit topology
of the spaces @Mjc (jf, L) as L increases monotonically to -|-oo; thus
@Mk {<&) is a*1 inductive limit space of a regular sequence of Banach
spaces. D
Now let fibean arbitrary open set in Rw.
Definition 1.3. $Mk (Q) denotes the space of infinitely differentiable
functions on Q, x^> <p(x), such that for every compact set Jf contained in Q,
the restriction of <p to X belongs to Q)Mu (jf).
If we let jf vary over the compact sets contained in Q, and if we
denote by r# the restriction of <p to Jf, then we can provide $Mk (Q)
with the projective limit topology of the topologies of the spaces @Mlc (jf)
with respect to the set of mappings r# (that is, the coarsest locally convex
topology which makes the mappings r# of SMlc (Q) into @Mk (jf)
continuous) . D
We note that no hypothesis on the Mk's has been used for the
definition of the spaces @Mk{^)' ^n tne other hand, using (1.3) and (1.4),
we see that @Mk [X) and SMu (Q) are stable with respect to
differentiation and are algebras.
2. Scalar-Valued Ultra-Distributions of Class Mk;
Generalizations
2.1 The Space &Mhi9)
Let Q be an open set in R* and {Mk} a sequence of positive numbers
satisfying (1.1), (1.2), (1.3) and (1.4).
Definition 2.1. &Mu{Q) denotes the dual of @Mlc(Q), provided -with
the strong dual topology.
The elements of ^}'Mh (Q) (continuous linear forms on <2jM]c (£?)) are
called ultra-distributions on Q of class Mk. If Mk is given by any one of
the formulas (1.5), then we shall call the elements of <2JrMk (Q) Gevrey
ultra-distributions or functionals of order s and we shall also denote
®'Mk{Q)hy®'s{Q). D
@'Mk {&) is a Frechet and a Schwartz space. D
Applying property e), Section 1.2, of the space <2)Mk(Q), we may
identify Q}' (Q) with a subspace of @'Mk (Q); therefore every distribution
on Q is an ultra-distribution on Q of class Mk and we have the
continuous injection
(2.1) 2'{Q)C2'Mk{Q). Q

2.2 Non-Symmetric Spaces of Class Mk 7
Taking into account property b), Section 1.2, of @Mk (Q), we see
that, as for the distributions of L. Schwartz [1], we can show the
localization principle for ultra-distributions and then define the support of an
ultra-distribution. D
Differentiation in @'Mk (Q) and multiplication by a function of
$Mk (Q) are also defined as for distributions, that is:
for u £ Q)'Mh (Q), we have
2.2) <J)"u, <p> = (-1)1"1 (u, D» V<p G 0Mfc(fi);
for u G 0^fc (£?) and ip £ <^Mfc (^) > we have
(2.3) <y>u,<py = <u)Wy vcpe@Mk{Q). D
The structure of ultra-distributions is specified by the following
theorem (see Roumieu [2], Theorem 10).
Theorem 2.1. Every ultra-distribution u £ &Mk (Q) can be represented,
in a non-unique fashion, by the form
oo
(2.4) «=2 EDX-
k=0\<x\=k
where the jua's are measures on Q such that
oo
(2-5) 2 2 MkL» f\d^\<+c*
fc=0|a|=fc #
for every L > 0 flw^ 00073/ compact set c/f contained in Q.
Conversely, if the jua's are measures on Q satisfying (2.5), then u given
by (2.4) (that is
<»>f>=Z 2 (-1)W/D><K)
£ = 0|<%|=& Q
defines an element of SffMk (Q). D
Remark 2.1. The same type of structure results, where the /u0's are
LP'(Q)-functions, can be obtained by using the (equivalent) definition
of 3)Mk (Q; jf, L) given in Remark 1.2, with \\p + \\p' = 1.
2.2 Non-Symmetric Spaces of Class Mk
For the definitions of the spaces 2iMk (Q) and $Mh (Q) we have
considered the variables xv ..., xn in symmetric fashion; but, in view of
certain applications, it is useful to treat them in non-symmetric fashion;
and this can be realized (see for example Roumieu [2]) by imposing,
instead of (1.6), bounds of the type
(2.6) sup | T>"<p(x) | < cL^Ma, Va,

8 2. Scalar-Valued Ultra-Distributions of Class Mk; Generalizations
where the sequence {Ma} depends on the multi-index of differentiation <x.
We shall restrict ourselves to a discussion of a particular case which
arises in connection with partial differential evolution equations. 0
Again let fibean arbitrary open set in RM and let ]t0, t±[ be an
arbitrary open interval, finite or infinite, in R1; consider the cylinder Q =
Qx]tQ,t±[ in R*+1 and two sequences of positive numbers {Mk} and
{Nh} satisfying conditions (1.1) —(1.4).
We then can introduce the spaces @Nh,Mk{Q)> @Nh,Mk(^) ffl a
compact set in Q) and SNhtMk ((?) in &n obvious way; for example, if we
denote by (x, t) the point in R"+1 (x = [xly..., xn) GRM,^ R1): @Nh,Mk ((?)
is the space of infinitely differ entiahle functions (x, t) —> <p[x, t) with compact
support in Q, such that there exist two positive numbers c and L [depending
on q>) with
(2.7) sup |D*DjV(*, t) \< cLk+hMkNh, \<x\ = kt k, h = 0, 1, ...
@Nh,Mk (Q) *s provided with a topology of inductive limit of Banach
spaces in a completely analogous way to what was done for @Mk [Q). Q
In this manner, we arrive at the notion of ultra-distribution on Q of
class {Nh, Mk}, as a continuous linear form on @Nh,Mk ((?) '> more precisely,
we define
[Q)= strong dual of ®NhtMk{Q). Q
In the following chapters we shall make use of the particular case
for which Mk = [k\y, Nh = [h\)s, with real r, s > 1; for this case, we
shall sometimes denote the spaces @Nh,Mk ((?) an(^ @Nh,Mk(Q) by @s.r(Q)
and @'Sir(Q) and call them Gevrey spaces of order [s, r).
2.3 Scalar Ultra-Distributions of Beurling-Type
Let us also call attention to another generalization of the notion of
distribution which is close to the one given in Section 2.1. We still
consider an open set Q in RM and a sequence {Mk} satisfying (1.1) —(1.4),
and, instead of @Mk [Q), we use as our fundamental space, &Mk [Q)
defined by
Definition 2.2. &Mk [Q) is the space of infinitely differentiable
functions x—> (p[x) with compact support in Q, such that for every L > 0 there
exists a positive number c [depending on L and 99) such that
(2.8) sup I D>(*) I < cLhMk, \<x \ = k, k = 0, 1, 2, ...
3SMk [Q) is provided with the following topology:
(2.9) @Mk [Q) = ind Urn (project Urn Q)Mk [Q; jf', L)),

3.1 The Spaces 3tf{X) and 3tf'{tf)
9
where Jf increases monotonically to Q and L decreases monotonically
to zero (compare with definition (1.8) of @Mk (£?)).
By applying Proposition 1.1., we see that &Mk (Q) is a strict inductive
limit of Frechet and Schwartz spaces, whose topological properties we
therefore know (see for example Yoshinaga [1]).
Evidently @Mk {Q) C @mk (&)•
By applying Lemma 1 on page 66 of Roumieu [1], we can show that
there exists another sequence {M*} satisfying conditions (1.1) and (1.2)
such that
(2.io) ®m(Q)c®MM;
from which we have the fact that properties a) and b) of Section 1.2
hold for <%Mk (Q) as well (see Remark 1.1).
Furthermore, applying Proposition 9, page 54 of Roumieu [1]
(@m% {&) is dense in <3tMk (Q)) we see that &Mk (Q) is dense in @Mk (£?).
Finally we see that 0&Mk{Q) is a stable algebra with respect to
differentiation. D
The space of ultra-distributions of Beurling-type relative to the
sequence {Mk} is defined by
(2.11) &Mk (<£) = stronS dual of @Mk (Q) •
From the properties of &Mk (Q), we deduce that @'Mk (Q) may be
identified with a subspace of 0&'Mk {&) \ for a fixed sequence {Mk}, we
therefore have
(2.12) 9'{Q)C9'm{Q)C^Mh^)- D
For the development of the theory of Beurling-type
ultra-distributions, see Beurling [1] and Bjorck [1] (see also the Comments to this
chapter). The notions given here are sufficient for the rest of this text. D
Remark 2.2. Clearly we can also consider non-symmetric Beurling
spaces with respect to xlt..., %n\ in particular we can consider the spaces
&Nh,Mk(P) and &Nh,Mk(®) (compare with Section 2.2).
3. Spaces of Analytic Functions and of Analytic Functional
3.1 The Spaces ^(jf) and ^'(jf)
Let ctf be a compact set in RM. In the sequel, we shall use the space
2tf{X) of analytic functions on jf. One of the usual ways to define 2tf{X)
is the following.

10 3. Spaces of Analytic Functions and of Analytic Functionals
We imbed RM in the ^-dimensional complex space Cn. For (9 an open
set in Cn, we denote by J^(0) the space of holomorphic functions on (9,
provided with the topology of uniform convergence on the compact
subsets of 6. Then we define 3/f pf) by
(3.1) 3tf (jf) = ind Km jf (0)
o
as 0 varies over the set of (complex) neighborhoods of Jf. ^(Jf) is
complete, separable, a Montel space (and therefore reflexive), the dual
of a Frechet-Schwartz space, nuclear (see for example Grothendieck
[1] [3]).
The space of analytic functionals on jf is by definition the space
(3.2) Jf'(jf) = strong dual of Jf(jf)
(see the Comments for references).
3.2 The Spaces tf{T) and ^'(jT)
For the sequel we shall require the particular case where Jf is an
(n — 1)-dimensional, real analytic variety r, the boundary of a bounded
open set Q in Rn (Q is considered as a variety with boundary, the
boundary being r).
For this case it is of interest to introduce ffl(T) without "leaving"
the variety F, by using the Laplace-Beltrami operator Ar on F (for the
definition of Ar, see for example de Rham [1]).
For fixed L > 0, we define ffljST) as the space of infinitely
differentiate functions q> on r, such that there exists a positive number c
(depending on q>) such that
(3.3) sup \AkMx) I < cLk{2k)! k =0, 1, ...;
ffljjj") is a Banach space for the norm
Akr<p(x)
(3.4) ||p || = sup
*€r,ft=o,i,...
L*(2ft)!|
It will follow from Chapter 8 (theorem on "elliptic iterates") that
(3.5) j#>(r) = ind lim Jf L(.T).
L->+oo
From which we deduce (see Lions-Magenes [1], Proposition 1.3) a
theorem on the structure of analytic functionals on T:

4.1 The Space 3>MlAJ'> F) H
Theorem 3.1. Every element u of $"(T) may "be represented, non-
uniquely by the form
oo
(3.6) <«.?>>= 2 /Aj^dj«»,
k=o r
where the fa's are measures on r such that
oo
(3.7) 2 Lk(2k)l \Pk I < + °° f°r every l>o.
k=0
Conversely, if the measures /Lik are given with (3.7), then u, defined by
(3.6), is an element of #e'(T). 0
Remark 3.1. The norm (3.4) may be replaced by the norm
k=o,i,... ^ \Ak\)
Another structure theorem for analytic functionals on r can be deduced
from this, by replacing the measures [ik in Theorem 3.1 with functions
gkeLp\r) (lfp +l[p' = 1). Similarly, #eL{T) becomes a Hilbert
space if we provident with the norm
/ oo 1 \ J/2
4. Vector-Valued Functions of Class Mk
4.1 The Space @Mh(J\ F)
For the applications to the boundary value problems studied in the
following chapters, the above definitions must be extended to the case of
functions and ultra-distributions with values in a topological vector
space. We shall restrict ourselves to the consideration of functions and
ultra-distiibutions of a single real variable, in an open interval of R1;
this is in fact the case of interest for the evolution equations we have in
mind.
Thus let F be a locally convex, separated, topological vector space.
Let J = ]t0, t±[ be an open interval, finite or infinite, in R1.
<2){J>; F) shall denote the space of infinitely differentiable functions,
t-^(p(t), on «/, with values in F, and with compact support, provided
with the topology of L. Schwartz [2] (this is the inductive limit topology
of the spaces Q){J, jf; F) as jT varies over the set of compact intervals
contained in J, where @{J, C/f\F) is the subspace of 2$(<f\ F) of func-

12
4. Vector-Valued Functions of Class Mk
tions with support contained in Jf, provided with the topology of
uniform convergence on Jf of 9? and of each of its derivatives). 0
Again, let {Mk} be a sequence of positive numbers satisfying (1.1) —
(1.4).
Definition4.1. Q)Mh{J\F) denotes the space of functions t-><p(t)
defined on J and with values in F, such that
(4.1) <pe@(J?)F)
and such that there exists a number L > 0 and a bounded set 38 in F [both
depending on cp) such that
<P{k)(t)
(4.2) i^lle^fytrzJ) £ = 0,1,2,...
LrMh
If F = C (scalar case) we recover (algebraically) the space @Mjc [J>)
defined in Section 1.2.
The topology on @Mlc {J\ F) is defined as follows. Let c/f be a compact
interval contained in J and let L > 0 be fixed; we consider the subspace
®Mk {<$> C/f ,L\F) of <2)Mk («/; F) made up of the elements q> with
support in Jf and such that (4.2) holds for this fixed L (the bounded set &
depending on 99); and we provide <2)Mk (</, jf, L; F) with the topology
defined by the fundamental system of neighborhoods of the origin given
by
^/e-rF,vM = o,i,2,... ,
I LkMh J
where i^F describes a fundamental system of neighborhoods of the origin
in F.
As for the scalar case (see Section 1.2), we provide <2jMic [J\ F) with
the inductive limit topology of the topologies of the spaces <2jMic (J', Jf, L; F)
as c/f increases monotonically to J and L increases monotonically to
+ 00; therefore
(4.3) &tl1t{J;F)=mAlim 9m (J, X, L; F).
Then, if F = C, we have <2)Mk {J, C) = Q)Mh (<?) algebraically and topo-
logically.
4.2 The Spaces @Mk(3f; F) and SMh{J\ F)
Let X be a compact interval in R1.
Definition 4.2. <2)Mk {C/f \ F) denotes the space of infinitely differentiable
functions, t-xp(t), defined on jf, with values in F, such that there exists
r
-{,

4.3 The Spaces ^±)Mk(jr'' F)
13
a positive number L and a bounded set g& in F (both depending on (p) such
that
(4-4) lTTT^a> V*€ Jf, A = 0, 1, 2, ...
L*Mh
@Mk {$f > F) is provided with the following topology. For every fixed
L > 0, we consider the subspace $)Mk (Jf, L; F) of <2)Mk (Jf; i7) made
up of the elements q> for which (4.4) holds for this fixed L, provided with
the topology defined by the fundamental system of neighborhoods of
the origin given by
r = L\-^-erF,vte*r,k = 0,1.2 ...J,
where *V'F describes a fundamental system of neighborhoods of the origin
in F.
We then provide @JMk (Jf; F) with the inductive limit topology of the
spaces @Mk (jf, L; F) as L increases monotonically to + oo. Q
Now let J be an open, finite or infinite, interval in R1.
Definition 4.3. $Mk (J\ F) denotes the space of infinitely differentiable
functions t-xp(t)y on Jy with values in F, such that for every compact
interval Jf contained in J the restriction of cpto C/f belongs to <3tMk (jf; F).
By letting Jf vary over the set of compact intervals contained in «/,
we provide $Mk («/; F) with the projective limit topology of the
topologies of the spaces @Mk (X; F) with respect to the set of restrictions r# of
<p to Jf.
4.3 The Spaces ®±,Mh(f\F)
We still let J> = ]t0, ^[ be an open, finite or infinite, interval in R1.
Definition 4.4. ^+tMk (^)F) denotes the space of infinitely differ
entiable functions t-> (p(t), defined on J>, with values in F, and with support
bounded on the left in J (that is, zero in a neighborhood of t0, depending on (p)
and such that
(4.5)
for each b with t0< b < tv there exists a
number L > 0 and a bounded set & in F (both
depending on (p and b) such that
<pW(t)
LkMk
ea, vte]t0,&], a = o, 1,2
The topology of @+Mk {<$'> F) is defined in the following way.

14 4. Vector-Valued Functions of Class Mk
For each closed sub-interval [a, b] of / (t0 < a < b < ^) and for
each fixed L > 0, we consider the space
(4.6)
@a,Mk ^-a b] L\F) = (closed) subspace of Q)Mk ([a, b], L; F)
made up of the functions (p such that q>^k\a) = 0,
* = 01, 2, ...
Next we define the space
(4.7) 9aMk ([a b];F) = ind lim 9aMk ([a, b], L;F),
L->+oo
where L increases monotonically to + oo, and the space
(4.8) @a>Mk ([«, ^[; F) = project lim 9aMk ([a, b];F),
where b increases monotonically to t± and where the projective limit is
taken with respect to the mappings rb>p, the restrictions of the q>'s
defined in [a, b"] to the interval [a, b'] for b' < b".
Finally, if we identify the functions q> of 3fatMk {la> hl>F) w^n tneir
extensions by zero in ]t0> a[, we can provide @+tMk(Jr; F) with the
inductive limit topology of the topologies of the spaces 3fa,Mk ([a> ^i [* F)
as a decreases monotonically to t0] that is
(4.9) @+tMk (J; F) = indlim @a>Mk {[a, ^[; F).
a->t0
In short, we have defined @+tMk (<?'> F) by
(4.10)
S>+tMk (S: F) = in(i Him (project lim (ind Km 9 M {[a, 6], L; F))\. Q
a->t0 b->tx L->+oo
We define @-tMk [*f\F) in a completely analogous way (we
interchange the roles of t0 and ^; the functions of @-tMk i^'> F) nave their
supports bounded on the right in«/, that is they vanish in a neighborhood
of y. D
Remark 4.1. The spaces ®Mh[f\F)t SMk{J)F) and @±:Mk{J\F)
are stable with respect to differentiation. By using the known properties
of the space of continuous functions on J with values in F (see Bourbaki
[3], Chapter III, § 1, No. 1) we also see that the spaces under
consideration are separated. Q
Remark 4.2. In connection with the spaces @±tMk {^''> F) and
$Mk (</, F), we note that in Lions-Magenes [3], [4] we have defined the

4.4 Remarks on the Topological Properties
15
topologies in a slightly different manner. The definitions given here are
easier to use, and in any case the topologies are equivalent1.
Remark 4.3. We shall also use the space 0+(«/; F) of infinitely
differentiable functions on «/, with values in F, and with support bounded
on the left in J, provided with the topology of L. Schwartz [3]; in the
same notation as for the definition of ^+>Mfc {<#\F)t this means that
@+ (J\F) = indlim (project \im^a{[ayb\yF))y t0<a<b<tv
where @a([a, &]; F) is the space of infinitely differentiable functions cp
on [a, b], such that q>W(a) = 0, k = 0, 1, 2, ..., provided with the
topology of uniform convergence on [a, b] for q> and each of its derivatives.
Similarly, we shall consider the space Q}_{f\F) of functions with
support bounded on the right in J. Q
4.4 Remarks on the Topological Properties
of the Spaces @Mk(J?; F), &Mh{S\ F), 9±,mh(S; F)
Several questions may come up in connection with the spaces
@Mk(/'>F)> $Mk{<?'>F) and @±iMk(J\ F): what are the topological
properties of these spaces relative to the properties of the space F7
For example, are these spaces complete if F is complete? Fuithermore:
what is the relation between these spaces and the spaces defined by using
topological tensor products (for example @Mk {J) ® F, provided with
one of the topologies of Grothendieck [1]) ?
These are questions which we shall avoid in so far as it is possible.
As we go along, we shall give the topological properties required for the
applications, for the concrete spaces which will be used.
Here we just call attention to a problem which will come up in
Chapters 10 and 11: let F = indlim Fn, where the Fn's are separated,
w->oo
topological vector spaces; is the space @Mk («/; F), ... the inductive
limit of the spaces Q)Mh («/; Fn), ...?
1 Let us verify this for ^+,Mfc {J\ F), for example. In Lions-Magenes [3], [4],
J> = R and^+,Mfc (R; F) is denoted by ^+,Mfc {F). From the definition given in
Lions-Magenes [3], [4] we immediately deduce the fact that the topology of @+,Mk(F)
is finer than the topology of ^+,Mfc (R; F). We must therefore show that it is also
coarser, i.e. that the identity is continuous from ^+tMfc (R; F) into ^+;Mfc {F), and
for this it is sufficient that the identity be continuous from @a>Mk (R, F) (space
obtained by extending the functions of @a,Mk ([a, + oo[; F) by zero for t < a)
into #(F) (space of infinitely differentiable functions on R with values in F,
according to the notation of Lions-Magenes [3], [4]), which is obvious from the definitions.

16 5. Vector-Valued Ultra-Distributions of Class Mk; Generalizations
In Chapters 10 and 11, we shall make use of the following two results
due to G. Geymonat [1], [2]:
(4.11)
if F and Fn are (££#")-spaces in the sense
of Grothendieck [2], and if they are complete, then
Km, ([<*> 6] i F) = indlim ® Mjt {[a, 6], L; Fn)
(see Geymonat [2], Corollary to Theorem 4.2).
(if F and Fn are complete nuclear spaces, then
9a>m ([a, b], L; F) = indlim 2a>m {[a, b], L; Fn)
w->+oo
(see Geymonat [2], Corollary to Theorem 4.1).
We note that (4.11) applies with F = ^{F) and F = @Mk (Q), and
(4.12) with F = @(Q), because of the properties of ffl[F) (see Section 3)
and of !£}(Q) (see Grothendieck [1], Schwartz [1]). D
Remark 4.4. From (4.11) and (4.12) we can also deduce conditions
to obtain Suslin-type spaces (see Geymonat [2]); for example @atMlc (la> b] >
<#?(r)) is a Suslin-type space. D
5. Vector-Valued Ultra-Distributions of Class Mk; Generalizations
5.1 Recapitulation on Vector-Valued Distributions
From now on we shall assume that
(5.1) F is a reflexive, separated, locally convex vector space.
We still let J be an open interval, finite or infinite, of R1. We denote
by &{Jy F) (resp. @+(J; F), resp. 3>L{f\ F)) the strong dual oi@(J] F')
(resp. Q}J^J>'; F')y resp. ^+{J\ F')) (where F' = strong dual of F) and
we call it space of distributions on J> with values in F. D
Remark 5.1. These definitions are different and more restrictive than
the usual definitions of L. Schwartz [3]: according to L. Schwartz, the
space of distributions on J with values in F is the space ££{&(<$); F)
of continuous linear mappings of @)(J) into F.
Similarly, one introduces the spaces S£{Q)-{J); F) and ££{$)+{<$); F)
which correspond to @'+{<f',F) and $)'_{J\ F) respectively.
However, the above definition will be more convenient for the
applications we have in mind. In any case, we have
(5.2)
&{ff;F)C&[®{S);F)

5.1 Recapitulation on Vector-Valued Distributions
17
(resp. ®'+{f\ F) C &{®-{J); F), resp. ®'_(S\ F) C &(®+{S); F)) and
under certain conditions on F (see Schwartz [3], Lions-Magenes [2],
§ 4.2) the first and second terms of (5.2) coincide, for example if F is
a reflexive Banach space. Q
Differentiation in <&'{J\ F) and multiplication by a (scalar) function
ip of ${J) are defined in the usual manner by the formulas:
(5.3) /^fV^±-/u,^\,ue0'{S;F),(pe®{S;F'),
(5.4) <y«, y} = <u, yip}, u G @\J\ F)} ip G g(f), y G @{f\ F'),
where the brackets denote the duality between Q}'{J\ F) and^(«/; F'). 0
The following property holds:
(5.5)
can be identified with
the subspace of distributions u^Q)'(J\F) with
support bounded on the left [resp. on the right).
Indeed the space Q}{J\ F') is dense in 0_(«/; F') and consequently
S}'+{<f)F) may be identified with a subspace of Q}'{f\F). Next, if
u G ^+(«/; F), then u vanishes for t < tut for a suitable tu\ for otherwise,
there would exist a sequence of intervals [an, bn] contained in/ = ]tQ) tx[
{J> finite or infinite), with bn -> t0, and a sequence of functions yn G
Q)(J\F') with
(5.6) support of yn C K, bn]
such that <u, yn} 4= 0. Then, by replacing yn with Kn(pn, Kn£C,we
could always assume that
(5.7) <«,pn> = l.
But (5.6) implies 99^-^ 0 in 0_(«/; i7'), which contradicts (5.7).
Finally, let u^&(J\F) and have its support bounded on the left;
let 0 G S{J\ C), with 0(t) = 1 in a neighborhood of the support of u and
0(£) = 0 in a neighborhood of t0. Then 0w = u and, if 99 G i^(«/; i7')* we
have
(5.8) O, <p> = <>, 0<p>.
If <p-+ 0 (in 0(,/; i7')) for the topology of ®_{f\ F')9 then 0y-+ 0 in
Sf(J\ F') and therefore <w, dy)-^ 0; therefore, thanks to (5.8), u defines
a continuous linear form on^(/; i7') provided with the topology
induced by @J-(J>\ F'), so that u£ @+{J', F), which completes the proof of
(5.5). Q

18 5. Vector-Valued Ultra-Distributions of Class Mk; Generalizations
Remark 5.2. The analogue to (5.5) for 5£{3)_{J); F) may be incorrect;
for example if J = R, F = ^'(R), the mapping u £ JS?(0_(R); ^'(R))
defined by
(5.9) u = d(x — t) (i.e. u(<p) (x) =<p(x), x £ R),
does not have its support bounded on the left in t, since the support of u
in R2 = R^xR^ is the line x = t. This example also shows that in (5.2)
the inclusion may be strict, since u given by (5.9), cannot, according to
(5.5), also belong to @'+(R; ^'(R)), as its support is not bounded on the
left.
5.2 The Space @Mk(J; F)
Let J and F be defined as in Section 5.1 and let {Mk} be a sequence of
positive numbers satisfying (1.1), (1.2), (1.3) and (1.4).
Definition 5.1 Q)'Mu{J\F) denotes the dual space of <2)Mlc{J',F')
provided with the strong dual topology.
The elements of &Mlc {*?', F) are called ultra-distributions of class Mk
on J', with values in F. 0
Remark 5.3. The analogue to Remark 5.1 holds: by analogy with the
definition of vector-valued distributions of Schwartz [3], we could call
$£(<2)Mh {J); F) the space of vector-valued ultra-distributions of class Mk,
but definition 5.1, although more restrictive, is more convenient for the
sequel.
In fact, we have
(5.io) &*('••*) C#{®uAS);f)
and under certain hypotheses on F the equality holds in (5.10) (for
example if F is a Frechet space; G. Geymonat [2], Prop. 5.1). Q
<2)'{J', F) (defined as in 5.1) can be identified to a subspace of
@Mk (<? '> F) by using the fact that
(5.11) @Mjt (Jf; F') is dense in Q){J; F').
Indeed, let y £ 3){J\ F'); let qn be a regularizing sequence of scalar
functions, of class Mk (ew£^Mfc(R)> Qn> ®> f Qn{t) & = 1, Qn has
— oo
support in [ocn,f}n], ocn, /?»-> 0),; such a sequence exists (see Section 1.2,
property a)). But, according to (5.1), F' is quasi-complete, see for example
Bourbaki [2], page 8.8.
Then we can regularize q> by Qn: y * qn = <pn is defined by
y»W = / Qjf - a) <p(a) d<*

5.2 The Space &Mk (J\ F)
19
(which is well-defined for sufficiently large n); we have: cpn £ @Mk {<?'> F')>
the q>n's have support in a fixed compact set;
q>n-+q> in i7', uniformly in t, as well as each derivative; whence (5.11).
Thus we have
(5.12) ®'(J')F)C@'mM>F). D
Differentiation in @'Mk («/; F) and multiplication by a scalar
function y) of $Mk (J) are defined, as for ordinal distributions, by formulas
analogous to (5.3) and (5.4). D
The structure of ultra-distributions of Q}'M k («/; F) can be specified,
at least under the additional hypothesis:
(5.13) 3P{f\ F') is barrelled,
where £#°(J;F') is the space of continuous functions with compact
support in«/, with values in F', provided with the topology of L. Schwartz
[2] (that is, the inductive limit topology of the spaces 0°(«/, JT; Fr), as JT
varies over the set of compact intervals contained mJ>, where 0°(«/; Jf, F')
is the subspace of Q}\J; i7') made up of the functions with support
contained in X and provided with the topology of uniform convergence
on jf).
Then we have (see Lions-Magenes [3], Theorem 7.1, Chapter 1):
Theorem 5.1. Under hypotheses (5.1) and (5.13) every u £ 3f'Mk (J; F)
may be represented, non-uniquely, by the form
oo dk
(5.H) ^2-sft,
where
/^£ [Q)\J ;F'))' (measures on/ with values in F)
{for every continuous (scalar) function B with
compact support in «/, and every L > 0, the
1 °°
series ^ LkMkB[jik converges in[Q}\J\ F'))'
[ k=o
and where (5.14) means that
oo
(5.17) (u, <p> = £ (-1)* <**, 9>W>, y<P € 9Uh (J; F')
(the brackets in the summation denoting the duality between 0°(«/, F')
and {Q}\J',F')y and <w, 9?) denoting the duality between &Mk(J\F)
and ^Mfc («/; F')). Conversely, every u in the form (5.14), (5.17) defines an
element of @'Mk(J; F). Q
(5.15)
and
(5.16)

20 5. Vector-Valued Ultra-Distributions of Class Mk; Generalizations
Remark 5.4. Hypothesis (5.13) is satisfied if F' is a Frechet space
(see Bourbaki [3], Chapter III, § 1, no. 1) and therefore, according to
(5.1), if F is the dual of a Frechet space (this is the case for F = @Mjc (Q)
and F = J^(r)). Other cases have been pointed out to us by G. Gey-
monat: (5.13) is satisfied if F' is a barrelled, nuclear (^J^)-space or if F'
is a strict inductive limit of a sequence of nuclear Frechet spaces (thus
(5.13) holds if F = &Mk (Q) orF = ^'(T) or F = &(Q)). D
5.3 The Space 2f±tMh(f\ F)
The hypotheses on «/, F and {Mk} are the same as in Section 5.2.
Definition 5.2. $)\iMh(<$\F) (resp. @j'_fMk(J>\F)) denotes the strong
dual of @_tMk (J; F') '(resp. ®+tMh (J; F')). D
Remark 5.5. The analogue to Remark 5.3 holds. D
In the same way as for (5.5), it can be shown (see Lions-Magenes [3],
Theorem 9.1, Chapter I) that
(5.18)
rV,Mfc (•?'> F)(resp. @'_fMk (<$', F)) can be identified
the subspace of $)'Mk (J\ F) of
ultra-distributions of class Mk with support
bounded on the left (resp. on the right).
From which we deduce (see Lions-Magenes [3], Theorem 9.2,
Chapter I) the following theorem on the structure of the elements of ^'+,Mfc (<$\F)
(and analogously of @-tMk (<$'> F)):
Theorem 5.2. Under hypotheses (5.1) and (5.13), if u is given in
@'+tMk {<$'> F) with t* = the left endpoint of the support of u, then, for all
*i < t%, there exists a (non-unique) decomposition of u in the form
(5.19) u=Z^i*kt
(5.20) /**€(®V; *"'))'.
(5.21) ph = 0 for t < 1,
(5.22)
for every continuous (scalar) function %, with
compact support in J", and every L > 0, the
oo
series ^ LkMk%fjik converges in (@)0(J\ F'))',
k=0
and (5.19) is taken in a sense analogous to (5.17).

5.4 Vector-Valued Ultra-Distributions of Beurling-Type 21
Conversely, if u is given in <2j'M]c («/; F) and if for every t < t* there
exists a decomposition of u in the form (5.19), with (5.20), (5.21) and
(5.22), then u belongs to &+tMk {<#'> F) and vanishes for t < t*. D
Remark 5.6. UMk = (kl)s, s > 1, we shall write &S{J\ F), @'S{S\ F), ...
instead of @Mk {J\F), 9'Mh («/; F), ... and speak of Gevrey
ultra-distributions of order s, with values in F.
5.4 Vector-Valued Ultra-Distributions of Beurling-Type
The generalization of scalar ultra-distributions given in Section 2.3
can be extended to the vector case.
We still have J = ]t0, ^[, a finite or infinite, open interval in R1,
F a space satisfying (5.1), {Mk} a sequence of numbers satisfying (1.1),
(1.2), (1.3) and (1.4). As fundamental space, we take &Mlc (J>;F), defined,
in analogy to (2.9), by
(5.23) ®Mu (J; F) = ind lim (project lim @Mk (J, jf, L\F)),
where Jf (compact interval contained in J) increases monotonically
to J and L decreases monotonically to zero (compare with (4.3)).
Then the space of Beurling-type ultra-distributions on J', with values
in F, relative to the sequence Mk is by definition
®'Mk (S'>F)= strong dual of <%Mjc (J\ F') Q
Analogously, we define the space
(5.24)
^+,Mfc {S\F)= ind lim (project lim (project lim 9 Mh ([a, b],L; F))),
a^*t0 b^-tx L-s>»0
where t0 < a < b < tx and L > 0 (compare with (4.10)), and then the
space
(5.25) @'_tMk (J; F) = strong dual of <%+>Mjc (S\ F').
In the same way, we introduce the spaces ^_>Mk(J';F) and
a+tMM{S',F). D
We restrict our discussion to pointing out that
(5.26) @Mk (J; F) is dense in @Mk (J ;F)
(apply the remarks of Sections 2.3 and 5.2, see (5.11)), from which, after
suitable identifications, it follows that
(5.27) &V; F) C ®'Mk {/: F) C @'Mk V\ F) • D

22
6. Comments
Remark 5.7. UMk = (kl)s, s > 1, we shall write ®s{f\ F), @'s(jf; F),...
instead of ®Mlc (J\F)y &'m (JyF)y...
5.5 The Particular Case: F = Banach Space
Let us only consider the case of SiMlc {^\ F) and ^Mk {<&> F)
(analogous considerations are also valid for the other spaces).
Thus let J = ]t0, t±[be an interval in R1, {Mk} a sequence of numbers
satisfying (1.1), (1.2), (1.3) and (1.4), and F a Banach space. Then,
definition 4.1 of the space £&Mk (J"; F) is equivalent to: ^Mk (J'; F) is
the space of functions cp £ &(<#', F) such that there exist two positive numbers
c and L (both depending on y) such that
(5.28) sup \\cpW(t) \\F < cLkMk> k = 0, 1, 2,...
its
The space <3tMk (</, X', L\ F), which intervenes in (4.3) for the
definition of the topology of 3fMk (J>',F)y is now a Banach space with norm
(5.29) sup
LkMh
Therefore @Mk (J'; F) is an inductive limit of Banach spaces. 0
Another equivalent definition of the space 3fMlc {*?'> F) can be given
by using the space LP(J>\ F) (of classes) of pth-power (1 < p < + oo)
integrable functions on «/, with values in F, provided with the norm
/ tx \i/p
(5.30) \\v\y{w = y\\m&&) •
Condition (5.28) then becomes
\\<Pm\W;F)<0LkMk, V*.
Using this presentation of the space SfMlc (<? '> F) we obtain (see Lions-
Magenes [3], Theorem 2.1, Chapter I) a new theorem on the structure of
ultra-distributions of &Mk(^>F)y which is analogous to Theorem 5.1,
but where the measures /Lik are replaced by locally ^>'th-power integrable
(W-1)
functions gk, with values in F.
6. Comments
The notion of distribution on an open set Q in Rw introduced by
L. Schwartz [1] has been generalized in several ways. The main idea of
these generalizations has been to choose a suitable "fundamental space"
of functions, different from the space @(Q) of Schwartz, and to consider

6. Comments
23
the continuous linear forms on this space. In this manner, one has
obtained ''objects" which have been called generalized functions or
distributions, ultra-distributions, ...: see Gelfand-Shilov [1] and the
bibliography of this work.
It was Gevrey [1] who introduced the function spaces 0 Mk (Q) for
the case Mk = (kl)s, with s > 1 (for s = 2, functions of this type had
already been considered for the study of solutions of the heat equation
by Holmgren [1] and E. E. Levi [1]). The generalization to sequences
{Mk} satisfying (1.1), (1.2), (1.3), (1.4) takes its inspiration from the
theory of non-quasi-analytic functions and from the Denjoy-Carleman
theorem (see the book of Mandelbrojt [1] and its bibliography); under
hypothesis (1.1), condition (1.2) is necessary and sufficient for the space
<2>M]c (Q) not to be reduced to {0}. For classes of non-quasi-analytic
functions, we also call attention to the work of Friberg [1], Friedman [1],
Rudin [1], Boman [1], Talenti [1], Carleson [1], Dzanasija [1], [2],
Mityagin [2], Leray-Waelbroeck [1] ... For interpolation between
Gevrey spaces, see Goulaouic [1], [2].
For the theory of @Mk (Q) -spaces, we have followed and used the
work of Roumieu [1], [2], to which we refer the reader for developments
of the theory; see also Gelfand-Shilov [1] (see the 5-type spaces in
Volume 2 and in particular the space S0, which coincides with @Mk (R)
for Mk = kpk, /? > 1, and the Appendix to Volume 2).
The spaces of ultra-distributions @l'Mk (Q) were introduced by Beur-
ling in [1] (see Bjorck [1], Larsson [1] for the development of the theory
and applications), where the definition is given in a different, way
(instead of the estimates of type (2.8) on the derivatives, a condition on
the Fourier transform of q> is used). For the spaces &Mk (Q) with
Mk = (kl)s and their applications to the Cauchy problem, see Horman-
der [1].
Analytic functional were introduced by Fantappie (see [1], [2] and
also Pellegrino [1]). The bibliography on analytic functional and their
applications is very extensive: in addition to the studies noted in the
text and above, we indicate in particular the work of Leray [1], [2],
Silva [1], [6], daSilvaDias [1], Malgrange [3], Martineau [3], Mantovani-
Spagnolo [1], Kothe [1], Tillman [3] ...
Analytic functional and distributions are also closely connected with
the problem of boundary values of holomorphic functions: see Kothe [2],
Tillman [1], [2], Zerner [1], Martineau [2], Ehrenpreis [1], Beltrami-
Wohlers [1]. In fact, it is through his development of the studies on this
problem that M. Sato [1] arrived at his theory of hyper functions, which
generalize both distributions and analytic functional (roughly speaking,
according to the presentation of Martineau [1], a hyperfunction on Rn
is a locally finite series of analytic functional with compact support,

24
6. Comments
which "stick" together). On the theory of hyperfunctions, see also
Bengel [1], [2], [3], Harvey [1], Harvey-Komatsu [1], Komatsu [2],
Martineau [1], [2], Boutet de Monvel-Kree [1], Schapira [2], [3], [5],
Kant or [1]. Along similar lines, we draw attention to the theory of
tempered ultra-distributions of Silva [3], [4] (see also Hasumi [1], Yoshi-
naga [2]) of which an axiomatic generalization has recently been
announced by Sousa Menderes [1] (see also Silva [4]).
Finally, let us note the ultra-distributions introduced by Treves [2]
(see also Steinberg-Treves [1]) in the study of the Cauchy problem and
the axiomatic formalization of ultra-distributions given by Schapira [1].
The theory of vector-valued distributions was founded by L. Schwartz
in [2], [3]; an axiomatic theory appears in Silva [5]. We also note the
work of Yoshinaga [3], [4]. For the spaces of infinitely differentiable,
vector-valued functions, also see de Wilde [1], Garnir-de Wilde-Schmets
For the spaces of vector-valued functions of class Mk (Section 4), see
Geymonat [1], [2], Lions-Magenes [3], [4], [5].
Finally, the vector-valued ultra-distributions of class Mk and of
Beurling-type were introduced by Lions-Magenes [3], [4], [5].

Chapter 8
Elliptic Boundary Value Problems in Spaces of
Distributions and Ultra-Distributions
This chapter requires only the knowledge of the scalar-valued
distributions and ultra-distributions of Chapter 7. From the point of view of
boundary value problems, we assume the essential parts of Chapter 2
(elliptic problems) to be known.
1. Regularity of Solutions of Elliptic Boundary Value Problems
in Spaces of Analytic Functions and of Class Mk;
Statement of the Problems and Results
1.1 Recapitulation on Elliptic Boundary Value Problems
In Chapter 2 we have studied boundary value problems for linear
elliptic equations in the Sobolev spaces Hs (Q); as an immediate conse-
oo
quence of the fact that fl HS{Q) = Q)\Q) (see Corollary 9.2, Chapter 1),
5-0 _
the study of the same problems, in the space <2)\Q) of infinitely
differentiate functions on Q, follows. More precisely, from Theorem 5.2,
Chapter 2 and the first Remark of Section 8.2, Chapter 2, we deduce the
following result:
Theorem 1.1. Let:
(I) Q be a bounded open set in RM with boundary Fyan (n — 1)
-dimensional, infinitely differentiable variety, Q being locally on one side
ofT;
(II) A be a differential operator given by
(1.1) Au= 2 (-l)lPlVp(apq(x)Wu),
\p\,\q\<m
where apq£<2)\Q), A being properly elliptic in Q [in the sense of
Definition 1.2 of Chapter 2);

26 1. Regularity of Solutions of Elliptic Boundary Value Problems
(III) [B^f~Q be a system of boundary operators given by
(1.2) BjU= 2 y«)D*«,/ = 0,l,...,«-l,
\h\<mj
where b.jh £ 3f{r)t 0 < m3- < 2m — 1, the system {B$~q covering A
on r {in the sense of Definition 1.5 of Chapter 2).
Then the boundary value problem
IAu = f in Q,
B-u = gj on r, i = 0, ..., m — 1,
with f given in Q}\Ci) and the g-s given in @(r)y admits a solution u
belonging to 3f\Q) and determined up to addition of a function w of the space
N defined by: N = {w \w£ <2j[Q)< B3w = 0, / = 0, ..., m — 1, Aw = 0},
if and only if
(1.4) Jfvdx+ 2 /g^d<r=0
Q j=0 r
for every element 0 = {v; (p0, ..., q>m-i} of the space = [0 \v£ 3f\Q)t
q>j€®{r), j = 0,...,m — l, 0**0 = 0}, 0* being defined by (5.2),
Chapter 2.
Now we ask whether, if the "data" /, g and r are analytic or more
generally of class Mk, (the solution or) the solutions of problem (1.3),
furnished by Theorem 1.1, are also analytic or of class Mk in Q.
This is a new regularity question, which, as we shall see, can be
answered affirmatively.
1.2 Statement of the Mk -Regularity Results
We shall work with spaces of functions of class {Mk}. For the
applications we have in mind, we always have
(i.5) Mk = (k\y,p>i
(/? = 1 is the "analytic case", /? > 1 is the "case Gevrey of order /?").
However we do not necessarily introduce the sequence {M^ in the form
(1.5), in order to show the generality of the question. We assume {Mk}
to have the following properties (see also Remark 2.6):
(1.6) Mk>0 Vk>0,
(1-7) Mt<Mk_,Mk+1 VA>1,

(1.8)
(1.9)
(1.10)
(1.11)
1.2 Statement of the ikf^-Regularity Results
there exists a constant cx such that
f \ Mk_tMt < c±Mk, 0 <t<k, Vk> 0,
Mh<Mh+1 Vk>0,
I there exists a constant d such that
Mt+s<dt+sMtMs V*,s>0,
[ there exists a positive dm (depending on m) such that
I \M-2ms) \^2ms+2m
»' < ^Z''(M2ms+fm Vs >0, 0 < * < 2m.
Let us verify that sequence (1.5) has these properties.
It can immediately be seen that (1.6), (1.7), (1.8) and (1.9) hold;
condition (1.10) is satisfied with d=$y since (t + s)! < 2t+st\ s\;
finally (1.11) is satisfied with dm = (2m)2mp, since
(1.12)
'((2ms + 2m) \)1
{(2ms + t)!/ J
=. [(2ms + t + 1) ••• (2ms+ 2m)ft <
< (2sm + 2mf{2m-i) <
< 2m2m^m-i\2ms + lf(2m~V <
< 2m2l^{2m-t) [(2ms + 1) • • • (2ms + t)]^2m-t] <
\((2ms +t)^'\2m-i
< 2m2m^2m~t)
{(2ms)!)"
which is equivalent to (1.11). Q
Note that hypotheses (1.2), (1.3) of Chapter 7 on the sequence Mk
will only be imposed from Section 4 of this chapter on.
Also note that from (1.7) we deduce
(1.13)
whence
(1.14)
¥l±l < M*±L for t < k and Vs > 0,
Mt+S Mh+i
< ——- for t < k, s < i.
MtMs - MhMt
Furthermore from (1.8) we deduce
(1.15)
kMh_x <^rMh Vk > 1

28 1. Regularity of Solutions of Elliptic Boundary Value Problems
and therefore also
(1.16) Mh_t < (Aj {A^31 Mkiovt<k. Q
For jf a compact set in Rw and {Mk} a sequence of numbers
satisfying (1.6), ..., (1.11), we shall use the notation <2)Mk pf) (see
Definition 1.2 of Chapter 7).
If Mk = k\, then we recover the space 3^(C^) of analytic functions
on Jf (see Section 3.1, Chapter 7).
As we have already recalled in Section 1.2, Chapter 7, the
(multiplicative) product of two functions of @Mlc (jf) still belongs to @Mjc (jf),
thanks to (1.8), and the composition / ° g of two functions still belongs
to $)Mh pf), again thanks to (1.8). D
A bounded open set Q in Rw will be called of class {Mk} if its boundary r
is an (n — 1)-dimensional, infinitely differentiable variety, whose "local
maps" are given by functions of class {Mk}, Q being locally on one side
ofr.
In the next Section we shall prove the following theorem.
Theorem 1.2. Let Mk = (k\)p, with real /? > 1 (or more generally
{Mk} satisfying (1.6),..., (1.11)).
Assume the hypotheses of Theorem 1.1 to be satisfied and that
apq £ @Mk {&)y bjh £ @Mk {r) an^ finally that Q is of class {Mk}.
Then, if u £ @(Q) and if there exist two constants c0 and L0 (depending
on u) such that
(1.17) II^ILW<co4^2» Vi>0,
m—1
(1.18) 2 ||fiy(^«)||«2«+2*»-^-l/2(^<^^<+1M(A+,+1)2m V*,A>0,
j = 0
(where the differential operator A1 is the i-th,iter ate of A and A°u = u),
the function u belongs to <3tMk (Q), and more precisely u £ <3tMk (Q; %(£0))((1))>
where %(£0) ^s a function of L0 independent of u and c0, and %(L0) > L0. D
This theorem, which we shall call the theorem on {<elliptic iterates",
and which is useful for several points, as we shall see in the following
chapters (see also the Comments), contains as a corollary the answer
to the question posed at the end of Section 1.1; indeed the following
corollary can be deduced from it.
«3') For the definition of 3>uh {Off \L) see Chapter 7, Section 1.3.

1.3 Reduction of the Problem to the Case of the Half-Ball 29
Corollary 1.1. Let Mk= (ktf, real f} > 1 (or more generally {Mk}
satisfying (1.6), ..., (1.11)); under the hypotheses of Theorem 1.1, if
furthermore Q is of class Mk and apq G @Mk (Q), bjh G @Mk (r), then every
solution u of problem (1.3) belongs to @Mk (Q) if f G @Mk {&) an<^ gj£@Mk (-0 I
*«<* more precisely u G 0Mfc (A; %(Lo)) */ / € ^Mfc (A; £0)> & € ^Mfc (-T; L0),
z^0f0 %(L0) is a function of L0, with %(L0) > L0.
1.3 Reduction of the Problem to the Case of the Half-Ball
Via "local maps" and by using the property, already noted, that the
composition / ° g of Gevrey type functions (of class Mk with (1.6), ...,
(1.11)) is still a function of the same type, the proof of Theorem 1.2 can
be reduced to proving Theorems 1.3 and 2.4 below.
We denote by x = (x',y) = (xlf ..., xn_lt y) the point in Rn and by
QQ (g > 0) the half-baU {(*', y) \ x\ + • • • + x\_x + y2 \ < q\ y > 0} and
furthermore by TQ the subset of its boundary such that y = 0; ~D"u,
T)*u, DyU denote respectively an arbitrary derivative with respect to all
the variables, the variables xlt ..., xn_lt the variable y. We have
Theorem 1.3. Let q0 be fixed with 0 < q0 < 1 and Mk — (k\)p, with
real f} > 1 (or more generally {Mk} satisfying (1.6), ..., (1.11)); let
(1.19) stu = 2 ap{x)T>pu
\p\<2m
be a properly elliptic operator in QQo with ap G @Mk (&Qo) '> ^
(1.20) ^u= 2 bjh(x')Dhu,j = 0)...)m~l,
\h\<.mj
be m boundary operators with bjh£ <3}Mk(ret), 0 < m^ < 2m — 1, the
system {<%$Jq covering s$ on ret. If u£ @(&Q) cind if there exist two
constants c0 and L0 (depending on u) such that
(1.21) || s?u\\L1{Qeo) < c0LlM2mi Vi > 0,
m — 1
(1.22) 2 2 II D^WM) hfim-ftj-WF,) <
< c0Ll+^M(h+i+1)2m Wi, k > 0,
then there exists q < q0 (depending on L0, sf, <%•) such that u G @uk {QQ) '>
and if c0 remains bounded, u belongs to a bounded set of <3Mk (QQ>).
Remark 1.1. Noting the properties of functions of @Mk (QQo) and the
ellipticity of sf, we may assume that the coefficient of the term DyWu
is 1 in (1.19), that is
(1-23) a(0,0,...,0,2m) = !•

30 2. The Theorem on "Elliptic Iterates": Proof
We may also assume, without loss in generality, that the coefficients
bjk ar the restrictions to reo of functions, still denoted by bjh, defined on
Q6o and belonging to <3tMk (QQo); and we may also assume that
(1.24) sup 2 |D*M*) I ^ cLkMk V^ > 0,
(1.25) sup 2 \V«bjh(x)\<cLk-*™Mk_Zm V£>0,
*ZnQQ\<x\=k
where c and L are two suitable positive numbers with L > 1 and where
we set M_] = M_2 = • • • = M_2m = M0.
2. The Theorem on "Elliptic Iterates": Proof
2.1 Some Lemmas
We introduce the following notation: for 0 < q < g0, & <m<Z s =
0,1,2,...,
II«IU= 2 2 ||D*(D»||L2
m —1
lll?*»«llkC=Z 2 \\<P^PA^)\\H^- ™i-l,2{r),
j=0 \P\=k e
where (p is a fixed function belonging to <3}(reo).
Let us also agree to set
\\Hs,-s>Q = \\Hw,Q{=\\Hu(nJ)>
so that
(2-1) HL-m-k,<II«IU.
for s, £, & such that the two terms of (2.1) are defined.
The following lemma adds some precision to Theorem 16.3,
Chapter 1:
Lemma 2.1. Let integer s > 0 be fixed and t be an integer with 0<£<s
and 0 < q < q0; there exists a constant cs (depending only on s and q0) such
that for every e > 0 and every u£Hs (QQ) we have
(2.2) II^IU<e|l«IUe + ^"V(S"')ll«llo,o,e-
Proof. 1) First we note that it is sufficient to show (2.2) for q = q0,
since by homothetic mappings it can easily be deduced for 0 < q < q0
as well.
2) Now let us prove (2.2) for q = q0. Applying the extension methods
introduced in Chapter 1 (see in particular Section 8.1) we are led to the

2.1 Some Lemmas 31
case of the entire space RM; u£ Hs(Q6c) is extended to a function v£ #S(R+)
(where R+ is the half-space of x's with %n > 0); next we extend the v's
to functions w G Hs(Rn). In the course of these extension operations, the
semi-norms of type
(2.3) / / |D%|2 d*V'2, / / |D*v |2 d*\1/2, / / |D% |2 d*V'2, |? | < s
remain equivalent, the constants of equivalence depending only on s and
q0 (it suffices to look at the construction of the extensions as given in
Chapter 1, in particular formula (8.4)).
But then if w G Hs(Rn) we have, by Fourier transformation (w
denoting the Fourier transform of w and f = (fx,..., fw) the dual variable of
(2.4) / |Tflw{x) |2 dx = J |f2*^(f) |2 df
Rw Rn
and the result then follows from the inequality: for every e> Owe have
(2.5) |f \< < e |f |s + yse-t/{t-s) Vf G Rw and 25 < s, with constant ys,
which is a consequence of the inequality
If I' = e6 |f f a-6 <— |f \ip +4*-^, # = s, e# = 1,- + ^ = 1. D
P P P P
(2.6)
By an application of Lemma 2.1 we obtain
Lemma 2.2. Z^^ £ fo an integer with 0 < £ < 2m #w^ Zstf @ < £0; 2/^0
£:mfo a constant cm (depending only on m and q0) such that for every s > 0
am£ every function u G @(&Q) w& have
(2-7) ||«|U, < s MLa,e + V-^-'> ||M||0,M V£ > 0,
(2-8) ||«||w,e < e \\u\\h>2me + cme-i«2»-» ||«||Mi<f V£ > 0,
(2.9) ||D>||Mje < 8 ||Df «|lo,*,e + cm£-«2-" ||«||Wje V£ > 0. Q
Remark 2.1. Let s be an integer > 0; applying (2.7) to Dpu with
\p | = 2sm and summing up, we obtain
(2-10) \\H2sm+t)k>Q < e \\u\\2{s+1)m>k)Q + cms-^~* \\u\\2sm>kQ
Vs, &> 0, 0 <*< 2m.
Analogous inequalities can be deduced from (2.8) and (2.9). D

32 2. The Theorem on "Elliptic Iterates": Proof
We shall also use
Lemma 2.3. For every e > 0 there exists cm(e)y depending on e {and
on m and q0) such that for every u £ @(Q6o) and 0 < q < Q0we have
2m
(2.11) 2 ll^*"-'«llw.e < « l|D?"«lloo.e + '*(«) ll«W
Proof. Following the same methods as for the proof of Lemma 2.1
(parts 1 and 2), we are led to the case of the entire space RM and then,
by Fourier transformation (this time denoting the dual variable of
(xlf..., xn_ly y) by (glt ..., £n_lt rj)), it all comes down to showing that,
for s > 0 and 0 < t < 2m, we have
V*m-2t\£\2t <*\v\im + yn>{e)\£\im> V^R1 andf = (fr..fn_1)GR»-1,
with ym(e) depending on m and e, which is well-known.
Remark 2.2. As for Remark 2.1, it follows from (2.11) that
2m
(2.12) 2 IIDf^-'HU, < « l|Dr+*"*«llo,*.e + '*(«) IID^IU+m
vs, &>o. Q
We also note
Lemma 2.4. L^ <p £ 0(£? ) fo fixed, then for every integer k > 0:
(2.13) ||^||^(o ^^(maxmaxlDVWDll^ll^o ), V«€ff*(0 ),
(2.14) ||^||^+i/2(r 0)<c*(max max |D£p(*) |) ||«||ff*+V2(r ,,
v^e#*+]/2(rj
mta 4 #w^ 4' constants depending only on k.
Proof. The proof of (2.13) is immediate, since kis an integer; (2.14)
can be obtained, for example, by interpolation of the mapping u-^<pu
between Hk(rj and #*+] (rj. D
Remark 2.3. (2.14) can be tightened by not including the derivatives
of q> up to the order k + 1 in the second member; but (2.14) is sufficient
for the sequel. For the study of the multipliers q> in #5-spaces see, for
example, Hormander [1], Peetre [2].
2.2 The Preliminary Estimate
Let q and d be given positive numbers such that 0<£<@ + (5<£0;
let %{t) be a fixed infinitely differentiable function on R1 such that
%(t) = 1 for t < 0 and x(t) = 0 for t > 1.

2.2 The Preliminary Estimate 33
Set
(2.i5) 9^{x)=xf\j^y xeoec.
Thus we have a function <PQ)d£@{QQ)> with support contained in
£2e+d and with cpQ}8{x) = 1 in Qe. Furthermore, there exists a y^j, which
depends only on \p\, such that
(2.16) |Dfyft«(*)|<yW«-W V^€^eo.
Theorem 2.1. Under the hypotheses of Theorem 1.3, tf^re &m£ ^°
positive constants @x and C1 such that, if 0 < o < £ + d < ^ <mdf w
^G @(Qeo)} we have
(2-17) |WU,e< ^{ll^^ll
l\0,Q + 6 "t
2w-l ^ 1
+ 2 ^2«'-/ ll^lko.e+df
Proo/. Thanks to the hypotheses on s/ and Jy we may apply the a
priori estimates (4.39) of Chapter 2 (see Theorem 4.3 and Section 8.3
of Chapter 2) to the function v = (pe>su; thus there exists a qx < q0 such
that, if 0 < q < q + d < £x, we have
(2.18) U^ll < \WQ)6U\\h^{Qq + 8) < CWWi<PQ,6U) ll0,0,e + <5 +
+ |||*(^«)|||ol(? + « + H^,^IIh2^-1(^0 + (5)}
where the constant C depends only on si and the J^'s (and by looking
at the proof of Theorem 4.3 of Chapter 2, we easily see that q1 and C
depend on the values of ap(0) for \p\ = 2m and bjh(0) for \h\ = mjt
on the moduli of continuity of ap(x) for \p\ = 2m and fy^M for \h\ = mj
at the origin, and on the quantities max |^(#)| and max |&jA(y)| for
\p\ < 2m and \h\ < m3),
But we have
l<|?|<2m
where the j^'s are differential operators constructed in an obvious way
from si and of order < 2m — \q\; and therefore, thanks to (2.16), we

34 2. The Theorem on "Elliptic Iterates": Proof
have
(2.19) W(cpeAu) ||0Ae+, < C'{\\<pe^u\\0Ae+6 +
\p\<2m-l
where C" depends only on m and on max \ap(x) | for \ft\< 2m. In ana-
logous fashion, we have
(2.20) \We^\\H^- < 2 l|D%M«)llo,o,e+*<
|?|<2m-l
< C" 2 s2m-|^|-l II D W Ho,0,e + «'
\p\<2m-l °
with C" depending only on m.
Finally, we have
l<\q\<mj
where the ^y,/s are differential operators of order < m;- — \q\,
constructed in an obvious way from the ^-'s, with infinitely differentiable
coefficients in Q6q (see Remark 1.1).
From the above, also applying the (trace) Theorem 7.5 of Chapter 1,
we deduce
(2.21) U^(^«)||H2«-^-i/2(rc+d) < \\<Pe,d^ju\\H^-^-V2{re^6) +
+ 2 II (Vq<Pe,6) @j,qU ||fl2m-m,-ly2(r > <
l<|?|<mi ^
< ll^,(5^l^-^-]/2(re+(5) +
+ C" 2 || D%^) *,. « ||*2m-m,(0 , <
+ c" 2 2 l|DaP^%^llo,o,,+.<
] <\<l\<m5 \<x\<.2m — mj
< \WQ,6®iUWH*™-™o~V2{r > + CIV 2 -p^^ HD^llo,0,e + 4
e^° |/>|<2m-l 0 ' ' J
with CIV depending on m and on max |D^(:x;) | for \h\ < 2m — my.
(2.17) then foUows immediately from (2.18), (2.19), (2.20) and (2.21).

2.3 Bounds for the Tangential Derivatives 35
2.3 Bounds for the Tangential Derivatives
Let us first show
Lemma 2.5. Let a be a given function with a £ ®Mk (Q6q) and
(2.22) sup 2 \V"a{%) I < cUMr Vr > 0.
*^e0l<*l=>
Then, for every r > 0, s > 0, 0<£<£+(5<£0 and u £ @(QeQ), we
have
(2-23) 2 2 l|D"k> D> - V«Aau))} ||0,0_e+, <
|/J|-s|«|-r
< c? 2 i 2 if+,^+i StAt- n«ii,-,-«e+».
with the constant Cf depending only on s and c.
Proof. We recall that (Leibniz formula)
D'w-ste)-fe)Dw""
where y = (ylt ..., yn), r\ = (rjlf ...,rjn) and rj < y means r\x < ylt ...,
We also recall that
/n + ••• + yw
Then we obtain
2 2 ii^k,^D>-D:'M)]iko,e+«<
iPi-*i*i-f
<t(;)ifjiif;1i:i: 222 m^ix
1=0 \'/«=l\?/l=0 \ * /|y|_J|»|_j|A|=r-j|,,|_/^|_s-J-/
x|D"D>||D"D>|||0,0>e+,<
^t^ittsf'rlS 2 2 2 2 max|D>M|x
Xmax|D"D>|||D"D>||0i0ie+,<
:c;|(;)t(;)|f7^-«,
<c:y.uy.i iy.i . \^cL*+'Mq+l\\u\\s_l_ttr_q!e+s

36 2. The Theorem on "Elliptic Iterates": Proof
<
C"k%i%{rlt^'M,
+t jj +t^ WuWs-l-t,i,Q+*
r+t
< (thanks to (1.8)) < Cf £ 2 i £'+'M,+( 2 i II«||,-,_w,+,- D
Remark 2.4. Assume that the function a, instead of (2.22), satisfies
the estimate
(2.24) sup £ \Vaa{x)\<cLr-2mMr_2m W>0
x£QQo\<x\=r
where we set M_± = ••• = M_2m = M0; and also assume that s < 2m.
Then, with the same proof, we obtain
2 2 IId» d> - v-Aau))] || < c; 2 (*) £ (') x
x 2 (s 7 /)4^+'-2",^+(-2» Wl,-«-,,-M+, < C 2 2 4 x
X g (, 1 ;) L>-i+'-*»>Mr_i+t_Zm ||«||,_,_^e+, <
< (considering that t < s < 2m and that we have (1.9)) <
< C 2 sis (f 1 ;V-' x m,_, IMIs-^,e+* <
< (thanks to (1.8)) < Cs £ 2 4^2 T^F HI.-«-W*+a
1=0 t=0 ° i=o-Llvli
and therefore finally
(2.25) 2 2 l|D'k^D>-D:'M)]llo,o.e+*
\P\.-s\«\=r
with the constant Cs depending only on s, c and L. Q
Remark 2.5. If /3 = 0, then (2.23) is also satisfied setting (pe>6 = 1. 0
Lemma 2.6. Under the hypotheses of Theorem 1.3, ^'/ @x <mrf Cx are the
constants of Theorem 2.1 and 0 < @ < q + (5 < qv for every e > 0

2.3 Bounds for the Tangential Derivatives 37
exists a y(e) such that for every integer k > 0 and every u £ 3(Qeo) we have
(2-26) M|2m,2,w,e < Cf J||j/«||a2J^c + , +^25S |||*«|l|2*m,C + « +
+ _2mT2wll^llO,2ftw,e + <5 +£ll U \\2m,2km,Q + d + ^2&m Z-l t[7 \\U\\2m,2sm,Q + d +
£ 0 5 = 0 lvl2sm
"•" e2w Zj ~~^7 Il^ll2w,2s»:,e4«5
0 s=-l iKi2(s+])w
m£A £A# constant Cx independent of e and of u.
Proof. Let us apply (2.17) to D",u, with oc = 2km:
(2-27) || u \\2Mi2km>e= 2 \\^^km,o,e<C1{\\^u\\0^Q+d +
\<x\=2km
+ 2 II ^(D» - !WM) IUe+« + 11 K«*« 11 !»*,«+« +
|<%|=2m&
m—]
+ 22 ll?'ft«[^PJ«)-DJ(«,«)]||Jfa.-v-W(jPe+a) +
y=0 \a\=2km V
2w-l -J^
~f~ 2*1 &m-l \[UX[l,2km,Q + d}-
1=0 °
Then thanks to Lemma 2.5 and Remark 2.5 (with s = 0 and r = 2km)
we obtain
2 ||*/p»-DJ(J*«)||0>0>e+a =
|<%|=2&m
= 2 2 II^D:'D^-DJKD^)llo,o,e+.
|£|<2m|<%|=2fcm
2km—1 -j
<C*L2^M2ftM 2 2 T^IID^IIo,,-,,^
\p\<2m i=0 ^ 1V1i
m 2km —1 tut
<2 2 -^Dr^ll^lU^.
t = 0 i=0 1V1i
with the constant Dx > 1 and depending on C*, c and L. Now we apply
(2.7) with s = 1 in order to estimate the terms ||^||^>e+<5; we obtain
2km — 1 ^
(2.28) 2 ||^(D»-DJ(^)||0i0,e+,< 2 -^Df-'X
X {\\U\\2m,i,Q + S +WU\\o,i,e + d}>
with D2 > 1 and depending on C*, c and L.

