Series on Analysis. Application* and Computation - Vol.6
ISAAC
* M W Wong
An Introduction to
Pseudo-Differential
Operators
3rd Edition

Series on Analysis, Applications and Computation - Vol. 6
An Introduction to
Pseudo-Differential
Operators
3rd Edition

Series on Analysis, Applications and Computation
Series Editors: Heinrich G W Begehr (Freie Univ. Berlin, Germanyj
Robert Pertsch Gilbert (Univ Delaware, USA)
M. W. Wong (York Univ., Canada)
Advisory Board Members:
Mikhail S Agranovich (Moscow Inst, of Elec. & Math., Russia),
Ryuichi Ashino (Osaka Kyoiku Univ, Japan),
Alain Bourgeat (Univ. de Lyon, France),
Victor Burenkov (Cardiff Univ, UK),
Jinyuan Du (Wuhan Univ., China),
Antonio Fasano (Univ di Firenez, Italy),
Massimo Lanza de Cristoforis (Univ. di Padova, Italy),
Bert-Wolfgang Schulze (Univ. Potsdam, Germany),
Masahiro Yamamoto (Univ of Tokyo, Japan) &
Armand Wirgin (CNRS-Marseille, France)
Published
Vol. 1:	Boundary Values and Convolution in Ultradistribution Spaces
by R D Carmichael, A Kaminski & S Pilipovic
Vol. 2:	Complex Analysis
by MW Wong
Vol. 3:	Topics in Mathematical Analysis
edited by P Ciatti, E Gonzalez, M L de Cristoforis & G P Leonardi
Vol. 4:	Nonlinear Waves: An Introduction
by P Popivanov & A Slavova
Vol. 5:	Asymptotic Behavior of Generalized Functions
by S Pilipovic, B Stankovic & J Vindas
Vol. 6:	An Introduction to Pseudo-Differential Operators, 3rd Edition
by MW Wong

Series on Analysis, Applications and Computation - Vol. 6
ISAAC!
An Introduction to
Pseudo'Differential
Operators
3rd Edition
o M W Wong
York University, Canada
^ World Scientific
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Library of Congress Cataloging-in-Publication Data
Wong, M. W. (Man Wah), 1951-
An introduction to pseudo-differential operators / by M.W. Wong (York University,
Canada). ~ 3rd edition.
pages cm. -- (Series on analysis, applications and computation ; vol. 6)
Includes bibliographical references and index.
ISBN 978-9814583084 (hardcover: alk. paper)
1. Pseudodifferential operators. I. Title.
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515'.7242-dc23
2014003783
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Preface
There have been a lot of developments in pseudo-differential operators since
the first edition was published in 1991. The second edition, published in
1999, has served well as an introduction to pseudo-differential operators in
the capacity of a textbook. The prerequisites for the first two editions are
minimal and can be seen from the prefaces for the first and second editions,
which follow this one.
The third edition is intended to contain not only improvements of some
of the contents and additional exercises to some of the existing chapters in
the second edition, but also new chapters to make the book more useful
without losing the original intent of keeping the book as elementary as
possible. The new chapters notwithstanding, the whole book remains to be
a textbook primarily for beginning graduate students in mathematics. It is
also useful to mathematicians aspiring to do research in pseudo-differential
operators and related topics.
The new chapters are the last seven chapters, Chapters 17-23, of the
book. The focus of Chapters 17-21 is on the class of pseudo-differential
operators studied in the first two editions. The theme underlying Chap¬
ters 17-19 is Garding’s inequality, which is used to prove the existence
and uniqueness of solutions of pseudo-differential equations. In particular,
the Hille-Yosida-Phillips theorem on one-parameter semigroups is used to
prove the existence and uniqueness of solutions of initial value problems
for heat equations governed by pseudo-differential operators. After a chap¬
ter, Chapter 20, on the general theory of Fredholm operators that we need
for this book, the ellipticity and Fredholmness of pseudo-differential oper¬
ators are developed in Chapter 21. The ellipticity, Fredholmness, an index
formula and the spectral invariance for another class of pseudo-differential
operators, dubbed symmetrically global pseudo-differential operators in this

vi
An Introduction to Pseudo-Differential Operators, 3rd Edition
book, are studied in Chapters 22 and 23. The emphasis of the book, as in
the first two editions, is on the global theory of elliptic pseudo-differential
operators on Lp(Rn), 1 < p < oo.
As we are now well into the new millennium and moving forward with
increasing acceleration, many advanced topics in any area of science and
engineering in 1991 are now being taught in basic courses to students. The
prerequisites for a complete understanding of the book can be succinctly
described as a first course in functional analysis including the Riesz theory
of compact operators. The book contains ample material to be studied
leisurely and carefully for a two-semester course. One-semester courses can
be designed by omitting certain topics in order to fulfil the needs of the
students and the duration of the semester.
Preface to the Second Edition
The first edition of the book has been used as the textbook for the standard
graduate course in partial differential equations at York University since its
publication in 1991. The motivation for writing the second edition stems
from the desire to remove several deficiencies and obscurities, and to incor¬
porate the improvements that I can see through many years of teaching the
subject to graduate students and discussions of the subject with colleagues.
Notwithstanding the many changes I have in mind, I am convinced that the
elementary character of the book has served and will serve well as an ideal
introduction to the study of pseudo-differential operators. Thus, the basic
tenet of the second edition is to retain the style and the scope of the first
edition.
Notable in the second edition is the addition of two chapters to the
book. Experience in teaching pseudo-differential operators reveals the fact
that many graduate students are still not comfortable with the interchange
of order of integration and differentiation. The new chapter added to the
beginning of the book is to prove a theorem to this effect which can cope
with every interchange of order of integration and differentiation encoun¬
tered in the book. Another new chapter, added as the final chapter in the
second edition, is to prove a theorem on the existence of weak solutions of
pseudo-differential equations. The inclusion of this chapter, in my opinion,
enhances the value of the book as a book on partial differential equations.
Furthermore, it provides a valuable connection with the chapter on minimal

Preface
vii
and maximal operators and the chapter on global regularity.
Other new features in the second edition include a deeper study of
elliptic operators and parametrices, more details on the proof of the Lp-
boundedness of pseudo-differential operators, additional exercises in several
chapters of the book, a slightly expanded bibliography and an index.
Preface to the First Edition
The aim of the book is to give a straightforward account of a class of
pseudo-differential operators. The prerequisite for understanding the book
is a course in real variables. It is hoped that the book can be used in courses
in functional analysis, Fourier analysis and partial differential equations.
The first eight chapters of the book contain the basic formal calculus of
pseudo-differential operators. The remaining five chapters are devoted to
some topics of a more functional analytic character.
It is clear to the expert that the book takes up a single theme in a
wide subject and many important topics are omitted. It is my belief that
this approach is in fact a more effective introduction of pseudo-differential
operators to mathematicians and graduate students beginning to learn the
subject. Exercises are included in the text. They are useful to anyone who
wants to understand and appreciate the book better.
The actual writing of the book was essentially carried out and completed
at the University of California at Irvine while I was on sabbatical leave from
York University in the academic year in 1987-88. The preliminary drafts of
the book have been used in seminars and graduate courses at the University
of California at Irvine and York University.
Many colleagues and students have helped me improve the contents and
organization of the book. In particular, I wish to thank Professor William
Margulies at the California State University at Long Beach, Professor Mar¬
tin Schechter at the University of California at Irvine, Professor Tuan Vu
and Mr. Zhengbin Wang at York University for their stimulating conver¬
sations and critical comments about my book. I also wish to thank Mr.
Lian Pi, my Ph.D. research student at York University, who has worked
out every exercise in the book.

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Contents
Preface	v
1.	Introduction, Notation and Preliminaries	1
2.	Differentiation of Integrals Depending on Parameters	5
3.	The Convolution	9
4.	The Fourier Transform	17
5.	Tempered Distributions	27
6.	Symbols, Pseudo-Differential Operators and
Asymptotic Expansions	31
7.	A Partition of Unity and Taylor’s Formula	43
8.	The Product of Two Pseudo-Differential Operators	55
9.	The Formal Adjoint of a Pseudo-Differential Operator	61
10.	The Parametrix of an Elliptic Pseudo-Differential Operator	69
11.	Lp-Boundedness of Pseudo-Differential Operators	75
12.	The Sobolev Spaces Hs'p, —oo <s<oo, l<p<oo	87
ix

x	An Introduction to Pseudo-Differential Operators, 3rd Edition
13.	Closed Linear Operators	95
14.	Minimal and Maximal Pseudo-Differential Operators	99
15.	Global Regularity of Elliptic Partial Differential Equations	105
16.	Weak Solutions of Pseudo-Differential Equations	109
17.	Garding’s Inequality	113
18.	Strong Solutions of Pseudo-Differential Equations	121
19.	One-Parameter Semigroups Generated by
Pseudo-Differential Operators	129
20.	Fredholm Operators	143
21.	Fredholm Pseudo-Differential Operators	155
22.	Symmetrically Global Pseudo-Differential Operators	167
23.	Spectral Invariance of Symmetrically
Global Pseudo-Differential Operators	177
Bibliography	181
Index	183

Chapter 1
Introduction, Notation and
Preliminaries
Let W1 be the usual Euclidean space given by
En = {(#i, £2,..., xn) : Xj’s are real numbers}.
We denote points in En by x,y,£,rj etc. Let x = (#1,^2,... ,xn) and
y = (2/172/2, - - -,2fn) be any two points in En. The inner product x • y of x
and y is defined by
n
* • y = E
3=1
and the norm \x\ of x is defined by
On En, the simplest differential operators are , j = 1,2,..., n. We some¬
times denote by dj. For reasons we shall see later in the book, we usu¬
ally find the operator Dj given by Dj = —idj, i2 = — 1, better in expressing
certain formulas.
The most general linear partial differential operator of order m on En
treated in this book is of the form
E	1D? ■Dn”,	(i-i)
<*i+0!2H	\-an<m
where c*i, «2,...,an are nonnegative integers and aalya2,...,an (x) is an in_
finitely differentiable complex-valued function on En. To simplify the ex¬
pression (1.1), we let
Oi — (c^i, OL2, •••, Oifi)j
1

2
An Introduction to Pseudo-Differential Operators, 3rd Edition
M = ai
3=1
and
The a, given by an n-tuple of nonnegative integers, is called a multi-index.
We call \x\ the length of the multi-index a. With the help of multi-indices,
we can rewrite our differential operator (1.1) in the better form
aa(x)Da.	(1.2)
|a|<m
For each fixed x in Mn, the operator (1.2) is a polynomial in D\, D%,..., Dn.
Therefore it is natural to denote the operator (1.2) by P(x,D). If we re¬
place D in (1.2) by a point £ = (£i,£2, • • • ,£n) in Kn, then we obtain a
polynomial E|c*|<min where £a = £“'£22	Naturally,
this polynomial is denoted by P(#,£). We call P(x,£) the symbol of the
operator P(x,D).
In this book, we shall study the partial differential operators (1.2) and
their generalizations called pseudo-differential operators. To do this, we
find it convenient to introduce in Chapters 2-5 certain aspects of analysis
pertinent to our need.
The following list of remarks, notation and formulas will be useful to
us.
(i)	We denote the set of all real numbers by M and the set of all complex
numbers by C.
(ii)	All vector spaces are assumed to be over the field of complex numbers.
All functions are assumed to be complex-valued unless otherwise specified.
(iii)	We do not bother to distinguish a function / from its value f(x) at
x. In other words, we shall occasionally use the symbol f(x) to denote the
function / without any warning.
(iv)	Although the differential operator Da = D^D^2 • • • D%n is more useful
to us, we still use the differential operator da = d“1 d%2 • • • d%n very often in
the book. In case we want to emphasize the variable x (or £) with respect
to which we differentiate, we write d% (or d£) for da and D% (or D^) for
Da.
(v)	We denote the set of all infinitely differentiable functions on W1 by
C°°(Rn).

Introduction, Notation and Preliminaries
3
(vi)	The Lp norm of a function / in Lp(Rn), 1 < p < oo, is denoted by
II/IIp-
Let a = (ai,a2,...,an) and P = (A,ft,...,ft) be any two multi¬
indices.
(vii)	p < a means that ft < aj for j = 1,2,..., n.
(viii)	a - P is the multi-index (<*i - ft,c*2 - ft,... ,an - ft) whenever
P < a.
(ix)	a! = aila^! * * * oin\.
W (p) = (£) (¾) • • • (TJ whenever
(xi)	The formula
£“(/$)=£ (a(1-3)
/3<a W
is known as Leibniz’s formula. It is a special case of the following more
general Leibniz’s formula.
(xii)	Let P{D) = £w<m aaDa be a linear partial differential operator
with constant coefficients, and P(Q its symbol. Then
P(D)(fg) = £ 1(^(0)/)(0^),
\ti\<m
where PM (D) is the linear partial differential operator with symbol P^ (¢)
given by
^(0 = (^)(0. (etf*.
(xiii) Let /eC“(r). Then
n*/T|_Vr	(^/)(^/).-(^/)
where Ca<i)>a(2),...>a(fc)’s are constants and the sum is taken over all possible
multi-indices	..., a^k\ which form a partition of a. The formula
(1.4) is valid at all points x in Rn for which f(x) ^ 0.
(xiv) Let / be a measurable function on Rn x Rn. Then for 1 < p < oo,
{[ \[ f(x>y)dy\	< [ {[ \f(xiy)\pdx\ dy. (i.s)
URn I JRn	|	)	J R» t JR"	J
This inequality is the well-known Minkowski's inequality in integral form.

4
An Introduction to Pseudo-Differential Operators, 3rd Edition
(xv) The inequality
|xa| < |*|W	(1.6)
for all x £ Rn and multi-indices a, which will be used quite often in this
book, is an inequality in terms of the absolute value of a real number, the
norm of a point in Rn and the length of a multi-index. Its proof is left as
an exercise.
Exercises
1.1. Find the symbol of each of the following partial differential operators
onR2.
(i)
a2 , a2
dx\ ' dx\
(») 6-6
M ^ + 6
(iv)a§7 + *6
(v)
dxi
557+^
d
dX2
1.2.	For each of the partial differential operators in Exercise 1.1, find the
zero set of the symbol, i.e., the set of zeros of the symbol.
1.3.	Find the symbol of the partial differential operator
P{X'D) = ^+X
ldx\
on R2. For each fixed x € R2, find the zero set {£ £ R2 : P(x, £) = 0}.
1.4.	What is the analog of Minkowski’s inequality in (1.5) when p = oo?
1.5.	Prove inequality (1.6).

Chapter 2
Differentiation of Integrals Depending
on Parameters
The aim of this chapter is to prove a theorem on how to differentiate an
integral depending on parameters in order to justify every interchange of
integration and differentiation throughout the book. I hope analysts can
find the theorem, or some variant of it, useful. Other criteria can be found
on, e.g., p.288 in [Friedman (1971)], pp.94-95 in [Royden (1988)], and p.85
in [Wheeden and Zygmund (1977)].
Theorem 2.1. Let (F,/x) be a measure space and f : Rn xY -* C be a
measurable function such that
(i)	/(x, •) € LX(Y) for all x in Rn,
(ii)	/(*,?/) E C°°(Rn) for almost all y in Y,
(iii)	sup^Kn fY \(9%f)(x,y)\dii < oo for all multi-indices a.
Then the integral JY f(x,y)dn, as a function of x, is in C°°(Rn) and
for all multi-indices f).
We begin with a lemma.
Lemma 2.2. Let f : W1 x Y -+ C be such that the hypotheses of Theo¬
rem 2.1 are satisfied. Then for any multi-index a and j = 1,2,..., n, the
integrals
and
as functions of Xj, are continuous on R.
5

6
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof Using the mean value theorem and hypothesis (iii), we obtain
J	-..,Xj + h,...,xn,y)\ ~
for all x in Rn, where
Ma,j= sup [ \(d%dXjf){x, y) \ dp,
xeRnJY
and hence the lemma.	□
Proof of Theorem 2.1 In view of of the proof of Lemma 2.2, the theorem
is valid for the zero multi-index. Suppose that the theorem is valid for any
multi-index with length l and let 7 be a multi-index with length i + 1. If
we write 7 as /3 4- e, where /? is a multi-index with length l, and € is a
multi-index with length one and the only nonzero entry in the jth. position,
then, by the fundamental theorem of calculus, Lemma 2.2 and the Fubini
theorem,
d7
f L f(x,y)dy
= 9, J^(d%f)(x,y)dy
= £“ \JY{(^f)(xi,---,Xj	xn,y) -
1, • • •	• • • ,a;„,y)} dy
= h™ol,JY {/
= lim ^ /	... ,s,... ,xn,y) dy^ ds
= f (%lf)(x,v)dy, a; € M.".
Thus, by induction, the proof is complete.
From the proof of Theorem 2.1, we obtain the following result.
Corollary 2.3. The conclusions of Theorem 2.1 remain valid if hypothesis
(iii) is replaced by the hypothesis that, for every multi-index a, the integral
/y ($?/)(#> 2/) dfi, as a function of x, is continuous on Rn.

Differentiation of Integrals Depending on Parameters
7
Exercises
2.1.	Let / be a bounded function defined on the strip
Q = {(xi,x2) : x\ € M, 0 < x2 < 1}
in M2 such that for each fixed X\ in M, the function f(xi, •) is measurable
on [0,1]. Furthermore, suppose that (9f f)(x\, x2) exists for all nonnegative
integers k and all (xi,x2) 6 Q, and for each nonnegative integer k, there
exists a positive constant Ck such that
1(^1/)(^1^2)1 < C*,	(xi,x2)eQ.
Prove that
(^1) L ^Xl,X2')dX2 = J (^1/)(^1^2)^2
for all xi € R
2.2.	Let / € L1 (M.n) and let g € C°°(lRn) be such that dag 6 L°°(ln) for
all multi-indices a. Prove that the function h on K" defined by
Hx) = / f(y)g(x - y) dy, x e R\
J R"
is in C°°(Rn) and
(dah)(x) = [ f(y)(dag)(x — y) dy, x&W1.
J Rn
2.3.	Let / be the function on R defined by
oo
/(*) = 53 cne*nx, X € R,
n=—oo
where ..., c_2, c_i, Co, ci, C2,... are constants such that
oo
$3 lc"l < °°-
n=—oo
Prove that, if
oo
53 I0"!71* < 00
n=—oo
for all nonnegative integers k, then / € ¢70°^) and for k = 0,1,2,..., we
have
oo
/(fc)(*)= 53 Cn«*eina, areR

This page intentionally left blank

Chapter 3
The Convolution
In this chapter we introduce two important subsets of C°°{Rn), usually
denoted by Cq°(W1) and S. The aim of this chapter is to prove that they
are dense in Lp(Rn), 1 < p < oo. To this end, we need the notion of
convolution.
Theorem 3.1. (Young’s Inequality) Let f £ L1(Wl) and g € Lp(Rn),
1 < p < oo. Then the integral
exists for almost every i G In. // the value of the integral is denoted by
(/ * 9)(x)i then f *g e Lp(W1) and
II/*»IIp<II/IIiIWIp-
Remark 3.2. We usually call f * g the convolution of / and g.
Proof of Theorem 3.1 For p = 1, let
h(x)= [ \f{x-y)\\g(y)\dy.
J Rn
Then, by Fubini’s theorem,
= IMIill/lli-
9

10
An Introduction to Pseudo-Differential Operators, 3rd Edition
Hence h(x) < oo for almost every x e Rn. Using Fubini’s theorem again,
we have
II/* plli =/	/ f(x-y)a{y)dy dx
JRn \JRn
<[ (/ l/(^-y)l|p(p)l%)rfa;
JRn \JRn	/
= (/ 19(y)\dy) ( [ \f(x — y)\ dx\
\JRn	/ \JRn	J
= IIpIIiII/IIi-
This proves the theorem for p = 1. For 1 < p < oo, let
M*) = / 1/(1 - v) I Ip(p)Ip%-
«/Rn
Then, as in the previous case, hp(x) < oo for almost every x G Rn. Let p'
be the conjugate index of p. Then, by Holder’s inequality,
f \f(x-y)\\g(y)\dy
JRn
= / 1/(31 — y)|1/,p" 1/(^ — 2/)|1/p|p(2/)| dy
JRn
<	!/(* - y)l dpj 1/(^ - p)| |p(j/)lp<fyj
= ll/lli/p'{M*)}1/p-	(3.1)
Hence fRn f(x - y)g(y)dy exists for almost every x € Rn. Moreover, by
(3.1), we have
11/ * Pllp = JRnf(x~y)9(y)dy| <2xj
-(/ (/ l/(x_y)llp(y)lrfp) dx\
Ultn \JRn	J J
< ll/lli/p
<ll/lli/p'{ii/llillG||i}1/p,
where G(x) = |ff(x)|,> for all i £ t". Hence
ll/*p||P<ll/llilMIP.

The Convolution
11
To prove the theorem for p = oo, note that
[ \f(x-y)\\g(y)\dy <\\g\\oo f |/(x-y)|dy = IMIooll/lli. (3.2)
JRn	JRn
Hence the integral fRn f(x — y) g(y) dy exists for every x G Rn. Moreover,
by (3.2), we have
11/*0 Hoc < Mlooll/lll-
□
Proposition 3.3. (Lp-Continuity of Translations) Let f G Lp(Rn),
1 < p < oo. Then
lim ll/x-/llp = °,
|aj|—>0
where fx is the function defined by
fx{y) = f(x + y), yet".
Before proving Proposition 3.3, let us define Co(Rn) to be the set of
all continuous functions on Rn with compact support. The support of a
continuous function h is defined to be the closure in Rn of the set
{x G Rn : h(x) £ 0}
and is denoted by supp(h). We give one property of the set Co(Rn) in the
following proposition.
Proposition 3.4. Co(Rn) is dense in Lp(Rn) for 1 < p < oo.
Proposition 3.4 is a measure-theoretic result which I ask you to believe.
Otherwise, see p.69 in [Rudin (1987)] for a proof.
Proof of Proposition 3.3 Let S > 0 and / € Lp(Rn). Then, by Proposi¬
tion 3.4, there is a function g in Co(Rn) such that
11/-»llp <§•	(3.3)
Now, using the triangle inequality and (3.3),
6	6 S
Wfx — /lip ^ II fx — 9x\\p + \\9x ~~ 9\\p + II# “ /Up <"3"^3"^3=<^
if \x\ is small enough. This completes the proof.
Theorem 3.5. Let cp G Ll(Rn) be such that
□

12
An Introduction to Pseudo-Differential Operatorst 3rd Edition
l/P
For e > 0, define the function <p£ by
(pe{x) — £~n(f	, X G
Then for any function f G Lp(Wl)11 < p < oo, we have
f *(pe-+ af
in Lp{En) as e -* 0.
Proof Since
I <pe (x) dx — a
for all e > 0, it follows from Minkowski’s inequality in integral form given
by (1.5) that we have
11/ *ipe- af\\p = (jf ^ 1(/ * <Pe)(x) - af(x)\pdxj
=	J^n{f(x-y)-f(x)}<Pe{y)dy^ dxj
= (j^ |j^{/(* - ey) -	dxj
</ {/ \f{x - ey) - f(x)\p\<p{y)\pdx\ dy
= / \<P(y)\{ [ |/(x - ey) - f(x)\pdx\ dy
J Rn	UR"	J
= / kp(»)l 11/-^-/My.	(3.4)
J R»
By Proposition 3.3, ||/_e2, — /||P -»■ 0 as e -¥ 0. Also, by the triangle
inequality, \\f-ey — f\\p < 2||/||p. Hence an application of the Lebsegue
dominated convergence theorem to the last integral in (3.4) implies that
||/ *<pt- af ||p -)• 0
as £ -¥ 0.	□
i/p
i
i/p
We introduce two important function spaces. It is customary to denote
by C^E71) the set of all infinitely differentiable functions on Rn with com¬
pact support and by S the set of all infinitely differentiable functions ip on
Rn such that for all multi-indices a and /9,
sup \xa{D^p)(x)\ < 00.
x€Rn

The Convolution
13
The space S is usually called the Schwartz space in deference to Laurent
Schwartz. Obviously, C§° (W1) is included in S. That the inclusion is
proper can be seen easily by noting that the function e~W is in S but
not in Co°(Mn). We want to prove that Co°(En) is dense in Lp(Wl) for
1 < p < oo. To this end, we need two preliminary results.
Proposition 3.6. Let ip € S and f € Lp(En), 1 < p < oo. Then
En)
and
d«(f*v) = f*(d«v)
for every multi-index a.
Proof Let p G S. Then for every multi-index a, dacp e Lp (En), where p'
is the conjugate index of p. Hence, by Holder’s inequality, (/ * (dap))(x)
exists for every x G En. Therefore differentiation and integration can be
interchanged.	□
Proposition 3.7. Let f and g be continuous functions on En with compact
support. Then the convolution f *g also has compact support. In fact,
supp(/ *g)c supp(/) + supp(g).
Remark 3.8. Let A and B be subsets of En. Then the vector sum A 4- B
is defined by
A + B = {x + y:x€A and y € B}.
In fact, the vector sum can be defined for any two subsets of a vector space.
Proof of Proposition 3.7 Since
(/ * 9)(x) =/ f(x - y)g(y) dy,
Jnn
it follows that if (/ * g){x) ± 0, then there exists a ye supp(^) such that
x — y G supp(/). Hence, by Remark 3.8,
x e supp(/) + supp(^).
This completes the proof.
Theorem 3.9. Co°(En) dense in Lp(En) for 1 < p < oo.
□

14
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof Let (p £ Co°(Rn) be such that
I (p(x)dx = 1.
JRn
Such a function exists by Exercise 3.2. For e > 0, define <p£ by
<p£(x) = £~nip	, X £ En.
Then for all functions g £ Co (Rn), we have, by Propositions 3.6 and 3.7,
g*(p£ £ Co°(Mn). Also, by Theorem 3.5,
9*<Pe->9	(3.5)
in Lp(Rn) as e -¥ 0. Let S > 0 and / £ Lp(Mn). Then, by Proposition 3.4,
there is a function g £ Co(Mn) such that
11/-0llp <§•	(3-6)
By (3.5), we can find a function ip £ Co°(Rn) such that
llfl - V’llp <	(3.7)
Hence, by the triangle inequality, (3.6) and (3.7), we have
11/ - i>\\p < 11/ - slip + lb - V’llp <\ + \=5-
This proves that C7q°(IRT1') is dense in Lp(Rn) for 1 < p < oo.	□
Remark 3.10. An immediate consequence of Exercise 3.3 and Theorem
3.9 is that the Schwartz space S is also dense in Lp(Rn) for 1 < p < oo.
Exercises
3.1.	Let (p and (pe be the functions given in the hypotheses of Theorem 3.5.
Let / be a bounded function on Rn which is continuous on an open subset
V of Rn. Prove that f *<pe	a f uniformly on every compact subset of V
as e ->• 0.
3.2.	Let ip be the function on Rn defined by
Prove that tp £ Co°(Rn).

The Convolution
15
3.3.	Prove that every function in S is in Lp(Rn), 1 < p < oo.
3.4.	Is S dense in L°°(En)? Explain your answer.
3.5.	Determine whether or not the Lp-continuity of translations is true for
p = oo.
3.6.	Use Minkowski’s inequality in integral form in (1.5) to prove Young’s
inequality for 1 < p < oo.

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Chapter 4
The Fourier Transform
The Fourier transform will be used in Chapter 6 to define pseudo-differential
operators. Two important results in the theory of Fourier transforms are the
Fourier inversion formula for Schwartz functions in S and the Plancherel
theorem for functions in L2(Rn). They are very useful for the study of
pseudo-differential operators.
Proof By Theorem 3.1, f *g G L1(Mn). Then, using the definition of the
Fourier transform, we have
Let / G Lx{IT). We define f by
The function / is called the Fourier transform of / and is sometimes
denoted by Tf.
Proposition 4.1. Let f and g be in L1(Mn). Then
f*g = (2n)n'2fg.
= (27r)-"/	f f{x-y)g(y)dy]dx.
JRn	\JRn	/
17

18
An Introduction to Pseudo-Differential Operators, 3rd Edition
So, by Fubini’s theorem, we get
(27:)-^/¾)
= (2ir)~n f f e~i<'x~y^'if(x-y)e~iyig(y)dydx
Jun JRn
= (27r)_n f e~ty ig(y) ( f e~t(x~yHf(x - y) dx\ dy
jRn	\JRn	/
= (2tt)-"/2 ([Rne-ivMy)dy) m=mm-
□
Proposition 4.2. Let	Then
(i)	(Da(p)A(£) = €a<p(Q for every multi-index a,
(ii)	(D0<p)(£) = {{—x)^(p)A(^) for every multi-index /3,
(iii)	(f> € S.
Proof Integrating by parts, we get
(D»A(0 = (27t)-"/2 [ e-ix<(Day){x)dx
J Rn
= (2tt)-"/2 f Ce~ix^(x) dx = em-
JRn
This proves part (i). For part (ii), we have
(D*<p) (0 = (27r)~n/2D0 (jf e~ix ^(x)dxj
= (27r)-"/2 f (-x)t}e-ix *<p(x)dx= ((-x)0v)A (0-
JRn
The interchange of the order of differentiation and integration is valid be¬
cause (-x)^(f E 5. To prove part (iii), let a and /? be any two multi-indices.
Then, by parts (i) and (ii),
m^xoi = r((-*yV)A(o i = k^((-^)V)}a(oi-
Since Dot((-x)l3(p) is in 5, hence in L1(Wl), it follows that
sup 1^(^0)(01 = sup |{J?°((-*)M}A(0l
£6Rn	£€R"
< (277)-^11^((-0^)111 < 00-
□
That the Fourier transform turns functions in Lx(IRn) into continuous
functions vanishing at infinity is the content of the following proposition.