38 2. The Theorem on "Elliptic Iterates": Proof
Then let i = 2sm + t with 0 < s < k — 1, 0 <t < 2m; applying
(2.8) (and Remark 2.1), we have for every fixed s > 0:
(2.29) |M|2m,2sm+*,e + <5 — S \\U\\2m,2{s+l)mlQ + 6^' Cm\8 ) II u \\2m,2sm,Q + 8 >
(2-30) \\u\\et2sm+tte+a < s' \\u\\0t2{s+1)miQ+e + cje')-"^-* ||«llo>2^e+«.
where we note that s may be chosen arbitrarily and in particular different
for each s and t. Thus, having fixed e > 0 arbitrarily, 0 < e < 1, we
choose, for fixed s and t,
m2{s+l)m u2
Applying (1.11), it follows that
(>\-t\{2m-t) ^> * /M2(s+l)mV/(2ffl"% ^ 1 , M2sm+t n<
e \ iW2sw+* / 8 1V12sm
2
and therefore
M2sm+t M2sm
with yx(e) = ~Dse2m and D3 a suitable constant.
We also see that
(0 qq\ ■^2kmT^2km-2sm-tf _^-2kni /j\2m\k-s-l
[ ] ~M 2 ~ ~M [ 2 j
iKi2sm+* iKi2(s+l)w
Then, using (2.29), (2.32) and (2.33), we obtain
2km —1 jy/r
(2-34) 2 -^D?-' IMIa.Ae+, < 2^ |MU^+. +
k-2 jur
+ 2*« 2 t^-— Pi")*-' -1 IMU2,.+i,-,e+4 +
5 = 0 m2{s+l)m
k-1 ]\£
+ 2mCm 2 W^{YMk-SM2m,2sm,e+8 < to»B\\»\\s~&m,a+a
5 = 0 m2sm
»-i ^{8))*-.
i
where y(e) = 2weD|m + y^s) 2mcm.
~^~ M2km 2j VJT \\U\\2m,2sm,Q+d>
5=0 m2sm

2.3 Bounds for the Tangential Derivatives 39
Applying (2.30), (2.32), (2.33), we also have
2km—1 yr k-1 yr
(2.35) 2 ^ D*--* H|oAe+a < 2ms £ -^»=- Pi")*—1 X
*=0 iWi s=0 iW2(s + l)m
ft—1 jj£"
s = 0 iW2sw
ft-1 7Lf
< (thanks to (2.1)) < 2ms £ "T^- Pi")*—'^^^^ +
s = 0 iW2(s + l)m
+ ^ 2 tt^ (yi(«))*"' ll«lk£(,-i)»,e+a<
s = 0 iV^2sm
-^ lvl2km Zj ii/r \\u\\2m.2sm,Q + d>
s=-l iW2(s + ])m
stiU with y(e) = 2weD|w + 2mcmy1(e).
Thus finally it foUows from (2.28), (2.34), (2.35) that
(2.36) 2 K(D» - DJ(j/«) ||0i0|8+d < 2^ IMU^-* +
|«| =2ftm
*-1(y(e))*"s *_1 yie)*-3'1
+ M2km2j—— ll^ll2w,2sm,e + <5 + M2km 2j ~Tjr IIU \\2m,2sm,Q + d
s=0 lvl2sm s=-l m2{s+l)m
with y(s) a function of e, c, cv L which goes to +ooase->0.
Let us now study the term \\\<pe,d&u\\\2km,Q+d in (2.27). Applying
Lemma 2.4 and formula (2.16) we immediately obtain
(2.37) IIK^IILtaM-n ^ ^ lll««lll«-,e+«
with the costant D4 depending only on y{ with i = 0, ..., 2m (see (2.16)).
Next we have
m — 1
y=0 \<x\=2km
< (thanks to Theorems 7.5, Chapter 1)
m —1
<2 2 ii?u^ro«)-Dwii*^-'»«<w
j=0 \a\=2km
m—1 2m —m.-
<2 2 2 2 2 IID%,^D:'D^-^(^DM))llo,o,e+^
j=0 s = 0 Jj8|=s|«|=2ftm|A|<my
< (using Remark 2.4)

40 2. The Theorem on "Elliptic Iterates": Proof
m — 12m — mj s 1 s—l 2km — 1 -t
<2 2 2 C2J2itt"^ 2 WHD*MIU-^
j=0 5 = 0 |A|<«y Z = 0 ° t = 0 i = 0 ^ lvli
<
2km — 1 i
cS 2 2 n7i2%. 2 t^I
y = 0 5 = 0 z,=0Z=0*=0 ° i = 0 ^ lvli
< (using (2.7) with £-=1)
M 2m i 2km —1 -t
<C^—L^Mhm £ ^{ML-m.^ + IMIo,^}-
1=0 ° i=0 ^lvli
Ans therefore we finally have
m— 1
(2.38) 2 2 ll^^(D»-D^M]llHa«-«,-i/2(re+,) <
j=0 \oc\=2km y_t"
m 2km — 1 71^ -f
<2 2 -^^""Vfl^l^-^+' + ^llo.*.^
Z=0 i=0 iWi 0
with the constant D5 depending on m, c, L and cv
Now we note that, thanks to (2.7), for every s > 0 we have
Hk-^+a < e' \\u\\2mM+s + cm(e')-*m-')l1 \H\oM+e
and therefore, taking s = dl, we have
(2.39) -j\\H2m-l,i,Q + 6< \\U\\2m,i,Q + 6 + °™ ^ II U llo,*,e + <5 •
1 1
Let us use (2.39) in (2.38) and note that -^- < -^ for 0 < I < 2m; we
obtain
m — 1
(2.40) 2 2 ll^[^(D»-D^M]||H2«-^-i/2(r ,<
y = 0|«|=2Jfcm ^
2fcm —1 7Lf 1
with the constant D6 depending on m, c, cv L.
We note that the second member of (2.40) differs from the second
member of (2.28) only by the presence of the factor ljd2m. Thus, following
the same procedure as was used to estimate (2.28), we easily see that we

2.3 Bounds for the Tangential Derivatives 41
obtain the following estimate (which is the analogue to (2.36)):
m — 1
(2-41) 2 2 ll9>M[^P»-D:'(^)]|lH^-^-i/2(re+5)
k~1 (y(e))k~s
< 2me \\u\\2m,2km,Q + 6 + ^2fcm 2 j|7 II U II
5 = 0 iKi2sm
"T" s2m Zj ti^ ll^l^w^sw.e + d"
0 s=-i ^2(5 + l)m
As far as the last term in (2.27) is concerned, we have, applying (2.7),
that for every e > 0:
\\U\\l2km,Q + 6 < S' ||«||2«l2*«lC + d + Cm^)~mm~l) IMIo,2*mlC + d
and therefore, if we take s = s d2m~l, we have
Il«llz9.^«4.ji<e|
-l({2m-l)
spm-l W^WlgkmtQ + d -^ c llwll2«»>2ft«»>e + «5 ~ um c2m c H " H0,2ftw,e + <5»
from which we deduce
2m_] 1 2wc
(2-42) 2 ~^T \\U\\l,2km,Q + 3 < 2me \\U\\2m,2km,e + d + ^£ II U Wo,2km,e + 6 •
The Lemma then follows from (2.27), (2.36), (2.37), (2.41) and (2.42). D
We now introduce the following definitions: for every real X > 0 and
real R such that 0 < R < qv
(2.43) J{u,l,R)= 1 sup (R-Q){k+1)2m\\u\\2m,2km}et
lvl2kmA Rf2<Q<R
k> -1;
1
^2{k-l)m^ R/2<q<R
(2.44) tf»(,u>JR)=xr__-ji sUp (JR-e)2*mll«llo,2»..,e. ^>o;
% —1
(2.45) y*(«,A>fi)= 1 sup (fi - e)2km\\\®u\\\2kmie,k > 0.
lvl2{k-l)mA Rj2<p<R
Note that we have
(2.46) o\(uyly R) <(/">, A, R) Vk>0.
Lemma 2.7. Under the hypotheses of Theorem 1.3, there exists a Xlf
independent of u, such that for every integer k > 0, every X > Xv every

42 2. The Theorem on "Elliptic Iterates": Proof
R < — q1 and every u £ <3){QQ^, we have
(2.47) g»(«, A, i?) < 4- ^-'''V-V«, A, J?) + -?—V»(«,A,2?) +
4 5=—]
(2? - o)2<*+]""
Proo/.'First, assume that & > 1. Multiply (2.26) with ,*+i—
and take the upper bound for R[2 <q<R choosing <5 = (R — g)/(ft + 1).
For the first member we obtain <r*(w, A, i?). The second member is equal to
C*(7, + 7g+ 78 + h + 76 + /,). with
(2? _ g)8(*+l)m |
h = „ ih+1 SUP 7^ 111 ^ ! I U,8 + a .
1 (# _ g)2(*+])»
^3 = jj^ ^t+1 SUP e2m(32« llMllo,2*m,e + «'
1 *~] ^(e)^-5-1 _ (R - e)2(*+«*
^ +15=-1 ^2(s+l)fl
^6 — Qft+i Zj ^ SUP s2m IMl2m,2sm,e + <5 >
where the sup is taken for R/2 < q < R and (3 = (R — Q)j{k + 1).
Then we have
to a* t Mnk-i)m (R - e)*k+1)m (R-q- s?km ., . ..
(i \2fcm\
1+t) )
(ie _ e)»(*+1)« /D ^2W
<<V-^^oV*a,*),
with the constant Gx depending on qx and w.

2.3 Bounds for the Tangential Derivatives 43
Next we have
^2(*-i>« „ (K - e)2<*+«" (R-q-d)
/2=1^-i)«sup
2km
AM2ftm " d2m(R - q - dfh™ M2^1}J
2km i\/r 7k SW^^ \\\2kmtQ
<
M<
2{k-
AM2km
But, thanks to (1.16), we have
(i \2km
k i
M«>-i)m lu , ^ ^ / cx \2- (2£m - 2m)!
*-»- (* + 1)- < (A.)*" ^ ~ ^ (* + I)2"* <
M2km -\MX) (2km)\
ct \*»/ k + 1 Ym<G
tfj \(k-l)2m+~i) ~ 2
with G2 depending only on clt Mx and >w, and therefore
(2.49) I2<^y>k(u,lR)
with Gs depending on cx, Mx and m.
Similarly, we have
Also
(2.51) /4<fiG4a*(«,A,2?),
with G4 depending only on m.
Next:
5 A ^ \ 1 ) F (2? - e - 3)*(«+D*
with G5 depending only on @x and m.

44 2. The Theorem on "Elliptic Iterates": Proof
And finally we have
*-i /w.U*-«-i M2sm(R - e)*(*-+i>»
'^t^x) sup
a^.+W* -e - <5)2(s+1)*
x
M 25+1 H^ll2m,25m,e + <5
=JL v1 /yW~'_1
^ s = -l \ ^ /
^2*. v
X
•^2(s-1)»
x sup 7 , Xg,s+1)>B X
(R - e)2(*+J)»' (As + lfm
<-_! v1 /^MV
/ 1 V
X sup (R - q)W-s-V (k + l)2m 1 + —J
(i? - Q - (5)2<S+1>
V2ftm
X
yr 05+1 Il'*'ll2m>2sm>e + d
X
1 t? Me)]*-8-X ^ /A\2
(1 \2&m
3+—I (TSKU).
But the expression
Mo /J? \2m(fc-5-l)
(t) {k + 1)2m
^2{s+l)m
is bounded by a constant G2 as & = 1, 2, ..., and — 1 < s < k — 1.
Indeed it suffices to note that J? < 1 and that:
a) for s = —1, 0, 1,
Mo / 1 \2m(k-s-i)
——(i) (* + d-
M2(s+l)m \ * J
is bounded as k varies from 1 to + oo;
b) if s > 2 (and therefore k > 3), then, thanks to (1.16), we have
M2sm ( 1 \2™&-s-V fCl \2m (2sm)\ ( 1 ^m(*-.-i)
^2(5+i)w \ 2 / " \MJ (2sm + 2m)\\ 2

2.3 Bounds for the Tangential Derivatives 45
but for the same s's and k's the function of s:
_ (2sm)\ I 1 \2«(*-*-D
X^ ~ (2sm + 2m) \ \Yj
is increasing, since
im < 2sm + 2 and therefore 2(s + 1) m + 2m < 2(2sw + 1),
[25+1(2(s + 1) w + 2w)]2m < [25+2(2sw + l)]2m,
22m(5+1)[(2(s + 1) m + 1) (2(s + 1) m+ 2) ••• (2(s + 1) m + 2m)]
< 22w(5+2)(2sw + 1) (2sw + 2^ ••• (2sm + 2w) X
1 / i \2w(& —s —1)
X
<
(2sm + 1) • • • (2sm + 2m) \ 2
^ / ^ \2m{k—s-2)
: (2(s + 1) m + 1) • • • (2(s + 1) m + 2w)
which yields exactly: %(s) < %(s + 1).
Then we have
(4T
[Mj \ 2(k-l)tn + l J ~ 2'
<
Consequently we may write the inequality
(2.53) J6 <^-Xi (^J"5"1 °S[U>l R)'
Using (2.48), ..., (2.53), it thus follows that
G* M
(2.54) a»(«, A, 1?) < 8G*c*(«, ht, R) + vr^- 0o(j*«. *. R) +
a M2km
G* G*
+ (i+y(e))^ I? (^T) ^(«,a,i?),
with a suitable constant £* depending on Cx and on the G/s

46 2. The Theorem on "Elliptic Iterates": Proof
But Gq(u, X, R) < ak~1(u, 1, R) and therefore we have
C* M
(2.55) o»(«, X, R) < eG*tf*(«, X,R)+- g^ <f-\stu, X, R)
X M2km
+ — xp\u, X, R) + —<?-\u,X, R) +
G*(l + y(e)) "-1 Me) ^~s~1
X
|(f p,,,,*,.
Let us now fix s = 1/2G*; then y(e) is fixed and we can determine Xx
such that for X> X1 (2.47) is satisfied for k > 1.
Finally let us show (2.47) for * = 0. Using (2.37) and (2.42) (which
also hold for k = 0) we deduce from (2.17) that
||2w,o,e < C* J || j/« ||0t0iC+, + -^ 111««11|0>e+tf
:m 2mll^llo,0,e + <5 f
We multiply this inequality with (R — g)2ml{M0X) and take the
upper bound for R/2 < q < R choosing d = (R — g)/2. We obtain
{R 22m
2m )tfi ll^ltem.O.c + d
\(R-g- d)2m MQX
2m 1 1
+ ^^WIo,o,,}<
< ~ o°0(rfu, A, 22) + -y /(«, A, 2?) + sG*g°(u, X,R)+^ a°0(u, X, R)
with G* a suitable constant; thus we have
o°(«, A, i?) < eG*a°(u, X, R) + — o-1{s0u, X, R) +-—y>°(u, X, R) +
A A
Choosing s = 1/(2G*), we can determine X1 such that for X > X1 we have
exactly (2.47) for * = 0 (recall that M0 = M_± = • • • = M_2m). Q

2.3 Bounds for the Tangential Derivatives 47
We are now ready to prove
Theorem 2.2 Under the hypotheses of Theorem 1.3, if X and R are fixed
as in Lemma 2.7, for every u £ @{Qe<) satisfying (1.21) and (1.22), we have
(2.56) (?VV IR) < M(*+*+])2m c0(L* + 2)k+i+1 Vk>-1, i>0,
M2ktn
with L* = L0 d2m.
Proof. If k = —1 and i is arbitrary, (2.56) holds thanks to (1.21)
and to the definition of cr\u, 1, R). We prove the theorem by induction
on k; thus we assume (2.56) to hold for k — 1 (and arbitrary i) and apply
(2.47) to s/*u with i > 0, * > 0; we have
(2.57) aVV IR) < 4" M?*~1)m ^"] K'+V I R) +
+-t^-v\^> *>R) + 4~ 2 aS^';- ^ •
VW X, R) < —^— 111^(^)111^,, < ^+i+1 -^±^
According to (1.22), we have
L2(k~l)tn lvl2m{k-l)
< CoL*+'+1 M(»+<+D8g M«»» < (thanks to (1.10))
0 0 -nr TUT \ v '/
lvI2km lvl2m{k-l)
S= C0^0 ^ ^ M2m S= C0V^* l Zi ^T iW2m
iW2m& iW2mft
and therefore
(2.58) aVV ^> R) < 4" ^?~1)m <?*" V+V ^, *) +
1 7l/f 1 ft_1
+ 4-c0(L!lt+2)^+1^f±i±«+4 2 <T%^U,#)-
4 ^2wfc 4 5=-l
From which, by induction, it follows that
" U,#) <^*%^c0(£* +2)*+^ +-L^(£, + 2)*+^ X
71/f 1 k — 1 TUT

48 2. The Theorem on "Elliptic Iterates": Proof
< (thanks to (1.13)) < -L^%±^c0(£* + 2)*+'+* +
1 M k~] 1
^ r (T _1_ 9\&+*+l ^2w(fe+i+l)
< C0^* + *) jjT
iW2w*
and the theorem is proved.
2.4 Bounds for the Normal Derivatives
First of all we note that, according to the Leibniz formula,
(2.59) 2 |djd;(W)| < ± ± (?)(*) 2 |d;d>| x
H = s 1 = 0 r=0 W V/|i7|=Z
x 2 iDr^D>i
|r|=5-z
for s and q integers > 0 and every couple of functions u and v. Q
Next, for ft and q integers > 0, X > 0, 6 > 0, 0 < R < qv set
^(«,A,fl,2?)=— • 1 a,+gfl, sup (2?-e)«*+^||D^«||oi2^ff
1VI{k + q-l)2mA u R/2<q<R
and note that
(2.60) o£°M, 0, #) = ^oK <M, #) < ct*_1(«, A0,2?), ft > 0. Q
We prove
Theorem 2.3. Under the hypotheses of Theorem 1.3, Z^ u£<3(Qeo)
and satisfy (1.21) <m<Z (1.22); £Aew ^r0 &m£ A0 <m<Z 0O, independent of u,
such that for R < q±[2 we have
(2.61) <#{jfu, X0, 00, R) < ^+?+'>2" c0(L* + 2)*+*+* ,
M(k + q-l)2tn
Vq,k ,i> 0.
Proof. Thanks to Remark 1.1, we may assume that
2m t
stu = v2ymu + s 2 S W- y) Df "'D>;
<=i;=o|/5|=y

2.4 Bounds for the Normal Derivatives 49
and therefore, for q > 0 and \<x | = 2mk, we obtain
D2m+2mgD^ = D2w*D£,(^) -
2m t
- 2 2 2 Dfq)^,^*', y) D*-'Dj»«).
<-iy=o|/3|=y
Applying (2.59) we have
(2.62) 2 |DyM+2m?D>|< ^ |Df*D^M)| +
\<x\=mk \<x\=2mk
+l"!f2i2(2r)(r)2iD;D>-<»'-)ix
i=o »-=o f=iy=o |/s|=y \ * / \ r I \n\=i
X 2 |If*",+2""lD^«|<
\y\=2km— I
(thanks to the fact that 2 I DJD>/y,/j(*'» y) I < cLl+''Mi+r)
\v\=i
< s ii^dm*,i + s£ss(T)(T)^'<
\<x\=2km i = 0r=0t = lj=0\ ' / \ ' /
Xtf,+, 2 2 |D*"-,+2"-'DJ,+"«|< 2 |D*"DJ(.jtf«)| +
|j8|=j \y\=2km-l \a\=2km
+ 2 £ £ 2 (2ti (2?i i,+,Af«+r s i dsd*-^2"-'" i.
It follows that
l|D*"+2"«IU„ < ||D*"W«)llo.ab« +
+ e ^ g-1 ^ I2km\ i^\Ll+rMi+f ||I^-H-»-'-'«||Wtal_/+A,.
z=o?=o *=iy=o\ v I \ r I
Applying Lemma 2.2 (and Remark 2.1) with e = 1, we have
2 iiD,2p"+2"-'-f«iio,flto,_J+,,e < 'iiDr+2~~'~'«iu»-;+,.. +
+ fcJ|D*»+2-'- '«||0,2ikw_,,,.
Again by Lemmas 2.2 and 2.3, we have for arbitrary s > 0:
2m *
2 2 l|Dr+2m"f"^Ho,2^-i+y.c < eym \\VT+2m-ru\\0,2km_hs +
t=lj=0

50 2. The Theorem on "Elliptic Iterates": Proof
where ym depends only on m, and c'm{e) depends only on e and m (q0
being fixed). Thus we have
(2.63) ||D^+2^||0,2ta>e < ||Df^||0>2^e +
+ «|| (2kf) (T) Ll+rM^y™ ^ ^+lim-»k».-K, +
+ cm(e) \\^m-'u\\mk+1)m_hs + c'Je) ||D2/-'W||0,2ta_;jJ
<(by(1.10))<||Df^||0>2^e +
+ cm(£) ||D*-'«IU*+i)..-«,e + 4(S) ||D*"~"«IU.-«,,>
where Lx > 1 depends on c, L and ^.
Let us now seek bounds for the terms of type || \\o,2km-i,Q> I =
0, 1, ..., 2km, by an expression containing only terms of type || \\0)2sm,Q
with s = 0, ..., k. In fact we have
(2.64) 2 ( ; ) LiMi II»Ik**.-,,, = Mo II»IU«,e +
2ftm-l / QyL \
+ Z (2jbw _ ■JLf'-'M^ \\v\\0ii>e < (using (1.8))
2km — 1 c
Now let i = 2sm + a, with 0 < s < k — 1, 0 < a < 2m; by Lemma 2.2,
we have
IMU.+a,e < «' Ho,2(S+l,*,e + ^(e')-0/(2W-a) IMIo,25«,e
and, choosing e' = — ^m— for each fixed s and c, we obtain,
thanks to (1.11),
(f\-al[2m-a) ^ jiyj-2sm+o JG
and therefore:
(2-65) ^ Z»-' || „||0,,e < -£**- !»--«-«- „„ |l0>2(5+1)m;e
+ ^^f-s,^ll^llo.25B,,e.

2.4 Bounds for the Normal Derivatives
0 0,q
Then from (2.63), we deduce
(2.66) 2 T L>M> ll'IU--*. ^ c« 2 ^f ^ IMIo,^
/ = 0 \ * / 5 = 0 M 2sm
with c^ depending on m and on 7kf0.
In analogous fashion,
(2.67) 5(2T)ii^ll^m",«l
^- '" V1 2?m r2qm-2sm \\T\2sm„.\\
s = 0 iK/2sm
with c'£ depending on m and on M0.
Using (2.66) and (2.67), we deduce from (2.63):
(2.68) l|D^m+2^llo,2^e < Pr^llo)2^e +
* « M2mfeM
Zj Zj yr tut
p = 0s = Q ±v± 2pmlvl 2sm
+ cje) \\I>?mu\\0,2ip+1)m,e + c'js) \\D^u\\0i2pmJ
with L2 depending on Lx, m and M0.
Now multiply (2.68) with
(^ _ ^2(*+?+l)m
I V1 V 1V12mkm2qm T2[k-p+q-s)m /_ n t^25w + w-. m
+ 2j 2j U jj7 ^2 1£ ll-^y ^llo,2£m,
and take the upper bound for R/2 < q < R; then, J? being < 1,
obtain
o*-'+1(u, X, B, R) < M^±i=}>L ^u> X, d, R) +
^M2{k+q)m
, y y L2»-*+,-.)m [A'+H*. A, 6, R) M^p+s)mM^mMnm
.t^o 2 1 A*-*+'-'0*-* M2(ft+?)MM2,MM2sw "•"
ffg+^K A, 8, 2?) M2(^+,)OTM2AOTM2?W
+ cm(s) }k-P+i-sQk-P-i M2{k+i)mM2pmM2sm +
, qfr*(«, A, 9, fi) M2(^+s_1)mM2^M2gOT|
< (using (1.14) and (1.9)) < Jf'<*+'-1)mo*-«(J/«, A, 6, R) +
^M2(k+q)m
I V V T2(k-p)m+2(q-s)m f ^Wj^)
£=0S=0 I A ^
qg+i'fr, A, 9, ft) 4(e) <#*(«, A, 0, fi)|
T~ cm\8) ik— 64-a —snk — * — 1 ' i oft— £+g —sf)k—p f "
tf-p + q-SQk-p-1 ' 4 ^

52 2. The Theorem on "Elliptic Iterates": Proof
FinaUy, setting L22mf(W) = f and L|m/A — <p, we obtain
(2.69) c#'+1(«> I 0, R) < MW+*-Um <%q{s4u, I 0, R) +
AM2(k + q)m
£=0s=0 [
]-4-\u,i,e,R)
+
-0 [
c'Js)
Now choose e = 1/40 and, for this choice of s, take 10 and 0O such that
(2.70)
1 c'Je) ^ 1
L2T 1 Z~ 1 _1_ _1_
^<T> "I^IT' ~^<ru>'
Then we have
(2.71) ±-<tf+1(u,*o,60,K) <±-M*k+<-J)mo*-^u,l0,d0,R) +
* 1U M2(k + q)m
+ ^ I] 5*"M,I+1(«. J* 0O,«) + 4 2 ?2 «*-V- x
xog-»+1(«, 4>, M)+in f*-V~sK+1'>. K 0o.«) +
ft-1
where, if & = 0, the sum ^ must be deleted.
Let us apply (2.71) to ^u\ we obtain
(2.72) itf+Wu, X0, 60, R) < \ ^?+9"1"'' o^+V Ao, fl0, 2?) +
+ -L *2 l*"M'?+1K«. K %. R) +
+ ^22 5*-V_,<'+1(^«. ^o. «o. *) +
+ 4 2 2 5*-V_K+1^(^«. *o- 0Q. R) + <#W A0, 60, R)}
6yjp=os=o
k-1
with the same convention as before for the sum ^ •
p=0

2.4 Bounds for the Normal Derivatives 53
We denote by p(q, i, k) the property (2.61) for the values q, i, k of
the parameters and by P(q, i, k) the property p(q', if, k'), 0 < q' < q,
0 < V < i, 0 < kf ^ k. From (2.72), we shall verify the following
implications :
(a) P(q, i + 1, 0) \J P(q, i, 1) =» p{q + 1, *, 0)
k) \J P{q + 1, i, k-l)\J P{q, i, £ + !)=»
J if ft>l,P(y,* + l,,
Indeed, according to (2.72), we have
1 M9j(k + q-l)ni mM2{
^2(k + q)m ^2(k + q-l)m
<%<+1{jfu, X0, 60, R) <-=- //+g~1"B • ^J^l+i+J^ c0{L* + 2)*+<+*-+1 +
-j ft —1 71f
I _y £k-P 1Y±2{p + g+i + l)m ,£ .^XP + q+i+l _j_
"• OA Zj ^ 71//" 0\ * ' / "•
*Vp = 0 lvl2(p+q)m
^U^ = 0s==0 ^2(p + 1)m
-[kg [M
M
2(p + s+i)m
cQ(L,l+2)*+'+'\< (using (1.13;
m2(p + s-l)m J
^ ~-^(ft + g + i + l)**
^2{k+q)m
f 1 1 *"* / £ V~p
+ 2o20,?oli»+2J U.+2J +
i i A (_J_\h+p(_v_\'-'\
>'■>,* + wM
Therefore (2.61) also holds for o^+1(^, A0, 0O, #).
We shall now prove the theorem by induction. According to (2.60)
and (2.56), since AO0O > Ax, we know that P(0, i, k) holds, Vi, k > 0.
We assume P(qf, i, k), q' < q, yi and k and show P(q + l,i, k), \/i and &.
By induction and (a), we have P(q + 1, *, 0), Vi. Let & be fixed
arbitrarily; assume P(q + 1, *', & — 1); this, together with the induction
hypothesis and (/?), implies p(q -\-l,i,k)\ this in turn shows P(q + l,i, k)
and ends the proof of Theorem 2.3.

54 2. The Theorem on "Elliptic Iterates" : Proof
2.5 Proof of Theorem 1.3
From Theorem 2.3, we can deduce
Proposition 2.1. Under the hypotheses of Theorem 2.3, there exists an
N depending only on stf, $j and L0 such that, if q' = ^x/4, we have
(2.73) 2 II D"M \\v(D ) < coN$Ms Vs > 0.
Proof. We use (2.61) with * = 0 and R = 6l/2; then
\ * I 1V12{k + q-l)m
from which, thanks to the definition of o%q(u, X, 6, R), we deduce
(2.74) IID^IIo,^, < Mnh+q)m X\+%{Q')-(k+q)2m *„£* + Z)k+q
< M2{k+q)mc0N\+'
where N-^ depends on A0, 0o, q , L* and therefore on L0, stf, ^-.
Let t and r be arbitrary integers > 0 and assume that t = 2qm + I
and r = 2km + / with 0 < Z < 2m, 0 < / < 2m\ then, applying
Lemma 2.2 with s = 1, we have
(2-75) IID^Ho,., < l|D^+1^||0,,,e, + cm \\D^u\\0>rje,
< ||D^+]>»V||0|2(,+1Ke, + cm ||D2«'+«"«||0>abB>e, +
+ cM \\-D?mu\\0i2hmie, + cl ||D^M|U»,e' < (by (1.9) and (2.74)) <
< Mnh+q+2)mcQQ + 2cm + cl) N*+<+* < (again by (1.9)) <
< M{t+f+2m)c0(l + 2cm + cl) iV*+<+2 < (using (1.10)) <
<c0N^Mt+r,
where N2 depends on Nlt d, cm, m.
Finally, we have
2 WHu^) = 22 IID^ILwi = 2 l|D>||0,s_(>e,
\<x\=s t=0\p\=s-t t = 0
<c0N*(s+l)Ms<c0(2N2yMs
whence (2.73), with N = 22V2. D

2.6 Complements and Remarks
55
We are now able to prove Theorem 1.3; in fact, applying the Sobolev
theorem (see Theorem 9.8, Chapter 1), from (2.73) we obtain
sup sup | ~Dau j < c*c0N%Ms
with c* depending on the dimension n of Rn and N* depending on q' and
on N. Therefore the theorem is proved.
2.6 Complements and Remarks
The proof of Theorem 1.3 also yields the theorem on "elliptic iterates'
in the interior of the domain, and more precisely:
Theorem 2.4. Let Mk = (k\)P, with real (3 > 1 (or more generally {Mk}
satisfying ((1.6),..., (1.11)). Let Q be an arbitrary bounded open setinW,
A a linear elliptic differential operator on Q given by (1.1), with aPq £ @Mk (Q).
Then, if u £ $(Q) and if there exist two constants c0 and L0 (depending on u)
such that
(2.76) ||^||L2(D)<c04M2m,,V,->0,
we have u £ SMjc (Q) (£Mjc (Q) = 3tf(Q) if Mk = k\).
Indeed, we reduce the problem to the case in which Q is a ball in W1
and use the same arguments as for Theorem 1.3 (bounds for the tangential
derivatives, Sections 2.2 and 2.3), starting from the a priori estimate
(3.6) of Chapter 2 (with l = r=2m), instead of (4.39) of Chapter 2. The
proof is also simpler than the proof of Theorem 1.3, not only because we
no longer need to distinguish between the (tangential and normal)
variables, but also because we do not have to study the estimates of the
boundary data 38jU D
Of course, we also have a corollary to Theorem 2.4 as an analogue
to Corollary 1.1:
Corollary 2.1. Under the hypotheses of Theorem 2.4 on Mk, Q and A,
if ueS'(Q) and if Au = f, with f £ S>Mje (Q), then u £ SMk (Q) (£Mk (Q) =
jr{Q),ifMk = k\).
Remark 2.6. It may be possible, with the same type of demonstration,
to eliminate conditions (1.9), (1.10), (1.11) on the sequence Mk, keeping
(1.6), (1.7), (1.8). This is evident for (1.9), since at least for sufficiently
large k, (1.9) follows from (1.15) (which is a consequence of (1.7)).
We have not done any further work on this point, since hypotheses
(1.6), ..., (1.11), as we have seen, cover the most important case of Gevrey
sequences, Mk = (k\)P, /? > 1. D

56
2. The Theorem on "Elliptic Iterates": Proof
Remark 2.7. The analogue to Theorem 2.4 and to its Corollary 2.1
for Beurling-type spaces (see Section 2.3, Chapter 7) has recently been
given by O. John [1] using similar techniques. Here we note one of these
results of which we shall make use later on:
let Q be a bounded open set in Rn and A an
elliptic differential operator on Q given by (1.1) withapq£ Jf (Q);
let Mk = (k\)p, with p > 1. Then, if u £ ®(Z»)
and satisfies:
VZ, > 0, there exists a c > 0 such that
(2.77)
\Wu\L,{Q)<cVM2mi^iy
it follows that u is of Beurling type in Q, i.e.
for every compact set c/f (^ Q and every L > 0, there exists
a c' > 0 such that
sup \D«u{x)\ <cLkMk,
\<x\ =k, Vk. 0
otZtf
Remark 2.8. We shall now indicate a procedure which, at least in
certain cases, allows the reduction of "elliptic iterates'' type theorems
to regularity results of the type given by Corollaries 1.1 and 2.1 for a new
operator constructed from A.
At first, we assume Mk — k\ (analytic case) and, in order to simplify,
we further assume homogeneous Dirichlet boundary conditions; in other
words, under the hypotheses of Theorem 1.2, for Mk = k!, we let u £ @(Q)
satisfy
(2.78) \
[ y^A'u) = 0 Vi > 0, j = 0, ..., m — 1.
Let us introduce an additional variable t (t£ R1) and the (vector)
function
oo j.2mi A i^
(2.79) wW = S(_ir+«__,
for which we shall verify the following properties:
' t^w{t) is an analytic function on ]—10, t0[
(suitable i0) with values in H2m(Q);
in the cylinder Q0 = Qx ]—10, t0[, we have
d2mw
(2.80)
(2.81)
(2.82)
y-w{t) = 0 on r and for t£ ] —t0, t0[, j = 0, ..., m — 1.

2.6 Complements and Remarks
57
Indeed, let us first recall that, for every v£H2m(Q) with y3v = 0,
j = 0, ..., m — 1, we have (use the estimate (5.3) of Chapter 2 and
Theorem 16.3 of Chapter 1)
(2.83) \MH2m[Q) < ^(H^H^ + M|L,(0)) (c, constant).
Applying (2.83) to v = A'u, it follows that
M*«ll*2*(fi) < c*c0[D(2im)\+Li+1(2(i + 1) m)\] V*,
whence
(2.84) ||4*«||ff2n.(fl) < ~cL\%m) \ (c, L constants) Vi.
Then (2.80) follows immediately from (2.79) and (2.84).
The verification of (2.8J) is as immediate calculation. Finally, since
yj is a continuous linear operator of H2m(Q) into H2m~j~ll2(r), we have
oo ±2mi
^^?0(-1)i<M+1W^'M)
(convergence in H2m~3~ll2(r) for every t£ ]—t0, t0[) and therefore
according to (2.78) we obtain (2.82).
Let us now assume that A is not only properly elliptic in Q, but
furthermore satisfies the relation
(2.85) {-l)mA0{x,£) 4= - \A0{x,£)\ VxeQ and VfeR", £4=0;
(note that in particular this hypothesis is satisfied if A is strongly elliptic
infij,
Then, in the cylinder Q0, equation (2.81) is properly elliptic in the
variables x1, ..., xn and t.
Thus applying Corollary 1.1 with Mk = k\ for the cylinder Q0 and
equation (2.81), (we shall obtain a global result on x, but local in t), we see
that w(x, t) is analytic in (x, t) in a cylinder (^ = Qx] — tv ^[, with
t± < £0; and therefore w(#) = w(#, 0) is analytic on Q.
Of course, we must have demonstrated Corollary 1.1, which can be
done directly, without using Theorem 1.2, by the same type of
arguments as for the proof of Theorem 1.2, but much simplified.
The same idea can also be developed in the the case of non-analytic
Gevrey classes (Mk =£ k\), but then the problem is reduced to a
regularity problem of the type of Corollaries 1.1 and 2.1, for more complicated
operators than elliptic operators, for example for quasi-elliptic operators
(and then the advantage of such a method is almost non-existent!).
Here is a simple example; take Mk = (k\)p with rational /? > 1, /? = pjq,

58 3. Application of Transposition; Existence of Solutions in the Space Q)\Q)
p and q integers, still with homogeneous Dirichlet conditions; thus
[||^IIl^)<^o4((2^)!)^ V*>0
(2.86) <
[y<J(4*«) = 0 V*>0, j=0, ...,m — 1.
Let us try to prove that u £ ^^(Q). We introduce
oo j.2mp
(2.87) w(t) = 2 (-l)^+]) ——- A«u
i=0 (2mtp)!
and verify, as for (2.84), that
\\Aiqu\\H2m{Q) <^cLi(2pmi)\>
so that we still have (2.80). Next, we verify that w is a solution of
d2mpw
(2-88) (_!)*-__+4% = 0
in Q0 = Q x ] —z0>20[, and that
?jw(t) = yj(Aw(t)) = .- = y^"1^)) = 0, *£ ]~^ tl
j = 0, ..., m — 1.
Now if ^4 is strongly elliptic or if A is not only properly elliptic, but also
satisfies
(2.89) (-ir Ai(x, f) 4= - |^(*> f) | v*e fi, vf e rm 4= o,
then equation (2.87) is quasi-elliptic in ()0; by using a theorem of Caval-
lucci [1], which is a generalization of Corollary 1.1 to quasi-elliptic
operators (see also Matzusawa [2]), it follows that w(x, t) is of Gevrey type
of order /? in x and analytic in t on Qx]— tv ^[, ^<^0; therefore
w(x, 0) = w(«) e ®me{Q). _
Remark 2.9. C. Goulaouic [2] has shown that the spaces Q)Mk (Q)
are not in general "interpolation spaces" between the space of analytic
functions on Q and the space Sf{Q); Here, we have an essential difference
with Volumes 1 and 2, where interpolation was directly applicable.
(Still according to Goulaouic [1], [2], interpolation could be used for
certain homogeneous problems.) Q
3. Application of Transposition; Existence of Solutions
in the Space &(Q) of Distributions
3.1 Generalities
Following the ideas developed systematically in this book, we shall
now see how the regularity results of the preceding sections can be used,
by applying the method of transposition.

3.1 Generalities
59
Our aim is the study of elliptic boundary value problems in the spaces
@f'(Q) and <3'Mk(Q)', we shall first examine the case of distributions,
®'(Q). D
Thus, let us again consider the boundary value problem (1.3), adding,
to hypotheses (I), (II), (III) of Theorem 1.1, the hypothesis of normality
of the system {B^~l (this hypothesis is essential to the method of
transposition, as we have seen in Section 6 of Chapter 2; see also
Section 8.3 of Chapter 2) and the hypothesis that the data of the problem are
analytic; more precisely, we assume that:
i) Q,A, {BjYf-J satisfy hypotheses (I), (II), (III) of Theorem 1.1;
ii) r is an analytic variety;
iii) the coefficients apq belong to 2tf (Q) and the coefficients bjh belong to
SPIT);
iv) the system {Bj}J~q is normal on r.
Under these hypotheses we can introduce a (formal) adjoint problem
of (1.3), as we have seen in Chapter 2, by applying Theorem 2.1 of
Chapter 2; we choose the system {S^JZq of "boundary" operators so as to
have Green's formula:
m — 1
(3.1) J (Au) vdx— f uA*v dx = £ j S^uC.v da -
q q j=o r
m — l
- 2 / BjUTjV dor Vw, v £ ®{Q),
i=o r
the operators Cj and Tj, j = 0, ..., m — 1, depending on A, [B^Zq
and {SjYJITq , according to Theorem 2.1 of Chapter 2.
We also recall that N and iV* denote the spaces
(3.2) N = {w \w£$){Q)}Bjw = 0, / = 0, ..., m — 1, Aw = 0},
(3.3) iV* = {w \w£9{Q)t C3w =0, / = 0, ..., m — 1, A*w = 0}.
We know that N and iV* are finite-dimensional subspaces of <&{Q)
(see Chapter 2, Section 5).
Thanks to Corollary 1.1 to Theorem 1.2 of the present chapter, we
may state that, under hypotheses i), ..., iv), N and iV* are made up of
analytic functions on Q. D
Let us now consider the homogeneous adjoint problem:
IA*v = <p, with <pe@(Q),
Cjv = 0>j = 0> 1, ...,w-l.
We can apply Theorem 1.1 of this chapter, of course changing A to
^4* and Bj to Cy, which is permissible (see Theorem 2.2, Chapter 2); we
obtain the following result.

60 3. Application of Transposition; Existence of Solutions in the Space 3>'{Q)
We introduce the space:
(3.5) X = {v | v e ®(Q),; Cjv = 0, / = 0, ..., m - 1; A*v £ @{Q)}
provided with the inductive limit topology of the spaces
(3.6) X{v) = {v \ve@{Q), C3v = 0,j = 0, ..., m - 1, A*ve@(jtrv',Q)},
where {jf„} is an increasing sequence of compact sets contained in Q
whose union is Q and Q)[XV\ Q) is the Frechet space of functions of
S){Q) with support contained in Jf „, (see Chapter 7), X{v) being provided
with the natural Frechet space topology; thus X is a strict (J£fF)-space.
We further denote by {<2}{Q))N} the closed subspace of Sf{Q)f
defined by
(3.7) {®(Q);N} = {(p\<pe ®(Q); f <pw dx = 0 Vwe N).
Q
Then, by application of Theorem 1.1 and by the definition of X, we
have (still denoting by A * the operator deduced from A * by passage to
the quotient by 2V*):
Proposition 3.1. The operator A* defines an [algebraic and topological)
isomorphism of X[N* onto {2)(Q); N}. D
Let us now transpose the isomorphism^* obtained by Proposition 3.1;
we obtain
Proposition 3.2. For every continuous antilinear form v"->L(v") on
X[N*, there exists one and only one element u belonging to the space
{@{Q); N}' {strong dual of {@(Q); N}) such that
(3.8) <um,A*vm> = L(v), VveX/N*,
where the brackets denote the duality between {&{£}) \ N}' and {@(Q); N};
u depends continuously on L [for the strong dual topologies). D
Note that
(3.9) {®(Q);N}' = ®'{Q)IN
Indeed it is sufficient to note that for every non-empty subset P of a
separated locally convex space E, the bipolar set P00 is identical to the
smallest balanced (circled) and closed convex set for the weak topology
which contains P.
We also note that (use (3.1))
J wA*v d^ = 0, VweN and Vv £ X.

3.2 Choice of the Form L; the Space S(Q) and its Dual 61
Thus we may also express Proposition 3.2 as follows:
given a continuous antilinear form L on X/N*, there
exists u £ S)\Q)} determined up to addition of a function
(3.10) \ of N, such that
O, A*v} = L[v), Vv £ X/N* and Vv <G v ,
[ the brackets denoting the duality between Q'(Q) and Q(Q). Q
3.2 Choice of the Form L; the Space 5(£2) and its Dual
Now we have to "separate" the equation fiom the boundary
conditions in (3.10), by choosing L in an appropriate way, according to the
procedure followed in this text (see for example Section 6 of Chapter 2.)
Formally we choose L to be in the form
m — l
(3.11) L(v)=<J,i>+ 2 <^7>>,
3=0
where / and gj are suitable given "functions" on Q and on T respectively:
then we have the equation
_ m—l
<«, A*v} = </, v> + 2 <gy 7»> Vw <E X,
3=0
from which, still formally, thanks to the Green formula, we deduce that
Au = f in Q, B-u = g. on r.
Now the problem is to choose / and gj and to justify the preceding
formal considerations. Q
Let us first study the problem as far as the equation Au = f in Q is
concerned. For this purpose, we introduce a space K{Q) of distributions
on Q such that
(3.12)
X C. K (Q) C L2(Q) with continuous injections;
K(Q) is reflexive;
Qj{Q) is dense in K{Q) [and therefore K(Q) is
a normal space of distributions on Q).
Spaces K{Q) with the properties (3.12) exist: for example K(Q) =
L2(Q).
But (as we have already seen in Chapter 2) in (3.11) we shall take /
in K'(Q), dual of K(Q). The theory will therefore be the more general,
the "greater" K'[Q) is, i.e. the "smaller" K(Q) is.
We do not know whether there exists a "smallest possible" space
K(Q) with properties (3.12). If it exists, the optimal space K(Q) depends
on the boundary conditions (since X C K{Q)). Here, we shall (conforming

62 3. Application of Transposition; Existence of Solutions in the Space <2>'{Q)
to the setting of Chapter 2, Section 6.3) construct a particular space
K(Q) = 5{Q), which is independent of the boundary conditions, but
already seems to be "sufficiently small".
Let q(x) be a function of Sf{Q)t positive in Q, vanishing on r of the
o(%\
same order as the distance d(x, T1) of x to r (i.e. lim = d 4= 0) ;
x-^x^r d(x, r)
we have already introduced and used a function of this type (Chapter 1,
Section 11.2, Chapter 2, Section 6.3, ...).
Definition 3.1. S(Q) denotes the space
E{Q) = {« | eW D"« e L2{Q), Voc},
provided with the topology defined by the family of semi-norms
lleND«M||LW v*.
It is easily verified that S(Q) is a complete space; therefore it is a
Frechet space.
This space is made up of infinitely differentiable functions in Q (but
not necessarily in Q); of course, we also have
(3.13) X C ®(Q) C S{Q) C L2(Q).
Proposition 3.3. @(Q) is dense in S{Q).
Proof. Let dv(x) be a sequence of functions of 9){fi), v = 1, 2, ...,
such that dv(x) = 1 for d(x} T) > 2[v and dv(x) = 0 for d(x, T) < \\v
and
| d(x, r) |'al | D" dv{x) | < ca (ca depending only on <x).
Now let u£ S(Q); then dv u£ @}(Q) and the proposition will be proved
if we can verify that
(3.14) dvu -^ u in 5(Q) , as v -> + oo.
To this end, we must show that ^ D"((J„m) -> gW Dau in L2(Q); but
(5^'al Dau-^ qW ~Dau in L2(fi); therefore it is sufficient to show that
(3.15) eW(D'(J,) (D^)->0inL2(fi), |/?| > 1, |/?| + \y\ = |*|.
Now £H(D/?<y (Dy^) vanishes for d(z, T) > 2v (since |/?| > 1), so that,
according to the Lebesgue theorem, we have (3.15), noting that
le1*1^,) (D^) I = (l^'P^J) (c|y| Dy^) I < ^ c|ylDy^ e £2(£). D
We denote the (strong) dual of 3(Q) by 5'(Q). Thanks to
Proposition 3.3, E'(Q) can be identified with a space of distributions on Q.

3.2 Choice of the Form L; the Space S(D) and its Dual 63
We obtain
Proposition 3.4. Every element f of the space E'(Q) may be represented
(non-uniquely) by the form
(3.16) / = 2 D"(eWa «*** /„ 6I2(£) •
finite
Indeed, we only need to note that every continuous linear form on E(Q)
may be written
M(q>) = S / gja{ D> <**, with fe e L2(Q),
finite £?
and, that according to Proposition 3.3, it is determined by its values for
each q> £ Q)(Q) (from which we obtain (3.16) by setting fa = (—l)1*' ga).
Proposition 3.5. The space E{Q) is reflexive.
Proof. It is sufficient to show that every bounded and weakly closed
subset in S(Q) is weakly compact. Now if {%} is a bounded sequence in
E(Q), there exists an Ma such that
lleWD"«<llL.(fl)<AfB, V*.
Then using the "diagonal" process, we can extract a subsequence {uv}
from {%} such that q^ T>auv converges weakly in L2(Q), for every fixed
oc, to a function ipa which depends on oc. But, since uv converges weakly
to a function u in the sense of L2(Q), q\"\ Dauv converges weakly to
gl*l J)"U in the sense of <3'(Q). If /G B'(Q), we have, thanks to
Proposition 3.4,
/= 2 D>H/J,
with a suitable integer / and fa G L2(Q); and therefore
</,«,> = 2 <D"(ew/„),«,>= 2 (-i)H<4,ewDX>
|«|<J |«|<1
and finally
um </,<> = 2 (-i)H<teHD"«)
i.e. uv converges weakly to u in S(Q). D
Therefore we see that, thanks to (3.13) and to Propositions 3.5 and
3.3, E(Q) may be chosen as the space K(Q). And we can take / in Ef{Q)
for the choice of L in (3.11).

64 3. Application of Transposition; Existence of Solutions in the Space 3i\Qi)
3.3 Final Choice of the Form L; the Space Y
For the choice of the boundary data gj, we must first note
Proposition 3.2.
v^&v = {T0v,..., Tm_^}
is a continuous linear mapping of X into the space [J>f(r)]m.
Proof. If v £ X, then by definition, A*v vanishes in a neighborhood
of r and CjV = 0 on r, j = 0, ..., m — 1. Therefore from Corollary 1.1
to Theorem 1.2 (the local character of this result is evidently a
consequence of Theorem 1.3), it follows that v is analytic in a neighborhood of
r and therefore T,v € J?(.T), so that &{v) <G [Jtf {r)]m iiveX.
Furthermore, the spaces X and &?{F) being of (^f^-type, we may
apply the closed graph theorem to <F (see Grothendieck [1]); now
v->(ev is, for example, a continuous mapping of X into [C°(r)]m and
therefore also of X into [jff(r)]m. D
Remark 3.1. In Section 3.5, we shall see that W is surjective. D
We can now choose gj £ 3%"{r) in (3.11) and thus make our final choice
of L; more precisely, we note that
(3.17)
(3.18)
if f€5'{Q), the form «-></, ?> (the bracket
denoting the duality between S'{Q) and 3(Q)) is
antilinear and continuous on X\
if gyejf'(r),the form *-* 2 <gy, t>> (the
bracket denoting the duality between 2tf '(jT) and Jf (7^))
is antihnear and continuous on X.
Then considering the form
m—1
(3.19) £(*>) = </,*> + £ <^7»,
making the convention that L[v') = L(v) for every v£X, element of
the class v of X[N*, we see that (3.19) defines a continuous antihnear
form v -> £(t>") on X/iV* if and only if
(3.20) </, ^> + *£ <gy, 7>> = 0, Vve N*.

3.4 Density Theorem
65
In this way we finally obtain
Theorem 3.1. Let hypotheses i), ii), iii), iv) of Section 3.1 be satisfied)
let feS'(Q) and g^tf'^F), j = 0, ..., m — 1, with (3.20). Then there
exists u£ &(Q), determined up to addition of a function in N, such that
m — 1
(3.21) <«, A*vy = </, vy + x (gj, 7», vv e x.
3=0
Furthermore {/; g0, ..., gw_i}-> u is a continuous linear mapping of the
closed subspace of S'(Q)x[3tf"(r)]m, made up of the elements satisfying
(3.20), into 3'{Q)IN. D
Writing (3.21) for every v £ 2f{Q) (which is contained in X), it follows
that
<«, ^> — </, v}, Vv e ®{Q),
and therefore u satisfies the equation
(3.22) Au = f
in the sense of distributions on Q.
There remains to interpret the boundary conditions which are "contained"
in (3.21); and for this purpose we must have a trace theorem for the
solution u.
Thanks to (3.22), we see that u belongs to a space Y defined in the
following way:
(3.23) Y = {u\ue ®'{Q), Au e 5'{Q)}.
We shall provide Y with the coarsest locally convex topology which
makes the mappings u-^u and w-> Au of Y into Q)'{Q) and S'(Q)
respectively, continuous.
We must therefore give trace theorems for the elements u of Y, as we
shall do in the following sections. Q
Remark 3.2. Theorem 3.1 still holds if, instead of 2'(Q)t we take the
dual Kf{Q) of a space K(Q) satisfying conditions (3.12). Indeed, in order
to prove Theorem 3.1, we only used the first and third "abstract''
properties (3.12) of the space S{Q). D
3.4 Density Theorem
Let us first show
Theorem 3.2. Let Q be an open set satisfying hypothesis (I) of Section 1.1
and let A, given by (1.1), be properly elliptic in Q with apq£ 34?(Q); then
Sf{Q) is dense in the space Y, defined by (3.23).

66 3. Application of Transposition; Existence of Solutions in the Space 3)'{Q)
Proof, Let w-> M(u) be a continuous antilinear form on Y; it may
be written, the intervening spaces being reflexive:
(3.24) M(u) = </, u} + <£, Ati>, with / £ 0(fi), g £ S(fl).
Assume that we have M(<p) = 0, V<p £ ^(-0),' and let us show that it
then follows that we have M(u) = 0, Vu £ Y.
Denote by / and g the extensions of / and g to Rw by zero outside Q;
and let s/ be a linear operator of order 2w with infinitely differentiable
coefficients in Rw and analytic in a neighborhood (9 of £}, which coincides
with the operator A in Q and which is properly elliptic in 0. Then noting
that /£ ^(Rw) and g £ L2(RW), (3.24) may be written
(3.25) Mfo>) = </, 0} + <£,^0> = 0, V<Z> £ ^(Rw),
where the brackets are taken in the sense of distributions on Rw and
cp = restriction of 0 to Q (and therefore y £ S){Q))'} therefore, if jrf*
denotes the (formal) adjoint of si, we have, in the sense of distributions
onR",
(3.26) sf*g = -/.
But then, according to the hypoellipticity of ja/* in 0 (see Theorem 3.2
of Chapter 2), g is infinitely differentiable in 0; and since / is analytic
(since zero!) in the complement of its support, which is a compact set jf
in Q, g is also analytic in (9 — Ctf', thanks to (3.26) and to Corollary 1.1
of Section 1; but since g vanishes in Q — Q, we have: g = 0 in a
neighborhood of r, therefore g £ £2>{Q).
Furthermore, by restriction of (3.26) to Q, we have
(3.27) A*g=—f\nQ.
But then, in (3.24), we have <g, Any = <A*g3 u>, since g£ 9)(£i) and
M[u) = </ + A*g, u} (duality between &{Q) and ®'{Q))
According to (3.27) we therefore have M(u) = 0. D
3.5 Trace Theorem and Green's Formula in Y
Let us return to the study of the mapping v^&v and first show that
it is surjective from X onto [jf?(I1)]™:

3.5 Trace Theorem and Green's Formula in Y
67
Lemma 3.1. Let Q be an open set satisfying (I) and ii), A be given by
(1.1), properly elliptic in Q, with apq£ 34f(Q), Bj be given by (1.2), with
bjh £ Jf (jT), the system {B^J~^ being normal on jT; let L be a fixed positive
constant. Consider the space ffljXT) {defined in Section 3.2 of Chapter 7).
There exists a continuous linear mapping q> = {q>0, ..., (pm-i}-> vL((p) of
WL(r)T into X, such that
(3.28) 7>L((p) =<p., j = 0, ..., m - 1,
(continuous "right-inverse" of <F).
Proof. 1) Let q> = {q>0, ..., g^-i} be given in [J-fL(jT)]w; we need to
construct v£<3t(Q) such that
(3.29) Cjv = 0} Tjv = <pj,j = 0,...,m-1,
(3.30) A*v£@{Q).
Applying Lemma 2.1 and the arguments of Lemma 2.2 of Chapter 2,
using the fact that {Cy, Ty} is a Dirichlet system of order 2m on T with
analytic coefficients on jT, it all comes down to constructing v £ <3{Q)
satisfying (3.30) and
(3.31) yjv =y)j} j = 0, ..., m — 1,
where y>j belongs to J-fM(jT) (M depending on L).
2) Let us then consider the Cauchy problem
\A*u = 0 in a neighbor
17ju = Wj on r, / = 0, ...,
[ghborhood of r,
2m — 1.
By the Cauchy-Kovalewska theorem (jT being compact and all
coefficients being analytic), problem (3.32) admits a unique solution in a
suitable neighborhood of r, this neighborhood depending on M, and
therefore on L, but not on the choice of <pj in JfL(jT). Thus, if we denote
by IQ the set of points of Q such that d(x, r) < q, q sufficiently small, we
see that there exists a q(L)} such that (3.32) admits a unique solution
in Ie{L). '
Let «be a function of 3}{Q) such that oc = 0 in Q — I^L) and
tx(x) = 1 in IQ(L)i2> f°r example; this function obviously exists. Finally,
set
(3.33) v=ocuin IQ[L)} v = 0 in Q — IQ{L).
Then (3.31) holds and therefoie (3.29) holds as well; furthermore v £ Q){Q)
and, in Ie{L)/2, A*v = <xA*u = 0, therefore A*v £ 0(fi).

68 3. Application of Transposition; Existence of Solutions in the Space 3>\Q)
3) The construction of v given in 2) defines, L being fixed, a linear
mapping q)->vL(<p) of [J>fL(r)]m into X (q(L) and oc being determined
byL).
There only remains to show the continuity, but the spaces [«#l(.T')],w
and X being in particular {J£3F) spaces, we may apply the closed graph
theorem. And then it suffices, for example, to note that, by the Cauchy-
Kovalewska theorem, ip = [y>0, ..., y^w-i}-** u = solution of (3.32) is
a continuous mapping of \0M^)fm into C°(IQ{L)). D
We can now prove the trace theorem:
Theorem 3.3. Under the hypotheses of Lemma 3.1, the mapping
w-> Bu = {B0u, ..., Bm_1u] of @(Q) into [@(r)]m extends by continuity
to a continuous linear mapping, still denoted by u^ Bu, of Y into [j^f(r)]m;
furthermore, for u^Y and v G X, we have the "Green formula":
_ m — 1
(3.34) (Au, v} - <u, A*v> = - £ <BjU, 7>>,
j=o
the first bracket denoting the duality between E'(Q) and S(Q), the second
between <2)'(Q) and S){Q) and (BjU, T^v} the duality between 3tf"(D and
Proof. 1) Let u be given in Y and let cp = {(p0, ..., <pm-i} be given in
[34f(r)]m. Then <pe [3>fL(r)]m for a suitable L; we choose v = vL[<p) as
in Lemma 3.1 and introduce
(3.35) Z(vL(<p)) = <u, A*vL(<p)> - (Au,^j>,
the first bracket denoting the duality between 3)'{Q) and Sf{Q) and the
second between 3'{Q) and S(Q) (note that vL(<p) G X C B(Q)).
Let us first verify that Z(yL((p)) does not depend on the "right-
inverse" used, but only on op. Indeed, if v± and v2 are two such "right-
inverses", then x = vi — v2 satisfies the conditions
CjX = 0, TjX = 0, / = 0, ..., m — 1,
and therefore, by Lemma 2.1 of Chapter 2:
7jX = °> j = 0,...,2m — l,
and since A*% G ^(-0), by the uniqueness of the Cauchy problem (3.32),
it follows that % £ ^(-0). But then we have
(u, A*%} = (Au, x) and therefore
Z{v±) = Z(v2).
Thus (3.35) depends only on <p and we may write Z{q>) instead oiZ(vL((p)).

3.5 Trace Theorem and Green's Formula in Y
69
2) The form y-> Z{(p) is antilinear on [Jf (jT)]w. Indeed, let us verify
that Z{cup1 + bcp2) = aZ(cp-^ + bZ (<p2); now, we can find an L such that
(pi and q>2 G ^ikF) and then use expression (3.35) for Z(cp) and the anti-
linearity of Z(cp) follows from the linearity of the mapping q>-> vL(q>).
Let us now show that q>-> Z{q>) is continuous on [Jf (jT)]w. It suffices
to verify that <p->Z{<p) is continuous on [J^L(r)]m, for every fixed L.
But then we use expression (3.35) for Z{cp) and the continuity follows
from Lemma 3.1.
3) Consequently, we have
m — 1
(3.36) Z(q>)= 2 <V#,^->,
where r^G Jf?'(r).
Still using expression (3.35) for Z(q>), it is easily seen that u-^xu =
{r0^ ... Tw_j^} is a linear mapping of Y into [j-f (jT)]w.
Let us now show the continuity of r. It is sufficient to show that,
given a bounded set & in [Jf (jT)]w, there exists a neighborhood of zero in
Y, say t^*, such that
I (Ju> <p} |
m — 1
J7=0
< 1, Vue-f" and 9? £,
But ^ is necessarily a bounded set in [jfL(jT)]w for a suitable Z, (see
Chapter 7, Section 1.2 and 3.2); choose vL(qj) for this L as in Lemma 3.1;
then
(3.37) <xu, cp> = <«, i\(^)) - (Au, vL(<p)>
and vL(g?) belongs to a bounded set of X, therefore of X[v) with suitably
fixed v and therefore, in particular, vL(q>) belongs to a bounded set 081
in Sf{Q) and ^4*?;L(<p) to a bounded set 0&2 in ^(.Q); note that 081 is also
bounded in S(Q). Then, E° denoting the polar set of E, we can take
<r = {u \u€\£l,Au€\£\},
noting that JfJ C S'(Q). And the desired result follows.
4) Now, taking u£ Sf(Q)t we deduce from (3.37) and the Green
formula (3.1) that
m — \
(Tu, £> = 2 J Bjuyj dcr, Vcp G [^(r)]^ and VZ,,
whence
xu = Bu.
Also note that, thanks to Theorem 3.2, r is the extension by
continuity of B.

70 3. Application of Transposition; Existence of Solutions in the Space 3}'{Q)
5) Finally the Green formula (3.34) results from the preceding
considerations: for, if v £ X, then cpj = TjV £ ffl(]T) and we can take vL(qj)=v
in (3.35) and therefore (3.34) foUows from (3.35) and (3.36). D
3.6 The Existence of Solutions in the Space Y
We are now ready to give a more precise interpration of Theorem 3.1.
Indeed, for the solution u obtained in Theorem 3.1, we can write either
formula (3.21) or formula (3.34); but we already know that An = f (see
(3.22)), therefore it follows that
m — 1
2 <Bju-gj,Tjv} = 0,VveX,
and also, thanks to Lemma 3.1,
m—1
2 <b3u - gj, ^> = o, v% e jr{r),
and therefore BjU = gj,- j = 0, ..., m — 1.
We have therefore shown
Theorem 3.4. Under hypotheses i), ii), iii), iv) of Section 3.1, the
operator
(3.38) 0>:u->0>u = {Au, B0u}...} Bm_^u}
defines an (algebraic and topological) isomorphism of YfN onto the space
{S,(Q)x[^,(T)'\m\ N*,&} of elements of S'(Q) X [JT(r)]m, satisfying
(3.20). D
In other words, the boundary value problem
(3.39) Au = fm the sense of 9'(Q),
(3.40) BjU = g3 in the sense of the (trace) Theorem 3.3,
/ = 0,..., m — 1,
admits a solution u £ Y, determined up to addition of a function of N,
for every f£S'{Q) and every gj£2%"(r), satisfying the compatibility
relations (3.20).
In particular, P is an indexed operator in Y and its index is given by
X(P) = dim N — dim N*.
3.7 Continuity of Traces on Surfaces Neighbouring r
In this section we add a complement to the trace theorem of
Section 3.5, along the same lines as Theorem 8.1 of Chapter 2.

3.7 Continuity of Traces on Surfaces Neighbouring r 71
Thus, let {r6}, 0 < q < q0 < 1, be the family of parallel surfaces to
r, tending towards J1 as q-> 0, which was introduced in Section .8.1 of
Chapter 2. In the same notation, we may now assume the homeomorphism
d{ and J{j (i, j = 1, ..., N) interverning in (8.1), Chapter 2, to be
analytic (since now Pis an analytic variety, hypothesis ii) of Section 3.1).
Consequently, we may also assume that the homeomorphism
(3.41) x-+ip[x, q) of re onto T,
defined with the help of ft{ by (8.2) of Section 8.1, Chapter 2, is analytic,
as well as its inverse yr1', also recall that ip and ip~l are bounded, togtther
with each of their derivatives, by constants which are independent of q
(but depending on the order of the derivative).
Still denoting by QQ the open set with boundary re contained in Q,
we may consider the spaces X(Qe) and Y(Qe) to be constructed from QQ
in the same way as the spaces X and Y. For this purpose, we assume,
as we are allowed to do, that the coefficients of the operators {Bj}, {C},
{Sj], {Tj] are defined and analytic not only on r, but in Q — QQq} so
that the systems {Bj}, {C;}, {5;}, {Tj} are normal on re for every q such
that 0 < q < q0.
And we shall set
X{QQ) = {v\ve®{Qe), Cjv\rQ = 0, 7 = 0, ...,w-l, A*ve@{Qe)},
Y(Qe) ={u\ue ®'{QQ), Aue 3'{Qe)},
with topologies analogous to those of X and Y.
The problem is to define Bju\Fe, if possible, for u given in Y, and
then to see if Bju\Fe converges (in a topology which remains to be
defined) to BjU.
But in fact it is not possible to define B3u\Fq for arbitrary u in Y\
indeed, the mapping "restriction of Q to Qe" does not map £'(Q) into
E'(QQ) (the transposed mapping is
q> -> <p = extension of <p from QQ to Q by 0 outside QQ,
and this, mapping does not send 3(Qe) into E(Q)).
This leads us to considerably restrict the class of u's for which we
shall solve the problem. We introduce:
(3.42) yx = Y^Q) ={u\ue 2'{Q)9 Au £ L2(Q) + £'{Q)}
(where $'{Q) = dual of ${Q) = space of distributions having compact
support in Q); in definition (3.42), we have provided L2(Q) + S"(Q) with
the dual topology of L2(Q) A ${Q) and Y1 with the coarsest topology
which makes the mappings u-^u and w-> Au of Yx into ^r(/3) and Yx
into L2(fi) + $f(Q) respectively, continuous.