The Fourier Transform
19
Proposition 4.3. (The Riemann-Lebesgue Lemma) Let f e L^K71).
Then
(i)	f is continuous on Rn,
(ii)	limkKoo /(0 = 0,
(iii)	fj -+ f in L1(Rn) =½ fj f uniformly on Rn.
Proof Let fj -+ f in L1(Rn). Then
1^(0 - /COI <	- /|U.
Hence fj —> f uniformly on Mn. This proves part (iii). To prove parts (i)
and (ii), let p € <S. Then, by part (iii) of Proposition 4.2, p £ S. Hence
parts (i) and (ii) are satisfied for functions in S. Let / G L1(Rn). Since S is
dense in Ll(Rn), it follows that there is a sequence {pj} of functions in S
such that Pj -+ f in L1(Rn). By part (iii) which we have proved, p] -+ f
uniformly on Rn. This proves parts (i) and (ii).	□
Let / be a measurable function defined on Rn. For any fixed y e Mn,
we define functions Tyf and Myf by
(Tvf)(x) = f(x + 2/), x € Rn,	(4.1)
and
(Myf)(x)=eixvf(x), ier.	(4.2)
Let a be a nonzero real number. Then we define the function Daf by
(Daf)(x) = f(ax), x € Rn.	(4.3)
Proposition 4.4. Let f e L^E"). Then the functions Tyf,Myf and Daf
defined by (4.1), (4.2) and (4.3) respectively are in Ll(Mn). Moreover,
(i)	(Tvfno = (Mjm ir,
(ii)	(Myfno = (71,/)(0, e g Rn,
(iii)	(Daf)A(0 = \a\ n(D1/o/)(^),	£ e Kn.
Proof Obviously, Tyf,Myf and Daf axe in L1 (Rn). By a simple change

20
An Introduction to Pseudo-Differential Operators, 3rd Edition
of variable, we have
(Tyf)A(0 = (27T)-"/2 f e~ix't(Tyf)(x)dx
JRn
= (2tr)-"/2 / e~ix<f{x + y)dx
jRn
= (2tt)-”/2 / e~^-^f(x)dx
= eiy i(^)~n/2 f e~ix<f(x)dx
J Rn
= (^y/)(0-
Also,
(Af,/)A(0 = (2tt)-”/2 f e-ix<{Myf)(x)dx
JRn
= (2tt)"”/2 / e~ix ieiv xf{x)dx
JRn
= fit - y)
= (r_,/)«).
Finally, by another change of variable, we have
(A./m = (2tt)-"/2 / e-ix<(Daf)(x)dx
J Rn
= (2tt)~n/2 f e-ix*f(ax)dx
J Rn
= (2x)“n/2 / e-i(5)'€/(*)|orn<fa
= |o|-"(i?1/a/)(0.
Proposition 4.5. Le£ </>(#) = e l®l2/2. T/aen <£(£) = e l£l2/2.
Proof We first compute
(2?r)"n/2 f e~ix't-\Xfdx.
J Rn
□

The Fourier Transform
21
Note that
r	i n	r°°
(2?r)-n/2 / e-ix<-^dx = T\(2n)-^2 e~ix^-xidxj. (4.4)
JRn	jJl	J —	OO
Hence it is sufficient to compute
/OO
e~lt<>~t2dkt, (, e (—00,00).
-00
But
/00	pOO
e-it(-t2 (ft = I e-V +^dt
-00	J—OO
/00
e-(t2+it(-(,c2/i))dt
-00
= e"^4 /°° e-(‘+i«/2))2dt
J —OO
= e~(*2^ f e~z*dz,	(4.5)
where L is the contour Imz	= | in the complex z-plane. Using Cauchy’s
integral theorem and the fact that the integrand goes to zero very fast as
\z\ -+ 00, we have
f e~z2dz= j	e~t2 dt = \pK.	(4.6)
J L	J—oo
Hence, by (4.5) and (4.6),
/oo
e-^dt = 2-1^^4.	(4.7)
-OO
By (4.4) and (4.7), we get
(27r)~n/2	[	e~ix't-W2dx = 2-n/2e_|(Jl2/4.	(4.8)
J Rn
Now, note that
(27r)-"/2 e-to-€-d*l2/2>dx = (D1/y^p)A (¢),
where ¢(x) = e"^2. Therefore, by (4.8) and part (iii) of Proposition 4.4,
we get
¢(0 = (27r)-n/2 f e-ix-t-W2Mdx = e”^2/2,
JRn
as asserted
□
Proposition 4.6. (The Adjoint Formula) Let f and g be functions in
L\Rn). Then
/ f(x)9(x) dx = / f(x)g(x) dx.
JRn	JRn
(4.9)

22
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof By Proposition 4.3, the Fourier transform of a function in L1(En)
is bounded on Rn. Hence the integrals in (4.9) exist. Moreover,
f f(x)g{x)dx = (27r)_n/2 [ (f e~lx'vf(y)dy\ g(x)dx
JRn	JRn \JRn	J
= (2tt)-"/2 f f(y) ( f e-ix vg(x)dx) dy
JRn \J Rn	J
= f f(y)g(y) dy.
JRn
The interchange of the order of integration can obviously be justified by
Fubini’s theorem.	□
We are now prepared to prove the first important result in the theory
of the Fourier transform.
Theorem 4.7. (The Fourier Inversion Formula) (/)v = f for all func¬
tions f € <S. Here, the operation V is defined by
g(x) = (2tt)-”/2 f eix tg(Odt, g&S.
J Rn
Remark 4.8. The function g is usually called the inverse Fourier trans¬
form of g.
Proof of Theorem 4.7 We have
(/)v(x) = (2x)-"/2 f eix<mdt.
JRn
Let e > 0. Define
Ie(x) = (2ir)-n/2 f ^-(^1=72)/(^.
J R"
Let
g(0 = e^€-(-2i€l2/2) = (MxDeym,
where
Then, by Propositions 4.4 and 4.5,
g(V) = (T-x£~nDi/e<p)(r)) = £-ne-^2^
(4.10)
(4.11)
(4.12)
(4.13)

The Fourier Transform
23
Hence, by (4.10), (4.11), (4.13) and Proposition 4.6,
Is(x) = (2tt)-"/2 f g(0f(0 (%
J Rn
= (271-)-^/ g(T])f(ri)dri
JRn
= 6-^(277)-^2 / e-l”-*l2/(2£2>/(77)d77
JRn
= (27r)-"/2(/*¥>,)(*),	(4.14)
where ip£(x) = £_rV ( f) • Since / G 5, it follows that / is in Lp(Rn),
1 < p < oo. Therefore, by (4.12), (4.14) and Theorem 3.5,
Ie -► (27r)-n/2 (^ e-l*'2/2^) / = /
in Lp(Mn) as e -» 0. Hence there exists a sequence {en} of positive real
numbers such that I£n{x) f(x) for almost every x G Rn as en 0. By
(4.10) and Lebesgue’s dominated convergence theorem,
I'(x)->(27T)-n/2 f e*<f(0dt
J Rn
for every x G Mn as e -» 0. Hence
(2^)-"/2/ eto«/(0 # = /(*)
«/Rw
for every x € Rn, and the proof is complete.	□
Remark 4.9. An immediate consequence of the Fourier inversion formula
is that the Fourier transformation / —> f is a one to one mapping of S onto
<S. If we define / by
/(*) = /(-*)>
then the Fourier inversion formula is equivalent to the formula
/=/,	/€5.
The next important result is the Plancherel theorem.
Theorem 4.10. (The Plancherel Theorem) The mapping f / de¬
fined on S can he extended uniquely to a unitary operator on L2(Rn).

24
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof Using the fact that S is dense in L2(Rn) and the Fourier inversion
formula, it is sufficient to prove that
ll^lb = IMh, e S.
Let *ip be the function defined by
^p(x) = </?(-#), x£Rn.	(4.15)
Then rp eS and
= (2tt)-"/2 / e^^xjdx
J Rn
= (2tt)“n/2 / eix S^x)dx
J Rn
= m-	(4.16)
Thus, by (4.15), (4.16), the adjoint formula and the Fourier inversion for¬
mula,
\mi= [
JRn
= /
J Rn	JRn
= / v(OV’d) <*£ = / ¥>(-£M-£)
J Rn	J Rn
= / <P(OW)d(, = Ml
JRn
and this completes the proof.	□
Remark 4.11. The Plancherel theorem states that the Fourier transform
of a function in L2(Rn) can be defined. If / G L2(Rn), then we shall denote
its Fourier transform by / or Tf. The inverse of T : L2(Rn) -¥ L2(Rn) is
of course denoted by : L2(Rn) -> L2(Rn).
Exercises
4.1.	Here is another elegant proof of Proposition 4.5.
(i) Let ip be the function defined on R by
<p{x) = e-*2/2
, xeK.

The Fourier Transform
25
Let y = <p. Prove that
v'(0 + ftK0 = o, sen
(ii) Use the result in part (i) to prove that
£(S) = e-«2/2, Sen
4.2.	Let {ipn} be the sequence of functions defined on R by
for all x G R and n = 0,1,2,	We call ipn the Hermite function of order
n.
(i) Prove that
for all x e R and n = 0,1,2, —
(ii) Prove that = i~n<pn for n = 0,1,2, —
4.3.	Prove that if /6 L1(Mn) and / G Ll(Rn), then (/)v = / a.e.
4.4.	Find a function / in Ll{R*) such that / is not in L1(En).
4.5.	Prove that for all functions / in Lx(Mn),
tp0(x) = e *2/2
and
<Pn+1(*) = *y>n(*) - ¥>»(*)
/(0 = /(-0,
4.6.	Prove that for all functions / in L1(Mn),
/ = /•
4.7.	Prove that T : 5 ->> 5 is bijective.

This page intentionally left blank

Chapter 5
Tempered Distributions
We only give the rudiments of the theory of tempered distributions in this
chapter. More details on this subject will be introduced in later chapters
as the need arises.
Definition 5.1. A sequence {cpj} of functions in the Schwartz space S is
said to converge to zero in S (denoted by (pj —> 0 in S) if for all multi-indices
a and /3, we have
sup \xa(D0<fij)(x)\ -»• 0
x€Rn
as j -¥ oo.
Definition 5.2. A linear functional T on S is called a tempered distribution
if for any sequence {ipj} of functions in S converging to zero in S, we have
T(<Pj) 0
as j —» oo.
Definition 5.3. Let / be a measurable function defined on Rn such that
l/(*)l
/
Jr*
rdx < OO
/r- (1 + M)"
for some positive integer N. Then we call / a tempered function.
Proposition 5.4. Let f be a tempered function defined on W1. Then the
linear functional Tf on S defined by
Tf(<p)= [ f(x)ip(x)dx, <p€S,
J Rn
is a tempered distribution.
27

28
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof Let N be a positive integer such that
!/(*)!
(i + |z|)"
dx < oo.
Then for all functions ip e <S, the integral fRn f(x)ip(x) dx exists. Indeed,
we have
f \f(x)\\<p(x)\dx
J Rn
= X-(iTRF(1 + w)''l,’w|,ix
s (/,. (TTSF*) ,s6Tf(1 + W’''w,)l) < “•
Let {ifj} be a sequence of functions in S converging to zero in S as j -+ oo.
Then obviously,
Since
sup {(i + IsD^W*)!} ->o.
\Tf(<fij)\< [ \f(x)\\<pj(x)\dx
J Rn
(5.1)
(5.2)
for all j, it follows from (5.1) and (5.2) that Tf(ipj) -+ 0 as j -+ oo. □
Proposition 5.5. Let f G Lp(En), 1 < p < oo. Then the linear functional
Tf on S defined by
Tf(>p) = f f(x)<p(x)dx, tpeS,
JRn
is a tempered distribution.
Proof Note that / is a tempered function. See Exercise 5.1.	□
Remark 5.6. It is customary to identify the tempered distribution Tf with
the function / and to say that such tempered distributions are functions.
Definition 5.7. Let T be a tempered distribution. Then the Fourier trans¬
form of T is defined to be the linear functional T on S given by
TV) = T(£), tpeS.

Tempered Distributions
29
Proposition 5.8. T is also a tempered distribution.
Proof Let {ipj} be a sequence of functions in S converging to zero in S.
We need only prove that the sequence (¾} also converges to zero in S. To
this end, let a and /3 be any two multi-indices. Then, by Proposition 4.2,
sup \C{D09j)(O\
€eRn
= sup i£“((-zyVj)A(oi
= Sup nx>“((-*yVj)}A(oi
< (27r)-"/2||D“((-x)Vi)lli-	(5.3)
Since (fj —> 0 in S as j oo, it follows that for any positive integer N, we
have
sup {(1 + 1^)^1(^((-^))(3)1} -*• 0	(5.4)
xeRn
as j —► oo. Since for any positive integer N greater than n, we have
11(^((-^)11,
< sup {(1 + |x|)w|(D“((-x)V))(3)l} /(1 +	(5.5)
xeRn	JRn
for all j. Hence, by (4.3), (4.4) and (4.5), we conclude that (pj -> 0 in S as
j -¥ oo.	□
Theorem 5.9. (The Fourier Inversion Formula) Let T be a tempered
distribution. Then
T = T,
where T is defined by
T(<p) = T{<p),	<p€S.
Proof Let <p € S. Then, by Definition 5.7 and the Fourier inversion formula
for S, we have
t{<p) = Ttf) = T(<p) = T(0) = T{fp).
□

30
An Introduction to Pseudo-Differential Operators, Zrd Edition
Exercises
5.1.	Prove that any function in Lp(Mn), 1 < p < oo, is a tempered function.
5.2.	Let S : S -> C be the mapping defined by
S(ip) = <p(0),	(p£S.
(i)	Prove that 5 is a tempered distribution.
(ii)	Prove that S is not a tempered function. (See Remark 5.6.)
5.3.	Let f £ Ll{Wl) and T be the tempered distribution which is equal to
/. Prove that T is equal to /.
5.4.	Do Exercise 5.3 again for f G L2(Mn).
5.5.	Find the Fourier transform of the tempered distribution S defined in
Exercise 5.2.
5.6.	Prove that there exist positive constants C\ and (¾ such that
¢71(1 + If!) < (1 + |d2)1/2 < C7a(l + |d), £ € K*.

Chapter 6
Symbols, Pseudo-Differential
Operators and
Asymptotic Expansions
In this chapter we give the definition and the most elementary properties
of a pseudo-differential operator and its symbol.
We begin by recalling that a linear partial differential operator P(x, D)
on Rn is given by
P(x,D)= £ aa(x)Da,	(6.1)
| a | <ra
where the coefficients aa (#) are functions defined on Rn. If we replace the
Da in (6.1) by the monomial in Rn, then we obtain the so-called symbol
p(*,o = £ M*)r	(6.2)
I O'| <771
of the operator (6.1). In order to get another representation of the operator
P(x,D), let us take any function ip in S. Then, by (6.1), (6.2), Proposition
4.2 and the Fourier inversion formula for Schwartz functions, we have
(P{x,D)tp)(x) = £ aa(x)(Datp)(x)
| O'| <777
= aa(x)(D°v)v(x)
\a\<m
= £ oa(*)(r£)v(s)
I a I <777
= £ oa(a:)(27r)-n/2
|a:|<m
[ e***Fmdt
JRn
= (2tx)~n'2 f eix*P{x,t)mdt.
JRn
(6.3)
31

32
An Introduction to Pseudo-Differential Operators, 3rd Edition
So, we have represented the partial differential operator P(x, D) in
terms of its symbol by means of the Fourier transform. This represen¬
tation immediately suggests that we can get operators more general than
partial differential operators if we replace the symbol P(x, £) by more gen¬
eral symbols <r(x,£), which are no longer polynomials in £. The operators
so obtained are called pseudo-differential operators. We shall do this in
due course. Meanwhile, we should point out that in order to get a useful
and tractable class of operators, it is necessary to impose certain conditions
on the functions cr(x,£). Many different sets of conditions have been pro¬
posed, resulting in many different classes of pseudo-differential operators.
Our discussion in this book is first restricted to the following class.
Definition 6.1. Let m G (—00,00). Then we define Sm to be the set of
all functions <t(x,£) in C°° (Rn x Rn) such that for any two multi-indices
a and /?, there is a positive constant (?<*,/?, depending on a and P only, for
which
\(D%Dl*)(x,01 < ¢7^(1 + |f|)TO-M € Kn.
We call any function a in UmGRSm a symbol
Definition 6.2. Let <7 be a symbol. Then the pseudo-differential operator
Ta associated to a is defined by
(!»(*) = (27T)-n'2 [ eix<o(x,0<p(0dt, <peS. (6.4)
JRn
We give some examples.
Example 6.3. Let P(x, D) = £|o|<m aa (x)Da be a linear partial differen¬
tial operator on W1. If all the coefficients aa(x) are C°° and have bounded
derivatives of all orders, then the polynomial
p(x,o= £
|a|<m
is in Sm and hence P(x, D) is a pseudo-differential operator. (See formulas
(6.3) and (6.4).)
Proof Let 7 and S be multi-indices. Then
\(D2D*P)(x,0\ < £ <?a,7|3fn
|a|<m
for all 1, { € ln, where
Ca,7 = sup |(L>7aa)(a:)|.
*€Rn
(6.5)

Symbols, Pseudo-Differential Operators and Asymptotic Expansions
33
(6.6)
It can be shown easily that
<*<<*,
otherwise,
for all £ G Mn. The proof of (6.6) is left as an exercise. See Exercise 6.5.
Hence, by (1.6), (6.5) and (6.6),
\(D2D*P)(x,Q\< £ Ca,75!(“WH«l
< <?;,*(!+lar-141
for all x, £ G Mn, where
c;,»= E
|a|<m	V 7
Example 6.4. Let <r(£) = (1 +|£|2)m/2, — oo <m< oo. Then cr G Sm and
hence Ta is a pseudo-differential operator. It is sometimes advantageous
to denote Ta by (/ - A)m/2, where I is the identity operator and A the
Laplacian, i.e.,
A A 92
A
j=i
Proof We need only prove that for any m G (—oo, oo) and multi-index /?,
there exists a positive constant Cm,p such that
K^)(oi <cm^(i+Kir-i«	(6.7)
for all £ G Rn. (6.7) is obviously true for the zero multi-index. Suppose
that (6.7) is true for all m G (—oo, oo) and multi-indices /3 of length at most
equal to l. Let 7 be a multi-index of length l +1. Then DT = D^Dj for
some j = 1,2, ..., n and some multi-index /3 of length l. Hence
|(I>7ff)(OI = \(d%a)(0\ = 1(0^)(01	(6.8)
for all £Gln, where
r(0=n^(l + Kla)(m/2)_1
for all £ e Mn. Therefore, by Leibniz’s formula,
(d^m = m £ (f) (d%)^~s{(l + Ifl2)*"*/2)-1}	(6.9)
d<0 ' 7

34
An Introduction to Pseudo-Differential Operators, 3rd Edition
for all £ G Rn. Hence, by (6.9) and the induction hypothesis, there exists a
positive constant Cm,p such that
i(0V)(s)i < cm3 £ (f) (i++Kir-2-w+i*i
= cJ}(i+Kir-171
for all £ € I71, where
cm,0 = cm,0Yl
6</3
Thus, by (6.8) and the principle of mathematical induction, the proof is
complete.	□
We give two very simple properties of pseudo-differential operators.
Proposition 6.5. Let a and r be two symbols such that Ta — Tt. Then
a = r.
We first prove a lemma.
Lemma 6.6. Let f be a continuous tempered function such that
Tf(<p) = 0,	(6.10)
Then f is identically zero on Rn.
Proof Without loss of generality, we assume that / is real-valued. Suppose
f(xo) 0 for some Xo € Rn. Then there is an open ball B(xo,r) with
center xo and radius r on which / is strictly positive (or negative). Choose
a nonzero function <po € Co°(Rn) such that <po(aO > 0 for all x 6 Rn and
supp(<£o) C B(xo, r). Such a function exists by Exercise 3.2. It is clear that
ipo € S and
Tf(<Po)= / f{x)cpo(x)dx
«/Rn
is strictly positive (or negative). This contradicts (6.10).	□
Proof of Proposition 6.5 By hypothesis and Definition 6.2, we have
[ eix<{<x(x,(,)-t(x, €)}$(£) d£ = 0, <p 6 S.
JRn
Since T : S —» S is one to one and onto by the Fourier inversion formula,
it follows that
f eix<{a(x, 0 - t(x,	= 0,	<p&S.
JR”

Symbols, Pseudo-Differential Operators and Asymptotic Expansions
35
Now, for any fixed x € W1, etx ^{a(x, £) - r(x, £)} is a continuous tempered
function in the variable £. This follows easily from the definition of a symbol
in Definition 6.1. Hence, by Lemma 6.6,
e**^{<r(x,4) — r(x,£)} = 0
for all £ £ Rn. Since x is fixed but arbitrary, it follows that
<r(a;,£) -t(x,€) =0
for all x, £ £ W1. This proves that a = r.	□
Proposition 6.7. Let a be a symbol. Then Ta maps the Schwartz space S
into itself.
Proof Let <p € S. Then for any two multi-indices a and /?, we need only
prove that
sup \xa(D^(Ta(p))(x)\ < oo.
xeRn
But, using integration by parts and Leibniz’s formula, we have
x^D^T^ix)
= x“(27r)-"/2 f D0{eix<a(x,O}mdt
JRn
= xa(2n)'~7
= (27T)-n/2
= (2tt)-"/2(-i)|“| f y,
= (27T)-n/2(-l)W f £	(f\eixHD^-SDt^)(x,0
Dl(t?m)dt.	(6.11)
Using (6.11) and the fact that a is a symbol, say a £ Sm, we can find
positive constants	depending on a, /?,7 and <5 only, such that
sup \xa{D0{Taip)){x)\
x€Rn
< E E (!) (/ <'+Kir-'*1'1
\DKem)\dt.
E r)ceixHDt^)(x, mtw
7</3 '''
/„ E (J)rpfefa-€)(^-v)(*,f)0(Ode
(6.12)

36
An Introduction to Pseudo-Differential Operators, 3rd Edition
Since ip € <S, it follows from (6.12) that
sup \xa(D0(Taip))(x)\ < 00,
and hence the proof is complete if we can justify the interchange of the order
of differentiation and integration in (6.11). But, using the same argument
as in the derivation of (6.12), we see that the last integral in (6.11) is
absolutely convergent. This completes the proof.	□
Remark 6,8. In general, a pseudo-differential operator does not map
Co°(Rn) into Co°(Rn). For a way of showing this, see Exercise 6.4.
An important notion in the theory of pseudo-differential operators is
the asymptotic expansion of a symbol.
Definition 6.9. Let a e Sm. Suppose we can find gj € Sm’, where
m = mo > mi > m2 > • • • > rrij -> -oo, j oo,
such that
N-1
a - £ Oj € Sm"	(6.13)
j=0
for every positive integer N. Then we call YlJLo °j an asymptotic expansion
of a and we write
oo
a ~ ^crj.
j=o
An important result in this connection is the following theorem.
Theorem 6.10. Let mo > vn\ > m2 > * * • > nij -+ -00 as j 00.
Suppose oj e Smj. Then there exists a symbol a G 5m° such that
00
3=0
Moreover, if r is another symbol with the same asymptotic expansion, then
0 — t £
Proof Let if € C°°(En) be such that
0<V>(£)<1> £eEn,
^(0 = o.

Symbols, Pseudo-Differential Operators and Asymptotic Expansions
37
and
¢(0 = 1, 10 >2.
That such a function exists will be proved at the end of this chapter. Let
(¾} be a sequence of positive numbers such that
l > £q > £\ > £2 > * * * > Sj y 0
as j ->• 00. Define the function 0 on Rn x Mn by
00
<r(,x, 0 =	¢(^0^-(*, 0, *, I e Mn.	(6.14)
1=0
Note that for each (x0,£o) E ln x f, there exists a neighborhood U of
(#o j £0) and a positive integer N such that
(*,0 = 0
for all (x,£) G U and j > N. Hence a G C00^71 x Rn). Furthermore, for
any e G (0,1] and nonzero multi-index a,
¢(¢0=0, 10 <\,
¢(¢0 = 1, 10 >§,
^{¢(¢0) = e|a,W)(e0 = 0,	|0 < -e or |0 >
and
|afWeO}l<^W, £6®",
where
Ca = SUp | (^¢)(01-
«eRn
If i < |£| < |, then e < -||| < Hence, for any nonzero multi-index a,
we have
i^rwe0}i<^iai(i+ior|o|=ci(i+ior|01, £€»% (6.15)
where C'a = Ca4laL It is obvious that (6.15) is also true for the zero multi¬
index a. Now, using (6.15), Leibniz’s formula and the fact that crj G Sm>,

38
An Introduction to Pseudo-Differential Operators, 3rd Edition
we can find positive constants Ca 7 and C^7 such that
= |0?MeO(i?jto)(*,O}|
<
7<a V /7
E + l£l)-,a|+,7|Cj,0,7(1 + KIH-W
y<a V
E	+ior-|a|
= CW(1 + I0)_1(1 + I0)mi+1-|a|	(6.16)
for all x,£ 6 E", where £,>,/? = £7<Q (“JCa^C^,,. Now, we choose et
such that
Cj,c,,0£j < 2~j	(6.17)
for all multi-indices a and /3 such that |a +/?| < j. By the definition of ip,
we have
</>(e,0 = 0	(6.18)
whenever 1 + |£| < ej1. Hence, by (6.16), (6.17) and (6.18),
1^-^(^0^,0)1 < 2_i(l + |^|)mi+l-|a|	(6.i9)
whenever i,(ef and \a + fi\ <j. Now, for any two multi-indices a0 and
A), we take j0 so large that jo > |a0 + Po\ and mjo + 1 < mo- Write
JO — 1	oo
*(*>0 = E ^(ejOo'iCaf.O + E ^0^,0
J=°	J=J0
= /(®,0 + J(*,0.	(6.20)
Since 7(x,0 is a finite sum, it follows that / e Sm». By (6.19),
|(£>^°£>f0J)(x,01 < E 1^^(6,0^,0)1
3=30
oo
<	52 2_j(! + k|)m»'+1-l“ol
3=30
oo
<	52 2_i(i+io)m<>-i“oi
3=30
= 2_-7o+1(1 + |0)«o-l*ol.
(6.21)

Symbols, Pseudo-Differential Operators and Asymptotic Expansions
39
So, J is also in Sm°. Hence, by (6.20), <j G iSm°. We need to verify condition
(6.13). To do this, we write
TV-1
- Y <7j(x,0
3=0
oo	TV— 1
=	- y
j=o	j=0
N-1	oo
= Y	+ Y
i=o	i=TV
As before, we can show that
oo
Y	€ Smiv.
j=N
Since
- 1 = 0, j < AT - 1,
if l£l >	, it follows from Exercise 6.3 that
TV—1
Y - ftofoO € nm€R5m
i=o
and consequently
TV-1
v-Y°i€SmN-
j=o
Finally, if r is another symbol such that r ~	then
£ gmN
for every positive integer N. Since tun —> — oo as iV oo, it follows that
¢7 — r G nmeR5m.
This completes the proof of the theorem.	□
We have used the following result in the proof of Theorem 6.10.
Proposition 6.11. There exists a function ^ G £7°°(En) such that
0 < V>(0 < 1,
^) = 0, iei < l.
and
m = i, ki>2.
TV-1
TV-1
a — r =
«■ - S <*i
-
T~Yai
©
II
©
II

40
An Introduction to Pseudo-Differential Operators t 3rd Edition
Proof We need only construct a function <£o £ Co°(Mn) suc^ that
o<MO<h ^eRn,
M0 = h Kl<i,
and
M0 = o, Kl>2.
For then the function ?/> = 1—<po will satisfy all the conditions of Proposition
6.11. To construct <po, let / be any continuous function on Rn such that
0 </(*)<!, t e Rn,
and
/(*) = !.	1*1 < 2»
/(*)= 0. 1*1 >4-
Let <p € Co°( Rn) be a real-valued and nonnegative function such that
Ifi(s) = 0,	\s\ > -.
and
/
J l«l<i
<p(s) ds = 1.
(6.22)
That such a function ip exists is an immediate consequence of Exercise
3.2.	Let (p0 = f * (p. Then, by Propositions 3.6 and 3.7,	€ Co°(Rn).
Furthermore, using the location of the supports of / and <p, we see that
Mt) = 0, \t\ > 2.
Finally, for \t\ < 1,
(po(t) = / f{t- s)ip(s)ds = / f(t- s)ip(s)ds.
JRn	J\«|<4
(6.23)
JRn	J\s\<\
Since, for \t\ < 1 and \s\ < we have \s - t\ < f and hence f(t — s) = 1.
Therefore, by (6.22) and (6.23),
Mt) = 1, 1*1 < 1-
Again, by (6.22) and (6.23), we can prove easily that
0<<po(*)<l, *eRn.
This completes the proof of the proposition.	□
Remark 6.12. For another proof of Proposition 6.11, see Exercise 6.6.

Symbols, Pseudo-Differential Operators and Asymptotic Expansions
41
Exercises
6.1.	Prove that if a E Smi and r E 5m2, then err E Sm 1+m2.
6.2.	Prove that if cr E Sm, then D^D^a E	for all multi-indices a
and /?.
6.3.	Let a be any symbol and p any function in S. Prove that the function
r defined by
r(x,0 = <r(x,Ov>(0» x’t e E”>
is a symbol in nm€R5m.
6.4.	Let a be the symbol defined by
<r(0 = e-'«l2/2
for all £ € ]Rn. Show that the pseudo-differential operator Ta does not map
Cg°(«”) into CjFiW1).
6.5.	Let a and 5 be any two multi-indices. Prove that for all £ G R",
S < a,
otherwise.
6.6.	(i) Let p E Co°(E) be such that p(x) > 0 for x E M, p(x) = 0 for
x £ [-2, -1], and
/oo
p(x) dx — 1.
-oo
Define a function \ on (—oo,0] by
X(0=f <p(x)dx, £ € (—oo,0].
Prove that x is infinitely differentiable on (—oo,0),
O<X(0<1,	£ € (—oo,0],
X(0=O, I <-2,
and
x(0 = i, Ce[-i,o].
(ii) Let p E Cq°(E) be such that p(x) > 0 for x E R, p(x) = 0 for x ¢ [1,2]
and

42
An Introduction to Pseudo-Differential Operators, 3rd Edition
Define a function x on [0? °°) by
x(0 = j <p(x) dx, £ € [0, oo).
Prove that \ is infinitely differentiable on (0, oo),
O<X(0<1,	¢€[0,00),
x(0 = 0, £>2,
and
x(0 = i, ¢€[0,1].
(iii) Construct a function ip € C°°(Kn) such that
0 < ^(¢) < 1, ¢6^,
¢(0 = 0, 10 <1,
and
¢(¢) = 1, 10 >2.
6.7.	Let a £ C°°(Rn x W1) and let m £ (—00,00). Prove that a £ Sm if
and only if for all multi-indices a and /?, there exist positive constants Ca,p
and Ra,p such that
\(D%Dfa)(x,Q\ < 0,3(1 + Kir"1*1, 10 > Ra,0-

Chapter 7
A Partition of Unity and Taylor’s
Formula
It is convenient to devote a chapter to several technical results which will
be of particular importance for us in the next two chapters. In Theorem
7.1 we construct a partition of unity. Then we use this partition of unity
to decompose a symbol a{x,£) into a family {0* (#,£)} of symbols with
compact support in the £ variable. We are able to obtain good estimates
on the partial Fourier transforms (with respect to the £ variable) of all
the symbols (Jk{x, £). The precise estimates are given in Theorem 7.2. In
Theorem 7.3 we prove a multi-dimensional version of Taylor’s formula with
integral remainder.
We begin with the construction of a partition of unity.
Theorem 7.1. There is a sequence	of functions in Co°(Mn) such
that
(i)	0<Wk(£)<l,	£ E Rn, fc = 0,1,2,...,
(ii)	E2Low(f) = i, 4er\
(iii)	at each £ E Mn, at least one and at most three of the tp'ks are nonzero,
(iv)	supp(^o) C {£ € R" : |0 < 2},
(v)	supp(^) C {£ € R” : 2k~2 < |0 < 2fc+x}, k = 1,2,...,
(vi)	for each multi-index a, there is a constant Aa > 0 such that
sup KdVjfeHOI <AQ2~kW, k = 0,1,2,....
$€Rn
Proof We pick i/io to be any function in Co°(Rn) such that
O<*>(0<1, £ G
*>(0 = 1, I0<1,
and
*>(0=o. 10 >2.
43

44
An Introduction to Pseudo-Differential Operators, 3rd Edition
For the existence of such a function, see Proposition 6.11 and Remark 6.12.
Let ip be any other function in C§°(Mn) such that
m = i < i€i < 2,
and
¢(0 = 0, |0 < ^ or \0 > 4.
The proof that such a function ip exists is left as Exercise 7.1. Now, for
k = 1,2,..., define ipk by
and then define by
oo
¢(0 = £^*(6, ^€Rn-
k=o
Obviously, we have
supp(V>o) c {£ € E” : |£| < 2}
and
supp(V»fc) c {£ € E" : 2*-2 < |£| < 2*+1}
for k = 1,2,	For each £ € K", the series defining ¢(0 contains at most
three nonzero consecutive terms. This implies that for each 0 £ Rn, there
exists a neighborhood U of 0 and a positive integer N such that ipk(0 = 0
for all £ € U and k > N. Hence V £ C°°(Rn). It is easy to see that for
each £ £ 1", the series defining ¢(0 contains at least one nonzero term.
Hence ¢(0 # 0 for all £ 6 IT. Now, for fc = 0,1,2,..., we define ipk by
¥>*( 0 =
ipkiO
¢(0 ’
f € Rn.
It is easy to see that <pk £ C{j°(Rn) for k = 0,1,2,... and the sequence
{VfcJtLo satisfies the first five conditions. Hence it remains to prove that
the last condition is true for <pk, k = 1,2, — To do this, we note that for
each multi-index a, we have, by Leibniz’s formula,
(ffv*)(0
= E (¾ {«* (¢) ®}	(2^)	<”>

A Partition of Unity and Taylor's Formula
45
Now, for each multi-index /?, by formula (1.4),
d*3
(i)- £
73^),...,0(1)'
¢/+1
(7.2)
00)+-+/3(0=/3
where ¢73(1) ^(0 is a constant and the sum is taken over all possible multi¬
indices /3(1),..., /?(*) which partition /3. Next, for every multi-index 7, there
exists a constant C1 > 0 such that
l(97*)(£)l < C72“*W	(7.3)
for all £ £ supp(<£fe) and k = 0,1,2,	Let us assume (7.3) for a moment.
Then, by (7.2) and (7.3), there exist positive constants Cp(i),..., Cpw such
that
(i)
tt)
- 1^0(0,...,0(01
^(1)+...+^0)=^ 1 vs/l
(7.4)
< C£2"*l/J|
for all £ G supp(^ft), where
=	1^0(0 ,...,0(01 |C^d) ---^0(1)1-
/3(1)+...+0(0=0
Hence, by (7.1) and (7.4), there exist positive constants Ca,p such that
|(9<V*) (¢)1 < 5] (2)^2-^1(7^/,2-^-1)^ =i4a2-fcl“l
0<a V*/
for all fc = 0,1,2,... and all £ € E", where
0<O!
It remains to prove (7.3). To do this, we consider three cases.
Case 1: Suppose k = 0. Then for all £ € supp(<po), £ is in supp(^) for
some j = 0,1,2 and hence
2
¢(£) = ^o(0 + 53 ^f(0-
3=1
Therefore
(9^)(0 = (9^o)(£) + 53^) (2P1) 2
j=i v y
-(i-i)lil