72 3. Application of Transposition; Existence of Solutions in the Space 3>'{Q)
By the same type of proof as for Theorems 3.2 and 3.3, we verify
that
I ^(^) *s dense in Y1 and we have, for Yv a trace
1 theorem analogous to Theorem 3.3.
We shall now solve the problem for the elements of Y\.
First of all, if u £ Yv then (Au)Qq = restriction of An to QQ} satisfies
(3.44) (Au)QQe L2(QQ) + £'{Q^ for sufficiently small q
and therefore
(3.45) (Au)ae = A(uae)eS'(Qe).
Therefore, in particular, we have uQe £ Y(Qe).
We can therefore define B3u\Fq (element of J-f'(i^,)), by the trace
theorem of Section 3.5 applied to Y(Qe), and then, by transfer of structure
with (3.41), we define
(3.46) B\Q)u = image of Bp \Fq under (3.41).
We have: Bfu ^^"(T). We shall now show:
(3.47) Bfu-> B3u in ^e'[T) as q-> 0, Vm£ Y-^
For this purpose we must show that
(3.48) <B{e,f#-> B,u9 Vj>-> 0 as e-> 0,
uniformly for <^ in a bounded set & of Jf (jT), and therefore (see
Chapter 7) in a bounded set g% of J^L(r) for suitable L.
Thanks to (3.43), we can approximate u in Y1 with a sequence {^v}
in Q){Q). Then we have
<BW« - Bp, Vjy = (Bfu - B<fuvi Vjy + (Bfuv - BjUv, %> +
(3.49) +<BJuv-BJu,<pjy.
Thanks to (3.43), the last bracket in (3.49) goes to zero as v-> + oo,
uniformly for cpj in 38.
The second bracket on the right-hand side of equation (3.49) goes to
zero as q -> 0, uniformly for <^ in 0&, for each fixed v.
We must therefore show that
(3.50) (B\Q)u - B®u0, Vj>-> 0 as v-> + oo,
uniformly for q)j in & and for q in [0, q0[, / = 0, ..., m — 1. After transfer
of structure by (3.41), this amounts to showing that
(3.51) <5i(« - u%) |rc, ?cJ>->0asv-> +oo,

3.7 Continuity of Traces on Surfaces Neighbouring r 73
uniformly for (pQJ in the bounded set 38 Q of J«f (jTJ, the set corresponding
to 38 by (3.41), and for q in [0, q0[, j = 0, ..., m — 1.
But we set
and let v((pe) G X(Qe) be the right-inverse of cpQ, analogue to the one of
Lemma 3.1, but passing from re to QQ.
Green's formula (3.34) applied to Qe yields
m—l
2 <B3(u - «,) 1^, ^ei> = <(« - uv),A*v(fe)y -
(3'52) 3~° ' -<A(u-uv) M^)y,
where on the right-hand side, the first (resp. second) biacket denotes the
antiduality between @'(Qe) and @[Qe) (resp. S'(Qe) and 3{QQ)).
Let wQ = extension of A*v(q>e) to Q by 0 outside Qe; then wQ G @(Q)
and remains in a bounded set of Q){Q) as £-> 0, therefore
<(« - *Ofi * ^*%e)> = <« - «„ V-> 0.
On the other hand:
A{u-uv) ={gv) + (/,) ,
"e "e "e
where gv -> 0 in <T (fi), /„ -> 0 in L2{Q).
The distributions gv have their suppoit in a fixed compact set Jf in
Q\ let 0 G &{&)> 0 = 1 in the neighborhood of Jf; then, /or sufficiently
small q :
where: e
v(<pQ) = extension of v(<pe) by 0 outside Q.
Therefore
(3.53) (A (u - «,) ^)> = <&, 0%J> + / fX^) dx
and it can easily be seen that each term on the right-hand side of (3.53)
goes to zero; and this completes the proof of (3.47). D
Remark 3.3. Of course, the condition "u G Y^', Yt defined by (3.42),
is not the "optimal" condition which guarantees the correctness of (3.47).
We note in fact that the results of this chapter can be extended to the
spaces Sp(Q), 1 < p < oo (constructed as S{Q) but replacing L2 with
LP) and to their duals S'p\Q) = (Sp(Q))f.
Taking An in LP(Q) + $'{Q), 1 < p < oo, we obtain a result which is
analogous to (3.47) (and for 1 < p < 2, this yields a less restrictive
condition than u G Y^. D

74 4. Existence of Solutions in the Space 3f'^k{Q) °f Ultra-Distributions
4. Existence of Solutions in the Space &Mk (Q)
of Ultra-Distributions
4.1 Generalities
The method used in Section 3 also applies to an even more general
setting, if we consider the equation Au = f in the sense of
ultra-distributions.
Let us again assume that the open set Q and the operators A and Bj}
j = 0, ..., m — 1, satisfy hypotheses i), ii), iii), iv) of Section 3.1.
Further, let
(4.1) Mk = (klf, fixed real /? > 1
(or more generally, let {Mk} satisfy (1.6),..., (1.11) and (1.2) of Section 1.1,
Chapter 7; then thanks to (1.10), {Mk} also satisfies (1.3) of Section 1.1,
Chapter 7).
The idea is to use Q)Mk (Q) instead of Sf(Q) (and therefore 3fMk (Q)
instead of &(Q)) in the theory developed in Section 3.
First, we introduce the space
XMk ={v\ve®Mk{Q)', C,-i; = 0, /=0,...,m-l,
(4.2)
A*v£@Mk{Q)},
where Q)Mlz (-0) and @mjc (^) are ^he spaces defined in Chapter 7
(Definitions 1.1 and 1.2), the space XMjc is provided with the inductive limit
topology of the spaces X$k defined as follows:
X{t ={v\ve@m{Q;%{Lp)),Cjv = 0>j = 0>...,m-l,
where {Lv} is an increasing sequence of numbers which tends towards
+ oo, {jf„} is an increasing sequence of compact sets contained in Q and
whose union is Q, %(L) is the function introduced in Corollary 1.1 to
Theorem 1.2, A* and Cj replacing A and Bj} respectively, and where the
spaces @Mk(&'> %(LV)), @Mk (@'> 2^v>Lv) are the Banach spaces defined
in Sections 1.3 and 1.2 of Chapter 7, respectively.
It is then natural to provide X$k with the norm of the graph, which
makes it a Banach space.
Thus XMk is an {££<F)-space (and even better, according to the
properties of the spaces @Mk (Q) and @Mk (Q)\ cf. Chapter 7, Sections 1.2
and 1.3).
We also introduce the space
{@Mk(Q)',N} = t<P \<pe@Mk(Q), J(pwdx = o vweN\
closed subspace of S)Mk (Q).

4.2 The Space SMh(Q) and its Dual 75
Applying Theorem 1.1 and Corollary 1.1 to Theorem 1.2, we obtain
(4.4)
{A* defines an [algebraic and topological)
isomorphism of XMk [N* onto {@Mk (Q); N}.
If, as we have done in Section 3.1, we transpose (4.4) and if we note
that
(4.5) {®Mlc{Q)',N} = ®'Mlc{Q)IN,
we obtain the following Proposition (see (3.10)):
given a continuous antilinear form L on XMk /N, there exists
u£@'Mk(Q), determined up to addition of a function of N,
such that
^ ' <u, A*v} = L[v) Vv' e XMk /N* and v £ v,
the bracket denoting the duality between @'Mjc (Q) and @Mlc {&).
Furthermore u depends continuously on L.
As in Section 3.2, we thus arrive at the problem of choosing the form
L; in particular, we are led to introduce the spaces KMjc (Q) of functions
such that (see (3.11))
(4.7)
XMje C. KM]c (Q) C. L2(Q), with continuous injections,
KMjc (Q) is reflexive,
[ @Mle (Q) is dense in KMjc (Q).
Such spaces always exist, for example L2(Q), since <3(Q) is dense in
L2(Q) and since @Mk (Q) is dense in <2){Q) (see Section 1.2 of Chapter 7).
Our goal is to introduce "the smallest possible" space KMjc (Q) (see the
comments at the beginning of Section 3.2 which are still valid in the
present setting). For this purpose, the next Section introduces a space
SMje (Q) which is analogous to E(Q). D
4.2 The Space SMjt (fi) and its Dual
Again let q(x) be the function introduced in Section 3.2 for the
Definition 3.1 of S{Q). For fixed L > 0, we first define the space
^mjc (-0) = {w | w G S{Q) such that there exists a c,
(4.8) depending on u, such that
2II^d^IIlW<^m,.v^>o};

76 4. Existence of Solutions in the Space @Mk(Q) of Ultra-Distributions
provided with the norm
it is a Banach space.
Next we define
(4.9) SMk(Q)=ind\imSLMk(Q),
L-^+oo
L increasing monotonically to + oo.
Proposition 4.1. The space SMk (Q) is reflexive.
Proof. It is sufficient to note that EMk (Q) may be equivalently
defined as the inductive limit of a monotonically increasing sequence of
Hilbert spaces and to apply the results of Chapter 7, Section 1.2,
Remark 1.3.
Indeed, let us consider the space
(4.io) *~LMk (Q) = L | ue s{Q), f; —j-j 2 \\ek d^III^) < + ool
for fixed L > 0.
Provided with the norm
(411) l£,wh£t'm,e~<T-
it is a Hilbert space.
Thanks to the properties of the sequence {Mk}, it is easily seen that
(4.12) 3Mk(Q) =ind Km *S%k(Q),
n->(x>
where Ln is a positive, monotonically increasing sequence which tends to
+ oo. D
It follows directly from the definitions that
(4-13) ®Mk{Q)CSMk{Q)
and therefore also that
(4-14) XmcSMk(Q). D
Therefore, in order to verify that SMk (Q) satisfies conditions (4.7),
there only remains to show
Proposition 4.2. The space Q)Mk (Q) is dense in EMk (Q).
Proof. Let u be given in 3Mk (Q), therefore
(4.15) u G S^k (Q) for a suitable L (and we may assume L > 1).

4.2 The Space SM]C(Q) and its Dual 77
For the time being, let us assume that
there exists a sequence dv of functions of @(Q),
v = 1, 2, ..., such that
(4.16) \ d'{x) = 1 if d{%'F) ~ V ' dv {%) = ° if d{%' F)-T'
\d(x, r) \k |D" dv(x) | < c2L2Mk for \oc \ = k and k > 0,
c2 = constant.
Then the function dvu 6 @M]e (Q) and Proposition 4.2 will be
demonstrated if we can show that dvu -> u in 3Mle (Q).
More specifically, we shall prove that
(4.17) dvu->u in Sjj+^fi), with arbitrary r] > 0.
Indeed, let
We have
where
2ll(<5,-l)^D««||L2(Q),
(i +
Next, set
H=*-i
Yv = sup y,^, s„ = sup s,|jfc, #„ = sup iT„>ft.
« « «
Then (4.17) is equivalent to Y„-> 0, which will be shown if we can
show that
(4.18) s,->0,
(4.19) ar,->o.
But thanks to the fact that u £ S^fc (12), we have
1
Zv,k<
therefore
(L+ri)*Mh
cLkMk,
zv < sup zv>k + c
0<ft<iV
U+J

78 4. Existence of Solutions in the Space @Mk(Q) of Ultra-Distributions
and this proves (4.18) if we can verify that
zvk->0, for ?-> + oo, fixed k,
which is immediate.
Let us now show (4.19). We have
k
^WVWM*k (/) L'M'L,-'M^VTwk LV" s
— c "^ 77—;—\k (c'' °"' c " suitable constants).
Therefore, for sufficiently great N
-*&r
&v < sup %Vyh + C
0<k<N
and we have the desired result if we show that
2£v k -» 0, for v -> + oo, fixed k.
Now for fixed k this is immediate, since, for |A | > 1, @'A' DA(5„ is bounded
on Q and vanishes except on a set ,QV whose measure tends to zero as
?-> +oo.
Thus there remains to show (4.16). Via "local maps" we are led to
show the existence of a sequence 6v(t) of functions of one variable t in
[0, 1] such that
6v(t) = 1 if t > 2[v, 6v(t) = 0 if * < — , 0 < 0,0) < 1,
v
\te?\t)\<c2i}Mk,vk,
and again, at the risk of having to translate and change v to 2v later, it is
sufficient to show the existence of %v with
(4.20)
*,(*) = ! if t> — , 0<&(*)<1,
V
;^)(0) = o v/,
(4.21) |^«(Q|<c£*M» V*.
To this end, we start with a function q g 2iMlc (R) with compact support
in [0, 1],
Q > 0, jedt = 1, | <><*>(*)1 < c' (y) *f» V*.

4.3 The Space YMjc and the Existence of Solutions in YMjc 79
Next we set
0 if t <0,
&(*) = Mvt)>%(t)
vt if 0 < t < -
1 if * > — ,
and define
Conditions (4.20) hold; and for (4.21) we have
«l
/ %(' - <*) ei*' W d<r < / / ! o(*» (w); v da
= v* J\QW(0)\d0<c'vk
L
Mh
D
therefore, if 0<t< 2/v.
I*****>W | < (yj »-»(y V M», whence (4.21).
"Mfc (^) sna^ denote the (strong) dual of EMk(Q). Thanks to
Proposition 4.2, E'Mk (Q) can be identified with a subspace of @'Mk (Q).
More precisely, we have:
Proposition 4.3. Every element f of E'Mjc (Q) may be represented (non-
uniquely) in the form
(4.22)
/=2 2DV/J>
k=0 \a\=k
with fa e L2(Q) and £ 2 LhMh ||/„||L.(fi, < + oo, VL >0.
4.3 The Space Ymu and the Existence of Solutions in Ymu
Still in analogy with Section 3, we now introduce the space
(4.23) Ym = {u | u e ®'m (fl), Au e S'm (fl)},
provided with the coarsest, locally convex topology such that the
mappings w-> u and w-> Au of YMfc into £^fc (.0) and E'Mk (Q)
respectively, are continuous.
We shall study a trace theorem for the space YMje and show that
the space of traces {BjU}fSQ of the u's£ YMjc is again [j-f (jT)]w (as for
Y). D

80 4. Existence of Solutions in the Space i^lffc(i2) °* ultra-distributions
Indeed, we first show, as for Proposition 3.6, that v-> &v is a
continuous linear mapping of XMjc into \3tf (r)]m (note that the hypotheses
on A and Bj are the same as in Section 3 and that if v £ XMfc, A *v still
vanishes in a neighborhood of jT).
This and the properties of 5Mk (Q) allow us to choose the form L in
(4.6) in the following manner:
m—1
(4.24) L(v) = </, v} + 2 <$, ?>>, / £ 5^ (fi), g. £ Jf'(J1),
the first bracket denoting the duality between B'Mk (Q) and BMk (Q)
and the others between #e'(T) and 3tf{T).
As for (3.22), it then follows that the solution u of (4.6) with L given
by (4.24) satisfies
(4.25) An = f in the sense of Sf'Mk (Q).
It can again be shown that (analogue to Theorem 3.2)
(4.26) 3{Q) is dense in YMk .
Indeed, it is sufficient to reproduce the proof of Theorem 3.2 with
the obvious formal modifications: in (3.24), f^^Mk{^)> S^^mjc(^)>
but (3.25) —(3.27) are again understood in the sense of distributions;
we only have to add that, thanks to (3.27) and to Corollary 1.1 to
Theorem 1.2 (note that since the coefficients apq of A are analytic, they also
belong to Q)Mk (Q)), g belongs not only to Sf{Q) but also to Q)Mk (Q),
and then we have
M{u) = </ + A*g, u} (duality between S'Mh {Q) and S)Mk (Q)). Q
Remark 4.1. With the same type of proof, it can be shown that
@Mk (£) is dense in YMje. D
Next, we again have a lemma analogous to Lemma 3.1, with XMje
replacing X; indeed, we again apply the Cauchy-Kovalewska Theorem
in the same way as in Section 3.5, and the only thing to modify in the
proof is the choice of the function oc in (3.33), now taken in Q)Mlc (Q),
which is always possible according to Section 1.2 of Chapter 7.
We are thus led to the following trace theorem.
Theorem 4.1. Under the hypotheses of Lemma 3.1, the mapping
u-> Bu = {B0u, ..., Bm_1u] of S){Q) into [@(r)]m extends by continuity
to a continuous linear mapping, still denoted by w -> Bu, of YMjc into
[3tf"{r)]m] furthermore, for u £ YMje and v £ XM]e, we have:
m — 1
(4.27) (Au, v> - (u, A*v> = -Z <BjU, 7»,

4.3 The Space YMk and the Existence of Solutions in YMjc 81
where the first bracket denotes the duality between E'Mk (Q) and SMk (Q),
the second between <3'Mk{Q) and S)Mk{Q), and (BjU, T3v} the duality
between #e'{T) and #e{T).
The proof is completely analogous to the one of Theorem 3.3. D
Finally, still with the same arguments as in Section 3.6, we obtain the
following existence theorem.
Theorem 4.2. Under the hypotheses i), ii), iii), iv) of Section 3.1, the
operator &, given by (3.38), defines an isomorphism of YMk[N onto the
space {S'Mk{Q)x[3tf"{r)]m; N*, &} of elements of E'Mk{Q)X[3tf\r)]m
satisfying (3.20). D
Remark 4.2. For all results obtained in this Section, we can take,
instead of S'Mk (Q), a "general" space K'Mk (Q), dual of the space KMk (Q)
satisfying (4.7); indeed only properties (4.7) intervened in the proofs;
the space Ymjc is then defined by
ym,c = {« I« e ®'Mk (Q), Au e K'Mk (Q)}. q
Remark 4.3. Assume that we do not have the optimal regularity
theorem in the classes {Mk} up to the boundary of Q (Theorem 1.2), but
only1 regularity in the interior of Q in the classes {Mk} and analytic
regularity up to the boundary of Q: more precisely, we only have
Theorem 1.1, Corollary 1.1 with Mk = k\ and Corollary 2.1 with arbitrary
Mk. Then it is still possible to develop the preceding theory; we only need
to adapt the definition of the space XMk by taking:
xm ={v\ve@(Q), CjV = 09 /=0,...,w-l, A*v£0Mk{Q)}.
If K%k (Q) is a space satisfying (4.7) (but with X%k instead of the space
XMk), we replace the space Y-Mk with
Y*Mk = {« | « e ®'m (Q), Au e Kti (Q)} {Kti (Q) = dual of K*b (Q));
to obtain the following result:
& is an isomorphism of Y%k jN onto the space {K^k (Q) X [Jf '(r)]m;
N*, &} of elements of K%k (Q) X [Jff\r)]m satisfying (3.20). D
Remark 4.4. The preceding Remark and Remark 2.8 of Section 2.6
show the possibility of a new generalization of the results of this Section
to spaces of ultra-distributions &'Mk (Q) of Beurling type (see Section 2.3,
Chapter 7).
Still under hypotheses i), ii), iii), iv), set
Xmk = & \ve®{Q), Cjv = 0, / = 0, ..., m - 1, A*vtaMk (Q)}
A situation of this type appears in Remark 4.4.

82 4. Existence of Solutions in the Space @Mk{&) of Ultra-Distributions
and
where K'Mk (Q) is the dual of a space i£Mfc (.0) satisfying the relations
(see (4.7))
XMk C_ KMk (Q) d L2(Q) with continuous injections,
KMk (Q) is reflexive,
I @Mk (Q) is dense in KMk {Q).
Then we still have a trace theorem analogous to Theorems 3.3 and 4.1
with the same trace space and we still have the result:
(4.28)
the operator 0* is an isomorphism of YMk jN
onto the space {K'M]e {Q) X [Mr'{r)]m; N*, 0}
of elements of K'Mfh{Q) X [Jf?'(.T)]m satisfying (3.20). Q
4.4 Application to the Regularity in the Interior
of Ultra-Distribution Solutions of the Equation Au =f
In the preceding Sections, we have seen that the different spaces Y,
Ymic YMk have the same space "of traces" (see Theorems 3.3 and 4.1,
Remark 4.4). Now this leads to an interesting consequence on the
regularity in the interior of Q of solutions of the equation Au = /, in the
sense of ultra-distributions. Indeed, we have
Corollary 4.1. Let Q be a bounded open set in Rn with boundary T,
an (n — 1)-dimensional variety, Q being locally on one side of r, and A a
properly elliptic linear operator on Q, with analytic coefficients in Q\ then
every ultra-distribution u of @'Mk (Q) or of &'Mk («0) which is a solution of
Au = /, with [for example) /£ L2(U), is a distribution of @f'(Q).
Proof. Taking into account the fact that the Dirichlet conditions
{7ju}T=o cover all properly elliptic operators (Remark 1.3, Chapter 2),
we may apply the (trace) Theorem 4.1 (or Remark 4.4) and we see that u
admits a trace yu = {g0, ..., gm_^ with g3 £ 3^f[F); and of course / and
go> -•> gm-i satisfy the compatibility conditions (3.20) of the Dirichlet
problem
(4.29) Au = f,yju = gJfj = 0,...,m — 1.
Therefore thanks to Theorem 3.4, problem (4.29) admits at least one
solution w £ Q)'(Q); then the difference u — w, also taking into account

5. Comments
83
the fact that 9'{Q) C @'Mk («Q) (or £'Mk [Q)), satisfies the conditions:
u-we&Mk(Q) (or@'Mk(Q)),
A(u — w) = 0, yj(u — w) — 0, / = 0, ..., m — 1.,
and therefore, thanks to Theorem 4.2 (or to (4.28)), u — w£N and
therefore ue@' (Q). D
This Corollary thus reduces the problem of the regularity in the interior
of Q, Q now being an arbitrary open set in Rw, of solutions of a properly
elliptic equation in the sense of ultra-distributions of Gevrey or of Beurling
type to that of ordinary distributions; it follows for example, also applying
Theorem 3.2 of Chapter 2 and Corollary 1.1 of this Chapter, that if f is
analytic in Q, then u is also analytic in Q. D
5. Comments
Analyticity "in the interior" of solutions of linear elliptic equations
with analytic coefficients (Corollary 2.1 with (3 = 1) is a classical result,
shown by E. Picard and S. Bernstein for second-order equations and by
Petrowski [1] for equations of arbitrary order; for other proofs, see also
F. John [1], Morrey-Nirenberg [1]. For operators with constant
coefficients this property is characteristic for elliptic operators (see Petrowski
[i])-
Analyticity up to the boundary of solutions of elliptic boundary value
problems (Corollary 1.1 with f} = 1) is due to Morrey-Nirenberg [1].
For the extensions to Gevrey spaces, either "in the interior" or "at
the boundary" (Corollaries 1.1 and 2.1 with f} = 1 or {Mk} arbitrary), see
A. Friedman [2, 3], Murthy [1]. For Beurling spaces, regularity in the
interior has been shown by Bjorck [1] and O. John [1].
The theorem on elliptic iterates was first considered for the analytic
case and in the interior of the domain (Theorem 2.4 with f} = 1);
particular cases have been studied by Aronszajn [1], Nelson [1], Komatsu
[1, 3], the general result being given by Kotake-Narashiman [1]. The
extension to spaces of class Mk and up to the boundary in the general
form of Theorem 1.2 is shown here for the first time; the method of
proof takes its inspiration from that of Morrey-Nirenberg [1] and we also
use techniques of Kotake-Narashiman [1]; for the case of Dirichlet
boundary conditions, see Lions-Magenes [3,4] (see also Roumieu [2],
Theorem 5, where the theorem in the interior is shown for the Laplace
operator). For result (2.7) (theorem on elliptic iterates in the interior in
Beurling spaces), see O. John [1].
For a more abstract point of view ("analytic domination" of operators
of a Hilbert or Banach space), see Nelson [1], Goodman [1].

84
5. Comments
Let us finally call attention to the numerous generalizations of
regularity results in analytic or Gevrey classes, which have recently been
obtained for operators which are "close" to elliptic operators (but which
we have not studied in this text):
a) elliptic differential systems, see Morrey-Nirenberg [1], Morrey [1],
Petrowski [1], A. Friedman [2];
b) hypoelliptic, and in particular quasi-elliptic, differential operators
and hypoelliptic convolution operators, see Hormander [1, 2], Friberg
[1], Pini [1, 2], Volevic [1], Cavallucci [1, 2], Matsuzawa [1, 2, 3], A.
Friedman [1], Shilov [1], ... (see also Chapter 10); in particular, let us call
attention to the work of Matsuzawa [1, 2], who generalizes Theorem 1.2
on elliptic iterates to quasi-elliptic operators;
c) pseudo-differential operators, see Hormander [3], Boutet de Monvel
[1, 2], Boutet de Monvel-Kree [1], Kree [2], ...
The results of Section 3 (application of transposition and
non-homogeneous boundary value problems in !3'{Q)) are given by the authors in
[1]; for the generalization to the spaces @'Mk (Q) given in Section 4,
see Lions-Magenes [5] (and also Magenes [3, 4]). The property of
continuity of traces on surfaces neighboiing r (Section 3.7) takes its pattern
from the formulation of boundary conditions given by Cimmino [1, 2],
who characterizes the "traces" on r of harmonic functions in Q by
different methods; for the Dirichlet problem for the Laplace operator,
see also Johnson [1, 2], Simon [1], J. Douglas [1].
The "traces" of solutions of the homogeneous equation Au = 0 and
the boundary value problems Au = 0, BjU = gj, have been studied
recently, within the framework of the hyperfunctions of Sato, by Ko-
matsu [4] for equations with constant coefficients and by Shapira [4, 5]
for equations with variable coefficients.
A generalization of the theory of non-homogeneous boundary value
problems in the space !3f(Q), when the boundary of Q is not analytic
and may even be highly irregular, is given by D. G. Schaeffer [1].
We also call attention to the work of Kree [1], who, by use of pseudo-
differential operators, studies non-homogeneous boundary value problems
with distribution boundary data on r.
Non-homogeneous boundary value problems for analytic pseudo-
differential operators have been studied by Boutet de Monvel [1, 2].
Concerning regularity in the interior of ultra-distribution solutions
of elliptic equations (Section 4.4), see Lions-Magenes [5] (and also
Magenes [3, 4]) for Corollary 4.1; by more direct methods, without using
the boundary value problems, the problem has been studied for operators
with constant coefficients by: Chou [1] for Gevrey ultra-distributions,
Bjorck [1] for Beurling ultra-distributions, Bengel [2, 3], Haivey [1],
Komatsu [2, 4] for Sato hyper functions, Silva [3] for Silva ultra-distri-

6. Problems
85
butions; Boutet de Monvel and Kree [1] have solved the problem for
operators with variable coefficients and also for certain
pseudo-differential operators. Recently Schapira [4], using the boundary value problems,
has also shown the regularity in the interior of hyperfunction solutions
of elliptic equations with analytic coefficients.
An application of the results of Section 3 to singular perturbations
has been given by D. Huet [1].
Let us finally point out a different approach to the problems studied
in this Chapter, which is due to Baouendi and Geymonat [1]: viewed as
an operator of Sf{Q) ->- 3}{Q), A is evidently not an isomorphism; but it
can be completed with an operator K of (jtf (r))m^- @(Q) so that the
operator {A, K} defined by
{A,K} {u,<p} =Au+K<p
is an isomorphism of 0(fl)x(J?(J1))1*-* S){Q).
The isomorphism {A, K] is then transposed and the problem is to
interpret the transposed isomorphism. If one would Hke to obtain a
concrete interpretation (with trace theorems) of this transposed
isomorphism, then one must of course introduce spaces analogous to the
spaces K of Section 3.2; for details, see Baouendi and Geymonat, loc. cit.
6. Problems
6.1. As we have noted in Remark 2.6, is it possible to suppress
conditions (1.9) —(1.11) on the sequence {Mk} in the hypotheses of
Theorem 1.2?
6.2. In what way can Theorem 1.2 on elliptic iterates be extended to
the case Mk = (kl)P, with 0 < /? < 1? This problem is also related to
the analytic characterization of the spaces D(^4°°; Mk), with for example
Mh = k\, which we shall discuss in Chapters 9, Section 7 and 10,
Section 3; for the one-dimensional case, see Lions-Magenes [2], Section 7.
This problem is also related to polyharmonic functions in the sense of
Aronszajn [1, 2].
6.3. Generalization up to the boundary of the result of 0. John [1]
in Beurling spaces, see Remark 2.7.
6.4. Examples of spaces of the type K(Q), KM]e (Q) (see (3.12) and
(4.7)), different from S(Q) and EMle (Q), also as a function of the given
boundary operators [B3]. It would also be of interest to study the union
of the spaces K'(Q) for all "admissible" K(Q) (see Baouendi and
Geymonat [1]).
6.5. Is it possible to interpolate between the results of Section 1
and those of Sections 3 and 4 ? For interpolation between spaces of

86
6. Problems
infinitely differentiable functions and spaces of analytic functions, see
C. Goulaouic [1, 2] and Remark 2.9.
6.6. Non-homogeneous boundary value problems in unbounded open
sets, for example in Q = R+. This topic offers the opportunity to develop
a very interesting entire chapter (see Schapira [5], and Manaresi [1] for a
very particular case).
6.1. Non-homogeneous boundary value problems in spaces of ultra-
distributions different from those considered in this text, for example,
for the hyperfunctions of Sato with / 4= 0.
6.8. Can the normality condition on the operators {Bj} be eliminated ?
(certain results are already given in Kree [1], Boutet de Monvel [1] and
Komatsu [4]).
6.9. Generalization of this Chapter's theory to elliptic systems (see
Problem 2.1 of Chapter 2).
6.10. The case of open set's whose boundaries are varieties of
dimension less than n — 1 (see Problem 2.9 of Chapter 2).
6.11. The study of transmission problems in spaces of
ultra-distributions.
6.12. To which classes of elliptic operators which degenerate or have
singularities on the boundary or in the interior can the theory of this
chapter be extended ? For results related to this type of operators, in Sobolev
spaces, see Baouendi [1], Baouendi-Goulaouic [1], Derridj [1], Oleinik
[1], Shimakura [1, 2], Vishik [1] and Zuili [1].
6.13. Let A be an elliptic operator of order 2m with C°°-coefficients
but not of class {Mk), the boundary r being only of class C°°. What can
be said about the functions u such that (1.17), (1.18) hold? If Mk = k\,
is the class of these functions quasi-analytic ? For one-dimensional cases,
questions of this type have been studied by M. K. Fage [1] and V. G.
Kriptun [1].

Chapter 9
Evolution Equations in Spaces of Distributions
and Ultra-Distributions
This Chapter makes use of the vector-valued distributions and ultra-
distributions of Chapter 7. From the point of view of evolution boundary
value problems, we assume the knowledge of the essentials of Chapter 3
and of the beginning of Chapter 4.
The aim of this Chapter is to see whether the solutions of linear
evolution equations belong to M^-classes, if the data belong to Mk-
classes. (Once in possession of such results, by transposition and trace
theorems, we "automatically" obtain well-posed non-homogeneous
problems in spaces of ultra-distributions.)
In order to perform this study, we reconsider the various methods for
evolution problems:
variational (or energy) methods (Sections 1, 2, 4, 5, 6, 9, 10), the Laplace
transform method (Section 8), the semi-group method, in the usual sense
(Section 7) or in the sense of distributions and ultra-distributions
(Section 11), the method of transmutations (Section 3), which allows for
the study of problems with singularities in M^-classes.
1. Regularity Results. Equations of the First Order in t
1.1 Orientation and Notation
We first consider the variational setting corresponding to Chapter 3
(Volume 1) and give regularity in t results and, in particular, Mk-regularity
in t results.
Let V and H be two Hilbert spaces on C, with
(1.1) V(ZH, V is dense in H, the injection V^H being continuous.
Identify H with its antidual so that, if V denotes the antidual of V,
we have
(1.2) VCHCV.

88 1. Regularity Results. Equations of the First Order in t
We denote by || ||, | | and || ||* the norms in V, H and V respectively.
In case of ambiguity || \\x shall denote the norm in X. D
Let
(1.3) u,v->a(t;u,v), t£R,
be a family of sesquilinear forms which are continuous on VxY and
such that we always have
(1.4) Vu, v£V, the function t->a(t; u, v) is infinitely differentiate.
We also assume (coerciveness):
(1.5)
for all T <oo, there exist X(T) = X and
oc(T) = (x > 0, such that
Re a{t\ v,v) + A \v\2>a \\v\\2, V v G V and t < T.
Let A(t)e^{V] V') be the operator defined by the triplet
{a(t; u, v), V, H] (see Chapter 2, Section 9); thus
(1.6) a(i;u,v) =(A(t)u,v), Vu,veV,
where (/, v) denotes the scalar product (linear in /, antilinear in v) between
/ G V and v G V.
We aim to study the ^-regularity properties of the equation
(1.7) A®u+^ = f,
where the supports of / and u in t are bounded on the left.
1.2 Regularity in the Spaces ^+
In general, X being (for example) a Banach space, we denote by
^+(R; X) the space of functions
t^cp{t)
which are infinitely differentiable mappings of R -> X, and whose support
is bounded on the left, i.e.
(p(t) = 0 for t < ty, 1^ depending on op.
If we set
(1.8) @a([a,oo[;X) = {<p\<pe@+(R;X), <p{t) = 0 if t < a},
provided with the topology of uniform convergence in X on every
compact set [a, T], variable T, of the functions and each of their deri-

1.2 Regularity in the Spaces @+
89
vatives, then
(1.9) 0+(R; X) = ind lim #.([«, oo [; X)
<Z—>—OO
(see Remark 4.3, Chapter 7.) D
Our first regularity in t result is
Theorem 1.1. Assume that hypotheses (1.4) and (1.5) ^o/i. 77^ £/&£
d^
u-> A(l) u + —
d£
w aw isomorphism of ^+(R; F) -^ ^+(R; F').
d^
Proof. 1) It is immediate that w ->- A (t) u + —- is a continuous
at
linear mapping of ^+(R; F)^^+(R; F)'. Therefore we only need to
show that if in (1.7) the function / belongs to ^+(R; V) then u belongs
to ^+(R; V) and depends continuously on /.
We assume that /£ ^o([0> °°[i V) (which is not restrictive, since 0
plays no special role on R$). We shall show, and this will be sufficient,
that u £ ^0([U, oo[ \V) and depends continuously on / in this space.
Now / belongs, in particular, to L2(0, T] V), arbitrary finite T;
therefore, according to Chapter 3 (Volume 1), there exists a unique u
which is a solution of (1.7) and satisfies
(l.io) ue l2(o, T\ v), u' =^el2(o, t- v), «(0) = o,
at
and
(1-H) \\U\\l>(0}T;V) < C ll/|lL"(0,r;n
(where c depends on T).
Since u(0) = 0, we may identify u defined for t > 0 with its extension
by 0 for t < 0, equation (1.7) being valid on t > 0 or on R.
2) Now we apply the method of differential quotients (which we have
already used previously on several occasions). In general, we set
(1.12) gh(t)=j[g(f)-g(t-h)l h>0;
it follows from (1.7) (valid on R) that
d
(1.13) A{t-h) u{t-h) + —-u{t-h) =f{t-h),
at
hence that
(1.14) A (t) uh+u'h=fh- Ah(t) u(t - h),

90 1. Regularity Results. Equations of the First Order in t
and (1.11) applies to (1.14) (considered as an equation in %); it follows
that
(1.15) |K <c\\fhLH0,T;V')+C\\AhU(t-h) \\tf{0,T',V) *
But the c's denoting various constants, it then follows from the fact
that fe %([0, oo[; V) and from (1.4) that
!I/JIz,2(0,:T;F') ^ C 11/ \\L*{0,T;V')>
\\Mt)y(v;vi<o\\Af(t)yiV;V1>
so that (1.15) yields:
n 1$) J II%IIl2(0;t;F) < c>
1 c = constant, independent of h.
Thus % remains in a bounded set of X2(0, T; V) as A-> 0, and since
%-> m' in 0'(]O, T[; V), it follows thatw' £ £2(0, T; 7) and that (passing
to the limit in (1.14)):
(1.17) A{t)u' +A'{t)u+u" =f} m
equality on R (or on {t > 0}, noting that u'(0) = 0).
Furthermore
(1.18) \\u'
\\L2(0,T;V)
Of course we can iterate on this procedure. It follows that
u e %([0, °°[; V) and that, for all k, we have
(1.19) «<*+1> + 2 f^U*"^) u®(t) = /<*>(*)
i=o V /
I where 99^ = —-99).
Since finally we have
(1.20) \\U^\\L,{0}T;V) < ^r(||/||L2(0,r.n + - + ||/(A/)||L2(0,T;n)^
we see that
/-> w
is a continuous mapping of %([0, oo[; 7') -> ^0([0, oo[; 7) <™. D
(d)) Indeed (see Remark 4.3, Chapter 7), %( [0, 00 [; X) = proj lim @0( [0, T]; -X"),
and we can provide ^0([0, T]; X) with the norms 2 | [ 9?^*) | |z,a(o, r;JK") -
i=0

1.3 Regularity in the Spaces @+ Mk 91
1.3 Regularity in the Spaces @+fMk
We now make the hypothesis
(1.21) t->A[f) belongs to the class #Mfc (R; JS?(F; V')),
SMlc (R; F) being defined in Chapter 7.
The sequence Mk satisfies the usual hypotheses (see (1.1), ..., (1.4),
Chapter 7), but we recall them here:
(1.22) the sequence Mk is logarithmically convex,
(1.23) the sequence Mk is non-quasi-analytic:
(1.24) Mk+,<HkMk Vft,
and
(1.25) (k\Mk_jMj<c^Mk Vk and Vj with 0<j<k.
We shall prove the theorem on Mk-regularity in t:
Theorem 1.2. Assume that a(t]u,v) satisfies (1.5) and (1.21) and that
the sequence {Mk} satisfies (1.22), ..., (1.25). Then the mapping
(1.26) «->«' + A(t)u
is an isomorphism of ^+>Mfc(R; V) onto ^+>Mfc(R; V).
Proof. 1) We shall first show (and this is the essential point) that for
arbitrary finite T, if fe®OtMk{[0,T],&',V') (see (4.6), Chapter 7),
then u£ ^o,Mfc([0» T], B;V) (for a suitable B which we shall specify),
where u is the solution of (1.7).
Let us make this precise.
We are led (by the change of variable u -> e*'u, which does not affect
the eventual property of belonging to a class {Mk}) to the case where
(1.27) 'Rea(t;v,v)>oc\\vf, Vv £ V, Vie [0, T].
Generally, for q> £ L2(0, T] V) (resp. cp £ L2(0, T\ H), resp.
<p£L2(0, T\V'))t we set
(1.28)
vM =
j v0{<p) =
v-l(v)
= (/llvWII2
\o
= (/W
\o
= (/llvW
\l/2
\l/2
d* ,
\l/2
II d*j

92 1. Regularity Results. Equations of the First Order in t
By hypothesis (and according to Chapter 7, Section 5.5):
(1.29) v_1{fk))<d^kMk Vft.
According to (1.21):
(1.30) \\A{k\t) yiV;V,} < cLkMk Vk, Vt e [0, T].
We shall show that
(1.31) v1{u^))<6BkMk Vk,
with
(1.32) d= — , 5 = max/7l+—Ml,^].
It follows from (1.7) that (see Chapter 3)
t 1 T
j Re a (t; u(t), u(t)) di -{ \u(T) |2 = Re / (/(*), u{t)) &t
o 2 0
from which
ocv^u)2 <Vi(w) v_](/)
and therefore
(1.33) vM<-V-i{f)-
(X
Thus (1.31) holds for k = 0. Let us assume that it holds up to (k — 1).
Note that (1.19) may be written
(i.34) 4~u{k) + AW u(k) = /(ft)W - 2 (* W*_i)(*) «tf,(*)
so that, applying (1.33) to equation (1.34), we obtain:
(1.35) „><*>) <—*_!(/«*>) +- Z (k)v-Mlh-^
* * y-o V /
According to (1.29), we have
— »_, (/W) < — ^Mh < (by (1.32)) -A BX •
Therefore in order to obtain (1.31), we must show that
(1.36) i = - 21(*)»'-i(^,*-fl«W,)^4jB*M*-
oc j=0 \j j 6
But
v_^k~j)u{j)) < (according to (1.30)) cLk-jMh_.v_^) <
< (by the induction hypothesis) c dlS-'B'M^Mj,
>u{j))

1.3 Regularity in the Spaces @+}M]c 93
and therefore
*=4J'(*)*,-^-'b'<
so that
(1.37)
Now
cc d k_1
< (according to (1.25)) -±-Mk 2 Lk~jBj,
(1.36) holds if
L
and (1.37) foUows from B>ll + —-1)L.
2) We recall that the space @0)Mk ([0, T], B] V) (for example) is a
Banach space with the norm
sup—r— vAu{k)),
so that, by the closed graph theorem (or by direct inspection of the
estimates in 1)) the mapping
/-> u
is continuous from
%,m* (t°. 7]. iJ F')"> %,Mk ([0, T], B; F).
By passage to the inductive limit in B and then in L, we see that f->u
is a continuous mapping of
%,Mkd.°.Ty>v')^®o,Mk([0,T];V).
By passage to the projective limit in T, we obtain the continuity from
%,uu ([0. + °°[; n "► ^o,m* ([0, + oo[; V);
as we have already noted, 0 plays no particular role on R^; therefore if
f€@a,Mk ila> + oo[; V')> tnen w G ®a,Mk {[<*,+°°[l V) and the mapping
/-> u is continuous in these spaces. By passage to the inductive limit in
a (-> — oo), we finally obtain the continuity of the mapping /-> w from
^+,M,(R;n->^+)Mfc(R;n. D

94 1. Regularity Results. Equations of the First Order in t
1.4 Regularity in Beurling Spaces
We define (see Chapter 7, Section 5.4) the Beurling space
(1.38)
&+ Mk (R; X) = ind km /proj lim/proj Urn 9aM]c (j>, J], L; X)\) .
«-»-<» \ &-»+oo \ L-»0 ' //
A natural question to ask is whether Theorem 1.2 holds in
Beurling classes; i.e.: for f belonging to &+ Mk(R]V), do we have
^^+,Mfc(R;T0? D
We must of course modify the hypotheses on a(t] u, v). We do not
know the "optimal" hypotheses under which the desired result is true;
we shall only give sufficient conditions.
Let us introduce a second sequence {M*} with:
(1.39) Mft* satisfies hypotheses analogous to (1.22), (1.23) and (1.24).
Next we assume that the sequences {Mk} and {M*} satisfy
(*) Mt_jMj < ehMh, Vk, 0 < / < * - 1,
(1.40)
sk -> 0 if k-^oo.
We shall prove
Theorem 1.3. Assume that the sequences {Mk} and {M*} satisfy (1.22),
(1.23), (1.24), (1.39) and (1.40). Let a(t] u} v) satisfy (1.5) and
(1.41)
for every compact set K in R, VL, there exists
a c such that
UWm^v;v)<cLkMt, VteK, Vk.
Then u->A{f)u-\-u' is an isomorphism of <%+ Mlc (R; V) onto
a+,Mk<R;V).
Proof. We assume that we have reduced the problem to the case for
which (1.27) holds. We see (as in the proof of Theorem 1.2) that it is
sufficient to show: under the hypotheses of the theorem, and B being a
fixed positive number, there exists d such that
(1.42) Vl{uw) <dBkMk, Vk.
Now, by hypothesis, there exist c and d such that
(1.43) v_,(fk))<dBkMk, Vk,
and
(1.44) \\AW(t) ||^(F;n < c (jj M* Vk, t € [0, T].

1.5 First Applications 95
According to (1.40), we can find k0 such that
(1.45) {^\Mt_jMj <^Mk for k > k0, 0 < / < k - 1.
We shall show (1.42) with
(1.46) d = max
1 , ,«K U
max —r^rvi(u)> —
,L = B/2.
Of course, according to (1.46), (1.42) holds for / < k0. Let us assume
that it holds up to k — 1 (k — 1 > k0) and show that it holds for k. As
in the proof of Theorem 1.2, we have:
(1.47) vMk)) < -"-i(/W) + - k£(k-Wi(^"
"«">)
But according to (1.43),
1 . .,m. d
-v_!&*>) < - B»Mh < (by (1.46)) — BkMk,
OC OC a
and thus (1.42) holds if
(1-48) - *S (flv^-Wi) <i.B*Mft.
a J=0 X! J &
But
- 2 (fW-^-^) < (by (i.44))- *S (-)i*-^V-i(«w) <
< (by induction) —(5 2 • )Lk-jBjMt_jMj <
<* j=o W
< (since A = —) A £*M,,
""\ £2/2 *
whence (1.48). D
1.5 First Applications
Let H = L2(Q) and let V be a closed vector subspace of Hm(Q)
(notation of Chapter 1, Volume 1) with
(1.49)
H%{Q)CVCHm(Q).

96 1. Regularity Results. Equations of the First Order in t
Let aPyq(%, t) be given functions with \p\, \q | < m, x G Q, t G R,
satisfying
(1.50) *-> aM(., *) belong to SMk (R; L~(fi)).
For u, v£V, set
(1.51) a(t\ u,v) = £ f ap,q(x>l) D% D^ d#,
\p\,\g\<m Q
and assume (1.5) to hold.
From (1.50), it follows that for every compact set [t0, t±] C_ R, there
exist c and L such that
II3*
and consequently
|(AW(t) u,v)\< cLkMk £ HD?«IIlW ||D*t;||l.(0) < c£*M» ||«|| ||v||
\p\,\g\<m
(where || || denotes the norm in V), so that (1.21) holds.
Remark 1.1. If we assume that
(1.52) t-+ aPtq{., t) belongs to <%M%(R; L°°{Q)),
then (1.41) holds. D
Therefore, under hypothesis (1.50) {resp. (1.52)), Theorem 1.2 (resp.
1.3) AoWs.
Remark 1.2. Theorems 1.2 and 1.3 yield results on regularity (in
{Mk} classes) in the variable t. With hypotheses (1.49), (1.50), alone, there
are no regularity results in the variable % £ Q; see Chapter 4 and
Remark 1.4 below. D
Remark 1.3. Generally, for the applications, one takes "H = L2(Q)".
We could also consider other spaces, as for example H = H\(Q) (see
Chapter 3, Section 4.7.5). Thus Theorem 1.2, together with the remarks
of Chapter 3, Section 4.7.5, yields the following result: let / be given in
@+fMk (Ho(®)) and let u t>e the solution, with support in t bounded on
the left, of
(1.53) ^+A«a = /f
dAu
(1.54) fi = 0,-—=0 on TxR,;
on
then
< cLkMk V&, Vt G p0, *i] >
L°°(D)
(1.55)
ue®+>Mk(V),

(1.56)
1.5 First Applications 97
where
V = lv \veHl(Q), ^-Av e L2{Q), i = 1, ..., n\ . D
Remark 1.4. Assume that we have a result on the regularity in the
variable x of the following type:
iifeHs(Q), integer s > 0((1)) and if w(t) is the solution in V of
a(t',w[t),v) = (/,«), Vve F((2)),
then w(t) e Hs+2m(Q) A V and
I ll«'WllH*+2*(fi,<j85||/||J,.(fi), teio.T].
This hypothesis is satisfied if the coefficients ap>q and the boundary
r of Q are sufficiently regular and if V is defined by "regular" differential
boundary conditions (see Chapter 2, Section 9.7). We also assume that
(1.57) t->A(f) belongs to £Mlc (R; &(Hs+2m{Q); HS{Q))).
Then we have
Theorem 1.4. Assume that A(t) satisfies (1.5), (1.56) and (1.57) and
that the sequence {Mk} satisfies (1.22),..., (1.25). Then the solution u,
with support in t bounded on the left, of
(1.58) A(t)u+u' = f, feS>+:MjHs(Q)),
of which we already know that it belongs to @+>Mk (V) (according to
Theorem 1.2), belongs to S)+tMk (Hs+2m(Q)). Furthermore
f->u is a continuous mapping of
(1.59)
®+tMjH<(Q))->®+>Mk(Hs+2™(Q)).
Proof. 1) Let g £ %fM]c ([0, T]; HS(Q)) and w(t) be the solution of
(1.60) a(t\ w{t), v) = (g(t), v), t < T.
Then
(1.61) w£%tMk{[0,T];H>+*m{Q)).
Indeed, we first show (by the method of differential quotients, as in
the proof of Theorem 1.1) that t->w(f) is an infinitely differentiate
((*)) One could also take non-integer s, and, in certain cases, s < 0 (see
Chapter 2).
((2)) vVe assume that we have reduced the problem to the case (1.27) on the
compact set, say [0, T], on which t varies.

98 2. Equations of the Second Order in t
mapping of t < T-> V (or Hs+2m{Q)) and that:
(1.62) a{t'} w™{t), v) = (gW(0, v) - "£ (k)a<k->\t; w«\t), v).
y=o W/
(1.61) follows via estimates analogous to those made in the proof of
Theorem 1.2.
2) Let u, in S)+tMle (Ri V), be a solution of (1.5.8). Since then
«' € ^+,Mfc (R; *0, it follows that
(1.63) A(t)u = f-u'e ®+>Mk (R; #mi^>(fi)),
and applying (1.61) (with s replaced by min(s, m), as the result still holds
for this case) it follows that
ue%}Mk{[0,T];H™^+2™(Q)),
therefore u' belongs to the same space and (1.63) yields
f-u>e%tMk{[0,T];H™^s>»\Q))
and therefore
" € %,Mk ([0, T]; Hm^s>^+2m(Q))
and so on, until we obtain the desired result.
2. Equations of the Second Order in t
2.1 Statement of the main Results
The notation is the same as in the preceding Section. Again we assume
that
(2-1) a(t',u,v)=a(t;u,v) Vu,v£V.
We shall prove the following results:
Theorem 2.1. Assume that hypotheses (1.4), (1.5) and (2.1) hold. Then
the operator
u->A(t)u + —
is an isomorphism of <3+(R', V) onto ^+(R; V).
Theorem 2.2. Assume that hypotheses (1.5), (2.1) and (1.21) hold and
that the sequence {Mk} satisfies (1.22), ..., (1.25). Then the operator
d2u
u^A(t)u + —
is an isomorphism of @+tMk (Ri V) onio @+,Mk (RJ V)

2.2 Proof of Theorem 2.1
99
These theorems will be proved in Sections 2.2 and 2.3.
Remark 2.1. In each of the above theorems, (2.1) may be replaced
by a hypothesis of the type: "the principal part" of a(t; u, v) is hermitian,
see Lions [5]. Q
Remark 2.2. We could also consider the operator given by:
d2u *
(2.2) A(t) u + — + J N(t, a) u(a) da.
at 0
If we assume that
ty <j-*N(t,o)
is of class Mk in both variables t,o,o < t, with values in J£(V; V), then
Theorem 2.2 is still valid for the operator defined in (2.2). Theorem 2.1
holds if N{t, a) is of class C°°. D
2.2 Proof of Theorem 2.1
Let / be given in ^+(R; V), we may assume that
f(t) =0 for t <0;
e t u be the solution of
(2.3) A(t)u + u" = f,
with (see Theorem 9.3, Chapter 3), for arbitrary finite T,
ueL°°(0, T;H),
«'eL~(0, T\V')9
(2.5) ^(0) = 0, w'(0) = 0.
According to (2.5), we may consider that (2.3) holds for t > 0 or on
R, by extending u by 0 for t < 0.
We also have
(2-6) IMlL°°(o,:r;H) + ll^'IL^crjn ^ c II/IIl2(o,:t;fv
c depending on T.
We need to show that u e ^+(R; V).
For this purpose, we apply the finite difference1 method, as in the
proof of Theorem 1.1. In the notation of this proof, we obtain:
(2.7) A(t) uh + u'l =fh- Ah(t) u(t - h).
1 In order to avoid technical difficulties for the interpretation of (2.3) and
(2.7), we may also approximate, in the sense of ^+(R; V), f with. fn£ @+(R; V)\
since the estimates which follow are then independent of n, the result will follow.
We could also use the method of Faedo-Galerkin (see Section 4, for the case of
Schroedinger equations).
(2.4)

100 2. Equations of the Second Order in t
Considering (2.7) as an equation in uh, we may apply (2.6). We obtain
IKIIl~(0,:T;H) + H«AllL~(0,r;n ^ C II/Jb(0,T;F') + C WAltf) UV ~ k) ll^(0,T;F')
from which, thanks to the hypotheses on / and A(t), we obtain that
llwAllz~(o,r;fl) + ll«AllL~(o,r;n ^ constant,
and since uh->u'{ resp. u'h->u") in 0'(]O, T[; H) (resp. ®'(]0, T[; V-))
it follows that
u' e £°°(0, T;H), u" e L°°(0, T; V)
and passing to the limit in (2.7):
(2.8) A{t)u' + A'{t)u+u{s)=f,
with
w'(0) =m"(0) = 0.
But we also have:
A{t)u = f-u"eL*(0, T;V),
whence
wGl2(0, T\V).
By iterating on this procedure, we obtain the theorem. D
2.3 Proof of Theorem 2.2
2.3.1 Reduction of the Problem
Since "t = 0" plays no particular role, and on the other hand, the
topological problems being solved as at the end of the proof to
Theorem 1.2, it all comes down to showing that if / £ ^o,Mfc ([0» T]; V), then
the solution u of (2.3), which belongs to ^+(R; V), satisfies
We shall verify that everything then reduces to showing the
following: let / e @0iMk ([0, T]; V) and let u be the solution in ^+(R; V) of
(2.9) A{t)u +fi{t)u' +u" =/,
where
(2.10) |8e^(R),i8>0,
(2.11) (il(Q»,w)^«|||t;||»f «>0, w€F, *€[0,T],
(2.12) (A'(t) v, v) < - y ||v||2, y > 0, v£V, * € [0, T];
then
(2-13) »e%,Mk([0,T];V).

2.3 Proof of Theorem 2.2 101
Indeed, first setting u = ektw, equation (2.3) becomes
(A{t) + k2I) w + 2kw' + w" = e-ktf,
or, changing the notation, and denoting the operator A(t) + k2I by A(t),
(2.14) A(t)u +r]u' +u" = /,
where rj £ R+, and where (choosing k in an appropriate manner) (2.11)
holds. But (2.12) is not necessarily satisfied, and in order to obtain a
condition of this type (condition which plays an essential role in the
estimates which follow) we must make a change of variable.
Set
s = exp (Xt), X > 0 chosen further on,
v® ==u\Ylogs
On s > 1, equation (2.14) becomes:
(2.15)
d2 /I «\ d , 1 „/ 1 , \ ,, 1 J 1 , \
(2.16) w(l) = 0,-^»(l) = 0.
as
But
d
ds"\A2s2"\ X ~°~ll Ps3
(^(ylogs))=^
-Mfjlog.) + |Wj logs)
But by virtue of (2.11), (1.3) and (1.4) we can always choose X in such a
manner that
(2.17) ^-24(i-logs)+i-4'(i-logs)JT;,»j<-«||»|P V«€F
for
ylogse[0,T].
Replacing s with £ + 1 and still denoting by ^4 (t) the operator
;WLJ-^(Tl0g(; + 1))'
and changing the notation, we see that we are led back to (2.9) with
(2.10) (2.11) and (2.12) (thanks to (2.17)). Since the classes Mk are in-
variant with respect to the changes of functions and of variables we have
made we are indeed led to proving (2.13).

102 2. Equations of the Second Order in t
2.3.2 Proof of (2.13)
We use the notation introduced in (1.28).
The hypotheses can be interpreted as follows:
(2.18) v_l{f))<dSekMk Vk,
(2.19) ||^w(0 ||^(F;n < cLkMk Vk, t e [0, T],
(2.20) |j8w(0 | < d^\Mk Vk, * G [0, T].
For the proof it is sufficient to show that
(2.21) Vl(u{k)) < (5(25)* Mk Vk
where
f d ]/T d££ cc, , m c.cL]
2.22 5 = 64max<-=, ^ -, c.d.T,^—}
(2.23) B = (1 + «) J2?2, with J2?a = max {J2?f JSfff, L, J2?lf L#}.
Let us prove first that for k = 0, 1, 2, ..., we have the inequalities
(2.24)
(i^Vij r "i(^)}+^+i))_i/2 v°{uik+i))+"]((r *t)v2 uik)) -
< S(ocy(2k + l))-1'2!*.^) + *2(*)^-i(^(*"'W +
+ 8(r(2£ + 1))-V* It *2 (*) sup 1^*^(0 1v0(u^)\ +
+ 8(y(2ft + l))"1 L^T - t)^ /<*+«) +
+ *S f^-i((T - ^)1/2 ^(*~i+1) ^(i)) +
+ 22(*Vi((r - 01/2^("^')^'+1))].
We agree to replace by 0 those summations which extend over an empty
set of indices.
To begin with, let us assume that, taking the kth order t derivative of
(2.9) we obtain that
(2.25)
«(*+2) +i8f*(*+i) + 21(*)j8(*-y)«y+1) + *S lk)^k~j)^j) +Au^ = /<*>.
i=o W j=o W

2.3 Proof of Theorem 2.2 103
Taking the scalar product of both sides of (2.25) with m(*+1) we obtain
(2.26)
+ 2^W_'Vy+1), «(*+1)) + 0 |«l*+1,l2 + (w<*+2)< M<*+1)) =
y-o W
=i.(/(»)i„i»))_(/(»+i)i(<(«)).
at
Let us take twice the real part of (2.26) and let us use the formula
^ (A(t) v(t), v(t)) = 2Re {A(t) v(t), v'(t)) + {A'(t) v(t), v(t))
which follows from (2.1); we obtain
(2.27)
4 2Re "-£ lk) (Al*->W>, «<*>) - 2Re ^ (*) M<*->+%0, ««) -
d* y_o W y=o W
-2Re 2l7*) (4(*-fl«y+1), «w) - (2* +1) (A'u™, «<*>) +
y=o W
+ 4 (4«(*,,«(*,)+ 2,8 |«(*+1) |2 + 2Re 21(*)^<*"y)(,*0+1)' M<*+1>) +
<^ y-o v /
+ A.I «<*+'> I2 = — 2Re(/<*>, «W) - 2Re(/<*+1>, «<*>).
d£ d£
We now multiply (2.27) by T — t and integrate from 0 to T; we obtain
(2.28)
T T T
J {Auw, uw) tit + J \u^+V |2 d* - (2k + 1) I (T - t) (A'uwuw) &t +
0 0 0
+ 2/ (r-0|8|«(*+1,|2d* =
o
= 2Re J (fk\ u^) &t - 2Re / (T - t) (f+1\ «<*>) dt -
o o
- 2Re ]£ (*) / (A{k-j)u^\ u™) at +
j=o y/o

104 2. Equations of the Second Order in t
+ 2Re 21 lk) j (T - t) (,!<*-'+1>««f «<*>) d* +
+ 2Re S (*) / [T - t) (i4(*-fl««+», «W) cU +
j=0 V/0
+ 2Re S (*) / (r - *) |8(*-y)(«w+1), «<*+1)) d*.
y=0 Wo
By virtue of (2.10), (2.11), (2.12) we have therefore
(2.29)
a[v,(uW)f + [v0(uV>+V)f + y(2k + 1) [Vl{(T - <)1/2 «W)]2 <
[ j=0 \3 } 0<t<T J
+ Uv_1[{T - *)]/2 /(*+1)) + 2 21 (^-i((T - ^)]/2^(""i+1)^(J)) +
I j=o W
+2 S2 (*V-i((r - *)1/2 ii(*-J,«u+1,)Ui((r - *)]/2 «<*>).
i=o V/ J
In the same manner, we show (2.29) for k = 0, 1, ... Then we obtain
(2.24), using the fact that the inequality
ax2 + by2 + cz2 < dx + ey -\- fz with a, b, c > 0 and d, e, f> 0
ya -\/b 4d 4tf 4
For / — 0, 1, 2, ... we now set
(2.30)
x* = t^T^Hv'{uij) + ^ +1})"1/2 v°{uU+1)) + "l((T "t)V2 u^
and we are going to show
(2.31) Xk<dBkMk,
hence (2.21) and the theorem follows.
For k = 0 (2.31) is true, since (2.24) for k = 0 implies:
X0 < S(*y)-]/2 M(/) + *y-**-i[(T - 01/2/');

2.3 Proof of Theorem 2.2
105
therefore, by virtue of (2.18) and (1.24) we have
X0 < S{ocy)-V2 dM0 + 8y-] iff A2MX <
< Siocy)-1'2 dM0 + By"3 ff d£?M0< dM0.
Let us admit (2.31) up to ft — 1 and let us verify (2.31) for ft; by virtue of
(2.18), (2.19) and (2.20), we see that (2.24) implies
X»<8,
(«y(2ft+l))-^J,
*-i
dSekMh + 2 • cLh~'Mh
3=0 VI
»-/M«W)}
+
+ 8(y(2ft + l))-1'2 T V lk\ d^\-iMh_jV^^) +
3=0 V /
3(y(2ft + I))"1 W
T d£k+'Mh+, +
+ *2 (*) cl}-^Mh_j+lVl{{T - tf'2 ««>) +
i=o v/
+KAK'
"".-^.'.((r-')"2"'")
using (1.24) and
2k
—(■\)<lh-
for l<j<k and "ft > 1 <
81/ T rf C?
< 8(<xy(2ft + 1))"]/2 d<?*Mh + I; (<?#)* Mk +
y(2ft +1)
+ 2 (/) ^I^-y8(^(2ft + I))"1'2 ^L±^j12 x
(ywrij)1'2^
,K1
fc-1
+ S • rrf,J2*->Mw 8(y(2ft + I))"1'2 (y(2j + l))1'2 X
i=0 v/
X [(y(2/ + 1))-^0(«W+1))] +
+ 8(y(2ft + l))-1 "•£ ik) cL{LH?-iMh_]Vl{{T - *)1/2 «W)) +
+ 8y-1 2 r)cL(LH)*-Wft_^]((r-01/2«O',)< using (2.30) <

106 2. Equations of the Second Order in t
< 8(«y)-1/2 d£e"Mk + 8)/-1 ]/Jd&i&H)" Mk +
k~! ft lh\ lWT k~! Ih\ IVf
+,3 ^fL,-'M'-'x'w, +§ 8(f) ".^-«-^sf +
+|,7(f)ci(ifl),"'M-x4 +
■j|'^*)^(iH),-i*.-AS1 +
7=1 7 V / 1Y±1
< using hypothesis (1.25), (2.22) and (2.23) <
<£ j=0 OC Mj j=0 Mj
+ 2 —cc1L{LH)k-iMkgr<^-BkMk + £ ±&\-i±LMh<
< by the induction hypothesis < — 5ftMft -f-
^ ^-1 ^ ^ k-l -Dj
1
(0
= y B A + y ^M* b~ =
= — BkMk + — d^2Mh^-^ <
5 Bk
= using (2.23) = dBkMk. D
— 2 * ^ 2 2 k B — &2 2 M
Remark 2.3. Examples analogous to those in Section 1.5 (this time
letting A*(t) = A(t)) can be given. Likewise, there exists a result
analogous to Theorem 1,4.

3.2 Proof of Theorem 3.1 107
3. Singular Equations of the Second Order in t
3.1 Statement of the Main Results
We shall study the M^-regularity properties of the solution of the
singular equation:
(3.1) A{f)u+u"-{ —«' = /, t >0,
t
for Xe C with
(3.2) m(0) =0,m'(0) = 0. D
We shall prove the following theorem.
Theorem 3.1. Let the hypotheses of Theorem 2.2 hold) thus: (1.5), (2.1),
(1.21) and (1.22),..., (1.25) hold. Let A £ C, with
(3.3) A# -1,-2,-3,...
Furthermore assume that the function t-> A(t) is even.
Let f be given with
(3-4) fe%,Mk([0,oo[;V).
Then problem (3.1), (3.2) admits a unique solution u which satisfies
(3.5) ue%,Mk([o,°°[;V).
Furthermore f-^uisa continuous mapping of
%,m* ([°< °°[; v') -+ %,Ut (to. °°[; v) ■
Remark 3.1. The same result is true (the method of proof remaining
unchanged) for the equation
91-1-1
(3.6) A(f)u+ u" + ——-«' + M(t) u' = /,
t
where
(3.7) t->M[t) is an even function, £ #Mfc (R). D
3.2 Proof of Theorem 3.1
We use the transmutation operators (see Delsarte [1], Lions [8]). We
know that there exist operators Bx and 38% with the following properties:
if ^0 denotes the space of C°° functions on t > 0 which, together with all

108 3. Singular Equations of the Second Order in t
their derivatives, vanish at the origin, then:
tBxe £?(%;%),
(3.8) < A—> Bx being an entire holomorphic function of
[C^ <?(%;%);
' 2xe&{%;%), A =|=-1,-2,...,
(3.9) \X^» @)x being a meromorphic function of
C^ JS?(^0; %), withpolesat —1, —2,...;
d
(3.10)
(3.11)
D2BX = B}L% on %, if D
di!
and if
L, = D2
2A
D;
Bx3&x = @}BX = Identity in %, A 4= —1, —2,..
Further on, we shall prove the following results:
(3-12) B,6^0A;\AfS);
(3.13) at e &(®o,Mk; %,Mk), A # -1, -2,...;
BxA(t) <%x<p = A(t)? + f Nx(t, a) <p(a) d<r«]»,
0
(3.14) I where <P € %,Mk ([°> °°[; V) and ^ <*-> N(t> a)
is a function of class Mk in t and <r, or < t,
t £ R, with values in £{V\ V).
Thanks to the transmutation operators B^ and ^, Theorem 3.1
reduces to Theorem 2.2. Indeed (3.1) is equivalent (if A =f= — 1, ...) to
(3.15) BxA(t) ®x{Bxu) + D\Bxu) = Bxf.
Set Bxu = w\ according to (3.14), equation (3.15) is equivalent to
A(t) w + f Nx{t, or) w(a) dor + w"(t) = BJ.
o
(3.16)
Now, according to (3.12):
(3-17) ^/€^„,^([0,oo[;F')
(W In this formula, the operators B% and 39% are extended to vector-valued
functions —which is immediate. We assume that A =f= —1» —2, ...

3.2 Proof of Theorem 3.1 109
and according to Theorem 2.2 and Remark 2.2, we obtain:
(3.18) we®0,„J[0,oo[;V).
Then
u = @,we%>Mk([0,oo[;V)
and since w depends continuously on BJ, we have the theorem if we can
show (3.12), (3.13), (3.14).
Proof of (3.12). We use the explicit formulas for Bx.
For —1< Re X < —1/2, we have
l
(3.19) BxV(x) = bxf t2*+\l - ?)-*~m <p{*x\ &
o
where
(3.20) bx = —^= r(X + 1) r ( -X - — J (r = Euler function).
Thus
(B^)W (x) =bxj t2X+l+\l - t2)-*-*12 <pW(tx) &,
o
from which, for x belonging to a compact set [0, x0]:
(3.21)
| (Si9»)W (*) | < \bt | dkMk / ^+l + *(l _ ^-ReA-3/2 ft < ;LkMk wk_
0
Next, for
—1 < Re X < n — 1/2 (« positive integer),
B^ is given, by analytic continuation of (3.19), in the form (see Lions
[8], page 69):
R m(x\ — (-1)" h r ,2A-(2»-3) x
xVKX) ~ (2^ + 1) (2A - 1) • • • (M - (2» - 3)) J
X (1 _ ^-(2-«/8 Tn(p{tt x) dt>
with
T1?>(*, x)= — (t<p{tx)),
T»V(t.x)=-^{*Tn-Mt.*))-

110 3. Singular Equations of the Second Order in t
Then
^r„#,) = iry/»)((,x)
and we obtain relations analogous to (3.21) for — 1 < Re X < n — 1/2.
For — 1 — n\2 < Re X < —1/2, we use another expression for the
analytic continuation of B?; for X # —3/2, —5/2, ...:
B UA = (^T^A r1p + , + l(1_^2)A + (, + l)/2 x
m} {2X + 2)(2X + 3)---(2X+n + l)0J { }
X <ttn(p(t,x)dt,
with
*1?,(ff*) =^((i- tyv<p{tx)),
d
29,(^)=A((l_^)3/2^i9,(^)))
*.?>('. *) = ^ ((! - ^2)S/2 *.-!?>('. *)) •
Again, relations analogous to (3.21) follow.
Finally, we verify (as in Lions [8], page 74) that X = —3/2, —5/2, ...
are not exceptional values when op £ @0—and we again obtain estimates
analogous to (3.21). D
Proof of (3.13). For Re X > —1/2, the operator 0&x is given by
i
(3.22) &x<p{x) = fiX J (1 - *y-1/2 <p{t%) dt,
o
where
(B.2S) ft- ' r" + 1>
A j/rc r(A +1/2)'
For <p £ i^o.Mfc > ^ follows immediately that
i
(3.24) |(J»(ft) (*)| < |j8A| cLkMk J (1 -t2)ReA-V2?&<~cLkMk Vft.
o
We then carry out the analytic continuation of ^; by formulas (see
Lions [8], page 78) which preserve the properties analogous to (3.24), for
X^-l, -2, ... D
Proof of (3.14). For |ReX\ < 1/2, we verify (see Lions [5], page 265)
that
t
(3.25) BXA{t) aA-A{t) = t f gr j (t, r) dr,
d '

4.2 Proof of Theorem 4.1 111
where
VXA$> r) = 7x /sin"2;+2 0 cos"2A OA^t2 sin2 d +t2 cos2 d)1'2) dd,
(3.26)
i41(Q = — ,4'(£) (note that A is gw» in J),
t
7x
r(A + l/2)T(-A+l/2)
Property (3.14), for | Re A | < 1/2, follows.
We then pass to the general case, when X ^ —1, — 2,..., by analytic
continuation. D
4. Schroedinger-Type Equations
4.1 Statement of the Main Results
In this Section we shall study the regularity properties in Qj+ and in
@+,Mk of the solution of the Schroedinger equation:
(4.1) iA(f)u+u'=f, (i = \!^l) ,
where the support of u in t is bounded on the left.
We shall prove the following results:
Theorem 4.1. Let the hypotheses of Theorem 2.1 be satisfied, i.e. (1.4),
(1.5) and (2.1) hold. Then the mapping
(4.2) u^iA(t) u + u'
is an isomorphism of ^+(R; V) onto ^+(R; V).
Theorem 4.2. Let the hypotheses of Theorem 2.2 be satisfied, i.e. (1.5),
(2.1), (1.21) and (1.22),..., (1.25) hold. Then the mapping (4.2) is an
isomorphism of @+fMk (R) V) onio @+,Mk (R' V')'
4.2 Proof of Theorem 4.1
Changing u to exp (—iXt) u and choosing X appropriately, we reduce
the problem to the case for which
[A(f)v9v)>*\\vf, *>0, v£V, te[0, T]((1))
and by eventual renormalisation of V we are led to the case for which
(4.3) {A(f)v,v)>\\vf, v£V, te[0,T].
W> It is sufficient to reason on [0, X], arbitrary fixed finite X.

112 4. Schroedinger-Type Equations
Next, we make the change of variable (as in Section 2.3):
5 = exp (M).
Setting
v{s) = ul—logs\,
we obtain the equation, on s > 1,
(14) ;HtHw(8)+£b(8)=^tH-
Then
and (as in Section 2.3) we are thus led to consider the case for which
(4.5) (A'(t) v, v) < -y \\v\\2, y > 0, t£ [0, T], v£V. D
We shall now prove Theorem 4.1, assuming that (4.3), (4.5) hold.
We start with the foUowing result1: if feL2{0 T\ V) with-^e
at
L2(0 T; V) then there exists a unique u in L2(0 T; V) solution of (4.1)
with
(4.6) «(0) = 0.
Furthermore (again using the notation introduced in (1.28)):
(4-7) v,(u)<2(v_1(f)+Tv_1(n).
Indeed, for example applying the Faedo-Galerkin method (see
Chapter 3), it is sufficient to show the estimate (4.7) for regular functions
satisfying (4.1) or the equivalent equation
(4.1a) A(t) u — iu' = —if.
Taking the scalar product of the two sides of (4.1a) with (T — t) u'
and taking twice the real part of both sides, we obtain
(4.8)
d
a(t; u, u) — a'(t; u, u)
at
at = —2Re i J {T - t) (/, u') at.
0
Therefore
T
J a(t; u, u) dt — J (T — t) a'(t\ u} u) &t
o o
(T T \
J tf,u)&- J (T-t) {f, U) tit)
0 0 /
1 See also Pozzi [1] for other results of this type.

4.3 Proof of Theorem 4.2
113
from which, since we have (4.3) and (4.5), we obtain
v1(u)*<2(v_1V)+Tv_1V'))v1(u)
whence (4.7). D
This time, in order to show the regularity, we do not use the method
of differential quotients, but rather the Faedo-Galerkin method; the
approximate solutions being regular in t, it is sufficient to formally
differentiate equation (4.1a) with respect to t and then obtain an estimate
analogous to (4.7) for v-^u'). More precisely, we have, formally
(4.9) A(t) u' + A'(t) u + iu" = if.
We take the scalar product with (T — t) u". We obtain
/ (r-*)
d*
a{t\ u', it') — a'(t;uf, u')
& +
T I T \
+ 2Re J {T-t) (A'{t) u, u") dt = 2Re (-i / [T - t) (/', u") dt\
o \ o /
from which since a'(t\v v) < 0:
r t t
j a(t; u', u') dt + 2Re J A\t) u, u') dt -
0 0
I T T
(4.10) \ -2Re/ (T-t) [A'{t)u', u') & - 2Re f {T-t) (A"{i),u,u')&
o o
(T T \
-i J (/', u') dt - i J {T-t) (/", «') dt\ .
Once again applying (4.5), it follows that
vM'?< 2(v_i(n + Tv^if")) Vl(u') + cv^v^u')
whence (c denoting a constant):
(4.11) vt(u') < c{v_,{f) + v_tf) + v_tf')) ■
This justifies the formal differentiation with respect to t. By iterating
on this procedure, we obtain the desired result. D
4.3 Proof of Theorem 4.2
We consider the problem on the interval [0, T], on which we assume
(4.3) and (4.5) to be satisfied.
The hypotheses may be interpreted as:
(4.12) v_^)<d<£hMh, Vft,
(4.13) H^Wtf) ||^(F;n < cLkMk, Vk, te [0, T].

114
5. Stability Results in M^-Classes
We have to show that
(4.14) Vl{u{k)) < (5(25)* Mk Vk,
where d and B depend only on oc, y, d, ££, c, L, T, clt L, H.
The proof is analogous to the proof of formula (2.21) given in
Theorem 2.2. We show fiist the relation similar to (2.24), without the term
v0(u{k+1)) (we take the kth order ^-derivative of (4.1) and we take the
scalar product of both sides with — u^k+1) and then twice the real part
of the result; next, we multiply by (T — t) and integrate from 0 to T;
the terml^+1),— ^*+:l)J drops out). We then complete the proof as
for the proof of (2.21). D
5. Stability Results in Mfc-Classes
5.1 Parabolic Regularization
Continuing in the same notation, let us assume that
(5.1) a(t',u,v)=a(t',v,u) Vu,v£V,
(5.2) [A{t)v>v)>(x\\vf> te[o,T], <x>o, vev.
In Chapter 3, Section 8.5, we have seen that we can "approximate"
(in the spaces L2; see Theorem .8.3, Chapter 3) the problem
(5.3) A(t) u +u" = /, u(0) = 0, m'(0) = 0,
with the problem (called parabolic regularization of (5.3)):
(5.4) A(t) ue + eA{t) ue + < = /, ue{0) = 0, ue(0) = 0, e > 0.
A natural question to ask is whether, if / and A are of class {Mk},
the solution ue of (5.4) converges to u as e-> 0, in the sense of the spaces
The answer is yes; we have:
Theorem 5.1. Let the hypotheses of Theorem 2.2 be satisfied. Let f be
given in 2+tMlc (R» V)- Let u (resp. ue) be the solution in <2>+jMk (R; V) of
(5.5) A(t)u +u" + = /
(resp. of
(5.6) A{t) ue + sA(t) ue +u" + = /, s> 0^).
((1)) According to Theorem 1.2, the solution ue of (5.6) does belong to
&+ Mk (R» ^)» smce» according to Chapter 3, Section 8, equation (5.6) reduces to
the parabolic case of Section 1.

5.1 Parabolic Regularization 115
Then:
(5.7) ue^uin@+,Mk (R>' v) ass^O.
Proof. 1) Changing u to exp (kt) u, equations (5.5), (5-6) become
(5.8) (A + ft2) u + 2ku' + u" = f,
(5.9) (A + ft2 + £^4ft) ^e + (2ft + e4) < + u" = /,
respectively.
Assume that fe%tM]c ([0, T]; 7'); then «ee ®0>Arifc ([0, 7]; V) and
is sufficient to show that
(5.10)
■u in %>Mk([0,T];V).
Thus we consider the problem on the interval [0, T], and make the
change of variable (as in Section 2.3):
Setting
s = exp (M).
Hs) = u (y log s\ , ve{s) = ue I— log sJ ,
equations (5.8) and (5.9) become respectively, for s > 1:
(5.12)
ds2
sk2
d2v / 1 2£\du 1
(5-11)d^ + (T+Il)d7+^
f 1 2ft -
[ s As
(ylogs)
d«e
~d7
sk
+ ft2 - eft3
+
^(tHHHMtH-
By a new change of variable and function, we are thus led to consider the
following situation: u is a solution of
(5.13)
where
(5.14)
u" +fiu' +Au = f,
P>0, p£SMk,
(Av,v) ><x\\v\\2 VveV, te [0, T],
{A'v,v)<-y\\vf, y>0, v£V, t£ [0T].

(5.16)
116 5. Stability Results in M^-Classes
and ue is a solution of
(5.15) < + (j8e + est) ue + (Ae + eA4) **e = /
where
Pe = P + &P> <P£ ^Mk >
Ae = A +ea, ae^Mk >
^=TsA{Tl0*s) + k2>
(s*{t) v, v) > oc± ||t;f, «1>0>»e7J6 [0, T],
[ KM»,») < -ri II»II2, ri > o, * e v, te [o, r].
2) We take the kth order ^-derivative of (5.15) and we take the scalar
product of both sides with uf+l); we take twice the real part of the result;
next, we multiply by T — t and integrate over (0, T). Taking into
account (5.16) and by a calculation analogous to the one in Section 2.3,
it follows that, / being fixed, there exists B £ R+, B independent of s,
such that
(5.17) ue remains in a bounded set of @0yM]c ([0, T], B\ V), as e^ 0.
3) Setting
(5.18) we = ue — u,
it follows from (5.13) and (5.15) that
(5.19) w" + pw'e + stfwe = ege,
where
(5.20) ge = ~{cp +sfjue-{a + kA) ue.
According, to (5.17) we have:
I there exists «£?-, > 0 such that ge remains in a
bounded set of %>Mje ([0, T\&x\ V) as e-+0.
But then, according to Section 2.3, there exists BY > 0 such that
(5.22) —we remains in a bounded set of @0fMjc ([0, T], Bx\ V) as £-> 0,
therefore we^ 0 in ^o,Mfc ([0> T], B^, V), which proves (5.10). D

5.2 Approximation by Systems of Cauchy-Kowaleska Type (I) 117
5.2 Approximation by Systems of Cauchy-Ko waleska Type (I)
In this section and the one following it, we consider problems which
are not of Cauchy-Kowaleska type (see equations (5.29) —(5.30) and
(5.46) —(5.47) below). We shall indicate how they can be approximated
(in a sense to be specified below) by systems which are of
Cauchy-Kowaleska type, with convergence of solutions to the solution of the initial
problem in spaces of class Mk. (We also have convergence in Sobolev
spaces, as introduced in Volumes 1 and 2, the data belonging to Sobolev
spaces, but this is not discussed here.) Q
In a bounded open set Q in RM, consider the space
(5.23) 0 = {cp\(pe@{Q)n, div<p = 0},
and next
(5.24) H = closure of & in (L2{Q))n,
(5.25) V = closure of & in (Hl(Q))n,
the scalar product in H (resp. V) being
n
K v) = 2 / uh d*
j = l G
n
(resp. (u, v)v=% {ujf ^)Hi(D)).
j=\
For u, v £ V, we set
(5.26) a[u,v)=± f^d*.
Since Q is bounded and the elements of V vanish at the boundary of Q,
we have
(5.27) a{v,v) ><x\\vfv, <x > 0.
Theorem 1.2 is valid1; consequently, for f given in @+>Mk (R,* V')>
there exists a unique u belonging to @+tMk (R»* V)> solution of
(5.28) — («, v) + a{u, v) = (/, v), Vv e V. D
at
The interpretation of problem (5.28) is the following: Since (5.28)
is satisfied for all test functions with vanishing divergence, the equality
of functions (after integration by parts) holds "modulus a gradient";
1 In a very simplified setting, since a(u, v) is independent of t.

— -A« +grs.dp = f,
118 5. Stability Results in MrClasses
therefore
(5.29)
(5.30) div^ = 0,
and according to (5.29), we have
(5.31) gr^pe9+iMk[-R;{H-\Q)T).
The system (5.29) —(5.30) is not of Cauchy-Kowaleska type: it does
not contain a derivative dpjdt.
Remark 5.1. Other problems of this type are found among transmission
problems', see Problem 17.11, Chapter 4—See also Problem 17.12,
Chapter 4 and Section 5.3 below. D
Remark 5.2. The above system is none other than the linearized
Navier-Stokes system. D
The preceding problem is approximated by problems of Cauchy-
Kowaleska type in the following manner: we introduce
(5.32)
and consider the problem
H = L2(Q)nxL2(Q),
V = Hl{Q)nxL2{Q),
(5.33)
(5.34)
If we set:
dt
eTt
-Aue +gra.dpe = f,
+ div^e = 0, e> 0,
K&}€F((1)).
%={%>Pe}>
(djdt 0 \
e \ 0 e d/dt)
a(u, v) = a(u, v) + $ grad p. v dx + j div u. ~q dx,
Q Q
for u = {u,p}, v={v,q}£V, ?={f,0},
((1)) More precisely, function of t with values in V.

5.2 Approximation by Systems of Cauchy-Kowaleska Type (I) 119
we see that problem (5.33) may be written:
(5.35) (Aeue> v)+a (ue, v) =(£?),
where
\f>'v) = f f-~v dx.
We verify that
a\uy v] = a(u, v) + f p • div v dx + J div u -qdx,
so that u, 7 -> <z(3\ ?) is continuous on V and
«Kw)>«||w||fflti(fi))»
and therefore
"||L2(£)'
«(w, v) + cx \\v\\% > a \\v\$HW))n + c^ \\qf
The operator Ae behaves like
/d/d* 0 \«™
\ 0 d/d//
Consequently Theorem 1.2 is valid and yields:
if/€0+>^(R;(#-](£)r),
there exists a unique {ue, fie}, solution of
(5.36) | (5.33), satisfying
uee®+>Mk{R;(Hl(Q)r),
We shall now prove
Theorem 5.2. As e-> 0, the solution {ue, pe] of problem (5.33) given
by (5.36) satisfies
(5.37) ue -> u in 3j+yMlc (R; (Hl(Q))n) weakly2,
(5.38) grad pe -> grad # in @+>Mjc (R; (H-^Q))") weakly,
where {u, p} is the solution of (5.29) —(5.30).
((])) More precisely and more generally see Section 8 below.
2 The sense of (5.37) will be specified in the course of the proof.

120 5. Stability Results in M^-Classes
Proof. It follows immediately from (5.35) that, Vk,
(5.39) (A.u?\ v) + «(«<», v) = (/H ?) ,
from which, taking v = w(ft):
* / lK*> l!(W ^ < / \\f{k) II(h-^,)» ll«i*' \\{Hnr.,r &
0 0
and consequently, if we set
\\Ue \\L\0,T;(Hl(Q))n) ~ VliUe )>
11/ \\L*(0,T;(H-1(Q))n) = V-1 (/ ) »
we have
(5.40) v,(«<»>) <«-_!(/««).
Assume that
/e<\Ms([o,r],i?; (#->(£))")•
Then
v_i(/w) < dX*Mh
and therefore there exists J2?x such that
<»i> iii/iik-£iS£r-,<w<~.
If we denote the space of functions / satisfying Hj/Hl^ < °° by
^Mk([0,T],&1i(H-1[Q))n)=W-1, ^ is a Hilbert space; if Wx
denotes the space analogous to W-v obtained by replacing H-\Q) with
H\(Q)y we see that (5.40) entails that
(5.42) ue remains in a bounded set of XP1 as s-> 0.
Thus we can extract a subsequence, still denoted by ue, such that
(5.43) ue -> w in Wx weakly1.
But it follows from (5.35) that
(5.44) ]/e pe is bounded in L°°(0, T; L2(Q)) weakly.
We may therefore assume that
j/e fte converges (and necessarily to 0)
in L°°(0, T;L2{Q)) weak star;
1 Which makes (5.37) precise, after having shown that w — u.
(5.45)

5.3 Approximation by Systems of Cauchy-Kowaleska Type (IT) 121
Taking v £ V in (5.35) (i.e. div v = 0), v = {v, q}, we obtain:
(u'e, v) + e{p'e, q) + a{ue, v) = (/, v)
and passing to the limit in, for example, the sense of distributions on
]0, T[, it follows that
(w'f v) + a(w, v) = (/, v), Vv£V.
Since w(0) = 0 we see that w = u and (5.43) entails (5.37). Then
(5.38) simply follows from the first equation in (5.33) which yields
grad^ = /-—e+A**e. D
5.3 Approximation by Systems of Cauchy-Kowaleska Type (II)
We now consider the setting of Problem 17.12, Chapter 4, for which
we shall examine the properties of M^-regularity in t. Q
We seek a function u(x, t), x£ Q, 2 £ R, satisfying
(5.46) A,«(*, t) = 0, xe Q, te R,
3« du „
(5.47) _+__ = gonr,«€ll.
Assume that (notation of Chapter 1 for the spaces Hs(r)):
(5-48) ge®+iMk{R;H-W(D).
Then problem (5.46), (5.47) admits a unique solution which satisfies
(5.49) ue@+>Mk{R;W(Q)).
Indeed, let us introduce
(5.50) w = y0u = trace of u on r.
As in Problem 17.12, Chapter 4, we introduce the operator
(5.51) J1 <G Se{H^2{r); H-^{r)f1)]
by
(5.52) &h = — , heHll2(r),
dv
where
(5.53) Ao) = 0, a)|r = A.
((])) We assume the boundary i"1 of Q to be sufficiently regular.

122 5. Stability Results in M^-Classes
Then (5.46), (5.47) is equivalent to
dw
(5.54) @w+ — = g,
ot
equation on F\ since
(Ov, v)>p \\v\\%it*ir), p>0, ve H^iD,
we can apply Theorem 1.2 (for a simplified case, since 0$ does not depend
on t) with:
v = #1/2(r), A{t) = a, h = L2(r).
It follows that w exists and is unique, and that
(5.55) we®+>Mk{R;HV\r)).
Now let us denote by G the operator
defined by
A(Gq>) = 0 in Q,
y0(G<p) =(ponr.
Then
(5.56) u(t) = Gw(t)
and (5.49) foUows from (5.55) and (5.56). D
Problem (5.54) is an evolution equation on the variety r; in the form
(5.46), (5.47), the problem is not of Cauchy-Kowaleska type. But it can
be approximated by the following problems:
(5.57) e —- — Aue = 0, x^Q, z5€ R, e> 0,
ot
(5.58) ^ + ^g,,€r,*€R.
In variational form (see Chapter 3, Volume 1), problem (5.57), (5.58)
is written
(5.59) e(u'e, v) + (u'e, v)r + a(ue, v) = (g, v)r, Vv g ff1^),
where
(<P>V>)r= /wdr,
r
dcp dip
^^g/^^

5.3 Approximation by Systems of Cauchy-Kowaleska Type (II) 123
It can be seen immediately (simple variant of Theorem 1.21) that, if
g€ @+ Mk 0^'y Hll2(r)), problem (5.59) admits a unique solution
We have:
Theorem 5,3, As e-> 0, the solution ue of problem (5.59) satisfies:
(5.60) ue->u™@+lMk[*>H^Q))>
where u is the solution of (5.46), (5.47), (5.49).
Proof. It foUows from (5.59) that
(5.61) e(uik+'\ v) + {uf+1\ v)r + a(u[k\ v) = (g<*>, v)r,
from which, taking v = u{ek) and assuming that g£ ^o,Mfc ([0> T]; #_1/2(.T),
we obtain:
4r WW l!W + 4" H w^r) "i-cr, + / "(«.W. «?') d* = / (SW> "i^r d/
from which we further obtain2
« / II^W lllw ^ ^ / ll^'Wllr^n U^'W »hVV) d'
0 0
and since (Chapter 1, Section 8):
\\v\\H^{r)<ci\\v\\HHO)>
it follows that
(5.62) / ||«W(Q |&(fl) d* < c2 / ||g<*)(0 ll^/V) d'-
0 0
But on the other hand u satisfies
(«', v)r + «(«, v) = (g, v)r, Vv e fl1^),
therefore
(5.63) («<*+1>, v)r + a(«W, v) = (gW, v)r, Vv g fl1^);
if we set
we = ue — u,
then it foUows from (5.61), (5.63) that
(w{ek+1\ v)r + a(wf\ v) = -e{uf+1\ v),
1 Moreover valid for a(u, v) replaced by a(t; u, v) depending on t, so that A(t)
is of class Mfc.
2 We always first reduce the problem to the case for which
a(v, v) >a \\v\\ip{Q), <x> 0.

124
5. Stability Results in M^-Classes
from which we deduce that
* / II^W \\%n°) dt ^ e / ||«i*+,,W ||fw d* <
< (by (5.62)) css / ||g(A+1)WlllrV2(r)^ <
0
< cBe<?k^Mk+1 <
<c±e(J?H)kMk,
whence the desired result. D
Remark 5.3. Analogous results hold for the problem (see Problem 17.12,
Chapter 4):
(5.64)
which can be approximated by
Am = 0 in fl, — + — = g on r,
cv 3P
(5.65)
dp
due (Put
dv dt2
Aue = 0 in Q,
= g on r. D
Remark 5.4. One could also consider the "parabolic regularization"
of problem (5.65):
(5.66)
\eKto> V) + K*>> V)r + WKeS V) + a(UU> V) = (& V)r>
\yv£H}(Q)
and let e, ex -> 0. D
Remark 5.5. Analogous considerations for the problem of "Schroedin-
ger type on the variety J7":
(5.67)
Au = 0 in Q, teR,
i-+--gonr,^R. D
Remark 5.6. One obtains analogous convergence results for the
parabolic regularization (see Chapter 5, Section 12.2) of problem (4.1)
which is approximated by
5.68)
(e+i)A{t)uB+4 = f. D

6.2 The Parabolic Case
125
6. Transposition
6.1 Orientation
With the exception of the results of Section 3, all results obtained in
the preceding sections can be "transposed'' so as to yield solutions in the
form of distributions or ultra-distributions of class Mk, when the right-
hand side of the equation is a distribution or an ultra-distribution of class
Mk.
We shall give some results1 for ultra-distributions. Q
6.2 The Parabolic Case
The notation is as in Section 1. We introduce the adjoint A*(t) of
A (t) defined by
(6.1) (A*(t) u, v) = (u, A{t) v), Vu,ve V.
Hypothesis (1.21) entails (and in fact is equivalent to):
(6.2) t-> A*{f) belongs to the class SMlc (R; JS?(F; V')).
Replacing the spaces @+tMk w^h ^_,Mfc and "changing the sense
of time" (thus d/dt to —d[df), Theorem 1.2 yields
I the operator A*(t) — d/dt is an isomorphism of
0-,M,(R;Honto0_^(R;F').
By transposition of (6.3), noting that the adjoint of A*(t) — d/dt
is A(t) + d[dt, and recalling the definition of &+fMjc(R;X) =
= (@-tMk (R',X'))' (see Chapter 7, Section 5), we obtain
Theorem 6.1. Let a(t;u,v) satisfy (1.5), (1.21) and assume that the
sequence {Mk} satisfies (1.22), ..., (1.25); then the mapping u-> A(t) u + u'
is an isomorphism of 3>'+tMk (R>* H onio @'+,Mk (Rj V)- D
Example 6.1. Let us give an explicit application of Theorem 6.1 in
the setting of Section 1.5 with m = 1.
Let
(6.4)
^ d I du\
^»« = -Z^(s(M^).
2 ««(*, f) ££ > « |||2, a. > 0, x£ Q, t£ R.
1 The —in fact very simple! —method is general.

126 6. Transposition
Let / and g be given with
f / € 0+,m* (R> L^))> or more generaUy,
(6.5) ] / G ^'+>Mfc (R; S^1^)) (see Chapter 3, Section 4.7.3,
[with r0 = <t> for S^CQ)),
(6.6) g6^+,Mfc(R;^-1/2(r)).
For v G V = H\Q)y set
(6.7) L(v) = V,v) + te,yQv)r9
where (/, v) G ^+,Mfc (R) is defined by
<(/, v), <p> = </, <p ® ^>, 9> g ^_>Mfc (R)
and similarly
<(g> Mr* <P> = <£> 9? ® >V>>'
formula (6.7) defines L G ^'+,Mfc(R; ^') an(^ Theorem 6.1 shows the
existence and uniqueness of u£ ®'+tMh (Rj F) satisfying
(6.8) («', v) + a(zJ; «, v) = L(v), Vv G F,
/ ^ r dw dw \
where a(zJ; <p, y) = 2 J a*A *) F^ 5~ <** •
\ *j=ifl ^" ^ /
It follows from (6.8) that, in the sense of SJ'+yMlc(R\ ®'{Qj) in
particular, we have
(6.9) u' +A(t)u = f.
It then foUows from (6.9) that A{t) u = / - u' G ®'+tMk (R>* S^CQ));
thus w has the properties
(6.10) . ue®'+iMk{R;H\Q)), A{t) u£ ®'+>Mk (R; 3~\Q)).
du
It can be deduced from (6.10) that (extension by continuity of
OVA(t)
dw * dw —
~ v — 2j aij(x> *) F~ cos (v» **) f°r regular functions 99 in QxRt)
is well-defined and that
(6.11) ^e®'+,Mk(R;H-»Hr)).
The proof of (6.11) is rather lengthy, except for the case where
A(t) = A is independent of t, to which we shall limit our presentation.

6.2 The Parabolic Case 127
We first recall (see Chapter 7, Remarks 5.3 and 5.5) that when X is,
in particular, a Hilbert space,
Thus if u satisfies (6.10), then, Vy> £ ^-,Mfc (R)> uiw) satisfies
u{xp) e H\Q),
A[u{rp)) = (Au) (y) g E^Q)((1))
and the linear mapping
tp-> u(xp)
is continuous from @-tMk (R) -> ^> where
y = {^ | w g #i(£), ^ € 5^(0)} >
provided with the (Hilbert) norm of the "graph":
(IMIIw + !l^lll-W1/2
and this is equivalent to (6.10).
But, according to the results of Chapter 2, w -> dwjdvA is a continuous
linear mapping of Y -> H~ll2(r) and consequently
d
dvA
is a continuous linear mapping of ^_)Mk(1R)-> H~ll2(r) and thus
defines the ultra-distribution du[dvA, element of ^'+,Mfc (R; H~ll2(r)).
We can now complete the interpretation of (6.8). Taking ^ in ^_Mfc (R),
it follows from (6.8) (still for the case "A(t) independent of i") that
{u'(xp), v) + a(u{yj), v) = (/(y), v) + (g(y>), y0v)r;
on the other hand, taking the scalar product of both sides of equation
(6.9) with y) ®v,we obtain
(«», v) + (Au{y>), v) = (f{v), v),
from which it follows that
(Au(y>), v) = a(u{ip), v) - (g(y), y0v)r
and consequently
du
((1)) This is where the hypothesis "A(t) independent of t" intervenes.

128
6. Transposition
So that we have obtained the existence and uniqueness of u in
@'+,Mk (R; H^Q)) satisfying
u' + Au = /,
du n
dvA
Remark 6.1. We have transferred the initial data to the second
member, a classical procedure for the solution of the Cauchy problem. D
Example 6.2. Let ro C r, as in Chapter 3, Section 4.7.3. According to
G. Geymonat, we then take (in the notation of this Section and of
Chapter 1):
ft®'+tMJS-\Q)),
ge®'+,Mk(HooV2(Fo))->
then we see that there exists a unique u in ^'+,Mfc (Rj ^(Q)), solution of
(6.9), with, formally:
^- = gonr1xR< (r,=r-r0),
WA(t)
u = 0 on f0xRr
If A is independent of t, a precise interpretation, such as for Example 6.1,
can be given. D
6.3 The Second Order in t Case and the Schroedinger Case
We now consider the setting of Theorem 2.2. Then
A*(t) =A(t)
and we can invert the sense of time:
(6.12) u—>A(t)u-\- u" is an isomorphism of
#-.**(n-*#-.ift(n-
By transposition of (6.12), we obtain
Theorem 6.2. Let hypotheses (1.5), (2.1) and (1.21) hold, and assume
that the sequence {Mk} satisfies (1.22), ..., (1.25). Then the mapping
u-> A(t) u + u" is an isomorphism of @'+tMk IY) onio @'+,Mk (V)-
By transposition of Theorem 4.2 (and changing i to — i, which is
permissible) we obtain
Theorem 6.3. Under the hypotheses of Theorem 6.2, the mapping
«-> iA(t) u + uf is an isomorphism of <2Jf+yMlc{V) onto @+tMk(V). D
Remark 6.2. Examples of applications of Theorems 6.2 and 6.3
analogous to Examples 6.1 and 6.2 can be given. D

7.2 The Space of Vectors of Class Mk 129
7. Semi-Groups
7.1 Orientation
We shall now consider evolution equations of the form
du
(7.1) — + Au = 0, t> 0,
at
(7.2) «(0) = m0,
where —^4 is the infinitesimal generator of a semi-group in a Banach
space E.
7.2 The Space of Vectors of Class Mk
Let £ be a Banach space on C, with norm denoted by || ||. In order to
simplify the presentation, we assume that E is reflexive.
In the space E, define a continuous semi-group G(t), i.e.:
G{t)e^{E;E)}\ft> 0,
V£ £ £, the function t -> G(£)0 is a continuous
(7.3) | mapping of t > 0 -> E,
G(0) = /,
G{t)G(s) =G{t + s), V25, s> 0.
If we are willing to change G(t) to exp (—coif) G(t), suitable co, we may
always assume that
(7.4) \\G(t) || < ii < oo (||G(*) || = norm of G{t) in j?(£; £)).
—A denotes the infinitesimal generator of G(t). Then the solution u
of (7.1), (7.2) may be written
(7.5) u{t) =G{t)u0.
Remark 7.1. Instead of (7.1), we could, more generally, consider the
equation
du
(7.1a) +Au = f;
at
then, at least formally, the solution of (7.1a), (7.2) is expressed by
t
(7.5a) u(t) = G{t) u0 + J G(t - a) /(cr) dcr,
o
If, for example, / is assumed to be continuous from t > 0 -> E, this
formula can be justified in the following manner.

130 7. Semi-Groups
Introduce
\G(t) for t > 0
(7.6) <§=\
[0 for t< 0.
Then (see Chapter 4, Section 3.1):
d
(7.7)
_ + A9 = d(t) ® IE (Ix = identity from Z-> X),
d£
^(4+4)==aw ® w
T>(A) = domain oi A = {e \ ^_1(G(^) e — e) converges
in E as h-> 0},
40=limA-1(G(A)0 — e)
((D)
[0, 25 <0, [0, t <0,
problem (7.1), (7.2) may be written
du ~
(7.11) —+ i4«=/ + a(0«o,
of which JAs solution in ^'+(D(^4)) is
(7.12) u = <Z*(] + d(t)u0)>
which is a justification of (7.1a). D
We aim to study the properties of regularity and Mk-regularity of the
function t->u(t) and, more particularly, of the function t->G(t)u0.
Thus, we shall assume that / = 0. D
The function t->G(t)u0 is once continuously differentiate from
t > 0 -> £ if and only if w0 £ D(^4) and then
(7.13) ±G®u0=-G®Au0.
We iterate on this remark; we set
(7.14) B{Ak) = {e\eeD{A), Ae£ D{A)t ..., A^ee T>{A)}9
«5)) The operator A is closed (see Hille-Phillips [1], K. Yosida [1]); T>(A) is
provided with the norm of the graph ||e|| + ||-4«||, which makes it a Banach space.

7.2 The Space of Vectors of Class Mk 131
which is a Banach space with the norm
(7.i5) s II^11;
3=0
then
IT>[Ak) is the space of vectors e such that t^G(t)e
is Mimes continuously differentiate from t > 0 -> E. D
Next we set
oo
(7.17) D(ii°°) = PI V(Ak),
k=o
which is a Frechet space with the sequence of norms (7.15); we have:
I~D(A °°) is the space of vectors e such that t -> G(t)e
is infinitely differentiate from t > 0 -> E.
The following is a classical result.
Proposition 7.1. The space D(^4°°) is dense in E.
Proof. Let Qn£ 0(]O, +oo[), £n(*) > 0, £M with support in [an, j8n],
0 < <xM < j8n, /?w^ 0, f Qn{t) dt = 1 (gn is a regularizing sequence).
In general, if cp is a continuous scalar function on i > 0, with compact
support (or "sufficiently small" at infinity), one sets:
oo
(7.19) G{<p)e = f G(t)eq>(t)dt,
o
an integral with values in E.
If q> g 0(]O, + oo[), we can easily verify that, We g £, G(<p) ^D(i)
and
(7.20) i4G(y)«=G(^V.
We can iterate: G(q>)e£ D(^4°°) and, We £ E, we have:
(7.21) AkG((p)e = G(/)e, Vft.
Thus, for arbitrary 0 in E, G(£M)<? £ D(^4°°). Now
oo
G(q) e-e=f [G{t)e - 6(0)e) Qn(t) dt,
0
therefore
\\G{Qn)e — e\\ < sup ||G(J)e — e||-> 0 as w-^oo,
<€[0A,]
whence the desired result. Q

132 7. Semi-Groups
We now introduce
Definition 7.1. We call vector of class Mk> {Mk} being a logarithmically
convex sequence, every element e £ E for which there exist constants
c and L such that
(7.22) \\Ake\\<cLkMk, Vk.
Of course, it is not obvious (and in fact not true!) that such non-zero
vectors always exist. • Nevertheless, we shall set (for the time being an
algebraic definition):
ID(A°°; Mh) = space (maybe reduced to {0}) of
n
'Vectors of class Mk" in the sense of Definition 7.1. D
We easily verify that
D(^4°°; Mk) coincides with the space of e's £ E
(7.24) \ such that the function t^G(t)e is of class Mk
irom t>0^ E. D
We have the following: condition of non-triviality of D(^4°°; Mk):
Theorem 7.1. // the sequence {Mk} satisfies (1.22) and (1.23), the space
D(^4°°; Mk) is dense in E.
Proof. Let BMjc (]0,oo[) = {cp \ <p£ ^(]0, oo[), <p of class Mk on
t > 0}. Since Mk is not quasi-analytic, this space is different from {0},
and there exists (see S. Mandelbrojt [1], C. Roumien [1]) a sequence
Qn^^Mk (]0>°°D which satisfies the conditions given in the proof of
Proposition 7.1.
If e £ E, we have: G(qh) e^ e in E, and therefore we shall have the
desired result if we can show that
G^)^D(^°°;M,), VeeE,, V^£ ^Mfc (]0, oo[).
But we have (7.21), which shows that
^.G(t)G(f)e = G(cpM)e,
so that
dtk G® G(<p)e
<f*\\e\\f \qt»®\dt<
0
< CLkMk (since <p £ @Mje Q0, oo[)),
therefore t^- G(t) G(q>)e is of class Mk. Q
We note that the preceding proof also shows that D(A°°t Mk) is
dense in T>(A°°).

7.2 The Space of Vectors of Class Mk 133
Remark 7.2. The hypothesis of non-quasi-analyticity of the sequence
{Mk} is not always necessary for the non-triviality of D(^4°°; Mk).
Indeed, let Mk = 1, V*.
Assume that A admits a complete system of eigenfunctions:
(7.25) AwJ =ljWjt WjtDiA).
Then
wy€D(i4~;MA), V/,
thus D(^4°°; Mk) contains the space generated by the w/s and is therefore
dense in E. D
Remark 7.3. Here is an example for which
(7.26) D(^°°; Mk) = {0}, {Mk} quasi-analytic.
Let E = 2/(0, oo), 1 < p < oo, and let G(t) be the semi-group of
translations defined by
10 for x< t,
a.e. in*,/£2/(0,oo).
f(x — t) for x > t,
Then
B(A) ={f\f,fe Z*(0, oo), /(0) = 0}, 4/ = /'.
If we let Mk = k\, the elements of D(^4°°;Mft) must be analytic
functions / on x > 0 with /<*>(0) = 0, V*, therefore / = 0. D
Remark 7.4. Topology on D^00; Mk)
Taking into account (7.24), we introduce
(7.27) D^~;M,)=Ljsup^<ooJ,
which is a Banach space with the norm
|| e ||= sup
*- LkMh
We then provide D(^4°°; Mk) with the topology
(7.28) D(4°°;Mj) =indUmDi(^0°;Ms).
L-^oo
The mapping
(7.29) e^"t^G(t)e"
(by definition!) maps D(^°°; Mk) -> 3)Mlc ([0, T];E).
We easily verify, using Chapter 7, Section 4, that the topology defined
in (7.28) coincides with the topology such that the mapping (7.29) is an
isomorphism of D(^4°°; Mk) onto its image, provided with the topology
induced by 3JMk ([0, T] ;E). D

134 7. Semi-Groups
Remark 7.5. The preceding considerations extend to the case in
which E is not a Banach space (for semi-groups in locally convex spaces,
see K. Yosida [1]). D
We shall now prove
Theorem 7.2. If the injection o/ D(^4) —> E is compact, so is the injection
of DL(A00;Mk)->DL'(A00;Mk), L < V (and* then, see Chapter 7,
Section 1.2, the space D(A°°;Mk) is an inductive limit of a regular
sequence of Banach spaces).
Proof. For the purposes of this proof, let us write DL instead of
DL(A°°; Mk)y etc. For «, v 6 DL, we have w, v g Du and
„ ||^4*(«_ V)||
<suP "^r^'+sup"^-^
and therfore
\\Ak(u — v)\\ I L\N
(7.30) ||**-*;||dl'< sup " ±,k*, +\l7) IK
(rf
£'*M„
o<fc<iv -^ iKIfc
Now let % be a bounded sequence in DL, thus
\\AkuJ < CilSMt, V£,Vn.
In particular un is bounded in D(^4?), V#. Since the injection "T)(A) -> E"
is compact, so is the injection D(^4?+1) -> V>[Aq) and consequently we can
extract a subsequence «„ from un such that
uv->u in D(^4?) strongly, V^
and
M*«|| < ciLkMk> Vk> therefore u£T>L.
Let us show that uv -> u in DL' strongly. To this end, let e > 0 be given.
Choose N such that
-ffl*
Zc^-) <e/2.
Then
\\Ak(uv-u)\\ e
||«„-«||Di'< sup IL_l7^rJIL +
and since uv-> u in D(^45) strongly, we have
sup —w -T for V^"W'
whence ||«„ — «||Di/ < e for i> > v(e). D
Remark 7.6. The principle of this proof is analogous to the one given
for the spaces <3Mjc (Q); see Chapter 7, Section 1.2. D

7.3 The Semi-Group G in the Spaces D(A°°; Mk). Applications 135
7.3 The Semi-Group G in the Spaces D(A°°; Mk). Applications
7.3.1 General Results
Theorem 7.3. Let the sequence {Mk} satisfy (1.22), (1.23), (1.24).
Then, for every t > 0:
(7.31)
G(t)€&{D(A">;Mky,D(A<»;Mk)).
t->G(t) is an infinitely differ eniiahle function of t> 0 -> J2?(D(^4°°; Mk);
~D(A°°;Mk))] finally
(7.32) G'(0) = -A.
Proof. Let e £ ~DL(A°°; Mk). Then
(7.33) AkG(t)e = G(t)Ake,
therefore
\\AkG{f)e\\<cfiLkMk,
which shows that
(7.34) G(t) e ^{T>L{A°°; Mk)\ BL(A°°; Mk)),
whence (7.31).
Let us now that show
(7.35; t->G(t) is a continuous mapping of
t>0->^(DL(A°°]Mk); T>LH(A°°'yMk)y[1».
Indeed, if we set
||G(* + h) - G(t) || = norm of G(t + h) - G(t) in
^{DL(A°°; Mk):T>LH(A°°; Mk)),
we obtain:
\\G(t-+h) —G{t)\\ = sup sup
e k
< sup sup
e k
A\G(t +h)- G(t)) e
LkH»Mk
t + h
f G{a) (Ah+1 e) d<7
\e\\DL(A'x>,Mic)
<
LkHkMk
\ e\\DL(A'x',M)c) ^
M*+1e|| _x
< nh sup sup jj^j- II e \\dl{A co „„ <
<L[jih
((1)) Provided with the uniform norm of the operators.

136 7. Semi-Groups
(since M*+^||<.||*^
Therefore we have (7.35) and then
(7.36) t -> G(t) is a continuous mapping of
t > O^J^(D(^~;Mfc);D(^~;ikg).
But
(7.37) ^G(0 = (-1)*G(*M*,
and since Ak £ ^(D^00; Mk); D(A°°;Mk)) (thanks to (1.24)), it follows
from (7.37) and (7.36) that £->■ G(t) is an infinitely differentiable mapping
of t > 0^ ^(D(^°°; Mk); D{A°°; Mk)).
Finally (7.32) follows from (7.37). D
Corollary 7.1. If u0£D(A°°) Mk) and if the sequence {Mk} satisfies
(1.22), (1.23), (1.24), then the solution u of the problem
du
(7.38) — + Au = 0, u(0) = u0,
at
is a C°°-function oft>0^ T>(A°°; Mk). D
Remark 7.7. The solution u(t) of (7.38) is also (by definition of
T>(A°°] Mk)) of class Mk with values in E. D
Remark 7.8. With stronger hypotheses on the sequence Mk, we can
strengthen Theorem 7.2. For example, let there exist a constant d such that
(7.39) Mk+J<dk^MkMy Vk,j.
Then the semi-group t^G(t) is of class Mk from t > 0^ D(A°°;Mk).
Indeed, let e be given in B(Aco]Mk)> thus in a space BL(Aco; Mk).
We shall verify (and this is sufficient to prove our assertion) that
t^G{t)e belongs to 3jM]c ([0, 1]; DdL(A°°; Mk)).
Indeed
1 n^x,* „ 1 \\Glt)WA>e\\
i \\Ai+ke\\ Mj+k \
<
^ ^ SUP lT^r = v\\e Lha ~ Mfc). D
M,
Note that if {M^} is a Gevrey sequence ;
Mk = ((#) 0^ integer £, j8 > 1,
then (7.39) holds; see Chapter 8, Section 1.2.

7.3 The Semi-Group G in the Spaces D(A°°; Mk). Applications 137
7.3.2 Application to Parabolic Operators
Let
[ A = A (x, djdx) = elliptic operator of order 2m,
(7.40) < Bj} 0 < / < m — 1, be a system of boundary
I conditions covering A.
Assume that {A, B^ is a variational system (see Chapter 2, Section 9).
Thus we can find:
a space V = closed subspace of Hm(Q) of v's
such that BjV = 0, for / 6 a (possibly empty)
set of indices of [0, 1, ..., m — 1],
a continuous sesquilinear form u,v^- a(u, v) on V, such that
(7.41)
D(A) = {v | v g J^2w(i3), 5.v = 0, 0 < / <
w
1}
is equal to {v \ v £ H2m(Q) A V, a(v, w) = [Av, w),VweV}.
We shall assume that
Rea(v, v) + ?l0\v\2> <x\\v\\2, oc>0,
\\v\\ = norm oi v in V, \v\ = norm of v in H = L2(i3) ((1)). Then
Proposition 7.2. TA^ operator —A, with domain T)(A) defined in
(7.41), is the infinitesimal generator of a semi-group of contractions in H.
Proof. Indeed, the equation
Au+Au = f, feH, ue !>{A),
admits a unique solution for Re A > A0 and which satisfies
1
ll«ll<:
Re A + c '
|, ReA>A0.
The desired result then follows from the Hille-Yosida theorem (see
Yosida [1]). D
Now assume that
(7.42)
the boundary r of Q is of Gevrey class of
order s, s > 1, and the coefficients of A and jB;
are of Gevrey class of order s in Q or on r.
(0)) For the choice of other "pivot spaces", see Chapter 2, Section 9.

138 7. Semi-Groups
Let
(7.43) Mk = ((2km)\)s.
According to Theorem 1.2, Chapter 8, we have:
(7.44) D(A°°; Mh) = {v \ v £ ®S(Q), B^v = 0, Vft, 0 < / < m - 1},
where (see Chapter 8)
Sfs(Q) = space of Gevrey functions of order s in Q;
(7.44) can also be stated in the equivalent form:
(7.44a) T>{A°°;Mh) = T>{A°°) f\&t{Q).
Remark 7.8 then yields the following result:
Theorem 7.4. Let u0 belong toD(A°°; Mk)t defined by (7.44). Then the
solution u(xt t) = u of
(7.45)
/ 0\
I x, — I u = 0,
\ 9*1
—-+A[x, — )u = 0txeQ,t>0,
BjU^O, xeT, t > 0, 0 < j< m — 1,
[u(x, 0) = u0(x), x^Qy
is a Gevrey function in t of order 2sm for f> 0 with values in the space of
Gevrey functions of order s in Q. D
7.3.3 Application to Schroedinger Operators
Analogous results can be obtained for the equation
(7.46) ?£ + iAu = 0,
ot
if the system [Ay B^ is self-adjoint, the other hypotheses remaining
unchanged. D
7.3.4 Application to. Operators of the Second Order in t
Let V be given as in Section 7.3.2.; thus
[H^(Q)CVCHm(Q)9
(7.47) ] V being defined by differential boundary conditions
I with Gevrey coefficients of order s > 1,
and let A, T>(A), a(u, v) also be defined as in Section 7.3.2.

(7.48)
7.3 The Semi-Group G in the Spaces D{A°°; Mk). Applications 139
Assume that:
the coefficients of A and B^ are Gevrey functions of
order s and r is of Gevrey class of order s,
(7.49) for suitable A, a(v, v) + X ||w|||,(fl) > oc \\v\\2v, Vv £ V, oc > 0,
(7.50) a(u, v) = a(v, u), Vu, v£V.
We would like to study the regularity properties of the problem:
(7.51) u" + Au = 0, xeQ, t> 0,
(7.52) Bp = 0, xeT, t> 0, 0<j<m- 1,
(7.53) u(x, 0) = m0(#j, «'(#, 0) = ux(x)y x^Q. D
We introduce:
'E=VxHy (H = L2(Q)),
-c:)
D(rf) = D(4)x7.
(7.54)
Proposition 7,3. The operator —stf is the infinitesimal generator of a
semi-group {in fact a group) in E.
Proof. Changing u to exp (kt) u} equation (7.51) becomes
u" + 2ku' +(A + k2)u = 0;
k is chosen so that
{(A+k2)vyv)>oc\\v\\2v.
We are thus led back to
(7.55) u" +fiu' +Au=-0,
with
(7.56) /?> 0, (Avtv)>oc\\v\\2v.
(7.55) is written as a first order system in t\ setting
du1
u = u1,
6i
u2f u = {u1, u2},
(7.55) is equivalent to
(7.57)
d*
+ s4m = 0,

140 7. Semi-Groups
where
«,58, *•-(!""/)■
E is provided with the norm
(7.59) \\u\\E = («(«i, «i) + ||^|2)i/2 (where |/| = ii/n^).
We shall show that — st$ is the infinitesimal generator of a group of
contractions in E (provided with the norm (7.49)).
To this end, let us examine the equation
(7.60) jtfpu + Xu=Jt J= {f\ f2} given in E,
that is
(7.61) -u2 + Xu1 = f\ Au1 + (X+P)u2 = f2,
whence
(7.62) Au1 + X(X + j8) «i = /2 - (A + j8) Z1,
equation which admits a unique solution if Re X > 0 or if Re X < — /?.
Taking the scalar product of the first (resp. second) equation (7.61)
with Au1( resp. u2) and adding, we obtain
-(u2, Au1) + Xa(u\ u1) + {Au1, u2) + (X + p) \u2 \2 = (/*, ^w1) + (/2, w2),
whence
(Re X) ||2||| + j8 |^212 = Re («, jT)fi.
It follows that
1
ReA1
W\\e< Af\\E if ReA> 0,
W^p4^)l|7||£ifReA<^'
which proves our assertion. Q
Remark 7.9. The preceding calculation is of course very similar to
those of Chapter 3, Section 8.3; see also the Comments to Chapter 3. Q
If we now take
(7.63) Mk = {(km)\)st
with
fs> 1 if m> 2,
(7.64)
\ s > 1 if m = 1,

7.3 The Semi-Group G in the Spaces D(A°°; Mk). Applications 141
then we can apply Remark 7.8. Thus:
if u° = {u0, u-^ G D(ja/°°; Mk), the solution
(7.65) j j_^ u(t) of (7.57), with u(0) = u°, is of class
[ Mk with values in D(jtf°°; Mk). D
There remains to interpret the fact of belonging to D(j/°°; Mk). We
shall need
Lemma 7.1. Let {Mk} be a logarithmically convex sequence satisfying
(1.24). Let srf be the infinitesimal generator of a bounded semi-group G(t)
in a Banach space E. Then
(7.66) D(^°°; Mk) = {e \ e£ T>{sf°°)9 \\^2ke\\ < cL2kM2k, V£}.
(In other words, instead of considering all powers stfk, it is sufficient to
consider the even powers jtf2k ((1)).
Proof. If we are willing to change G(t) to exp (—cot) G(t), which does
not affect the space D^00; Mk) of vectors of class Mk, we may assume
that
(7.67) li^)||^(£;£)<^e-^ (co>0).
Let us for the moment admit that
I there exists a constant c, such that, Veg D(^/2),
\\de\\<Cl\\s42e\^\\ef\
Let e satisfy
(7.69) \\^2he\\<cL2kM2k, V*.
We have
ll^-^H = \\s#{s#%k-h)\\ < (by (7.68)) cx \\d2(s#2k-2e)f2 X
X||^2*-2e||1/2 < (by (7.69)) cx{cL2kM2h)112 X
X{cL2h-2Mu_^l2.
Ma
But M2k<H2kMu_-i, M2k_2< 7^ilf2J_i (since, according to the loga-
Mk+i Mk M-i
rithmic convexity, —r-=— > —— > • • ■ > ■—r), so that
y' Mk - Mk_x~ ~M0h
f^-hW < cc, l^f* L2h~xMhM.
*m
kxvx2k-l>
(0» More generally, D(j^°°; Mk) = {e \ e£ D(j/°°), || j**le\\ < cLklMkq, Vk), q
being any fixed positive integer.

142 7. Semi-Groups
therefore e£D(jrf°°; Mk). Thus we have the desired result under the
assumption (7.68).
Proof of (7.68). If e e D(jaf2), the function
t-+G(t)e = u(t)
satisfies
du d2u
thanks to (7.67).
Applying inequality (3.18) of Chapter 1 (with X = Y = E, m = 2,
7 = 1), we obtain:
Il^(0)||<^||^||^0,oo;£)ll^ni^0,oo;£).
NOW W'(0) = —J**, ||«||L-(0foo;£) < H IMI> ||«"||l»(0,oo;£) < ^3 ll^ll> ^0111
which the result follows. Q
We now apply Lemma 7.1 to the operator
-p
(i.e. in the form (7.54), without a preliminary reduction of the problem
by changing u to exp (kt) «). We have
/A* 0\
*?2k = (-1)* .
\0 4*/
and consequently, according to Lemma 7.1, we see that
D(ja/°°; Mj) = {u | « == {V, «2}, «' 6 D(4°°),
(7.70) || A V ||F < ci2*M2S V*.
Theorem 7,5. Assume that (7.47), ..., (7.50) Ao/<£ and that the sequence
{Mk} is given by (7.63) ze>#A (7.64). Then, in the notation of (7.54), if
u = {u\u2}eB(^00;Mk)}
we have:
(7.71) «*eD(4~) ASJfl).
Proo/. For ^2, this follows from (7.70) and Theorem 1.2, Chapter 8.

7.4 The Transposed Settings. Applications 143
For u1, we note that, in particular,
(7.72) \\Akul\\v>c\\Akui\\L,m,
so that the result is again a consequence of Theorem 1.2, Chapter 8.
Note that nothing has been lost by using the (crude) inequality (7.72),
since according to Chapter 2, we have in particular
\\Aht\\v<c\\A»+^\\L>{Q),
so that it does not matter whether we take the inequalities in L2(Q) or
in V. D
According to Theorem 7.4, we may now state (7.65) in the form:
Theorem 7.6. Let the hypotheses of Theorem 7.5 be satisfied. Consider
the problem (7.51), (7.52), where u0, ure D(^4°°), i.e.
u{ e 2{Q), BjAku0 = 0, BjA^ = 0, V&, 0 < / < m - 1.
Assume that u0 and ux are Gevrey functions of order s in Q. Then £ —>- u{. ,t),
solution of the problem, is a Gevrey function of order sm for f> 0 with
values in the space of Gevrey functions of order s in Q.
In this statement, s > 1 if m > 2 and s > 1 if m = 1. D
7.4 The Transposed Settings. Applications
7.4.1 The Space D(A*°°;Mk)'
Let G*(£) be the adjoint semi-group of G(t) in &(E';E'). If —A* is
the infinitesimal generator of G*(t), we see that A* is the adjoint, in the
sense of unbounded operators, of the operator A in E, with domain
T>{A).
We thus introduce D(^4*°°; Mk) as above, replacing A with ^4*.
We assume that the sequence {Mk} satisfies (1.22), (1.23), (1.24), so
that, according to Theorem 7.1, D(^*°°; Mk) is dense in D{A*°°), D{A**)
and E''. Therefore, by duality (or anti-duality), we can consider the dual
(or anti-dual) space D(^4*°°; Mk)'. We obtain:
(7.73)
D(^°°; Mk) C D(^°°) C V(Aq) C £ C D^**)' C T>{A*°°Y C D(^*°°; Mk)'.
The spaces D(A*q)' ... are provided with their strong dual (or anti-
dual) structure. D

144 7. Semi-Groups
By applying the Hahn-Banach theorem and some standard properties
of topological vector spaces, we obtain the following structural result:
every element / of D(^4*°°; Mk)' may be represented,
non-uniquely, by
oo
/= 2Akek, A0 = identity,
(7.74) i k=o
where
oo
ekeE, ZLkMk\\ek\\<oo,VL.
fc=0
oo
In (7.74), the equality "/ = 2 Akek" is understood in the sense:
k = 0
if ^D(i*°°;Mfc), then
oo
(7.75) </, e) (= value of / at e) = £ <eh, A*ke}.
oo
Conversely, if the e^'s are given as in (7.74), / given by / = 2 Akek
(in the sense of (7.75)) belongs to T>(A*°°; Mk)'. D
k = 0
7.4.2 The Semi-Group G(t) in D(A*°°; Mfc)'
In order to avoid difficulties of a "topological vector space'' nature,
we shall assume that
(7.76) the injection T)(A*)^E' is compact.
Remark 7.10. This hypothesis is realized in all the examples to be
considered, as long as Q is bounded and sufficiently regular.
Also note that the results to be obtained below are valid without
hypothesis (7.76) by replacing (which in fact may be unnecessary!) the
differentiability by scalar differentiability, etc. Q
Theorem 7.7. Let the sequence {Mk} satisfy (1.22), (1.23), (1.24) and
assume that (7.76) holds. Then G(t) is a semi-groups in D(^4*°°; Mk)'.
For every /£ D(^4*°°; Mk)', t^G(t) f is an infinitely differentiable
function oft>0^ D(^*°°; Mky.
By transposition of the property "A* g if(D(^*°°; Mk); D(^*°°; Mh))",
it can be seen that A extends by continuity to a continuous linear operator,
still denoted by A, of ~D(A*°°; Mk)' into itself. The infinitesimal generator
of G(t) in D(^*°°; Mk)' is -A.
If the sequence {Mk} satisfies (7.39), the function
(7.77) t^(G(t)f,v}t feD(A*°°;Mky, ^D^JIfJ,
is of class Mk.

7.4 The Transposed Settings. Applications 145
Proof. Under hypothesis (7.76), the space D(A*°°;Mky is (see
Remark 7.5) a Frechet space; it is therefore, in particular, quasi-complete
and, according to a lemma of Grothendieck [3] (see also L. Schwartz [2],
page 146), in order to show that the function
*->G(Q/, feD{A*~;MkY,
of {t > 0} -» D(^*°°; Mky is of class C°°, it is sufficient to show that the
function
*-><G(*)/,v>, ^D(i*°°;MJ
is of class C°°. Now
<G(f) /, v} = </, G*(t) v}
and the result follows from Theorem 7.2 (with G*, A*, ... instead of
GyA,...).
If (7.39) holds, then the function (7.77) is of class Mk, according to
Remark 7.8. D
The following is another property of the semi-group G(t) in the space
D(A*">;Mk)':
Theorem 7.8. Let the hypotheses of Theorem 7.6 be satisfied and assume
that (7.39) holds. Then, for every function cp £ 2)Mlc (]0, oo[) (see the proof
of Theorem 7.1), we have
(7.78) G{<p)£&\p{A*~\Mh)'\ D(A°°;Mk)).
Proof. Since, according to Remark 7.5, the space D(^4*°°; Mk)' is a
Frechet space, the closed graph theorem holds from D(^4*°°; Mk) '-^
T>(A°°'tMk). Now G(<p)GJ^(D(^*°°;Mfc)'; T>(A*°°] Mk)'), according to
Theorem 7.7, and therefore it suffices to show that
(7.79) for / given in D(^*°°; Mk)\ G{<p) f belongs to D(^°°; Mk).
But let us apply (7.74). Then
oo
(7.80) G(<p) f = 2 G(cpW) ek
k = 0
and we shall show that (7.80) converges in a space DL(^4°°; Mk) for a
suitable L.
By hypothesis
\<p{k)(t)\<cLkMk, Vk,Vt.
We shall verify that G{<p) f€ DdL(A°°; Mk). Indeed, subject to
convergence, we have:
oo

146 7. Semi-Groups
from which we obtain (the c's denoting various constants):
ll^'G(y)/||<cf L^Mk+j\\eh\\ <
< (by (7.39)) c d>V [m3 J] LkMk K||) <
< (by (7.74)) c djLjMj} whence the desired result. D
Remark 7.11. It is possible to go a little further if the semi-group G is
analytic, i.e. if \fe£E, the functiont->G(t)e is analytic from t> 0->Ei.e.
(7.81) \\G{k)(t) || < cLkk\, Vk, t e [t0, t]} 0 < t0 < tt < oo.
(For analytic semi-groups, see K. Yosida [1].) Then
Theorem 7.9. Let the semi-group G(t) be analytic; under the hypotheses
of Theorem 7.7 and assuming the existence of a constant d1 such that
(7.82) k\ <d]Mh9 Vk,
we have:
(7.83) G{t)€&(T>{A*°°;Mky; B(A°°;k\)), t > 0((1)).
Furthermore, for /£ D(^4*°°; Mft)', the function t->G(t)f is analytic
from{t> 0}->D(A°°;k\).
Proof. Indeed, in order to show (7.83), it is sufficient, as in the proof
of Theorem 7.8, to show that for / given in D(^*°°; Mk)', G(t) f belongs
toD(^°°;^!).
Now, according to (7.74),
G(0/=f(-1)*G,*)We*
k=0
and, subject to convergence:
A'G{tyf) f = J (—l)k+jG{k+j+f){t) ek.
Therefore, for t £ [t0, y, 0 < t0 < tx < oo, we have:
\\A*G(t)M f\\ < £ cLh+>+'(k + / + r)! K|| <
oo
< c 2 Lk+j+f 2k+j+fk\ j\r\\\ek\\<
oo
< (by (7.82)) c(2L)!+' j\ r\ £ (2^ Lf Mk \\ek\\ <
<c(2L)j+fj\rl,
which proves the theorem. Q
(W) If the semi-group is analytic, T>(A°°; k\)^ G(t0) • E, Vt0 > 0, therefore does
not reduce to {0}.

7.4 The Transposed Settings. Applications
147
7,4.3 Applications
All results obtained in Section 7.4.2 apply in the settings of
Theorems 7.4 and 7.6.
If, for example, in the setting of Theorem 7.4, u0 is given in T>(A*Y,
resp. D(^4*°°)', resp. D(^4*°°; Mk)', then there exists a unique solution to
problem (7.45), which is continuous, resp. C°°, resp. of class Mk, with
values in D(4*)', resp. D(4*°°)', resp. D(^*°°; Mk)'. D
Therefore, if, for u0 given in D(^4*°°; Mk)' (for example), the solution
£-> u(t) is a very regular function—of class Mk—with values in the 'Very
coarse'' space D(^4*°°; Mk)', it is also a "very general distribution in t"
with values in a "very regular space1; more precisely: for y £ @Mlc (]0, oo[),
oo
0
belongs to ~D(A°°;Mk).
Thus, for fixed t, u(t) does not satisfy the boundary conditions of the
oo
problem, but J u(t) cp(t) dt does. Q
o
It must also be carefully noted that, since 2f{Q) is not dense in
D(4*), D(4*°°), D{A*°°;Mk), the spaces T>{A*)', ... are not spaces of
distributions on Q, but are spaces which contain "transversal layers to
Fy\ respectively of finite order, arbitrary finite order and infinite order.
Here is a very simple example. We take
an unbounded open set (see Remark 7.10), and
d2
(7.84) A = - — , B(A) = H2(Q) A #J(fi).
Then A = A*, and we take w0GD(i*)' defined by
(7.85) <w0, v} = -v'(0), v £ H2(Q) A #J(fl).
The space (D(^4*))' can be identified with <F'(R)/[<5], quotient of
^'(R) by the line [d] generated by the Dirac mass 6 at the origin.
Then the solution of problem (7.45) is
(7.86) ^^-^fXPK)'i>0-
As £-> 0, u(.,t) does not converge in the sense of &(]0, +oo[),
but in the sense of D(^4*)'; if we extend u(.,t) to u(., t) by 0 for t < 0,
we obtain: u(., t)^» u0 in <^'(R)/[<5] (dual of the space of functions in
^(R) vanishing at 0).
1 This is one aspect of the "uncertainty principle", in the form encountered by
L. Schwartz in the theory of kernels.

148
7. Semi-Groups
Remarks of the same type hold if we take u0 in D(^4*°°)' (resp.
D(A*°°;Mk)') by
(7.87) <u0,v}= £ V^+])(0),
(resp. by
finite
(7.88) <%, v} = £ ^<2*+1>(0), 2 £X | c» | < oo, VL). D
fc=0 fc = 0
Let us at this point insist on the importance of the space D(A°°; k\)
which appears in the case of analytic semi-groups (see Theorem 7.9),
and of its dual. We shall again encounter these spaces on several occasions
in the case of parabolic equations (see Section 10 and Chapter 10). It
would therefore be of great interest to characterize the space D(^4°°; k\)
when A and D(^4) are given as in Section 7.3.2; see Problem 6.2,
Chapter 8; in the case of space dimension 1, D(^4°°; k\) is a space of certain
entire functions; see Lions-Magenes [2], Section 7.
7,5 Another Mfc-Regularity Result
Let us now consider the problem
(7.89)
(7.90)
^ + A(t)u = f(t), te]0,T[,
«(0) = 0,
where, for each t, —A (t) is the infinitesimal generator of a semi-group in E.
More precisely, we make the following hypotheses:
(7.91)
(7.92)
(7.93)
for each t £ [0, T], A(t) is closed, with domain
T>(A(t)) dense in E, X — A(t) being invertible
for X^Z, where 27 = {X \ d < argX < 2tz - 0},
0€]O,*r/2[;
\t^ Ait)-1 (which exists according to (7.91)) is an
1 infinitely differentiable mapping of [0, T] -> J2? (E; E);
there exist c and L such that
A{t))-A
<W]L»Mk,Vk.
Then the following result, which is due to Tanabe [1], holds:
Theorem 7.10. Assume that hypotheses (7.91), (7.92), (7.93) hold and
let the sequence {Mk} satisfy (1.22), (1.24), (1.25), (7.39) and Mk < Mk+1,
Vk. Lei f be of class Mk in [0, T] with values in E. Then there exist constants

7.5 Another M^-Regularity Result
149
c and L such that
(7.94)
f!*_
d^
u(t)
<cLkMJ?-k. D
We refer the reader to Tanabe [1] for the proof of this theorem; he
uses sharp estimates on the "kernel'' U(t, s) defined by
" d
— U(t, s) + A(t) U(t, s) = 0, 0<s<t<T,
ot
U{sts) = 1, 0<s<T. D
Remark 7.12. Here the sequence {Mk} may be quasi-analytic. In
particular, we can take Mk = k\. We then obtain the fact that the
solution u(t) is analytic in t > 0 with values in E. Q
Remark 7.13. The estimate (7.94) brings into evidence a singularity
at t = 0. This singularity really exists and is not due to the method of
proof. Indeed, assuming for the sake of simplicity that A (t) = A, it
formally follows from (7.89) and (7.90) that
: ^ (-l)k~j Ak-jf{j){0),
(7.95) w*(0)
so that there is a singularity at t = 0, unless we assume that
/(j)(0)GD(r), v/.
In fact this condition is not sufficient if we want that
d*
d^
u(t)
< cL*Mk, Vk and VzJ £ [0, T].
For the cases studied in the preceding sections, we have assumed that
/£ ^+,Mfc iy')> assuming that 0 is the left endpoint of the support, we
then had
/tf>(0) = 0, V/. D
Remark 7.14. Consider problem (7.89) with
(7.96) «(0) =«0, u0eE.
Then there exist constants clf Lx such that
(7.97)
d*
< cLkMktl~k + ^LfM^r*, VA.
Here again (as in Remark 7.13) the singularity at J = 0 really exists,
except if u0 and /(0), /'(0),... satisfy compatibility relations which we have
not given (see Problems 14.8 and 14.9). Q

150
7. Semi-Groups
Remark 7.15. Here is a very partial result concerning the preceding
remarks. In the setting of Section 1, consider the problem (7.89), (7.96);
assume that A(t) is given in S£(V; V) by a(t; u, v) and that the function
t-> a(t; u, v) is of class C1 in [0, T], \fu, v£V. Further, assume that
u^V, A(0)uoeV,
/il2(0J;F),/(0)GF.
at
(7.98)
(7.99)
Then
(7.100) u' £ L2(0, T\ V), u" e L2(0, T\ V).
Indeed, we can find a function 0 having the following properties:
0 is a twice continuously differentiable function
of t<0->V,
(7.101) \ 0(t) =0 if t < — 1 (for example),
<P(0)=«o,
<Z>'(0)=/(0)-,4(0)*v
Next, we introduce
(7.102) w(t)
\0{f), if t < 0, \A(t) 0(t) + 0'(t), t < 0((1)),
[u(t), if 0 < t < T, \f(t), if 0 < t < T.
Since, if u is the solution, according to (7.101), we have:
0(0) =«(0), A (0)0(0) +0'(O) =/(0),
we see that
'#', if t < 0
w = <
F' =.
[«', if 0 < t < T,
and therefore
A'0 +A0' +0", if t < 0
f',iiO<t<T
and
F^eL^-oo.T'.V)
F in — oo<t< T.
w + A(t) w ■
Furthermore, the supports of w and F are bounded on the left. Thus
we may apply the method of differential quotients (see Section 1), from
which we deduce (7.100). D
(U)) We assume, which is permissible, that t -> a(t\ u, v) is extended to a function
of class C1 on — oo < t <C T.