46	An Introduction to Pseudo-Differential Operators, 3rd Edition
for all £ € supp(<p0)- Thus, there is a constant C7 > 0 such that
|(d7$)(OI < Cy, £ € supp(v>o)-	(7.5)
Case 2: Suppose k = 1. Then for all £ G supp(v?i), £ is in supp(^) for
some j = 0,1,2,3 and hence
3
*(© = *>(© + !>*(©•
J=1
Therefore
(^¢)(0 = (d^0)(£) + E(»V) (^pr) 2-0-1)^
j=1 V 7
for all £ G supp((pi). Thus, there is a constant C" > 0 such that
|(d7*)(£)l < C;[2|T| + 1 + 2-1^1 + 2-2!7']
= c;[22'71 + 2^ + 1 + 2_|7|]2“|71	(7.6)
for all £ € supp(v?i).
Case 3: Suppose k > 2. Then for all £ € supp(^fe), £ is in supp(V’j) for
some j = k - 2, k - 1, k, k + 1, k + 2 and hence
¢(0 = £ *,(©.
j=k-2
Therefore
A+2	/ £ \
(^¢)(0 = £ (0V) (2pr) 2-(j_1)l71
j=fc-2	'	7
for all f € supp(^fc). Thus, there is a constant C" > 0 such that
< C"[2~(k~3^ + 2-^-2^7^ + 2“^-1^7' + 2~k^ + 2”(*+1^7!]
= (7"[23I7I + 22'71 + 2^ + 1 + 2"|7|]2"*|7I	(7.7)
for all £ G supp(<pife). Hence, by (7.5), (7.6) and (7.7), (6.3) follows and the
proof of Theorem 7.1 is complete.	□
Let a G Sm. For k = 0,1,2,..., we write
<r*(s,0 =0^,0^(0
(7.8)

A Partition of Unity and Taylor’s Formula
47
for all x, £ E Rn and
Kk(x,z) = (2n)~nV [ e^Or.fldf	(7.9)
for all i,2GEn, where {(£&} is the partition of unity constructed in Theo¬
rem 7.1.
Theorem 7.2. For a// nonnegative integers N, and multi-indices a and /3,
there exists a constant A, depending on m, n, N, a and /3 only, such that
f \z\N\{d%d?Kk)(x,z)\dz < A2^m+
jRn
for all k = 0,1,2,
In the proof of Theorem 7.2, we make use of the following inequality
given by
\z\2N <nN |*>f, z€ir.
M=JV
(7.10)
The proof of (7.10) is left as an exercise. See Exercise 7.2.
Proof of Theorem 7.2 Let 7 be any multi-index. Then, by (7.8), (7.9),
Plancherel’s theorem, Proposition 4.2, Leibniz’s formula, parts (iv) and (v)
of Theorem 7.1,
/'
JR*
zJ(d^d^Kk)(x,z)\2dz
=/
Jwk
7'<7 V ' '
2
di, (7.11)
where
W0 = {f 6 R" : |f | < 2}
(7.12)
and
Wk = {f € Rn : 2k~2 < |f| < 2*+1}
(7.13)
for k - 1,2,— Hence, by (7.11), part (vi) of Theorem 7.1 and the fact
that €a(d§<r) is a symbol in Sm+M, we get positive constants Ca,/3,y and

48
An Introduction to Pseudo-Differential Operators, 3rd Edition
C7>7/ such that
f \z1{d^Kk){x,z)\2dz
jRn
< j (y)cW(l + I^l)m+lal-l7,l^'7i7'2~fcl7-7,l | de-
(7.14)
Hence, by (7.12), (7.13) and (7.14),
f \z'r(dgd?Kk)(ic,z)\2dz
J Rn
< J (fycajt,Y2(fe+2)(m+l“l-IVI)(77i7(2-fc|7-Vl| dj
^7#<7
_ 2*(2m+2|a|-2|7l)
:(i8>y<77)y22(TO+l“HVI) ( (¾
2fe(n+2m+2| a | - 2171)
^yC77,y22(m+l“l-lV
’}•
(7.15)
where C„ is the constant with the property that the volume of the ball in
Rn with radius r is equal to Cnrn. Let Aa.dn.m.n be defined by
2
n2” jE (y)^./3.7'^,7'22(m+H-|VI)| •
Then (7.15) becomes
f |Z7(^0“^fc)(a:,2)|2d2<^>%m)n2*(n+2m+2|a|-2|7l).	(7-16)
J Rn
Let N be any nonnegative integer. Then, by (7.10) and (7.16),
f \z\2N\(d^Kk)(x,z)\2dz
JR"
<nN / \^(^Kk)(x,z)\2dz
< ^412^n+2m+2lal“2iV)

A Partition of Unity and Taylor’s Formula
49
for all k = 0,1,2,..., where
M=N
I X/2
-N)k
By taking square roots, we get
{/ \z\2N\{d^d^Kk){x,z)\2dzy < ,422(("/2)+m+|<*
for k = 0,1,2,..., where A2 = A}/2. Now write
/ kn(^a?^)(*,z)|&= /	+ /
JR"	V|z|<2-fc	«/|z|>2-fe
(7.17)
(7.18)
'|z|<2-fc	«/|z|>2-
By (7.17) and the Cauchy-Schwarz inequality, there is a constant A3 > 0,
depending on m, n, AT, a and /? only, such that
1/2
dz!
>l<2"fc
< A32((n/2)+m+lal~N)k2~nk/2
= A32(m+lal~N)k	(7.19)
f <{[ \zn(d0d?Kk)(x,z)\>dz\1/2 ( /
J\z\<2~k	U Rn	J [j\z\
for k = 0,1,2,	Next, by (7.17) and the Cauchy-Schwarz inequality
again, there is another constant A4 > 0, depending on m, n, N, a and jS
only, such that
L
|z|>2-
<	y^_k\z\-2ndz^
<	AA2«n/V+m+M-N-n)k |y £ r-2nrn_1drdcr) 7 ,
where da is the surface measure on the unit sphere 5n_1. Hence
1/2
< A42^n/2)+m+'a*-'/v-n)fe|Sn-1|1/2rc'"1/22nfe/2
= ^2(^+1«!-^)*
(7.20)
for k = 0,1,2,..., where A5 = ^4|Sn_1|1/2n_1/2 and |5n_1| is the surface
area of 5n_1. Hence, by (7.18), (7.19) and (7.20), we complete the proof of
Theorem 7.2.	□
The following version of Taylor’s formula with integral remainder plays
an important role in the study of pseudo-differential operators.

50
An Introduction to Pseudo-Differential Operators, 3rd Edition
Theorem 7.3. Let f £ ¢7^(Rn). Then for all positive integers N,
/(£ + v)
- £	+	£ 21 f\i-0)"-Hay)K + »v)<ie
|a|<iV	|l|=JV 7‘ J°
(7.21)
for all £, 77 £ Rn.
Proof The proof is by induction on N. For N = 1, we need to prove that
m+v) = m + E S / V/)«+h m (7.22)
171=17- Jo
for all £, 77 £ Rn. To do this, we define a function (p : R C by
V?(x) = /½ + xrj), x £ R.	(7.23)
Then, by the fundamental theorem of calculus,
V(l) =^(0)+ [\f(t)dt.	(7.24)
./o
Hence, by (7.23), (7.24) and the chain rule, we get the formula
m+v) = ho+E fvsftit+*•»)»»*. e E">
7 = 1 70
which is exactly the same as (7.22). We now assume that (7.21) is true for
the positive integer N. Then, by the induction hypothesis, we have
/<£ + *)- £
a\
= /(1 + 9)- E
a|</V+l
(dam)„ v (dafm„
a! v ^	ot\ V
a\<N	|a|=N
= N E Hf t (1 - 9)^(^m + 0v)de- E
|7|=iV	|a|=AT
for all f, 7] £ Rn. Now, using the fact that
iV [ (1-0)^40 = 1
Jo
(7.25)

A Partition of Unity and Taylor’s Formula
51
and the formula (7.21) for N = 1, the formula (7.25) becomes
/«+•>>- £
|o|<N+l
al
= E	0)"_1{(d7/)(£ + Ov) - (dy)(t)}de
ItHv7' 70
= £	E {f(^+,/)« + ^)
M=JV 7'	|«|=l	-	J
|7|=JV
Let p = </90 in (7.26). Then
/«+■>)- £ 2¾¾°
(7.26)
M<jv+i
a:
|7|=JV 1(51=1 r’ *	l 0	J
(7.27)
By interchanging the order of integration in (7.27), we have
/({+D- £
|a|<AT+l
= £ £^w/V+V)«+OT){/,(i-0)"_,<»}<if>
|-y|=iV |tf|=l 7’ ‘	^	^
7+<5	/»l
= E E ^r/0 (^/)½+w)(i-p)Ar4p.	(7.28)
|7|=JV|«S|=1
For all multi-indices 7 and <5 with |7| = N and |5| = 1, we have
(7 + &)'■
7W!
= 7-5 + 1,
(7.29)
where 7 • S is the inner product of 7 and 5. Hence, by (7.28) and (7.29),
(dafm.
fit+v)~ El
|a|<JV+l
al
-V
7,7+^	r1
= E E (7^(7 • <5+1) jf (9^/)½+^)(1-^.
|7|=JV|i|=l
(7.30)

52
An Introduction to Pseudo-Differential Operators, 3rd Edition
For each multi-index a of length iV + 1, we can write a in the form 7 + 5,
where |7| = N and |5| = 1 by picking
7= (<*1,£*2,- 1,...,On)	(7.31)
and
5 = (0,0,...,1,...,0)	(7.32)
in which S has 1 in the jth position whenever aj > 1. Then, by (7.31) and
(7.32), we have
7 • S = aj — 1
(7.33)
whenever ctj > 1. Hence, by (7.30) and (7.33),
/(4 + 1)- E
|a|<JV+l
= E E / (9“/)(£+wX1
|a|=iV+lQj>l a*
= (^ + 1) E ^j\dam + (ro)(l-p)Ndp.
(7.34)
|o|=N+l
Therefore (7.21) is also true for the integer N + 1. Hence, by the principle
of mathematical induction, the proof of Theorem 7.3 is complete. □
Exercises
7.1.	Construct a function ?/> in Co°(l.n) such that
0<lKO<l, £eRn,
m = h i<ki<2,
and
¢(0=0,	|{| < i or |f | > 4.
7.2.	For any nonnegative integer AT, prove that
Iz\2N<nN £ l*T
hl=jv
for all z G Mn.

A Partition of Unity and Taylor’s Formula
53
7.3.	Let / G C00^71) be such that
sup \(daf){x)\ < oo
xeRn
for all multi-indices a. Prove that for every positive integer AT, there exists
a positive constant Cn such that
/(*)- £
H<jv
(dvm,
a! *
<	x €

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Chapter 8
The Product of Two
Pseudo-Differential Operators
In this chapter we prove that the product (or composition) of two pseudo¬
differential operators is again a pseudo-differential operator. We also give
an asymptotic expansion for the symbol of the product. The main result is
the following theorem.
Theorem 8.1. Let a G Smi and r G S™2. Then the product TaTr of
the pseudo-differential operators Ta and Tr is again a pseudo-differential
operator T\, where A is in Sm 1+m2 and has the asymptotic expansion
A~£^^(^)(w	(8-1)
Here (8.1) means that
A- £ ^r-(%°)(dST)
is a symbol in 5mi+m2-JV for every positive integer N.
To motivate the proof, let us begin with an argument to find out how
we should proceed. For any function tp in <S, we have
(T*k<p)(x) = (2tt)-”/2 f e***akMmdt
Jrn
for all x G Kn, where
and {pk} is the partition of unity constructed in Theorem 7.1. Hence
f^T^Xx) = (2Tr)""/2 f eix< (£ ak{x,¢) 1 <p($) ^
*=o	■'R" lfe=o	)
= (27r)-"/2/ eix*cr{x,Ov(0<%
Jr*
55

56
An Introduction to Pseudo-Differential Operators, 3rd Edition
for all x G tn. The interchange of YlkLo and JK« can of course be justified
by Fubini’s theorem. Hence
oo
k=0
and the convergence of the series can be shown to be absolute and uniform
for all x e W1. Our goal is to compute the symbol of TaTr. But let
us first compute the symbol of T«kTr- You will see why in a minute. Let
(p € S. Then, by the definition of a pseudo-differential operator, the Fourier
transform, Fubini’s theorem and (7.9), we get
{Ta„Tr<p)(x)
= (2<Tn/2 [ eix^k(a:,0(TTv)*(Qdt
J Rn
= (2*)~n [ eix<ak{x,£)\ f e~*y (Tr<p)(y)dy\d£
JRn	lJRn	J
= (27T)-n f (f ei(x-^<ak(x,Odt\ (Trv)(y)dy
JRn UR»	J
= (27r)-"/2 f Kk(x, X - y)(Tr<p)(y) dy	(8.2)
J Rn
for all x e W1. Hence, by the definition of a pseudo-differential operator
and Fubini’s theorem again, the formula (8.2) becomes
0^(Tk^rP) (%)
= (27r)“n f Kk(x,x-y) ( f eiy^T{y,r})p(ri)dq\dy
JRn	UR»	J
= (2tt)~n [ etxr> ( f e~t{x~y'>T>Kk(x,x-y)T{y,T})dy\<p(ri)dr)
JRn	KJRn	J
= (277)-^/ eixr)\k{x,Tj)if>{ri)dr]
JRn
for all x £ Mn, where
A*(a?,77) - (2ir)~n/2 f e~t<'x~v'> ”Kk(x,x - y)r(y,i])dy.
JRn
By a simple change of variable,
•^*(x, »7) — (27r)-n/2 f e~tz ,,Kk(x,z)T(x - z,rj)dz (8.3)
J R»
for all 2, 77 6 Wl. This suggests that
(IVT^Xx) = (2t7)-"/2 f eix n(x, Tj)ip{rj) dn

The Product of Two Pseudo-Differential Operators
57
for all iGln, where
oo
A(®, »?) = £ A* (a:, rj)	(8.4)
k=0
for all x,rj £ Mn. Hence all we need to do is to show that A(x,rj), defined
by (8.3) and (8.4), is a symbol in Smi+m2 an(j satisfies (8.1).
Remark 8.2. Had we begun with (TaTTip)(x) instead of (TakTT(p)(x),
we would have the divergent integral fRn e^x~y^ t<j(x,£)d£ instead of
fRnez(x~y)'t<Jk(x,Qd£ in (8.2). This is why we cut off the symbol cr(x,£)
in the £ variable by the partition of unity {<£>&}.
Proof of Theorem 8.1 For k = 0,1,2,..., we define A* by
Afe(x,0 = (27r)“n/2 f e~tz<Kk{x,z)r(x - z,£)dz (8.5)
JRn
for all x, £ £ En. Now, by the Taylor’s formula with integral remainder
given in Theorem 7.3, we get
t(x-z,0= £	(8.6)
ImKWi
where
Rn1(x,z,0 = N1 £ ^r~ A1 -0)Ni~H^t)(x -6z,0<W	(8.7)
N=/Vi	Jo
for all x, z,£ £ En. Replacing r(x - z,£) in (8.5) by the right hand side
of (8.6), and using (7.8), (7.9), Proposition 4.2 and the Fourier inversion
formula, we get
A*(*,0 = £	(d^k)(x,0(d^T)(x,0+	(*,0,	(8.8)
ImK^i
where
T$(x,£) = (2n)~n/2 [ e iz<Kk(x,z)RNl{x,z,£)dz (8.9)
J Rn
for all ar, £ £ W1. For any positive integer N, the function A given by (8.4)
satisfies
A- £
= A_ £ (^W(^)(^r)+	£
(8.10)

58
An Introduction to Pseudo-Differential Operators, Zrd Edition
where Ni is any integer larger than N. Obviously,
£	^7-(d^a)(d^T) € Smi+ma-Jv.
N<\ti\<Ni
Hence, if we can prove that for all multi-indices a and /?, there exists a
constant Ca^ > 0 such that
\ DaxDl
1 (*,0
l
\p\<Ni ^
J
< Ca,p(l + |fl)-i+-»-*-l0l	(8.11)
for all a;,£ G Rn, then we can conclude that A is in Smi+m2 and has an
asymptotic expansion given by (8.1). To this end, we first note that, by
(7.8), (8.4) and (8.8),
A- £ tD^(^amr) = f)T«.	(8.12)
\p\<Ni	k=0
Then for any two multi-indices a and /3, we need to estimate D%D^T$
for all k = 0,1,2,	We have the following estimate.
Lemma 8.3. For all nonnegative integers M, there exists a positive con-
stant	such that
\(D%DfT$)(x ,01 < Cajt.tiM1 + \Z\)m*-2M2(m'+2M-Nl)k	(8.13)
for all x, £ G Rn and k = 0,1,2, —
Let us assume Lemma 8.3 for a moment. Then for all positive integers iV,
and multi-indices a and /3, we can choose a positive integer M such that
(1 + |£|)m2-2M < (i + |£|)mi+m2-AM/?l	(8.14)
for all £ G Rn. With this M fixed, we can choose another positive integer
N\ so large that
mi+2M-N1< 0.	(8.15)
By (8.12)-(8.15),
D^DflX- ^-^(^)(^))1(^0
W<Ni
< Ca,$M,Nl{l + |^|)mi+ra2-Af-|0l2(rai+2W-^)fc
k=0
= CaS{ 1 + (¢1)-1+-^-101

The Product of Two Pseudo-Differential Operators
59
for all x,£ G Mn, where
oo
Cajt = Ca,0,M,NlYi 2(m'+2M-N'^k.
□
It remains to prove Lemma 8.3. To this end, we need another lemma.
Lemma 8.4. Let iJjVi(x,2,£) be the function given by (8.7). Then for all
multi-indices a, /3 and 7, there exists a constant ¢^,/3,7 > 0 such that
for all x,Zj£ G Rn.
The proof that Lemma 8.4 implies Lemma 8.3 is by Leibniz’s formula,
integration by parts, Exercise 4.5 and Theorem 7.2. We leave it as an
exercise. See Exercise 8.1.
Proof of Lemma 8.4 By (8.7), we have
{d%d%RNl)(x,z,£)
= ¾ V	f\i -O)N'-l(dZ+»d0iT)(X -6z,0d6 (8.16)
N=w. Jo
for all x,z,£ G Rn. Hence, by (8.16) and Leibniz’s formula,
for all G En. So, by Exercise 4.5, (8.17) and the fact that r G Sm2,
there exist positive constants C7> and	such that
7'<7
(d2c%dfRNl)(x,z,o
(arr-y+a+M^T)(a, _ 0Z,
(8.17)
|(«afi?7Vl)(x,z,OI
M=Wi 7' <7
7;<7
7712 —|/31
(1 + KI):

60
An Introduction to Pseudo-Differential Operators, 3rd Edition
for all	where
Ss{G)cvc“Ay'}
M=Ni
and this completes the proof of Lemma 8.4.
□
Exercises
8.1.	Prove that Lemma 8.4 implies Lemma 8.3.
8.2.	Let
P(x,D)= Y1 a«(x)Da
\a\<mi
and
Q(x, D) = Y, ba{x)Da,
\ot\<m2
where the aa's and ba’s are in C°° (W1) and all their partial deriva¬
tives are bounded functions on Rn. Compute the symbol of the product
P(x,D)Q{x,D) directly. Compare the answer with the symbol obtained
by Theorem 8.1.
8.3.	Let q € C°°(Rn) be such that
sup \(Daq)(x)\ < oo
for all multi-indices a. Let a be the symbol defined by
o(x, £) = q(x)
for all x,f € K". Let r be any other symbol. Use Theorem 8.1 to compute
the symbols of the operators TaTr and TTTa.
8.4.	Let cr 6 Smi and r € Sm2. Prove that the symbol of the pseudo¬
differential operator TaTr — TTTa is in Smi+m2-1.

Chapter 9
The Formal Adjoint of a
Pseudo-Differential Operator
We begin with a notation. For any pair of functions ip and ^ in S, we define
by
Let a be a symbol in Sm and Ta its associated pseudo-differential op¬
erator. Suppose there exists a linear operator T* : S -¥ S such that
Then we call T* a formal adjoint of the operator Ta. It is very easy to see
that a pseudo-differential operator has at most one formal adjoint. Three
problems arise.
(1)	Does a formal adjoint exist?
(2)	If it exists, is it a pseudo-differential operator?
(3)	If it is a pseudo-differential operator, can we find an asymptotic expan¬
sion for its symbol?
The aim of this chapter is to prove that the formal adjoint of a pseudo¬
differential operator exists and is a pseudo-differential operator. Moreover,
we can obtain a useful asymptotic expansion for the symbol of the formal
adjoint. To be more precise, let us prove the following theorem.
Theorem 9.1. Let a be a symbol in Sm. Then the formal adjoint of the
pseudo-differential operator Ta is again a pseudo-differential operator Tt,
where r is a symbol in Sm and has the asymptotic expansion
(9.1)
(Taip,¢) = (<p,T*il>), ip,i>eS.
(9.2)
t(*,£) ~	^ ,?i	(^^)(^0-
U/,
II.	'
(9.3)
61

62
An Introduction to Pseudo-Differential Operators, 3r<* Edition
Here (9.3) means that
m<n v-
is a symbol in Sm~N for every positive integer N.
Before the proof, let us show how the symbol r can be constructed. For
& = 0,1,2,..., we define ak and Kk by (7.8) and (7.9) respectively. Then,
by the definition of a formal adjoint,
= (<P,T;^)	(9.4)
for all ip and ^ in S. By (9.1), Proposition 4.4 (ii), Proposition 4.6 and the
definition of a pseudo-differential operator,
where
= f {Takip)(x)il)(x) dx
J R»
= (2ir)~n/2 f {[ eix nak(x,Ti)<p(Tj) drA ip(x) dx
= (27r)_n/2 f if ai{x,y- x)<p(y) dj/| rp(x) dx,
JRn \JRn	>
(9.5)
&k(x,y) = (2ir)-n/2 f e~iy'v<7k{x,f?)dt)	(9.6)
J R«
for all x,y € En. Therefore, by (7.9), (9.5), (9.6) and Fubini’s theorem,
(^V* VO
= (2n)~n/2f if Ok(x,y-x)i>(x)dx\ip(y)dy
JR» l JR"	J
= (2ir)~n/2 f if Kk(x, x - y)ip(x) da:} <p(y) dy (9.7)
Jr* ur"	)
for all tpiip e S. Therefore, by (9.4) and (9.7),
CC*V0(z) = (2ir)_n/2 f ~K^(y,y - x)i>{y) dy	(9.8)
JR*
for all x € En. Hence, applying the Fourier inversion formula to the function
tf) on the right hand side of (9.8), Fubini’s theorem and a change of variables,

The Formal Adjoint of a Pseudo-Differential Operator
63
we have
(T;k4>)(x)
= (2*)-»
f Kk(y,y-x)\ I ety'nip(T]) dr] 1 dy
JRn KJRn J
= (27T)-"
[If eiy vK^(y,y - x) dyl ¢(7)) d7)
JRn URn )
= (2n)~n
f eix'n | f eiv (-y~x)~Kk(y,y — x) dyl ¢(7)) dr)
JRn URn J
= (2*)"»
f etX Tt{[ etTI zKk(x + z,z) dz\^(rj) drj
jRn lJRn )
for all x E En. It is clear from (9.9) that
T;k=TTk,	(9.10)
where
rk(x,7)) = (2n)~n/2 [ etv'zKk(x + z,z)dz.	(9.11)
J R»
Since
oo
(2^,^) = ^(2^,^)
k=0
for all (p, *0 £ £, it is clear from (9.4) and (9.10) that a good candidate for
r is given by
oo
r(x, v) = n (x, V)	(9-12)
k=0
for all x, f) e Mn. Hence it remains to prove that r is a symbol of order m
with an asymptotic expansion given by (9.3) and
= ivM)	(9.13)
for all </?, ^ £ S.
Proof of Theorem 9.1 For fc = 0,1,2,..., define T& by (9.11). Let N\ be
any positive integer. Then, by the Taylor’s formula with integral remainder
given in Theorem 7.3,
Kk(x + z,z) = ^ —x{d^Kk)(x,z) + R^l{x,z),
ImKJVi ^
(9.14)

64
An Introduction to Pseudo-Differential Operators, 3rd Edition
where, by (7.9),
r(n! (z. z)
N1 E 7T t(l-O)N'-\d>!Kk)(x + 9z,z)d0
ImNJVi -7°
iVi E / (1 — 6)Nl_1 (2,K)~n/2 f eiz Hd2ak)(x + dz,0dZM.
y° •/Rn
(9.15)
Hence, by (7.9), (9.11), (9.14) and Proposition 4.2,
n<*,,>= ^ (-ow"~—	- ■
l/*l<JVi ^
;re
T$(x, r,) = (2^)-n/2 f e^R^(x, z) dz
J Rn
all x,r] e En. For any positive integer AT,
r_ v
,,,
(9.16)
(9.17)
. £
n\ x V
I »\<N	^
= r- E ^rW + E	<»•"»
iVj is any integer larger than JV. Obviously,
E
we can prove that for all multi-indices a and /?, there exis
Cat& > 0 such that
i exists a
E {-ddd^A\(x,n)
{	M<* ^ JJ
. aC\ -I- Inh^-^-l^l
< ca,0( i + ii?ir-w-
Rn, then we can conclude that r € Sm and has an asymptotic
iven by (9.3). To this end, we first note that, by (7.8), (9.12)
*•- E t^w=E«.
I^KJVi	k=o
(9.19)
(9.20)

The Formal Adjoint of a Pseudo-Differential Operator
65
Let a and () be any two multi-indices. Then, by (9.17) and an integration
by parts,
(DZD0T$)(z,t,)
= (27T)-"/2J eir,'zZP	(x, z) dz
= (1 + M2r*(27r)-"/2 ^ e^(l - A,)* [z*	{x,z)}dz,
(9.21)
where K is any positive integer. Let P(D) = (1 — A)K. Then, by (9.15),
Leibniz’s formulas and an integration by parts,
(1-A2)*{z^[d{^) (x,z)}
r. ___	(3	/*1
= (1 - &z)KNx V —_ / (1-0)^-1(270-^2
M=*. Jo
f (_oMe--«(d"+"<Tfc)(z + 0z,() dttdB
JRn
= JVi s
|mI=JVi |«|<2K	' Jo
f (—)d£dO,
J R“
(9.22)
where
(.•.) = (-01°1 £ (fV* p](Dze iz i)(Di~pd2+,lak)(x + 9z,0- (9.23)
p<6
Let 7 be any multi-index. Then, by (9.22), (9.23), an integration by parts
and Leibniz’s formula,
(l-A,)*{*/» (DZR\#)(*,*)}
fll'-'lpw)-"/2 (***)<»,	(924)
where z7(* * *) is equal to
£.-V0M,_ i,w£(2)
{DJ (£pDsx~Pd*+,1a)}(x 4- 9z^)(Dy~'r2k)(0 d£.	(9.25)

66
An Introduction to Pseudo-Differential Operators, 3rd Edition
Using the fact that a is a symbol in Sm, (9.25) and Theorem 7.1 (vi), we
can find a constant C > 0, depending on 7',p,<S,a,p but not on k, such
that
|z7(***)l< / V ¢(1 HH£|)ro+'pHV'2-fch'-vl<i£) (9.26)
Jwk ^
where Wo and Wjfe, k = 1,2,..., are given by (7.12) and (7.13) respectively.
Let M be any positive integer. Then, by (9.26) and Exercise 7.2, there is
another constant C > 0, depending on M, p, 8, a, fjt but not on fc, such that
|(* * *)| < C\z\~2A*2fe(m+W-2M+n).	(9.27)
Hence, by (9.24), (9.27) and Exercise 9.5, there is a constant C > 0, de¬
pending on a, /?, K, M, iVi but not on k, such that
\a-Az)K{ze (d:<)(x,Z)}|
<	i E \z\m+Nr-\s\[2Hm+2K-2M+n)^	(9.28)
l|i|<2AT	J
We choose K so large that
(1 + \n\)-2K < (1 +	(9.29)
for all rj £ Rn. Then we choose M so large, say, equal to M\ that
m + 2K - 2M' + n < 0.	(9.30)
Then we choose Ni so large that
/ \z\-2M'< E	1
\6\<2K	)
dz < 00.
(9.31)
Hence, by (9.28) and (9.31), there exists a constant C\ > 0, depending on
a, /3, K, M', N\ but not on k, such that
J |(1 - A2)* {z0	(a;,z)}| dz < <712k(m+Mf-2M'+n>.
'M<i
Now, we choose M so large, say, equal to M", that
m + 2K- 2M" + n < 0
and
(9.32)
(9.33)
/	|z\-2M" | E M|/J|+JVl_|{| 1 dz < oo.
|j£|<2JC J
(9.34)

The Formal Adjoint of a Pseudo-Differential Operator
67
Hence, by (9.28) and (9.34), there exists a constant C2 > 0, depending on
a, /3, K, M", Ni but not on k, such that
J |tt - &z)K {z0 (¾¾ (*,2!)} | dz
< C2 2k(<m+2K-2M"+n).	(9.35)
Hence, by (9.21), (9.32) and (9.35), we get another constant C > 0, de¬
pending on a, /3, K, M', M", Ni but not on k, such that
<	_|_ |7/|2)“K{2*(m+2^-2M,+n)	(9.36)
Hence, for any two multi-indices a and /3, we have, by (9.20), (9.29), (9.30),
(9.33) and (9.36), a constant > 0 for which (9.19) is valid for all
x,rj £ Rn. Therefore the function r defined by (9.12) is a symbol in Sm
and has an asymptotic expansion given by (9.3). That (9.13) is true should
be by now obvious. At any rate, it is a simple consequence of Theorem 7.1
(ii), (7.8), (9.4), (9.10) and (9.12).	□
Exercises
9.1.	Prove that a pseudo-differential operator has a unique formal adjoint.
9.2.	Let a and r be any two symbols. Prove that
(t;y = ra
and
(TaTry = t;t;.
9.3.	Let P(x,D) = J]|Q|<maa(x)HQ, where the aa’s are in 0°°^) and
all their partial derivatives are bounded functions on Mn. Compute the
symbol of the formal adjoint of P(x, D) directly. Compare the answer with
the symbol obtained by Theorem 9.1.
9.4.	Let a and r be as in Exercise 8.3. Use Theorem 9.1 to compute the
symbols of the formal adjoints of Ta + Tr, TaTr and TrTa.
9.5.	In deriving (9.28) from (9.24) and (9.27), we use the fact that there
exists a positive constant (7, depending on /x, /3 and K only, such that
\5\<2K	\S\<2 K
for all z £ W1. Prove the fact.

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Chapter 10
The Parametrix of an Elliptic
Pseudo-Differential Operator
Among all pseudo-differential operators there exists a class of operators
which come up frequently in applications and are particularly easy to work
with. They are called elliptic operators. They are nice because they have
approximate inverses (or parametrices) which are also pseudo-differential
operators. Our first task is to make these concepts precise.
A symbol a in Sm is said to be elliptic if there exist positive constants
C and R such that
k(*,oi>c(i + i€ir, m>R.
Of course, a pseudo-differential operator Ta is said to be elliptic if its symbol
is elliptic.
Theorem 10.1. Let a be an elliptic symbol in Sm. Then there exists a
symbol r in S~m such that
TtT„ — I + r
(10.1)
T„Tt = I+ s,
(10.2)
where R and S are pseudo-differential operators with symbols in C\keR.Sk,
and I is the identity operator.
Remark 10.2. In other words, Theorem 10.1 says that if Ta is an elliptic
pseudo-differential operator, then it can be inverted modulo some error
terms R and S with symbols in n*€R5fe. In the theory of regularity of
solutions of partial differential equations, operators with symbols in
are called infinitely smoothing and can be neglected. (See Chapter 15 for
a discussion of infinitely smoothing operators and regularity theory.) For
these reasons, we call Tr an approximate inverse, or more often in the
literature, a parametrix of Ta.
69

70
An Introduction to Pseudo-Differential Operators, 3rd Edition
We first give a proof of (10.1). The idea is to find a sequence of symbols
Tj G S~m~i, j = 0,1,2,	Let us assume that this has been done. Then,
by Theorem 6.10, there exists a symbol r G S~m such that r ~ Tv
Then, by the product formula given in Theorem 8.1, the symbol A of the
product TrTa is in 5° and such that
A- £	S'"	(10-3)
|7| <N	7*
for every positive integer N. But r ~ Yl^Lo tj imP^es that
N-1
T-^Tje s~m~N	(10.4)
j=0
for every positive integer N. Hence, by (10.3) and (10.4),
A" E	EW>)e S~N (10-5)
ft\<N	j=0
for every positive integer N. But we can write
£
h\<N 1	j=0
N-l
= w + E
«=i
ncr +
+ £
l7|+i>AT
h\<N, j<N
|7| +3=1
3<l
{-Z^(dJrj)(d2a).
(10.6)
To simplify (10.6), we choose rh j = 0,1,2,..., in the following way. Define
T0 by
To(x,0
( *te)
J o(x£) ’
\o,
If I >
If I < Rl
(10.7)
where ip is any function in ¢7^(171) such that ip( f) = 1 for If I - 2R and
V’(f) = 0 f°r Ifl < R, and we define rj for / > 1 inductively by
|7| +i=l	r*
n~-{
>70.
(10.8)

The Parametrix of an Elliptic Pseudo-Differential Operator
71
Then it can be checked easily that Tj G	j = 0,1,2, — (See Exercise
10.7.) Now, by (10.7), r0<7 = 1 for |£| > 2R. The second term on the right
hand side of (10.6) vanishes for |£| > 2R by (10.7) and (10.8). As for the
third term there, we see easily that
€ S~N
whenever |7| + j > N. Hence, by (10.6),
£ 44	-1 € S~N
(10.9)
\y\<N	'	3=0
for every positive integer N. Thus, by (10.5) and (10.9),
A - 1 € S~N
for every positive integer N. Hence, if we pick R to be the pseudo¬
differential operator with symbol A — 1, then the proof of (10.1) is complete.
By a similar argument, we can find another symbol k in S~m such that
TaTK = I + &,	(10.10)
where R1 is a pseudo-differential operator with symbol in C\keuSk. (See
Exercise 10.8.) By (10.1) and (10.10),
Tk + RTk = Tt + TtR!.
Since RTk and TtR' are pseudo-differential operators with symbols in
HjkeRSk, it follows that
Tk = Tr + R",	(10.11)
where
R" = TrR! - RTk
is another pseudo-differential operator with symbol in C\keuSk. Hence, by
(10.10) and (10.11),
T'Tt^I + S,
where
S = R' - TaR”.
Since S is a pseudo-differential operator with symbol in flkeuSk, it follows
that (10.2) is proved.
The following theorem tells us that only elliptic pseudo¬
differential operators have parametrices.
Theorem 10.3. Let a G Sm be such that there exists are S~m for which
(10.1) or (10.2) is true, where R and S are infinitely smoothing pseudo-
differential operators, and I is the identity operator. Then a is elliptic.

72
An Introduction to Pseudo-Differential Operators, 3rd Edition
Remark 10.4. If (10.1) (or (10.2)) is true, then we call Tr a left (or right)
parametrix of 7^. Thus, a consequence of Theorems 10.1 and 10.3 is that if
a pseudo-differential operator Ta has a left (or right) parametrix Tr, then
Tt is also a right (or left) parametrix of Ta.
Proof of Theorem 10.3 Let us first assume that (10.1) is valid and let r
be the symbol of R. Then, by Theorem 8.1,
1 + r - or = <5,	(10.12)
where S is some symbol in S-1. Since r G S-m, we can find a positive
constant C\ such that
|r(*,fl| <Ci(l + |£|)-m, x,^P.	(10.13)
Thus, by (10.12) and (10.13),
|1 + r(*,0 - *(*,0| < Ci\o(x,01(1 + IO)-m, *,$€»*,
and hence
k(*.OI>T(i + |or(i-|*(*,oi-|r(*,oi),	(10.14)
Since * € S-1, it follows that there is a positive constant Ci such that
W*,0l<3»(l + I0)_1, 2,(6 1”.	(10.15)
Since r € C\k^ftSk, we can find a positive constant C% such that
|r(*,OI < ¢3(1 + Kir1.	€ R”.	(10.16)
So, by (10.14)-(10.16),
k(*.01 > T(1 + K|)ro{l - (Ca + C3)(l +Ifir1}, x,£€Rn. (10.17)
Now, let R be any positive number such that
1 - (C-2 + c3)(l + Kl)"1 >	10 > R.	(10.18)
Then, by (10.17) and (10.18), we get
k(x,OI> 2^(1+ I0)ro, I0>-R,
and this completes the proof under the assumption that (10.1) is valid.
The proof for the case when (10.2) is valid is similar and hence left as an
exercise. See Exercise 10.9.	□

The Parametrix of an Elliptic Pseudo-Differential Operator
73
Exercises
10.1. (i) Let P(x,D) = J2^<maa(x)Da, where the aa’s are in C00^71)
and all their partial derivatives are bounded functions on Rn. We call
S|a|=m aot(x)€a	principal symbol of P(x, D) and denote it by Pm(xi 0*
Prove that P(x, D) is elliptic if and only if there exist positive constants C
and R such that
\PmM\>C(l + \^\)m, 14 >R.
(ii) LetP(D)=E|a|<ma aDa be a linear partial differential operator with
constant coefficients. Let Pm(0 be the principal symbol of P{D). Prove
that P(D) is elliptic if and only if
Pm(0 = o, £ € Rn =» t = 0.
10.2.	Let Ta and Tr be elliptic pseudo-differential operators. Prove that
the product TaTr is also elliptic.
10.3.	Let Ta be an elliptic pseudo-differential operator. Prove that the
formal adjoint T* of Ta is also elliptic.
10.4.	Prove that any two parametrices of an elliptic pseudo-differential
operator differ by an infinitely smoothing operator.
10.5.	Is a parametrix of an elliptic pseudo-differential operator elliptic?
Explain your answer.
10.6.	Prove that, in the proof of Theorem 10.1, (10.3) and (10.4) imply
(10.5).
10.7.	Let {rj}^ o be the sequence of functions defined by (10.7) and (10.8).
Prove that Tj is a symbol in S~m~j for j = 0,1,2, —
10.8.	Prove that there exists a symbol k in S~m such that (10.10) is valid.
10.9.	Let a G Sm be such that there exists a r G S~m for which (10.2)
is true, where S is an infinitely smoothing operator and I is the identity
operator. Prove that a is elliptic.
10.10.	Find a parametrix of an elliptic linear partial differential operator
with constant coefficients on Rn.

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Chapter 11
Zp-Boundedness of
Pseudo-Differential Operators
Let a be a symbol. Then, by Proposition 6.7, the pseudo-differential opera¬
tor Ta maps the Schwartz space S into S. In fact, the following proposition
is true.
Proposition 11.1. Ta maps S continuously into S. More precisely, if
<Pk 0 in S, then ->• 0 in S as k ^ oo.
To prove Proposition 11.1, we need some preliminary results.
Lemma 11.2. If ifk -> 0 in S as k -» oo, then ipu -» 0 in Lp(Rn) as
k —y oo for 1 < p < oo.
Proof If (fk -* 0 in S as k -► oo, then <pk 0 uniformly on En as k -> oo.
This proves the lemma for p = oo. So, consider 1 < p < oo. Then for any
positive integer N with N >	, we have
Since (1 + |a?|) Np 6 L1(Rn), it follows from Lebesgue’s dominated conver¬
gence theorem that
Lemma 11.3. The Fourier transformation T maps S continuously into S.
More precisely, if tpk -> 0 in S as k ->• oo, then cpi —> 0 in S as k -¥ oo.
(1+ 1^1^(^)1-^0
uniformly on Rn as k oo. Hence for k large enough,
hk(*)l<(i + M)"N,
as k -> oo. This completes the proof.
□
75

76
An Introduction to Pseudo-Differential Operators, Zrd Edition
Proof Let a and /3 be any two multi-indices. Then, by Proposition 4.2,
IH^WkKOI = |{r>“((-*yV)}A(£)l
< (27r)-n/2||2?“((-*yv*)||i,	4eE”.	(11.1)
Since ipk —*■ 0 in S, it follows that Da((—x)^tpk) -4 0 in 5 as k -¥ oo. By
Lemma 11.2, ||2)"((—x)&tpk)\\i -*• 0 as k -* oo. By (11.1),
sup 1^(2)^)(01 -t 0
£eRn
as A; -¥ oo. This proves that ^->Oin<Sasfc^oo.	□
Proof of Proposition 11.1 Suppose a G Sm. Then for any two multi¬
indices a and /3, we have, by (6.12), positive constants	depending
on a, /3,7 and <5 only, such that
sup 1^(2^(1^))(^)1
xeRn
<
|2)f(ov*(0)|<*e
■M+M
(11.2)
Since -* 0 in <S as k —> oo, it follows easily from Lemma 11.3 that
(i+io2)(mHa|+|{|)/20|(rv*(O) -»■ o
in S as k oo. Hence, by Lemma 11.2, the integral on the right hand
side of (11.2) goes to zero as k -» oo. This proves that Taipk 0 in S as
k -* oo.	□
The pseudo-differential operator TV, initially defined on the Schwartz
space <S, can be extended to a linear mapping defined on the space <S' of
tempered distributions. To wit, take a distribution u in S' and define Tau
by
{Tau){v)=u{T&), <peS,	(11.3)
where T* is the formal adjoint of Ta introduced in Chapter 9.
Proposition 11.4. T'<? is a linear mapping from S' into S'.
Proof Let u€ S'. Then for any sequence {<£*} of functions in S converging
to zero in 5, we have, by (11.3),
(Tau)((pk) = w(T*^), k = 1,2,....
(11.4)

Lp-Boundedness of Pseudo-Differential Operators
77
By Proposition 11.1, T*Jpk 0 in S as k -» oo. Hence, using (11.4) and
the fact that u is a tempered distribution, we conclude that (Tffu){ipk) 0
as k -> oo. Hence Tau e S'.	□
To enquire whether Ta maps S’ continuously into Sf or not, we need a
notion of convergence in S'.
Definition 11.5. A sequence of distributions {uk} in S' is said to converge
to zero in S' (denoted by Uk -> 0 in S') if Uk{<p) -» 0 as k —> oo for all
<peS.
Proposition 11.6. Ta maps S' continuously into S'. More precisely, if
Uk 0 in S1 as k -» oo, then TaUk 0 in S' as k oo.
Proof We need only check Definition 11.5 for the sequence {TaUk}- To
do this, let ip e S. Then, using (11.3) and the fact that Uk -+ 0 in 5' as
k -¥ oo,
(TaUk)(ip) = Uk(T*Jp) -* 0
as k oo. Hence TaUk 0 in 5' as k oo, and the proof is complete.D
Let us recall that, by Proposition 5.5 and Remark 5.6, every function
/ in Lp(Rn) is a tempered distribution. Hence, by Proposition 11.4, Taf
is also a tempered distribution. What sort of a tempered distribution is
it? We answer this question in the following theorem and Theorem 12.9 in
Chapter 12.
Theorem 11.7. Let a be a symbol in S°. Then Ta : Lp(Rn) -> Lp(Rn) is
a bounded linear operator for 1 < p < oo.
The following result plays an important role in our proof of Theorem 11.7.
It is a special case of Theorem 2.5 in [Hormander (I960)]. Its proof is
outside the scope of this book and hence omitted.
Theorem 11.8. Let m € Ck(Rn — {0}), k > be such that there is a
positive constant B for which
|(0°m)(0| <	¢#0,
for all multi-indices a with \a\ < k. Then for 1 < p < oo, there is a positive
constant C, depending on p and n only, such that
II^VIIp < CB\\tp\\p, veS,
where
(T<p)(x) = (27r)-"/2 / efa-«m(fl0(0d£,	x€l".

78
An Introduction to Pseudo-Differential Operatorsf 3rd Edition
Now, we can give a proof of Theorem 11.7.
Proof of Theorem 11.7 Let Zn be the set of all n-tuples in Rn with integer
coordinates. We write Rn as a union of cubes with disjoint interiors, i.e.,
Rn = UmeznQm^ where Qm is the cube with center at m, edges of length
one and parallel to the coordinate axes. Let Qo be the cube with center
at the origin. Let rj be any function in Co°(Rn) such that rj(x) = 1 for all
x e Qo- For m € Zn, define am by
0Vn(z,£) = r]{x-m)cr(xy£), x,£eRn.
Obviously, T*m — rj(x — m)Ta, and
f \(T.<p)(x)\*dx< [ |(Ta„tp)(x)\*dx, <peS. (11.5)
JQm	JR"
Since crm(x,f) has compact support in x, it follows from Theorem 4.7 and
Fubini’s theorem that
CTrm¥>)(*) = (2ir)~n/2 f eix<om{x, £)<?>(£) d£
J Rn
= {2*)~n ( eix<\[ eixXa^(X,0dx\<p(0^
JRn	UR"	)
= (2ir)~n f eixX | f eix ^(X,Omd^}d\, (11.6)
JRn	URn	)
where
<^(A,£) = (2tt)-"/2 f e-iX x<rm(x,Z) dx, A, £ € Mn.
JRn
Lemma 11.9. For all multi-indices a and positive integers N, there is a
positive constant Ca,s, depending on a and N only, such that
|(£>^)(A,£)| < CaMl + KI)-|o|(l + |A|)-W, A,£ € 1".
The proof of Lemma 11.9, though easy, will be given later. This lemma
and Theorem 11.8 imply that the operator p i-» T\ip, defined on S by
(Tx<p)(x) = (27r)-"/2 f	(A, £)*>(£) d£,	(11.7)
JRn
can be extended to a bounded linear operator on Lp(Rn). Moreover, for
any positive integer N, there is a positive constant Cm such that
\\txAp<cn(i + 1^)-^11^, cpes.
(11.8)

Lp-Boundedness of Pseudo-Differential Operators
79
Using (11.6)-(11.8) and Minkowski’s inequality in integral form,
\\T*m>p\\P = (2tr)-"/2 {j(_	dAp*} ^
<	(2tt)-"/2 /(/ 1(^)(3:)1^)
JR™	J
= (27r)-"/2/ IMIpdA
JRn
<	^(277)-^2 (jT(l + |A|)-^a} |M|p,	^e5.
By choosing N sufficiently large, we can get another positive constant Cn
such that
l|IVmv>||P < CnIMIp, ip € 5.	(11.9)
Hence, by (11.5) and (11.9),
/ \(Tatp)(x)\pdx < C'^llyllp, <p€S.	(11.10)
J Qm
Now, we represent Ta as a singular integral operator. Precisely, we have
Lemma 11.10. Let
K(x,z) = (27r)”n/2 f elz’^a{x^)d£t
J Rn
in the distribution sense. Then
(i)	for each fixed x G Rn, K(x, •) is a function defined on W1 — {0},
(ii)	for each sufficiently large positive integer N, there is a positive constant
Cn such that
\K(x,z)\<Cn\z\-n, z? 0,
(iii)	for each fixed x G Mn and ip G S vanishing on a neighborhood of x,
(Taip)(x) = (27r)_n/2 f K(x,x- z)ip(z) dz.
J Rn
Let us assume Lemma 11.10 for a moment. Let Qbe the double of Qm,
i.e., Qm the same center as Qm and edges parallel to the coordinate
axes and twice the edge length of Qm. Let Q*m be another cube concentric
with Qm and Q™ such that Qm C Q*m C Qm- Furthermore, we assume
that there is a positive number S such that \x — z\ > 6 for all x G Qm

80
An Introduction to Pseudo-Differential Operators, 3rd Edition
and z € Rn — Q*m. The geometry is illustrated by the following figure.
Let ^ € Cq°(Rn) be such that
0 < ip(x) <1, xEf,
supp(^) c Q”
and
ip(x) — l
on a neighborhood of Write p = pi + P2, where ipi = 'tpp, and
P2 = (1 - Then
Tap = Taipi +Ta(p2-
Write
| (Ta<p)(x)\rdx
\(Tap2)(x)\Pdx.
and

Lp-Boundedness of Pseudo-Differential Operators
81
Then for any sufficiently large positive integer N, inequality (11.10) implies
that there is a positive constant Cn such that
Im= f \(Ta<pi)(X) + (Ta<p2){x)\*dx
JQm
<2p [ \(Ta(fi)(x)\pdx + 2PJm
JQm
<2pC&|Mp + 2pJm.	(11.11)
By Lemma 11.10, there is a positive constant C2n such that for all x € Qm,
I K(x,x — z)ip2(z)dz
J]R"
I K(x,x — z)ip2(z)dz
jRn-Q^
< C2N f \x- z\~2N\ip2(z)\dz.
JRn-Q*
|(7>2)(s)| = (27T)-”/2
= (2tt)-"/2
(11.12)
Let A > y/n + 1. Then there exists a positive constant C\yN, depending on
A and N only, such that
\x-z\~™	_ (A + |s - z\)2N ^ „
(A + |x - z\)~2N \x - z\™	~ K
,N
(11.13)
for all x 6 Qm and z6l“- QHence, by (11.12) and (11.13),
m<p2)(x)\
< C2nC\,n f (A + |ar - z\)~2N\<p2{z)\dz, x G Qm. (11.14)
Next, we note that for all x £ Qm and z £ Mn — Q*m,
\+\x-z\ = \ + \x-m + m-z\
> A + |m — z\ — |a: — m\
“ (A ~ + ~ z\
>H+\m-z\,	(11.15)
where ^ = ^ + 1. By (11.14) and (11.15),
m<P2)(x)\
<c,NcK„f
jR*-Q-m (» + \m-z\)N

82
An Introduction to Pseudo-Differential Operators, 3rd Edition
By Minkowski’s inequality in integral form and Holder’s inequality,
(jf |(r^2)(*)|Pd*y/P
(// + \x-z\)~N\<p2(z)\ 'P	' /P
<C2NCxN\f /
[jQm |./R”-
< C2NCx,N f (/
JUL'-O' Uq
-dz
p \ 1
dx >
= C2nC\n
X»-Qi
Q,m	(ji + \m- z\r
(fi+\x-z\)-Np\Mz)\Pjy/P
Q-m UQm	ip + \m- z\)NP
|ya(-g)l
Q.m (ii + \m-z\)N \JQ
v
dx > dz
{y*	(/x+|x-2;|)	dz
< C2nC\,n If (fi + \m- z\)
[JRn-Q'm	J
l<P2(z)|P
1/P
r dz
\JwL"-Q'm +	z\)np/2 j
Hence for any sufficiently large positive integer iV, there is a positive con¬
stant C\,N,p, depending on A, N and p only, such that
\<P2(Z)\P
Jm~Cx’N’PL-Ql,^+\m-z\)^
dz.
(11.16)
By (11.11) and (11.16),
Summing over all m in Zn, we get a positive constant <7, depending only
on n,p, N and A, such that
) dx
f l(7»(*)|P
J Rn
< 2pC% £ f \>p(x)\pdx
meZn ^
+
2PC\,
,jv.p 5Z /
jr»-c
l¥’2(z)|P
m6z„	(// + |m - zD^p/2
< c/r_ w.)p*+zi_Q_
=CLlvl:,)rd*+2'CA,iv-’)"p,td‘'
dz
(11.17)

Lp -Boundedness of Pseudo-Differential Operators
83
But using the same argument as in the derivation of (11.15), we get
// + \m - z\ > 1+ \m — l\	(11.18)
for all z € Qi and l ± m. By (11.18),
Y Y f	dz
Jk-l&h (/t + |m_zl)Np/2
=,£lteW^.5.a^
= .5
(11.19)
Hence, by (11.17) and (11.19),
L i»4 s {c+ro»-|.[i+M^} L i^r^-
Since S is dense in Lp(Rn) by Remark 3.10, it follows that Ta can be
extended to a bounded linear operator on Lp(Rn).	□
Remark 11.11. We leave it as an exercise to prove that the bounded
extension coincides with Ta : S' S' restricted to the space Lp(Rn). See
Exercise 11.2.
We now come to the proofs of Lemmas 11.9 and 11.10.

84
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof of Lemma 11.9 Let /? be an arbitrary multi-index. Then, by
integration by parts and Leibniz’s formula,
=	2n)~n'2 f e-ix X*m(x,Odx
J Rn
= (—iA)^D?(27r)_n/2 f e~tx Xri(x — m)(r(x1 £) dx
J R*
= D%(2ir)~n/2 f	- m)a(x,£) dx
Jun
= (2tt)-"/2 f {d^e~ixX}V(x - m)(D%a)(x,0 dx
JRn
= (—1)1^1 (2it)~n/2 f e-ix Xd%{n(x -m)(D%a)(x,€)}dx
JRn
= (-1)1^(2^)-^2 £ (13) f e~ix X(d2n)(x - m)(dt'lD%<j)(x, £) dx.
££ W Jk*
Using the properties of 7} and the fact that a E S°, we can find a positive
constant C^,/?, depending on a and /3 only, such that
|(-iA)^(^)(A,OI < CaA 1 + |f|)-l“l,	A,* e IT.
The lemma follows easily from this estimate.	□
Proof of Lemma 11.10 Let a be an arbitrary multi-index with length
greater than n. Then
(~iz)aK(x,z) = (27r)-"/2 f e*x(8?a){x,$dt	(11.20)
J Rn
in the distribution sense. Since a E 5°, it follows from (11.20) and Propo¬
sition 4.3 that (iz)aK(x, z) is a continuous function on W1 and there is a
positive constant Ca such that
\z*\\K{z,z)\<Ca
for all xy z E Rn. Hence part (i) follows immediately and part (ii) follows
if we use the inequality in Exercise 7.2. To prove part (iii), we define the
tempered distribution Lx by
Lz(il>)= [ <r(s.0^(£)d£> 1>eS.
J R"

Lp-Boundedness of Pseudo-Differential Operators
85
Then, by the definition of a pseudo-differential operator, Proposition 4.4
and the definition of the Fourier transform of a tempered distribution,
M(x) = (27r)-"/2 [
J Rn
= (2n)-n'2Lx(Mx<p)
= (2tt)-”/2L*((T^)a)
= (2w)-n'2Lx(Txlp).	(11.21)
By part (i),
Lx(ip) = f K(x,-z)i>(z)dz	(11.22)
J R"
for all ip e S vanishing on a neighborhood of the origin. Hence, by (11.21)
and (11.22),
(Taip)(x) = (27r) n/2 f K(x,-z)(Tx(p){z)dz
JRn
= (27r)"n/2 f K{pc, -z)(p(x + z) dz
J Rn
= (27r)”n/2 f K(x,x — z)(p(z) dz
J Rn
and the proof is complete.
Remark 11.12. The proof of the Lp-boundedness of pseudo-differential
operators for 1 < p < oo given in Theorem 11.7 is based on the Hormander
multiplier theorem, which is formulated without proof as Theorem 11.8.
All proofs of the alluded Lp-boundedness invoke some results and concepts
that go beyond the elementary nature of this book. For L2-boundedness,
we can provide a complete and self-contained proof. Indeed, for all positive
integers N, we obtain by Lemma 11.9 a positive constant Cjv such that
\^(\0\<Cn(1 + \\\)-N, A,|eRn.
By (11.7) and the Plancherel theorem, we get for all positive integers N, a
positice constant Cn such that
||I>||2 = ||5£(A,	< CN(l + lAD-^lh = CN{ 1 + |A|)-"|M|a,
which is (11.8) for p = 2. The use of Hormander’s multiplier theorem is
thus avoided.

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An Introduction to Pseudo-Differential Operators, 3rd Edition
Exercises
11.1.	Prove that the definition of Tau given for a tempered distribution u
given in (11.4) coincides with the definition of Tau for a Schwartz function
u given in (6.4).
11.2.	Let a G S° and 1 < p < oo. Then we have shown in the proof of
Theorem 10.7 that there is a positive constant C such that
\\TMp<C\Mp, <p€S.
Hence Ta can be extended to a unique bounded linear operator from Lp(Rn)
into Lp(Rn). Prove that the extension coincides with Ta : S' S' re¬
stricted to Lp(Rn).
11.3.	Let cr(x, ¢) G 5° be a nonzero symbol which is independent of f G I71.
Prove that the bounded linear operator Ta : LP(Rn) -¥ Lp(Rn), 1 < p < oo,
is not compact.
11.4.	Show that the limit in operator norm of a sequence of pseudo¬
differential operators Tak : L2(Rn) L2(Rn), where crG 5°, need not
be a pseudo-differential operator.
11.5.	Suppose that a G 5° has compact support in x. Prove that the
pseudo-differential operator Ta : Lp(Rn) —» Lp(Rn) is a bounded linear
operator for 1 < p < oo.

Chapter 12
The Sobolev Spaces
Hs,p, —oo < s < oo, 1 < p < oo
Theorem 11.7 tells us that Ta : Lp(Rn) -¥ Lp(Rn) is a bounded linear
operator for 1 < p < oo if a is a symbol in 5°. In order to find an analog of
Theorem 11.7 for an arbitrary symbol in Sm, we need to introduce a family
of spaces of tempered distributions.
For — oo < s < oo, we denote by Js the pseudo-differential operator of
which the symbol crs(£) is given by
M« = (i+i£i2rs/2, £€Rn.
It should be noted that the symbol of Js is in S~s. (See Example 6.4.)
The operator Js is often called the Bessel potential of order s.
It is an easy exercise to prove that for any u £ <S', the product asu of
a3 and u defined by
{<Jsu){ip) =u{a8(p), (p G«S,
is also in 5'. See Exercise 12.1.
The following proposition is an easy consequence of (11.3) and the def¬
inition of Js, and the proof is left as an exercise. See Exercise 12.2.
Proposition 12.1. Jsu = T^OsTu, u £ S'.
An easy corollary of Proposition 12.1 is the following proposition. Its
proof is also left as an exercise. See Exercise 12.3.
Proposition 12.2. Let u € S'. Then
(i)	JsJtU = Jg-j-tti,
(ii)	J0u = u.
For —oo < s < oo and 1 < p < oo, we define Hs'p to be the set of
all tempered distributions u for which J-8u is a function in Lp(Rn). It is
87

88
An Introduction to Pseudo-Differential Operators, 3rd Edition
obvious that H8,p is a vector space. It can be made into a normed vector
space if we equip it with the norm || ||s>p, where
|HU,P = II j-su\\p, u G Hs'p.
We usually call H8'p the Lp-Sobolev space of order s. It is obvious that
tf°’p = Lp{ Rn).
Theorem 12.3. H8iP is a Banach space with respect to the norm || ||s>p.
Proof We need only prove completeness. To do this, let {u&} be a Cauchy
sequence in Hs p. Then, by the definition of Hs,p, the sequence {J-sUk}
is a Cauchy sequence in Lp(Rn). Since Lp(En) is complete, it follows that
there exists a function u in Lp(Rn) such that
J-8uk -» u	(12.1)
in Lp(Rn) as k -> oo. Let v = J8u. Then, by Proposition 12.2,	= u.
Hence J-8v is in Lp(Rn), i.e., v G H8tP. That Uk -> v in H8,p as k -4 oo is
an immediate consequence of (12.1).	□
Proposition 12.4. Jt is an isometry of HStP onto H8+t'p. More precisely,
\\Jtu\\8+t,p = ||u||alP, U G H8'p.	(12.2)
Proof Let u G H8’p. Then, by Proposition 12.2,
||*7t^||s+t,p = II J-s-tJt'U’Wp = \\J— su\\p = ||^||«,p*
Let v G £P+t’p. Then, by Proposition 12.2, J-tv G H8'p and JtJ-tv = v.
This proves that Jt is onto.	□
Theorem 12.5. Let 1 < p < oo and s <t. Then HUp C Hs'p, and
IMUp — lklkp> u ^ Http,
Theorem 12.5 is usually known as the Sobolev embedding theorem. To
prove Theorem 12.5, we use a technical result which gives us an explicit
formula for the inverse Fourier transform of the function (1 4- |£|2)_s/2,
where £ G W1 and s > 0, in the distribution sense.
Proposition 12.6. Let s > 0. If we define the function G,
GJx) =	-	 f e“r/2e"|x,2/(2rV~(n_5)/2 —
2*/2r (f) Jo
then
(i)	Gs G L1(Wl)}
(ii)	lIG.Hx = (27r)n/2,
(iii)	G;(0 = (l + |^|2)“s/2,
on E" by
x € En,
^ € 1”.

The Sobolev Spaces H5’p, -oo < s < oo, 1 < p < oo	89
Proof By Fubini’s theorem,
I \Gs(x)\dx= I Gs(x)dx
JRn	JRn
_ (27r)n/2	f°°	/2 /2 dr
2s/2r (4)/0	7
(12.3)
if we recall that
(12.4)
(12.5)
[ e-^^dx = (27rr)”/2.
JRn
But for any e > 0 and a > 0, we have
e-°r(o)= f°° e~erra—.
Jo	r
Putting e = \ and a = | in (12.4), we get
jf°e-r/V/2y =2*/2r (I).
Hence, by (12.3) and (12.5),
/ |G»(x)|dx = (2ir)"/2.
JRn
This proves parts (i) and (ii). To prove part (iii), let <p e S. Then, by
Proposition 4.6,
[ Gs(0<p(t)d$= f GsiOviOdt
JRn	jRn
- 	I	 [ f r e-r/\-\i\VWr-(n-W-}md$.
2*/2r(f)yRnU	ri
Using Fubini’s theorem, we get
«/Rn
-	1	r°e-r/2 -(n~«)/2 / f	(12.6)
" 2*/2r (§) /„	Ur-	J r
But, by Proposition 4.6 again, we get
J <£(£)e_l€|a/(2r)# =	V»(€)^(0 <*£>
(12.7)

90
An Introduction to Pseudo-Differential Operatorsf 3rd Edition
where
(12.8)
V>(a;) = e-ll|2/(2r), x <E f.
By Propositions 4.4 (iii) and 4.5,
$(0 =rn/2e-r|«l2/2, £eRn.
Hence, by (12.7),
f 0(0e"l€l2/(2r)df = W2 / y>(0e-r|£|2/2d$.
«/Rn	./Rn
Therefore, by (12.6) and (12.8),
L S<<M0 « = 57½) f {/,.	*
Using Fubini’s theorem again, we get
[ GsitMOdti
J Rn
= 57¾
Putting e = (1 + |£|2)/2 and a = § in (12.4), we get
J G.MrtOdt
= f (i + KIT'/2¥>(0«-
^Rn
Hence, by Lemma 6.6 and (12.9), we can conclude that
Gs(0 = (1 + |£|2)-s/2, £€Kn.
(12.9)
The following consequence of Proposition 12.6 will be useful to us.
Proposition 12.7. Let s > 0 and 1 < p < oo. Then
II J.»||p < IMIr, « e Lp(Rn).
(12.10)

The Sobolev Spaces if3,p, —oo <s<oo,l<p<oo
91
Proof Let ip G S. Then, by the definition of Js,
(^)A(0 = (i + kl2)"s/V(a
On the other hand, by Propositions 4.1 and 12.6,
(g. *<pno = (2*)n/2Gs(om
= (2^(1 + 1^)-^(0, C€En.
Hence for all (p G 5,
JS(P = (27T )~n/2(Gs*if),
and, by Theorem 3.1,
\\JM\p < (2^)-^110.1^ IMIP = IMIp.
Since S is dense in Lp(Rn) by Remark 3.10, it follows that Js can be
extended to a bounded linear operator on Lp(Rn) satisfying (12.10).	□
Remark 12.8. As in Remark 11.11, we leave it as an exercise to prove
that the bounded extension coincides with Js : S' -> S' restricted to the
space Lp(Rn).
Proof of Theorem 12.5 Let u G Ht,p. Then, by the definition of HtiP,
we have J-tu G Lp(Rn). Hence, by Proposition 12.2, J-Su = Jt-8J-tu• By
the definition of H9'p and Proposition 12.7, we get
llwlls,p = ||«/— $u\\p = ||«/it—s«/—$u||p < ||t7_iix||p = ||u||t,p,
and hence Theorem 12.5 follows.	□
We can now give a more precise result generalizing Theorem 11.7.
Theorem 12.9. Let a be a symbol in Sm. Then Ta : Hs'p H8~m'p is a
bounded linear operator for —oo<s<oo and 1 < p < oo.
Proof Since Jm-gT^Js G 5°, it follows from Theorem 11.7 that there is a
positive constant C such that
llTfl-uHs—jT^p = ||«7m—= \\Jm—sTffJsJ~-su\\p
< C\\j-8u\\p = C\\u\\s,p, u G H8'p.
□
Remark 12.10. A weaker result than the Sobolev embedding theorem
formulated in Theorem 12.5 is one that asserts the existence of a positive
constant C such that
IMkp < CIMkp, n € H

92
An Introduction to Pseudo-Differential Operatorst 3rd Edition
where 1 < p < oo. The weaker inequality has a much shorter proof. Indeed,
for all u G H
IMU,p = ll*^-swllp= \\Jt—sJ—t^l Ip*
Since s < £, Jt-S is a pseudo-differential operator with symbol in 5°. So,
by Theorem 11.7, there exists a positive constant C such that
||= IIJt-sJ-tu\\p < C\\J-M\P = CIMkp, u € H
Thus,
IMkp < ll«lkp, « € HtlP.
Notwithstanding the apparently shorter proof, the Lp-boundedness of
pseudo-differential operators is invoked. So, the proof of Theorem 12.5
as given in this chapter is self-contained and elementary, and it gives a
better result.
Exercises
12.1.	Let s G (—oo, oo) and as be the function on Rn defined by
MO = (i+iei2rs/2> £ € Mn.
Prove that if u G S', then the product asu of as and u defined by
(cr8u)(ip) = u(as(p), ip G <S,
is also in <S'.
12.2.	Prove that for all s G (—oo, oo),
Jsu = T~lasTu, u G <S',
where as is the function on Rn defined in Exercise 12.1.
12.3.	(i) Prove that for all s,t G (—00,00),
JsJt'U’ — Js-\-t'U,i w G <S .
(ii) Prove that Jqu = u for all u G S'.
12.4.	(i) Prove that for all s G (—00,00),

The Sobolev Spaces Hs'p, — oo <s<oo, l<p<oo
93
(ii) Prove that for all s £ (-00,00),
IW* = y (1 + iei2)sl«(0l2rfe}1/2, t* € h°*.
12.5.	(i) Prove that if u £ if5’2, s > § + fc, where k is a nonnegative
integer, then u is equal to a Ck function on Rn almost everywhere.
(ii) Prove that if u £ i/5’2, s > then u can be modified on a set of
measure zero to a continuous function v on Rn such that
lim v(x) = 0.
|x|—>00
12.6.	(Erhling’s Inequality) Prove that if s < t, then for any positive
number e, there exists a positive constant C, depending on e, s and t only,
such that
IMI«,2 < e|Mlt,2 + CIMIo.2, € s.
12.7.	Let s > 0 and 1 < p < 00. Then we have shown in the proof of
Proposition 12.7 that
II JsvWp < IMIp, v e 5.
Hence Js can be extended to a unique bounded linear operator from Lp(Rn)
into Lp(Rn). Prove that the extension coincides with J8 : S' -¥ S' restricted
to Lp(Rn).
12.8.	Let cr be an elliptic symbol in Sm. Let u € Lp(Rn) be a solution of
the pseudo-differential equation Tau = /, where / 6 Lp( Rn). Prove that
u £ Hm'p.
12.9.	Let a £ Sm, m > 0, and / £ Lp(Rn), 1 < p < 00. An approximate
solution of the pseudo-differential equation Tau = / on Rn is a function u
in such that Tau - f modulo	i.e.,	Tau - f £ r\seuHs'p.
Prove that an approximate solution exists if a is elliptic.
12.10.	Find all real numbers s such that <5 £ H8'2.
12.11.	Let s £ (—00,00). Find Js5.
12.12.	Let s > 0. Find a solution u in Lx(Rn) such that (/ - A)8f2u = S
on Rn, where (/ - A)5/2 is the operator introduced in Example 6.4. (u is
termed a fundamental solution of the operator (/ — A)5/2.)
12.13.	Is DseRH8'2 = 5? Explain your answer.