8.2 MrRegularity Results
151
8. Mfc-CIasses and Laplace Transformation
8.1 Orientation-Hypotheses
We now consider operators A which are more general than the
(negatives of the) infinitesimal generators of semi-groups studied in the
preceding section.
More precisely, we consider the setting of Chapter 4, Section 3.
Let H be a reflexive Banach space, A an unbounded, closed operator
in H with domain D(^4) (provided with the norm of the graph).
Assume that
(8.1)
A + p is an isomorphism of T>(A) -> H for Re p > £0,
such that
II (A + P^W^Hm) < polynomial (\p\).
We shall see that under these conditions, the "basic results"
pertaining to Mfc-regularity (and the transposed results) remain valid.
Consequently, the Af^-regularity results are, in particular, valid for
the general parabolic systems with coefficients independent of t, studied
in Chapter 4, Section 4. We do not specify the corresponding results,
since they are immediate. Q
Remark 8.1. In no way have we tried to put the theory into the
framework of maximum generality. For example, one could consider the
framework of Lions [5], Chapter 11, Section 5. Q
8.2 MrRegularity Results
Theorem 8.1. Assume that (8.1) holds and that the sequence {Mk} satisfies
(1.22), (1.23) and (1.24). Then A + d/dt is an isomorphism of
@+yMh (R; T>(A)) onto ®+yMlc (R; H).
Proof. Reasoning as in the proof of Theorem 1.2 for the "topological"
aspects, it all comes down to verifying: if /G^+,Mfc (R>"#), then the
solution u (which exists and is unique in ^'+(R; ^(^4)), according to
Theorem 3.1 of Chapter 4) of
(8.2) Au + «' = /
satisfies
(8.3) «G0+>m*(R;D(^))-
Since if u£<2)+}Mlc (R;#), so does u' (thanks to (1.24)), and thus
Au = / — u' £ @+tMk (R> H), we shall have (8.3) if we can show that
(8.4)
ue®+yMk(R;H).

152
8. Mfc-Classes and Laplace Transformation
There is no loss of generality by assuming that u = 0 for t < 0.
According to equations (3.4), Chapter 4, the solution u of (8.2) is
(8.5) u = &*f,
where ^ (inverse Laplace transform of (A + p)"1) satisfies
[9e&(9-{R);&{H;I>{A))),
I ^ =, 0 for t < 0.
(8.6)
But then, according to the theory of L. Schwartz of distiibutions with
values in Banach spaces, <& is of finite order on every compact set; thus,
for arbitrarily fixed finite T, we can find 0(1) such that
(8.7) t^O(t) is a continuous mapping of [0, T] -> g[H; ~D(A)),
and such that, for suitable I,
(8.8)
Then (8.5) yields
(8.9)
But
(8.10)
and therefore
(8.11)
0 =,
^0 on]-oo,T[,
\0 for t> 0
0 for t< 0.
&*—;j for t< T.
it? '
km,(R;^).
u = &* g for t<T.
Thus there remains to be shown that if
||g<*>(*)||H <cL*Mft, Vk,t<T,
we have analogous inequalities for w given by (8.11), which is immediate,
since
«(*>(*) = 0 ■* g<*>(*) = J 0(t- a) g{k){a) da,
therefore
\»WMh < SUP \\0(t)\\<?{H;H) SUP ||g(*V)lltf < C.cL'M, Vk. Q
^[0,r] ' <r€[<M]

9.1 General Results 153
8.3 Transposition
We note that A* + p is an isomorphism of D(A*) onto H' for
Re p > f0, with \\(A* + p^W&iHtf') < polynomial (\p |), thus
(immediate variant of Theorem 8.1):
J A* — d/dt is an isomorphism of 2>_ M]c (R; D(4*))
(8-12) |onto^_,Ms(R;^)-
Similarly A* + /> is an isomorphism of #' onto D(-4)' for Re p > f0
and
|(il* +^)-1|l^(Dur;D(^))< polynomial (|£|),
so that
j A* — d/dt is an isomorphism of @J_ Mk (R; H')
By transposition, we obtain
Theorem 8.2. Under the hypotheses of Theorem 8.1, the operator
A + d/dt is an isomorphism of 3)\>Mlc (R; D(^4)) °^° ^V,m7<; (R; #)• D
Remark 8.2. It is easy to show that, under the same hypotheses,
^4 + d[dt is an isomorphism of
®+,Mk{R;D(A«))^®+iMk{R;I)(A*-% ?>1,
^+>m, (R; D(^*T) - ^+,m, (R; (D(^*9+1))'), ? > i
and analogous results hold if we replace ^+,Mfc with ^'+iMh ' D
9. General Operator Equations
9.1 General Results
We consider the setting of Chapter 3, Section 1. Thus we consider
the Hilbert spaces
(9.1) fC/Cf,
(/, v) denoting the scalar product between / £ V and v^V and || ||y,
|| Hjf, ... denoting the norm in f, ^, ...

154 9. General Operator Equations
Consider:
continuous semi-group G(s)
contractions in $?\
fa bounded, cont:
( ' ' [mir',Jfr,ir,of
f an operator si G Se{V\ iT\ wi
( ' \ (j*v, v)>oc \\v\$, <x>0, \fve-
with
In Chapter 3, Section 1, we have introduced the spaces
D{A, iT),D(il, JT), ...,
domain of ^1 in f, ^f,..., where —^1 is the infinitesimal generator of
the semi-group G.
We know (Theorem 1.1, Chapter 3) that
J under hypotheses (9.2), (9.3), the operator
^ ' ' [A + si is an isomorphism of i^ A V>{A\ V) onto i^'. D
We propose to give certain regularity results pertaining to the solution
u of
IAu + siu = /,
/Gf, ^fAD(4;f). D
We shall use the following notation (already used in Chapter 3,
Lemma 1.2 and in Section 7). Let <p £ ^QO, oo[); set
(9.6) G(p) = / G(t) q>{t) dt,
o
integral with values in $e{1T; tT), jS?^; jf), jS?(tT'; tT').
We shall use a regularizing sequence £w £ @o,Mk (see the proof of
Theorem 7.12), the sequence {Mk} being assumed non-quasi-analytic.
Thus:
(9.7) Qn £ ^0jMfc » ^w^ ^ in tne sense of measures on [0, + oo[.
We shall also make use of the usual bracket notation: if 0$x and ^2
are two operators, we set:
(9.8) [&v &2] = J^2 - J^. D
((])) stf plays the role of M in Chapter 3; we have changed the notation in order
to avoid any confusion with the sequence {Mk}.
2 For, for example Theorem 9.1, it is not necessary to take gn in ^o,Mfc J it is
sufficient to take Qn in ^(]0, oo[); but this choice permits us to pick@M once and for
all without loss in generality.

9.1 General Results
The first regularity theorem is
Theorem 9.1. Assume that (9.2), (9.3) hold and that
(9.9) \
[st £ £e{T>{A;r);T>{A; T')),
(c = constant independent of ri). Then, if
(9.10) f£D(A;r'),
we have
(9.11) x6D(i;-f)AD(i2;f).
Proof. Applying the operator G(q'h) to both sides of the equation
noting that [A, G(ip)] = 0, it follows from (9.5) that
(9.12) AG(e'n) u + ^G{q'n) u = G{q'n) / + [s/, G(e'n)} «.
But we also deduce from (9.5) (see Chapter 3, Section 1) that
(9-13) Hl*< —11/11*-;
(X
using (9.9), the same inequality applied to (9.12) yields:
(9.14) \\G(Q'n) u\\r < — (\\G(efn) /||r. + c \\u\\r).
(X
But
G(e'n)f^G(d')f=Af in r',
and consequently (9.14) implies:
(9.15) \\G(Qn) u\\r < constant.
It follows (see for example Chapter 3, Section 1) that
«6D(4; V)
and we deduce from (9.12) that
(9.16) A2u + s/Au =Af + \sf, A] u.
This implies that u £ B(A2; V"). D
Remark 9.1. If we take (in the notation of Section 1):
(9.17)
r == £2(0, T;V), X = L2(0, T; H),
s4u = "£->- A(t) u(t)",
A = d/d*. T>(A; Jf) = {/ I A /' € i2(0, T;H), /(0) = 0},

156 9. General Operator Equations
then hypothesis (9.9) means that
(9.18) \\^^)h(v;v') < constant, t g [0, T].
Then the above proof is a variant of the one given in 2), proof of
Theorem 1.1; indeed, if we take for Qn the function (evidently not in
^o.Mjc ••» kut as already pointed out, this hypothesis is in no way
indispensable for the moment):
In in [0, lift]
0 for t > l[n,
then
Gfe)/ = Ifn • D
Remark 9.2. Hypothesis (9.9) may also be formulated, in a somewhat
less rigorous fashion, as
(9.19> \\lj49A\v\\r.<c\\v\\r. Q
Remark 9.3. Of course the preceding result may be iterated; in this
way we obtain:
Theorem 9.2. Assume that (9.2), (9.3) are satisfied and that
| II [■* G(Qlk))l v\\r < ck(\\v\\r + \\Av\\r + .- + ||^-^|W,
(9.20) j V»€ Dfil*"1; 1T); ^GJ^D^', r);T>(Aj; r')),
[0 <j <k - 1.
(9.21) /GD(^;f),
(9.22) M € D(il*; 1T) A D(^+]; iT).
Corollary 9.1. Assume that (9.2), (9.3) ora satisfied and that (9.20)
/&o/is V& (without uniformity in k; the ckys depend on k). Then, if
(9.23) /GD(i°°;f'),
we have
(9.24) ^D(i°°;f). Q

9.1 General Results 157
Remark 9.3. In the setting of Remark 9.1, the conditions of
Corollary 9.1 are satisfied if t^ A(t) is an infinitely differentiable function of
[0, r]-*JS?(7; V). D
Let us now give a general M^-regularity result.
Theorem 9.3. Let (9.2), (9.3) be satisfied. Let {Mk} satisfy (1.22), ...,
(1.25). Assume that there exist c and L such that
\st, G(e<?>)] u\,. < c*£ (*) L'-'M^WA'uWr,
j=o v I
(9-25) *Vu£D{Aft-1\ir)9 and stf e JS?(D(;1'; r)\ T>(Aj ] *T)),
0 <j <k - 1,
inequality (9.25) being satisfied for all k. Then, if
(9.26) /GD(^°°, Mk\r')y
we have
(9.27) ue'D{Aco>Mk]i^).
Proof. The proof is in a certain sense an "axiomatisation" of the
proof of Theorem 1.2.
By hypothesis, there exist d, J2? such that
(9.28) MVlly <dJS?*Mft> V*.
By induction on k, we shall verify that
(9.29) \\Aku\\r < d&Mk, Vk,
where
(9.30) d = 2d/a, 5 = max (jl + ^), L sA .
Indeed we have:
(9.31) A{Aku) + j/il*« = [j/, Ak] u + il*/
and for u £ D(^; 1T), it follows from (9.25) that
(9.32) || [sft Ak] u\\r, < /£ lf\ V-'M^ \\Aiu\\r.
i=o y /
Then (9.13) applied to (9.31) yields:
(9.33) \\Aku\\r <- [\\Akf\\r, + c *£ (k\Lk-iMk_3 \\A^u\\r].

158
9. General Operator Equations
Relation (9.29) holds for k = 0. Admitting it up to (k — 1), (9.33) yields
1 (d<?kMk+cY(kX
\\Aku\W < -£- (d&Mh + c 2 n i*"X-i <^X) <
<Tif*Mft
<_B*Mi
-,. i k-i
£*Mt
5
<dBkMk. Q
9.2 Application. Periodic Problems
We consider the setting of Chapter 3, Section 6. Thus:
iT = L2(0, T; V) 2tf = L2(0, T; H),
(9.34) jtf = A(t) (i.e. j*m = "t^A(t) u(t)"), with
(,4(*K*>)>*H2, vev, te [0, r], *> o;
next we let
(9.35) A = d/dt}
D(A; JP)={f\ f, f e L2(0, T; H), /(0) = /(T)}.
Hypotheses (9.2) and (9.3) are satisfied. We are within the conditions
of applicability of (9.4). Therefore, for / given in L2(0, T; V')y there exists
a unique u in L2(0, T; V) with dujdt£ L2(0, T; F'), satisfying
(9.36)
^ + A(t)u = f, u(0)=u(T). G
at
Now, via an applications of Theorem 9.3, we shall prove
Theorem 9.4. Assume that (9.34) holds and that
U^ A(t) is a function of class Mk of [0, T] -> &(V\ V),
( ' [with A® (0) =A®(T), V/.
Further assume that the sequence {Mk} satisfies (1.22), ..., (1.25). Let f
be given and satisfy:
(9.38)
there exist d and ££ such that
T
I T vi/2
/W(0) =f{h)(T), Vft.
d&kMk, Vft,

9.2 Application. Periodic Problems
159
Then the solution u of (9.36) is a function of class Mk of [0, T] -> V
and satisfies u^(0) = u^k)(T)} VA.
Proof. Thanks to the hypothesis ",4^(0) = A^{T)9i V/, we have:
st e &(D(Aj; r); D(Aj; r')) V/
and the first part of (9.24) follows from the fact that t^ A(t) is of class
Mk from [0, T]^^(V; V). Then the theorem is a consequence of
Theorem 9.3. D
Example 9.1. Consider the setting of Example 6.1, with the
coefficients a^x, t) satisfying
d?
et*
Let us, for example, take V = H1^), H = L2(Q) and a(<p, ip) as in
Example 6.1.
Let fx and g be given with:
(9.39)
(9.40)
Vl/2
/ T V
U \f\k)(t)\2dt\ <^j?}M,VA,
f?)(0)=f?)(T) VA,
r / t \ 1/2
| \J U{k)(t) |||-i/2(r) d*j < d2<?\Mk VA,
y*)(0) = gw(r) va.
We apply Theorem 9.4, taking / given by
(9.41) (f(t),v) = ff^vdx + fg(t)y0v&r,v£lP(Q).
a r
Then the solution w of (9.36) satisfies:
(9.42) (u'{t), v) + a(t; u(t), v) = (f(t), v), Vt> e V,
where (/(£), w) is given by (9.41).
(9.43)
It follows from (9.42) that
du
dt
+ A
[X'*'l)U = fl-
u is a function of class Mk of [0, T] -> V, therefore, according to
(9.43), t^A(x,tdjdx)u(t) is of class Mk from [0, T] -> L2(Q) «»).
(CD) L2(Q) could be replaced with E-1{Q)i in the hypotheses on/.

160 9. General Operator Equations
Then, according to the results of Chapter 2, we can define
(9.44) *Leff-i/2(r)
WA(t)
and, from (9.42), (9.43), it is easy to show that
du
(9.45)
dv
A(t)
In short, for /x and g given with (9.39), (9.40), there exists a unique
u, of class Mk, mapping [0, T] -> V = H\Q) and satisfying (9.43),
(9.45) and
/(*>
(*, 0) = «<*>(*, T),Vk. D
9.3 Transposition
Under the hypotheses of Theorem 9.3, A + j^ is an isomorphism of
D(A°°, Mh\1T) onto D(^L°°, Mk; V). If we make "adjoint hypotheses',
then A* + <tf* is an isomorphism of D(A*cof Mk; 1T) onto D^*00, MA; V)
and therefore, by transposition:
fA+j* is an isomorphism of (D^l*00, Mh\ Y"))'
( } [onto (T>{A*°°,Mk;r)y. D
Example 9.2. Let us reconsider Example 9.1 formally; we obtain the
existence and uniqueness of u satisfying:
IT T T
f [-(«, <p') + <*> u> <P)~\ dt= f (fv <P)dt + J {g> Yo<P)r dt -
0 0 0
-(Ao^CO^AoG^Cfl),
for^GD^l*00, Mk;i^)} i.e.
f/ r \i/2
If/II^WIIr^l <cLkMkVkt
yW(0) =q>{h)(T) \fk.
In this fashion, we obtain w, solution—in a sense which can be made
precise1—of (9.43) with (9.45) and
(9.48) u(T) - u(0) = h0. D
1 By procedures analogous to those of Section 10 below.

10.1 Orientation. General Problems 161
10. The Case of a Finite Interval ]0, T[
10.1 Orientation. General Problems
Let us consider a family of operators A(t) £ J2? (V; V), as in Section 1
but with t e [0, T].
We make the hypotheses:
(10.1) Re (A{t) v,v) + h\v\2>oc\\vf, oc> 0,v£V,
(10.2) t^A(t) is of class Mk from [0, T] -> ^(V; V),
(10.3) the sequence {Mk} satisfies (1.22), ..., (1.25).
We consider the problem:
fi4*(*) v-*' = ?>,*€]0, T[,
(10.4)
[v(T) = 0,
where
(10.5) <pe^Mk(]0,T[;V).
We denote by X the space described by the solution v of (10.4) as cp
describes S)Mk (]0, T[; F').
According to Theorem 1.2, we know that
(10.6) X C @Mk ([°> T]; V) (algebraic inclusion).
Providing X with the topology translated from Q)Mlc (]0, T[; V) by
the mapping q> -> v, we have (done what was necessary to have):
J the operator A* — d/dt is an isomorphism of X
( * } {onto^Mfc(]0, T[;V).
By transposition of (10.7), we obtain:
Theorem 10.1. Let hypotheses (10.1), (10.2) and (10.3) be satisfied.
Let v^L(v) be a continuous antilinear form on X. Then there exists a
unique element u £ 2f'Mlt (]0, T[; V) such that
(10.8) <«, A*(t) v-v'}= L(v)y Vv£X,
(where in (10.8) the brackets denote the scalar product in the anti-
duality between &Mk (]0, T[\ V) and @Mk (]0, T[; V')). D
The problems are now completely analogous, in principle, to those
already encountered in Volumes 1 and 2 and in Chapter 8 and
Section 9.3:
(i) choice of L in (10.8);
(ii) interpretation of (10.8).

162 10. The Case of a Finite Interval ]0, T[
Formally we shall take
(10.9) L(»)=/</,»>cU + («0,»(0)),
o
where / is given as a "suitable"' functional with values in V', ^0 is given
in the dual to the space described by v(0).
Next, still formally, we have
\A(t)u + u' = f,
(10.10)
[«(0) = UQ.
Thus, in principle, this procedure yields "the most general" right-
hand members / and u0 for which problem (10.10) is well-defined and
admits a unique solution. D
Remark 10.1. Of course, in the examples of differential operators
where the coefficients are "regular in x", we can go further, as we shall
see in Chapters 10 and 11. D
Remark 10.2. Remarks analogous to the preceding ones can be made,
starting from the results of Sections 2 and 4 (second order equation and
Schroedinger equation).
We shall not develop these possibilities. D
Orientation
We shall successively examine points (i) and (ii), moreover subject to
additional hypotheses. D
10.2 Space Described by v(0) as v Describes X
We have completely resolved the problem of the characterization of
v(0) as v describes X only under two very restrictive types of hypotheses.
See Problem 14.12.
10.2.1 Case of an Analytic Semi-Group
We make the hypothesis:
(10.11) £-> a(t; u, v) is constant in the neighborhood of t = 0;
we set
(10.12) a(t; u, v) = a(u, v), A(t) = A, in the neighborhood of t = 0.
Note that thanks to (10.1):
J —A is the infinitesimal generator of an analytic
] semi-group in H (or in V),
and this, in fact, is the only hypothesis which will intervene.

10.2 Space described by v(0) as v describes X 163
Theorem 10.2. Under the hypotheses (10.1), (10.2), (10.3), (10.11) {or
more generally A(t) = A in the neighborhood oft = 0,A satisfying (10.13)),
v(0) describes the space D(^4*°°; k\) as v describes X.
Proof. Let v£X, therefore solution of (10.4). Since <p has compact
support in [0, T[, we have:
(10.14) A*(t) v — v' = 0 in the neighborhood of t = 0,
and, according to (10.11), we thus have
(10.15) A*v = v' in the neighborhood of t = 0,
and therefore
(10.16) A*kv(0) =vik)(0), V£.
But, according to (10.13) (and the same property holding for A*),
we see that £-> v(t) is analytic in the neighborhood of t = 0, with values
in, for example, H and therefore there exist c and L such that
(10.17) \vw{0)\<cLkkl, Wk.
From (10.16) and (10.17), it follows that v(0) g D(^*°°; k\).
Conversely, let v0 be given in D(^4*°°; k\), Then the function
(10.18) „,(*) = £ -—V,
k = 0
is analytic in [0, t0], t0 sufficiently small, with values in V, since \A*kv01 <
< cLkk\ and therefore ||^4*^0|| < cLkk\. If 0 is a function of class Mk in
[0, T], 0(0 = 1 in [0,*0/3], 0(25) = 0 for t > 2*0/3, and if we define
\0(t)w(t) in [0,*0],
v(t)={
[Oin p0, T],
we obtain
,l*v _ v> =v> we@Mk (]0, T[; V), v(T) = 0,
therefore v£X and since a(0) = v0, we have the desired result. Q
10.2.2 Case of a Group
Let A be independent of t. Thus we take
(10.19) a(t; u, v) = a(«, v), 25 g [0, T].
We assume that
(10.20) —A is the infinitesimal generator of a group in V.

164 10. The Case of a Finite Interval ]0, T[
Then we have:
Theorem 10.3. Under hypothesis (10.20), v(0) describes the space
D(A*°°;Mk).
Proof. Indeed A*v — v' = 0 in the neighborhood of t = 0; therefore
we have (10.16) again and since v is of class Mk with values in V, we have:
(10.21) tl^WII <cLkMk, Vk,
whence v(0) £ D(^*°°; Mk).
Conversely, if v0 is given in D(^4*°°; Mk), we have:
IA*w — w' = 0,
w(0) =v0,
a well-posed problem, thanks to (10.20); since v0£ D(^*°°; Mk), t-> w(t)
is a function of class Mk of [0, T] -> V; then we take v = 6w, 6 as in the
proof of Theorem 10.2, and we proceed as in this proof. Q
10.3 The Space SMk
Orientation
Our aim is to construct a space which contains X and "the smallest
possible", such that @Mlc (]0, T[;V) is dense in this space.
We shall only use the fact that we have (10.6).
More generally, we introduce a Banach space F and construct a space
3Mk (]0, T[; F) with the following properties:
(10.23) 2>Mk ([0, T];F)C 3Mk (]0, T[; F),
(10.24) ®Mk (]0, T{; F) is dense in 3Mk (]0, T[; F). D
First we define
S&={H»i
(10.25) 3^ak = \v | v is C°° from [0, T] -> F, v = 0 in [T — a; J],
i4»lle*»,*,llL.(o,T;n<c,v4,
where q is given by
(10.26) Q(t) =
Provided with the norm
(10.27) sup_J_||eV*)||mr.F)
it is a Banach space.

10.3 The Space Sj^k 165
Next we take
(10.28) SMk (]0, T[; F) = Bm = indlim (indlim EfrM .
We obviously have (10.23).
Theorem 10.4. Assume that (10.3) is satisfied. Then property (10.24)
holds.
Proof. The proof is analogous to the one in Section 4.2, Chapter 8.
Let v be given in 5Mk, thus
(10.29) v £ E]i\, for suitable L and a.
Let
(10.30) L1 = max(L, 1).
Later on we shall show that:
f there exists a sequence 6n(t) satisfying
(10.31) I 0 < 6Jt) < 1, en(t) = 1 for t > 2/w, 6n{t) = 0 for 25 < 1/w,
[l^fWl^^M,, VA.
Then 0nv £ ^Mfc (]0> ^[J F) and we shall nave the theorem if we can
show that 0nv -> v in S,Mfc. More precisely, we shall show that
(10.32) 6nv -» v in S^^, arbitrarily fixed rj > 0.
Indeed, let
We have:
where
^ = (^^ll(0»-i)A(*W,>>
i * /a
Next, we set
yn = sup yM, *n = sup *M, C„ = sup fM.
&>0 fc>0 £>0
To show (10.32) is equivalent to showing that
(10.33) yw->° as «-»oo.

166 10. The Case of a Finite Interval ]0, T[
Now yn < zn + Cn and we shall have (10.33) if we can show that
(10.34) zn^® as ^-^°°,
(10.35) C„->0 as n^oo.
But, according to (10.29),
therefore, the c's denoting different constants:
zn< sup z k +cl )
and this proves (10.34) if we can show that
Znk~> 0> W-> OO, & /^^,
which is obvious.
Let us now prove (10.35). We have
1 * Lft
< C r y ULk~j < ck r .
Therefore, for sufficiently large N:
CM< sup C^+cnI-^-Y
and the result follows if we can show that:
But, for fixed k,
k J 2fn \ 1/2
CM < (constant) £ j P{k~5) || v^M ||| d* ^ 0.
i=i \o /
Therefore the theorem is proved, subject to proving (10.31).
At the risk of having to translate and change n to 2n later, it is
sufficient to show the existence of %n with
[&,(<) = 1 for t>2[n,
(10-36) 0<^)<1,
Uj,(0) = 0, V/,
(10.37) \t*£\t)\<c2L\Mk,Vk.

10.4 Choice of L 167
To this end, we start with a function q G 3)Mlt (R), with support in
[0,1],
q > o, / eat = i, \e{k)(t) | < H(hjMk, v*.
Next, we set
e«W = Mnt)>
[0 if t<0,
rpn(t) =\ nt if 0 <t< 1/n,
[l if t> 1/n,
and define
Conditions (10.36) hold. Let us verify (10.37). We have:
o
whence
\%{n\t) I < nk f \Q{k\nc) I n da - nk f \QW{a) \ da < cnk[-±\ Mk,
0 0 \ 2 /
therefore if 0 < t < 2jn,
\W) I < c l—\ nk fc\ Mk, whence (10.37). D
Corollary 10.1. // (10.3) holds, the space (EMlc)' can be identified to a
subspace of 2'Mh (]0, T[; F'). Q
10.4 Choice of L
We now take in (10.8) the choice (10.9) of L(v), where:
(10.38) /Gi%fc(0, T; V') = {SMjc(0, T; V))',
\ u0 e (D(^*°°; k\)Y in the case of Theorem 10.2,
(10.39) \
[u0e (D(^*°°; Mk))' in the case of Theorem 10.3.
There remains to interpret the problem; in order to fix our ideas, we
shall consider the case where A (t) = A does not depend on t.

168 10. The Case of a Finite Interval ]0, T[
10.5 The Space Y and Trace Theorems
10.5.1 In the Setting of Theorem 10.2
Let (10.8) hold, with the choice (10.9), (10.38), (10.39). It follows that
(10.40) Au+u'=f in a>'m (]0, T[; V).
Consequently u belongs to the space Y defined by
(10.41) Y = {v | v e 9>'u% (]0, T[; V), Av + v' £ S'm (]0, T[; V')}.
We provide Y with the coarsest locally convex topology which makes
the mappings #-> 0 and #-> Av + v', of
y^®U(]0,T[;7) and Y^ 3'Mk (]0, T[; V),
continuous, each of these spaces being provided with the strong dual
topology. D
In order to state the density result which we are aiming at, let us
make the hypothesis in the setting of Theorem 10.2:
if in [0, a], tp satisfies
(10.42)
A*ip — ip = 0 with y(0) = 0
and if \p is of class Mk in [0, a] with values in D(^4*),
then tp(t) =0 for t£ [0, a].
In this case A* is said to have the retrograde uniqueness property.
This hypothesis is satisfied for parabolic operators (see Lions-Mal-
grange [1]).
Theorem 10.5. Assume that (10.3) and (10.42) are satisfied. Then the
space @Mk ([0, T]) ® V (where @Mk ([0, T]) is the space of scalar
functions of class Mk in [0, T]) is dense in Y.
Proof. Let I be a continuous linear form on Y, vanishing on
#jft(P>,r]) ®v.
We need to show that M = 0. Now M may be represented in the form
IM(u) = <w, u) + <w, Au + u"),
9£®Mk (]0, T[; V), y>e3MkQ0, T[; V) = 3m,
where the first (resp. second) bracket denotes the duality between
^m,(]0,J[; V) (resp. 3Mk) and &Mk (]0, T[; V) (resp. S'Mk).
Let <p and xp be the extensions of cp and ip to R, by 0 outside [0, T].
If w e gjMk (R/) ® V, its restriction u to [0, T] belongs to @Mlc ([0, T])®V,
therefore M(u) = 0 and therefore
(Spy W} + <^> ^^ + ^') — 0,

10.5 The Space Y and Trace Theorems 169
therefore
(10.44) £ + A*y> - -^ = 0 on Rt.
dw
But for example, <££L2(-1, T; V), and by (10.44), -^gL2(-l, T\V)
and therefore
(10.45) v>(0) = 0.
On the other hand, the support of <p is bounded on the right (by
T — a, suitable a > 0), so that (10.44) yields (since q> £ Q>M]e (R; ^')):
(10.46) ^+>Mfc(R;F).
But, in the neighbourhood of, 0 99 = 0, therefore
A*w-d^=0>
* dt
which, together with (10.45), (10.46) and hypothesis (10.42), yields
(p = 0 iw £/^ neighbourhood of 0.
Therefore
and consequently, for u £ Sj'mtc G^» ^T> ^0> we nave:
<^, 4« + «'> = (A*ip — y/, w>.
But then (10.43) yields
M(u) = (q>+ A*ip - ip', u} = 0 (by (10.44)). D
We are now in a position to prove the following trace theorem.
Theorem 10.6. Assume that (10.3) holds, A being given as in Theorem
10.21. The mapping u^ u(0) of QjM]c ([0, T]) ® F-> V extends by
continuity to a continuous linear mapping, still denoted by u->u(0), of Y
(defined by (10.41)) into (D(^*°°; &!))'.
Proof. Let u £ Y be fixed.
By the proof of Theorem 10.2, we can find a continuous linear
mapping v0^ &v0 of DL{A*°°; k\) -> X such that
(10.47) (K)(0)=V
Set
(10.48) Z(v0) = (A*Wv0 - (Sv0)', ^> - <&v0, Au + «'>,
1 Then (10.42) holds.

170 10. The Case of a Finite Interval ]0, T[
the first (resp. second) bracket denoting the scalar product between
#*» (]0, T[; V) T(resp. Sm) and 2>'m (]0, T[; V] (resp. S'Mk).
The number Z(v0) depends only on v0, i.e. if we replace &v0 by w £ X
with w(0) = v0, then
Z(^0) = (A*w — w', u) — (w, Au + w'>.
Indeed if ip = <Fv0 — ze>, this amounts to showing that
(10.49) <4«y — %p\ u) — <y, 4« + u'y = o.
But we have:
A *^ — xp = 0 in the neighbourhood of 0,
?(0) = 0,
therefore, by the retrograde uniqueness property, \p = 0 in the
neighbourhood of 0, whence (10.49).
The form
is continuous on DL(^4*°°; &!) and since it is independent of L, we see
that v0^- Z(v0) is a continuous linear form onD(i*°°;^!). Therefore
(10.50) Z(v0) = (ru, v0), ru £ (D(i4*°°; k\))'.
But if w £ @mjc ([^» ^]) ® ^> we immediately see that
rw = u(0).
Finally, if u -> 0 in Y, then Z(v0) -> 0 uniformly for v0 in a bounded
set of D(A *°°; k!) —indeed v0 then remains in a bounded set of DL(A *°°; k!),
suitably fixed L, and we can choose <Fv0 in a bounded set of X, from which
the desired result follows. Q
10.5.2 In the Setting of Theorem 10.3
We again have the definition of Y given by (10.41); hypothesis
(10.42) is evidently satisfied; the analogue of Theorem 10.5 holds and,
in the same way as above we arrive at
Theorem 10.6a. Under the hypotheses of Theorem 10.3, the linear
mapping u^u(0) of 3fMle{\.^y T] ® F-> V extends by continuity to a
continuous linear mapping, still denoted by u -> u(0), of Y-> D(^4*°°; Mk)'.
10.6 Non-Homogeneous Problems
We can now state the principal results:
Theorem 10.7. Assume that (10.3) holds, A being given as in
Theorem 10.2. Let f be given in E'Mh (]0, T[; V), u0 be given in D(^4*°°; &!)'.

10.6 Non-Homogeneous Problems 171
There exists a unique u in Y satisfying
(10.51) Au + «' = / in the sense of 3f'm (]0, T[; V),
(10.52) u(0) = w0 m the sense of Theorem 10.6. D
Similarly, from Theorem 10.6a, we obtain:
Theorem 10,7a, Under the hypotheses of Theorem 10.3, let f be given
in E'Mh (]0, T[; V), u0 be given in D(^4*°°; Mk)'. There exists a unique u
in Y, solution of Au + u' = / in the sense of @)'Mh QO, T[; V) and of
u(0) = ^0 m ^ sc^se of Theorem 10.6a.
Remark 10.3. Thus we see that in the parabolic case, we may take u0
in D(^4*°°; k\), whereas in the second order in t case (which is reduced to
the first order as in Section 7.3.4) or in the Schroedinger case (and in
particular in the hyperbolic case) we may take the initial data in
D(A*°°;Mky. D
Let us give some applications of Theorems 10.7 and 10.7 a.
Example 10.1. (Non-homogeneous Neumann problem; parabolic
case).
Let us consider the setting of Section 7.3.2, with A a second order
elliptic operator and V = H1(Q); for u, v £ H1(Q)t we set
(10.53) a(u, v) = 2 / ai3(x) ^ ^- Ax.
i)3=\ q OXj OX{
We apply Theorem 10.7, with / given by
(10.54) (f,v) = (f0,v)+(g,y0v)r,
where
(10.55) fo€B'„t(]0,T[;L*{£i)f™,
(10.56) g€E'Mk(]0,T[;H-^(D).
Then the interpretation of problem (10.51), (10.52) is the following
(without specifying the adequate trace theorems), u satisfies
(10.57)
du
du
}A v'vA i,j=l v*j
u(x, 0) = u0(x) on Q. D
—- == g on Z, where — = £ a{j — cos (w, *.),
((D) More generally, we could take f0£ E'MkQ0t T[; S^iG)), with E-\Q)
defined as in Chapter 2, Section 6.3.

172 10. The Case of a Finite Interval ]0, T[
Example 10.2. (Non-homogeneous Dirichlet problem; parabolic case).
If we take A as in the preceding example, but with V = H\(Q),
then Theorem 10.7 yields the existence and uniqueness of
ue®'Mk{]0,T[;Hl(Q))
such that
du JL d / du\
where
foe3'Mk{]0,T[;H~i(Q)),
with
(10.59) «(0) = 0 on Z
and
u(x, 0) = u0(x) on Q.
But we can consider non-homogeneous data on Z (as opposed to
(10.59)) in the following way.
Generally speaking, let A—independent of t—be in J£(V] V), such
that (10.1) holds. Then, as we have seen in Section 7, A satisfies for
example
(10.60) Ae&(H;D(A*)')
and —^4 is the infinitesimal generator of a semi-group in V
We may apply Theorem 10.7 by replacing V with H and V with
D(A*)'.
The space D(^4*°°; k\) becomes the space of v's belonging to V such
that A*v £ V', ..., A*kv £ V,... and such that there exists an L with
\\A*kv\\v,<cLkkl.
But thanks to the "coerciveness inequalities", this does not change
the space D(^4*°°; k\) Therefore the analogue to Theorem 10.7 is:
(10.61)
let / be given in £'Mk(]0, T[; T>(A*)') and u0
in D(^4*°°; &!)'. There exists a unique u in
9'Mk (]0, T[; H), such that Au + u' e 5^ (]0, T[; D(4*)'),
with ^ + «' -- /, u(0) = u0.
(Note that even more generally, we may consider A as belonging to
&(T>{A*m)'; D{A*m+1Y) and take / in 3'Mk (]0, T[; D(A*m+1)'), and
then find u in ^fc (]0, T[; D(A*m)')).
If we come back to the concrete situation, we may take / by
(10.62) (/, v) = (/„, v) - L, -£-) v e D(A*) = H*(Q) A H\{Q),

10.6 Non-Homogeneous Problems 173
where
(10.63) /0 e S'Mk{]0, T[; H~\Q)), g£~'Mk (]0, T[; H~^(r)).
The interpretation of the equation Au + u' = f is:
(«, ^*v) + («', v) = (/, v), Vv € D(4*)
and consequently w satisfies (10.58) and the conditions
Iu = g on 27,
u(x, 0) = u0(x) on Q. D
Remark 10.4. If we take / in 5^fc (]0, T[) D(,4*w+:1)')> then we may
take g e 5^ (]0, T[; ff-*-^^)) in (10.64). Q
Remark 10.5. Of course the preceding examples can be extended to
elliptic operators A of arbitrary order. Q
Example 10.3. If we new consider the operator given by
where (in addition to the ellipticity condition):
(10.65) «* = ^ Vm,
then we can apply Theorem 10.7a, as long as we write the equation
as a first order system in t.
If we first take V = H}-(Q) (as in Example 10.1), then we are solving
(without specifing the adequate trace theorems) equation (10.66) with
the boundary conditions:
^- = g on 27, g given in E'Mjc (]0, T[; #']/2(r)),
(10.67)
«(*, 0) = u0{x) on fi, ^0 £ D(^*°°; M,)' ((1)),
^{x,0) = u1(x) on fi, «1GD(i4*~;MJk)'.
era
// we take V = H\(Q), then we find—as in Example 10.2—u = 0 on27
as boundary condition.
((1)) We write u0(x) by an abuse of language. u0(x) is evidently an "entity" of a
very general nature ...

174 11. Distribution and Ultra-Distribution Semi-Groups
But, by the same principle as in Example 10.2, we can solve equation
(10.66) with ue &MkQ0, T[; L*[Q)) and
(10.68)
u = g on S, g given in 3'Mk (]0, T[; tf"1/2(r))
du
and u(x, 0) = u0(x), — (x, 0) = ttJx), w0, ux
ct
given in D(^*°°; Mk)'. D
Remark 10.6. Following the same principle, we can solve the
analogous problems for the Schroedinger equation. Q
11. Distribution and Ultra-Distribution Semi-Groups
11.1 Distribution Semi-Groups
Let E be a Banach space and A a closed unbounded operator in E,
with domain T>(A) dense in E. We provide T>(A) with the norm of the
graph, which makes it a Banach space.
We shall make use of the space
(11.1) JS?(»_(R);E)
of distributions according to Schwartz [3] on R with values in E.
If E is reflexive, then, as already noted in Chapter 7, Remark 5.1,
JS?(0_(R); E) = (0_(R; £'))' = ®V(R» £)- However in the sequel the
reflexivity of E plays no role. Thus, in order to avoid any confusion, we
shall use the notation (11.1).
The problem is the following: for / given in «£?(®_(R); E), find the
necessary and sufficient condition for the existence of a unique
distribution u having the properties:
(11.2) «€JS?(®_(R);D(i4)),
(11.3) Au + it' = f, («' = d^/d*),
(11.4) inf {support of u in t] < inf {suppoit of / in t},
(11.5)
/ -> w is a continuous mapping of
^ JS?(&-(R); £) -> JS?(®_(R); D(i4)). D
Theorem 11.1. (Chazarain [1]) In order that conditions (11.2),...,
(11.5) be satisfied, it is necessary and sufficient that A has the following
properties: there exists a set 0t C C, of the form
(11.6) « = {A|A = f+«?, i>oc\og\rj\ + 0, £ > f0, «,^06R},

11.1 Distribution Semi-Groups 175
such that A + X is an isomorphism of T)(A) onto E for A G ^ and that
(11.7) || (A + A)-1^.^ polynomial in |A|, X <G 31.
Proof of the necessity of (11.6), (11.7). We divide the proof into several
points.
1) It is easily seen that it is equivalent to assume (11.2), ..., (11.5)
or the existence of a distribution ^ ("elementary solution") with the
following properties (see Chapter 4, Section 3):
£€J2?(®_(R);J2?(E;D(i4))),
<3 = 0 for t < 0,
(11.8)
(£+*)"
**{*+*)■■
d (x) IE (where Ix = identity from X-> X),
d ® L
D(A)'
We shall show that (11.8) implies (11.6), (11.7).
2) Formally (X + A J"1 is given by ^(e_A/), but (and this is the main
difficulty) <&(e~M) may have no meaning for any X £ C.
The essential idea behind the proof is to "approach" &(e~M) with
^(0A), where 6^ is a truncation of e~M such that dx £ ^_.
More precisely, we introduce once and for all a function b £ ^(R)
with the properties:
(11.9)
&6 0(R), support of & in [t0, t{\, Q<t0<tlt
b>0, fbdt = l.
Next, we introduce:
(11.10) \ = e-i¥b (so that b0 = b),
(11.11) Bn(X) = f e%(t) dt=f tPb(t) dt = B(i).
Let 6} be the solution with support bounded on the right (by ^) of
1
(11.12)
ddJ+w^-m^)-
If Y is the Heaviside function (Y(t) = 1 for £ > 0, Y(*) = 0 for t < 0),
then we have:
(!-)«>
Y) e-*) =

176 11. Distribution and Ultra-Distribution Semi-Groups
so that (11.12) yields
1
'5(f)
((1-Y)e-*)*V
(11.13) (
or, expanding:
(11.14) 04=.e-"-_((e-«Y)*&).
Thus we see that
(11.15) 0, has support in p0, y, 0A G ®(R),
and 0$) = e~A/ for 25 < *0.
We now form &(0j), and we shall show that it is an "approximated
resolvent".
Noting that, according to (11.8), A<g((p) = g%') + q>(0) IE,
g> G ^(R), we obtain:
& + A)9(0J =9(0'x + MJ + IE,
from which, together with (11.12), we have:
t'-m*^'
(11.16) (A + A) 9(6J
and, in the same way,
(11.17)
W (A + ^)
■^,).
Now the plan of the proof is as follows: we shall successively show:
(11.18)
there exists a region 01 given by (11.6) such that
B(S)
9(bv)
1
< —,
&{E;E)
B(«
W)
^(DU);DU))
— 2
for A = g +irjedl;
then, by (11.6), (11.17), (11.18), (A + A)-1 exists for AG #, and we have
||(A + 4)-i <IIWII;
then we shall verify that we have the estimate:
(11.19) 110(0*) ||^(£;£) < polynomial in |A|, AG St.
3) Proo/ o/ (11.18). Since 0 vanishes for J < 0, there exists (see
L. Schwartz [5], Chazarain [1]) a constant c and an integer m such that
(11.20)
\\^)Ue'V<c\M

11.1 Distribution Semi-Groups
177
where
(11.21)
Then
(11.22)
HkllU = 2 sup I^WI*
k=Ol£[OA]
<p€@(R), <p = 0 for t> tt.
\B(i)
w„)
lif(£;E> -]B(^|
But, for | > 0, B(£) > e?'' / b & = e?(>, and (10.22) yields:
1 ii^)ii*(^<ce-4,,(i + i'?ir-
Thus we shall have obtained the first inequality in (10.18) (and the
second inequality can be verified in the same way) if we choose f, rj to
satisfy:
(11.23)
f > 0, Ce"*'(l + \rj\)m<
1
Then we choose M in (11.6) so that (11.23) holds.
4) Proof of (11.19). It foUows from (11.20) and (11.14) that:
IWIIw)<cllle~
+ ^\\\e-^((e^Y)^b)\\}m<
< (the c's denoting different constants) <
<c(i + m
B®
e-*'(l + \tl\
But, for A<E M, we have (11.23), and since B(|) > e{\ we obtain
whence (11.19). Q
Proof of the sufficiency of (11.6), (11.7). For A(E M, we set:
(11.24)
and, for^e^R),
(11.25)
R(Z) = (A + 4)-i,

178 11. Distribution and Ultra-Distribution Semi-Groups
We introduce a contour y as indicated in fig. 1; y is contained in
and has a "vertical"' part f = f0 and two arcs
-IL|«_
e * I rj | = constant.
>/,
•
0
k
J
/
/
k
To
^
«
(11.26)
Fig.l
Then, V<p£ ^(R)> and subject to verification of convergence, we define:
1
»(90=^/*W*WdA.
2m
But, by integration by parts:
(11.27) 0(X) = A-qJeM<p{q)(t) dt} Vq,
and if 9? has support in [r0, tJ , it follows that
(11.28) \0(X) I < ct(q>) e"« |A |-», c» = sup |^>(*) |.
But then, according to (11.7), we have
(11.29) ||2?(A) ®(X)\\x{e;E) < ct(V) (1 + \X\)m |A|"' ?*, X^0t.
Since, on y, eTl1 ~ |A|e, we see that (11.29) yields:
ii w) *w !!*<**> < cm (i + *r ia rq.
Therefore we can choose q so that the integral in (11.26) converges and
we have
(11.30) ||S%)|| <ccg(<p).
It follows that (11.26) defines ^ with
(11.31) 9£&[@(R);&{E;E)).

^ = 0 for
(^H=
**$+AY-
t<0,
= d ®IE,
= $ ® 7D(^)
11.1 Distribution Semi-Groups 179
But we also have
(11.32) ^ e J2?(®(R); &(E\ B(A))),
since 4fl(A) = I — AR{X), therefore
\\X(E;D(A))
<(l + \X\)\\R(Xi
\\£C(E)E)
and it is sufficient to replace q with q + 1 in the above argument to
obtain (11.32).
We still need to verify the properties:
(11.33)
(11.34)
Proof of (11.33). Let
yN = translate of y by +2V in the direction of £ > 0.
By Cauchy's Theorem, we have
(11.35) 9(ip)=-±rfR(X)0(X)dX.
*m yN
Let <p £ ^(R), with support in ]— oo, —e[, e > 0. It suffices to show
that 9(<p) = 0. Now, if £ > N + £0, we have:
|0(A)!<^)e-^|A|^,
from which, in the same way as above, using (11.35), we deduce that
11^)11^^ <^)e"^.
Letting N^ oo, it follows that 9(q>) = 0.
Proof of (11.34). For q> £ 0(R), we have
A 9(fp) = 9{_9>) = * J^(A) 0(A) dA
d* vr/ v T/ 2m
Y
(since JV'(— g/) dA = A0(A) and the integral J 1R(A) 0(Xj dA converges
(after a suitable choice for #)). As A,R(A) = IE — AR(X), it follows that
1«%) = ^M + IE± f0(X) dA.

180 11. Distribution and Ultra-Distribution Semi-Groups
Since — J0(2) (U = <p(0), the first equality in (11.34) follows. The
second equality is verified in the same way. D
Remark 11.1. The distribution ^ introduced in (11.8) has, in
particular, the property:
(11.36) 3% *• %p) = <&{<p) • 3%), Vy, xp e 0(R),
which is the basis for the definition of distribution semi-groups. D
Remark 11.2. Theorem 11.1 is a variant of the Hille-Yosida Theorem
(see Yosida [1]), characterizing the infinitesimal generators of the usual
semi-groups. It can be shown that, in an appropriate sense, —A is the
"infinitesimal generator" of the distribution semi-group <$. See Lions [4]. D
Orientation
The criterion (11.6), (11.7), yields the.most general conditions for the
"abstract Cauchy problem'' (11.3) to be well-posed, in the spaces
J2?(^-(R); E). We can pose a more general problem, by replacing
J2? (^-(R); E) with spaces of ultra-distributions. This is the aim of the next
section. D
11.2 Ultra-Distribution Semi-Groups
The data are the same as in Section 11.1.
We further consider a non-quasi-analytic sequence Mk, more precisely
satisfying (1.22), (1.23), (1.24).
In the same way as in Section 11.1, we seek the necessary and
sufficient condition on A for the existence of a unique ultra-distribution u
having the following properties:
(11.37) ue<?{®_tMk(R);-D(A)),
(11.38) Au+u'=f, f given in Se[<3_>m (R), E),
(11.39) inf {support of u in t} < inf {support of / in t},
I/-> u is a continuous mapping of
&{®-.m> (R); e) -> &{®-,Mk (R); dm) . D
In older to state the main result, we introduce the function X-> M(A)
by:
(11.41) expM(A)=sup(|A|*^).
For h, oc, /? £ R, we define:
(11.42) Xtf ={X\X = S+ iVt | > »M(k-hi) + 0}.

12.1 Statement of the Result
181
Then we have:
Theorem 11.2 (Chazarain [2]). In order thai conditions (11.37),...,
(11.40) be satisfied, it is necessary and sufficient that A has the following
properties:
(11.43)
{
VA > 0, there exists &°Mh such that [A + X) is
an isomorphism of T>{A) onto E, VA£ ^fh]
(11.44)
VL, h, Vf > 0, there exists M^ with (11.43) and
there exists a constant c such that
II (A + A)"1 \\<?{E.E) < c exp (m (A) +sSy X € 0tlh. Q
The principle of the proof is analogous to the one used for the proof
of Theorem 11.1. We refer the reader to Chazarain [2], for technical
details.
Remark U.S. The same type of problem can be posed by replacing
the spaces J2? (@-,Mk (R) '> F) witn tne Beurling spaces J?(B_>Mk (R); F).
We obtain the same type of result, replacing the conditions VL, VA
with: "there exist L0 and h0" such that the conditions contained in (11.43),
(11.44) are satisfied. See Chazarain [2]. Q
Remark 11.4. If we take Mk = (k\)s, s > 1, the regions M^f)h decrease
in size as s decreases—i.e. the hypotheses on A become more and more
general (as would be expected!). Q
12. A General Local Existence Result
12.1 Statement of the Result
Let s be a parameter £ [0, 1]. Let Es be a family of Banach spaces,
with norm denoted by || ||s. Assume that
(12.1) E1C'"CEsCE8fC'"E0t l>s>s' >0,
with
I the injection of Es -> E , is continuous with norm < 1, i.e.
IML-<IML veeE$,s>s'.
Let t be a real or complex parameter which varies in a closed
neighborhood (9 of the origin, thus:
(12.3) t e 0, & C R or ^CC, 0 neighborhood of the origin.

182 12. A General Local Existence Result
Consider a family of operators A (t), t £ &, with the following properties:
fV^ff, Vs, s', 0 < s' < s < 1, we have:
A(t)e<?(Es;Es,),
(12.4)
(12.5)
(cx = constant independent of t, s, s');
the function t -> A (t) of 0 -> &(ES; Es,) is:
continuous if ^CR,
holomorphic if 0£C. D
Remark 12.2. Examples of this situation will be given in Section 12.2. Q
Theorem 12.1 (Ovciannikov [1], Treves [4], [2]). Let hypotheses
(12.1), ..., (12.5) be satisfied. Let t^ f(t) be a given function of (9^EV
with:
(12.6)
f(t) is continuous (resp. holomorphic) from
(resp. G£C)^EX.
Let u0 be given with
(12.7)
u0eE1.
Then, for every s £ ]0, 1[, there exists one and only one function t^- u(t)
having the following properties:
[ u is defined in (9 A {t \ \t \ < K(l — s)} = 0S((1)),
(12.8) \ t^ u(t) is continuously differ entiable i/^CR
I (resp. holomorphic if (9 C Q from Gs -> Es,
(12.9) ^ + A(t)u(t)=f(t),te(9s,
(12.10) «(0)=«0.
(Of course, u is "independent of s").
Proof. The proof is an application of the successive approximation
method. Equations (12.9), (12.10) are equivalent to
(12.11)
u(t) = j f(a) da + u0 — j A (a) u(a) da,
((!)) K = constant to be specified in the course of the proof.

12.1 Statement of the Result
which by recurrence leads to defining
t t
(12.12) uk{t) = f f(a) da + u0 - f A(a) uk^{a) da,
o o
with u0(t) = u0.
Let us introduce constants c2 and L satisfying
183
(12.13)
(12.14)
We shall show that
*2>suPll/Wlli +CiH«olli,
L> c±e.
(12.15)
l«*+iW-«*WII.<^
i*+i
(1 - s)k+1'
Let us first verify (12.15) for k = 0. By (12.12) for k = 1, we have:
(12.16)
KW ~uol<
/ f(a) da
+
f A (a) u0 dff
<
<|<|sup||/WII.+ |<|sup||^(f)«0||,.
m
t£0
But by (12.4), || A(t) u0\\s < 1 l|^0llx and il follows a fortiori
from (12.16) that
KM - «oll. <r^- Tsupll/W"1 + ci Kllil <f
C2 N
Let us admit (12.15) up to k — 1 and prove it for k. It follows from
(12.12) that
t
«*+](*) - «*(*) = - f A(<r) («*(*) - «*-](*)) d<*>
o
whence
x JW0-«>WIL<t
(12.17) J £
! t
!o i
s > 0 such that s + £ < 1.
By induction, (12.17) yields
(12.18) \\uk+1(t)-Mt)l<-T^L'
k-1
v
Choose s such that
(1 - s - «)* A + 1
»(ft + 1) = 1 - s

184 12. A General Local Existence Result
(and then s + e < 1); (12.18) then yields
-C1C2 \f\k+1
"i" (1 - S)'
«.+j 0 - «»(*) IL < ^ TT^vm Lk (* + y)*
<c2
\T) (1 - s)*+>
from which we obtain (12.15) thanks to (12.14).
It follows that
oo
u(t) = £ (uh+1{t) -uk{t))
converges in Es and yields a solution of the problem, if
\t\L
<1,
1-5
i.e., taking into account (12.14), if
(12.19) |*|<JL(i -s).
We have thus shown the existence in Theorem 12.1, in which the
constant K must be chosen < 1/^e. The proof of uniqueness is
standard. D
12.2 Examples
Example 12.1. In a Banach space E, let A be a closed unbounded
operator. In the notation of Section 7.2, for 5 £ [0, 1], we define
(12.20) Es = DVs{A°°;k\),
i.e.
(12.21) Es = {e | e€D(4~), ||e||. =sup —M*e|| < oo}.
&>o
We assume that Es does not reduce to {0} (see Section 7). Then it is
easily verified that A satisfies (12.4). Q
Example 12.2 (Treves [4]). Let r^cp(r) be a given function such that
(12.22) r^cp(r) is non-decreasing > 0 on r> 0.
Let X be a Banach space. Define:
IE{cp; X) = space of entire functions 2-> f(z) of C-> X,
such that sup (exp(-<p(M)) ||/(*)y = \\f\\E{,;X) < oo,
zZC
which is a Banach space for the norm ||/||£(V;^).

12.2 Examples
185
For se [0, 1], set
(12.24)
and
(12.25)
Then we have
Es = E(cp;X).
Proposition 12.1. The operator A = d[dz satisfies (12.4), (12.5) in the
family Es defined by (12.25). In (12.4), we may take
(12.26) c3 = 2exp<p(l).
Proof. Let us first verify that
if v > 1, then, for \z\ = r, we have
(12.27)
exp (-<?»)
f»
j <^-4ll/lk- 0<s'<s<l.
\X 5 — 5
Indeed, according to the Cauchy theorem:
£<■>
<er* sup ||/(f) H^ < e-1 exp V,(r — e) ||/||£, .
X |*|-r + e
Therefore
(12.28) exp (-%,(,))
w
< e-1 exp (<ps(r + e) - <ps,{rj)
(s — s') r
Choosing e = — — , we have: q>s(r + e) = <pS'{r) and (12.28) yields
2-5
(12.27) (since r > 1).
For \z| = r < 1, we have
"0/
exP (-9VM)
0z
(*)
5(exP-^(|f|)||§(f)|l )^
<exp9>s,(.l) sup
* 1*1
< (by (12.27)) ~7exp ^(1)
from which the desired result follows. D
A particular case
Letting
(12.29) cp(r)=rrk} t > 0, integer k>0,
we obtain for E(q>, X), as t varies, the space of entire functions / with
values in X such that ||/(^)||x < M exp (M' \z\k) for suitable constants
M and M'. D

186 13. Comments
Extension to n complex variables
We introduce
(12.30)
(12.31)
each <pj satisfying the analogue to (12.22).
We define
E(q>; X) = spaces of entire function z-> f(z) of Cn -> X
such that
sup(exp(- t%(|^|)||/W||)) =
\\E{<p;X)
< oo,
and then we define Es as in (12.24), (12.25).
As for Proposition 12.1, we verify
Proposition 12.2. For every j, the operator d/dzj satisfies the analogues
to (12.4), (12.5), with c1=2 exp (%-(l)). D
Then we define
(12.32) A{t) = Yia3{t)^- + a0(t),
3 = 1 0Zj
where
/ * ~* ai^' ® — *— n' *s a continuous (resp.
( ' ' [ holomorphic) function of 0 -> jgf (Z; X).
We are within the conditions of Theorem 12.1. Thus we obtain the
existence and uniqueness of a function u(z, t), z£ Cn, t £ (9S, solution of
du " du
(12.34) — + £ *,(*) — + <z0(*) u = f(z, t),
(12.35) u(z, 0) = ^0(^), z e Cn.
If £->/(.,£) is a continuous (resp. holomorphic) function of 0->
£-^99; X) and if ^0 6 £1(9?; X), then the solution u(0, t) belongs to Es{y; X)
for 2 £ 0S and depends continuously (resp. holomorphically) on t. D
13. Comments
The results of Section 1 have been given in Lions-Magenes [3]. The
results of Sections 2 and 4 have been obtained in Lions-Magenes [4], but
under much stronger hypotheses on A(t). C. Baiocchi [2] observed that
it is possible to obtain "the good estimates" when (A'(t)v,v)<i — y\\v\\2
and that the problem can be reduced to this case by a simple change of

14. Problems
187
variables (this change of variables was introduced independently by
J. Lieutaud (personal communication) in order to reduce nonlinear
problems to the case (A'(t) v, v) < 0).
The results of Sections 3 and 5 are given here for the first time.
The results of Section 6 are given in Lions-Magenes [3, 4].
Sections 7.1 through 7.4 develop certain results contained in the
author's notes [6]. See also Lions [11], Magenes [5].
As stated in the text, the results of Section 7.5 are due to Tanabe [1].
For the case Mk = k\, see Agmon-Nirenberg [1], A.Friedman [7],
Kato-Tanabe [1], Kato [1], Komatsu [6], Kotake [1], S. Krein [1],
Shirota [1], Sobolevski [1], Yosida [1]. For the interpolation of the spaces
D(^°°; Mk), see Goulaouic [1, 2].
For the regularity of solutions of evolution equations by the method
of semi-groups, see P. Suryanarayana [1].
The results of Sections 8, 9, 10 are given here for the first time. Other
results of this type should exist, relative to other classes of operator
equations (see Problem 14.10).
The results of Section 11 are due to Chazarain [1, 2]. They extend
previous results of C. Foias [1] pertaining to the Hilbert case, the operator
A being normal (Foias used spectral decomposition). The distribution
semi-groups, mentioned in Section 10, were introduced by Lions [4],
numerous additional properties being given by Peetre [1], Da Prato
[1, 2], Da Prato-Mosco [1], D. Fujiwara [1], Fattorini [1], I. Cioranescu
Numerous other properties can be found in Chazarain [3, 4]. In
particular, he shows that certain mixed hyperbolic problems are well-posed in
certain spaces of ultra-distributions, while the corresponding problems
in the "usual" spaces of distributions are not (cf. Ikawa, Add.
Bibliography).
The results of Section 12 are due to Ovciannikov [1], Treves [4, 2]
and Treves-Steinberg [1]. We have followed the presentation of Treves
[4], where numerous other results can be found; Theorem 12.1 extends
to certain nonlinear pioblems, which Treves applies, in particular, to the
Cauchy-Kowaleska theorem.
14. Problems
14.1. The results of Section 1 most probably extend to the case where
the spaces V are replaced by families of spaces V(t) depending on t in a
"regular of class Mk" fashion (for example if the V(t)'s are closed sub-
spaces of a Hilbert space K, the orthogonal projection operator P(t) of
K onto V(t) satisfying
t^ P(t) belongs to SMh (R; 5£(K\ K))).

188
14. Problems
But we have not attempted to treat this point in detail. The analog
extension for operators of the second order in t or for Schroedinger
operators is certainly much more delicate.
14.2. We have not systematically studied what happens to the results
of this chapter if the classes $Mh are replaced with the Beurling classes
&Mk (except in Section 1.4).
14.3. It would be of interest to examine the stability properties in Mk
classes by elliptic regularization (see Chapter 3, Section 7).
14.4. In relation to Section 7, can one find examples where D(A °°; Mk)
does not reduce to {0} without being dense in E ?
14.5. Do the results given in Amerio-Prouse [1] peitaining to almost
periodic solutions in t of partial differential equations extend to almost
periodic solutions of class Mk ?
14.6. A number of considerations pertaining to the spaces D(A °°; Mk)
have meaning without —A being the infinitesimal generator of a semi-group;
how far can one go in this direction ?
14.7. "Perturbation" problems: for which operators B "small relative
to A" do we have
B((A +Br;Mh) = D(A°°;Mk)?
If we assume that E is a space of local type on an open set Q of Rn
and if A and B are differential operators, when do the spaces
D((4 + B)°°;Mk) and D(A°°;Mk) coincide on every open set 6 with
14.8. What are the necessary and sufficient conditions on f®(0) in
order to have
(14.1)
<cLkMk, Vk,Vte [0, T],
instead of (7.94) ? (See Remarks 7.13 and 7.14).
14.9. More generally, consider problem (7.94), (7.96). What
"compatibility relations" must u0 and /(0), f'{0), ... satisfy in order to have
(14.1) ? (See Remark 7.14).
14.10. Is it possible to extend Theorem 9.3, under suitable hypotheses,
to the operator equations
(14.2) Au+s/u = f,
studied by P. Grisvard [1] ? In the text, we have given ik^-regularity
results in A. It should be possible to give "abstract" M^-regularity in A
and Mjf-regularity in si results pertaining to (14.2).

14. Problems
189
14.11. The following is an example of an equation which fits
Section 9, but where hypotheses (9.20) or (9.25) are not satisfied. Let
1T
(14.3)
A = \
£2(0, T; V)xL2(0, T; V), M> = L2(0, T;H)xL2(0, T;H),
djdt 0 \
\ 0 -d/dtj
D(A;tf)={f\f = \fv 4}. U fi € ^(0, T; H),
./i(0) = 0,/2(r) = 0},
(A{t) +1 \
(14.4) s/ = [
Assume that
(14.5) Re (A (t) v, v) > oc \\ v ||2, oc > 0, Wv £ V, t£ [0, T].
Then hypotheses (9.2) and (9.3) are satisfied. In fact, we could more
generally consider jtf to be given by
Re(Ai(t)v9v)>»\\v\\\ * = 1, 2,
but (14.4) is the case which comes up in the applications to the optimal
control of systems governed by parabolic partial differential equations
(see Lions [5.])
Equation (9.5) becomes:
du-.
— +
dt
«i(0)
A*(t) u2 — t
= 0, u2(T) = 0.
■u.
But hypothesis (9.9) is not satisfied. Can one still obtain regularity results
—and in particular M^-regularity in t results ?
14.12. Characterization of the space described by v(0) as v describes
X, outside the cases given in Sections 10.2.2 and 10.2.3.
14.13. Problems analogous to Problem 6.4 of Chapter 8 come up in
connection with Section 10.3.

190 14. Problems
14.14. One can probably extend most of the results given in this
chapter to the case of equations with delay, for example of the type
du
(14.6) — + A(t)u+u(t- co(t)) = f(t),
where the function t-+ co(t) is of class Mk. For the study of (14.6) in L2-
spaces, see Artola [1], Baiocchi [3].
14.15. Do the results of Section 11 extend to families of operators
A(t), suitably dependent on t?
14.16. The results of Sections 1 and 7 apply to the operator
du
-1 J
(take V = Hy02(Q), Q = ]-l, 1[, H = L2{Q)). Thus if A is defined by
du
f5M,
Au = r>.v. I dy,
J x — y
-l J
one is led to study the space D(^4°°; Mk). Probably
(14.7) D(A~;Mk) ={v\\\v^\\L.<cLhMh9 V*. ww(±l) =0},
but this has not been shown.

Chapter 10
Parabolic Boundary Value Problems in Spaces
of Ultra-Distributions
This Chapter develops boundary value problems in spaces of ultra-
distributions for parabolic partial differential equations.
From the point of view of partial differential equations, we assume
the knowledge of the essential part of Chapter 4. Chapter 7 is also used.
1. Regularity in the Interior of Solutions
of Parabolic Equations
1.1 The HypoeUipticity of Parabolic Equations
The application of the results of Chapter 9 to the concrete case of
parabolic partial differential equations of evolution has given us mostly
information about the regularity of solutions with respect to the variable
t in Gevrey type spaces. We shall now study the global regularity, either
with respect to t or with respect to the space variables ; and to
start, we examine the regularity in the interior.
The first question which arises is the hypoellipUcity of parabolic
operators (see Chapter 2, Section 3.3 for elliptic operators). We shall of
course restrict ourselves to the parabolic evolution operators which we
have considered in this text (see the Comments for the geneial case) and
more precisely to the operators P introduced in Section 1.1 of Chapter 4.
Therefore, let us again consider the notation of Chaptei 4. Let Q be
a bounded open set in Rn and r its boundary (in this Section we do not
make any regularity hypothesis on r, since we are only concerned with
regularity in the interior). In the space Rw+1 = R^xRj, we consider the
cylinder Q = Qx]0, T[, 0 < T < oo and set S = Tx]0, T[. In Q, set
(1.1) Pu^.t.lj+l.

192 1. Regularity in the Interior of Solutions of Parabolic Equations
where A = A (x, t, djdx) is given by
(1.2) Au= Y, (-l)]p]Vpx{apq(x,t)Vlu)
\p\,\q\<n
and the coefficients apq are infinitely differentiable in Q, i.e.
(1-3) «*€<&«?).
We assume that
for every 6 £
(1.4)
71 70
and every £0£ [0, T], the operator
A
[XJ-Tx)
{-l)meidV2m
is properly elliptic in QxRy (see Chapter 2, Section 1).
Note that, under hypothesis (1.4), the operator P is parabolic in the
sense of Petrowski [2].
Then we have
Theorem 1.1. If P is given by (1.1), (1.2) with (1.3), (1.4), then P is
hypoelliptic in Q, i.e. if u £ 2f'(Q) and Pu £ g{Q), then u £ g(Q).
Proof. We shall give an indirect proof of the theorem, using the
results on boundary value problems for the operator P given in Chapter 4.
Let us first prove
Lemma 1.1. The operator
du
u-+—i = l,...,n.
dx{
is a continuous linear mapping of H2mr>r(Q) into H2mr~u~ll2m(Q), for
every real r with 2mr 4= 1/2.
Proof, a) If r > l[2m, then the result is a particular case of
Proposition 2.3 of Chapter 4.
b) If r < 0, the result follows by duality of a), since, if u £ H2mr>r(Q),
for every <p £ @(Q), we have
dcp I
\dx{ • 7
\U' dxj\
<||w||H2m,y(0)
dx.
j—Zmr. — r,
XQ)
(we recall that by definition H2mr>r(Q) is the dual of H'2mr'-r(Q)) and
therefore we deduce from a) (since — 2mr + 1 > 1) that
W7
< \\u\\H2mr,r,Q) \\<p\\H-2mr+l,-r+ll2tn{
XQ)

1.1 The Hypoellipticity of Parabolic Equations 193
and therefore
r Tj2mr — l,r — l/2m /q\
dxi
c) If 0 < r < 1/2m, we first note that, thanks to a), d[dx{ is a
continuous linear mapping of
(1.5) H^ll2m{Q) into H°'°{Q) = L2{Q)
and, thanks to b), of
(1.6) H°>°(Q) into H-l>-v>2m(Q)
and therefore by interpolation of
(1.7) [ff1'1'2*(0), H°>°(Q)]e into [#°^),#-lj-1/2mV O<0<1.
But (see Proposition 2.1, Chapter 4) we have
[#u/2w(<2), H°>°{Q)]d = Hl-0^-^l2m{Q)
and (duality theorem, Section 6.2, Chapter 1)
(1.8) [H°'°(Q), H-1--1**®)], = [Hl$l*>(Q), H°'°(Q)][_e.
Now, by an analogous proof to the one of Theorem 11.6, Chapter 1, we
have
(1.9) [H$2™(Q), ^°(0]1_9 = HW»(Q) if 6 #= 1/2.
Thus the right-hand side of (1.8) is equal to H-°>-°l2m{Q) and then (1.7)
proves the lemma for the case 0 < r < l/2w (set 1 — 6 = 2mr), under
the condition 2mr 4= 1/2. Q
Now let (x0, t0) be an interior point of Q and let q0 be the distance from
this point to the boundary of Q. For 0 < q < q0, let QQ denote the
cylinder {{x, t) | \x — x0\ < q, \t — t0\ < q}. Theorem 1.1 will be proven if
we can show that u £ @{Qe), 0 < q < q0.
Choose q-l and q2 such that q < q2 < q1 < £0. The distributions on Q
being locally of finite order, we can find a real r such that the restriction
of u to QQi (which we still denite by u) belongs to H2mr>r(Qei); in order to
, avoid the difficulties of "exceptional parameters" we also assume that r
is irrational, which is always possible. Let us show that
{if ue H2mr'r(Qo) and Pu = j G g(Q),
(1.10) \
[then ^e^2w(f+1/2w)'f+1/2w((?J.
Let (p€@{QQl) be fixed with cp = 1 in QQ%\ and set v = (pu. Then
0 g H2mr'r(Q6i) and has compact support in QQi. Furthermore, if ZQ and

194 1. Regularity in the Interior of Solutions of Parabolic Equations
ae denote respectively the lateral surface of QQ and the lower base of Qe,
we have
(1.11)
Pv=y in QQi,
0,7 = 0, ..., m — 1, v normal to 27gi,
dv3
*k = o.
with \p = <pf + [P{yu) — cpPu\. Thanks to Lemma 1.1 and (1.1), (1.2),
we have
Ir TT2tn(r-l + ll2fn),r-l+ll2m(S) \
with compact support in Q .
But then v appears as a solution of the boundary value problem (1.11),
to which we can apply the results of Chapter 4, for the system of Dirichlet
conditions {di/dv^JZo covers every properly elliptic operator and
therefore (1.11) satisfies the conditions of Section 1.1, Chapter 4.
The application of these results yields
(1.13) y ^ j£2m{r+ll2m),r + ll2m/Q \
Indeed:
a) If r — 1 + l/2m > 0, we can apply Theorem 6.2, Chapter 4 (note
that r is assumed to be irrational and thus the "exceptional" cases of
the parameter do not interfere, see (6.28) in Chapter 4);
b) if r — 1 + l/2w < —1, we can apply Theorem 12.1 of Chapter 4,
since, thanks to (1.12), we have
£&2tn(r-l + 1l2tn),r-l+ll2tn/Q \
(and the exceptional cases of the parameter are excluded);
c) there only remains the case
which fits the problems studied in Sections 13, 14, 15 of Chapter 4: in
the notation of these Sections (see Section 13.1) it corresponds to the
case where the data of problem (13.1) of Section 13.1, Chapter 4, are of
"regularity" 5, with — 1 < 5 < 0, and furthermore
(1.15) gj = 0, / = 0, ..., m — 1, and u0 = 0.
Now it is easy to see that, under conditions (1.15), the mapping
(1.16) 9:{f, 0,0}-*« = £(/, 0, 0),

1.2 The Regularity in the Interior in Gevrey Spaces 195
defined by (13.7) of Chapter 4, is a continuous mapping of
[H°'°(Q),{H2m-1(Q))'}e into [H^\Q), H°-°(Q)]e, 0<6<1,
and therefore, in the notation of Section 15.2, Chapter 4, of
(1.17) 34?-2dm'-d{Q) into H**1-®'1-'®),
without exception for 0, 0 < 0 < 1. Therefore it follows that (see (iii)
of Section 15.1, Chapter 4):
(1.18)
\if fe34f2ms's(Q), gj = 0, «0 = 0, then the solution u
of problem (13.1), Chapter 4, belongs to H2m{s+1)>s+1(Q)
for every s, with — 1 < 5 < 0.
We can now easily apply this result to problem (1.11) under conditions
(1.14), since, thanks to (1.12), we have
^ j0?2m(r-l+ll2m),r-l + ll2ni,Q \ ^
Thus we finally obtain the fact that (1.13) holds, even for the case (1.14).
But then it follows from (1.13), since cp = 1 in QQ9, that
u e H2m(r + ll2m)tr + H2m^Qj and thJs prQves ^^y
Let us now choose a sequence of numbers {qv}, v — 3, 4, ... such that
q < qv+i < qv < ^i f°r eacn ^> a proposition analogous to (1.10) applies
and it follows that
u ^ jj2m{r+vj2m),r+vl2m/Q \ yy
and therefore u € fl H2m^r+vl2^-r+vl2m(Qe) = @(Qe) and the theorem is
proved. Q
Remark 1.1. Hypothesis (1.3) may of course be replaced with the
hypothesis apq £ S{Q). Q
1.2 The Regularity in the Interior in Gevrey Spaces
Since the operator P is hypoelliptic, there exists a "characteristic"
Gevrey class for P (see for example Hormander [1]): this is the Gevrey
class to which each solution of the equation Pu = 0 belongs, under the
assumption that the coefficients of P are constants. In the present case,
it is the Gevrey class of order 2m in t and of order 1 (i.e. analytic) in x,
that is $2rn,\(Q) (notation of Chapter 7). As for the elliptic case (see
Chapter 8), we shall prove a more general theorem, by assuming variable
coefficients of P (apq£ ^2m,i(Q)) an(i by studying the regularity of the
solutions of the equation Pu = f in the spaces <f>s,AQ)> w^n s > 2m and
r > 1 (see also Remark 2.3 below and problem 6.2).

196 1. Regularity in the Interior of Solutions of Parabolic Equations
Thus let P be given by (1.1), (1.2) with (1.4) and
(1.19) a^e^iK?).
We aim to show that
(1.20)
if u £ £(Q) and Pu £ £Stf{Q), with s > 2m, r > 1, then u £ £s>r(Q).
To this end, we first prove some lemmas.
Lemma 1.2. There exists a constant c' > 0 such that for every e > 0
and every function v £ <3i(Rn+1) we have
(1.21) sW\\Dy\\Li{Rn+1)<c'\s^ 2 ||D^||l2(r^i) + II^IIlKr^^)1
for every oc with \oc | < 2m.
Proof. Applying the Fourier transform, the result is an immediate
consequence of the inequality
eH||jH<c'e2-(|||2- + l), £ = (£lt ...,|J, \oc\<2m. D
Now let (x0, £0) be an interior point of Q; let us again use the cylinders
Qe> 0< £< Qo> as *n Section 1.1; and assume, which is permissible,
that q0 < 1. Denote by ||^||e the norm of u in L2(Qe).
Then we have
Lemma 1.3. Let P be given by (1.1), (1.2) with (1.3) and (1.4); there
exists a constant C1 > 0, such that ifO<Q<Q + d<Q0 and if u £ @(Qet),
we have:
(1.22) £ eW ||D>||e + b2- ||D««||tf < Cx
\a\<:2m
X \s2m\\Pu\\e+d^^
82m £
|D>|
tfrn-WU^x^UQ + d
<x\<2mO ' '
+ ll«l
le+<3f
for every s > 0.
Proof. Let v be an arbitrary functions of <3>(Qe); then setting Pv = wy
we may consider vasa solution of the boundary value problem
(1.23)
Pv
djv
a7
»h
= w
Zq
= 0,
= 0, / = 0, ...,m — 1,

1.2 The Regularity in the Interior in Gevrey Spaces 197
to which, thanks to hypothesis (1.4) on P and to the fact that the
Dirichlet conditions cover every properly elliptic operator, we may apply
the results of Chapter 4; in particular Theorem 6.1, Chapter 4 yields the
existence of a constant c independent of v such that
(1-24) H|H2«,i(0,eo)<c||Pi,||eo.
Applying Lemma 1.2, we obtain the estimate
(1.25) £ eM || D>||fc + e2m || Dtv||fc < C{e2m || Pv||ft + ||v||J
\<x\<S.m
for every e > 0 and every v £ @(Qee), with C independent of e and v.
Also note (see Chapter 8, Section 2.2) that we can construct a function
<pQt6{x91) e ®(0J such that
(1.26)
VQ.dfa* 0 = 1 in (?e> w^t^1 suPPort contained in Qe+d
and satisfying
sup I D«xDfoeiS(X, t) | < ya>^-(l-l+2"»»,
X,t
yajk depending only on <x and k.
Now, if u(z@(Qeo), let us apply (1.25) to the function v given by
v = (pQy8u- Then we have
(1.27) £ *W HD>||e +£2M ||D4«|!8 < C{e2m \\P(<pe,eu)\\e+d + \\%,MQ>i}-
\a\<:2m
P may be written in the form
(1.28) P = X ^D^+Dr
|£|<2m
Then
where the c^'s denote suitable constants; therefore, by (1.26),
(1.29)
\\p(?e^)\\e+e<\\p^\\e+s+^ 2 2 iiDr'«iu+iiNu<
\p\<2,m qi<Pi 0™i 0
< II *Hh-«+<* 2 ^r^H^IU,
with suitable constants cx and c2. Then (1.22) follows from (1.27) and
(1.29). D

198 1. Regularity in the Interior of Solutions of Parabolic Equations
Now, for fixed real [jl and R with ^ > 0, 0 < R < q0, let us introduce
the following notation:
(1.30) N(liR(u)= sup (R-or\\u\\e.
0<q<R
The function u-^N^jR{u) is a norm which is analogous to the various
norms ak(u, X, R), Gkjq{u, X, d, R) introduced in Chapter 8 for the study
of the regularity of solutions of elliptic equations.
Note that N^R(u) (u fixed) is monotonically increasing in R and
monotonically decreasing in /j,.
Lemma 1.4. Under the hypotheses of Lemma 1.3, there exists a constant
K such that for every u £ @(Qe) and every R < q0, we have
(1.31) £ NHtR(D«u) + N2mtR(Dtu) < K{N2mjR(Pu) + ||u\\R].
\*\<2m '
Proof. Let us apply formula (1.22) with e = rjd, where rj is a positive
number to be chosen later on and 6 = (R — @)/2. Then
£ ,M 2-M {R - g)M ||D>||9 + rf 2~*m(R - Qfm ||D,«||e <
\<x\<2m
<C1\r,2m(R-Q-dfm\\Pu\\e+s+r12m £ («~ Q~ <5)W l|D>IU +
[ \*\<2m
+ II«lle+4 < ^2 {^(P*) + 2 N]xlR(D«u) + -11| M || J
and, taking the sup over q £ ]0, jR[ on the left-hand side:
(1-32) J^ (I)'"' ^|,*(D» + (t)W Xim.RPt") ^
< V2mC2N2m_R(Pu) + C2 ||«||s + ^C, 2 tfK*(D».
|a|<2w
If we now jfe 17 such that
(l)H_,?2MC2-T for 1*1 <2w»
(1.31) follows from (1.32), by moving the term
j«|<2*»
to the left-hand side. D

1.2 The Regularity in the Interior in Gevrey Spaces 199
Let us now make use of hypothesis (1.19) on the coefficients of the
operator P. Writing P in the form (1.28), it follows from hypothesis
(1.19) that
(1.33)
2 sup \D«D?ap(x, t) | < cZJ"l+*(|*| + h)^+2m\
\p\<2m QB
for eveiy oc, h and for R < q0, with suitable constants c and L.
Analogously, having fixed R < q0, it follows from the hypothesis
that
(1.34) sup |D2D*/(*, t) | < cL^+h{\oc\ + A)'W+S\ V<x, h,
where, i? fomg /^^, we may assume that c and L are the same constants
as in (1.33).
It follows from (1.33), if 3 > 0 and R — jd > 0, that
(1.35) dfW+sh 2 sup \Dffiap{x, t)\ <cLW+hU^—Y + ,
\P\ <2m QB.jd \ 1 I
for | tx | + A < j.
Indeed it suffices to note that jd < -R, since i? < 1 and to multiply
the left-hand side of (1.33) with (/<$)'M+S*; it follows that
(\ /y I 4_ h\a+2hm
y W +s* 2 sup I DJDf «,(*, Q | < cll-l +* ^-L+g_..__
|^| <2w Qb-36 1
from which, thanks to the hypothesis r > 1, s > 2m, we obtain (1.35)
for \oc\ + h < /.
Similarly, it follows from (1.34) that
(1.36) <$'!"!+s* sup |DJD*/(*, *) | < cL1*^ for (|a| + ft) 6 < 1.
Lemma 1.5. Under the hypotheses of Lemma 1.3 am? m'2A (1.33) and
(1.34), if u £ @{Qr) with Pu = /, then there exist two positive constants c0
and B such that
f 0"/"+'V 2 ^|a|,fi-(l^l+w(D:+',DfM)+iV2roiR_(|,|+,)d(D^Df+1M))
j \|«|<2»» /
<c0BM+ht
for every /?, h and 6 with R — (|/? | + h) 6 > 0.
Proof. We prove the lemma by induction on \fi\ + h. The inequality
(1.37) is evidently true for \/}\ + h = 0; let us assume that it holds for
every f} and & such that |/? | + /&< / and prove it for /? and A such that
|j8|+A = />0.
(1.37)

(1.38)
200 1. Regularity in the Interior of Solutions of Parabolic Equations
Using Lemma 1.4 applied to D^Df^, we obtain
[d"«+«*J £ NHiR_,aM+tyu) +N^R_ji(B&+1u)\<
I |>|<2m J
( < K{d^+shN2mtR_js{P(D^Dfu)) + 3W+'» ||DfD*«L_,a}.
There are two cases: either h > 1 or h = 0 and /? = (ft, ..., ft) with at
least one ft > 1.
In the first case (& > 1), we have, thanks to the induction hypothesis,
d'M+^N2m>R_{j_1)e (D£D» < c0BW+*-i = c^1
and therefore, by definition (1.30):
(1.39) <JfM+s(*-1,(fl — (/ — 1) d — ep ||DfD*«||c < c^'1
for 0 < £ < R — (/ — 1) 6. Take q = R — jd] then we have
^|+.*-.0*. ||D£D*„||B_ja < CoB^-i;
thus, recalling that s > 2m and 6 < 1, we have
(1.40) ^l+^IID^II^^c^-1.
In the second case (h = 0, ft > 1 for at least one i), set ft =
(ft,...,ft-i,ft-l^m---ft);thusi8=ft+e,,e,= (0,...,0,l,0,...,0)
and | ft | = / — 1; still by the induction hypothesis, we have
6r^NliR_u_1)6(D^'u)<c0B'-1
and therefore
0*l/>l-U(fi _ (,- - 1) 3 - e) ||D^||e < c0Bi~\ 0<Q<R-(j-l)3
whence, taking q = R — jd and recalling that r > 1 and 6 < 1,
(1-41) ^l||D^||fi_.,<Coi^\
which is precisely inequality (1.40) in the case h = 0; therefore (1.40) is
still valid.
Now note that
(1.42) P(D£D*«) = DfDfP« + [P(D£D» - D^DfP«] = DfDf/ + g.
First, thanks to (1.36), we have
(1-43) ^m+shN2m,R^M^f) =
= ^W +** sup (R-jd- Q)2m || DfD»/||e <
o<e<^-;<3
<d'W+sh sup |DfD*/(*,*)| sup (R - / 6 - q)2m (meas. <?e)1/2 <
Or-;<5 o<e<^-i<3
< yl^l+sA sup |DfD*/(*, 0 | < cL^+A = cl/.

1.2 The Regularity in the Interior in Gevrey Spaces 201
Next, taking into account (1.28), we have:
(1.44) 6^»N2mtR_jd(g) < 2 2 (f) &)-&) X
X(5fN+s* sup |D*D{a,(*, *) I ^(l^-^l)+s(A-^iV2Wji,_;6(D^+^-^Df-^).
Qb-jS
But
since | /? — 771 -\-h — I <j, thanks to the fact that I + | rj | > 0.
Therefore, using (1.35) and the induction hypothesis, we have
J+hl>o
i+\n\>o
Noting that
s(£)-W)- °^'"-
hi-*
we obtain (recalling that |/?| -{-h = j and that if rj<P, |/?— ^| =
I/* I -Nl)
(1.45) ^^l+s^2,,^;,(g) < £ Q (yj cLV*'"' <
A/* V^^-i
< we haver. II— < e*"1 if 1 <»'</]<
i /eLV'-1
< cc^LB1-1 2 l-g) < cc0LBi-x
for B > 2eL.
Let us now fix (which is permissible) c0 and 2? so as to satisfy
(1.46) B > 2eL, B > K + LK + cLK, c0 > c.

202 1. Regularity in the Interior of Solutions of Parabolic Equations
It then follows from (1.38), (1.40), (1.42), (1.43) and (1.45) that
[ \<x\<J2,m J
< K^B*'1 + cl> + cc^B*-1) < c0B> [^ + ^ + ^j < c0B.
and we thus obtain (1.37), for \p\ + h = j. Q.E.D. D
Finally, we have
Lemma 1.6. Under the hypotheses of Lemma 1.5, for every R1 < R,
there exist c% and L* such that
(1.47) sup |D£D*«(*, t) | < c+Lf+k(\p\ + /*)^l+sA, V£ and VA,
and therefore u£ 3fStfiQR^.
Proof. Indeed, it follows from Lemma 1.5 that
(1.48) *lfl+'* ||DfDf«||JJ.(W+A)a < ^1+*
for R— (\p\ +h)d>0.
R- Rt
Let d — . . in (1.48); we obtain
IP I + "
R|0I+^
^' — ■ ■ -r\fi\+sh
\^M^\\Rl<oo{R_RiYm^h(m + h)
whence, applying the Sobolev inequalities (see Chapter 1, Section 9.4),
we deduce (1.47), with c* and L* depending on c0, B and Rv And we know
that (1.47) is equivalent to saying that u £ ^sAQrj)- D
In summary, we have
Theorem 1.2. Let P be a parabolic operator given by (1.1), (1.2) with
(1.4) and (1.19); then if u £ g(Q) and if
Pu<E&Sff(Q) with s>2m and r>l,
we have
u£gv{Q). 0
Remark 1.2. Hypothesis (1.19) may of course be replaced with the
hypothesis: apq £ ^2ml(Q). D
Remark 1.3. If u£3f'{Q) and Pu£gSjf(Q), it still follows that
u £ gSt (Q), thanks to Theorems 1.1 and 1.2.' D

2.1 The Regularity in the Space 9{Q)
203
2. The Regularity at the Boundary of Solutions
of Parabolic Boundary Value Problems
2.1 The Regularity in the Space @(Q)
In this Section we shall study the regularity up to the boundary of
solutions of parabolic boundary value problems in the non-variational
formulation which we have already studied in the spaces H2mr>r(Q) in
Chapter 4. In this manner, we obtain a very extensive class of boundary
value problems. We note that many of the results to be obtained are still
valid for other problems, which fit different formulations (variational
theory, semi-groups, ...) and which we have discussed in the preceding
Chapters. Q
Let Q and P be the cylinder and the operator introduced in Section 1.1,
P being given by (1.1) and (1.2), with (1.3) and (1.4). Furthermore we
shall assume that
(2.1)
the boundary r of Q is an (n — 1)-dimensional,
infinitely differentiable variety, Q being locally
on only one side of r.
We shall also assume (see Chapter 4, Section 1.1) that a system
{B^JJq of "boundary" operators, defined by
(2.2) B,u= X h>P>,
\h\<mj
is given, where the functions bjh = bjh(x, t) satisfy
(2.3) bjh£®(E) [E = lateral boundary of Q),
with the hypotheses:
(2.4) 0<m- <2m — 1;
(2.5)
(2.6)
for every t0e [0, T], the system {B3-(x, t0, DJ}^'1
is normal on r (see Chapter 2, Section 1.4);
for every 0£ [—?z/2, tc/2] and every t0£ [0, T], the
system {B (%, t0, DJ}^1 "covers" the operator [considered
in the space R^xRj) A(x, t0, DJ + (-l)m eid Df* on TxRj.
Then Theorem 6.2 of Chapter 4 applies for every integer r > 0. But
oo
fl H2wr>r(Q) = @(Q), thanks to the Sobolev inequalities (see Chapter 1,
Section 9.4) which easily extend to the case of the cylinder Q. Further-

204 2. Boundary of Solutions of Parabolic Boundary Value Problems
more, we are given /, u0 and gjf j = 0,..., m — 1, with
(2.7) f£®{Q),u0£®{Q),gje®{Z)
and with the compatibility relations (0t • #) such that there exists a
w £<2)(Q) with
IB,W = &» / = 0, . • •, w — 1, on E, w(x} 0) = «0(#) on Q,
D*[A(x, t, DJ w + D^l_0 = Df/(*f 0) on Q, k = 0, 1, 2, ...
Thanks to Whitney's theorem (see Malgrange [2]), the (0t • #) are the
conditions for linking gj, u0 and / on r so that (2.8) holds.
Thus we have
Theorem 2.1. Let P be given by (1.1), (1.2) and {BjffjJ by (2.2) under
the hypotheses (1.3), (1.4), (2.1), (2.3), (2.4), (2.5), (2.6); and let f, u0 and
gJ} j = 0, ..., m — 1, fo given with (2.7) awdf ^ compatibility relations
(0t • #); #&0W ^ problem
{ Pu = f in Q,
(2.9) j £;.^ = gy ow 27, / = 0, ..., m — 1,
\u(x, 0) = uQ(x) in Q,
admits a unique solution u belonging to @(Q).
2.2 The Regularity in Gevrey Spaces
Let us now study the regularity of the solution of problem (2.9) when
the data (/, gj, u0, Q, aPq) belong to Gevrey classes. As we have seen for the
problem of regularity in the interior, the characteristic Gevrey classes for
the parabolic operatois of the type of operator P are analytic classes in
x and of order 2m in t. Therefore, to the hypotheses on Q, on P and on
the B/s used in Section 2.1, we add the following hypotheses:
(2.10) r is an analytic variety,
(2.ii) «*e*W0).
(2-12) bjhe%mj(Z)-
With these conditions, we have
Theorem 2.2. Let P and Bjt j = 0, ..., m — 1, be given by (1.1), (1.2)
and (2.2), under the hypotheses (2.1), (2.10), (2.11), (2.12), (2.4), (2.5), (2.6).
Let /, uQ, gj, j = 0, ...,m — 1, be given with
(2.13) / e ®*JQ), ^ € %m,^), «0 € X{0) (= S^Q))

2.2 The Regularity in Gevrey Spaces 205
and with the compatibility relations [M • #) such that there exists a w with
W € @2m,l(Q)> B3W = gj 0n Z> / = 0, . . . , W — 1,
(2.14) <j w(#, 0) = «0(#) on Q,
D*(A(x, t, DJ w + D^) |/=0 = D*/(*, 0) o» £, VAj.
Tfow ^^ 0#isfc o?^ a^^ only one solution u of problem (2.9), belonging to
the space @2nt,i(Q)- D
Remark 2 J. The problem of specifying the compatibility relations
(2.14) still seems to be open; it would require a Whitney theorem in the
space @2tn,i(Q)> tnat is to show the existence of w£ @2m,i{Q)> satisfying
(2.14), /, gj and u0 being given with (2.13) and the relations (0t*(€)i of
Section 2.1. D
Proof. Let us introduce the function v = u — w, where w satisfies
(2.14). Then
Pv = cp in Q (where cp = / — Pw),
(2.15)
with
(2.16)
(2.16')
BjV = 0 on IT, / = 0, ..., m — 1,
v(x, 0) = 0 on .Q,
Dfo(*,0) = 0, £ = 0,1,...
Then it is sufficient to show that *>€ ^^iK?)- We may assume1
to have extended the coefficients apq and bjh in a cylinder §' = flx]^', T[,
with t' < 0 and sufficiently small so that the operators A and £y, thus
extended to Q', still satisfy the same hypotheses in Q' as in Q. Then, if
we also extend the functions cp and v to Q', setting cp(x, t) = v(x, t) = 0
for t < 0, we see that v £ @{Q'), thanks to Theorem 2.1, and that
(2.17)
Pv = cp in 0', with 9>e02mfl((?')>
5^ = 0 on rx]*', T[,
*,(*,£') =0 on fl,
and the regularity problem for v in Q thus reduces to two problems: that
of the regularity of v in the interior of Q', which is already solved by
Theorem 1.2, and that of the regularity of v in the neighborhood of the
lateral boundary rx]f, T[ of Q'
1 For example using Carleson [1], pages 197 — 201.

206 2. Boundary of Solutions of Parabolic Boundary Value Problems
This second problem is solved by
Theorem 2.3. Let P and Bjy j = 0,..., m — 1, be given by (1.1),
(1.2) and (2.2), under the hypotheses (1.4), (2.1), (2.10), (2.11), (2.12),
(2.4), (2.5), (2.6); let {x0,t0) be a point of Z {x0er, 0<t0< T) and V0
be a neighborhood of (x0, t0) in Rn+1. If v £ @{V0 A Q) and satisfies the
conditions
(2.18) Pv = cp in Vor\Q, with <p £ @2mjl(V0 ^ Q),
(2.19) B3v =VjonZnv0,j = 0,...,tn-l, with Vj e %m>1(Z ^ V0),
then there exists another neighborhood Vq of (x0, t0), Vq C VQt suc^ that
In order not to burden our presentation unnecessarily, we shall not
give the proof of this theorem here and refer the reader to the original
work of Cavallucci [1] for the case of Dirichlet conditions {Bj = yj) and
of Matsuzawa [2] for the case of general Bj; in these studies, Cavallucci
and Matsuzawa more generally examined the regularity at the boundary
of solutions of quasi-elliptic equations in Gevrey spaces and Theorem 2.2
is a particular case of their results.
Let us just note that the methods of proof make use of techniques
analogous to the ones we have used in Chapter 8, for the regularity of
solutions of elliptic equations, and in this Chapter, for the proof of
Theorem 1.2.
Finally, Theorem 2.2 follows from Theorems 1.2 and 2.3. Q
Remark 2.2. A simple proof of Theorem 2.2 can be given in the
particular case of operators A and Bj with coefficients independent of t.
Indeed, the problem reduces to (2.15) with (2.16) and (2.16') and one
can show that v £ ^2w,i ((?) in the following way. Under the given
hypotheses we can apply Theorem 8.1 of Chapter 9, from which it follows that
ve®2w([0, T];L*(Q)).
Then, differentiating the equation Pv = q> with respect to t and
applying the operator A, it follows that
i-l
(2.20) A'v = {-iy T>iv + 2 (-I)'-*-1 Ah{T>\-h-1(p), i = 1, 2, ...
and
(2.21) B^A'v) = % (-I)'-*"1 £y(^(Dp*"V)), * = 1, 2, ...,
h=0
j — 0, ..., m — 1.

2.2 The Regularity in Gevrey Spaces 207
Thanks to the fact that v£ 02m([O, T]\ L2(Q)) and to (2.16), it follows
that there exist c0 and L0 such that, for 0 < t < T,
(2.22) \\A*v{x, t) ||t.(fl) < CoI* (2»m) !> * = 0, 1, 2, ...,
m —1
(2.23) 2 ||BJ(i4t"i;)||ff2m+2m*-m,-i/2(r) < c0L*+i+1(2m(; + k + l))\
j = 0
i, k = 0, 1, 2, ...;
thus, applying the theorem on elliptic iterates (Theorem 1.2, Chapter 8),
we obtain that v(x, t) remains in a bounded set of jf(Q) (— Sf-^Q)) for
every t of [0, T],
Bat from (2.20) we obtain
|d*dm%, t) | < |d£4m*. 0| + 2 |D^*(Dr*-V(*, 0) I
A = 0
and then it can be seen that there exist c% and L% such that
|DfDM*, *) | < c.l)*\+i(2mi)\\p\\, (x, t)eQ
and consequently we have
Remark 2.3. The above proof, for the case of coefficients independent
of t, can easily be extended to Gevrey spaces of the type ^Nh,Mk ((?)>
more general than ^2m,i{Q)- Let us agam start fr°m problem (2.15),
assuming that
(2.24) V^SNhiMjb(Q)9 with (2.16'),
instead of (2.16).
And let us assume that the sequence {Nh} satisfies hypotheses (1.22),
(1.23), (1.24) of Chapter 3 and that the sequence {Mk} satisfies conditions
(1.6),..., (1.11) of Chapter 8.
Then Theorem 8.1 of Chapter 9 again applies to v and it follows that
v£@N ([0, T];L2(Q)). We continue as in the preceding Remark and
find (2.20) and (2.21).
Thanks to (2.24) and to the fact that v £ @Nh ([0, T]; L2(Q)), we deduce
from (2.20) that
(2.25) WAtyx, t) ||Li(fl) < c^N, + £ c.L^M^N^^
h = 0
with suitable constants c0 and L0.
Assume that
(2-26) Nh < M2mh, Vh.

208 3. Application of Transposition: The Finite Cylinder Case
Then it follows that
i-l
II^IIl-w < ^M2mi + C.U-1 £ M2hmM2m{i_h_iy
h=0
But, thanks to the hypotheses on {Mk}, we have
^■2hm^-2m{h-i-l) ^ ^■2hm^2m(h-i) ^ °Y^2mi
and therefore
(2-27) II^IIl^)<^4^2^.
with suitable c% and L% and for every i = 0, 1, 2, ...
Similarly, we have
(2.28)
m — 1
2 II Ww) ll^+2^-^-i/2(r) < o^Lk+i+1M2m{k+W), ft, * = 0,1, 2,...
; = 0
and therefore we can apply the theorem on elliptic iterates of Chapter 8,
and continuing as in the preceding Remark, we find that
In particular, we can take
(2.29) Nh = (h\)s, Mk = (k\)r, with r > 1, l<s<2mr.
Thus this Remark poses the problem of finding under what conditions on
the sequences {Mk} and {Nh} and on the coefficients of P and of the B/s,
Theorem 2.2 is still valid in the space ^Nh)Mk (Q)»tne conditions used here,
in particular (2.26), are not necessary (compare for example (2.29) with
the conditions 5 > 2m, r > 1 of Theorem 1.2 and with the result of
Friedman [4] on the regularity "in the interior"). Q
3. Application of Transposition: The Finite Cylinder Case
3.1 The Existence of Solutions in the Space @'(Q):
Generalities, the Spaces X and Y
We shall now apply the regularity results obtained in the preceding
Sections in.order to study, by the transposition method, the boundary
value problem
IPu = f in Q = Qx]0,T[, T finite,
Bju = gj on Z, 7 = 0 m — 1,
u(x, 0) = u0 on Q,

3.1 The Existence of Solutions in the Space 3>'(Q) 209
in spaces of distributions, and of ultra-distributions on Q. We shall first
examine the case of distributions.
The notation and the hypotheses on Q, P and {B3}f~Q are those of
Section 2.1 of this Chapter.
We further introduce (see Section 1.3 of Chapter 4) the operator P*,
formal adjoint of P:
(3.2) P* = A*-Vt= £ (-l)W V*{apq(x,t) D*) - D,.
\P\,\g\<m
We recall Green's formula:
(3.3)
m — 1 m — 1
J (Pu) v dx dt — j uP*v dx dt = ^ J S-uC^v da — ^ j B^l^v da
Q Q 3=0 Z j=0 Z
+ / u(x, T) v(x, T) dx - J u(x, 0) v(x, 0) dx, u,v£ Sf{Q),
Q Q
where C;-, Sj, T;, j = 0, ..., m — 1, are suitable "boundary" operators
(see Chapter 2, Section 1 and Chapter 4, Section 1.3); we note that the
coefficients of Cj, S1 and T3 will also belong to @2m,i{£)> thanks to the
hypotheses on apq and bjh. Q
We start from the adjoint problem:
' P*v = q> in Q, <p £ 9>{Q),
C-v = 0 on 27, / = 0, ..., m — 1,
v(x, T) = 0 on Q,
(3.4)
to which we can apply Theorem 2.1 (with the obvious changes of operators
and of time direction). We obtain the following result.
Let
X = {v | v e ®[Q), v{x, T) = 0, C3v = 0, / = 0, ..., m — 1, P*v £ @{Q)}
provided with the inductive limit topology of the spaces
X{v) = {v | v e @{Q), v{x, T) = 0,CJv = 0,j = 0,...,tn-l,P*ve 2^ (Q)} ,
where {jf„} is an increasing sequence of compact sets contained in Q, of
union Q, ($}%■ (Q) being the Frechet space of functions of 2{Q) with
support contained in Jf„, X^ being provided with the natural Frechet
space topology); thus, X is a strict {££3F)-space.
Then it is easy, by an application of Theorem 2.1 and of the closed
graph theorem, to obtain
Proposition 3.1. The operator P* defines an [algebraic and topological)
isomorphism of X onto 2(Q). D

210 3. Application of Transposition: The Finite Cylinder Case
By transposition of Proposition 3.1, we deduce
Proposition 3.2. For every continuous antilinear form v —> L(v) on X,
there exists a unique distribution u £ &(Q) such that
(3.5) (u,P*vy =L(v), VveX
and u depends continuously on L [for the strong dual topologies). Q
It is now required to choose the form L in an appropriate way and to
interpret (3.5); formally L should be in the form (see Green's formula (3.3))
(3.6) L(v) = </, v} + <u0, v(x, 0)> + 2 <g,, ?>>,
where /, u0, gj are "functions" given on Q, Q and S in such a way as to be
able to deduce the solution to problem (3.1) from (3.5). Q
Concerning the choice of /, we can proceed along considerations
analogous to those of Section 3.2 of Chapter 8 for the elliptic case. Choose a
space K{Q) of distributions on Q, such that
(3.7)
X C K{Q) C L2(Q), with continuous injection,
K(Q) is reflexive,
@J{Q) is dense in K(Q).
Then, if we pick / in K'(Q), dual of K(Q), we easily see (write (3.5) with
(3.6) for every v £ S){Q)) that
(3.8) Pu = fin the sense of 3>'{Q). Q
Among the possibilities for K(Q), we shall choose one, which seems
reasonable and is valid for all boundary conditions of the problem (i.e.
for all Bj). Similarly as for the elliptic case (see Section 3.2, Chapter 8)
we shall take for K(Q), the space S(Q) defined in the following way. Let
q(x, t) be a function of @{Q), positive in Q, zero on the boundary oiQoi
the same order as the distance from (x, t) to this boundary, then
(3.9) 3{Q) = {u | eW D*« € L2(Q), Woe}
is provided with the Frechet space topology given by the semi-norms
lleHD"«ll^(0,. v«.
The space S(Q) satisfies conditions (3.7); by the same proofs as for the
space 3(Q) (see Propositions 3.3, 3.4, 3.5 of Chapter 8) we show that
@{Q) is dense in S(Q) and that S(Q) is reflexive, and we obtain a
representation theorem for the elements / of the dual S'(Q) of E(Q). Q

3.1 The Existence of Solutions in the Space @'(Q) 211
Having fixed the choice of S(Q), we now introduce the space
Y = {u\ue@'(Q),PueB'(Q)}>
provided with the coarsetst locally convex topology which makes the
mappings u^u and u-> Pu of Y into 3f\Q) and E'(Q) respectively,
continuous.
As for the elliptic case, we have the density theorem:
Theorem 3.1. The space <2){Q) is dense in Y.
Proof. Let u^M(u) be a continuous antilinear form on Y; it may
be written, all the spaces introduced being reflexive,
(3.10) M(u) = </, uy + <g, Pu} with fe@(Q) and geS(Q).
Assume that we have M(q>) = 0, Vcp £ 9>(Q) and let us show that M(u) = 0,
vue y.
Note that, thanks to the hypotheses on A, there exists an s > 0 and
an open set Qe such that Q C A* and a linear operator s£{%, t, Dx)
extension of A to Qe = QeX]—e, T + e[, with coefficients belonging to
®2m,i{Qe) and such that the operator
J*(x,t0,Dx) + {-ire6v»*
is properly elliptic in QexRy for every 6 6 [—tz/2 ,tt/2] and every
^o £ [~~e> ^ + £] (see also the footnote to the proof of Theorem 2.2 of this
Chapter).
Let / and g denote the extensions of / and g to Qe by 0 outside Q. Let
0 be arbitrary in @(Qe); then we have
(3.11) </, <£> + <l s40 + D^> = 0,
the brackets being taken in the duality between @'(Qe) and @{Qe);
indeed, this expression is equivalent to M(<p), q> = restriction of 0 to Q,
element of @(Q). It follows, s/* denoting the formal adjoint of s/t that
(3.12) j/*g — T>tg = —/ in the sense of 2>'{QB).
But then, thanks to the fact that / £ @(Qe) and to the hypoellipticity of
jaf* — D^ (see Theorem 1.1), it follows that g is infinitely different]'able
in Qe\ furthermore, it follows from Theorem 1.2 that for each t,g is
analytic in x in Qe — Kt (Kt support of #-> f(x, t)); now, by definition
g vanishes in Qe — Q, therefore it vanishes for each £ in a neighborhood of
r and for every x £ Qe, if t > tx or t < t0 (where 0 < t0 < t± < T, in
such a way that /(#, t) = 0 for £ < £0 and for £ > tltx €Q). Therefore

212 3. Application of Transposition: The Finite Cylinder Case
Then (3.12) implies
P*g = -/ in Q
and therefore in (3.10) we have
M(u) = -<p*g, uy + <g, Pu~>
and so M(u) = 0,Vue Y, since g e 2&{Q). D
Remark 3.1. For Theorem 3.1 to be valid, it is sufficient that P
satisfies the hypotheses of Theorem 1.2 of this Chapter.
3.2 Space described by Tdv as v Describes X
We now have to choose the boundary data in (3.7) and prove a trace
theorem for the elements of Y. To this end, the essential point is the study
of the space described by
&v = {v{x,0), 7>, ..., Tm_xv)
as v describes X.
We have completely solved this problem only under more restrictive
hypotheses on A and the Bj's, the problem being already technically very
complicated in that case. We have met a similar problem in Chapter 9,
Section 9.2; similarly as for that Section (see the hypotheses (9.11),
(9.13)) we shall make the following hypotheses:
(3.13)
(3.14)
the operator A and B-t j = 0, ..., m — 1, do not depend
on tin a neighborhood oft = 0 (i.e. apq(x, t) = apq(x),
[bjh(x,t) = bjh(x) for 0<*<g;
—^4(0) = —A(x, 0, DJ, considered as an operator in L2(Q),
unbounded, with domain
D(il(0)) = {v | ve H2™(Q), Bj(x, 0, DJ v = 0, / = 0, ..., m - 1},
is the infinitesimal generator of an analytic semi-group.
Note that (3.13) (choosing, which is permissible, the operators Sj
independent of t) implies that Cj and Tj are independent of t for
0 < t < t0.
Also note that (3.14) implies that — A*(0) = —A*(x, 0, Dx),
considered as an unbounded operator in L2(Q), with domain
D(A*(0)) = {v\veH2m(Q), CjV = 0, j = 0, ..., m - 1},
is also an infinitesimal generators of an analytic semi-groups.

3.2 Space described by £Lv as v describes X
213
Finally, let us note that (3.14) is surely satisfied if the problem
[A (0), Bj(x, 0, Dx)] is variational and coercive (see Chapter 9, Section 9). D
Thus, let us study the image of X under (p. First of all, in a
neighborhood of T, v(x, t) vanishes, for in this neighborhood we have P*v = 0
(thanks to P*v £ Sf{Q)) and v(x, T) = 0, C-o = 0, / = 0, ..., m — 1;
therefore
(3.15)
T-v = 0 for x^r and t in a neighborhood of T.
Next, applying Theorem 2.3 and the fact that P*v = 0 in a neighborhood
of 27 (stiU because of P*v £ 2{Q)) and Cp = 0 on 27, / = 0, ..., m — 1,
we see that in a neighborhood of 27, v(x, t) is a Gevrey function of order
2m in t and of order 1 (i.e. analytic) in x\ and therefore
(3.16)
there exist c and L, depending on v, such that
d*(2»
d^
l*x[r)
<cLk(k\)2m, 0<t<T, / = 0, ...,m- 1,
k = 0, 1, 2,
Next, using Remark 7.11 of Chapter 9, thanks to the fact that P*v £ @{Q)>
0^ = 0, / = 0, ...,m — 1, and thanks to (3.13) and (3.14), we find
that v is analytic in t with values in D(^4*°°(0); k\), in a neighborhood of
t = 0 and therefore there exist c and L such that
(3.17)
d*(2»
d^
and
(3.18)
*l01
<cLkk\, 0<t< 1/L, / = 0, ..., m - 1, A = 0, 1, 2,...
v(*, 0)GDL(^*°°(0);^!).
Finally, we have the compatibility conditions on r, still thanks to the
fact that P*v £ 9{Q):
(3.19)
d*(2>)
<«*
*=0
2}(*,o)(^**(o)v(x,o)), *6r,/ = o,...,»-i,
A = 0,1,...
It now seems natural to introduce the following space: for every
fixed L > 0, let
^L = {^;^--->^-i}|^€i)L(^*oo(0);^!),
Vi e 9[[0, T], JTL{r)), (pj(t) = 0 if T -l[L<t<T,

214 3. Application of Transposition: The Finite Cylinder Case
and there exists c, depending on cpjt such that
HvfP) yL(D < cL»(k\r», ljL<t< T, Vk;
W^m^D < cLkk\, 0 < t < 1\L, Vk;
ff (0) -= Tfr, 0) (A*\0) V(x)), Vk}.
Provided with the norm
m— 1
ll{-"}lk= IMIdV°°«»;*I> + 2 SUp X
;/=0 *
( sup ''^»W | sup I'^WIIwl
[l/L<*<T £(£!)*" 0</<l/L ^ ^' J
^L is a Banach space.
By the same method as for Theorem 7.2, Chapter 9, it can be also
shown that
(3.20) fLCfL' if L < L', with compact injection.
Finally set
y = ind lim i^L.
In this manner, we obtain an inductive limit space of a regular sequence of
Banach spaces (see Chapter 7, Section 1.2).
From (3.15), ..., (3.19) and from the closed graph theorem (see Gro-
thendieck [1]), we then deduce:
Theorem 3.2. Under the hypotheses of Section 2.1 and furthermore
with (3.13) and (3.14), v^-Vv is a continuous linear mapping of X into
r. D
We shall now see that this mapping is also surjective. More precisely:
Theorem 3.3. Let the hypotheses of Theorem 3.2 be satisfied. Let L > 0
be fixed) there exists a "right-inverse" of &, continuous from i^L into X,
i.e. there exists a continuous linear mapping {ip, cp0, ..., 9?m_i}-> v =
&(ip, (p0, ..., (ptn-i) of VL into X such that
(3.21) v(x, 0) = y(x), TjV = <pjt j = 0, ..., m — 1.
Proof. Let {^;^0, ...,^m_1} be given in *fL. We first consider the
function
(3.22) w(x,t) = fl[A*k(0)f(x))^

3.2 Space described by J2v as v describes X
215
and show that it is analytic in t with values in D(^4*°°(0); k\) in a
neighborhood of 0. Indeed the series (3.22) converges uniformly on every
interval 0 < t < d with d < 1/L, as a series of functions with values in
L2(Q), since yj£ DL(^*°°(0); k\). Furthermore, we have
,4**(0) w\
L\Q)
fk — h oo +1
<^*)T=1?/*"+"+"(0)',w7r
< 2 cLl+h+i(l + A +*)! — < c0Z,J;+1'/W!, 0 < * < d
dh °°
^■(A*t(0)w)=^A^+i>(0)y,(x)
and therefore
1=0
with suitable c0 and L0; and therefore z# is analytic in t with values in
DL<^*°°(0);£!).
By differentiation, we immediately verify that
(3.23) P*w =0, xeQ, 0<t<d
and
(3.24) w(x, 0) = ip(x), xeQ.
Next, we consider the Cauchy problem:
f p*z = 0 in a neighborhood of 27,
(3.25) ICjZ = 0 on 27, / = 0, ..., m - 1,
[ 7^2 = ^ on 27, / = 0, ..., m — 1.
This is indeed a Cauchy problem, for {C0, ..., Cw_-,, T0, ..., r^-J
is a Dirichlet system of order 2m on 27. Then we can apply the results of
Talenti [3] on the Cauchy problem in Gevrey classes and find that there
exists a unique solution z of (3.25) in the space ®2»»([0> ^]>«^G?Cn)>
@(r) being a suitable neighborhood of r (which depends on L).
Furthermore, thanks to the fact that y^ is analytic in (x, t) for x £ r,
0 <t < 1/L, we find that z is also analytic for x £ i2, 0 < £ < 1/L.
Also note that, thanks to the compatibility relations between ip and
q>j (i.e. yj*)(0) = !}(%, 0) (4**(0)y(*)), VA)> and to the definition (3.22)
of w, we have, in a suitable neighborhood of 0, say for 0 < t < d:
2}(*, *) w(x, t) = 2 ry(*» °) (4**(°) V(*)) ^T
(3.26)
A=0
**
= S^*,(0) ^ = ^(0,
ft = 0
C(#, t) w(x, t) = 0, j = 0, 1, ..., m - 1;

216 3. Application of Transposition: The Finite Cylinder Case
therefore, by uniqueness of the Cauchy problem
(3.27) w(x,t) =z(x,t)
in a suitable neighborhood of rx [0, <5] (we may assume in @(r) X [0, <5]).
Finally we note that
(3.28) z(x, t) = 0, x e Q, T -l[L<t<T,
since %•(*) = 0 for T - 1/L < t < T.
We can now construct the "right-inverse" v = M(^\ cp0, ..., (pm-i)
by setting:
'v{x,t) = 0, in Qx[T~l[L, T],
v{x, t) = w{x, t) <x(x, t) in Qx [0, <5],
v(x, t) = z(x, t) <x{x, i) in ^T) x [0, T],
v{x, t) = 0 elsewhere in Q,
(3.29)
where a is a fixed function of @(Q), with oc(x, t) = 0 in (Q — ^(i1)) X [6, T]
and a(*, t) = 1 in fix [0, 5/2] and in^- X [0, T].
Finally, using for example the closed graph theorem, it is easy to
see that the right-inverse 0t is continuous from ifL into X. Q
Corollary 3.1. The image of X under the mapping V is the space if.
3.3 Trace and Existence Theorems in the Space Y
Let us now see how it is possible to define the traces u(x, 0) and BjU
for the elements u of Y.
Here we meet a similar difficulty as in Section 12.3 of Chapter 4 for
the case of the spaces Dp(f~^(Q), and which does not come up in the
elliptic case (see Section 3.5 of Chapter 8): we shall extend to the space
Y the operator u-> au = {u(x, 0); B0u, ..., Bm_1u], but not the
operators u-> u(x, 0) and u-> BjU separately (see Remark 3.2). Q
Let if' denote the (strong) dual of if\ then if' is the dual space of
an inductive limit space of a regular sequence of Banach spaces (see
Chapter 7, Section 1.2) and therefore if' is a Frechet-Schwartz space.
We first note the following: let u0£ <2){Q) and gj£ @(Z), and verify
the compatibility relations (2.8); then the form
m— 1
(3.30) {xp; <p0, ■ ■ ■, ?V_i} -> / «<>(*) y(«) dx + £ / SfPj &*
a 1=0 z

3.3 Trace and Existence Theorems in the Space Y 217
is continuous linear on irf therefore it may be written in the form:
{V>'><P0>'~><Pm-l}>
the bracket denoting the duality between y' and rT, and L{u,;gc,...,gm^}
belonging to V. Later on, we shall identify {u0; g0, ..., gm^} with
^0,go,-,gm-i}' wmcn is legitimate, because, as we shall see, the mapping
{uo}Jo> •■■»ft«-i}-> L{^;g0,-,gm-i} is one-to-one from a subspace of
®{Q) X [®{Z)]m of elements satisfying the {St • Tj into iT'. Indeed, if
LK;g.,.,^} = °> then
m — 1
(3.31) fu0y>dx + 2 /gj%dcr = 0, Vfy; <p0, ..., ym_x).
Q j=0 E
Let u £ £^((?) be the solution of
P^ = 0, u(x, 0) = w0(#), BjU = gjt j = 0, ..., m — 1,
which exists thanks to Theorem 2.1. Then for arbitrary v in X, applying
Green's formula (3.3), we have
m — l
— J uP*v dx dt = — J u0(x) v{x, 0) dx — ^ j g-Tp da
Q Q j = 0 E
and therefore, thanks to (3.31) and Theorem 3.2,
JuP*vdxdt = 0, WveX.
Q
But for arbitrary 0 in @(Q) there exists a.v£X with P*t; = 0 and
therefore
/ ud dx dt = 0, V0 £ 0((?) and therefore u = 0 in Q
Q
and ^0 = 0, g. = 0, / = 0,..., m — 1,
which proves our assertion.
Therefore in the sequel, we shall always assume that this identification
has been made. Q
Theorem 3.4. Under the hypotheses of Theorem 3.2, the mapping
u^au = {u(x,0); B0u, ..., Bm_xu\ of 3>(Q) into 3>{Q)x[@(Z)]m
extends by continuity to a continuous linear mapping, still denoted by u^ au,
of Y into "T''; furthermore, for u^Y and v £ X, we have the "Green's
formula"
(3.32) (Pu, v} — (u, P*v} = —<au, ¥v>,
where the first bracket denotes the duality between E'(Q) and B(Q), the
second between &{Q) and @{Q) and the third between V and V.

218 3. Application of Transposition: The Finite Cylinder Case
Proof. 1) Let u be given in Y and {up; <p0, ..., cpm-^} in if\ then
{wI <Po> • • • > <Pm-i} £ ^l f°r suitable L; choose v = S&{ip;<p0,..., 9?w_i}
as in Theorem 3.3 and introduce the form:
(3.33) £{v) = <u, P*v> — (Pu, v>,
the first bracket denoting the duality between S#'(Q) and 3l{Q) and the
second between S'{Q) and S(Q) (note that X C E{Q))-
Let us first verify that ££{v) is independent of the ' 'right-inverse''
used and only depends on {y>',(po> --->(pm-i}- Indeed, if vx and v2 are
such right-inverses, then % = v1 — v2 satisfies the conditions
X(x, T) = 0, X{x, 0) = 0, CjX = 0, TjX = 0, / = 0, ..., m - 1
and since P*# £ 3f{Q) (and therefore = 0 in a neighborhood of the
boundary of Q) it follows that
(3.34)
(3.35)
X vanishes in a neighborhood of Z (thanks to
the uniqueness of the Cauchy problem);
X vanishes in a neighborhood of t = 0 for x £ Q
(thanks to (3.34) and the analyticity in % for
sufficiently small t > 0)
We already know that x vanishes in a neighborhood of t = T for
x^Q, since # £ X, therefore
But then we have
and therefore
<2TK)=i2r(W2).
Thus the form (3.33) depends only on {ip; <p0,..., (pm-i} and may be
written £f(y),<p).
2) It is easy to see that £f(ip,<p) is continuous and antilinear on*V; it is
sufficient to prove it in yL for every fixed L; but then we can use the
expression (3.33) for 2£ and the proof results from Theorem 3.3.
3) Consequently, we have
(3.36) #(y,£)=<w, {?,£}>>
where au £ V.
Thus, we have defined a mapping u-> au of Y into if1 and we
immediately see, still using expression (3.33) for i2f, that au is antilinear.
Let us now show the continuity of au. It is sufficient to show that,
given a bounded set Si in *V (and therefore in 1fL for suitable L), there

3.3 Trace and Existence Theorems in the Space Y 219
exists a neighborhood °U of zero in Y such that
\<pu, {ip, ip}> | < 1, Vu e <%, {if, (p}e@.
Let us choose the right-inverse v of {ip, 99} given by Theorem 3.3; we have
(aut {ip, <p}} = —(Pu, v} + (u, P*v}
and v belongs to a bounded set of X and therefore to a bounded set of X{v),
fixed v, and therefore in particular v belongs to a bounded set 08x of
@J(Q) (which is also a bounded set in S(Q)) and P*v to a bounded set of
3}# (Q). Then we may take
w = \u\ue — &l,Pue — al[,
where E° denotes the polar of E, and the result follows.
4) Now \iu€3}(Q)y using Green's formula (3.3) and the identification
given earlier in this Section, we see that
au = {u(x, 0); B0u, ..., Bm_1u}.
5) Finally Green's formula (3.32) results from the above, for if
v £ X, then ffvd't" and in (3.33) we may take exactly this v and then
(3.32) follows from (3.33) and (3.36). D
We now come to the existence theorem. In Proposition 3.2, we can
choose the form L(v) in the following way:
(3.37) L(v)=<f,v>+<g*,&>>>
where /£ 5'(Q) and g* 6 if', the brackets denoting the duality between
B'(Q) and S(Q) and if' and ^ respectively. Therefore, for every / £ £"(@)
and g* G if'', there exists a unique u in ^'((?) such that
(3.38) <«, PV> = </, ^> + <g*, 2^>, Vv e X.
It follows that
(3.39) Pu = f in the sense of 9'{Q)
and therefore u£ Y; but then, thanks to Green's formula (3.32), we see
that
<au — g*, tfv} = 0, VveX.
Therefore, the image of X under To being if (see Corollary 3.1), we have
(3.40) au = g*.

220 3. Application of Transposition: The Finite Cylinder Case
Thus we have shown
Theorem 3.5. Under the hypotheses of Section 2.1 and if furthermore
(3.13) and (3.14) are satisfied, the mapping ^-> {Pu, au} is an (algebraic
and topological) isomorphism of Y onto S/(Q)Xir/.
Therefore the problem
(3.41) Pu = f in the sense of 3f'{Q),
(3.42) au = g* in the sense of Theorem 3.4
admits a unique solution uinY, for every / £ 5'(Q) and g* (0^' ,u
depending continuously on / and g*. Q
Remark 3.2. As we have already, pointed out in Theorems 3.4 and 3.5,
the boundary conditions, BjU = gj and u(x, 0) = u0, are "united" in
condition (3.42). Now, a natural problem would be to separate (if possible)
in au the operators BjU on 27 and u(x, 0) on Q. We have met a similar
problem in the case of the spaces D^~(f-1)(()) in Section 12.3 of Chapter 4;
in this case the pioblem can be resolved either by interpreting the space
analogous to i^' as a product of distribution spaces on QxUxT (see
Baiocchi [1]), or as we have done in Chapter 4, by giving meaning to the
operators BjU and u(x, 0) separately in larger spaces than the "optimal
spaces" and by taking, for the existence theorem, the data gj and u in
smaller spaces than the "optimal spaces".
Here, the problem is appreciably more complicated; we shall restrict
outselves to showing how the operators BjU for u £ Y can be defined in a
space of ultra-distributions on 27; we have the following Proposition:
(3.43)
the mapping w-> Bu = {BQu, ..., Bm_1u} of &(Q)
into [^(27) ]m extends by continuity to a continuous
linear mapping, still denoted by u-> Bu, of Y
[into [^;M(]o,r[;^'(r))r.
The proof of this Proposition is similar to the proof of Theorem 3.4,
taking the following two remarks into consideration:
a) thanks to the results of Geymonat [2] recapitulated in Chapter 7,
Section 4.4, the space ^2w(]0, T\_\ 3^{r)) may be considered as an
inductive limit space of a regular sequence of Banach spaces (see
Chapter 7, Section 1.2):
(3.44)
where </T = compact set contained in ]0, T[;
®am(]0, T[: #(r)) = inductive lira ^2m(]0, T[; Jf, L\ ^(i1)),
1 jr->]0,7,[,L->+oo,M->+oo '

3.4 The Existence of Solutions in the Spaces @'s,r[Q) 221
b) The same construction as in Theorem 3.3 (if one takes ip = 0 and
VjeS2JJi09T[;Mr{r))9 therefore in a ®2m(]0, T[; JT, L; ^M{T)))
yields a right-inverse v =■■ ^(0,q>0, ..., q>m_1), continuous from
®2m(]0, T[; JT, L; JfMCO) into X, with *;(#, t) = 0 in a neighborhood of
* = 0, for x e &. D
Remark 3.3. It would be of interest to give a structure theorem for
the elements of the space V (see Problem 6.7).
In any case, they are rather general ultra-distributions on Z \J Q;
for example, we easily see that the form
m — 1
;=0
where u0e (D(,4*°°(0), A!))' and gj £ ^'+,2m(]°> T[; Jf'(r)), the brackets
denoting the duality between (D(A*°°(0), &!))' and D(,4*°°(0), k\) and
^'+J2m(]0, T[; Jf'(r)) and 0_,2m(]O, T[; Jf(r)) respectively, defines an
element of ^'.
In this regard, note the differences between the preceding results
and those obtained in Chapter 9, Section 10.6, Examples 10.1 and 10.2.
In Chapter 9, we could take / and gj in spaces
3'm{]0, T[; 3-HQ)) and S'Mk{]0, T[; H-S(r))^,
with an "arbitrary" sequence {Mk} (i.e. only subject to the ''general''
conditions (1.22), ..., (1.25), Chapter 9); thus, the ultra-distributions
considered in Chapter 9 are.more general in t; on the other hand, they are
less general in the space variables, since here, we can take the g/s with
values in $"(T). Concerning the initial conditions, the generality of the
result is the same in both cases.
Remark 3.4. For all results of this Section, one can take, instead of
the space S'(Q), a general space K\Q), dual of a space K(Q) satisfying
(3.7); indeed, only the properties (3.7) intervened in the proofs; the space
Y then being defined by
Y = {u\ue&'(Q), PueK'(Q)}.
3.4 The Existence of Solutions in the Spaces ^,r((?) of Gevrey
Ultra-Distributions, with r > 1, s ^ 2m
In this Section, we shall investigate how the same methods as in the
preceding Section can be used to study problem (3.1) in the spaces
®s,r{Q) °f Gevrey ultra-distributions on Q, of order 5 in t and r in x, with
(3.45) s>2m, r>l.
((!)) We have restricted our discussion to s "integer +1/2", but s may in fact
be chosen arbitrarily.

222 3. Application of Transposition: The Finite Cylinder Case
Still under the hypotheses of Section 2.1 of this Chapter and furthermore
(3.13) and (3.14), we consider the spaces @Sjr{Q) and &Str{Q) instead of
@J(Q) and &(Q) respectively, and we follow the same reasoning as
before.
We thus start from the adjoint problem (3.4) with <p£@s,r(Q) an(^
introduce the space
(3.46)
Xs>f = {v\ve®(Q),v(x,T) = 0, Cjv = 0,j=0,...,m-1, P*v£@8it(Q)}.
The operator P* defines a one-to-one mapping of XSfY onto <2)Str{Q),
thanks to Theorem 2.1, we can therefore define on XSfT the image
topology of @s,r(Q) un(ler (P*)_1. Then XSfY is an inductive limit of a regular
sequence of Banach spaces XQ (Chapter 7, Section 1.2) and
(3.47) P* is an isomorphism of Xsr onto @Sjf{Q)-
Therefore by transposition:
for every continuous antilinear form v-^ L(v) on Xsr,
there exists a unique u in @'sr{Q) such that
(3.48)
[<u,P*v> = L{v),VveXSjr.
Still following the method of Sections 3.1—3.3, we introduce a space
Ksr(Q) of ultra-distributions on Q such that
(3.49)
XSjf CKsr(Q) C L2(Q), with continuous injection,
KSjr(Q) is reflexive,
[@sr(Q) is dense in Ksr(Q).
In order to fix our ideas and taking as a pattern the spaces S introduced
previously, we shall take, for the space KSf(Q), the space SSjf(Q) defined
in the following way: let q(x) be the function introduced in Section 3.2
of Chapter 8 and let d(t) be an analogous function defined on [0, T]
(continuous, positive on ]0, T[, d(t) = t (resp. T — t) in the neighborhood
of 0 (resp. T)).
For every L > 0, we define:
S^r(Q) = {u\u£${Q), such that there exists a c
with £ ||^d^D>||LW<^+^!r(^!)s, Wk and A};
it is a Banach space for the norm
M=MP|£"^+W(*0- '

3.4 The Existence of Solutions in the Spaces @sr(Q) 223
then we define
S,Sff(0)=indlim5'i;f(0)>
L increasing monotonically to + oo.
Through methods entirely analogous to those used for the space
SMk(Q) in Section 4.2 of Chapter 8, it can be shown that 3SiT(Q) is
reflexive and that ^s,r{Q) ^s dense in ESjY(Q). Concerning the condition
XVC3V(Q).
we note that the elements of Xsr belong to $Stf{Q), thanks to
Theorem 1.2.
Therefore Ss>r(Q) satisfies conditions (3.49). Q
We then introduce the space
Y..r = {^\ue&,>r{Q), Pue<,(<?)},
provided with the coarsest locally convex topology which makes the
mappings u->u and w-> Pu of Y into @'St,(Q) and S'S}f(Q) respectively,
continuous. D
Next, we show that
(3.50) 2(Q) is dense in Ys>f.
We follow the same steps as for Theorem 3.1; this time we have / £ @s,r{Q)
and g £ 5SfT(Q). Formulas (3.11) and (3.12) are again taken in the sense of
3f'(Q^ and it follows again that g£ @(Q); finally using Theorem 1.2 of
this Chapter, it follows that g£ @SjY{Q) and therefore we have
M(u) = - <P*g, u} + <& Pu~}
in the sense of &Str(Q), and therefore again M(u) = 0. D
The study of the image of XSmT under the mapping V is the same as for
X and the image of Xs>r is again given by V, Theorems 3.2 and 3.1 being
still valid with XSjY replacing X; indeed the only point which requires
modification in the proofs of Section 3.2 is the function oc(x, t) (used in
formula (3.29)). It now has to belong to 2>s,r{Q) as we^> which is possible
since r > 1 and 5 > 2m.
We note that it is precisely at this point that the hypothesis "5 > 2m"
intervenes, for the function z belongs to £^2»»([0, ^1> <^(£?(-0)) an(^
therefore cannot be extended to the entire cylinder Q in such a way as to
belong to @Stf{0) if 5 < 2m. Q
Finally the trace and existence theorems (Theorem 3.4 and 3.5)
extend to the present case; thus, we have:
Theorem 3.6. Under the hypotheses of Section 2.1 and if furthermore
(3.13), (3.14) and (3.45) hold, the mapping u-^au={u(x} 0) ',B0u,..., Bm_1u}

224 4. Application of Transposition: The Infinite Cylinder Case
of @(Q) into 3l{Q) X [Q(U)]m extends by continuity to a continuous linear
mapping of Ys>r into V, and we have
(3.51) (Pu, v} - (u, PV> = — <tr«, ¥v\ Vu e Ysr, w £ Xs>r,
the first bracket denoting the duality between B'sr{Q) and 3SJ{Q), the second
between <3>'S}r(Q) and <3>s,r{Q) and the third between V and *V', the space i^
being defined in Section 3.2.
Theorem 3.7. Under the hypothesis of Theorem 3.6, the mapping
w-> {Pu, <ru} is an isomorphism of Ysr onto B^iQ)X'f". Q
Remark 3.5. Remarks entirely analogous to Remarks 3.2, 3.4 are
valid for the spaces Ysr. [)
Remark 3.6. It is also possible to apply the results of this Section to
the regularity of Gevrey ultra-distribution solutions of the equation Pu = f
in a similar way as was done for elliptic equations in Section 4.4 of
Chapter 8. Indeed, the spaces Y and YSjf have the same space yf of
traces on Z \J Q.
Thus let u be an ultra-distribution belonging to @'SiT{Q) with 5 > 2m,
r > 1 and Pu = f, / £ L2(Q) (for example) and let us assume that there
exists a system of operators {B3}f~Q such that problem (3.1) satisfies
the hypotheses of Section 2.1 and (3.13), (3.14); then, thanks to
Theorem 3.6, u admits a trace au which belongs to V and therefore, applying
Theorem 3.5, it follows that u £ &(Q)- This situation occurs, for example,
if A(x, t, Dx) = A(x, Dx) does not depend on t and if it is strongly
uniformly elliptic in Q with analytic coefficients in Q; then the hypotheses
of Section 2.1 and (3.13), (3.14) are satisfied with the Dirichlet system
Therefore in this case the problem of regularity in the "interior^of
Gevrey ultra-distribution solutions of order s > 2m and r > 1 of
equations Pu = /, reduces to that of the regularity of distribution solutions,
which we have already studied in Section 1. Q
4. Application of Transposition: The Infinite Cylinder Case
4.1 The Existence of Solutions in theSpace^ (R; ^(&)); the Space X_
In this Section, we shall study non-homogeneous boundary value
problems for the parabolic operator P in an infinite cylinder. We shall
still use the transposition method, starting from regularity results which
can be deduced from Sections 1 and 2.