94
An Introduction to Pseudo-Differential Operators, 3rd Edition
12.14.	For —oo < s < oo and 1 < p < oo, let u G Hs,p and v G H~s'p'. Let
{<^j} and {^j} be sequences in S such that ipj u in JTS’P and ^ t; in
H~s,p' as j -¥ oo.
(i)	Prove that \imj^oo((pj^j) exists and the limit is independent of the
choice of the sequences {ipj} and {^}. (The limit is denoted by (u,v).)
(ii)	Prove that
\{u,v)\ < IM|s,p|M|_s,p/, u G H°>p, v G H~s’p'.

Chapter 13
Closed Linear Operators
In this chapter we give a brief account of the theory of closed linear oper¬
ators on Banach spaces. The choice of topics is dictated by what we need
for the theory of minimal and maximal pseudo-differential operators in the
next chapter.
Let X and Y be complex Banach spaces with norms denoted by || ||x
and || ||y respectively. We are interested in linear operators A mapping a
dense subspace of X, usually denoted by V{A), into Y. We call V(A) the
domain of the operator A.
Definition 13.1. The operator A is said to be closed if for any sequence
{xk} of vectors in V{A) such that Xk ->■ x in X and Axk —> y in Y as
k -¥ oo, we have x £ T>(A) and Ax = y.
Definition 13.2. The operator A is said to be closable if for any sequence
{xk} of vectors in V(A) such that Xk ->* 0 in X and Axk y in Y as
k —► oo, we have y = 0.
Obviously, a closed linear operator is closable.
Definition 13.3. Let A and B be linear operators from X into Y with
domains V(A) and V(B) respectively. We call B an extension of A if
V(A) C V(B) and Bx = Ax for all x e P(A).
Proposition 13.4. Let A be any linear operator from X into Y with do¬
main V{A). Then A has a closed extension if and only if A is closable.
Proof Let B be a closed extension of A. Let {#&} be a sequence of vectors
in V(A) such that ->* 0 in X and Axk y in Y as k -)- oo. Since B is
an extension of A, it follows that Xk € V(B),Xk 0 in X and Bxk y
in Y as k -¥ oo. Since B is closed, we have y — 0. Hence A is closable.
95

96	An Introduction to Pseudo-Differential Operatorsf 3rd Edition
Conversely, suppose that A is closable. We define an operator Ao as follows.
V(Ao) is the set of all vectors x G X such that there is a sequence {xu} of
vectors in V{A) with the property that x& ->> x in X and Axk -» y in Y
for some y G Y as k oo. For any x G V(Aq), we define Aox to be equal
to y. We have to check that the definition of Ao does not depend on the
particular choice of the sequence {x*;}. Indeed, if {zk} is another sequence
of vectors in V(A) such that 2* x in X and Azk -> w for some other
w eY as k -> 00, then x* — 2* ->■ 0 in X as k ->• 00, x/. - Zk G £>(A), and
A(x& -	-> y - w in T as A; ->• 00. Since A is closable, it follows that
y - w = 0, i.e., y = w. Obviously, Ao is an extension of A. It is closed.
For let {xk} be a sequence of vectors in V{Aq) such that Xk -> x in X and
A0xk y in Y as k -> 00. For each fc, there is a sequence {x^ } of vectors
in V(A) such that x^. ->> Xk in X and Ax^ AoXk in Y as j 00. Hence
for each k, there is a 2¾ G X>(A) such that
ll*fc - zfclU < ^
and
- Azk\\y <
Hence for each k,
IN* - ®||x < IN* -	+ |\Xk - x\\x < ^ + ||Xfe - x||x*
Therefore IN* — x||x ->■ 0 as fc 00. Similarly, ||A&* — 2/||y -f 0 as A; -* 00.
This proves that x G V(Ao) and Aox = y. Hence Ao is closed.	□
Let us study the operator Aq constructed in the proof of Proposition
13.4.
Proposition 13.5. Aq is the smallest closed extension of A. This means
that if B is any closed extension of A, then B is an extension of Ao.
Proof Let x G V{Ao) and Aox = y. Then, by the definition of Aq, we
can find a sequence {x*} of vectors in A) such that Xk x in X and
Axk ->• y in Y as k ->* 00. Since B is an extension of A, it follows that
Xk G T>(B), xu ->* x in X and Bxk ->■ y in Y as k 00. Since B is closed,
we can conclude that x G T>(B) and Bx = y. This proves that B is an
extension of Aq.	□
Remark 13.6. In view of Proposition 13.5, we call Ao the minimal operator
of A.

Closed Linear Operators
97
Let X be any complex Banach space with norm \\\\x- We denote by
X' the dual space of X. Let us recall that X1 is the Banach space of all
bounded conjugate linear functionals on X. The norm || \\x> in X' is given
by
ll/ll*'= supInP. /ex'.
*ex IfIIx
x^O
Let X and Y be complex Banach spaces. For any linear operator A from
X into Y with domain V{A) dense in X, we define an operator A1 :Y' —► X1
as follows.
V{At) is the set of all functionals y' in Y* for which there is a functional
x* in X* such that
yf(Ax) = x'(x), x G 'D(A).	(13.1)
Lemma 13.7. Let y1 GY*. Then there is at most one xf G X' for which
(13.1) holds.
By Lemma 13.7, we can define Aty* to be equal to x* for all y' G V(At).
We call A1 the true adjoint or simply the adjoint of A.
Proof of Lemma 13.7 Let yf G7'. Suppose x' and z* are functionals in
X' for which
y\Ax) = x’(x), x G V{A),
and
y\Ax) - z\x),	x G 'D(A).
Obviously, x' = zf on V(A). Since V(A) is dense in X, a simple limiting
argument will show that x1 = zf on X.	□
Proposition 13.8. At is a closed linear operator from Yl into X'.
Proof Linearity, as usual, is easy to check. To prove that At is closed, let
Wk} l>e a sequence of functionals in 1^{A1) such that y'k -* yr in Y* and
A% -»• x' in X' as k oo. Then, by (13.1) and the definition of At1
y'k(Ax) = (A* y'k)(x)
for all x G V(A) and k = 1,2,	Let k -¥ oo. Then
y* (Ax) = x'(x), x G T>(A).

98
An Introduction to Pseudo-Differential Operators, 3rd Edition
This proves that y' G V{At) and Atyf = x'. Therefore A1 is a closed
operator.	□
Another observation about the adjoint of a linear operator we shall use
is given in the following proposition.
Proposition 13.9. Let A be any linear operator from X into Y with do¬
main V(A) dense in X. Then for any extension B of A, the operator A1
is an extension of Bl,
Proof Let y' G Vi^B1). Then we can find a functional x1 in X' such that
y\Bx) = x,{x), xeV(B).
Since B is an extension of A, we have
y1 (Ax) = x;(x), x G V(A).
This proves that yf G V{At) and Aty/ = x' = Bty*.	□
Exercises
13.1.	Prove that for -oo < s < oo and 1 < p < oo, the dual space of H8'p
is H~8'p', where
i + i-i.
P P
13.2.	Let A be a linear operator from Lp(Rn) into Lp(Rn), 1 < p < oo,
with dense domain V{A). Prove that T>(At) consists of all functions u in
Lp> (Rn) for which there exists a function / in Lp (Rn) such that
(u,Av) = (f,v), v G V{A),
where
(9, h) = / g{x)h{x) dx
JRn
for all g G Lp (Rn) and h G Lp(Rn).
13.3.	Let Z be a complex Banach space. A subset S of Z'
total if
{z G Z : z'(z) = 0 for all z* G S} = {0}.
Let A be a linear operator from a complex Banach space
complex Banach space with dense domain V(A). Prove that
if and only if the domain T>(At) of At is a total set.
is said to be
into another
A is closable

Chapter 14
Minimal and Maximal
Pseudo-Differential Operators
Let cr be a symbol in Sm. Then the pseudo-differential operator initially
defined in Chapter 6 on the Schwartz space S, has later been extended in
Chapter 11 to the space S' of all tempered distributions using the formal
adjoint T*. By Theorem 12.9, Ta : HSiP -> Hs~m'p is a bounded linear
operator for — oo < s < oo and 1 < p < oo. As a matter of fact, when
m > 0, the operator Ta can also be considered as a linear operator from
Lp(Rn) into Lp(En), 1 < p < oo, with domain S. We denote this operator
simply by Ta. It is not closed in general. Fortunately, it is closable. Hence,
by Proposition 13.4, it has a closed extension.
Proposition 14.1. The operator Ta is closable.
Proof Let {</?*} be a sequence of functions in S such that ipk ~+ 0 and
Taifk -+ f in Lp(Rn) as k oo. Then for any function ^ in 5, we have
(Ta^ki ip) = Tp'ip), k = 1,2,...,
where T* is the formal adjoint of Ta. Let k —> oo. Then (/,^) = 0 for all
functions ^ eS. Since S is dense in Lp(Rn), it follows that / = 0. Hence,
by Definition 13.2, Ta is closable.	□
Remark 14.2. A consequence of Proposition 14.1 is that the minimal op¬
erator Tffio of Ta exists. (See Remark 13.6.) Let us recall that the domain
ViTafl) of Tv# consists of all functions u in Lp(Rn) for which a sequence
{ifk} in S can be found such that tpk u in Lp(Rn) and Taipk -+ f in
Lp(Rn) for some / G Lp(Rn) as k ->> oo. Moreover, Ta$u = /.
Definition 14.3. Let u and / be functions in Lp(Rn), 1 < p < oo. We say
that u lies in ©(T^i) and Ta^u = / if and only if
{u,T*ip) = (f,<p),	<p£S,	(14.1)
99

100
An Introduction to Pseudo-Differential Operators, 3rd Edition
where T* is the formal adjoint of Ta.
Proposition 14.4. Let u G X>(T^i). Then Ta^u = Tau in the distribution
sense.
Proof By Definition 14.3,
(7V,lU, if) = (U, T*(p), (peS.
Hence, by considering u and Ta^u as tempered distributions, we have
(T^u)(jp)=u(T^), (peS.	(14.2)
On the other hand, by (11.3),
cTau){Tp) = u{T{*£),	(14.3)
Hence, by (14.2) and (14.3), Ta^u = Tau in the distribution sense. □
Proposition 14.5. T^i is a closed linear operator from Lp(Rn) into
Lp(Rn) with domain V(T^i) containing S.
Proof That S C £>(1^1) is obvious from (14.1) and the definition of the
formal adjoint T*. Linearity is again easy to check. To prove that T^i is
closed, let {uk} be a sequence of functions in V(Ta>i) such that uk u in
Lp(Rn) and Ta^uk f in Lp(Rn) for some u and / G Lp(Rn) as k ->■ oo.
Then, by (14.1),
{uklT*(p) = (Ta,iuk,(p)	(14.4)
for all (p G S and k = 1,2,	Let k -4 oo in (14.4). Then
(u,T» = (/,</?), ip G S.
Hence, by Definition 14.3, u G ^(T^i) and Ta^u = f. This proves that
is closed by Definition 13.1.	□
Proposition 14.6. S C V(T*tl), where T* ± is the adjoint ofTati.
Proof Since T^i is a closed linear operator from Lp(lRn) into Lp(Rn), for
1 < p < oo, with domain containing S, it follows from Proposition 13.8
that T*x is a closed linear operator from Lp> (En) into Lp'(Rn), where pf is
the conjugate index of p. Let ^ eS. Then for all functions u G V(Tati),
(1>,Tatlu) = (T>,n)
by Definition 14.3. Hence, by the definition of ^ £ T>(T£i) and
T*Ai> = T;ip.	’	'	□
Proposition 14.7. is an extension ofT^0.

Minimal and Maximal Pseudo-Differential Operators
101
Proof Let u G V(Ta$) and T^oii = /. Then, by Remark 14.2, there is a
sequence {tpk} of functions in <S for which <pk u and Ta(pk ->■ / in Lp(Rn)
as k —> oo. Hence, by the definition of T*, we have
for all ip G S and k = 1,2,	Let k -¥ oo. Then
(u,r» =
So, by Definition 14.3, u G £>(1^1) and = /.	□
Remark 14.8. Using Propositions 13.9 and 14.7, we see that 0 is an
extension of T* v Since, by Proposition 14.6, the domain of T*x contains
the space <S, it follows that the domain of T* 0 contains S as well.
Proposition 14.9. is the largest closed extension of Ta in the sense
that if B is any extension of Ta such that S C D(R£), then T^i is an
extension of B.
We first prove the following lemma.
Lemma 14.10. T*(p = T*ip for all ip G S. In other words, the true and
formal adjoints coincide on the space S.
Proof Let (p G S. Then, by the definition of T*, we have
So, by the definition of T* and the duality of Lp(En), ip G D(T*) and
= □
Proof of Proposition 14.9 Let u G V(B). Then for all ^ G 5, we have
^ G V(Bt). Hence, by the definition of Bl,
(V>,£u) = 0B^,u).	(14.5)
Since B is an extension of TV, it follows from Proposition 13.9 that T* is
an extension of Bl. Hence, by (14.5),
(^Bu) = (T^u).	(14.6)
By Lemma 14.10, T* = T* on S. Hence, by (14.6), we have
(^bu) = (t;^u), ii>es.
Therefore, by Definition 14.3, we have u G P(Taji) and Ta^u = Bu. □
Remark 14.11. Because of Proposition 14.9, we call Tay\ the maximal
operator of Ta.

102
An Introduction to Pseudo-Differential Operators, 3rd Edition
By Proposition 14.7, we know that T^i is an extension of T^o- The aim
of this chapter is to prove that T^o = T^i if a is an elliptic symbol in
5m,m > 0. (See Chapter 10 for the definition of ellipticity.) We need some
preparation.
Theorem 14.12. Let m > 0 and a be an elliptic symbol in Sm. Then
2>(2V|0) = Hm*
To prove Theorem 14.12, we use the following estimate, which is the
analog of the Agmon-Douglis-Nirenberg estimate in [Agmon, Douglis and
Nirenberg (1959)] for pseudo-differential operators.
Proposition 14.13. Let m > 0 and a be an elliptic symbol in Sm. Then
there exist positive constants C\ and C2 such that
Ci||w||7Ti,p ^ (||T<7u||o,p H" |M|o,p) <	u E H 'p.
Proof By Theorems 12.5 and 12.9, there is a positive constant C' such
that
\\T*u\\o,p + ||u||o,P < C'||u||m,p, u E Hm'p.
Next, by (10.1) in Theorem 10.1, we have
u = TTTau -Ru, u E	(14.7)
where r E S~m and R is a pseudo-differential operator with symbol in
Hence it follows from Theorem 12.9 and (14.7) that there is a
positive constant C such that
C\\u\\m,p < (|M|ofp + ||u||o,p), U E Hm'p.
This proves Proposition 14.13.	□
Proposition 14.14. S is dense in H8yP, -00 < s < 00, 1 <p < 00.
Proof Let u E H8'p. Then, by the definition of Hs'p, J-Su E Lp(Rn). Since
S is dense in Lp(Wl) by Remark 3.10, it follows that there is a sequence
{ifk} of functions in S such that ipk -» J_su in Lp(W1) as k -»> 00. Let
ipk = Js<Pk, fc = 1,2, — By Proposition 6.7, E S, k = 1,2, — Also, by
the definition of H8'p again,
W'lfik ~~ u||s,p = ||»7— s'lpk “ J-s^Wp — \\<Pk ~ J-sU\\p 0
as k ->• 00. This proves that 5 is dense in Hs'p.	□
We can now prove Theorem 14.12.

Minimal and Maximal Pseudo-Differential Operators
103
Proof of Theorem 14.12 Let u E iTm,p. Then, by Proposition 14.14, we
can find a sequence {</?*.} of functions in S such that <fk u in Hm'p as
k -+ oo. By Propositions 14.13 and 14.14, {Ta(fk} and {<pk} are Cauchy
sequences in Lp(En). Hence (pk -+ u and Ta<pk -+ f in Lp(En) for some u
and / in Lp(Rn) as k -+ oo. Hence, by the definition of 7V}o, u E £>(7V50)
and Tv^u = f. On the other hand, if u E V(Ta>o), then, by the definition
of 7V,0 again, we can find a sequence {(} of functions in S for which
(fk -+ u in Lp(Rn) and Ta(pk -+ f in Lp(Rn) for some / E Lp(Rn) as
k -+ oo. Hence {(pk} and {Ta(fk} are Cauchy sequences in Lp(Rn). So, by
Propositions 14.13 and 14.14, {<} is a Cauchy sequence in Hm'p. Since
Hm'p is complete by Theorem 12.3, it follows that -+ v in Hm'p for
some v E Hm'p as k -+ oo. Then, by Theorem 12.5, ifk —► v in Lp(Rn) as
k-+ oo. Hence u = v and consequently u E Hm,p.	□
Finally we come to the main result of this chapter.
Theorem 14.15. Let m > 0 and a be an elliptic symbol in Sm. Then
T„,0 =
Proof Since is the smallest closed extension of Ta, it follows from
Proposition 14.7 and Theorem 14.12 that it is sufficient to prove that
©(T^i) C Let u E	Then, by (10.1) in Theorem 10.1,
u = TrT'U - Ru,	(14.8)
where r E 5~m and jR is a pseudo-differential operator with symbol in
CfeeR#*. By Proposition 14.4, Ta,\u = Tau in the distribution sense. Thus,
by the definition of IV,i, Tau E Lp(Rn). Since r E 5_m, it follows from
Theorem 12.9 that TTTau E Hm'p. Since u E Lp(En) and has symbol
in 5”m, it follows from Theorem 12.9 again that Ru E Hm'p. Hence, by
(14.8), u E ifm’p.	□
Exercises
14.1.	Let a be any symbol in Sm, m < 0. Prove that the minimal operator
Ta>o oi Ta : S -+ S in Lp(En), 1 < p < oo, is a bounded linear operator
from Lp(En) into Lp(En).
14.2.	Let a be any symbol. Consider TV as a linear operator from Lp(En)
into Lp(En), 1 < p < oo, with dense domain S. Prove that T* = (TV,o)*-
14.3.	Let a and Ta be as in Exercise 14.2. Prove that = (T*)i, where
(T*)i is the maximal operator of the pseudo-differential operator T*.

104
An Introduction to Pseudo-Differential Operators, 3rd Edition
14.4.	For any closed linear operator A from a complex Banach space into
itself with domain V(A), a subspace V of T>( A) is called a core of the
operator A if the minimal operator of the restriction of A to V is equal to
A. Prove that if a is any elliptic symbol, then Co°(Mn) 1S a core °f ^<r,o and

Chapter 15
Global Regularity of Elliptic Partial
Differential Equations
Let P(x,D) = £,a,<m act(x)Da be a linear partial differential operator of
order m such that
sup \(D^aa)(x)\ < oo, \a\ < m,	(15.1)
for all multi-indices /?. Then we have observed in Example 6.3 that P(x, D)
is a pseudo-differential operator with symbol P(x,f) in Sm, where
P(x,0=	a<* (*)£“•
\a\<m
The purpose of this chapter is to use the theory of pseudo-differential
operators we have developed to prove the following Lp analog of a result in
[Hess and Kato (1970)].
Theorem 15.1. Let P(x,D) = 2|a|<m aa(x)Da be a linear partial dif¬
ferential operator of order m satisfying (15.1). Suppose that there exists a
point xq G Rn such that we can find positive constants C\ and (¼ for which
and
M*o)C
\a\=m
(15.2)
(a<*(x) ~ aa(xo))C
\a\=m
< C*\£T
(15.3)
for all x and $ G In, and C% < C\. If u G H8*p, P(x,D)u = f and
f G H8*, then u G H8+m*.
105

106
An Introduction to Pseudo-Differential Operatorsf 3rd Edition
Remark 15.2. If P(x, D) is a linear partial differential operator of order
m satisfying the hypotheses of Theroem 15.1, and if / is any tempered
distribution in #5,p, then Theorem 15.1 asserts that any solution u in H8,p
of the partial differential equation P(x,D)u = / on ln lies in a more
“selective” or “regular” space iJs+m>p. If we recall, by Theorem 12.5, that
jjs+m,p £ jjs+m—ltp q ... q fls>P
then the solution u can be thought of being m steps more “selective” or
“regular” than the given source data / defined globally on Rn. For this
reason, we call Theorem 15.1 a global regularity theorem.
To prove Theorem 15.1, we use the following lemma.
Lemma 15.3. Let P(x,D) = Yl\a\<ma<x(x)Da be a linear partial differ¬
ential operator of order m satisfying the hypotheses of Theorem 15.1. Then
there exist positive constants C and R such that
\p(x,o\>c(i+\s\r, ifi >r.
Remark 15.4. Lemma 15.3 tells us that under the hypotheses of Theorem
15.1,	P(x,D) is a pseudo-differential operator with symbol P(x,$) in Sm
and P(x,£) satisfies the ellipticity condition defined in Chapter 10.
Proof of Lemma 15.3 By (15.2) and (15.3), we obtain
|a|=m
Y - Mxo))fa + y a*(xo)£a
| a |=77i
| a |=7Ti
>
Y a<*(xo)fa
| a | =777
>{Cx-c2mm,
- Y (“oW ~ Oo(*o))f“
|a|=m
x,i € K".
(15.4)

Global Regularity of Elliptic Partial Differential Equations
107
Next, by (15.4), we can find positive constants C", C" and R such that
Y °a(®)£“
\a\<m
Y aa(x)Za+ Y ««(*)£“
\a\=m	|a|<m
>
Y o«(*)€“
—
^ o0(x)r
|o: 1=771
|a|<77l
> c'(l+Kir - c"(i+Kir-1
= (1+ 1^(^-^(1 + 1^1)-1). Kl >R-
Obviously, we can find another positive constant Ri > R such that
c' - c"(i + Kl)_1 > y, Kl > Ri-
Hence, by (15.5) and (15.6),
(15.5)
(15.6)
Y Maor
| a | <ra
>y(l + kl)m,
This proves Lemma 15.3.
□
Proof of Theorem 15.1 By Lemma 15.3, P(x,D) is an elliptic pseudo¬
differential operator with symbol P(x, £) in 5m. Hence, by Theorem 10.1,
we can find a symbol r E S~m and a pseudo-differential operator R with
symbol in OkeRSk such that
TrP{x,D) = I + R.	(15.7)
Hence, by (15.7), u = Tr/ — Ru. Since r E S~m and / E H8'p, it follows
that Tr/ E	Also, Ru E Hs+m'p because P has symbol in 5~m
and u E Hs'p. Hence, by (15.7), u E JIa+m*	□
Exercises
15.1.	Let P(x, D) be a linear partial differential operator of order m satis¬
fying all the hypotheses of Theorem 15.1. Prove that if / E HSyP, then any
solution u in UteRHtiP of the partial differential equation P(x, D)u = f on
Mn is in Hs+m'p.

108
An Introduction to Pseudo-Differential Operators, 3rd Edition
15.2.	Let m > 0 and a £ Sm be an elliptic symbol. Let / £ Lp(Rn), for
1 < p < oo and u be any solution in Ut&iHttP of the pseudo-differential
equation Tau = / on Rn.
(i)	Prove that u £ Lp(Rn).
(ii)	Prove that (u,T£ip) = (/, p) for all ip £ S.
(iii)	Prove that there exists a sequence {(pk} 0f functions in S such that
<Pk -> u in Lp(Rn) and ^ / in Lp(Rn) as Ar oo.

Chapter 16
Weak Solutions of Pseudo-Differential
Equations
We give in this chapter a result on the existence of weak solutions in
Lp(En), 1 < p < oo, of pseudo-differential equations on En. We begin
with the definition of a weak solution.
Definition 16.1. Let a G Sm, m > 0, and / G Lp(En), 1 < p < oo. A
function u in Lp(En) is said to be a weak solution of the pseudo-differential
equation T^u — f on En if
(u,t;<p) =	<pes,
where T* is the formal adjoint of Ta introduced in Chapter 9.
From the definition of the maximal operator Tffii of Ta given in Chapter
14, it is obvious that the following proposition is true.
Proposition 16.2. Let a G Sm, m > 0, and f G Lp(En), 1 < p < oo.
Then a function u in Lp(En) is a weak solution of the pseudo-differential
equation Tau = f on W1 if and only ifue ^(T^i) and Ta^u = /.
For any a G Sm, m > 0, the following theorem characterizes the func¬
tions / in Lp(En), 1 < p < oo, for which the pseudo-differential equation
Tau = / on En has a weak solution u in Lp(En).
Theorem 16.3. Let a G Sm, m > 0, and f G Lp(En), 1 < p < oo. Then
the pseudo-differential equation Tau — f on W1 has a weak solution u in
Lp(En) if and only if there exists a positive constant C such that
1(/, ^)1 <c\\t;v\\p,, ves,	(16.1)
where p1 is the conjugate index of p.
109

110
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof Suppose that Tau = / on ln has a weak solution u in Lp(Rn).
Then, by Definition 16.1,
(f,y) = (u,T;<p),	<p<=S.
Hence, by Holder’s inequality,
1(/,01 <IMIplir>||P-,
and the inequality (16.1) holds with C = IM|P. Conversely, suppose that
the inequality (16.1) is true. Let W be the subspace of Lp (W1) defined by
W = {T> :<peS}.
We define the linear functional F :W —» C by
Fw = (<£,/), w e W,
where <p is any function in <S with the property that T*tp = w. To see that
the defintion of F : W —> C is independent of the function <p, let and (p2
be functions in S such that T*p>i = w and T*<^2 = w. Then, by (16.1),
Hence (ip\,/) = (<^>2, /) and this proves that the choice of the function ip is
irrelevant to the definition of F : W -»■ C. Since, by (16.1),
l-FH = \{<P,f)\ < C\\T*M\P, = CIHIp-, wew,
it follows that F : W -» C is a bounded linear functional. Hence, using the
Hahn-Banach theorem and the Riesz representation theorem, we can find
a function u in Lp(Rn) such that
Fw = (</>, /) = (w,u), we W,	(16.2)
where ip is any function in S satisfying T*<p = w. Since {T*ip : ip e S} is
obviously a subspace of W, it follows from (16.2) that
iv>, f) = (T*ip,u), tpeS,
and hence, by Definition 16.1, u is a weak solution in Lp(Rn) of the pseudo¬
differential equation Tau = f on Rn.	□

Weak Solutions of Pseudo-Differential Equations
111
Exercises
16.1. Let <r G Sm, m > 0, and let u and / be in Lp(Rn), 1 < p < oo. Prove
that u is a solution of Tau — f on Rn in the distribution sense if and only
if u is a weak solution of Tau = / on Rn.
16.2.	Let <j G Sm, m > 0, be an elliptic symbol, and let / G Lp(Mn), for
1 < p < oo. Prove that every weak solution u in Lp(Rn) of the pseudo¬
differential equation Tau — f on tn is in Hm'p.
16.3.	Let s > 0 and J-s be the pseudo-differential operator defined in
Chapter 12. Let q be any real-valued and nonnegative function on Rn such
that
sup \(Daq)(x)\ < oo
xeRn
for all multi-indices a. Prove that the pseudo-differential equation
J-Su + qu = f
on Rn has a weak solution u in L2(Rn) for every function / in L2(Rn).
16.4.	Let cr(x,£) G Sm, m > 0, be such that a is independent of x G Rn.
Prove that the pseudo-differential equation Tau = / on Rn has a unique
weak solution u in L2(Rn) for all functions / in L2(Rn) if and only if there
exists a positive constant C such that
\*m >c, £ £ Rn.