4.1 The Existence of Solutions in the Space 2\ (R; 2'{Q)); the Space X 225
We again denote by Q a bounded open set of Rn whose boundary r is
an (n — 1)-dimensional analytic variety, Q being locally on only one side
of r. In the space Rn+1 = R^x R], we consider the infinite cylinder Q =
QxR] an(i its boundary H = rxR}. The operator P is given by
d
P = A + — , where
ot
(4.1) Au = A(x,t;Dx)= 2 (~1)W D?KM) DH
|^|,kl<w
under the hypotheses:
(4-2) apqeS^;^(Q)).
We are also given a system of boundary operators:
(4.3) 5y# = B^x, t; D,) « = 2 &,»(*• 0 Dj«, order B, = mjt
with
(4.4) 0<mi<2m — l, (j = 0,..., m — 1);
(4-5) ' bihZSjp.',XiT));
(4.6)
(4.7)
/or every 0 £ [—tc/2, tc/2] and every t0 £ R, the
operator A(x, t0; Dx) +J-l)m eid Df
[ is properly elliptic in Q x Rj;
for every t0e R, the system {B^x, t0; DJ}™^1
is normal on 71; for every 0£ [—tc/2, tc/2] and
t0€ R, the system {B-{x, t0] T>x))f~Q covers the
[ operator A{x> t0; Dx) + (-1)m eid T>2ym on Tx R].
We consider the boundary value problem:
(4.8) Pu = f in Q,
IBm = & on Z, j = 0, 1,..., m — 1,
f, gj, u with support in t bounded on the left.
It seems natural to consider the problem in spaces of vector-valued
distributions or ultra-distributions on Rj. Thus we shall first seek the
solution u in the space ^'+(R; 2'(Q)).
We start from the adjoint problem:
IP*v = m in Q,
CjV = 0 on Z, i = 0, ..., m — 1,

226 4. Application of Transposition: The Infinite Cylinder Case
with ye0_(R;0(fi)) ,(we recall that by definition @'+(R;@'{Q)) is
the dual of £^_(R; &(&)), see Section 5.1, Chapter 7), the C/s still being
given by Green's formula (see (3.3) for the finite cylinder case)
m — 1
(4.11) / (Pu) vdxdt - j uP*v dx dt = £ / S-uCp do —
Q Q j=0 E
m — 1
— 2 JBjuTjVdo, u, v £ ®(R; @{Q))
j = l E
(note that, thanks to (4.2) and (4.5), the coefficients of Sj, C;, Tj also
belong to <f2m(R; 3P{T))).
We introduce the space
X_ = {v\ve@_(R;@{Q)), Cjv= 0, / = 0, ...,m — 1,
P*v e ^-(R; ®(Q))} •
Thanks to Theorem 2.1, we can easily see that P* defines a one-to-one
mapping of X__ onto ^_(R; @}(Q)y, we can therefore define on X_ the
image topology of ^_(R; ^(.Q)) under (P*)-1. We thus obtain:
\ P* is an [algebraic and topological) isomorphism
( * } \ofX_onto@_(R;@(Q)). Q
We now have to study the space described by
Vv = {T0v,..., Tm_lV}
as v describes X-; we have
Theorem 4.1. v -> &v is a continuous linear mapping of X_ into
[#_i2„(R;jr(r))]".
Proof. We recall that by definition (see Remark 4.1 of Chapter 7) we
have
(4.13) 0_(R; 9{Q)) = ind lim®6(]- oo, b];9(Q)),
&-» + oo
where @bQ— oo, 6]; @(Q)) = proj lim 2>h(\a, b] \ @(®)) and where
a-> — oo
^&([a, b]; @(Q)) is the space of infinitely differentiable functions q> on
[a, b] such that cp^k){b) = 0, V£, space provided with the topology of
uniform convergence on [a, b] for cp and each of its derivatives.
We also recall that
(4.14) 9b[[a,by,9(Q))=md1iM9b[[at b];SXi(Q))t

4.1 The Existence of Solutions in the Space @'+(R; 3>'{Q)); the Space X
227
where {jfY} is an increasing sequence of compact sets contained in Q, of
union Q, and Qj^^Q) is the (Frechet) space of functions of Q)(Q) with
support in Jf*; the proof of (4.14) follows, for example, from the
Corollary in Chapter 1, page 47, of Grothendieck [1] and from the Proposition
in Chapter 2, pages 84—85, of the same work.
Let us introduce
r XJfc b]; JQ = {v | v G Sb([a b]; S(Q))
Cjv = 0 on rx [a, 6], / = 0, ..., m — 1,
(4.15) j X_{[a, b]) = {v \ve@b([a, b]; @(Q)), Cp = 0 on
| Tx [a, 6], / = 0, ..., m — 1, P*v £ @b{[a, b]; @{Q))},
X-(6)={w|w6®6(]—cx>,6];^(fi)), C^ = 0 on
[rx]—oo,b], / = 0, ...,m — 1, P*^€^&(]-oo, 6];0(fi))},
each space being provided with the image topology of @}b( [a, b]; ^^.(Q)),
resp. 3fh(\_a, b]\9{Q)), resp. 06(] — oo, &]; 0(fi)) under (P*)"1, which is
possible since P*, thanks to Theorem 2.1, defines a one-to-one mapping
of X-([a, b]\Xi)t resp. X_([a, &]), resp. Z_(6), onto 2)b([a, b]\3)^.{Q))y
resp. 06([>, b];@(Q)), resp. 06(] — oo, 6]; 2>{Q)). Also note that:
(4.16)
X-[[a, b]) = ind Hm X_{[a, b]; jf,-)
and therefore, thanks to (4.14) and to the fact
that 3fh[\_at b]; &#■ (Q)) is a Frechet space,
X_([a, b]) is an (JS?#")-space.
Furthermore, we have
X_(&) - proj UrnX_[[a, b]); X_ = ind lim X_(6).
Let us show that
(4.17)
v-^ffv is a continuous linear mapping of
X„{[a,b]) into [@bt2m{[a, 6]; ^{T))\m (notation
of Section 4.3 of Chapter 7).
Indeed, we note that if v £ X_([a, b]), then P*v vanishes in a
neighborhood of rx [a, b] and Cp = 0, / = 0, ..., m — 1, on Fx {[a, b];
therefore we can apply Theorem 2.3 and deduce from it that v is of

228 4. Application of Transposition: The Infinite Cylinder Case
Gevrey class of order 2m in t and analytic in x in a neighborhood of
rx[a,b] and therefore TjV belongs to @b,2m([a>b] '> ^(-H) ior j =
0, ..., m — 1. (4.17) then follows from the closed graph theorem of
L. Schwartz [4], thanks to (4.16) (X_([a, 6]) then is a space of type (/?)
according to Grothendieck [1], Chapter 1, page 17) and to the fact that
@b,2m([&>b]m> 3^{r)) is a Suslin space (see Chapter 7, Section 4.4,
Remark 4.4).
We can now pass to the projective limits with respect toa->- oo,
in (4.17); we obtain that v-^&v is a continuous linear mapping of
X-{b) into
[»6iftw(]«-~fj];jr(D)rf
and finally, by passage to the inductive limits with respect to b -> + oo,
the Theorem follows. Q
We now have to show that To is surjective. To this end, let us prove
Theorem 4.2. For every fixed positive L and M and every fixed interval
[a, 6], there exists a continuous linear mapping (fright-inverse" of ft)
<p = tyo> • • > <Pm-i} ^v = &$) of
into X^([a, b]; Jf;), where the compact set Jf { depends on L, M and [a, 6],
such that
(4.18) TJv=tpj, j = 0, ...,m- 1.
Proof. Let <^ £ @b,2m([a> b], L; 3ffM[r))t j = 0, ...,m — 1. Consider
the Cauchy problem
IP*w = 0 in a neighborhood of rx [a, b],
Cj-10 = 0, T-w = <pj} j = 0, ..., m — 1, on Tx [a, 6].
Thanks to the results of Talenti [2, 3], there exists a neighborhood (9
of r, depending on L and M, and in this neighborhood one and only one
function w £ @b,2tn([a> b]; 3^(0)) (and more precisely, w £ @2b,m( [a> b];
Z/, .#V(0)) with U and M' depending on L and M) solution of (4.19);
furthermore w depends continuously in @b,2m([a>b]'t L', ^M>((9)) on
q>jt as q>j varies in %2m(l>, 6]; £, ^mCO)-
Next, let #(#, £) be a function of Q){Qx [a, b]) with:
*(#, *) = 1 in a suitable neighborhood & X [a, b] of .Tx [a, 6], 0' C 0,
oc(x, t) = 0 in (fl — 0) X [a, 6].

4.2 The Existence of Solutions in the Space @'+(R; 9\Q)) 229
Then, the function v defined by
Ioclx, t) w(x, t) in OX [a, 6],
_
0 in {Q - (9) X [a, b],
provides the required right-inverse ^(9?). Q
Remark 4.1. If <p(x, t) = 0 for % £ F and t £ [a', b'], a < a' < b' < b,
then the given construction of v guarantees that v(x, t) = 0 in
Qx[a',b']. D
Corollary, v-^ftv is a surjective mapping of X- onto
[0_>2M(R;^(r))r.
Proof. It is sufficient to slighlty modify the given construction for
the right-inverse of Theorem 4.2: let y = (q>0, ..., q>m-i) belong to
[^_,2w(R; ^{r))]m', there exists a b such that ty (or better, its restriction
to ]'— 00, b]) belongs to ^&j2w(] — 00, b]; ^(i1)), / = 0, ..., w — 1.
Therefore for every a <. b, there exist La and Ma such that
9ye®6faii([a,6],ifl;^Mfl(r)).
It will suffice to consider a sequence an which decreases to — 00 and to
assume that the corresponding sequences Lan and Man increase. For
each ny we then make the construction of Theorem 4.2, in which we may
assume that the neighborhood (9n of T contains the neighborhood Qn+i,
so that the function wn+1(x, t) coincides with wn(x, t) in ^+1X [an, b];
and we may also assume that ocn(x, t) is the restriction to Q X [an, b] of
an infinitely differentiate function oc(x,t) in Qx~\— 00, b], Q
4.2 The Existence of Solutions in the Space Q)\ (R; @\Q)):
The Space Y+ and the Trace and Existence Theorems
Now, let us again consider Proposition (4.12); by transposition, we
obtain:
(4.21)
for every continuous and antilinear form v-+L{v)
on X-, there exists a unique u in ^'+(R; &'{&))
[ such that (u, P*v> = L(v),v £ X_.
Formally, we shall take L(v) in the form:
m = </, *> + 2 <gP TjVy
J=0

230 4. Application of Transposition: The Infinite Cylinder Case
and seek the "best" spaces for / and gj. We shall follow the general
procedure of this text, in particular of Section 3. But, in order to avoid
topological difficulties which might overburden the presentation, we shall
make use of the weak dual topologies in the spaces intervening in the
trace and existence theorems.
For the choice of /, we consider a topological vector space i£L such
that
IX_ C K- C £?oc(R; ^2(^))> with continubus injection,
0_(R; 2f{ii)) is dense in K_.
Such spaces exist; for example ^_(R; L2(Q)). In order to fix our
ideas and in analogy with the choices made in the preceding Sections,
we take for iC the space
(4.23) 0-{R;E(Q)),
where B(Q) is the space defined in Section 3.2 of Chapter 8. Indeed, the
space (4.23) satisfies conditions (4.22): in particular, we can easily see
that ^_(R; 2f{Q)) is dense in 9-{R) B(Q)), using the fact that 9{Q)
is dense in B(Q) (see Proposition 3.3 of Chapter 8). Q
Remark 4.2. One can even show that ^_(R; B(Q)) is reflexive (see
Lions-Magenes [2], Sections 4.2): but we shall not make use of this
property here. []
We shall use the spaces
(4.24) ®'+(R; B'(Q)) = dual of #_(R; B(Q)),
(4.25) ®'+(R; 9'{Q)) = dual of 0_(R; 9(0)),
(4-26) 9+jJjL; *'(r)) = dual of @_>2m{R; Jf (i1)),
all provided with the weak dual topology.
Note that, thanks to (4.22), ^'+(R>* s'[®)) maY be identified with a
subspace of 9\{R) 9'{Q)). Q
We introduce the space
(4.27) Y+ ={u \u£9'+(R;9'(Q)); Pu£ #'+(R; 5'(Q))},
provided with the coarsest locally convex topology which makes the
mappings u-^u and w-> Pu of Y+ into ^+(R; &'(&)) (weak) and
3f'+(JL\ B'(Q)) (weak) respectively, continuous.
First we have
Theorem 4.3. The space ^(R; 3f{Q)) is dense in Y+.
Proof. Let u-> M(u) be a continuous antilinear form on Y+; it may
be written, the spaces @+(R; 2f'(Q)) and @'+(R; B'(Q)) being provided

4.2 The Existence of Solutions in the Space $'+(R; 9\Q)) 231
with the weak dual topologies:
(4.28) M(«) = </,S> + <g,Pii>,
with / £ ®-(R; 0(G)) and g £ 0_(R; ^(fi)).
Assume that we have M(qj) = 0, V<p£ ^(R; ^(£)) and let us show
that M(«) = 0, "in £ Y+.
Thanks to the hypotheses on A, there exists a linear operator
si{x, t] DJ defined in a suitable neighborhood V((?) of the cylinder Q,
coinciding with A on Q with coefficients which are Gevrey functions
of order (1, 2m) in V(Q) (i.e. analytic in x and of Gevrey class of order 2m
in t), and such that the operator
^(x,t0:Dx)+(-ire*dD2y™
is properly elliptic in V(Q) A {(*, *), * = *0} x Ry for every 6 £ [—rc/2, rc/2]
and every £0£ R} (see also the footnote to the proof of Theorem 2.2 of
this Chapter).
Let / and g denote the extensions to V(Q) of / and g by zero outside Q.
Then if 0 £ ®(V{Q)), we have
(4.29) </, 0} +<l^0 + B^y = 0,
the brackets being taken in the sense of distributions on V(Q), for this
expression is equivalent to M{cp), cp = restriction of 0 to Q (and therefore
<p £ 0(R; 2&{Q)). $i* denoting the formal adjoint of si, it follows that
(4.30) j/*g — Dtg = —fin the sense of Sf\V{Q)).
But then (Theorem 1.1), / belonging to @(V(Q)), it follows that g is
infinitely differentiable in V(Q); furthermore, it follows from Theorem 1.2
that for each t0, 'g is analytic in x in
V(Q) A {(x, t), t = t0}~ K,
(Kt support of %-> f(x, t)); now by definition g vanishes in V(Q) — Q,
therefore it also vanishes in a neighborhood of 27and therefore g£ £^_(R; @(Q)).
Consequently (4.30) implies
(4.31) P*g=-f in Q
and therefore we have
(4.32) M(u) = -<P*g, u} + <g, Pu>;
therefore M(u) = 0, Vu £ Y+, since g £ ^L(R; ^(fi)). Q

232 4. Application of Transposition: The Infinite Cylinder Case
Next, we have the trace theorem:
Theorem 4.4. The mapping u-> Bu = {B0u, ..., Bm_1u} of
@(R;@(Q)) into [^(R; @(r))]m extends by continuity to a continuous
linear mapping, still denoted by u-^Bu, of Y+ (weak) into [@'+)2m(R;
J^f(r)]m (weak); and we have Green's formula
(4.33)
m — 1
<J>u, v} — (u, P*v} = — 2 (B^, T3vy, Wue Y+ and *> £ X_,
j=o
the brackets denoting the duality between &+(R;E'(Q)) and @-(R;E(Q)),
@'+(R;@'(Q)) and@-(R;@(Q)),@'+}2m(R; #\r)) and@_>2m(R; ^(P)),
respectively.
Proof, 1) Let fixed u £ Y+; then u has support in t bounded on the
left by t0. Let [a, b] be an interval with a < t0 and let d be fixed in
@2tn(]a> &[)■ For given positive L and M, consider the mapping
y-+Zd(y),
defined for y = (Vo, ..., y^) e m)2m([a, b], L; 3fM(r))]m by
(4.34) Ze(q>) = (u, P*v(6<p)} - <Pu, v(0<p)>,
where the first bracket denotes the duality between i^'+(R; @>'(Q)) and
^L(R; ®{Q)) and the second between @'+(R; E'(Q)) and ^L(R; E(Q)),
and v(dq>) is the right-inverse of <T defined by Theorem 4.2 and extended
in Q so that v(6ql) = 0 for t > b and v(dcp) £ ^L(R; 2{Q)) with P*v£
^L(R; ^(fi)) (note that: 1) 6 being fixed, dy belongs to
with suitable L' and ikfr depending on L and M, and 2) the construction
given in the proof of Theorem 4.2 guarantees, see Remark 4.1, that
v(dq>) vanishes in neighborhoods of t = a and t = b, since dq> vanishes in
neighborhoods of t = a and t = b; therefore, we can for example extend
v(Qcp) by zero outside [a, b], obtaining a function of ^_(R; &(&)) with
Let us verify that Ze((p) does not depend on the right-inverse of £T and
on the extension into Q used for the function v(dcp), but only on <p (6
being fixed). In fact, if vx and v2 are two such functions v(dcp), then
w = vx — v2 satisfies
(P*w =w with y€0jR;0(fl)),
CjW = 0, TjW = 0, ] = 0, ..., m — 1,

4.2 The Existence of Solutions in the Space @'+(R; ®'(Q)) 233
and we have
<u, P*w} — <Pu,w} = 0,
whence
<u, P%> - (Pu, v±y = <«, p^> - <p^, ^2>.
This being set, it follows immediately from Theorem 4.2 and (4.34) that
<p -> Ze(<p) is a continuous antilinear form on [^&j2w([<z, &]; L, ^M(r))]w.
2) We can now extend the definition oiZ6((p) ([a, b] and d being fixed)
to the space [^&j2w(|>, b], J^(r))]m by using the fact that (see Chapter 6,
Section 4.4)
(4.36) ind lim 9bfim[[a, b], L; jfM(D) = @b>2m{la> b^> -W)) •
Indeed if
£e[0M„([M]..*(T))r,
then there exist L and M such that
^[^>2M([a,&],I;^M(r))r
and therefore we can define Zd(<p) by (4.34). Using (4.36) we then see
that Zd(<p) is continuous on
3) Now let 9? be arbitrary in
[0_>2w(R; *(r))T.
Then the support of <p in £ is bounded on the right by a number ^ and we
may assume that t0 < tv Consider a fixed interval [a, b] with a < t0 <
^ < 6; and let d £ ^2w(]<z, b]) with 0 > 0 and 0 = 1 in a neighborhood
of p0, y. Then 09? (or better, its restriction to [a, b]) belongs to
and furthermore dcp = 0 in the neighborhood of a and b.
Therefore, we can define <p->Z(<p) by
(4.37) Z$)=Ze$),
where Ze(fp) is defined in 2).
We see that Z(y) is independent of 0. In fact if d± and 02 are two such
functions, we can find L and M such that, 9? being fixed, dtf and 02^ both

234 4. Application of Transposition: The Infinite Cylinder Case
belong to
[<3bi2m([a,b-],L;tfu(r))Y.
Then we can use the same right-inverse to define Z6i((p) and Zd2((p) by
(4.34) and we shall therefore have
zS) - ZS) = o> pS(ei - e2) v)> - <p™> »((0i - %) ?)>■
But 6± — 02 = 0 in a neighborhood of [t0, JJ, therefore ^((0-l — 02) q>) = 0
in this neighborhood (this follows from the same construction as in
Theorem 4.2, see Remark 4.1).
Then, u (resp. <p) having support bounded on the left by t0 (resp. ty,
it follows that
zoS<P) = ze$)-
Therefore Z((p), defined by (4.37), is independent of 0.
4) We easily verify that Z(cp)t defined in 3), is an antilinear form on
[^_>2M(R;^(r))r.
Let us now show that it is continuous. For this purpose, it is sufficient
to show the continuity of Z on
[#»,*»(]-~.»];-W)):r.
for each fixed b. Consider a partition of unity in ] — oo, b + s] (s > 0
fixed) by 6{ £ S2t„(R), such that
oo
(4.38) £ 0,-W = l if t<h-
* = 1
Note that
(4.39) Z(9) = ZIZ(0#),
i
where the sum ^ is taken over a finite number of indices i only, thanks
i
to the properties of the supports of the 0/s and of u.
Again recall that
(4.40) 06>2W(] - oo, b], jfT(D) = proj lim ^2m( [a,, b]; jf(r)),
di^*— oo
where, for ^, we may take the left endpoint of the support of %{.
Consider the mapping
of

4.2 The Existence of Solutions in the Space Q>'+ R; 3>'{Q)) 235
into the topological product
ft [#»*.([«* *];-w)]"-
* = 1
oo
This mapping is one-to-one since 2 8i<p = (f> in ]— oo, 6]. Furthermore,
i = l
thanks to (4.40),
is provided with the topology of
oo
*=i
Thus, according to (4.39), it suffices to show that the mapping
y-^*Z(d(p) is continuous on
[3W[<*.*];-W)r
for fixed [a, b] and fixed 0 £ @2*n([a> &])» which we have seen in 2).
5) Therefore we can write Z{cp) in the form
m — 1
(4.41) Z(y) = 2 <t,«, ^>, V^ = (<p0, ..., ym_!)
i-o
with TjW £ ^V,2w(Rl &"{r)), u-^ru = (t0w, ..., tw_x^) evidently being
a linear mapping of Y+ into
[0'+i2M(R;,?nr))r.
Let us show that u —> to is continuous in the z^^a^ topologies; to this
end, we need to show that, for fixed <p in [^_j2w(R; ^(Jri))]m,
— W —1
j=0
is continuous on Y+. But this follows from (4.41) and the definition of
Z{cp), by using the representation (4.34). of Z.
6) Finally, using Green's formula (4.11) and making an
identification analogous to the one in Section 4.3, we verify that if u £ £^(R; @{Q)),
then we have tjU = BjU, j = 0, ..., m — 1, and we have thus shown the
existence of the traces Bu for u £ Y+.
7) Whereas for Green's formula (4.33) it is sufficient to note that if
v £ X-, then
yy = rJ»€^_,ai,(R;jr(r)),
/ = 0,..., m — 1, and then the construction of Z(q>) maybe given directly
by _
Z(£) = <«,P*»>-<«,»>. D

236 4. Application of Transposition: The Infinite Cylinder Case
Now, choosing
feS'+[R;S'(Q))
and
and applying (4.21) and Theorem 4.4, no further obstacles remain to
showing the following existence theorem:
Theorem 4.5. Under the hypotheses (4.1), ..., (4.7), the boundary value
problem
IPu =f in the sense of Sf' (R; @'(Q)),
BjU = gj} ] = 0, ..., m — 1, in the sense of Theorem 4.4,
admits a unique solution in Y+ for every /£ ^+(R; Ef(Q)) and gy£
^+,2w(R; Jff'IJ1)); furthermore, (/; g0, ..., gm^) -> uis a continuous
mapping of
S'+[R; E'{Q)) X \9'+t2M[R\ X\r))T onto Y+ . Q
Remark 4.3. Thanks to (4.21), the mapping (/; g0, ...,gw_i)-> w in
Theorem 4.5 is continuous even if the intervening spaces are provided
with the strong dual topologies. [)
Remark 4.4. For all the results obtained in this Section, one can
take, instead of the space ^'+(R; E'(Q)), a general space K'+, (weak)
dual of a space K+ satisfying (4.22). Q
4.3 The Existence of Solutions in the Spaces ^'+,S(R; ^(fi)),
with r > 1, s > 2m
The methods and results of Sections 4.1 and 4.2 can be extended to
the study of problem (4.8), (4.9) in spaces of ultra-distributions of
Gevrey type ^+jS(R; 9f'r{fi)) with r > 1 and 5 > 2m.
We shall restrict the present discussion to indicating the necessary
changes to be made to Sections 4.1 and 4.2 to this effect.
The space X- will be changed to the space
Xv = {v\ve9_(R',9(Q))> CjV = 0, j = 0,...tm- 1,
P*w€0_,,(R;0,(0))}
provided with the image topology of <2)_}S(R) @r(Q)) under (P*)""1; we
still have:
(4.43) P* is an isomorphism of Xs* onto ^_>S(R; @r{Q)).

4.3 The Existence of Solutions in the Spaces ^'+>S(R; @'r{Q))
237
By a proof completely analogous to the one given for Theorem 4.1,
we show that
(4.44)
v-^ftv {defined in Section 4J) is a continuous
linear mapping of
Xtl' into [S>_i2m{R)^(r))r.
It must only be pointed out that (4.14) must be replaced with
(4.45) @b>5([a, b];®r(Q)) = ind lim 06>s([a, b], L; 9r(Xit Q)),
L-^+oo
which is valid (see (4.11) of Chapter 7); the spaces (4.15) must be replaced
with
2Cf([a, b], L;Xt) = {v\ v£®b[[a, b]; ®{Q)),
CjV = 0 on rx [a, b],j = 0,...,m — l,
P*ve®biS{[a,b],L; 0,(^,0))},
Xs^([a, b]) = {v | v€ 2b{[a, b]; 9(0)),
(4.46) | ctv = 0 on Px [a, b], j = 0,..., m - 1,
P*ve®b,s{[a,b];®r(Q))},
Xtr (b) = {v\ v € 9b[] - oo, b]; ®{Q)),
CjV = 0 on rx]—oo, J], / = 0, ...,m — 1,
I P*« €#»,,(]-oo, 5] ;0f(fl))},
each of^these spaces still being provided with the topology carried over
by(P*)-!. Q
Next, we show that <F is surjective and more precisely that:
for every positive L and M and every fixed interval [a, b],
there exists a continuous linear mapping {^'right-inverse"
of$)f^v = R(f) of
(4.47) \ [3Wt«, b], L; tfM(r))Y into X'f{[a, b]; V, jf,),
where ihe number L' and the compact set X{ depend
on L, M and [a, b], such that
[ tjv = <Pj> i = °>
-1.

238 4. Application of Transposition: The Infinite Cylinder Case
The proof is the same as for Theorem 4.2; we only have to point out
that the functional, t) used in (4.20) must now belong to @s([a, b]; @r{Q))
(and not only to <2){Qx [a, 6])), which is possible since r and 5 are greater
than 1. (Also note that the hypothesis s > 2m intervenes in order to
guarantee that the function v(x, t) given by (4.20) belongs to Sfs{\a, b];
3fr{Q)) thanks to the fact that w, solution of (4.19), belongs to Q)2m{ [a> b],
jf(tfj)). D
The space Ys£ (which replaces Y+) is introduced in analogous fashion.
The space i£_ is replaced with a space Ksf_ such that
IXs? C Ks' C ^focfRj L2(Q)) with continuous injection,
@_}S(R;@r{Q)) is dense in Ks'.
For Ks^ ,we can choose the space
(4.49) ®_tS{K;Er(Q)),
where Br(Q) = S^y(Q) is defined in Chapter 8, Section 4.2 (note that
in this case the condition Xs' C Ks- is a consequence of Theorem 1.2).
Then we have
(4.50) yy = {u I ue &'+pL\ 9'M)> Pue®'+(R; s's(Q))},
provided with the coarsest locally convex topology which makes the
mappings «-> u and «-> Pu of Ys{ into ^'+,s(R ; @'r (fi))and ^+,s(R > Sr{®))
respectively, continuous, these two spaces being provided with the weak
dual topologies.
The density Theorem 4.3 extends without difficulty, by the same
proof; it is sufficient to note that this time g £ £^_jS(R; <2jy{Q)), thanks to
Theorem 1.2; then (4.32) may be taken in the sense of the duality
between 0+fS(R; 0,(fi)) and ^_|S(R; 9r{Q))\ therefore
(4.51) 0(R; 2(D)) is dense in Ys/ .
We also have the trace theorem:
the mapping u -> Bu extends by continuity to a
mapping of Y% [weak) into [&'+i2m(R) ^'{r))]m
[weak) and we have Green's formula:
(4.52)
m — ]
(Pu, vy - <u, p*vy = - 2 <Bj"> tjv>>
Wu £ YX and v £ X? .
The reasoning follows the proof of Theorem 4.4, the brackets in (4.34)
now denoting the duality between 0+,s(R; @'r{Q)) and ^_,S(R; 3ff{Q))

4.4 Remarks on the Existence of Solutions and Trace Theorems . 239
and between @'+tS(R; S'r(Q)) and ^_jS(R; Sr{Qj), and using (4.47)
instead of Theorem 4.2 and (4.51) instead of Theorem 4.3.
Finally, we obtain the following existence theorem:
Theorem 4.6. Under the hypotheses (4.1), ..., (4.7) and if s > 2m and
r > 1, the boundary value problem
r Pu = f in the sense of &' (R; Sf'r{Q))
(4.53) \ '
[ BjU = gj, ] = 0, ..., m — 1, in the sense of (4.52),
admits a unique solution in Ys_j* for every f £ &+}S(R; £,(&)) and g3 £
3>'+tS(R', 3%"{r)), and (/; g0, ..., gm_1) -> u is a continuous mapping of
^;^)x[^(R;/(il)r onto Ys/
(aw^ gv^w for the strong dual topologies). [)
Remark 4.5. As in Remark 4.4, instead of ^+jS(R; S'r(Q)), one could
take the dual of a space i£Y satisfying (4.48).
Remark 4.6. In Sections 4.1 and 4.2 we studied problem (4.8), (4.9)
in the space @'+(R; @'(Q)) and in Section 4.3, in the space @'+}S(R;@'r(Q))
with s> m and r > 1. It is now evident that we could study problem
(4.8), (4.9) by the same methods in the spaces
9+(R;9'r(Q)) and ®'+fS(R;0'(fl)), s > 2m, r>l.
We do not specify the results. Q
Remark 4.7. A corollary to Theorems 4.5 and 4.6, which is the
analogue to Remark 3.6, concerns the regularity in the interior of ultra-
distribution solutions of the equation Pu = f in the cylinder Q; we
easily see that: under the hypotheses (4.1), ..., (4.7), each ultradistribution
u£ ^+jS(R; S)'Y{Q)) with s > 2m and r > 1, solution in Q of Pu=f
with (for example) /£^'+(R; L2(Q)), is a distribution of @+(R; &(&)),
and therefore (see Chapter 7, Section 5.1) is a scalar distribution on Q,
i.e. belongs to <2>'{Q).
4.4 Remarks on the Existence of Solutions and Trace Theorems
in other Spaces of Ultra-Distributions
In Sections 4.1, 4.2 and 4.3 we have characterized, by the space
[^V,2m(R; J?'(r))m], the "traces" on E (the B3u's) of solutions u of
the equation Pu = 0, or more generally of Pu = f, with suitable /,
which are distributions or ultra-distributions whose supports in t are
bounded on the left (the spaces @'+(R; &'{&)) and @'+)S{R; 0f(fi))).
Now, it seems natural to pose the more general problem of
characterizing the £ttraces" on E of solutions of Pu = f which are distributions or ultra-
distributions with arbitrary support in t.

240 4. Application of Transposition: The Infinite Cylinder Case
In fact, one could also pose the problem, tied to the preceding one,
of studying, if possible, the boundary value problem (4.8), (4.9) without
the condition that the support in t of f, gj and u (or at least of gj and u)
he hounded on the left; the equation Pu = f being a parabolic evolution
equation, one can foresee the existence of ''conditions" for t = — oo
on the data / and gj and the solution u.
The methods which we have studied can be adapted to these two
problems, with, as a matter of fact, rather great technical difficulties.
We shall restrict this discussion to giving an idea of the situation and to
putting into evidence the main difficulties (see Problem 6.13). Q
Let us, for example, study the problem in the space £^'(R; @'(Q));
analogous considerations can be developed in ^(R; ^(-0)) with 5 > 2m,
r> 1.
The starting point is the space
(4.54)
X = {v | v e ®_(R; 9(Q)), Cjv = 0,j = 0,.,.,m-1, P*v e ®(R; 9{Q))}
and the adjoint problem (4.10), with q> £ 3f(R\ 3f{Q))\ providing X with
the image topology of ^(R; 3f{Q)) under (P*)"1, we see that P* is an
isomorphism of X onto ^(R; 3f{Q))
The essential point then (as always) is the concrete characterization
of the image &(X) of X under the mapping & defined in Section 4.1 (for
the space of "traces" which we seek will be the dual of this image). Since
X C X- (where X- is defined in Section 4.1), we have, thanks to
Theorem 4.1:
(4.55) &(X)C[@-,2m{R;3>?(r))r.
But in this case &(X) is a strict subspace of [^_j2w(R; J^(r))]m- Let us
just consider the case where A(x, t, Dx) = A(x, Dx) and Bj(x, t, ~DX) =
Bj(x, T>x) do not depend on t and where furthermore —A, considered as an
unbounded operator in L2(Q) with domain
D(A)={v\veH2m(Q), B,i> = Of 7 = 0, ...,m-l},
is the infinitesimal generator of an analytic semi-group (this is the case,
for example, of coercive variational problems; compare with hypotheses
(3.13) and (3.14) of Section 3).
Then, applying Remark 7.11 of Chapter 9, we find that for t < t0 the
function v is analytic in t, with values in D(^4*°°; k\) and therefore TjV
is, for t<t0, analytic in t with values in the space Tj(T>(A*°°; k\)) of
"traces" on r of the elements of D(^4*°°; k\), j = 0, ..., m — 1.
Furthermore, we must study the behavior of v as £-> — oo; we
obtain, for example, that \\v\\Lt^ tends to zero exponentially for £->■ — oo,

4.4 Remarks on the Existence of Solutions and Trace Theorems 241
which shows that there is a condition "for t = — oo" on the elements of
W(X).
Thus, we see that the problem of the characterization of &(X) involves
rather great difficulties; for a particular case (the classical heat equation
in two variables on a strip) we refer the reader to Lions-Magenes [2],
Section 8.
But we can show the existence of a "trace" on Z without giving an
explicit characterization of the space %>(X) described by this "trace".
Indeed, let us consider the kernel N^ of the mapping V\ it is a closed
subspace of X. Let £" be the quotient mapping of ^ by N^, which
operates from X' = X/N^ onto &{X) in one-to-one fashion; then we can
provide &{X) with the topology such that v —>- Vv is a topological
isomorphism of X' onto &(X). Denote by ^, the space &(X) provided
with this topology.
This much being set, we can introduce the space Y (analogue of the
space Y+ of Section 4.2) in the following way:
(4.56) Y = {u\ue@'(R;@'{Q)), Pu£ ®'+(R; S'(Q))},
provided with the coarsest locally convex topology which makes the
mappings u-+ u and u-+ Pu of Y into @'(R; &(Q)) and @'+(R; E'(Q))
respectively, continuous, these two spaces being provided with the weak
dual topology.
By the same proof as for Theorem 4.3, we show that:
(4.57) 0(R; @{Q)) is dense in Y.
Then we can state the trace theorem for Y:
the mapping u-+ Bu = {B0u, ..., Bm_xu} extends by
continuity to a continuous linear mapping u —>- Bu
of Y into the space <&'', weak dual of $.
(4.58)
To prove (4.58), let u be given in Y; for (p in <g, 9? = {(p0, ..., 9V-i}>
set
(4.59) v ={r)~1y
and for arbitrary v in v' (v £ X) set
(4.60) Z(<p) = <«, P*v} — (Pu, v>,
where the first bracket denotes the duality between £^'(R; &{Q)) and
@(R;@(Q)) and the second between @'+(R; S'{Q)) and ^_(R; S{Q)).
As for Theorem 4.4, using the uniqueness of the Cauchy problem, we
verify that Z(<p) depends only on 9?.

242 4. Application of Transposition: The Infinite Cylinder Case
Thus, we have defined an antilinear form on ^. It is continuous;
indeed the form v-+&(v) defined by
@{v) = <u, P*v> — (Pu, u}
is continuous on X and null on N^; therefore the form v' -> @'(v') = &(v),
v&v', is continuous on X' and therefore
is continuous on ^.
Consequently
(4.61) Z(<p) = (xu, y >, w^'= dual of ^,
and we have in this manner defined a linear mapping u -> xu of Y into
^'. It is also continuous for the topologies of Y and <3' (weak); indeed,
thanks to (4.61) and (4.60),
(xu, ^> = (u, P*v> — <P^, ?>, Vv e X,
and if «-> 0 in Y, then «-> 0 in ®'(R; 0'(fl)) (weak) and Pu-+ 0 in
0'+(R» £"(&)) (weak), whence the result.
Finally, applying Green's formula, we see that xu = Bu if u £
0(R; 0(£T))- And therefore (4.58) is proved. Q
There are no difficulties in showing the existence theorem:
[for every /£ ^'+(R; B'\Q)) and g£ <&', there exists
a unique u in Y, solution of
(4.62) \ \Pu = fin the sense of 0'(R; ®'{Q)),
I Bu = g in the sense of (4.58),
I and u depends continuously on {/, g}.
Thus we can say that every solution in @)f(JL', &(&)) of equation Pu = f
with / = 0, or more generally / £ £^'+(R; E'{Q]), admits "traces" Bu on H
in the space <&'.
And we have also solved problem (4.8), (4.9) by imposing only on f
[and not on gj) the condition of having the support in t bounded on the left, [)
Remark 4.8. According to Chapter 7, Section 5, the space 3ff(R\ &(Q))
may be identified with a subspace of &{Q), but this subspace does not
coincide with &{Q). Thus, we can ask the questions of Section 4.4, taking,
instead of 3l'(JL', <3>'(Q)), the space Q}\Q) of scalar distributions on Q
(and we could also take Sffsr{Q) instead of ^(R; S}'r(Q)), with 5 > 2m,
r>l).

5. Comments
243
The methods of this Chapter still apply. Indeed, it is sufficient to
take as a starting point, instead of the space X defined by (4.54), the
space
X+ = {v\ve 0_(R; 9(Q)), C3v = 0, / = 0, ..., m - 1, P*v e 9{Q)}
and the theory can be developed as in Section 4.4, with the same type of
difficulties; we find, in particular, that every solution of equation Pu = 0
in the sense of scalar distributions on Q, admits traces Bu on U, in a
suitable space.
But we shall no longer insist on this point or on other possible
generalizations to different spaces of distributions or ultra-distributions
(see for example Lions-Magenes [2] and Problem 6.13 below).
5. Comments
The hypoellipticity of parabolic operators in the sense of Petrowski
[2] has been proven by Mizohata [1], Eidelman [3] and Browder [1]
(see also Friedman [6]); Theorem 1.1 is a particular case of these results,
but the type of proof given here, which makes use of boundary value
problems in the spaces H2mr,r(Q), seems to be new.
The regularity in the interior in Gevrey type spaces is due to Friedman
[4, 5] (see also Petrowski [2] and Eidelman [2, 3], for the analyticity in
the space variables). Our proof of Theorem 1.2 follows Cavallucci [1].
The result for the heat and second order equations is classical:
Holmgren [1], Levi [1], Gevrey [1].
For the theorem of Whitney [1] in the form used in Section 2.1
(compatibility relations (2.8)), see Malgrange [1, 2].
We have already cited Cavallucci [1] and Matsuzawa [1, 2] in
connection with Theorem 2.3 on the regularity at the boundary, in these works,
the authors more generally, study the regularity of solutions of quasi-
elliptic equations.
We must again note here the regularity results for hypoelliptic
operators already cited in the Comments to Chapter 8: Hormander
[1, 2], Friberg [1], Pini [1, 2], Cavallucci [2], Volevich [1], Shilov [1],
Friedman [1], ...
Concerning the boundary value problem (2.9), we must also recall
the results on the regularity of the solution from the point of view of the
analyticity with respect to the variable t, which we have mentioned in
Chapter 9. For the analyticity results with respect to all the variables, see
Tanabe [2]; for the case of solutions of a second order parabolic
equation, the following is a method due to K. Masuda [1]: let u be a solution of
(5.1) %+A®u = 0.

244
6. Problems
We already know the analyticity in t; therefore du/dt = 0, so that
(5.1) implies
du d2u
(5-2) ¥ + ^-^ = 0'
an elliptic equation, from which the result will follow.
If A(t) is of order 2m, consider, instead of (5.2), the equation
, x du d2mu
(M -pj + A{t) u + {~l)m —j=r = 0.
The results of Section 3 (finite cylinder case), announced in Magenes
[4], are proven here "in extenso" for the first time; less precise results
had been given in Lions-Magenes [2].
The regularity in the interior of ultra-distribution solutions
(Remark 3.6) in the case of hypoelliptic equations with constant coefficients
has been studied by different methods by Bj orck [1].
The results of Section 4 (infinite cylinder case) extend and specify the
results of the authors [3, 5].
6. Problems
6.1. General study of the differentiation operator in the spaces
Hrs(Q) (see Lemma 1.1 of this Chapter and Proposition 2.3 of Chapter 4).
6.2. What are the "optimal" hypotheses on the sequences Mk and
Nh and on the coefficients of P and Bj for Theorems 1.2 and 2.3 still to
be valid in the spaces SMk,Nh ((?) and @Mjc,Nh ((?) respectively ? (see
Remark 2.3 and Friedman [4, 5]).
6.3. To find the suitable extensions of the elliptic iterates theorem
(Chapter 8) to parabolic, or more generally quasi-elliptic operators in the
spaces @Mk,Nh(Q) (see Matsuzawa [1, 2] and Nelson [1], Goodman [1]).
6.4. Regularity theorems in Beurling type spaces (see Bjorck [1]).
6.5. Examples of spaces of type K{Q), Kr>s(Q), K_, Ksf, different
from the ones given in the text; see Problem 6.4, Chapter 8.
6.6. Study of the traces u(x, 0) and Bu, separately, for the u's in Y
(see Remark 3.2).
6.7. Structure of the elements of the space V.
6.8. Non-homogeneous boundary value problems in unbounded open
sets (see Cavallucci [2], Pini [2]).
6.9. Non-homogeneous boundary value problems in other spaces of
ultra-distributions, for example the hyperfunctions of Sato.
6.10. Generalization of the theory to parabolic systems.

6. Problems
245
6.11. Is Theorem 4.4 valid if the spaces are provided with the strong
dual topology? (it is possible for Theorem 4.3 by using Remark 4.3).
6.12. Do there exist counter-examples to the validity of the regularity
in the interior of ultra-distribution solutions (in @'s>r(Q)) of equation
Pu = 0, in the case 1 < 5 < 2m, 1 < r (see Remark 3.6 and Bjorck
6.13. The Remarks of Section 4.4 pose several problems in relation
with the various spaces of distributions or ultra-distributions which can
be chosen (see Lions-Magenes [2], for an example).
6.14. Non-homogeneous boundary value problems for quasi-elliptic
operators.
6.15. Non-homogeneous boundary value problems in non-cylindrical
open sets.
6.16. Problem of the compatibility relations in Gevrey classes; see
Remark 2.1.
6.17. To see to what extend the theory of this Chapter can be
extended to the settings of Problems 17.6, 17.7, 17.9, 17.11, 17.12, 17.13 of
Chapter 4.

Chapter 11
Evolution Equations of the Second Order in t
and of Schroedinger Type
This Chapter extends, in so far as possible, the problems studied in
Chapter 5 in the Hilbert space setting, to spaces of distributions or ultra-
distributions.
1. Equations of the Second Order in t;
Regularity of the Solutions of Boundary Value Problems
1.1 The Regularity in the Space @(Q)
We shall reconsider, in the setting of spaces of distributions and of
Gevrey functionals, the boundary value problems for the equations of the
second order in t, studied in Chapter 5, with the same notations and
hypotheses (Section 1.1, Chapter 5).
Thus, let Q be a bounded open set in Rn, with:
[ the boundary r of Q is an(n — 1) dimensional,
(1.1) < infinitely differentiable variety, Q being locally
on only one side of r.
In Rn+1 = R*xR?, we consider the cylinder
Q = QX]0, T[, with27 = rx]0, T[, T< +oo.
We propose to study the problem:
Pu = A(X,t,-ju+- = f1nQ,
(1.2) \ BjU = gj. j = 0, ..., m — 1, on S
duix, 0) _
u(x, 0) = u0(x), —-— = u±(x) inQ,
at

1.1 The Regularity in the Space 3(Q)
247
where
(1.3)
with
(1.4)
(1.5)
and where
(1.6)
with
(1.7)
l*tl?l<»
^€S(Q),
A symmetric i.e. A = A* or apq = at
Bju = 2 V (x, t) Dj,
|A|<»y
Jrte#(£),
(1.8)
{:
0 < mj < 2m <m^ ^ system {£•}, /or £^ry £ € [0 T],
is normal on r.
Furthermore we assume that
r the _ boundary conditions B,u = gj, j = 0,
m
1,
correspond, according to Chapter 2, Section 9.4, to a form
a(t; u,v)=^ f apq(x, t) Dqx uDpxv dx
\P\,\q\<f»o
(1.9) \ an^ a dosed vector subspace V of Hm(Q), with
the form a(t;u, v) being V-coercive, uniformly
with respect to t, i.e. there exists <x > 0 independent of t such that
I a(t;vtv)>ot\\v\\%m{Q),VveV,Vte [0, T].
We are given /, gj} u0, ux with
(i. io) / e mQ), gj e @(f), u0 and Wl e @(Q)
and with the compatibility relations (see Whitney [1], Malgrange [1]
implying the existence of w £ @(Q) with
(1.11)
BjW = gj on 27, / = 0, ..., m — 1,
dw
w(x, 0) = w§{%), -r- (%, 0) = ux(x) on Q,
ct
T>ktPw{x, 0) = D*/(*, 0) on Q, k = 0,1, ....
We seek w in ^(Q), solution of (1.2).

248 1. Regularity of the Solutions of Boundary Value Problems
To this end, it suffices to solve the problem
f Pv = f — PwmQ,
I B.v = 0 on Z, j = 0, ..., m — 1,
(L12) dv
v(x,0) = —(x,0) = 0 in Q;
[ ot
using Theorem 7.1 of Chapter 5, we obtain v£ @(Q), solution of (1.12),
and then u = v + w solves (1.2) and u £ ^((?). Furthermore u is unique,
thanks to the results of Chapter 3 (Theorem 8.1 applied to the present
situation).
Therefore, we have
Theorem 1.1. Under hypotheses (1.1), (1.2), ..., (1.11), there exists a
unique solution of (1.2) in the space @(Q). D
Remark 1.1. As we have already said in Chapter 5 (see Remark 1.2),
instead of (1.5) and the F-coerciveness of a(t\ u} v), it would be sufficient
that the principal part of A be symmetric and that there exist oc > 0 and X
such that :
principal part of a(t; v,v)> oc \\v\\%m^Q) — ). ||tf|||«(fl), Vv G V, t£ [0, T].
1.2 The Regularity in Gevrey Spaces
Let us now study the regularity of the solution of problem (1.2) when
the data (D, /, gj} u0, u, apq, bjh) are in Gevrey classes. Here, the situation
is more complicated than for the parabolic problems studied in Chapter 10,
for the operator P is not hypoelliptic and therefore there is no Gevrey
class "associated" to the operator; in particular we do not have local
regularity results of the type of Theorems 1.2 and 2.3 of Chapter 10.
Nevertheless, it is possible for the boundary value problem (1.2) to
prove global regularity results in certain Gevrey spaces, using the
regularity in t studied in the abstract case in Chapter 9, Section 2 and
the Theorem on "elliptic iterates" of Chapter 8, according to an idea
previously developed for the parabolic case in Remarks 2.2 and 2.3 of
Chapter 10.
We seek regularity results in the spaces @mS}S{Q), with s > 1 (for a
generalization, see Remark 1.4 below). Therefore, we shall first make the
hypothesis:
(1.13) Q is of class {Mk} with Mk = (k !)s, real s > 1.

1.2 The Regularity in Gevrey Spaces
249
We shall also assume that the coefficients of the operators A and B3,
given by (1.3) and (1.6), are independent of t and of Gevrey class of
order s, i.e.
(1.14)
(1.15)
Then we have
Theorem 1.2. Let P and Bj} j = 0, ...,m — 1, be given by (1.2),
(1.3), (1.6) with (1.13), (1.14), (1.15), (1.5), (1.8), (1.9) and
(1.16) s>l, sm>l)
let /, gjt Uq,u1}j= 0, ..., m — 1, be given with
(1.17) / 6 ®ms,s(Q), gj € ®ms<s{Z), u0 e ®M> «i € #, (fl),
aw^ the compatibility relations which guarantee the existence of w such that
(1.18)
dw
w(x} 0) = u0(x) and — (x} 0) = ux(x) on Q,
at
Bjw = gj> J = 0> • • • m — 1> on Z>
D*Pw{x, 0) = D*/(*, 0) on Q, V£.
77&0W £/&0r0 msfc cw^ *m<i on/y one solution of problem (1.2) belonging to
the space @ms>$(Q)>
Proof. As for Theorem 1.1, we reduce the problem to (1.12). Then it
is possible, noting that D*(/ — Pw) |/=0 = 0, V&, and by an obvious
extension into the cylinder Dx]— oo, + oo[, to apply Theorem 2.2 of
Chapter 9; therefore the solution v of (1.12) belongs to @stn([0, T]; L2(D))
(and more precisely, the extension of v belongs to <2)+]Sm(R\ L2(D))).
Differentiating the equation Pv = q> (q> = f — Pw) with respect to t
and applying the operator A, it follows that
*-i
(1.19) A*v = {-iyD*v + 2 (-l)*"+*+1 il*(D^-*-«y), * =1, 2, ...
and
i-1
(1.20) B^A'v) = 2 (-l)'+*+1 B}{Ak(D^-h-»<p)), i = 1, 2,...;
*=0 /=0, .... w —1.

250 1. Regularity of the Solutions of Boundary Value Problems
Thanks to the fact that *>6^sw([0, T];L2(Q)) and that <pe@smjQ),
it follows that there exist c0 and L0 such that, for 0 < t < T,
(1.21) \\Alv{x f)\\L%w < c0Li((2mi)\)st i = 0, 1, 2, ...,
m— 1
(1.22) £ ll^(^)llH2-+2^-Wj-l/2(r) < CoL^^1^^ + k + 1))!)S,
i«o
*,* = 0, 1, ...
Therefore, applying the theorem on elliptic iterates (Theorem 1.2,
Chapter 8), we obtain that v(x, t) belongs, for each t £ [0, T], to a bounded
set of 9S{Q).
By the same reasoning, we also find that T>tv(x, t) belongs, for each
t £ [0, T], to a bounded set of 3fs{Q); indeed we only need to note that
T>tv = z is a solution of the problem:
[ Pz = T>tq> in Q,
< BjZ = 0 on 27, / = 0,..., m — 1,
[ z(x, 0) = T>tz(x, 0) = 0 on Q
and that D,<p again belongs to 0m([O, T]; L2(i3)), with D?(D#>) |,=0 = 0,
V*.
Then, from (1.19) and the analogous equality for Dtv:
*-i
A'(Dtv) = ( —1)* D*+1v + 2 (_l)'+*+1 i4*(Df(*-*-1)+V), * = 0,1,...,
/i=0
we deduce the existence of c# and L% such that
|D*D|Dt;(*, t) | < c^+'tf!)' (Zip, (x, t)eQ
and therefore
Remark 1.1. Condition (1.16) implies that for the hyperbolic case
(m = 1) we must have s > 1 ««<? therefore we do not prove the analyticity
in x. D
Remark 1.2. As we have already stated in Remark 2.1 of Chapter 10
for the parabolic case, we do not know whether the "natural" explicit
compatibility conditions on /, gj, uQ, ux are sufficient for the validity
of (1.18). D
Remark 1.3. The hypothesis that the operators A and Bj do not
depend on t is used only in point (1.19) and (1.20). It would be of interest
to generalize the result to the case of time-dependent coefficients, and

2.1 Generalities 251
more precisely to the case
(1.23) «* 6^,(0. ^6^.(1?). D
Remark 1.4. The method used for Theorem 1.2 can be generalized
to spaces @Mk,Nh ((?) i*1 the same way as for the parabolic case; Remark 2.3
of Chapter 10. Instead of the condition (2.26) of Chapter 10, for the
sequences {Nh} and {Mk}, we have the relation
N2k<M2km, Vh,
therefore we can take Nh = (h \)s, Mk = (k\)r with r > 1 and 1 < s < mr
(and therefore u G ®,,,((?)). D
2. Equations of the Second Order in t;
Application of Transposition and Existence
of Solutions in Spaces of Distributions
2.1 Generalities
The notation and the hypotheses on Q, P and {B^JITq1 are as in
Section 1.1 (hypotheses (1.1), ..., (1.9)).
We recall Green's formula (see Chapter 5, Section 1.1):
m—1
(2.1) / (Pu) vdxdt — J uP*v dx dt = £ / SjuCjV da —
- 2 I BjuTjV da + I y8t ' v(x, T)dx- —^-i v(x, 0) dx -
E Q Q
C , m dv(x, T) J r , M dv(x, 0) ,
- I u(x, T) —^— d* + J u(x, 0) -^~1 dx,
Q Q
where P* is the formal adjoint of P and therefore, under hypothesis
(1.5), coincides with P (but we shall keep the notation P*, because, as we
have already stated in Remark 1.1, we could, for the sequel, assume that
only the principal part of A is symmetric).
We propose to study problem (1.2), i.e.
(2.2) Pu = fmQ,
(2.3) Bju = gj on 27, / = 0, ..., m — 1,
du
(2.4) u(x, 0) = uQ(x), — (x, 0) = uJx) on Q
dt
in certain spaces of distributions on Q.

252 2. Transposition and Existence of Solutions in Spaces of Distributions
Again, the starting point is the adjoint problem:
P*v = cp in Q,
C,v = 0 on Z, j = 0,..., m — 1,
(2.5)
dt
The "most general" situation for problem (2.2), (2.3), (2.4) (i.e. to find
u in &(Q)), would lead us to study problem (2.5) for
(2.6) <pe&{Q).
Thanks to Theorem 1.1, we would obtain the existence and uniqueness
of the solution v of (2.5) belonging to the space
— dvlx T)
(2.7) X = {v | v 6 9{Q), v(x, T) = -^—; = 0, C.v = 0,
ot J
and we would then have to study the space described by ($v as v describes
X, where
(2.8) Wv = J v(x, 0), ^p. ;T0v,..., TmJ
(see for example Section 3.2 of Chapter 10). But here we meet great
technical difficulties which lead to the elimination of the case (2.6):
for example (compare with (3.25) of Chapter 10) we would have to study
the Cauchy problem:
IPv = 0 in a neighborhood of 27,
C-v = 0, Tp = (pj on U, j = 0,..., m — 1,
where ty is infinitely differentiable on £', but P* is not hyperbolic with
respect to S (except for the very particular case corresponding to the
wave equation in one space variable); and therefore (see Hormander [1],
page 130, Courant-Hilbert [1], page 759) the space of <p/s (i.e. the space
described by {T0v,..., Tm_1v} as v describes X) depends on the operator
P* is an essential way. Furthermore, there are difficulties due to the
dv
"linking" of v(x, 0),— (x, 0) and TjV\t:=0 on the variety r.
Therefore, in the following Sections, we shall seek another possibility,
which will lead to the study of problem (2.2), (2.3), (2.4) in a still very
general sufyspace of &(Q).

2.2 The Space 0_>y([O, T]; @Y(Q)) and its Dual 253
2.2 The Space ®_tY( [0, T]; @y(Q)) and its Dual
The space @y(Q).
We define
(2.10) @y(Q)={9\<pe®{Q), 7j(p = 0 on T, V/= 0, 1, ...}
(0))
This is a closed subspace of Qiifi). If we note that, thanks to Corollary
9.2 of Chapter 1, Q)(Q) can be considered as a Frechet space with respect
to the family of semi-norms:
(2-11) l|D^||i2(Q), Vj>,
then we easily see that
I3iy{fi)t provided with the family of semi-norms (2.11),
is a Frechet space.
Also note that
(2.13) 2y(Q) = closure of 2)(Q) in 3(Q). D
The space @f'Y(Q).
We denote by &Y{Q), the (strong) dual of @Y(Q). Thanks to (2.13),
(2.14) ®V{Q) is a space of distributions on Q.
By applying the Hahn-Banach theorem, we also obtain a
representation fo the elements / of Q)Y (D):
(2.15)
every element f of @Y(Q) can be written in the form
/=2 D%fpeL*(Q).
finite
As in Proposition 3.5 of Chapter 8, it follows that
(2.16) the space @y[Q) is reflexive.
Finally, comparing 2Y{Q) with the space S{Q) introduced in
Section 3.2 of Chapter 8, we have the following inclusions:
(2.17) 9{Q) C ®y[G) C 3{Q) C 5{Q)
and
(2.18) S'(Q)C&y(Q)C®'(Q). Q
((!)) These spaces should not be confused with the spaces of Gevrey functions.

254 2. Transposition and Existence of Solutions in Spaces of Distributions
The space 0_,y([O, T]\ 3y{Q)).
First, for fixed a such that 0 < a < T, we define
(2.19) 0!fy([O, T];9y(Q)) = {cp \cpe®{[0, T]\9y{Q)), <p®(0) = 0, V/
and <p(t) = G for t> T — a],
where £^([0, T]; @y{Q)) is the space of infinitely differentiable functions
t-xp(t) on [0, T], with values in 3fy{Q)t provided wth the natural Frechet
space topology. The space £^L>y([0, T]; @y{Q)) provided with the
topology of ^([0, T]; &Y{Q)) is a Frechet space.
Then we define
(2.20) ^->y([0, r]; ^y(fl)) = indjjm ^_,y([0, T]; #y(fl)).
We thus obtain a strict {££&*)-space; and we easily see that
(2.21) 9(Q) is dense in ^_,y([0, T\)9y(Q)).
The space ^'+,y([0, T] ;@'Y{Q)).
By definition
(2.22) #'+iy([0, T]; ^(fl)) is the dual of ^_,y([0, T]; 9y{Q)).
In order to avoid any topological difficulties, we provide
£^'+,y([0, T]] <3'y(Q)) with the weak dual topology.
Note that, thanks to (2.21), 0'+>y([O, T]; 3fY{Q)) can be identified
algebraically with a subspace of distributions on Q. [|
2.3 The Spaces X and F
Let us now come back to the adjoint problem (2.5) and take, instead
of (2.6),
(2.23) ye^-fy([0,r];^y(fl)).
Thanks to Theorem 1.1 (applied to the adjoint problem), we see that
there exists a unique solution of (2.5) belonging to 3)(Q) for every
9>€#_,„([0, T];9y(Q)).
Let us introduce the space
(2.24) X = {v\ve &[Q), v(x, T) = ^%^-} = 0, C,v = 0,
j = 0,...,m-l,P*ve 9_iV{ [0, T]; #y(flj)}.

2.3 The Spaces X and Y 255
The operator P* defines a one-to-one mapping of X onto
£^_>y([0, T]; £^y(i2)); therefore, we can define on X the image topology
of 0_>y([O, T]; 0y(fi)) under (P*)'1; and we have:
IP* 2s an [algebraic and topological) isomorphism
-
of X onto 9_„{[*, T];9y(Q)). Q
Note that, as 0_>y([O, T]; %(£)), X is a strict (jS?^)-space; more
precisely
X = indlimX",
where
X*=L \ve®(Q), v(x, T) = ^§-^ = 0, CjV = 0,
/ = 0,...,m-l, P*^G^_,y([0, T];%(£))j,
provided with the image topology of Q)a_ >y ([0, T]; ^y (£>)) under (P*)"1. []
By transposition of (2.25), we obtain
(for every continuous antilinear form v-> L(v) on X, there exists
a unique u in ^'+ ([0, T]; &y(Q)) such that
<«, P*^> =L(v), VveX.
Formally, we choose L(v) in the form
dv(x, 0)
(2.27) L(W) = </, *> + <%, *(*, 0)> - <u0, -^-L > + £ <gy, Z»
the brackets being taken in the sense of spaces which we shall specify. []
For the choice of /, we can consider a topological vector space K such
that:
IX C. K C. L2(Q), with continuous injections,
-
^_,y([0, T\\ @Y{Q)) is dense in K.
Such spaces exist; for example K = L2(Q). In order to fix our ideas
and in analogy with the choices made in this text for similar situations,
we shall take for K the spaces
(2.29) E-(0,T;E(Q)),
defined in the following way. Let t-> d(t) be an infinitely differentiable
function on [0, T] such that
Id(t) = 1 in the neighborhood of 0,
d(*) > 0 for t > 0.

256 2. Transposition and Existence of Solutions in Spaces of Distributions
For integer k > 0 and real a with 0 < a < T, we define the (Hilbert)
space
(2.31) 5{k, a)={v\ d¥j) 6 L2(0, T; 5k-j(Q)), 0 < / < A,
v(t) = 0 for t > T — a},
provided with the norm
/ * \l/2
(2-32) (Slld^lli^r^-ydi,,]
(see Chapter 2, Section 6.3, for the definition of 3k~j(Q)).
Next, let
(2.33) 5(a) = n 5(k,a),
a Frechet space for the family of norms (2.32), k = 0, 1, ... Finally, let
(2.34) 5JO, T; 5{Q)) = ind lim3(a).
Since the functions v of X vanish in the neighborhood of I — T, we
see that
XC2-(0,T;5(Q))CL*(Q).
And, by the same type of proof as for Proposition 9.1 of Chapter 4
and Proposition 3.3 of Chapter 8, we see that 3l(Q) (and therefore also
0_>y([O, T]; %(Q)) is dense in £L(0, T\ 3(D)).
Therefore S»(0, T\ 3(D)) satisfies (2.28). D
Now we denote by
(2.35) 5+(0, T\ 3'(Q)) the dual of 5L(0, T;3{Q)),
provided with the weak dual topology. D
Having set this, let us introduce the space
(2.36) Y = {u | ue ^'+,y([°> T];9'Y(Q)),Pue3+(0, T; 3'(£)))},
provided with the coarsest locally convex topology which makes the
mappings u-> u and u-> Pu of Y into 3f\ >y([0, T]; 3i'Y(Q]) (weak) and
3+(0, T\ S'(Q)) (weak) respectively, continuous.
First, we have
Theorem 2.1. The space Sf(Q) is dense in Y.
Proof. Let u-> M(u) be a continuous antilinear form on Y; all the
spaces being provided with the weak dual topologies, it may be written
(2.37) M(«) = </,5>+<g,P^>,
with / 6 ®_,y([0, T]; 3y(Q)) and g £ 5L(0, T; S(£))).

2.3 The Spaces X and Y
257
Assume that we have
(2.38) M(<p) = 0,V(pe@(Q).
Introduce the open set 0 = RMx]— oo, T[ and denote by /, g the
extensions of / and g to 0 by zero outside Q and by P the extension of
P to 0 having "the same properties" as P (actually, it is sufficient to
extend A into RM in such a way that the extension A has infinitely
differentiable coefficients in RM and that the operator P* = A* + d2/dt2
satisfies the conditions for the "regularity" of the Cauchy problem in the
sense of (2.39) and (2.40) below; this is always possible).
Then, by (2.38), we have
<A ®> + <g, p$> = oty0e ®(0)
and therefore
(2.39) P*g + / = Ointf,
with the "initial" conditions for t = T:
(2.40) g(x, T) = 0, J (*, T) = 0,Vxe R\
ot
Then, according to the regularity of the solution of the Cauchy problem
(see Hormander [1]), we have
(2.41) g is infinitely differentiable in (9.
But since g has support in Q, (2.41) implies
y.'g — 0 on £ for every /,
^— = 0, x £ Q, for every k.
Therefore
g€^_,y([0,IMy(£))
and consequently, for u £ Y, we have
so that
M(«) = </ + P*g, u}
and therefore, according to (2.39), M(u) = 0,VmGY. D

258 2. Transposition and Existence of Solutions in Spaces of Distributions
2.4 Study of the Operator V
We now have to choose the "boundary data" in (2.27); and, as
always, the essential point is the studv of the space described by (2.8), i.e.
by
&v = <v(x, 0), ^ ; TQv, ..., TmM
as v describes X.
If v G -X', then z; vanishes in a neighborhood of t = T, therefore
(2.42) Fy e &(&) X@{Q) X [^_([0, T]; S[T))\m,
where £^_([0, T]; 2i(r)) is the space of infinitely differentiable functions
on [0, T] with values in $)(F) and vanishing in a neighborhood of T,
provided with the usual strict (SPSF)-space topology:
^L( [0, T]; ^(r)) = ind Km @a{[0, T]; ^(r)),
®a(P>, ^] ;®(r)) being the closed subspace of 0([O, T]; &{r)) of
functions vanishing for T — # < t < T, a < T.
Of course, the mapping <F is not surjective from X onto
3(Q) xB{Q) X [0_([O, T]; ®(r))T,
for the images v(x,0), dv(x,0)ldt, T0v, ..., Tm_1v must satisfy the
compatibility relations (which we denote by M • #), i.e. linking conditions
on the variety F, for 2 = 0; these are the pointwise differential conditions
which can be "formally" specified by taking into account the fact that,
for every v £ X, we must have
(2.43)
v e @{Q)> Cjv = 0 on 27, / = 0, ..., m — 1
^(P*y)
yA(P*w) = 0on2,f V*, V '
= 0onQ,Vk.
For 0 < a < T, we set
(2.44) ira = {(Xo,Xl;<p0,..., y^j/feo,zi;%. •■■■ v—i) e W)x
X®[Q) X [^([0, T]; @{r))]m, satisfying the 01 •
we provide i^a with the topology induced by the topology of

2.4 Study of the Operator & 259
and then
(2.45) -T = ind lim-T„.
Thus if is a strict [S£^)-space. Thanks to the closed graph theorem,
we have
(2.46) v->Wv is a continuous linear mapping of X into if. n
We shall now show that to is surjective:
Theorem 2.2. The mapping v->(ev is surjective from X onto if'.
Proof. Let^,^, ',<p0 • • -, <pm-i} be given in if. We seek v£<3(Q)
satisfying
(2.47) P*ve&_tY{[0,T];9Y(Q)),
(2.48) C^ = 0on27,/ = 0, ...,m- 1,
(2.49) ty = ^ on 27, / = 0, ..., m — 1,
dv
(2.50) v(x, T) = 0, — (x, T) = 0, *G &,
ot
(2.51) »(*, 0) = Xo (x), ^|^ = Zl(*), xe Q.
First, we note that, the system {C;, ^};=o being of Dirichlet of order
2w, the conditions (2.48), (2.49) are equivalent to (see Chapter 2, Lemma 2.1):
(2.52) yjV =ipj on 27, / = 0, ..., 2m — 1, y. given in
0_([o, r];®(r)).
Since the ^>/s vanish for T — a < t < T, we shall take (also taking
into account (2.50) and (2.47)) v = 0 for T — a < t < T.
Furthermore condition (2.47) decomposes as follows:
(2.53)
yk(P*v) = 0 on 27, V*,
dk{p*v)
a*
= 0 on Q, \fk.
t=o
Since P* = A* + d2[dt2 and since ^4* is elliptic in Q (this is a
consequence of hypothesis (1.9)) and we have (2.51) and (2.52), we easily see
by induction (see Theorem 10.2, Chapter 4) that (2.53) is equivalent to
Iy,v = tp on 27, / = 2m, 2m + 1, ..., tp- determined
uniquely in 0_([O, T]; 2(1")) and zero for t > T — a,

260 2. Transposition and Existence of Solutions in Spaces of Distributions
and
dkv(x 0) —
(2.55) ~— = %k, k = 2, 3, ..., Xk uniquely determined in 2f{fi).
ot
Furthermore, according to the ^?# # (this can in fact be the definition
of the 0t. #), we have
t=0
Conversely, if v £ 9{Q) satisfies (2.51), (2.52), (2.54), (2.55) with (2.56)
and if furthermore
oft
(2.57) —^ (x, T - a) = 0, Vft and v(#, *) = 0, for t > T - a,
ctr
then v is the element we seek.
Now, by Whitney's theorem (see Malgrange [2]), such a v exists, and
this proves the theorem. D
By passage to the quotient with respect to the kernel Ker(F) of &,
which is a closed subspace of X, denoting by <F" the quotient mapping
and recalling that X/Ker(F) is an {<£SF)-space (see Grothendieck [1])
and therefore that the closed graph theorem applies, we have:
IF" is an algebraic and topological isomorphism
n
of X/Ker^) onto Y. D
2.5 Trace and Existence Theorems in the Space Y
Let y' denote the dual of i^, provided with the weak dual topology.
We are now ready to prove the trace theorem:
Theorem 2.3. The mapping
[ du(x, 0) 1
(2.59) u->nu = -^-J, -u(x, 0); B0u, ..., Bm_xu\
of @(Q) into @(Q)X!&(Q)X[!3(U)]m extends by continuity to continuous
linear mapping, still denoted by u = nu, of Y into *V; and we have Green's
formula:
(2.60) <«,p*v> — <j>u,vy = (nu,fv>, vueY, vex,
the brackets denoting the duality between S'+o,([0, T]; £^,(42)) and
0_,y([O, T]; %{£})), E'+(0, T; E'(Q)) and S_(0, T; 3{Q)), and ir' and
TT, respectively.

2.5 Trace and Existence Theorems in the Space Y 261
Proof. 1) As in Section 3.3 for the parabolic case, we first note that
we can identify (which we shall do) an arbitrary element (u^, —u0;
go,.--,gm-i) of ®{Q)X&{Q)X[@(Z)]m satisfying the compatibility
relations (1.11) of Section 1.1, with an element of if' (by the same type
of arguments as in Section 3.3, using Green's formula (2.1)).
2) Let u be given in Y. For every %p = {^, #0 \cp0, ..., tpm_$ £ if, we
set
(2.61) v = (r)-1^, v 6 X/Ker(gT),
and for arbitrary v in v' (v £ X), we set
(2.62) Z(y) = O, P*v} — <Pu, v},
the brackets denoting the duality between ^'+>y([0, T]; @'Y{Q)) and
®-,y([0, T];9Y[Q)) and between 5+(0, T; S'(Q)) and 51(0, 71 £(£)),
respectively.
Let us verify that Z(ip) depends only on ip)letw be another element
of the class v'; then w £ X and satisfies
te/(#, i) = = 0, C-ze> = 0, i ze> = <p,
3t
'j>
dw
j = 0, ..., m - 1, «;(*, 0) = X0(x)>^ (x> °) = Zi (*)
and therefore we obtain
(2.63) (u, P*{v — w)y —(Pu, v — w} = 0
(using Green's formula (2.1), we first prove (2.63) for u £ £^(0 and then
by passage to the limit, since £&{Q) is dense in Y).
3) The mapping y> ->Z(y>) is continuous antilinear on if\ indeed, if
we set
&{v) = <u, P*v} - (Pu, w>,
we obtain a continuous form on X, vanishing on Ker(£°) (according to
2)); and therefore the form
v^>&\v) =Z (v) {v£v)
is continuous on J£/Ker(£°). Now
Z(y)=^'((F-)-1v),
whence the result.

262 2. Transposition and Existence of Solutions in Spaces of Distributions
4) Therefore
Z(ip) = (nu, ip}, nu £ if1
and we have in this way defined a linear mapping of Y into *V".
Let us show the continuity of n for the weak topologies; we need to
show that, for every ip£i^, u-> (nu, ip} is continuous on Y. Now
(nu, y)} = Z(y>) = (u, P*v} — (Pu, v}, v fixed in X,
whence the result.
5) Finally, applying Green's formula (2.1) and the identification made
in 1), we see that, if «t @(Q), we have
nu = nu.
From which we have the first part of the Theorem, according to the
density Theorem.
6) Finally, Green's formula (2.60) follows from the above, for if
v£ X, then lov^i^ and we can take precisely this function v in (2.62). []
Finally, we have the existence theorem :
Theorem 2.4. Under hypotheses (1.1), ..., (1.9), the problem:
Pu = f in the sense of ^'+ ([0, T]; 3f'Y(Q)) [and therefore
(2.64) \also of @\Q))t
nu = g* in the sense of Theorem 2.3,
admits a unique solution in Y for every /£ ^+(0, T; Sr{Q)) and g* £ i^',
u depending continuously on f and g*.
Proof. Given fe 5+(0, T; S'(Q)) and g* e T', take
L(v) = </, v> + <£*, ^>
in (2.26).
It follows that the solution u of (2.26) satisfies
Pu = f in the sense of ®'+>y([0, T]; @'Y{Qj)
and therefore u£ Y; but then, thanks to Green's formula (2.60) and
Theorem 2.3, we have
(nu — g*, ¥v)y = o, vve x}
whence nu = g*; and the Theorem follows. Q

2.6 Complements on the Trace Theorems 263
Remark 2.1. It is very likely that the results of Theorems 2.1, 2.3
and 2.4 remain valid if the spaces @'+3y([0, T]; %(Q))9 £+(0, T\ Sf(Q))
and V1 are provided with the strong dual topologies (for an analogous
situation, see Lions-Magenes [2], N°. 4). D
Remark 2.2. Particular elements of V can be constructed in the
following way. Define the space £1(0, T; 21(F)) in analogous fashion to
the space 5L(0, T; 5(D)): first introduce the Hilbert space
S{k, a)={v\ djvU) e L2(0, T; Hk-j(r)), 0<j<k, v(t) = 0
if t > T - a}
for integer k > 0 and 0 < a < T. Next, set
5(a) = fl 5(k, a) (Frechet space)
k = 0
and finally define
5 (0, T;2(r)) = ind lim 5(a).
1 a->0
We see that Q)(E) is dense in 5L(0, T; 2(1")); therefore the dual of
S_(0, T; &(rj), which we denote by ^(0, T; ®'(r)) and provide with
the weak dual topology, is an algebraic subspace of Q)'(Z).
If
g, e S'+(0, T; &(D)9 u0 and u, e E'(Q),
the form
m — 1
<"i> Xo>~ <uo> Xi> + 2 <8j> %>
j=o
is continuous linear on i^, according to (2.46) and the inclusions:
S(Q) C S(Q), ®_([0, T]; S(r)) C S_(0, T; ®(r)).
Remark 2.3. Results which are more general in the time variable and
somewhat less general in the space variables can be obtained for this
type of problem by application of Chapter 9, Section 10.6. D
2.6 Complements on the Trace Theorems
The (trace) Theorem 2.3 is of a type analogous to Theorem 3.4 of
Chapter 10 for parabolic equations (see also Section 12.3 of Chapter 4
and the same question in Chapter 5): the boundary conditions, BjU = g;,
du
u(x, 0) = u0(x),— (x, 0) = u±(x), are "contained" in the condition

264 2. Transposition and Existence of Solutions in Spaces of Distributions
"nu = g*". Now the problem is to "separate", if possible, the operators
du
BjU on 27and u(x, 0) and— {%, 0) on Q. For the parabolic case the mat-
Gt
ter is delicate (see Remark 3.2 of Chapter 10). In the present case, the
situation is somewhat simpler1: it is rather similar to the one encountered
in Volume 2, Chapters 4 and 5, in the case of the spaces Dp('~ ^(Q) and
Indeed, we can give meaning separately to the operators BjU, u(x, 0)
du
and — (x} 0) in spaces which, in a certain sense, are probably larger
dt
du
than the "optimal" spaces. Let us first try to define u(x, 0) and — (x, 0)
oz
for u £ Y. To this end, we introduce the space
(2.65) X1 = {v\veX} T3v = 0, / = 0, ..., m - 1},
which is a closed subspace of X.
The mapping
(2.66) w^L*>0),|!(*,0)J
then is a continuous, linear and surjective mapping ofX1 onto 3fy(Q) X @Y{Q).
In order to prove this, it is sufficient to go through the proofs of
Section 2.4 again, adapted to the new situation (the mapping ^ is
replaced by (2.66) and X by X±): in particular we note that in (2.52) and
(2.54) we now have ^ = 0 for every / and therefore the 0t% c€. (2.56)
reduce precisely to
By a proof analogous to the one for Theorem 2.3, still replacing <F with
(2.66) and X with Xv we obtain:
3i{Q)x3i(Q) extends by continuity to a continuous
linear mapping of Y into Q}' (Q) X Q}'y (Q). D
(2.67)
1 We have the same simplification in the parabolic case if we consider a
functional setting analogous to the one chosen for this Chapter; but the setting of
Chapter 10 leads to more general results. In fact, see Section 4.3 below.

2.7 The Infinite Cylinder Case
265
Concerning the operators BjU, we follow an analogous procedure by
introducing the closed subspace of X:
(2.68)
x2 = Iv | vex, v{x, o) = ^ (*, o) = oj
and the mapping
(2.69) v->{T0v}...,Tm_1v}.
We see that (2.69) is continuous and surjective from X2 onto
[0_>y([O, T]; 9{r))]m, where 0_>y([O, T]; 9{r)) is defined in completely
analogous fashion to £^_jy([0, T]; @Y{Q)) (we just replace the space @Y(Q)
with ^(i1) in the definition given in Section 2.2).
It follows that
(2.70)
the mapping u-> Bu = {B0u, ..., Bm_1u} of 2i{Q)
into [@(U)]m extends by continuity to a continuous
linear mapping of Y into [^'+>y([0, T]; ^'(i1))]™ = weak
dual of [®_iy([0, T]',9{r))]M. D
Of course, these results do not completely solve the problem of
du
"separating" B3u, u(x} 0) and — (x, 0) in the mapping nu\ the question
still remains of whether the space *V" can be interpreted as a product of
spaces of distributions on Q, H and T in analogous fashion to what can
be done in the parabolic case relative to the spaces D~^_1)((2) (see
Baiocchi [1] and Section 12.3 of Chapter 4). D
2.7 The Infinite Cylinder Case
We now consider the cylinder Q = Q x Rt> with boundary E = rx Rt,
and the operator
where A is given by (1.3) with (1.5) and
(1.4') apq,££(Rt;®(Q)).

266 2. Transposition and Existence of Solutions in Spaces of Distributions
The operators B}u are again given by (1.6) with (1.8) (where R, replaces
[0, T]) and
(1.7') bjhe®{Z).
We again make the hypothesis (1.9), still replacing [0, T] with R4.
Then, "formally", the problem we wish to study is
\Pu =/in<?,
(1.2')
|^ B-u = gj on 27, ] = 0, ..., m — 1.
Now the methods of Sections 2.1, ..., 2.6, completed by some of the
methods of Section 4, Chapter 10, are also applicable to the study of
problem (1.2'). We shall restrict the present discussion to giving a sketch
of the main points.
We introduce the space ^_(R; @Y(Q)) (note that it is a strict (&&)-
space) and its dual ^'+(R; @v(&)) according to the definitions of
Chapter 7. We provide @'+(R', 2fy[Q)) with the weak dual topology. Note that,
algebraically,
®'+(R; ®;(fl))C<(R;®'(£>)).
We consider the adjoint problem:
\p*v = <p in g,vdthye®-(R;®y(fi)),
I C-v = 0 on 27, 7 = 0, ..., m — 1
and the space
(2.72) X_={v\v£2j^\2(Q)), C.v = 0, / = 0, ...,w —1,
provided with the image topology of £^_(R; 3f[yQ)) under (P*)_1.
Then we are required to study the mapping <F defined (compare with
(2.28)) by
(2.73) Vv = {T0v,...,Tm_1v}.
Applying the closed graph theorem (X_ and ^_(R; ^(jT)) being strict
[S£^) -spaces) we show that
(2.74) v-^&v is a continuous linear mapping of
X^into [9_(R; 9[r))]m
and then using Whitney's theorem in analogous fashion to Theorem 2.2
(but now we do not have the compatibility relations on r for t = 0!) we

2.7 The Infinite Cylinder Case
267
show that
(2.75)
Fv is a surjective mapping of X_ onto
[9~(R;9{r))]m.
Taking, instead of the space S'+{0, T\ S'(Q)), the space 9'+(R\ S'{Q))
(dual of ^_(R; S(Q)) provided with the weak dual topology as in
Section 4 of Chapter 10) we can introduce the space
(2.76) Y+ = {u | ue ®'+(R; 9y[Q% P«6 ®+(R; B'{Q))},
provided with the coarsest locally convex topology which makes the
mappings u -> u and u-> Pu of Y+ into 9\(R; @'Y[Q)) and ^'+(R> &'{&))
respectively, continuous.
Then, by the same method as for Theorem 2.1, we show that
(2.77)
®(R; @[Q)) is dense in Y+.
Finally, it is easy to adapt the arguments used for Theorems 2.3 and
2.4 to obtain the trace and existence theorems:
the mapping « -> Bu = {B0u, ..., Bm_1u} of ®(R; 9{Q))
into [£^(R; @(r))]m extends by contuinuity to a
continuous linear mapping of Y+ into [^'+(R; ^'(J1))]™;
and we have Green's formula
(2.78)
and
m — 1
(2.79)
<«, P*v} - (Pu,v y = Yu <Bju> Tjv>> *u €Y+> v£x->
j=0
the boundary value problem
Pu = / in the sense of ®'+(R; 2'y{Q)),
BjU = gj} j = 0, ..., m — 1, in the sense of (2.78),
admits a unique solution in Y+ for every
/€ ®'+(R; S'(Q)) and gje ®'+[R; &{r)),
u depending continuously on f and g,.
Remark 2.4. The preceding arguments are simpler than those of
Chapter 10, Section 4.1; this is due to the fact that here we know that
we can use the closed graph theorem, whereas the analogous question in
the setting of Chapter 10, Section 4.1 seems to be open.

268 3. Transposition and Existence of Solutions in Spaces of Ultra-Distributions
3. Equations of the Second Order in t;
Application of Transposition and Existence
of Solutions in Spaces of Ultra-Distributions
3.1 The Difficulties in the Finite Cylinder Case
We shall now attempt to apply the regularity results of Section 1.2
to the study of non-homogeneous boundary value problems in spaces of
ultra-distributions, for equations of the second order in t.
We have the regularity hypotheses of Section 1.2 on the data, and
furthermore the following hypotheses:
(3.1) Q satisfies (1.1) with F an analytic variety,
(3.2)
P and Bj satisfy the hypotheses (1.2),..., (1.9) in
the cylinder Q = I2x]0, T[ and furthermore
api{x,t)=apq(x)etf{Q),
M*'*) =&/*(*) e*W
The starting point is still the adjoint problem (2.5). The regularity results
of Section 1.2 lead us to take in (2.5):
(3-3) <pe®sm,M> s>i,
and to introduce the space
(3.4) Xs = {v\ve ®ms,M> v(x, T) = 3^r) = 0, CjV = 0,
7- = o, ...,w-i, p+ves^Q)}.
But by considerations analogous to those of Section 2.1 ontheCauchy
problem (2.9), we are led to eliminate the case (3.3), since the results on
the Cauchy problem in Gevrey classes of order sm with respect to t and s
with respect to % are not valid for s > 1 (see Talenti [2]). Q
Thus we are led to study the adjoint problem (2.5) in spaces analogous
to the space ^_>y([0, T]\ @y(Q)), in order to develop a theory of non-
homogeneous problems in spaces of Gevrey ultra-distributions in parallel
with the theory of Sections 2.2, ..., 2.6. We can introduce the space
@S}Y{Q) = {<p \q>e®s(Q), 7j<p = 0 on r, / = 0, ..., m - 1}
and then the space
(3.5) 0_W([O, T\\ ®StV{Q)) ={9\VZ9j^9 T];9,tY(Q)),
q>M(0) = 0, / = 0, ..., m — 1, <p(t) = 0 in a
neighbourhood of T)

3.2 The Infinite Cylinder Case for m > 1 269
and, in problem (2.5), take op belonging to
^-,-»,y([0. T];9tiV{Q)).
Then the essential difficulty is the analogue to Whitney's theorem in the
Gevrey spaces @mS}S(Q) (see Section 2.4, Theorem 2.2), which we do not
know to be valid. Now the use of this theorem is essential here, in order
to know the space analogous to ir.
Thus we cannot at this time develop the analogue of the theory of
Sections 2.2, ..., 2.6 in Gevrey spaces. D
Finally, let us point out another possibility: in the adjoint problem
(2.5), take cp belonging to
0_,M(]O, T[; &(Q))
with m > 1, which leads to the study of homogeneous problems in ultra-
distributions of
In this case, we can avoid the difficulties tied to the Cauchy problem
(2.9) (we can apply Talenti [2, 3]). But we shall have the difficulties due
dv
to the linking of v(x, 0), — (x, 0), TjV\i=0 on the variety r. We shall
ct
develop this idea for the infinite cylinder case in the next Section.
3.2 The Infinite Cylinder Case for m > 1
Under*hypothesis (3.1) on Q, we consider the infinite cylinder
(3.6) Q = QxRt
and again make hypothesis (3.2), replacing [0, T] with R,, and
furthermore assume that
(3.7) m>l.
We shall study the problem
\Pu = / in Q,
(3.8)
[ BjU = gj on Z, 1 = 0, ..., m — 1,
in the space of ultra-distributions ^+>W(R; 3>'{Q)) (the Remarks of the
preceding Section lead us to eliminate the spaces ^+(R; @'(Q)) and
^+,sw(R; ®s(£)) a^d the case m = l).

270 3. Transposition and Existence of Solutions in Spaces of Ultra-Distributions
We start from the adjoint problem
f P*v = v with v e ^_ JR; 9{Q)),
(3.9) \
yCjv = o, i = o,...,w-i,
and introduce the space
(3.10) X_ = {v | v e ®-(R' ®(^))» C,-« = 0, / = 0, ..., m - 1,
P*ve9_tm(R;9{Q))}.
Thanks to Theorem 1.1, we easily see that P* defines a one-to-one
mapping of X_ onto $) _ W(R; 0 (£?)); therefore, on X_, we can define
the image topology of ^_>W(R; @{Q)) under (P*)"1; so that we have
IP* is an [algebraic and topological) isomorphism
ofX_onto®_jR)®{Q)). Q
Let us study the space described by ffv as v describes X_, where
(3.12) Fv = {T0v,...,Tm_1v}.
Applying Theorem 2.2 of Chapter 9 (see also Remark 2.3 of Chapter 9)
we see that (for example):
(3.13) ve®_jR;L2(Q)).
But we have
d2v
(3.14) P*v = A*v + —- = 0 in a neighborhoodof 27,
ct
(3.15) Cyu = 0, on 27, / = 0,...,w-l.
Then, by the same type of arguments as in the proof of Theorem 1.2,
it follows from (3.13), (3.14), (3.15) and the Theorem on "elliptic iterates"
of Chapter 8, that, for each t, v(x, t) is analytic in x in a neighborhood of
r and finally that
(3.16) Vve[9_jR;je(r))]m. D
Next, by a proof completely analogous to the proof of Theorem 4.1
of Chapter 10, we obtain that
(3.17) v-> (ov is a continuous linear mapping of X_ into

3.2 The Infinite Cylinder Case for m > 1 271
Still following Chapter 10 (see Theorem 4.2 and Corollary) we find
that
(3.18) v->&v is a surjective mapping of X_ onto
[#_,„(R; Jtf(r))T.
In this regard, we note that the application of the Theorem of Talenti
[3] to the Cauchy problem analogous to (4.19) is possible even for the
operator of the second order in t, when the data are in Q)_ m(R; ^(T1)). D
Now we introduce the space
(3.19) Y+ = {u | u e 0'+iH,(R; ®'{Q)), Pu e ®'+>„(R; B'(£}))},
provided with the coarsest locally convex topology which makes the
mappings u-> u and u-> Pu of Y+ into ^'+jW(R; 3t'(Q)) and @'+ttn(R;
S' (Q)) respectively, continuous, these two spaces being provided with
the weak dual topologies (resp. of ^_jW(R; 2f{Q)) and ^_ W(R; S(Q)).
Then we show that
(3.20) the space @(R;@(Q))isdenseinY+.
The proof of (3.20) is slightly different from the proof of Theorem 4.3
of Chapter 10, since now the operator P is not parabolic (and not even
hypoelliptic). Let u-> M(u) be a continuous linear form on Y+; it may
be written
M(u) = </, «> + <g, Pliy
with /£ ^_>W(R; 2{Q)) and g<E ^_>W(R; S^fl)). Assume that we have
M(q>) = 0 V<p€ ®(R; ®(J2)) and let .«/ be an extension of A into a
suitable neighborhood 0 of 42, with analytic coefficients in 0 and elliptic
in 0 (which is always, possible since A is elliptic in Q). Then let / and g be
the extensions to 0 X R* of / and g by zero outside @.
Then if & £ 9{(9 X R,), we have
</~ #> + <g, rff^) = 0,
the brackets being taken in the sense of distributions on^xR/.
Therefore we have
(3.21) j/*g + Dfg = / in 0XR,,
with, in particular, g<E ^_,™(R; £2(0)) and /~<E ^_,™(R; 9{Q)). By an
obvious application of Theorem 1.1, it follows that g£ ^_(R; ^(0)). Let
us now consider a fixed interval [a, b] of R^; then / vanishes in °ll x [«, 6]

272 3. Transposition and Existence of Solutions in Spaces of Ultra-Distributions
(^ a suitable neighborhood of 71) since /£ ^_^(R; @(Q)) and therefore
in ^X [a, b] it follows from (3.21) that
s/*g+V?g = 0.
Differentiating and applying the operator s/* as in the proof of
Theorem 1.2, we obtain
' =(-1)* Dfg
and therefore, since g£ Q) _>W(R; £2(0)), we have
\***%l\um < c0li(2m*)!,* = 0, 1, ...,
and therefore (see Theorem 2.4 of Chapter 8)
g is analytic in # in ^ for every 2 £ [a, 6].
But g vanishes outside $ and therefore g vanishes in <%X]a,b[. Also,
thanks to the fact that g £ ^_)W(R; £"(fl)), it follows that g £ 0_.m(R;
0(12)) and then (3.21) implies '
A*g+Dfg=-f in Q
and therefore M(w) = —(P*g, u} + <g, Pu} and therefore M(w) = 0,
VueY+. D
Finally, by the same methods as for Theorems 4.4 and 4.5 of
Chapter 10, we have no difficulty in showing the following trace and existence
theorems:
(3.22)
and
(3.23)
the mapping u-> Bu = {B0u, ..., Bm_1} of 0(R; @(Q))
into [0(R; @(r))]m extends by contuinuity to a
continuous linear mapping of Y+ into [£^'+)W(R; ^'(i1))]*";
and we have Green s formula
m — l
<«, p*v> - (Pu, vy = % <b,u, Tjvy, vu e y+, vex_,
3 = 0
the boundary value problem
Pu = f in the sense of ^'+>W(R; @'{Q)),
B-u = gy, / = 0, ..., m — l, m the sense of (3.22),
admits a unique solution u in Y+ for every
/€ ^+>r(R; £'(£>)) «** § € 0'+>M(R; M\r)\ «
I depending continuously on f and g-. D

4.1 Regularity Results for the Schroedinger Equation 273
4. Schroedinger Equations;
Complements for Parabolic Equations
4.1 Regularity Results for the Schroedinger Equation
Results comparable to those obtained in the preceding Sections for
equations of the second order in t, can also be obtained for Schroedinger's
equation, by the same type of proofs. We shall therefore restrict this
discussion to the brief statement of results, and refer the reader to the
preceding Sections for the proofs. D
Concerning the regularity results, we have either regularity in the
space 3t(Q), or regularity in Gevrey spaces (Q = Qx]0, T[).
On the open set Q and the operators A and BJ} let us make the hypotheses
(1.1), (1.3), (1.5), (1.6), (1.8) and (1.9) and let us replace (1.4) with
(4.1) apq(x,t) = apq(x)e@(Q)
and (1.7) with
(4.2) bjh(x,t) = bjh(X)emr),
so that the form a(t;u, v) in (1.9) does not depend on t.
We are given /, gj, u0 with
(4.3) fe&(Q), g,-e9(Z), «o€ 9(Q), / = 0,...,m-l,
and with the compatibility relations which guarantee the existence of
we@(Q) with
IBiw = & on Z> i = 0, ..., m — 1, w[x, 0) = u0 (x) on Q,
T>ktPw{%, 0) = D*/(*, 0) on Q, Vk,
where
(4.5) P = iA+^.
Applying Theorem 12.2 of Chapter 5, we obtain in analogy to
Theorem 1.1:
Theorem 4.1. Under the preceding hypotheses, there exists a unique u
in Q){Q)y solution of
Pu = f in Q,
(4.6) ] BjU = gj on 2, j = 0, ..., m — 1,
u(x, 0) = u0(x) on Q. D

274 4. Schroedinger Equations; Complements for Parabolic Equations
Now assume that D satisfies (1.13) and that (4.1) and (4.2) are replaced
with (1.14) and (1.15), with
(4.7) s>l.
Let /, gj} u0 be given with
(4.8) / 6 @2ms,s(Q), g3 6 <WA "o 6 W)
and the compatibility relations which guarantee the existence of
we @2mS,s(Q) satisfying (4.4). Then
Theorem 4.2. Under the above hypotheses, there exists a unique u in
^W(<2)> solution of (4.6).
After reduction of the problem to the case gj = 0 and u0 = 0, the
proof is exactly the same as in Remark 2.2 of Chapter 10 (parabolic
case); and by the same considerations as in Remark 2.3 of Chapter 10,
we can replace in the Theorem the spaces @2?ns,s{Q)> • • • ™fth @Nh,Mk,{Q)> •••
under the same hypotheses on the sequences {Nh} and {Mk} as in
Remark 2.3 of Chapter 10.
4.2 The Non-Homogeneous Boundary Value Problems
for the Schroedinger Equation
The analogy with the case of equations of the second order in t is
complete for the application of the transposition method. It will be
sufficient to state the main results.
Let D satisfy (1.1), Q be the cylinder Q = Qx]0, T[, A and B- the
operators satisfying (1.3), (1.5), (1.6), (1.8), (1.9), (4.1) and (4.3).
We introduce the following spaces:
X = {v \ve@{Q), v(x, T) = 0, Cjv = 0, / = 0, ..., m - 1,
P*v(= -iAv - ^U^_,y([0, T];9y{Q))},
Y = {u | ue ®;>y([0, T]; &y(Q)), PueS'+(0, T; 5'(£)))}
with topologies analogous to the topologies of X and Y in Section 2.
We consider the mapping
v -> &v = {v{x, 0); T0v, ..., T^v)
and show that it is a continuous, linear and surjective mapping of X onto
the subspace tT of ^(Z?)x [^L([0, T]; @(r))]m characterized by the

Non-Homogeneous Boundary Value Problems for the Schroedinger Equation 275
compatibility relations on r, for t = 0:
f v £ @(Q), C^ = 0 on Z, / = 0, ..., m - 1,
yA(P*u) = 0 on Z, Vk, K } \ = 0 on Q, Vk.
[ & \t=0
if can be provided with a strict {SPfF} -space topology in analogous
fashion as in Section 2.4. Then if' is the {weak) dual of if.
This being set, we show, following the methods of Sections 2.1, ...,
2.5:
Theorem 4.3. The space @{Q) is dense in Y and the mapping
u-> au — {u(x, 0); B0u, ..., Bm_1u]
of @(Q) into Q}(Q) X [!3(Z)]m extends by continuity to a continuous linear
mapping of Y into if'\ furthermore, for every f^Sr+(0, T\E'{Q)) and
g* £ «fr'f the problem
\Pu = f in the sense of 0+,y([O, T]; @'Y{Q))
\ [and therefore also of &{Q)),
ou = g* in the sense defined above,
admits a unique solution u in Y; and u depends continuously on f and g*. Q
Remark 4.1. We also have complements analogous to those of
Section 2.6: u-> u(x, 0) is a continuous linear mapping of Y into 2f'Y{Q)
and u-> Bu = {B0u, ..., Bm_1u] is a continuous linear mapping of Y
into
[®'+jr([0, T];2'{r))T. D
There are no difficulties in extending the results of Section 2.7 for the
infinite cylinder case to Schroedinger's equation. We do not specify the
results. D
Remark 4.2. As in Remark 2.3, we see that the application of the
results of Chapter 9, Section 10.6, leads to problems which are more
general in t and less general in the space variables. D
Finally, we can also adapt to Schroedinger's equation all that we have
seen in Section 3 for the equation of the second order in t; we have the
same technical difficulties and the same type of results. Let us only note
the fact that now ms must be replaced with 2ms and therefore 2ms > 1
if m > 1 and s > 1. Therefore, in particular, the results of Section 3.2
are valid, for every integer m > 1, in the space £^'+j2w(R; 3f'{Q))\ more
precisely, once having introduced the space

276 4. Schroedinger Equations; Complements for Parabolic Equations
we have:
the mapping u-> Bu = {B0u, ..., Bm_1u} extends by continuity to a
continuous linear mapping of Y+ into [£^'+)2m(R>* ^'(/7))]w cmd the
problem
yBjU = gj,i = 0, ...,m — 1,
admits a unique solution in Y+ for every
fe&+ttm(R;B'(Q))
and
u depending continuously on f and gj.
4.3 Remarks on Parabolic Equations
The idea introduced in Section 2 for equations of the second order in t
(and in Section 4.2 for Schroedinger's equation) to use the spaces
®-,y([°. TV>%(9)) and 0+>y([O, T]\3j'y(Q)), may also be useful for
parabolic equations.
One obtains results in less general spaces for the solution u than
those obtained in Chapter 10, but under more general hypotheses for the
operators P = A + d/dt and {BjftrJ.
It is not difficult to see that by the methods of Sections 2.2, ..., 2.5
and applying the regularity theorems of Chapter 10, Sections 1 and 2,
we arrive at the following results.
The notation and the hypotheses on Q, P = A + #/#*, {Bj}J[Tq are
those of Section 2.1 of Chapter 10; note that we shall not need hypotheses
(3.13) and (3.14) of Chapter 10. We introduce the spaces (different from
X and Y of Chapter 10):
X7 = {v\ve9(Q),v{x, T) = 0, C3v = 0, / = 0,...,m-l,
Yy = {u | u e ®;.y([0, T]; &y(Q)), PueS'+(0, T; S'(Q))},
with topologies analogous to the topologies of the spaces X and Y of
Section 2.
We consider the mapping
v->&v = {v{x,0)i 7>, ..., T^}
and show that it is a continuous, linear and surjective mapping of Xy
onto the subspaces i^y of ®{Q) x [®-([0, T]; 9{r))]m of elements satis-

6. Problems
277
fying the compatibility relations on T and for t = 0 (obtained "formally''
from the definition of Xy). Next, we set i^y = (weak) dual of i^y.
Then we can prove
Theorem 4.4. 77ze space <2j(Q) is dense in Yy and the mapping
u->ou = {u(x, 0); B0u, ..., Bm_^u)
extends by contimtity to a continuous linear mapping of Yy into i^y. The
problem:
Pu = f,
ou = g*,
admits a unique solution u in Yy for every f £ £"+ (0, T; S' (Q)) and g* £"Ty\
u depends continuously on f and g*. Q
We can also, in opposition to what we have done for the space Y in
Chapter 10 (see Remark 3.2), define u(x, 0) and Bu separately for the
elements u of Yyy by following the method of Section 2.6; we obtain:
1) u-> u(x, 0) is a continuous linear mapping of Yy into 3f'Y(Q);
2) u-> Bu is a continuous linear mapping of Yy into [^'+>y([0> T];
®'{r))]m. D
Finally, the results of Section 2.7 for the infinite cylinder case can be
extended to parabolic equations; but in this case the hypotheses on A
and Bj will be the same as in Section 4 of Chapter 10 and the spaces
will be less general (&+(R; @>Y{Q)) being contained in 0'+(R; &'(&)) and
®;(R; &{r)) in ^+,2w(R; #"(r)).)
5. Comments
The regularity in Gevrey spaces of solutions of boundary value
problems for equations of the second order in t and Schroedinger
equations (see Sections 1.2 and 4.1) has been given in Lions-Magenes [4].
In this same work, we study non-homogeneous problems for the case
of the infinite cylinder and the spaces £^'+>w(R; &(Q)) (see Section 3.2
and the end of Section 4.2).
The other results of this Chapter are given here for the first time.
For the space 3fy{Q) of Section 2.2, see also Triebel [1].
6. Problems
6.1. We recall the problem (already noted in Remark 1.2 and in
Section 3.1) of specifying the compatibility conditions in Gevrey spaces.
6.2. Is Theorem 1.2 valid under hypothesis (1.23) (case of coefficients
depending on t, see Remark 1.3)? Same problem for Schroedinger's
equation.

278
6. Problems
6.3. If m = 1 (hyperbolic case), is Theorem 1.2 still valid for s = 1 ?
6.4. Reflexivity of the spaces ^_>y([0, T]; 0y(J2))and£L(O, T;S(Q))
(for a similar case, see Lions-Magenes [2], Section 4.2).
6.5. Structure of the various spaces V (see Remark 2.2, Sections 4.2,
4.3).
6.6. Do Theorems 2.1, 2.3, ... remain valid if the spaces ^'+>y([0, T];
^ (42)), 5+(0, T; S'(Q)), ... are provided with the strong dual topologies ?
(See Remark 2.1 and Problem 6.4).
6.7. "Separation" of the components u(x, 0), du(x, 0)/dt} T0v,.. ,,Tm_1v
in the operator nu of Section 2.5.
6.8. Same problem as above for the operator an which comes up for
Schroedinger's equation (Section 4.2) and parabolic equations
(Section 4.3).
6.9. In opposition to the case of elliptic and parabolic equations, we
do not have ' 'optimal" regularity results for the case of equations of the
second order in t or of Schroedinger. Thus, there are other possible choices
than the spaces ^'+>y([0, T]; @'y{Q)); it may be of interest to study them
in a more systematic way.
6.10. In particular, note the case of the operator
d2u d2u
Pu =
dt2 dx2
(wave operator in two variables) which is hyperbolic either with respect
to t or with respect to x; this should allow for a study of non-homogeneous
boundary value problems in spaces other than £^'+jy([0, T]; @'Y(Q)); see
Section 2.1.
6.11. It is very likely that a theory analogous to the one given here is
valid for first order hyperbolic systems (see M. S. Agranovich [1],
K. O. Friedrichs and P. Lax [1], P. Lax and R. S. Phillips [1], C. Bardos
[1]); but this remains to be made explicit.
It would be of great interest to study the analogous problems for
higher order hyperbolic equations; see S. Agmon [1], S. Miyatake [1],
S. Mizohata [2], T. Shirota and K. Asano [1].
6.12. In Chapter 9, Section 5, we have seen how it was possible to
"approximate" hyperbolic systems or systems of the second order in t
with parabolic problems; then it is very likely that the solutions obtained
in this Chapter can be "approximated" with solutions of parabolic
problems (solved in the preceding Chapter).
Same remark concerning the approximation by systems of Cauchy-
Kowalevski type (see Chapter 9, Section 5).
6.13. Extension of the theory of this Chapter to the settings of
Problems 14.8, 14.9, 14.10, 14.12, 14.13, 14.14, 14.17 of Chapter 5.

Appendix
Calculus of Variations in Gevrey-Type Spaces
1. Generalities
1.1 Duality Operators in Gevrey-Type Spaces
Let Q be a bounded open set in RM. We are given
(1.1) s > 1
and a number L > 0.
For q> given in @(Q), we set
/ 1 \1/2
which in general is infinite, and define the space E by1
(1.3) E = {<p\?e9(Q),\ME<oo}.
Provided with the norm \\<p\\E, E is a Hilbert space.
Next, we introduce
(1.4) E0 = {y | <p e E} D> = 0 on T, Woe)
(where, as usual, r denotes the boundary of D).
We verify that:
(1.5) ^s(^) E is dense in E0,
\9tj(Q)CE0iil<Sl<s,
(1.6)
[ Q)Si (D) is dense in E0.
1 The space E (which depends on L) coincides with the space ^^s(Q) L),
in the notation of Chapter 7, Section 1.3.

280
Calculus of Variations in Gevrey-Type Spaces
Then the dual E'0 of E0 can be identified with a subspace of &Si(Q),
arbitrary s1 with 1 < s1 < s.
Since E0 is a Hilbert space, there exists a canonical isomorphism A0
of Eq onto E'0, given by
The operator A0 is the "duality operator" of E0 onto E'G.
Every element / of E'Q can be represented, non-uniquely, by
(1.8)
(X
2 (|«|i)2,i2Wll/Jl!.(fl)<«»-
Then, for f given in the form (1.8), there exists a unique u in E0, solution
of
(1.9) \(u) = f.
This is a [homogeneous) Dirichlet problem of infinite order. Q
Remark 1.1. The duality operator A of E onto E' solves an infinite
order Neumann problem. Q
Remark 1.2. It is possible to construct spaces analogous to E and E0,
replacing L2 with Lp, 1 < ^> < oo, p ^ 2. Then, the duality operators
A (and Aq) of £ onto £' (and E0 onto £q) are nonlinear operators of
infinite order. []
1.2 Orientation
The remarks made in Section 1.1 show to what type of problems we
are led when we consider calculus of variations problems1 in Gevrey
spaces.
We shall examine more closely what one obtains in optimal control
theory in the setting of Gevrey spaces or of analytic functions. The
following results are taken from notes by the authors (see Lions-Magenes [6]).
We shall distinguish two cases:
(i) elliptic systems;
(ii) evolution systems.
In either case, one obtains boundary value problems of infinite order.
1 Indeed (analogue of the Dirichlet principle) problem (1.9) is equivalent to the
search for the minimum of \\v\\\ — 2(f,v) on E0, (/, v) — scalar product between
/££J and v£E0.

2. Elliptic Systems
281
2. Elliptic Systems
2.1 Notation
Let A be the second order operator
and assume that D is of class {&!} (see Chapter 8, Section 1.2), that the
coefficients ai} are real analytic in Q and that A is elliptic in Q ((1)).
Suppose that the state of a physical system is given by
y = y[%\ v), x£Q, v = control,
solution of
(2.1) Ay{x;v) = 0inQ,
(2.2) y(x; v) =vonr.
(we shall also denote the function x-> y(x; v) by y{v)).
We shall assume that v belongs to a space of analytic functions on F.
More precisely (see Chapter 7, Section 3.2) we introduce:
Ar = Laplace-Beltrami operator on 71,
J^L{r) = {space of functions cp such that
JoZ^w l,A^,li2(r) = M%*r) < ~}((2))-
(2.3)
We know (see Chapter 8) that ^L{T) is a space of real analytic
functions on F\ provided with the norm H^H^r), &iXF) is a Hilbert space. Q
In the sequel, we shall assume that
(2.4) v e <% = JeL[T) [L > 0 fixed).
Then
dy
(2.5) — e J^L {r) {vA normal to r with respect to A),
dvA
where Lx depends on L, on the coefficients of A and on r.
((])) We have taken A to be a second order operator (and of a particular type)
and we consider the Dirichlet problem relative to A, but only in order to simplify
the presentation; what is to follow can be extended to regular problems for elliptic
operators of arbitrary order with analytic coefficients and, more precisely, under
the hypotheses of Chapter 8, Section 3.1.
((2)) por the sake of simplicity, all the functions are taken to be real-valued.

282 Calculus of Variations in Gevrey-Type Spaces
Therefore, for every v £ %, we can define the function:
(2.6)
where:
M =
10y(f)
\ Sva 1
2
zd is given in ^Li[T),
v is a given positive number.
2.2 Control Problem
Now we consider
(2.7) <%ad = closed convex set C
and we seek
(2.8) Inf. /(^)((1))
It is easy to see that there exists a unique element u £ ^ad such that
/M</KV*,£^ad.
u is called the optimal control.
We aim to give the system of equations or inequalities which
characterize the optimal control.
2.3 Necessary and Sufficient Conditions for Optimality
2.3.1 First Characterization
If J'(v) denotes the derivative of / (the existence of which is easily
verified), then the optimal control u is characterized by
(2.9) J'{u)[v-u)>0, Vv£Wad,
which is equivalent to
(2.io) m -^™_m\ + v{u, v - u)w >ov,6V
\ dvA dvA dvA /xLi{r) L{ )
((])) Thus we consider problems analogous to those studied in Chapter 6 (Vol. 2),
but the Sobolev spaces are replaced with Hilbert spaces of Gevrey type or of analytic
functions.

2. Elliptic Systems 283
2.3.2 Adjoint State
We know (see Chapter 7, Section 3.2) that the series
k
represents a continuous linear form on J^(-T), i.e. an element of the space
ffi"{F) of analytic functional on F. And then, according to the results of
Chapter 8, Section 3.6, the Dirichlet problem
(2.11) A*p(u) = 0,
where A * is the formal adjoint of A, admits a unique solution p(u) £ Q}' (Q),
condition (2.12) being taken in Jff"(r). The distribution p(u) is called the
adjoint state of the control problem.
2.3.3 Transformation of (2.10)
Applying Green's formula (see Chapter 8, Section 3.5), we can then
write the following relation:
(2.13) - (^, y(v) - y(u)\ + <fi{u), -£- (y[v) - y(«))> = 0
where the brackets denote the duality between ffl'(F) and ffl(F). It
foUows from (2.13) and (2.12) that
, V — U ;.
(2.14)
Set:
(2.15)
Then
(2.16)
(dy{u)
\dvA "
dy(v) 8y{u)\ /dp{u)
dvA dvA )jfLl{r) \dvA*
AL —E \2k
A "2,£«*((2ft)!)^r"
(u, v — u)^^ = (ALu, v — u),
so that, with (2.14) and (2.16), condition (2.10) is equivalent to
(2.17) /-|p + vALu, v -u)>0 Vve <%ad.
\ dvA*

284 Calculus of Variations in Gevrey-Type Spaces
2.4 Conclusion
We summarize the results in
Theorem 2.1. The optimal control u is given by the resolution of the
system:
IAy{u) =0inQ,
A*p(u) = OinQ,
[y{u) = u onT,
(2.19) i T (dylu) \
U{u)=AL>(^-zdynr,
(2.20) /^^ + vALu, v-u\>0, Vv£ ^ad((])).
\ cvA* /
(2.18)
2.5 Applications (I). Unconstrained Case
The problem is said to be unconstrained when ^ad = <%. Then (2.20)
reduces to (since 2tfL(T) is dense in Jf?(r)):
(2.21)
— (u) + vALu = 0.
dvA*
u can be eliminated from (2.18), (2.19), (2.20); we obtain:
\Ay =0,
(2.22)
(2.23)
A*p = 0,
inQ,
dp
dvA*
+ vALy = 0 on T,
-AL
3y
8va
z, on r.
Therefore the optimal control is obtained by:
(i) solving system (2.22), (2.23)2;
(ii) setting u = y \r.
(W) (2.18), (2.19), (2.20) is a "unilateral, infinite order" problem.
2 The system (2.22), (2.23) is a coupled elliptic system of infinite order, with
the particularity that the component y of the solution (y, p) is "very regular",
whereas the component p is "very irregular". The components y and p are "in
duality".

2. Elliptic Systems
2.6 Applications (II). Constrained Case
285
Now let
(2.24) ^ad = {v | v e °U = 3rL{r), v>0onr}.
Then (2.20) is equivalent to (since ^ad is a cone with apex {0}):
(2.25)
(2.26)
p(u) +vALu, /7>0Vv^
^dvA* /
But if v e Jf?(r), v > 0 on r, then there exists v, € #iXT)> ^ > °>
v3-> v in ^(i^), and therefore (2.25) implies (and is equivalent to) the
same inequality Vv £ ffl{T), v > 0 on 71. Therefore
0£
(2.27)
0^*
(^) + vALu is an analytic functional > 0.
But, by arguments analogous to those used by Schwartz [6] in order
to show that every distribution > 0 is a measure > 0, we see that
(2.27) is equivalent to:
(2.28) p(u) + vA^u is a measure > 0 on T.
Sva*
Then condition (2.26) is equivalent to
(2.29)
u (-— p(u) + vALu) = 0 on r.
\dvA* J
Finally we obtain: the optimal control is provided by the resolution
of the system:
[Ay =0,
I in 12,
[A*p = 0,
(2.30)
(2.31)
Then
AL
\dvA
y > 0 on T,
dp
zAon r,
dvA*
+ vALy > 0 on T,
y\-^- +vALy\ = 0onT.
\dvA*

286 Calculus of Variations in Gevrey-Type Spaces
3. Evolution Systems
3.1 Generalities
It is possible, in the setting of Gevrey and analytic function spaces,
to study optimal control problems for systems governed by evolution
operators like those considered in Chapter 4, 5 and 9, 10, 11 of this text.
In order to fix our ideas, we consider the setting of Chapter 9,
Section 7.
We take the notation of Chapter 9, Section 7, but with E a Hilbert
space on R.
Let A be an operator such that —A is the infinitesimal generator of a
semi-group G(t) in E.
We introduce (there is a slight modification with respect to the
definition (7.27), Chapter 9, and this so as to be able to benefit from the
Hilbert structure of E):
f DL(A°°; Mk) = {e\ee D(^°°), \\e\\DL(A~.m) =
(3.1) \ /co -, W/2 I
provided with the norm ||£||JDL(,400;Mfc)> DL(^4°°; Mk) is a Hilbert space.
Note that
(3.2) G{t) e ^{DL{A°°; Mk); DL{A°°; Mh)),
and therefore that we have: let v be given in DL(^4°°; Mk); there exists a
unique solution y(t, v) =y{v), with values in DL(^4°°; Mk), of
(3.3; d^;y) +Ay(t'>v) = 0>t>°>
(3.4) y(0;v)=v.
We shall consider v as the "control"1 and y(v) as the state of the
system2.
3.2 The Optimal Control Problem
Setting
(3.5) qt = DL{A°°;Mk),
we consider the cost function
(3-6) Av) = \\y(T;v)-zd\\%+v\\v\\%,
1 Actually this is more like a "filtering" problem.
2 One could also consider systems governed by equations of the second order
in t (and therefore, in particular, hyperbolic equations).

3. Evolution Systems 287
where
T > 0 is fixed,
zd is given in DL(A°°; Mk),
v > 0 is fixed.
Next, we are given
(3.7) ^ad = closed convex set in °U,
and we seek to "characterize'' the unique element u of <%ad (the optimal
control) such that
(3.8) Inf/(»)=/(«).
3.3 Necessary and Sufficient Conditions for Optimality
3.3.1 First Characterization
The control u is optimal if and only if
f [y{T; u) - zdi y{T; v) - y{T; u))dl{Aoo.Mk) +
(3.9) i
[ + v{u, V - u)DL{Aoo;Mk) >0Vve®ad.
Let us introduce the operator
(3.10) p^=^_l_^4*.
We easily see that
(3.11) t/Le^{DL{A°°;Mk); (DL(A°°; Mk))').
Then (3.9) is equivalent to
f (FL(y(T; u) - zd), y(T; v) - y(T; «)> + v<pLu, v - u} > 0
(3.12)
where the brackets denote the duality between DL(^4°°; Mk) and its dual.
3.3.2 Adjoint State
The adjoint state p(u) is defined by the solution of (apply Chapter 9,
Section 7.4):
(3.13) - A p{u) + A*p{u) = o in ]G, T[,
at
(3.14) f(T;u) = VL(y(T;u)-zd) ,
(3.15) p{t;u)e{VL(A~;Mk))'.

288 Calculus of Variations in Gevrey-Type Spaces
The solution is given by
(3.16) p(t; u) = G*(T - t) pL(y(T; u) - zd).
Then we see that
<pL(y(T; u) - zd), y(T; v) - y(T; «)>
= <#(0; «), y(0; v) - y(0; «)> = <^(0; «), v - **>
and therefore (3.12) is equivalent to
(3.17) <p{0]u) +v\7Lu}v -u}> 0 Vve<%ad.
3.4 Conclusion
In summary, we have:
Theorem 3.1. The optimal control u is given by the resolution of the
system:
— y(u) +Ay(u) = 0,
(3.18)
(3.19)
(3.20)
dr
--£(«) +ii*0(«)=O,
*»]0, T[
fy(0;«) =«,
{^(r;^)=:FL(3;(r;^)-^),
<£(0; «) + rfV u - u) > 0, V^ 6 ^ad-
3.5 Application
We only give one application. We take D and A as in Section 2 and
£ = L2(Q). Then we have
D(4) = {y> | ^G H*{Q),7o<ip = Oonr}.
We are within the conditions of applicability of the preceding theory.
If we take
(3.21) M, = ((2£)!)*(s>l),
then, thanks to Theorem 1.2 of Chapter 8:
(3.22) T>L(A °°)Mk)C ®s{®) = {Gevrey functions of order s in Q}.
We take
(3.23) <%ad = {v\ve T>L{A°°)Mk)t v>0mQ}.

3. Evolution Systems
289
(Applying Theorem 3.1) we end up with the following problem:
" B
(3.24)
(3.25)
(3.26)
dt
y + Ay = 0,
-g + ^ = o,
inQ = Qx]0, T[
y = 0, p = 0 on Z,
\p(x,T) = VL[y(x,T)-zd(x)),
y{x,0)>0,
p{x, 0) + v\7Ly{x, 0) > 0 in 42((1)),
y(x,0) [j>(x,0) + v\7Ly(x,0)]=0.
This is a nonlinear problem (of "unilateral" type) containing
differential operators of infinite order, and admitting a unique solution.
((1); We write the functional p(T)£ (T>L(A°°; Mfc))'like a function. This is a
symbolic notation.

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Additional Bibliography
Chapter 7
For Gevrey and Beurling classes of functions and ultra-distributions, additional
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Bjorck, G.
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Cerofolini, L.
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Dostal, M. A.
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Komatsu, H.
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Sato, M.
1. Regularity of hyperfunction solutions of partial differential equations. Actes
Congres Intern. Math. Paris: Gauthier-Villars, Vol. 2, 1970, 785 — 794.
A new class of "ultra-distributions" is introduced in:
Aronszajn, N.
1. Traces of analytic solutions of the heat equation; relation to distributions
and hyperfunctions. Symp. C.N.R.S Paris, Sept. 1972.
Chapter 8
For elliptic boundary value problems in unbounded domains and in spaces of
hyperfunctions (see also Problems 6.6 and 6.7) cf.

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305
Schapira, P.
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France, 99, 113—141 (1971).
For questions related to. trace problems and to Problem 6.4 we refer to
Baouendi, M. S., Geymonat, G.
1. Transposition des problemes aux limites elliptiques. 1st. Naz. Alta Math.,
Rome, Symposia Math., Vol. VII, 421 — 441 (1971).
Boutet de Monvel, L., Geymonat, G.
1. Solutions irregulieres d'un probleme aux limites elliptiques. 1st. Naz. Alta
Math., Rome, Symposia Math., Vol. VII, 381 — 402 (1971).
For the regularity of hyperfunction solutions for elliptic and other operators,
cf. M. Sato, already quoted above.
For a theorem similar to Theorem 1.2 for degenerate elliptic operators, and
applications to interpolation theory and to approximation theory, cf.
Baouendi, M. S., Goulaouic, C.
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Analysis.
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Amer. Math. Soc. 78, 3 (May 1972).
Baouendi, M. S., Goulaouic, C, Hanouzet, B.
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Pures et appliquees, 1973.
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Appliquees. 1973.
For other results on degenerate operators, we refer to the book
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1'. Equations du 26me ordre dont la forme caracteristique est non negative.
In Russian, Moscow 1971.
Chapter 9
For general semigroups and applications to Cauchy problems, cf.
Beals, R.
1. Semi groups and abstract Gevrey spaces. To appear. J. Funct. Analysis.
2. On the abstract Cauchy problem (to appear).
Ouchi, S.
1. Hyperfunction solution of the abstract Cauchy problem. Proc. Jap. Acad.
47, 541-544 (1971).
Simon, B.
1. The theory of semi-analytic vectors. A new proof of a theorem of Masson and
McClary. To appear.
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For Problem 14.2, cf.
Bernardi, M. L.
1. Equazioni differenziali lineari astratte del secondo ordine e di Schroedinger
in spazi di Beurling. 1st. Lombardo (Rend. Sc.) 106 (1972).
Chapter 10
For parabolic problems on compact varieties in Gevrey classes, cf.
Bernardi, M. L.
already quoted in the additional bibliography for Chapter 7 and
Cerofolini, L.
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varieta. Boll. U. M. I. IV (1969) 65-70.
2. Sulle tracce sulla varieta M delle soluzioni in Mx(0, + oo) di una equazione
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Chapter 11
For Problem 6.10, cf.
Pozzi, G. A.
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distribuzioni. Rend. 1st. Lombardo Sc. Lett. (A) 104, 534-579 (1970).
2. Ulteriori osservazioni sui problemi ... Id. 105, 306—316 (1971).
For mixed problems for general hyperbolic operators, cf.
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112 (1971).
Beals, R.
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Chazarain, J., Piriou, A.
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Kreiss, H. O.
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(to appear).
Sakamoto, R.
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1. On necessary and sufficient conditions for L2 well-posedness of mixed
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Baouendi, M. S.
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Bony, J. M., Schapira, P.
1. Probleme de Cauchy, existence et prolongement pour les hyperfonctions
solutions des equations hyperboliques non strictes. C. R. A. S. Paris 274,
188-191 (1972).
2. Solutions hyperfonctions du probleme de Cauchy (to appear).
Kawai, T.
1. Construction of elementary solutions of /-hyperbolic operators and solutions
with small singularities. Proc. Japan Acad. 46, 912 — 915 (1970).
Komatsu, H., Kawai, T.
1. Boundary values of hyperfunction solutions of linear partial differential
equations. Publ. R. I. M. S. Kyoto Univ. 7, 95-104 (1971/1972).
For the global existence of real analytic solutions of general partial differential
equations and the properties of analyticity of the solutions, cf.
Anderson, K. G.
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de Giorgi, E., Cattabriga, L.
1 Una dimostrazione diretta dell'esistenza di soluzioni analitiche nel piano
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4, 1015-1027 (1971).
Hormander, L.
3. Uniqueness theorems and wave front sets for solutions of linear differential
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(1971).
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1. On the global existence of real analytic solutions of linear differential
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For finite element method approaches in numerical analysis using variational
methods of partial differential equations, we already gave some basic references in
the Additional Bibliography of Vol. I, p. 354. To these references, we wish to add
here, for elliptic problems:
Babuska, I.
1. The finite element method with Lagrangian multipliers. Technical Note
BN-724. Inst. Fluid Dyn. and Appl. Maths. Univ. of Maryland, January 1972.
Babuska, I., Kellog, R. B.
1. Numerical solution of the neutron diffusion equation in the presence of
corners and interfaces. Technical Note BN-720. Inst. Fluid Dyn. and Appl.
Maths. Univ. of Maryland, December 1971.
Bramble, J. H.
1. Variational methods for the numerical solution of elliptic problems. Lecture
Notes Chalmers Inst. Technology 1970.
Bramble, J. H., Zlamal, M.
1. Triangular elements in the finite element method. Math. Comp. 24, 112
809-820(1970).
Brezzi, F.
1. Sull 'analisi numerica del problema di Dirichlet per le equazioni lineari
ellittiche. Publicazione 18, Laboratorio di Analisi Numerica, Pavia, 1971.

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Bibliography
Ciarlet, P. G., Raviart, P. A.
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Strang, G.
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For evolution problems, we mention
Douglas, J., Dupont, T.
1. Galerkin methods for parabolic equations. SI AM J. Num. Anal. 7, 4, 575 — 626
(1970).
Dupont, T.
1. L2-estimates for Galerkin methods for second order hyperbolic equations (to
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1. The use of numerical integration in finite element methods for solving
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Dublin, 1972.
721/8/73

Die Grundlehren der mathematischen Wissenschaften
in Einzeldarstellungen
mit besonderer Berucksichtigung der Anwendungsgebiete
Eine Auswahl
67. Byrd/Friedman: Handbook of Elliptic Integrals for Engineers and Scientists.
2nd edition
68. Aumann: Reelle Funktionen. 2.Aufl.
74. Boerner: Darstellungen von Gruppen. 2.Aufl.
76. Tricomi: Vorlesungen iiber Orthogonalreihen. 2.Aufl.
77. Behnke/Sommer: Theorie der analytischen Funktionen einer komplexen Ver-
anderlichen. Nachdruck der 3.Aufl.
78. Lorenzen: Einfiihrung in die operative Logik und Mathematik. 2. Aufl.
86. Richter: Wahrscheinlichkeitstheorie. 2. Aufl.
87. van der Waerden: Mathematische Statistik. 3. Aufl.
94. Funk: Variationsrechnung und ihre Anwendung in Physik und Technik.
2. Aufl.
97. Greub: Linear Algebra. 3rd edition
99. Cassels: An Introduction to the Geometry of Numbers. 2nd printing
104. Chung: Markov Chains with Stationary Transition Probabilities. 2nd edition
107. Kothe: Topologische lineare Raume I. 2. Aufl.
114. MacLane: Homology. Reprint of 1st edition
116. Hormander: Linear Partial Differential Operators. 3rd printing
117. O'Meara: Introduction to Quadratic Forms. 2nd printing
120. Collatz: Funktionalanalysis und numerische Mathematik. Nachdruck der
l.Aufl.
121./122. Dynkin: Markov Processes
123. Yosida: Functional Analysis. 3rd edition
124. Morgenstern: Einfiihrung in die Wahrscheinlichkeitsrechnung und
mathematische Statistik. 2. Aufl.
125. Ito/McKean jr.: Diffusion Processes and Their sample Paths
126. Lehto/Virtanen: Quasiconformal Mappings in the Plane.
127/ Hermes: Enumerability, Decidability, Computability. 2nd edition
128. Braun/Koechler: Jordan-Algebren
129. Nikod^m: The Mathematical Apparatus for Quantum Theories
130. Morrey jr.: Multiple Integrals in the calculus of Variations
131. Hirzebruch: Topological Methods in Algebraic Geometry. 3rd edition
132. Kato: Perturbation Theory for Linear Operators
133. Haupt/Kiinneth: Geometrische Ordnungen
134. Huppert: Endliche Gruppen I
135. Handbook for Automatic Computation. Vol. 1/Part a: Rutishauser:
Description of ALGOL 60
136. Greub: Multilinear Algebra
137. Handbook for Automatic Computation. Vol. 1/Parth b: Grau/Hill/Langmaack:
Translation of ALGOL 60
138. Hahn: Stability of Motion
139. Doetsch/Schafke/Tietz: Mathematische Hilfsmittel des Ingenieurs. l.Teil
140. Collatz/Nicolovius/Tornig: Mathematische Hilfsmittel des Ingenieurs. 2.Teil
141. Mathematische Hilfsmittel des Ingenieurs. 3.Teil
142. Mathematische Hilfsmittel des Ingenieurs. 4.Teil
143. Schur: Vorlesungen iiber Invariantentheorie
144. Weil: Basic Number Theory

152. Hewitt/Ross: Abstract Harmonic Analysis. Vol.2: Structure and Analysis
for Compact Groups, Analysis on Locally Compact Abelian Groups
154. Singer: Bases in Banach Spaces I
155. Miiller: Foundations of the Methematical Theory of Electromagnetic Waves
156. van der Waerden: Mathematical Statistics
157. Prohorov/Rozanov: Probability Theory
158. Constantinescu/Cornea: Potential Theory on Harmonic Spaces
159. Kothe: Topological Vector Spaces I
160. Agrest/Maksimov: Theory of Incomplete Cylindrical Functions and Their
Applications
161. Bhatia/Szego: Stability Theory of Dynamical Systems
162. Nevanlinna: Analytic Functions
163. Stoer/Witzgall: Convexity and Optimization in Finite Dimensions I
164. Sario/Nakai: Classification Theory of Riemann Surfaces
165. Mitrinovic: Analytic Inequalities
166. Grothendieck/Dieudonne: Elements de Geometrie Algebrique I
167. Chandrasekharan: Arithmetical Functions
168. Palamodov: Linear Differential Operators with Constant Coefficients
169. Rademacher: Topics in Analytic Number Theory
170. Lions: Optimal Control of Systems Governed by Partial Differential Equations
171. Singer: Best Approximation on Normed Linear Spaces by Elements of Linear
Subspaces
172. Biihlmann: Mathematical Methods in Risk Theory
173. F. Maeda/S. Maeda: Theory of Symmetric Lattices
174. Stiefel/Scheifele: Linear and Regular Celestial Mechanics. Perturbed two-
body Motion —Numerical Methods — Canonical Theory
175. Larsen: An Introduction of the Theory of Multipliers
176. Grauert/Remmert: Analytische Stellenalgebren
177. Fliigge: Practical Quantum Mechanics I
178. Fliigge: Practical Quantum Mechanics II
179. Giraud: Cohomologie non abelienne
180. Landkof: Foundations of Modern Potential Theory
181. Lions/Magenes: Non-Homogeneous Boundary Value Problems and
Applications I
182. Lions/Magenes: Non-Homogeneous Boundary Value Problems and
Applications II
183. Lions/Magenes: Non-Homogeneous Boundary Value Problems and
Applications III
184. Rosenblatt: Markov Processes. Structure and Asymptotic Begavior
185. Rubinowicz: Sommerfeldsche Polynommethode
186. Wilkinson/Reinsch: Handbook for Automatic Computation II, Linear Algebra
187. Siegel/Moser: Lectures in Celestial Mechanics
188. Warner: Harmonic Analysis on Semi-Simple Lie Groups I
189. Warner: Harmonic Analysis on Semi-Simple Lie Groups II
190. Faith: Algebra: Rings, Modules, and Categories I
191. Faith: Algebra: Rings, Modules, and Categories II
192. Marcev: Algebraic-Systems
193. Polya/Szego: Problems and Theorems in Analysis. Vol. 1
194. Igusa: Theta Functions
195. Berberian: Baer*-Rings
196. Athreya: Branching Processes
197. Benz: Vorlesungen iiber Geometrie der Algebren
200. Dold: Lectures on Algebraic Topology