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Chapter 17
Garding’s Inequality
We are interested in a subset of the set of all elliptic pseudo-differential
operators introduced in Chapter 10. These operators satisfy an important
inequality in the study of pseudo-differential operators.
Theorem 17.1. (Garding’s Inequality) Let a € S2m be such that there
exist positive constants C and R for which
Re<r(*,0 > <?(1 + |£|)2"\ Id >R.
Then we can find a positive constant C' and a constant Cs for every real
number s > | such that
R*C> c'Mtz ~ c.Mt-.* v e s.
A symbol satisfying the hypothesis of the theorem is said to be strongly
elliptic. In order to prove the theorem, we need two lemmas.
Lemma 17.2. Let F be a C°° function on the complex plane C. Then for
every a in S°, F o a £ S°.
Proof We need to prove that for all multi-indices a and /3, there exists a
positive constant Ca,/3 such that
|(0“af(Fo<r))(x,0| <Cajj(l + |d)“l'J|, z,£eir. (17.1)
(17.1)	is true for all multi-indices a and /3 with \a + P\ = 0. Indeed, there
exists a positive constant C such that
K*.0I<C,	x,£ € 1".
Thus, F o a is in fact a bounded and C°° function on r xln. Hence there
exists another positive constant C' such that
|(F o cr)(a;,£)| < C',	1,^1".
113

114
An Introduction to Pseudo-Differential Operators, 3rd Edition
Now, suppose that (17.1) is valid for all C°° functions F on C, a in S° and
multi-indices a and /? with \a + 0\ = l. Let a and /3 be multi-indices with
\a + /?| = / + 1. We first suppose that
=
for some multi-index 7 and some j = 1,2,..., n. Then, by the chain rule,
(3?qf (F 0 <r))(*,0 =	0 <T)d(ja + (F2 o
for all x and £ in Rn, where F\ and F2 are the partial derivatives of F with
respect to the first and second variables respectively. Now, by Leibniz’s
formula and the induction hypothesis, there exist positive constants Cpj
and CatPntsj such that
SI7
= ^.7,i(i + ieir(l7l+1)> *,£eRn,
where
Cp,5Ca,p,'Y,5,j-
Similarly, there exists a positive constant Cfa/y j such that
mdim O a)^a}(*,0l < C'a^(l + |6-(M+1\
for all x and £ in W1. Therefore
m%{F 0 *))(*, 0| < (Ca,Jtj + C'a^){ 1 + K|)-I*l
for all x and £ in Rn. Now, we suppose that
= d2dXidl
for some multi-index 7 and some j = 1,2,..., n. Then, as before, there
exists a positive constant Canj such that
m$(F 0 *))(*, 01 < <7)j( 1+ki)-m

Garding’s Inequality
115
for all x and £ in Mn. Thus, by the principle of mathematical induction,
(17.1)	follows.	□
Lemma 17.3. Let a be a strongly elliptic symbol in S2m, m > 0. Then
there exist positive constants 7 and k such that
Rea(x,£) > 7<£>2m -	€ 1",
where ( ) is the function on Mn defined by
<£) = (i + I£I2)1/2> feir*.
Proof By strong ellipticity, there exist positive constants C and R such
that
Re<7(x,£)>C<£>2m, Kl >R-
Since a G 52m, we can find a positive constant K such that
k(*,0l<*(02m. *>£eir.
Therefore
|Re<r(x, ^)| < K(02m < K(1 + R2)m,	\£\ < R.
Hence there exists a positive constant M such that
Re a(x, £) > -M,	|£| < R.
Since ^ is continuous on the compact set {£ € W1 : |£| < R}, we can
find a positive constant k such that
Re<r(a;,£)
^2m-l
>
Kl < a
Therefore
Re<7(x,£) + K<02ro-1 >0, \(\<R.
Since Re7^.ir~ - is a positive and continuous function on the compact
set {(el": |£| < R}, there is a positive constant S such that
Re<r(x,£) + «(02m-1
(f>
2m
>J,	|£| <R.
So, the lemma is proved if we let 7 = min(C',<5).
□
Proof of Theorem 17.1 Let Tt =	where Jm is the Bessel
potential of order m. In fact, Jm =	)-*». Then, using the asymptotic

116
An Introduction to Pseudo-Differential Operators, 3rd Edition
expansion for the product of two pseudo-differential operators in Theorem
8.1,
where
Tn-Jm — TT
Similarly,
and
ri - ( )'ma e Sm-\
Tr = JmTTl
(17.2)
r-()-m7i GS"1.	(17.3)
Multiplying (17.2) by ( )“m and adding the result to (17.3), we get
r - ()~2ma e S'1.
Therefore
r = ( )~2ma + r,
where r € S~l. So, by Lemma 17.3,
Rer = ()“2mRecr + Re r > 7 — k( )_1 + Rer > 7 — /c'( )_1,
where k! is another positive constant. Therefore r satisfies the conclusion
of Lemma 17.3 with m — 0. Let us suppose for a moment that Garding’s
inequality is valid for m = 0. Then we can find a positive constant C’ and
a positive constant C8 for every real number s > \ such that
Re ip) = Re (J-mTr <p)
= Re (TrJ-mtp,
— & ll^-m^|lo,2 “ Cs\\J-m(P\\‘LSi2
for all ip in S. We are now ready to prove Garding’s inequality for m = 0.
By Lemma 17.3, we have positive constants 7 and k, such that
Re<r 4- k( )-1 > 7.
Let F be a C°° function on C such that

G&rding’s Inequality
117
Let r be the function defined on Rn x Rn by
t(x,0 = F(2(R.e<r(x,£) + k(£)-1 —7)),	x,£ € Kn.
Then, by Lemma 17.2, r € 5°, and for all x and £ in Rn,
f(*,0 =	+2Reff(*,f)+ 2^(0-1-27
= ^2Rfi<r(x,0 +2k(£>-1 - |t-
Using the asymptotic expansion for the formal adjoint of a pseudo-
differential operator in Theorem 9.1, we have
where r* G 5° and r - r* G S”1. Using also the asymptotic expansion for
the product in Theorem 8.1,
t;tt = rA,
where
A-tVgS"1.
If we let ri and r[ in 5_1 be such that
r* = r + r\
and
A =
*T + rl,
then, with r2 = nr + r[ G S"1,
A = (r + ri)r + 7^ = 2Recr + 2k( )-1
So, if we let r3 = 2k( )_1 + r2 G S-1, then we get
3
A = 2Recr — -7 + r3.
z
But
3
27 + r2-
2Re 0- = 0- + 0 = 0- +0*+7*4
for some r4 in S 1. Therefore
A = 0- + 0* - -7 + r5
(7 + 0-* = A + -7 - r5.
for some 7*5 in S Thus,

118
An Introduction to Pseudo-Differential Operators, 3rd Edition
Since
(T\(p,tp) = (TT(p,TT(p) >0, <p G <S,
it follows that
2Re (Tfip, (p) — (Tpipiip) + (Tpipytp) —	cp)
= (T\<P, <p) + ^711^110,2 “ (Trs<P,<p)
> i\Mh + {%Mh - MINIMI-*,*} •
Using the L2-boundedness of pseudo-differential operators in Theorem 11.7,
we get a positive constant // such that
2Re (T„<p,(p) >7lMlo,2 + {|lMlo,2-A*IMllj,2}» V e S-
But
= J ^0^10(012 d£ = I + J,
where
I=[ p(Q-l\m\2dt
and
j=f
Obviously,
i <1 JRnmo\2dt=l\Mi2-
To estimate J, we note that
^(0-1 > | =*• <0 < y •
So, for /t(£)_1 > 2> we Set> f°r every real number s >
Hit)-1 = M(O2'-1<0-a*
^rv*.
Then for every real number s >
j<»(-)2° 1 f <0-2*i^(0i2de=c;iMii.,2,
\ 7 / JRn
/o \ 2s—1
where Cfs =	. Therefore
2Re (Tffip,tp) > Tlbll2i2 - CSlM|l.,a, V € S,
and the proof of the theorem is complete.
□

Gdrding’s Inequality
119
Exercises
17.1.	Prove that a strongly elliptic pseudo-differential operator is elliptic.
17.2.	Give an example of an elliptic pseudo-differential operator that is
not strongly elliptic.
17.3.	Let a e 52m, m > be such that a is strongly elliptic. Prove that
there exists a constant C such that
Re <p) > -CIMI*, V € S.

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Chapter 18
Strong Solutions of
Pseudo-Differential Equations
Let a G Sm, m > 0, and let / G Lp(Mn), 1 < p < oo. From Chapter 16 we
see that a function u G Lp(Mn) is a weak solution of the pseudo-differential
equation Tau = / on Mn if u G V(Ta,i) and TV,iu = /. In this chapter we
give another notion of a solution of the equation. A function u G Lp(Rn)
is said to be a strong solution of the equation = / if u G X^T^o) and
XV,	= / •
Remark 18.1. For the pseudo-differential equation T^u = / on Rn, it
should be clear that strong solutions axe weak solutions. If a is elliptic,
then it is also true that weak solutions are strong solutions. These simple
facts are best left as exercises. See Exercises 18.1 and 18.2.
We begin with strong solutions in L2(Rn).
Theorem 18.2. Let a G S2m, m > 0, be an elliptic symbol such that there
exists a positive constant C for which
Re(2>,¥>)>C|M|^2, ¥>€5.	(18.1)
Then for every function f in L2(Rn), the pseudo-differential equation
Tau = / on Rn has a unique strong solution u in L2(Rn).
We need the following lemma.
Lemma 18.3. Under the hypotheses of the preceding theorem, there exists
a positive constant C such that
Re (T„u,u) >C|H|^2,
Proof Let u G Hm'2. Then, by Proposition 14.14, there exists a sequence
{(fk) of functions in S such that	u in Hm,2 as k oo. By Theorem
121

122
An Introduction to Pseudo-Differential Operators, 3rd Edition
12.9, Ta : Jfm>2 H~m'2 is a bounded linear operator. Hence, by Exercise
12.14, there exists a positive constant C' such that
\{Ta(pk,<Pk) - {TaU,u)\
—	1CTaVkitpk) ” {TGU,(pk) + (TgU, (fk) — ti) |
= |(2V(<p* - ti),^fc) + (Tau,(pk - w)|
—	— 11—fn,21|^/j11771,2 H" |I^V^II —m.,21— ^||m,2
^ C7 ||^fe	^11m-,21111m-,2 “1“ II^Hra^H^fc	0
as k —> oo. Therefore
Re(Tau,u) = lim Re(T^ip^tpk) > C lim \\<Pk\\m,2 = IMIm,2-
k—tOO	ft—HX>
□
We also need the following fact, which is a generalization of the Riesz
representation theorem.
Theorem 18.4. (Lax—Milgram Lemma) Let X be a complex and sepa¬
rable Hilbert space with inner product and norm denoted by (,)x awd || \\x
respectively. Let B be a bilinear mapping on X such that there exist positive
constants C\ and (¾ for which
\B(x,y)\ < Cillxllxllyllx, x,y € X,
(18.2)
(18.3)
Then for every bounded linear functional f on X, there exists a unique
vector y e X such that
f(x) = B(x,y), x£X.
Proof For a fixed vector y in X, we see from (18.2) that B(-,y) is a
bounded linear functional on X. Using the Riesz representation theorem,
there exists a unique vector z(y) G X such that
B(x,y) = (x,z(y))x, x€X.
We observe that the mapping
XBy^ z{y) € X

Strong Solutions of Pseudo-Differential Equations
123
is linear. Indeed, let y\ and y2 be vectors in X and let ci and C2 be complex
numbers. Since B is bilinear,
(x, z(ciyi + c2y2))x = B(x,cxyi + c2y2)
= c^B(x,yx) +c^B(x,y2)
= ci(x,z(yA))x +ci(x,z(y2))x
= (x^xziyi^x + (x,c2z(y2))x
= (x,c1z(yi)x + c2z(y2))x, x € X.
Therefore
z(ciyi + c2y2) - ciz(yi) + c2z(y2)
and linearity is established. Let M be the subspace of X given by
M = {z{y): y £ X}.
Then M is a closed subspace of X. Indeed, let {z(yk)} be a sequence in M
such that
z(Vk) -*• z
as fc —^ oo. Then for j,k = 1,2,...,
B(x,yj - Vk) = (x, z(yj) - z(yk))x, x e X.
By (18.3) and the Cauchy-Schwarz inequality, we get for j,k = 1,2,...,
(¼||Vj ~ Vk\\x < \B(Vj ~ Vk,Vj ~ Vk)\
= \{Vj-Vk,z{yj)-z(yk))x\
< \\Vj ~ Vk\\x \\z(yj) - z(yk)\\x-
So, for j, k = 1,2,...,
CzWVj - VkWx < ||z{yj) - z{yk)\\x-
Thus, {yk} is a Cauchy sequence in X and hence
Vk^y
for some y in X as k -¥ 00. By (18.2),
\B(x,yk -y)| < CiUarllxlly* -y||x -► 0
as k —¥ 00 for all x € X, which gives
B(x,yk) -¥ B(x,y)

124
An Introduction to Pseudo-Differential Operators, 3rd Edition
as k oo. Furthermore, for all
(z>2(2/fc))x -* 0*>*)x
as k ->* oo. Since
B(x,yk) = (a?,z(i/*))x, x6l,
for fc = 1,2,..., it follows that
B(x,y) = {x,z)x, xeX.
Therefore z E M and this proves that M is closed. Now, M = X. To see
this, let us assume that M is a proper (and closed) subspace of X. Then
there exists a nonzero vector x G X such that
B(x, y) = (x, z(y))x = 0, y € X.
If we let y = x, then by (18.3), x = 0 and this contradiction implies that
M = X. By the Riesz representation theorem, there exists a unique vector
w G X such that
f(x) = (x,w)x, xeX.
Since X = Af, we can find a vector y € X such that w = z(y). Therefore
/Or) = (x,ti7)x = (®,3(y))x =	xeX.
The uniqueness of y follows again from (18.2).	□
Proof of Theorem 18.2 Let B : Hm,2 x Hm>2 -> C be the bilinear
mapping defined by
B(u,v) = (u,Tav), u,v € if771’2.
Then for all u and v in if771’2, we see by means of Exercise 12.14 that
\B(U,V)\ < |M|m,211^11^,2 < |M|mf2|M|m,2
and, by Lemma 18.3,
\B(u,u)\ > \(T„u,u)\ > C\\u\\2m,2, u € Hm’2.
Let / € L2(En). Then we define the linear functional F : Hm’2 -¥ C by
F(w) = (w,f), w e Hm’2.
It is a bounded linear functional because, by Exercise 12.14 and Theorem
12.5,
\F{w)\ = 1(1«,/)| < |MU,2||/ll-m,2 < ||/||2||tl>||m,2,
W € Hm’2.

Strong Solutions of Pseudo-Differential Equations
125
So, by the Lax-Milgram lemma, we get a unique function u in Hm'2 such
that
F(w) = B(w,u), w G Hm'2,
or equivalently,
(w, f) = (w,Tau), w G Hm'2.
So, u is a weak solution in L2 (Rn). Since a is elliptic, it follows from Exercise
18.2 that u is a strong solution in L2(Rn).	□
Another Proof of Theorem 18.2 By Theorem 12.5, the inequality (18.1)
and the Cauchy-Schwarz inequality, we get for all functions <p € S,
M\l < IMIm.2 < £lMhl|T>||2
and hence
IMk < 1||T>||2.
Let f € L2(R"). Then
m>p)\ < 11/H2IMI2 < 1||/ii2||i;vii2i vg<s.
So, by Lemma 18.3, the pseudo-differential equation Tau = / on En has
a weak solution u in L2(Rn). Since a is elliptic, it follows that u is also
a strong solution in L2(Rn). Let v be another strong solution in L2(Rn).
Then, by Theorem 18.3,
llu - Him,2 ^ ^Re (T^(u ~ V), U - V) = 0.
So, u = v and u is the unique solution.	□
We can now give sufficient conditions in terms of the symbols a for
the existence and uniqueness of strong solutions in L2(Rn) for pseudo-
differential operators Ta.
Theorem 18.5. Let a G 52m, m > 0, be a strongly elliptic symbol Then
there exists a real number Ao such that for all f in L2(Rn) and A > Ao, the
pseudo-differential equation (Ta -I- AI)u = / on Rn, where I is the identity
operator on L2(Rn), has a unique strong solution u in L2(Rn).
Proof By Garding’s inequality, there exist constants A and Ao such that
A > 0 and
Re (Tm p) > A|M|*,>2 - AolMli, p € S.

126
An Introduction to Pseudo-Differential Operators, 3rd Edition
Then for A > A0,
Re ((T„ + \I)<p, ip) > A\\<p\&a + (A - A0)|M|| > A\\<p\&ti,	€ <S.
Thus, by Theorem 18.2, the proof is complete.	□
The following theorem is a result on the existence and uniqueness of
strong solutions in Lp(Rn).
Theorem 18.6. Let a G 5m, m > 0, be an elliptic symbol such that a is
independent of x in Rn and
£git.
Then for every function f G Lp(Rn), 1 < p < oo, the pseudo-differential
equation Tau = / on Rn has a unique strong solution u in Lp(Rn).
Proof By Exercise 18.3, it is sufficient to prove that there exists a positive
constant C such that
IMIp' <	^ s.
Let r — 1/(7. Then for all multi-indices a, we use (1.4) to obtain
daT = Y Cam
><*(*° ^+1
where C^u),...,<*(*) is a constant depending on ... and the sum is
taken over all multi-indices ,..., that partition a. Thus, we can
find positive constants Ca(i),..., Ca(k) such that
ctt(D- -caW(i + |£l)*ro_w
k(0lfc+1
for all f G Mn. Since a is elliptic, we can find positive constants C and R
such that
k(0l>c'(i + ior, Kl >R-
Since	is a continuous and positive function on the compact set
{£ G Rn : |£| < i?}, it follows that there exists a positive number 6 such
that
k(0i>*(i + ifi)m, iei < «-
Therefore there exists a positive constant C' such that
Koi>c'(i+Kir, ^ g Rn.

Strong Solutions of Pseudo-Differential Equations
127
So, there exists a positive constant C" such that
|(fl*T)(0| < c"( 1 + I£|)"m“w, £ € Kn.
Therefore r E S~m. By Theorem 12.9 and the Sobolev embedding theorem
in Chapter 12, we get a positive constant Cm such that
Wvb = \\TtT*p\\p' < C"'\\T^p\\p'. V e 5.
It remains to prove uniqueness. Let u and v be strong solutions in Lp(Rn).
Let w = u — v. Then w E ifm,p and T^w = 0 on Rn. Thus,
aw = 0
in the sense of distributions. So,
w{ap) = (aw)(ip) = 0, ip G S.	(1S.4)
For all if) £ 5, we can find a function ip G 5 such that ap = That this
can be done is Exercise 18.5. Thus, by (18.4),
w(ip) = 0, ip € S.
So, w = 0 and hence w = 0. Therefore u = v and uniqueness is proved. □
Exercises
18.1.	Let a € Sm, m > 0. Let / € Lp(Rn) for 1 < p < oo. Prove that a
strong solution u in Lp (En) of the equation Tau = / on ln is also a weak
solution in Lp(Rn).
18.2.	Let a € Sm, m > 0, be an elliptic symbol. Let / G Lp(Rn), where
1 < p < oo. Prove that every weak solution u in Lp(Rn) of the equation
Tau = / on Mn is also a strong solution in Lp(Rn).
18.3.	Let cr E Sm, m > 0, be elliptic and such that there exists a positive
constant C for which
M\p>< c\\t;<p\\p,, <pes.
Prove that the pseudo-differential equation Tau = / on Rn has a strong
solution u in Lp(Rn) for every function / € Lp(Mn), 1 < p < oo.
18.4.	Let a be as in Exercise 18.3 and such that {T*cp : ip € S} is dense
in Lp' (Rn). Prove that the equation Tau = / on Rn has a unqiue strong
solution u in Lp(Rn) for every function / G Lp(Rn), 1 < p < oo.
18.5.	Let a G Sm, m > 0, be an elliptic symbol such that a is independent
of x in Rn and
*(0^0,	£ E Rn.
Prove that for all functions xp G <S, there exists a function ip G 5 such that
CT(£ = Xp.

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Chapter 19
One-Parameter Semigroups
Generated by
Pseudo-Differential Operators
Let A be a closed linear operator from a complex Banach space X into X
with dense domain V(A). Let / € X. Then we are interested in the initial
value problem for the heat equation governed by A given by
f u'(£) = A(u(t)), t > 0,
l u(0) = /,
where u : [0, oo) —> X and
u'(t) = lim
v '	h->o
u(t + h) — u(t)
h
if the limit in X exists. The questions to be answered are related to the
existence and uniqueness of a global solution u : [0, oo) -> X.
It is intuitively clear that the solution u : [0, oo) -* X is given formally
by
u(t) = etAf, t > 0.
What is etA when A is a closed linear operator densely defined on a complex
Banach space X? The aim of this chapter is to answer this question.
A family {T(t) : t > 0} of bounded linear operators on X is said to be
a one-parameter semigroup if
(i)	T(0) = /, where I is the identity operator on X,
(ii)	T(s)T(t) = T(s + £), s, t > 0,
(iii)	T(t)x —> x in X as t -¥ 0+ for every x in X.
Let {T(t) : t > 0} be a one-parameter semigroup on X. We denote by
T>(A) the set of all elements x in X such that lim*_*.o+ T^*~x exists in X.
Proposition 19.1. V(A) is a dense subspace of X.
129

130
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof Let W be the subspace of X given by
W= jxs = i^ T(S)xdS : x 6 X, s > o|.
Then W is dense in X simply because xs -+ x in X for all x € X as s -+ 0+.
For t > 0, let
Let s and t be positive numbers. Then for t < s,
= T(t) fg T(6)xdS — J0a T(S)xd6
1 s	st
_ j‘ T(t + 5)x d6 - /0® T(8)x d6
st
f"+t T(5)x d5 - Jq T(S)xdS
st
_ Ss+tT{S)xd8 - /q T(5)xdS
st
= Asxt.
Thus, Atx8 -+ Asx in X as t -+ 0 +. Therefore W C V(A) and hence V(A)
is dense in X.	□
The linear operator A from X into X with domain V(A) defined by
Ax = lim	—x ? x e 'D(A),
t—>o+	t
where the limit is again understood to take place in X, is called the in¬
finitesimal generator of the one-parameter semigroup {T(t) : t> 0}.
Proposition 19.2. The infinitesimal generator A of a one-parameter semi¬
group {T(t) : t > 0} on a complex Banach space X is a closed linear oper¬
ator.
We need two lemmas to prove Proposition 19.2.
Lemma 19.3. Let A be the infinitesimal generator of a one-parameter
semigroup {T(t) : t > 0} of bounded linear operators on a complex Ba¬
nach space X. Then for all t € [0, oo) and x G T)(A), T(t)x € T>(A) and
AT(t)x = T{t)Ax, x e V(A).

One-Parameter Semigroups Generated by Pseudo-Differential Operators
131
Proof Let x G V(A). Then for all t G [0, oo),
T(t)AhX = AhT(t)x, h > 0.
Therefore
lim AhT(t)x = lim T(t)AhX = T(t) lim AhX = T(t)Ax.
h—>-0	h—>0	h—>0
Thus, T(t)x G V(A) and
AT(t)x = T{t)Ax,
as asserted.	□
Lemma 19.4. Let x G V(A), where A is the infinitesimal generator of a
one-parameter semigroup {T(^) : t > 0} on a complex Banach space X.
Then for all t > 0,
T(t)x-x= f T(s)Axds.
Jo
Proof Let / be a bounded linear functional on X. Then we define the
function F : [0, oo) -> C by
F(t) = f (r(t)x - x - j T{s)Axds^ , t > 0.
So, for t > 0, the right-hand derivative F'(t-h) of F at t is given by
F'(t+) = f(AT(t)x-T(t)Ax)
in view of the definition of the infinitesimal generator and the fundamental
theorem of calculus. By Lemma 19.3,
F'(t+) = 0, t G [0, oo).
Thus, F(t) is a constant for all t G [0, oo). Since F(0) = 0, it follows that
F(t) = 0, t > 0.
Thus, using the Hahn-Banach theorem, we see that
T(t)x-x= f T(s)Axds,	£ > 0.
Jo
□
Proof of Proposition 19.2 Let {Xj} be a sequence in V(A) such that
Xj x

132
An Introduction to Pseudo-Differential Operators, 3rd Edition
T(t)x — x = lim (T(t)xj - Xj) = lim / T(s)Axj ds =/ Tsyds.
Therefore x € T>(A) and Ax = y. Thus, A is a closed linear operator. □
When is a closed linear operator A from a complex Banach space X into
X with dense domain V(A) the infinitesimal generator of a one-parameter
semigroup on X? The answer is given as the content of the celebrated
Hille-Yosida-Phillips theorem.
We first introduce some notation and terminology. Let A be a closed
linear operator from a complex Banach space X into X with dense domain
V{A). We define the resolvent set p(A) of A to be the set of all complex
numbers A such that XI — A : T>(A) X is bijective, where I is the
identity operator on X. If A £ p(A), then the bounded linear operator
(AI — A)~l : X X is called the resolvent of A at A and is often denoted
by R(A; A). The norm a bounded linear operator A from a complex Banach
space X into another complex Banach space Y is denoted by ||A||.
Theorem 19.5. (Hille-Yosida-Phillips Theorem) Let A be a closed
linear operator from a complex Banach space X into X with dense domain
V(A). Then A is the infinitesimal generator of a one-parameter semigroup
{T(t) : t>0} on X if and only if we can find a positive number M and a
real number u such that
for all A in (u, oo) and all positive integers n.
Proof Let us first prove the sufficiency. For A > a;, we define the bounded
linear operator B\ : X -* X by
j—+oo
Hence
{A 6 1: A > Go*} C p(A)
and
Bx = -X(I-XR(X;A)).

One-Parameter Semigroups Generated by Pseudo-Differential Operators
133
Then for alH G ®,
etB, =e-Xtf^&lR{X.A)
n\
n=0
and therefore
So, for every in (a;, oo),
||etBx|| < Met(Jl
for all sufficiently large A. Now, we claim that
lim B\x = Ax, x G T>(A).
(19.2)
(19.3)
Indeed, let x G T>(A). Then
XR(X; A)x - x = XR(X; A)x - R(A; A) Ax 4 R(A; A)i4x - x
Since V(A) is dense in X, it follows from a standard limiting argument that
for every x in X,
XR(X; A)x -4 x
in X as A -4 oo. But for all x in V(A),
5ax = -Ax 4* X2R(X; A)x
= -Ax + A#(A; i4)Ax - XR(X; A)Ax 4 XR(X; A)Ax
= XR(X-,A)Ax
and hence we get
= R(A; A)(XI - A)x 4 #(A; A) Ax - x
= x 4 #(A; A)Ax — x
= R( A; A)4x
and hence
M
\\XR(X;A)x-x\\x = \\R(X;A)Ax\\x < t	\\Ax\\x -4 0
A — CO
as A -4 oo. Moreover, for all sufficiently large A,
||Ai?(A; A)|| < M-r-^— < 2M.
X — oj
B\x = A.R(A; A) Aa: -»■ Aa:

134
An Introduction to Pseudo-Differential Operators, 3rc^ Edition
as A -* oo. For A > cj, let
S\(t) = etB\ t € R.
Using the first resolvent formula in Exercise 19.4, we know that
R(A;	A) = R(»; A)R(A; 4), A, /x > w.
Thus,	= B^Bx and we get
= SA(t)BM
for all A and fi in (w, oo), and all t in M. Hence for all x G B[A) and
t € [0, oo),
Sx(t)x - SM(t)s = J ^[SM(< “ *)$*(*)*] ds
= [* 5„(* - s)Sx(s)(Bx - B^x ds. (19.4)
Jo
By (19.2) and (19.4), there exists a positive constant A0 such that
||Sx(t)x - SM(t)x||x < M2tetu'\\Bxx - B»x\\x, t > 0,	(19.5)
for all A and p in (A0,oo). By (19.3) and (19.5), we see that for all x in
V(A) and all t in [0, oo),
||Sx(t)x - S„(f)x||x -> 0
as A and n tend to oo. So, in view of (19.2), we see that for all x in X and
all t in [0, oo),
||S\(t)x-S/t(x)||x
as A and n tend to oo. Therefore for all x in X and all t in [0, oo), we define
T(t)x by
T(t)x = lim Sx(t)x.
A—yoo
It is then easy to see that T(t) is a bounded linear operator on X and
||T(t)|| < MetUl	(19.6)
for all t in [0, oo). Let x € V{A). Then, by (19.5), Sx{t)x -> T{t)x uniformly
with respect to t on compact subsets of [0, oo) as A —> oo. Therefore for all
x in X, the mapping
[0, oo) T(t)x G X
is continuous. To check the semigroup property, we first note that
T(0) = lim 5A (0) = /.
A-»oo

One-Parameter Semigroups Generated by Pseudo-Differential Operators 135
Moreover, for 0 < s, t < 00 and all x in X,
T(s + t)x = lim Sx(s + t)x = lira Sx(s)Sx(t)x
\—>00	A—>00
and, by (19.2),
\\Sx(s)Sx(t)x -T(s)T(t)x\\x
= ||5a(«)5a(*)* - Sx(s)T(t)x + Sx(s)T(t)x - T(s)T(t)x\\x
<	||SA(s)|| ||5A(t)* - T{t)x\\x + ||(5a(«) - T(s))T(t)x\\x
<	Me°“'\\Sx(t)x - T(t)x\\x + ||(5a(«) - T(*))r(t)*||x -* 0
as A oo. It remains to prove that A is equal to the infinitesimal generator
B of the one-parameter semigroup {T(t) : t > 0}. Let x 6 X. Then
S\(t)x - x — f 4~S\(s)xds = f S\(s)B\xds. (19-7)
Jo as	Jo
For all x in T>(A) and all s in [0, oo),
\\Sx(s)Bxx-T(s)Ax\\x
= \\S\(s)B\x - S\(s)Ax + S\(s)Ax - T(s)Ax\\x
< IISawil \\Bxx - Ax\\x + \\(Sx(s) - T(s))Ax\\x -> 0
uniformly with respect to s on compact subsets of [0, oo) as A —> oo. Thus,
by (19.7),
T(t)x — x= f T(s)Axds, x G 'D(A).
Jo
Therefore
Bx = lim	—- = lim f T(s)Axds = Ax, xeV(A).
t—>o+	t t-^o+ J0 v 7
Hence B is an extension of A. So, we only need to prove that V(A) = V(B).
To this end, we claim that there exists a real number Ai such that
\ep(B), A>Ai.	(19.8)
Assuming the claim for a moment, then there exists a real number A such
that
Thus,
\ep(A)nP(B).
(AI - B)V(A) = (AI - A)V(A) = X

136
An Introduction to Pseudo-Differential Operators, Zrd Edition
and
(AI - B)V(B) = X.
So, if x G V(B), then there exists an element z in V(A) such that
(AI - B)z = (AI - B)x.
Since A/ — B is one-to-one, z = x € 'D(A). It remains to prove the claim,
i.e., (19.8). For A > cji, we define R(X)x for every x in X by
That the integral exists is a consequence of (19.6). It also follows from
(19.6) that R(A) is a bounded linear operator on X. Now, for A > ui and
all x in X,
BhR{X)x
T(h)R(X)x - R(X)x
h
-» XR(X)x - x
as h ->• 0+. Thus, for A > oji and all x in X.
R(X)x e V(B)
and
BR(X)x = XR(X)x - x,
which is the same as
(XI - B)R(X)x = x.
So, if A > ui and x e V(B), then
R(X)x G V(B)
(19.9)

One-Parameter Semigroups Generated by Pseudo-Differential Operators
137
and
‘OO
BR(X)x = B e~xtT(t)xdt
Jo
*oo
e~xtBT(t)xdt
0
rOO
e~uT{t)Bxdt
R(X)Bx.
(19.10)
By (19.9) and (19.10),
R(X)(XI - B)x = x.
So, for A > wi, (XI - B)_1 = R(X) and the proof of the sufficiency part of
the theorem is complete. As for the necessity, we let g : [0, oo) —> R be the
function defined by
g(t) = In ||T(t)||, t 6 [0, oo).
Then g is subadditive. Indeed, for all s and t in [0, oo),
g(s +t) = In ||T(s + t)\\ = In (||T(s)T(t)||)
< In (||T(s)|| ||T(t)||) = In ||T(s)|| + In ||T(f)|| = g(s) + g(t).
Let t0 > 0. Then for all t in [0, oo), we write t = nt0 + s, where n is an
integer depending on t and 0 < s < t0. By the subadditivity of g, we get
Let S > infto>0 Then there exists a positive number R such that
g(t) < ng(t0) g(s) g(t0)
t ~ t	t	to
as t -¥ oo. So,
and hence
t>R t
g(t) c
sup < 8.
Thus,
ln||T(0ll<«. *>R-

138
An Introduction to Pseudo-Differential Operators, 3rd Edition
or
l|T(t)||<e« t>R.
Since	is a continuous function of t on the interval [0, iJ], there exists
a positive constant C for which
\\T(t)\\ < Cest, t € [0, i?].
Hence there is a positive constant M such that
||T(i)|| < Mest, t € [0, oo).	(19.11)
Now, for A > S and for all x in X, we define R(\)x by
/»oo
R(X)x = / e~xtT(t)xdt.
Jo
Since A is the infinitesimal generator of the one-parameter semigroup
{T(t) :t> 0}, we can repeat the analysis given above with A for B and S
for to conclude that (J, oo) C p(A) and
pOO
R(X; A) = / e~xtT(t)xdt	(19.12)
Jo
for A > S and for all x in X. For all A and p in (5, oo), the first resolvent
formula in Exercise 19.4 gives
R(A; A) -	A) = (p - A)JI(A; A)R(p; A).
Thus,
±R(X;A) = -R(X;A)2
and, by induction, we get
^R(\;A) = (-irn\R(X;A)n+1
for all A in (5, oo). Differentiating both sides of (19.12) n — 1 times with
respect to A and using the preceding formula for ^rR(A; A), we get
R(A; A)n =	/ e~xttn~1T(t)x dt	(19.13)
(n - 1)! J0
for A > S and for all x in X. So, by (19.11) and (19.13),
»*<* ^11s =
for all positive integers n.	□
Remark 19.6. We leave it as Exercise 19.6 to prove that the function
u : [0, oo) -> X defined by
u{t) = T(t)/, t > 0,
is the unique solution of the initial value problem (19.1). Thus, etA is the
natural notation for T(t) for t > 0.

One-Parameter Semigroups Generated by Pseudo-Differential Operators
139
As an application to pseudo-differential operators, we give the following
theorem.
Theorem 19-7. Let a G S2m, m > 0, be a symbol such that we can find a
positive constant C and a constant Ao for which
Re	<p) > C\M|^>2 - Ao|M|i VeS.	(19.14)
Then the operator Tc$ is the infinitesimal generator of a one-parameter
semigroup of bounded linear operators on L2(En).
For a proof of Theorem 19.7, we need the following lemma.
Lemma 19.8. Let A > A0. Then for every f G L2(Rn), there exists a
unique solution u G if2m»2 of the equation (AI — T<r$)u = /, where I is the
identity operator on L2(Rn). Moreover,
||(A/ - T,,o)u||a > (A - Ao)|M|2, u € H2m'2.	(19.15)
Proof Let A > Ao- Then, by (19.14),
Re ((AI - Ta)(p,ip) = Re((A0J - Ta)<p,<p) + (A - A0)|M|\
>C||*2 + (A-Ao)IMIi
> (A — A0)||<^||i, ¥>€5.
So, by a limiting argument,
Re ((AI - 7Vto)u, u) > (A - A0)||«||i, « € H2m'2.
Thus, for every / G L2(Rn), we can use Theorem 18.2 to obtain a unique
solution u G #2m’2 of the equation
(AJ-^,0)14 =/■
Moreover,
IKAi-r^oMil
= (A — Ao)2H^lll + 2(A — Ao)Re((AoI — Ta$)(p,(p) -I- ||Ao/ —
> (A — Ao)2|Ml2? <p € S.
Thus, by a standard limiting argument again,
ll(AI - T,to)«||a > (A - A0)|M|2, u 6 H2m’2.
□
Proof of Theorem 19.7 Ta,o is a closed and densely defined linear oper¬
ator from L2(Rn) into L2(IRn). By Lemma 19.8, (XI - To)”1 exists for
A > Ao* Let A > Ao* Then, by (19.15),
ll(AJ - T\,o)_1|| < (A - Ao)-1-

140
An Introduction to Pseudo-Differential Operators, 3r<* Edition
Thus,
||(AI - Ta%0)-n|| < ||(A7 - TA,o)-1||n < (A - A0)-n, n = 1,2,....
Hence, using the Hille-Yosida-Phillips theorem, the proof is complete. □
The following theorem gives a familiar class of symbols a for which —Ta
is the infinitesimal generator of a one-parameter semigroup of bounded
linear operators on L2(Rn).
Theorem 19.9. Let a G S2m, m > 0, be a strongly elliptic symbol
Then —Tato is the infinitesimal generator of a one-parameter semigroup
of bounded linear operators on L2(Rn).
Proof By Garding’s inequality, i.e., Theorem 17.1, we can find a positive
constant C and a constant Cs for every real number s > \ such that
Re p) > C\\p\\2m>2 - Cs\\p\\h-,,2, p€S.	(19.16)
If Cs < 0, then
Re(T^,<p) > CWpW^, p&S.
Now, suppose that Cs > 0. Let e G (0, C/Cs). Since m - s < m, it fol¬
lows from Erhling’s inequality in Exercise 12.6 that there exists a positive
constant C£ for which
+ C'M\l p€S.	(19.17)
Thus, by (19.16) and (19.17),
Re (Trp, p)>(C- Cse)||vC,2 - C.C,M\\, p € 5.
So, by Theorem 19.7, the operator Tc$ is the infinitesimal generator of a
one-parameter semigroup of bounded linear operators on L2(Rn).	□
Exercises
19.1.	Let A be a bounded linear operator from a complex Banach space
X into X.
(i)	Prove that for t > 0, the series YlkLo *~W~ converges absolutely. (We
denote the sum by etA.)
(ii)	Prove that for all x G X,
e(t+h)Ax __ etAx
lim 	:
h—¥ o+	h
AetAx, t > 0.

One-Parameter Semigroups Generated by Pseudo-Differential Operators
141
(iii)	Prove that for all s, t € [0, oo),
esAetA = e(s+t)A'
(iv)	Prove that e0A = J, where I is the identity operator on X.
(v)	Prove that for all t € [0, oo), AetA = etAA.
19.2.	Let {T(t) : t > 0} be a one-parameter semigroup on a complex
Banach space X. Prove that the subspace W of X defined by
is dense in X.
19.3.	Let A be a closed linear operator from a complex Banach space
X into X with dense domain. Prove that p(A) is an open subset of the
complex plane C.
19.4.	Let A be a closed linear operator from a complex Banach space X
into X with dense domain. Prove that for all Ai and A2 in p(A),
fl(Ax; A) - R(\2;A) = (A2 - X1)R(X1;A)R(X2;A).
(This is known as the first resolvent formula.)
19.5.	Prove (19.4) in the proof of the Hille-Yosida-Phillips theorem.
19.6.	Prove that the function u : [0,00) ->• X defined by
is the unique solution of the initial value problem (19.1).
19.7.	Let X be a complex Banach space. Let A be a closed linear operator
from X into X with dense domain P(A). Prove that A is the infinitesimal
generator of a one-parameter semigroup {T(t) : t > 0} on X with
u(t) = T(t)f, t > 0,
I|T(*)||<1, t>0,
if and only if
(0,00) C p(A)
and
ll*(A,A)||<i

142
An Introduction to Pseudo-Differential Operators, 3rd Edition
for all A G (0, oo). (A one-parameter semigroup {T(t) : t > 0} on a complex
Banach space X with the property that
I|T(*)II<1, t>0,
is called a one-parameter contraction semigroup.)
19.8.	Let X be a complex and separable Hilbert space in which the inner
product is denoted by (, )x- A closed linear operator from X into X with
dense domain T>{A) is said to be dissipative if
Re (Ax, x)x <0, x e V(A).
Prove that A is the infinitesimal generator of a one-parameter contraction
semigroup if and only if A is dissipative and the operator XI- A is surjective
for all A G (0, oo).

Chapter 20
Fredholm Operators
Among the basic questions in the study of equations are the existence and
uniqueness of solutions. However, it is in the nature of the subject that we
cannot always expect to have both for a given equation. So, we have to seek
the second best. To see what is sensible to ask for, let us note that existence
and uniqueness are intimately related to, respectively, the range and the
null space of the operator associated with the equation under investigation.
Intuitively, we naturally want the range to be big and the null space to be
small To make these ideas precise, we introduce Fredholm operators in
this chapter. Since there are very lucid accounts of Fredholm operators in
the literature, we give only the results that we need about these operators.
Details and proofs can be found in [Schechter (2002)].
Definition 20.1. Let A be a bounded linear operator from a complex
Banach space X into a complex Banach space Y. Suppose that
(i)	the range R(A) of A is a closed subspace of Y,
(ii)	the null space N(A) of A is finite-dimensional,
(iii)	the null space N(A1) of the adjoint A£ of A is finite-dimensional.
Then we call A a Fredholm operator.
Definition 20.2. Let A be a Fredholm operator from X into Y as in
Definition 20.1. Then the index i(A) of A is defined by
i(A) = dim N(A) - dim iV(A£),
where dim N(A) and dim N(At) are the dimension of N(A) and the dimen¬
sion of AT(A£) respectively.
Remark 20.3. The requirement that N(A) be finite-dimensional is what
we mean by saying that the null space of A is small. It is well known that the
quotient space Y/R(A) can be identified with N(At). Thus, the condition
143

144
An Introduction to Pseudo-Differential Operators, 3rd Edition
that N(At) is finite-dimensional can be translated into the requirement that
the range R(A) of A is big in Y.
Remark 20.4. Let K be a compact operator on X. Then it is well known
that I — K is a Fredholm operator on X and
%{I - K) = 0.
This fact is the Riesz theory of compact operators on Banach spaces.
The following theorem is the main tool that we use to study Fredholm
pseudo-differential operators. It is known as Atkinson’s theorem.
Theorem 20.5. (Atkinson’s Theorem) Let A be a bounded linear op¬
erator from X into Y. Then A is Fredholm if and only if we can find a
bounded linear operator from Y into X, a compact operator K\ on X and
a compact operator K<i on Y such that
BA = I - Ki
on X, where I is the identity operator on X, and
AB = I -K2
on Y, where I is the identity operator on Y.
We give a necessary and sufficient condition for a bounded linear oper¬
ator to have a closed range.
Theorem 20.6. Let A be a bounded linear operator from a complex Banach
space X into a complex Banach space Y. Suppose that A is injective. Then
the range R(A) of A is closed in Y if and only if there exists a positive
constant C such that
Mx < C\\Ax\\y, xeX.
Proof Suppose that R(A) is closed in Y. Then R(A) is a Banach space
with norm inherited from Y. So, A : X —> R(A) is a bijective and bounded
linear operator. Thus, by the bounded inverse theorem, A~l : R(A) X
is a bounded linear operator. So, there exists a positive constant C such
that
p-'yllx < C’llvllr, y € R(A).
This means that
WA-'AxWx < C\\Ax\\y, xgX.

Fredholm Operators
145
Thus,
INIx < C\\Ax\W,
Conversely, suppose that there exists a positive constant C such that
\\x\\x<C\\Ax\\y, xeX.
Let {yk} be a sequence in R{A) such that
Vk^y
in Y as k —► oo. For k = 1,2,..., let Xk G X be such that
Axk — yk-
Then
11¾ - Xk\\x < C\\Axj - Axk\\y = 11¾ " Vk\W ->• 0
as j, k -t oo. So, {xk} is a Cauchy sequence in X. Since X is complete, it
follows that
Xk -> x
for some x in X as oo. But
as fc —>• oo. Therefore
-> y
Ax = y.
Hence y £ R(A) and this proves that R(A) is a closed subspace of Y. □
That the index behaves like the logarithm converting multipication into
addition is made precise in the following theorem. Since the index is com¬
puted only for Fredholm operators on Hilbert spaces in this book, the the¬
orem is stated and proved in a Hilbert space setting.
Theorem 20.7. Let X, Y and Z be complex, separable and infinite¬
dimensional Hilbert spaces. Let A\ : X ->• Y and A2 :Y Z be Fredholm
operators. Then A2A\ :X ->• Z is a Fredholm operator and
i(A2Ai) = i(A2) + i(A\).
Before giving a proof of Theorem 20.7, it is helpful to give a formula for
the null space N {A1) of the adjoint A1 of a Fredholm operator A from X
into Y, where X and Y are complex and separable Hilbert spaces.
Proposition 20.8. Let X and Y be complex and separable Hilbert spaces.
Let A be a Fredholm operator from X into Y. Then
Y = N(At)®R(A).

146
An Introduction to Pseudo-Differential Operators, 3rd Edition
The proof of Proposition 20.8 is left as an exercise. We note here that
each of N(At) and R(A) is an orthogonal complement to the other. Of
particular importance here is the fact that
N{At)=R(A)±,
which we also write as
N(At)=YeR(A).
Proof of Theorem 20.7 That A2 Ai : X	Z is a bounded linear operator
is trivial. By Atkinson’s Theorem 20.5, there exist bounded linear operators
B\ : Y X and B2 : Z Y, and compact operators K\ on X, K2 and
Kz on Y and K4 on Z such that
B1 Ai = I-K 1
on X, where I is the identity operator on X,
AlBl=I-K2
on y, where I is the identity on Y,
B2A2 = I — Kz
on y, where I is the identity operator on Y and
A2B2 = i-k4
on Z, where I is the identity operator on Z. Thus,
BxB2A2Ax = £1 (I - Ks)Al = BxAi - B\KzA\	BxKzAi
on X and
A2A\B\B2 — A2(I — K2)B2 = A2B2 — A2K2B2 — I — K4 — A2K2B2
on Z. Thus, by Atkinson’s Theorem 20.5 again, A2 A\ : X —> Z is Fredholm.
Now, let Mi be the finite-dimensional subspace of Y given by
Mi = R(Ai) fl N(A2),	(20.1)
and we write
R{A\) = Mi 0 M2,
N(A2) = Mi 0 Mz	(20.2)
and
y = R(Ai) 0 Mz 0 M4,
(20.3)

Fredholm Operators
147
where M3 and M4 are finite-dimensional subspaces of Y and M2 is a closed
subspace of Y. Now, let X\ be the subspace of X defined by
Xi = N{A2A1) © N{Ai).	(20.4)
Then
Ai(X1) = M1.	(20.5)
Indeed, let xi G X\. Then Axx\ G R(A\). Also, by (20.4), A2A\X\ = 0.
Therefore, by (20.1), A\(X\) C Mi. Conversely, let m 1 E Mi. Then, by
(20.1), mi = A\x for some x E X and, by (20.1) again,
Azmi = A2A\x = 0.
Therefore x E N{A2A\). If we write x = xo - w, where xo E N(A2Ai) and
w G N(A\), then
Aix = Aix0 - Aiw = Aix = mi.
Thus, Mi C Ai(Xi). Let Z4 be the subspace of Z given by
Z± = R(A2)eR{A2Ai).	(20.6)
Then
Z4 = A2(M4).	(20.7)
To prove (20.7), we just note that for all y G T, by (20.3),
A2y = A2Aix0 + A2m3 + A2m4,
where xq G X, m3 G M3 and 7714 G M4. Thus,
A2y = A2Aix0 + A2m4,
and the proof of (20.7) is complete. Let dj = dim Mj, j = 1,3,4. Since
Ai : Xi Mi and A2 : M4 Z4 are bijective, it follows from (20.5) and
(20.7) that
dim Xi = d\	(20.8)
and
dim Z4 = d4
(20.9)
respectively. Thus, by (20.4)-(20.6),
i(A2Ai) = dimiV(A2Ai) - dim(Z © R{A2A\))
= dimiV(Ai) + d\ — dim (Z © R(A2) 0 Z4)
= dimiV(Ai) + di — dim (Z © R{A2)) — d*.

148
An Introduction to Pseudo-Differential Operators, 3rd Edition
On the other hand, by (20.2) and (20.3),
i(A2)+i{Ai)
= dimN{A2) - dim (Z © R(A2)) + dimN(AX) - dim (Y © R(Ai))
= d\ + c?3 — dim (Z © R(A2)) + dim iV(Ai) — ds —
= d\ - dim (Z © R(A2)) + dim N(Ai) - d4.
Thus,
i(A2Ai) = i(A2) + i(A\),
as claimed.	□
Given a Fredholm operator, the problem of computing its index is usu¬
ally an important but difficult problem. We give in this chapter a formula
for computing the index of a Fredholm operator from a Hilbert space X
into a Hilbert space Y. As the formula is expressed in terms of traces of
trace class operators on Hilbert spaces, we first recall without proofs the
basic facts on trace class operators and traces, which can be found in, for
instance, [Lax (2002)] and [Reed and Simon (1980)].
Let A be a compact operator on a complex and separable Hilbert space
X. Then A1 A is a compact operator on X. Moreover, A1 A is a positive
operator on X in the sense that
(AlAx,x)> 0, x € X.
Then it is well known that the operator A1 A has a positive and compact
square root, which we denote by y/AlA. This simply means that y/A*A
is a positive and compact operator on X such that its square is the same
as A1 A. By the spectral theorem, there exists an orthonormal basis {<pk}
for X consisting of eigenvectors of y/AtA. For k = 1,2,..., let Sk be the
eigenvalue of y/AlA corresponding to the eigenvector <pk. Then we say that
A is a trace class operator on X if
oo
Sk < 00
k=i
and we say that A is a Hilbert-Schmidt operator on X if
oo
5^4 <00.
k=1
Remark 20.9. Let Si be the set of all trace class operators on X and let
S2 be the set of all Hilbert-Schmidt operators on X. Then it is well known
that S\ and S2 are two-sided ideals in the set B(X) of all bounded linear
operators on X.

Fredholm Operators
149
Let A : X X be a bounded linear operator. Suppose that
oo
$3l(A¥’fc>Vfc)x| <00
k=l
for all orthonormal bases {y>k} for X. Then it can be proved that A is
a trace class operator. Let A be a trace class operator on X. Then it
can also be proved that YlkLi(A<Pki<Pk)x is absolutely convergent for all
orthonormal bases {<£&} for X and the sum is independent of the choice
of the orthonormal basis. This sum is termed the trace of the trace class
operator A on X and is denoted by tr(A). It is easy to prove that if A and
B are trace class operators on X and c is any complex number, then A + B
and cA are also trace class operators on X. It is also easy to prove that
We need the following property of the trace.
Theorem 20.10. Let A be a trace class operator on a complex and sepa¬
rable Hilbert space X and let B be a bounded linear operator on X. Then
AB and BA are also trace class operators on X. Moreover,
A proof of Theorem 20.10 can be found on page 334 of [Lax (2002)].
As examples of Hilbert-Schmidt and trace class operators on L2(Rn), we
give one for Hilbert-Schmidt operators and one for trace class operators.
They are useful to us for studying Fredholm pseudo-differential operators
on L2(Rn).
Theorem 20.11. Let A : L2(Rn) L2(Rn) be a bounded linear operator.
Then A is a Hilbert-Schmidt operator if and only if there exists a function
K E L2(Rn x Rn) such that
for all f £ L2(Rn).
The function K is usually called the kernel of the Hilbert-Schmidt op¬
erator A.
tr (A + B) = tr(A) + tr(J5)
and
tr(cA) = ctr(A).
tr (AB) = tr (BA).
(Af)(x)=	K(x, y) f(y) dy, xeRn,

150
An Introduction to Pseudo-Differential Operators, 3rd Edition
Theorem 20.12. Let A be a bounded linear operator on L2(Rn) given by
for all f € L2(Rn), where K is a Schwartz function on W1 x Rn. Then A
is a trace class operator on L2(Rn) and
Theorem 20.12 is a special case of the more general results in [Brislawn
(1988)].
We can now give the formula in Section 30.7 of [Lax (2002)] for the
index of a Fredholm operator from one Hilbert space into another Hilbert
space.
Theorem 20.13. Let X and Y be complex, separable and infinite¬
dimensional Hilbert spaces. Let A : X Y be a Fredholm operator. Sup¬
pose that we can find a bounded linear operator B :Y X, a trace class
operator T\ on X and a trace class operator T2 on Y such that
where I is the identity operator on Y. Then the index i(A) of A is given by
(Af)(x) = K(x,y)f(y)dy, iGP,
BA = I-TU
where I is the identity operator on X, and
AB = /- T2,
(20.10)
(20.11)
(20.12)
and
Y = R(A)®Y0,

Fredholm Operators
151
where Xo and Yq are the orthogonal complements of N(A) in X and R(A)
in Y respectively. Let P be the projection of X onto Xo along N(A). Then
AP = A.	(20.13)
Indeed, let x G X. Then we can write x = z + xo, where z G N(A) and
xo G Xq. So,
APx = Axq
and
Ax — Ax o.
Thus, APx = Ax, which gives (20.13). By (20.12) and (20.13), we get
APTi = T2A.	(20.14)
It is obvious that A : Xo	R(A) is a bijection. Since P : X	Xo is a
projection, it follows that PT\ maps Xo into X0. By (20.12), T2 maps R(A)
into R(A). Now,
tr(FTilx„)=tr(T2|fl(jl)).	(20.15)
Indeed, by Exercise 20.11, there exists a surjective isometry U of R(A) onto
X0. Then, by (20.14),
UAPTX = UT2A = UT2U~lUA
and hence
(U A)(PTi)(U A)~x = UT2U~x.
Since each factor on the left hand side and the operator on the right hand
side of the preceding equation maps Xo into Xo, we can invoke Theorem
20.10 to conclude that
tr(PT1|Xo)=tr([/r2t/-1|Xo).	(20.16)
Now, we note that
tr	(20.17)
Indeed, let	{xj-}	be an orthonormal basis for Xo.	Then {U~lx*•} is an
orthonormal basis for i?(A). Thus,
oo
tr(i/ratr1|jro)
3=1
= ^T2U-lx%U-lx])Y = tr (t2|r(/4)) .

152
An Introduction to Pseudo-Differential Operators, 3rd Edition
So, (20.17) is proved and (20.15) follows from (20.16) and (20.17). Let
be an orthonormal basis for N(A) and let {#*•} be an or¬
thonormal basis for Xq. By (20.10), T\ — I on N(A). It follows that
dim N(A)	oo
tr(2i) =	53	(TlZj,Zj)x + Y,(TiX0j,x°j)x
j= 1	3=1
dim N(A)	oo
= E (zi>zi)*+ E(ri xj>Pxj)x
j=l	j=1
dim N(A)	oo
= E
i=i	i=i
= dim N(A) + tr (PTi |Xo).	(20.18)
Let {j/jA*4 ' be an orthonormal basis for lo and let {wj} be an or-
thonormal basis for R(A). Then
dim N(At)	oo
tr(T2) =	53	(r2j,9, y0)Y + 53(^-,Wj)y
j=i	i=i
dim N(At)
=	53	(r22/o,2/°)y+tr(T2|iJ{A)).	(20.19)
3=1
Since #(/ - 7¼) C -R(A), we see that
i?(/ - t2) _l y0.
Therefore
((/-7^,^=0, j = 1)2,
So, by (20.19),
dim N(Al)
tr(T2) =	5Z (y%Vj)Y + *r (T2ln(A))
J =1
dim JV(j4*)
=	E	^(¾))
j=1
= dim JV(A*) + tr (r2|fl(jl)) .	(20.20)
Thus, by (20.15), (20.18) and (20.20),
i(A) = dim N(A) — dim N(At) = tr(Ti) - tr(T2).

Fredholm Operators
153
Exercises
20.1.	Can a compact operator from a complex Banach space X into a
complex Banach space Y be Fredholm? Explain your answer.
20.2.	Can a Fredholm operator from a complex Banach space X into a
complex Banach space Y be compact? Explain your answer.
20.3.	Let X and Y be complex, separable and infinite-dimensional Hilbert
spaces. Let A : X -+ Y be a Fredholm operator and let K : X -+ Y be a
compact operator. Prove that A + K : X -* Y is Fredholm and
i(A + K)=i(A).
20.4.	Let A be a bounded linear operator on a complex and separable
Hilbert space X. Prove that A* A is a positive operator on X.
20.5.	Let A be a bounded and positive operator on a complex and separable
Hilbert space X. Prove that A is self-adjoint, i.e., A1 = A.
20.6.	Let A : L2(Rn) -+ L2(Rn) be a Hilbert-Schimdt operator given by
the kernel K. Prove that the kernel Kl of Af : L2 (Rn) -+ L2(Rn) is given
by
Kt{x,y) = K{y,x), x,y€Rn.
20.7.	Prove that Si is a subspace of B(X).
20.8.	Prove that tr : Si -+ C is a linear functional.
20.9.	Prove that if A is a Fredholm operator on a complex ansd separable
Hilbert space X, then
X = NiA1) © R(A).
(This exercise says that N(At) and R(A) are orthogonal to each other.)
20.10.	Let M be a closed subspace of a complex and separable Hilbert
space X. If we write X = M 0 M1- and let P be the projection of X onto
M along M-1. Find Pt.
20.11.	Find a surjective isometry U : R(Ai) -+ Xo used in the proof of
Theorem 20.13

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Chapter 21
Fredholm Pseudo-Differential
Operators
We begin with proving that Fredholm pseudo-differential operators on
Lp(Mn), 1 < p < oo, with symbols in Sm, —oo < m < oo, are elliptic.
To this end, we need some technical preparations.
Definition 21.1. For A > 0, r > 0 and #o,£o Gln,we define the operator
R\Axo,£o) : Lp(Rn)	Lp(Rn), 1 < p < oo, by
(«A,r(®b,&)«)(*) = XTn/peiXx<au(\T(x - x0)), X € ®n,
for all u € Lp(»n).
Proposition 21.2. The operator R\^(xo,£q) : Lp(Rn) -> Lp(Rn) *5 a
surjective isometry and the inverse is given by
(®A,r(xo,£o)-1u)(x) = A-Tn/pe-a(*0+A'T:c)-«0w(xo + A~Tx), x € IT,
/or aM u € Lp(Rn).
Proof We get
This proves that -Ra,t(#o,£o) is an isometry. Now, let v € Lp(Rn). We need
to find a function u in Lp(Rn) such that
||#a,t(zo,£oHI£
f |(ftA,r(#0,fo)u)(z)|Pdz
Xrn\u(XT(x - x0))\pdx, ti e Tp(Mn).	(21.1)
Let y = Ar(# — #o) in (21.1). Then we have
||flAfr(®o,&)u||p = ||u||p, u € Lp(Rn).
R\,t(xo,£o)u = v.
155

156
An Introduction to Pseudo-Differential Operators, 3rd Edition
Let u be the function in Lp(E71) given by
u(x) =	+ \~TX), x € Rn.
Then
#A,r(tfo,fo)u = v.
□
Proposition 21.3. For all u E Lp(Rn) and v E Lp' (E71), where ^ 4- y = 1,
1 < p < oo,
(R\,r(Xo^0)u,v) -+ 0
as A -» oo.
Proof Let (^(E71). Then we have
|(tfA,r(£0,fo)u,v)|
<	Arn/p [ |i/(Ar(x-aro))||v(a:)|dar
J Rn
= A™tp f A”rn|u(2/)||v(ar0 + A“Ty)| dy
jRn
= X~Tn/p' f Kx0 + A-Tj/)|Ky)|dy
J Rn
<	X~Tn/p' sup \v(y)\ f	(21.2)
y£Rn	JRn
So, by (21.2), (R\^t(xo£o)u’v)	0 35	00 f°r u>v € Co^E71). Let
u E Lp(Wl) and v E Lp>(E71). By density, we can find sequences {<pj} and
{'ipj} in Co°(En) such that
and
(pj -+ u
ifrj -+ v
in Lp(En) and Lp (Mn) as j -+ oo respectively. Therefore for every positive
number e, there exists a positive integer J such that
\(R\,t(x0,£o)u,v) - (R\,T(xO,&>)Uj,Vj)\ < £
for all j > J. Then, again using the fact that (R\tT(xo,£o)uj,Vj) -t 0 as
A -+ oo for Uj, Vj E Co°(En), the result follows immediately.	□
Proposition 21.4. Let a E Sm. Then
R\,t(xqi£o) TaR\yT{xo-,Co) = Tax r,
(21.3)

Fredholm Pseudo-Differential Operators
157
where
<J\,T(x,rj) = a(x0 + A"rx,Afo + ATrj), x,rj G Mn. (21.4)
Moreover, if a G 5°, A > 1, 0 < r < 1/2 and fo i= 0, then for all multi-
indices a and /3, there exists a positive constant Cp such that
l(3?3S,r)MI < C'^aJ,(a)^A-rl“lA-<1-aT)l',l, x,V € R»,
(21.5)
where pa,p denotes the corresponding norm in S°.
An ingredient in the proof of Proposition 21.4 is the following Peetre’s
inequality.
Lemma 21.5. (Peetre’s Inequality) For all t G (-00,00) and for all
x,y G Mn,
(Tw)'£2l‘l(1 + |l','|i)'‘1'
Proof The inequality is obviously true if t = 0. For all y and z in Rn,
1 + \y - *\2 = 1 + (y - z) ■ (y - z)
= l + \y\2-2yz + \z\2
<l + \y\2 + 2\y\\z\ + \z\2.
Since
2\y\\z\<\y\2 + \z\2,
it follows that
l + \y-z\2<l + 2\y\2 + 2\z\2 < 2(1 + |y|2)(l + \z\2)
for all y and z in Rn. Replacing z by y — x, we get
1 + \x\2 < 2(1 + |j/|2)(l + \x - y\2).
If £ > 0, then
(1 + |ar|2)* < 2*(1 + |»|2)*(1 + \x-y\2Y,
as asserted. If t < 0, then — t > 0 and by what we have proved,
(1 + |y|2)-‘ < 2-*(l + |*| V(1 + I* - y\2)-\
which is the same as
(l +1*!2)* < 2l*l(l + |*r|a)*(l +1* -

158
An Introduction to Pseudo-Differential Operators, 3rd Edition
as required.	□
Proof of Proposition 21.4 We first note that for all ip € <S,
(!»(*) = (2tt)-"/2 f eix'^a(x, ¢)0(0 ^
= (2n)~n f f et(-x~yH<T(x,£)u(y)dyd£,
J]Rn «/Rn
where the integral JRn JjRn is understood to be an iterated integral in which
the integration is first performed with respect to y and then £. Let u e S.
Then for all x £ W1,
(R\,T(xo,£o)~lTaR\,r(xo, £o)«) (x)
= e-iA(*o+A-^Ko(2,r)-« f f e<A(*o+A-’-*-»)-€<T(aro + A_Tar,0
«/Rn JR"
eiXy^u(\T(y-xo))dyd^
= \-Tn(2ir)~n f [ eiX~T^-^x-^a(x0 + A-r*,0«(z)dz<%
JR» JR»
= (27r)_n f j e^x~z^Tla(xo + A_Tar, A£0 + ATri)u{z) dz dq.
JRn «/Rn
Thus, we get (21.3) and (21.4), as asserted. Now, using (21.4), the chain
rule and Peetre’s inequality,
m^x,r)M\ =	+A-r®, a&+\rn)\-T'a'\rm\
<	paj,{<r)( 1 + |A£0 + Ar.f|)-MA-T|o|ATM
<	C0pa,0(o)(Ko +
<c?pa,?{<!){ Ae0)-|/J|(Ar»?)l's|A-T|a|
<	C0pa,0(<r)\$>\-m{v)WaP’-WA-’-W (21.6)
Hence (21.5) is proved.	□
The following theorem is one of the main theorems of this chapter.
Theorem 21.6. Let a £ S° be such that Ta : Lp(Rn) ->* Lp(Wl) is a
Fredholm operator for 1 < p < oo. Then there are positive constants C and
R such that
k(*,£)l > c
for all x e W1 and |£| > R, i.e., a is elliptic.

Fredholm Pseudo-Differential Operators
159
Proof Since Ta is a Fredholm operator, it follows that we can find a
bounded linear operator S on Lp(Rn) and a compact operator K on Lp(Rn)
such that
STa = I + K.
Let M be the set of all points £ in Mn such that there exists a point x in
W1 for which
W*'01 S 2pii'
Now, if M is bounded, then there exists a positive number R such that
Thus, for each point £ G Rn with |£| > R, we get, for all iGln,
|<r(ae,OI > 2jjsj|>
which is the same as saying that a is elliptic. So, suppose that M is not
bounded. Then there exists a sequence {(#&,£&)} m x such that
l&l -> 00
as k oo and
|£*)|	2\\S\\ ’	^	1,2,....
Thus, there exists a subsequence of {(#&, £&)}, again denoted by {(#&, £&)}>
such that
^ ^oo
for some complex number (Too as k -» oo. Therefore
l<To01 - 2pii-
For k = 1,2,..., let Afc = |£*|. Then, by Proposition 21.4, we have
where
(21.7)
<?\= <r{xk + Aferx, £^ + A*//), x,rj G Rn.
Let a and /3 be multi-indices. Then, in view of (21.5), there exists a positive
constant Cp such that
(21.8)

160
An Introduction to Pseudo-Differential Operators, 3rd Edition
For k = 1,2,..., let <t£° be given by
<r*° = cta*,t(0,0) = a{xk,ik)-
Using Taylor’s formula given in Theorem 7.3 and the estimate (21.8), we
get
W\„AX,V) ~0fc°|
= kA»,r(*,»?) -^Afc,r(0,0)|
E	f maXk,r){6xMdB
+#*l=i
= E w|7|nImI/1 ^,#.k)^)l/i|Arw^(1-2T)M^
17+/.1=1	Jo
(21.9)
uniformly for (x,rj) on every compact subset K of Rn xln as k —» oo. Let
u £ S. Then
(7VX ,«)(*) - ^u(ar) = (2#r)-n/2 / eixt,(axk<r(x,v) ~	dV
k'	jRn
for all x € Rn. By (21.9), the assumption that cr € S° and Lebesgue’s
dominated convergence theorem,
(T,Xk.ru)(*)->o?u(x)	(21.10)
for all x £ Rn as fc oo. Moreover, for all l € N, using (21.8) and an
integration by parts, we can find a positive constant C^ for each p, with
| //1 < 21 such that
\{x)2l(TaXk,ru)(x)\
(x)21 (2n)~n/2 [ eix T><jXk<T(x,V)u(v)dv
J Rn
(2ir)-”/2 f ((/ -	£)«(£) <*£
Jr*
= (2ir)~n/2 f E
/r» , .T7l, M!
l#»l<a
for all x € K", where />(£>) = (/- A)1. So, there exists a positive constant
C such that
1(^.^)(^)1^(7^)-^, *€Mn.

Fredholm Pseudo-Differential Operators
161
Now, if 2Ip > n, then (x) 2lp € L1(Rn). So, there exists a positive constant
Ci such that
l(^fc.r - <r?)«)(*)| <	x e Rn. (21.11)
Thus,
T<ry. U (TonU
in Lp(Wl) as k -» oo. Let u be a nonzero function Lp(Rn). Since
R\kyT ||jj^ is an isometry, it follows that
0 < Hip =
RXk,r (**>|^|)W
{STa-K)RXk<r (^>|||)
(**’ i&t)
STaR\
+ \\KRx,
i|i) 1^,,(..,,1,).1
Ml
Now, using the fact that K is a compact operator and Proposition 21.3, it
follows that
*Kw(*‘’iti)“ll^0
(21.13)
as k -¥ oo. Then by (21.12),
IHIP< 11-511 KolNIp.
Then (21.7) and (21.13) give the contradiction that
1 I l 1
M -l<To01 - 2pi’
which completes the proof.	C
The preceding theorem can be generalized to the following theorem.
Theorem 21.7. Let a € 5m, -oo < m < oo, and let Ta : H8iP H8~m'p
be a Fredholm operator for some s € (—00,00). Then Ta is an elliptic
operator.

162	An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof The operators : H8* -> H8~m>p, J_s : Hs* Lp(Rn) and
Jm-s ' H8~m'p Lp(Rn) are bounded linear operators. Here Js, the
Bessel potential, is a pseudo-differential operator with symbol as £ S“s,
where
MO =	£^n-
Let
Jm—sTaJs
= Tr.
Then
Tt : Lp(En) -> Lp{Rn),
where r £ 5°. Since Js is bijective, it follows that Js is Fredholm and
elliptic for all s £ (-oo, oo). So, Tr is elliptic. By the fact that Js, s £ Mn,
is bijective, it follows immediately that Ta is elliptic.	□
For pseudo-differential operators with symbols in 5m, -oo < m < oo,
that ellipticity does not imply Fredholmness can be seen from the following
example.
Example 21.8. Let P(D) — £|a|<m aa Da be a linear partial differential
operator with constant coefficients on Mn such that P(D) : Hm'p -+ Lp(Rn)
is injective and there exists a sequence {£&} in Rn such that
P(Sk) -»• o
as k —y oo. Then P(D) : Hm,p —> Lp(Rn) is not Fredholm. More precisely,
the range R{P) of the operator P(D) : Hm'p -+ Lp(Rn) is not a closed
subspace of Lp(Rn). To this end, let <p be a function in S such that ||^||p = 1.
Let {ek} be a sequence of positive numbers such that
4^(6)^0
for 0 < \ii\ < m as k -+ oo. Let {ipk} be the sequence of functions in S
defined by
<Pk(x) = e^n/pip(skx)etx'ik, x € M",
for k = 1,2, — Then
ll^fcllp — k — 1,2,....

Fredholm Pseudo-Differential Operators
163
By Leibniz’s formula, we get for fc = 1,2,...,
(P(D)<pk)( x)
= e~knlp E -i{P(*\Dy**'‘)(.D»v){ekx)£M
M<m ^*
= Hn/P E -}P{,i)(^)eixik(D^)(skx)£[ri
\p\<m
= er/pP(^)eix^<p(ekx) + er/p E ^P{l,)(^)eix^(D^)(ekx)e^
l<\p\<m
for all x € Rn. So, for fc = 1,2,...,
\\P(D)<fk\\P < |P(&)| + E 7^1^(6)1 WD^Wp -+ 0
l<|^|<m
as fc -» oo. Thus, we cannot find a positive constant C such that
\\Vk\\p < C\\P(D)<pk\\p,	& = 1,2,—
So, the range R(P) is not closed in Lp(Rn) by Theorem 20.6.
For a positive result, we have the following proposition.
Proposition 21.9. Let P(D) be a linear partial differential operator with
constant coefficients and of order m onln. Then P(D) : ifm’2 ->• L2(Rn)
is Fredholm with zero index if and only if
0£{P(0:£eR”},
where {• • • } is the closure in W1 of the set {•••}.
Before giving a proof of Proposition 21.9, we need two lemmas.
Lemma 21.10. Let P be a polynomial on W1 given by
p(o=Ea^°’ £eRn-
\a\<m
Let Z(P) be the zero set of P, i.e.,
Z(P) = {£ £ Rn : P(0 = 0}.
Then m(Z(P)) = 0, where m is the Lebesgue measure on Rn.

164
An Introduction to Pseudo-Differential Operators, Zrd Edition
Proof The lemma is certainly true for n = 1. Suppose that the lemma is
true for all polynomials on Rn_1. Write
En = Rn_1 x E
and any point in En is of the form (£', £n), where £' £ Rn 1. Any multi¬
index a = (ai, a2,..., an) can be written as (o', an), where
ol — (ai,a2,...,an-i).
Now,
P(0=P(Z'An)=	£
\a'\<m-an
t/oc
Thus,
Z(P) =
j(£',£«)€lT
^	( y aa',an^nn
|a' |<m—\an=0
Let Xz(P) be the characteristic function of Z(P). Then, by the induction
hypothesis, we get
m(Z(P))=r f XZ(P)(£',£n)«n
J—oo iR"-1
= /°° (J Xzwti,U)#)dL = 0.
□
Lemma 21.11. Lei P(D) =	6e a /mear partial differential
operator with constant coefficients and of order m on En. Then the op¬
erators P(D) : Hm>2 -> L2(En) and P(D)* : L2(En) if"™’2 are both
injective.
Proof Let u £ Hm,2 be such that
P(D)u = 0.
Then u £ L2(Rn) by the Sobolev embedding theorem in Chapter 12. Taking
the Fourier transform on both sides, we get
W = 0
for almost all f £ En. By Lemma 21.10, u = 0 a.e. on En. Therefore u = 0.
This proves that P(D) : Hm,2 ->• L2(En) is injective. Now, let u £ L2(Mn)
be such that
P{D)lu = 0.

Fredholm Pseudo-Differential Operators
165
Then, by Exercise 13.1, Exercise 13.2 and Plancherel’s theorem,
(P(D)tu,v) = (u, P(D)v) = (ii,Pv) = (Pu,v) = 0, v € Hm’2.
Therefore
(Pu, ip)= 0, <p £ S.
Since S is dense in L2(Rn), it follows that Pu = 0. By Lemma 21.10, u = 0
a.e. on Rn. Therefore u — 0. So, P(D)1 : L2(Rn) ->> #“m’2 is injective. □
Proof of Proposition 21.9 Suppose that P{D) : Hm’2 -> L2(Rn) is
Fredholm with zero index. Then the range R{P) of P(D) : Hm,2 —>• L2(Rn)
is closed in L2(Rn). By Theorem 20.6, there exists a positive constant C
such that
IMka < C||P(Z?)tt||a, u € Hm'\
So,
0¢{P(¢) :€€«»}.
For if this were not true, then there would be a sequence {&} in Rn such
that P(£k) -> 0 as k oo, and the analysis in Example 21.8 would apply
to give a contradiction. Conversely, suppose that
0 i {P(0 :*€**}.
Then there exists a positive constant C such that
^ € Bn.
So, for all u € Hm,2, by Plancherel’s theorem,
||P(D)w||2 = HF&lla > C||fi||2 = C||ti||2, t« 6
Therefore the range R(P) is closed in L2(Rn). The rest of the proof follows
from Lemma 21.11.	□
The following result follows from Proposition 21.9.
Theorem 21.12. Let P(D) be a linear partial differential operator with
constant coefficients and of order m on Mn. Then, for —oo < s < oo, the
operator P(D) : Hs’2 -» Hs~m,2 is Fredholm with zero index if and only if
0

166
An Introduction to Pseudo-Differential Operators, 3rd Edition
Proof We only need to prove that P(D) : H8'2 -+ H8~~m,2 is Fredholm
with zero index if and only if P(D) : Hm'2 -+ L2(En) is Fredholm with
zero index. Suppose that P(D) : H5,2 -+ Hs~m'2 is Fredholm with zero
index. By Exercise 21.5, P(D) = Jm-SP(D)JS-Tn : Hm'2 -+ L2(En) is
Fredholm with zero index. Similarly, the converse is true.	□
Remark 21.13. In fact, in Theorem 21.12, if 0 ^ {P(£) : £ E E71}, then
P(D) : H5,2 -+	is a bijection for — oo < s < oo.
Exercises
21.1.	Give an example of a linear partial differential operator P(D) on En
such that P(D) is elliptic, injective and there exists a sequence {£*} in En
such that
^(6)^0
as k -* oo.
21.2.	Prove that a linear partial differential operator P(D) with constant
coefficients on En is elliptic if
0£{P(ChiTE^}.
Is the converse true?
21.3.	Use the definition of 5° to give another formula for pa,p((r) in Propo¬
sition 21.4.
21.4.	Use the definition of S° and Peetre’s inequality to fill in the details
in deriving (21.6).
21.5.	Let P(D) be a linear partial differential operator with constant
coefficients and of order m on En. Prove that for all s E (-00,00), P(D)
and Js commute in the sense that
P(D)JS : H*’2 -+
and
JSP{D) : tf*’2 ->	2
are equal for all t € (—00,00).

Chapter 22
Symmetrically Global
Pseudo-Differential Operators
The pseudo-differential operators studied in this book are global operators
on Rn. The decay estimates on the associated symbols due to differentia¬
tions have hitherto been imposed only on the ^-variable. We introduce in
this chapter another class of pseudo-differential operators on Mn of which
the symbols satisfy similar decay estimates due to differentiations with re¬
spect to the ^-variable and the ^-variable. They are appropriately called
symmetrically global pseudo-differential operators. These operators are also
known in the literature as SG operators with SG standing for symbol-global,
scattering operators and operators with exit at infinity. For the sake of con¬
venience and the close match with the title of this chapter, we simply call
these operators SG operators and their symbols SG symbols.
Let mi, m2 € (—00,00). Then we let Smi,m2 be the set of all functions
in C00 (Rn x En) such that for all multi-indices a and /3, there exists a
positive constant Ca,p for which
\(D%Dl<T)(x,t)\ < Ca,^)m2~laiiOmi~W, € Rn.
A function in Smi ,m2 is said to be a SG symbol of order (mi, m2). It is clear
that if a G Sm 1,7712 and m2 < 0, then a G Smi, where Smi is the class of
symbols of pseudo-differential operators studied before this chapter in the
book. Let a G 5mi,m2. Then we define the SG pseudo-differential operator
Ta with symbol a by
(7»(z) = (2tt)-"/2 / eix<a(x,0<p(0dti, * € M",	(22.1)
JRn
for all functions (p in the Schwartz space S. It can be proved easily that
Ta : S -» S is a continuous linear mapping. (See Propositions 6.7 and
li.i.)
We begin with the product formula.
167

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An Introduction to Pseudo-Differential Operators, 3r<i Edition
Theorem 22.1. Let a € Sm'-m2 and r € S'11-"2. Then TaTT = Tx, where
A € Smy+<*i ."»*+<** and
M **'
iTere the asymptotic expansion means that for every positive integer M,
there exists a positive integer N such that
A- ^	6 5"»1+Mi-M,m2+M2-M
ImI <n
The next important result is the formula for the formal adjoint.
Theorem 22.2. Let a G Smi,m2. Then the formal adjoint T* of Ta is a
SG pseudo-differential operator Tt, where r G Smi,m2 and
T ~
E
(—*)|M|
d$d»a.
Here the asymptotic expansion means that for every positive integer M,
there exists a positive integer N such that
■ _	(	QV’QV'tf q ^mi-M,rri2-M
^ n\ x S
As in Chapter 11, we can extend the definition of a SG pseudo¬
differential operator from the Schwartz space S to the space <S' of all tem¬
pered distributions. Let a G Smi,m2. Then for all u in S', we define the
linear functional Ta : S C by
(Tau)(<p) = u(T*Jp), ipeS.
As an analog of Proposition 11.6, Ta maps S' into S' continuously. The
more interesting result is the following L^-boundedness theorem. It follows
from the fact that each symbol in 50,0 is in 5° and Theorem 11.7.
Theorem 22.3. Let a G 50,0. Then Ta : Lp(Rn) -» Lp(Rn) is a bounded
linear operator for 1 < p < oo.
We now come to the ellipticity of SG pseudo-differential operators. Let
a G Smi,m2, -oo < mi,m2 < oo. Then cr is said to be SG elliptic if there
exist positive constants C and R such that
k(*,0| > C'<*>TOa(0m‘, l*l2 + Kl2 > R2-

Symmetrically Global Pseudo-Differential Operators
169
The following theorem tells us that SG elliptic operators have SG para-
metrices.
Theorem 22.4. Let a G 5mi,m2, —oo < mi,m2 < 00, be SG elliptic.
Then there exists a symbol r in S~mi’~m2 such that
TTTa = I + R
and
TaTr = / + S,
where R and S are infinitely smoothing in the sense that they are SG
pseudo-differential operators with symbols in	'
The SG pseudo-differential operator Tr in the preceding theorem is
known as a SG parametrix of Ta.
The next development of the theory is to determine the domain of a
SG elliptic pseudo-differential operator. We need Sobolev spaces for SG
pseudo-differential operators for this task. For Si,$2 G (—00,00), we let
Jsi,s2 be the Bessel potential of order (si,S2) defined by
J$1,S2 =	31,82 ’
where
ctsus2M = (x)-82(0'91^
Obviously, cr8uS2 G S~Sl~82. It is left as Exercise 22.6 to show that the
mapping J5l,52 : S' S' is a bijection for —00 < «i, $2 < 00, and
J.-U =	(22-2)
and hence, by Theorem 22.1, J~\St is a SG pseudo-differential operator of
order (-^1,-^2)-
For 1 < p < 00, and -00 < si,s2 < 00, the U’-Sobolev space HSl'S2’p of
order (*’i, s2) is defined by
tf81'*2’3’ = {u € S' : J-Sl,-s2n <E Lp(Rn)}.
Then H8uS2’p is a Banach space in which the norm || ||3l,S2lP is given by
IMI»1,*2,P =	^ H ’ 2’P-
Obviously,
Ap = Lp(En).
Theorem 22.3 can now be improved to the following result. Its proof is
left as an exercise.
Theorem 22.5. Let a € Smi’m2, -oo < mi,m2 < oo, be a SG symbol.
Then T * HSliS2iP ff8^~~Tn^*82~rn2,p is a bounded linear operator for
1 < p < oo and -oo < s\, $2 < oo.

170
An Introduction to Pseudo-Differential Operators, 3rd Edition
We have the following Sobolev embedding theorem for SG pseudo-
differential operators. (See Theorem 12.5 and Remark 12.10.) In Theorem
12.5,	we have a bounded linear operator of which the norm is at most one.
There is a short proof if we relax the requirement of the norm. The proof
of the following theorem is also not difficult if we just need the inclusion to
be a bounded linear operator. This is a good exercise.
Theorem 22.6. Let si,S2,ti,t2 £ (—00,00) be such that si < t\ and
S2 < Then Htl,t2jP C H81,S2,P and the inclusion i : Htl,t2iP c—HSliS2,p
is a bounded linear operator.
The most important feature of the Sobolev spaces for SG pseudo¬
differential operators is the following compact embedding theorem, which
is not true for the Sobolev spaces in Chapter 12.
Theorem 22.7. Let Si,S2,h,t2 £ (-00,00) be such that Si < t\ and
S2 < Then the inclusion i : Htl,t2'p	Hs 1,S2,P is a compact operator.
A proof of Theorem 22.7 depends on the following result in [Wong (1983,
1994)]. It is a result in [Grushin (1970)] at least for the L2-case and we
skip the proof.
Theorem 22.8. Let a G Sm, m G (-00,00), be such that
lim Cad3(3)=0
| x | —^00
for all multi-indices a and /3. Then for every positive numbers, the operator
Ta : Hs+m'p —► Hs~£,p is compact for — 00 < s < 00 and 1 < p < 00.
The following corollary follows immediately. We leave it as an exercise.
Corollary 22.9. For every positive number e, Je,£ : Lp(Rn) —► Lp(Rn) is
a compact operator for 1 < p < 00.
Proof of Theorem 22.7 Let e be a positive number such that
t\ — si — e > 0
and
*2 - 82 - e > 0.
Then, by Theorem 22.1 and (22.2), J~£j-Sli-S2 is a SG pseudo-differential
operator of order (51 + e, S2 + ¢). So,
j—l j	. jjtifaiP jjti—si—e,t2—S2—e

Symmetrically Global Pseudo-Differential Operators
171
is a bounded linear operator, the inclusion
i: Hl	Lp(Rn)
is a bounded linear operator, the operator
Je,e : Lp(Rn) -+ Lp(En)
is compact, and the operator
JSu82 : Lp(En) -+ H8"82'p
is a bounded linear operator. Thus,
i =	: Htl,t2,p -» HS1'S2’P
is a compact operator.
The first main result in this chapter is the following theorem.
Theorem 22.10. Let a € Smi’m2,-oo < mi,m2 < oo, be SG elliptic.
Then for all sus2 € (-00,00), T<, : H*l'32'p ->■ HSl~mi'32~m2’p is a Fred-
holm operator for 1 < p < oo.
Proof Since a is SG elliptic, it follows from Theorem 22.4 that there exists
a symbol r in S~~m 1,-m2 such that
TrTCT = I + #
and
71* Tr = / + 5,
where and 5 are infinitely smoothing in the sense that they are SG
pseudo-differential operators with symbols in	^or
positive numbers t\ and t2, the linear operator R : Lp(Rn) -+ Lp(En) is the
same as the composition of the linear operators R : Lp(En) -+ 7ftl,t2,p and
i. jjtut2,p	£,p(En). Since 1? : Lp(En) -+ Htut2'p is bounded by Theorem
22.5 and i: HUM'P *-+ Lp(Rn) is compact by Theorem 22.7, it follows that
R : Lp(Rn) -+ Lp(Rn) is compact. Similarly, 5 : Lp(En) -► Lp(En) is
compact. So, by Theorem 20.5 of Atkinson, TV,o is Fredholm.	□
We give an analog of Theorem 21.7 for SG pseudo-differential operators.
Theorem 22.11. Let a G 5mi’m2, mi,m2 G (-00,00), be such that the
operator TV : HSl'82'p -+ Hs^ Fredholm for si,s2 G (-00,00)
and 1 < p < 00. 77&en o* «5 SG elliptic.

172
An Introduction to Pseudo-Differential Operators, 3rd Edition
We first establish an analog of Theorem 21.6 for SG pseudo-differential
operators.
Theorem 22.12. Let a G 50,0 be such that Ta : Lp(Wl) —> Lp(Rn) is
Fredholm for 1 < p < oo. Then we can find positive constants C and R
such that
for all x and f in W1 with
K*iOI > o
\x\2 + |£|2 > R2
Proof By Atkinson’s theorem, i.e., Theorem 20.5, we can find a bounded
linear operator S on Lp(Wl) and a compact operator K on Lp(Rn) such
that
STa = I + K.
Let M be the subset of W1 x Rn given by
{(*>£) e En : |ff(ar,OI < 2p||} '
We first suppose that M is bounded, i.e., there exists a positive number R
such that
So,
\(x,0\<R, (x,0eM.
X2 +£2 > R2 => (x,£) ¢ M => \cr(x, ^)| >
2iisir
Thus, the theorem is proved. If M is unbounded, then there exists a se¬
quence {(xfe,^)} in M such that
|(Xfc,6)l -t oo
as k -»• oo and
1
2||5||
, A: = 1,2, —
So, there exists a subsequence of {(xft,£jt)}, again denoted by {(xfc,£*)},
such that
k(x*,&)| -*■ <7oo
as fc -t oo, where <Too is some complex number. Thus,
1

Symmetrically Global Pseudo-Differential Operators
173
Now, for k = 1,2,..., let A& = |(^A5?Cfe)|- Since (21.3) is equally valid for
S0,0, it follows that
RXk>T (**• ill) T°Rxk'T (**’ ill)= T°k'r’
where
(*>»?) = ° (** + AfcTl’Aft]|j + A*»l) ’	€
By Exercise 22.13, (21.5) is valid also for 50,0 if we replace pa,p by the
corresponding norm qa^ in 50,0. So, for all multi-indices a and /?, there
exists a positive constant Cp such that
m^aXk,T){x,v)\ < C0qa,0{ri)W^TlaXil'2Tm^
and the rest of the proof is exactly the same as that of Theorem 21.6. □
Using Theorem 22.12, the proof of Theorem 22.11 is similar to that of
Theorem 21.7 and is best left as an exercise.
We end this chapter with an index formula for SG pseudo-differential
operators. We need a lemma.
Lemma 22.13. f\kltk2eRSkl'k2 = <S.
Proof It is easy to check that S C n&^eRS*1’*2 and is left as an exercise.
Let a G nkuk2&tSkuk2. Then for all multi-indices a, /?, 7 and S, let k\ and
/¾ be real numbers such that
M + k2 - H < 0
and
101+*i-i*i <0.
Since a G Skuk2, there exists a positive constant	such that
sup \xat?(D2D%<r)(x,0\
®,€€Rn
< sup
z,£6Rn
<CklM,^5 sup ((*)H+**-^<Oi<5,+*1_|4|)<oo.
z,£€Rn
Therefore a G S.
Let a € 5°’° be a SG elliptic symbol. Let r € S°'° be such that
TtT„ = I — Ti
□

174
An Introduction to Pseudo-Differential Operators, 3rd Edition
and
T,Tt=I-T2j
where Tj is a SG pseudo-differential operator with symbol tj in
j = 1,2. Therefore for j = 1,2, Tj e S by Lemma 22.13.
Let <p e S. Then for j = 1,2,
(Tj<fi)(x) = (2tt)-"/2 f
= {2ir)~n/2 f T-x[T2Tj)(x, y) <p{y) dy
J Rn
for all x £ Kn, where denotes the Fourier transform with respect to the
second variable. So, for j = 1,2,
(Tjtp)(x) = (27r)“n/2 f {?2tj)(x,y-x) <p(y) dy
Jrn
= (27r)“n/2 f {T2lTj){x, x-y) cp{y) dy
Jr»
for all ar € Mn. For j = 1,2, (^_1'rJ)(a;, ar - y) is a Schwartz function of
(*,y) in Kn x IT. So, by Theorem 20.12, Tj : L2(Rn) L2(Rn) is a trace
class operator with
tr(Tj) = (27r)-n/2 / (^^)(^0) <fc
«/Rn
= (2tt)-" / f e^rjfaQdxdt
Jrn Jr*
= (2?r)"n / f Tj{x,t)dxd¢,
«/Rn ,/r»
for j = 1,2. By Theorem 20.13,
i(7V) = tr(Tj) - tr(T2) = (27r)“n f f (n(*»0 ~ r2(x,0)dxd^.
Jr* «/Rn
Finally, let cr € S7"1'™2 be such that the corresponding SG pseudo¬
differential operator T„ : H3^3^2 -+ H*i-™-82'7"2’2 is a Fredholm operator
for -oo <mi,m2 < oo and —oo < «i, «2 < °°- Then, by Theorem 22.11, a
is elliptic. So, there exists a symbol r € 5'7”1’ m2 suc^
TtT„ = I-T{	(22-3)
TaTr = I-T3,
and
(22.4)

Symmetrically Global Pseudo-Differential Operators
175
where T[ and
tively, symbols
Jm\—s\tms “^2
n
axe SG pseudo-differential operators with, respec-
and t'2 in f\uk2&tSkl>k2.	Now, the operator
,/ is a SG pseudo-differential operator with symbol in
50,0 _ By the analysis carried out in the preceding paragraph, Theorem
20.7 and the fact that JS1)S2 is a surjective isometry for all «i and s2 in K,
	 	 L2(Rn) is a Fredholm operator such that
)=i{Ta).	(22.5)
Jmi-sum2-S2T<rJsi,s2 • L )
,m2—
By (22.3),
(J_Sl,_S2Tr JSl-mi,«2-m2)(^mi-si,m2-S2^T^si,S2)
= J— Sis — 32^^0 JsiyS2
= J_Sl}_S2(I — T[)JSl,S2
= I — J-81,-82^1 J81,82*
Moreover, by (22.4),
(Jmi-si,m2-S2^crJsuS2)(J-si-82,^TJsi-muS2-m2)
= Jmi—si,m2-82^'^'T^8i-mi,S2-fn2
— Jfji\—S\,m2 — S2	-^2 ) ^81 7711, S2 m2
— / Jm\— s\,m2—828\—m\,82—^2 -
By Theorem 20.12, the equation (22.5) and Theorem 20.7, we get
i(T<r) = tr(J_Sl,_S2T/JSl>S2) — tr(Jmi-si,m2-«2^2^»i-mi,S2-m2)
= tr(I?)-tr(22).
Using the same analysis as in the preceding paragraph, we conclude that
i(Ta) = (2<Tn [ f (T[(z,t)-TiM)dxdt.
Exercises
22.1.	Prove that every linear partial differential operator with constant
coefficients on W1 is a SG pseudo-differential operator.
22.2 What is the SG order of the linear partial differential operator
2|a|<ma<*Da with constant coefficients on Rn?
22.3.	Prove that every linear partial differential operator P(D) on Rn is
SG elliptic if and only if
0*{P(fl :*€»>}•

176
An Introduction to Pseudo-Differential Operators, 3rd Edition
22.4.	Let a G Sm 1,m2, -oo < mi,m2 < 00. Prove that Ta :	-> «5 is a
continuous linear mapping.
22.5.	For -00 <mi,m2 <00, prove that if cr G Smi,m2 and m2 < 0» then
a G Smi.
22.6.	Prove that for -00 < si,s2 < 00, the mapping J8liS2 : 5' -> 5' is a
bijection and
22.7 Prove that for 1 < p < 00 and -00 < s < 00, #s’0,p =
22.8.	Prove that for 1 < p < 00 and —00 < si,$2 < 00, the mapping
J-Sl-S2 : H8l'S2'p -» Lp(Rn) is a surjective isometry.
22.9.	Prove Theorem 22.5.
22.10.	Let Si,S2,ti,t2 £ (-00,00) be such that si < t\ and s2 < t2.
Prove that Htlit2tP C HSl'S2'p and the inclusion i : Htlit2iP ^ H81 yS2'p is a
bounded linear operator.
22.11.	Prove Corollary 22.9.
22.12.	Prove that, for —00 < si,s2 < and 1 < p < 00, the Schwartz
space S is dense in i/5l,S2,p.
22.13.	Prove (21.5) for symbols in S0,0, where pa$ is now replaced by the
corresponding norm qa,p in 50,0.
22.14.	Prove that if Ta and Tr are SG elliptic pseudo-differential operators,
then TaTr is a SG pseudo-differential operator.
22.15.	Prove that if Ta is a SG elliptic pseudo-differential operator, then
T* is also a SG elliptic pseudo-differential operator.
22.16.	Prove Theorem 22.11.
22.17.	Prove that S C ,k26RSkl'k2.
22.18.	Let A : L2(Rn) -> L2(Rn) be an operator of finite rank such that the
range R{A) of A is contained in S. Prove that A is a SG pseudo-differential
operator with symbol in n^^RS*1’*52.

Chapter 23
Spectral Invariance of Symmetrically
Global Pseudo-Differential Operators
This chapter contains an application of the equivalence of the ellipticity and
Fredholmness of SG pseudo-differential operators on Lp(Mn), 1 < p < oo.
To motivate the topic, let us first observe that, in view of Theorems 22.2
and 22.5, the set of SG pseudo-differential operators with symbols in 50,0
is an algebra of bounded linear operators on Lp(Rn) for 1 < p < oo. In
the case when p = 2, the algebra is in fact a *-algebra. Suppose that
a E 50,0 is such that the corresponding SG pseudo-differential operator
Ta : Lp(Rn) Lp(Rn) is bijective, where 1 < p < oo. The problem is
to determine whether or not the inverse is also a SG pseudo-differential
operator with symbol in 50,0. This is known as the spectral invariance
problem. See Exercise 23.4 for an explanation of the terminology.
We begin with the L2-case.
Theorem 23.1. Let a E 50,0 be such that the corresponding SG pseudo¬
differential operator Ta : L2(Rn) —>• L2(Rn) is bijective. Then its inverse
T~l : L2(Rn) -> L2(Rn) is also a SG pseudo-differential operator with
symbol in 50,0.
Proof By bijectivity, Ta : L2(Rn) —» L2(Rn) is Fredholm and
i{Ta)= 0.
By Theorem 22.12, Ta is SG elliptic. By Theorem 22.4, there exists a
symbol r in 50,0 such that
TaTr = / + 5,
where R and S are infinitely smoothing in the sense that they are SG
pseudo-differential operators with symbols in C\kiMeRSkl,k2. By Atkinson’s
177

178
An Introduction to Pseudo-Differential Operators, 3rd Edition
Theorem 20.5, Tr is Fredholm and hence by Theorem 22.4, Tr is elliptic.
Also, by Remark 20.4 and Theorem 20.7,
So,
i(Tr) + m = i{TrTa) = i(I + R)= 0.
i(Tr) = 0.
Now, let u e N(Tr). Then
Tru = 0 => TaTTu = 0 (J + S)u = 0 => u = —Su.
Since n*lj*2eRfr*1,tf2'2 = <S, it follows that
u = —Su 6 S.
Similarly, the null space N(T'*) of the true adjoint T* : L2(En) ->• L2(En)
of Tt : L2(En) -A L2(En) is also a subspace of S. Now, we write
L2(Rn) = N(TT)®N{TT)±
and
L2(En) = N(T*) 0 -Rx,2(Tr),
where Rl^{Tt) is the range of Tr : L2(En) -+ L2(En). Let P = iFn,
where 7r is the projection of L2(En) onto N(Tr), F is an isomorphism
of N(Tt) onto N(T*) and i is the inclusion of AT(T*) into L2(En). Then
the operator Tt + P : L2(En) -+ L2(En) is a bijective parametrix of Ta.
Therefore, without loss of generality, we may assume that the parametrix
Tt : L2(En) ->■ L2(En) is bijective. So, I + R is bijective. In fact,
(J + P)-1 =I + K,
where K maps L2(En) into S. Indeed, there exists a bounded linear oper¬
ator K : L2(Rn) -+ Z,2(En) such that
(I + R)(I + K)=I.
So,
K = -R - RK
and
K* = -R* - K*R*.
It is obvious that K and K* map L2(En) into S. Thus, by Exercise 23.2,
the kernel of K : L2(En) -+ L2(En) is a Schwartz function on En x En,

Spectral Invariance of Symmetrically Global Pseudo-Differential Operators 179
and hence, by Exercise 23.3, K is a SG pseudo-differential operator with
symbol in C\kltk2euSklM- Thus,
T~1T~l=I + K
or equivalently
T-1 = (I+ K)Tr
and this completes the proof.	□
We can now give the Inversion of the spectral invariance of SG pseudo¬
differential operators.
Theorem 23.2. Let a G 50,0 be such that the pseudo-differential operator
Ta : Lp(En) ->• Lp(En) is invertible, where 1 < p < oo. Then its inverse
T~l : Lp(Rn) ->• Lp(Rn) is also a pseudo-differential operator with symbol
in 50,0.
Proof Since Ta : Lp(Rn) ->■ Lp(En) is bijective, it follows that it is Fred¬
holm. So, by Theorem 22.4, a is an elliptic symbol in 50,0. Thus, by
Theorem 22.4, Ta : L2(Rn) —»• L2(En) is Fredholm. So, there exists a
symbol r in 50,0 such that
TaTr = / + 5,
where R and 5 are infinitely smoothing. Let u G L2(En) be such that
Tau = 0. Then
TrTau = 0 => (/ + R)u = 0 =½ u = —Ru G 5.	(23.1)
So, u is also in the null space of Ta : Lp(Rn) -> Lp(Rn). Since
Ta : Lp(En) —> Lp(En) is injective, it follows that u = 0. Therefore
Ta : L2(En) -> L2(En) is injective. To show that Ta : L2(En) -> L2(En)
is surjective, let u be a function in the null space N(T'*) of the adjoint
Tl : L2(En) -> L2(En) of Ta : L2(En) -> L2(En). Then
T^u = 0 => Ta*u = 0,
where a* in 50,0 is the symbol of the formal adjoint of Ta. Since a* is
elliptic, we can use a left parametrix of Ta* as in (23.1) to conclude that
u = 0. This proves that Ta : L2(En)	L2(En) is surjective and hence
bijective. So, by Theorem 23.1, the inverse T”1 : L2(En) -t L2(En) is a
SG pseudo-differential operator Tt with symbol r in 50,0. Since
77 V = I>, f €5,
and S is dense in Lp(En), it follows that T”1 = Tr on Lp(En).	□

180
An Introduction to Pseudo-Differential Operators, 3rd Edition
Exercises
23.1.	Prove that DSl>S2eRHSl’S2’2 = <S- (Compare this exercise with Exer¬
cise 12.13.)
23.2.	Prove that if A : L2(Rn) L2(Mn) and A1 : L2(Rn) -> L2(Mn) are
bounded linear operators such that A and A* map L2(Rn) into <S, then the
kernel of A is a Schwartz function on W1 xln. (Hint: Use Exercise 20.6.)
23.3.	Let A : L2(Rn) L2(Rn) be a Hilbert-Schmidt operator such that
its kernel is a Schwartz function on Mn xln. Prove that A is a SG pseudo-
differential operator with symbol in CikiMeR*^1’*2-
23.4.	Let B(X) be the Banach algebra of all bounded linear operators on
a complex Banach space X. Let A be a sub-algebra of B(X) containing
the identity operator I on X. Then for every element A in A, we define
the resolvent set Pa(A) of A with respect to A to be the set of all complex
numbers A such that A — XI has an inverse, which also lies in A. The
spectrum E^(A) of A with respect to A is defined to be the complement in
C of the resolvent set pa(A).
(i)	Prove that
Eb(x){A) C E^(A), A e A.
(ii)	A sub-algebra A of B(X) containing the identity operator I on X is
said to be nontrivial if A ± B(X). Give an example to show that there
exist complex Banach spaces X and nontrivial sub-algebras A of B{X) for
which
%)(A) = E^(A), A€ A.	(23.2)
(A nontrivial sub-algebra A of B(X) containing the identity operator I on
X and satisfying (23.2) is said to be spectrally invariant)

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Index
adjoint, 97
adjoint formula, 21
Agmon-Douglis-Nirenberg estimate,
102
approximate solution, 93
asymptotic expansion, 36, 55, 61, 168
Atkinson’s theorem, 144
Bessel potential, 87, 169
boundedness in L2(Rn), 85
boundedness in Lp(Rn ), 77, 85, 91,
168
closable linear operator, 95, 99
closed linear operator, 95
continuity of translations in Lp (Rn),
11
convolution, 9
core, 104
dissipative operator, 142
distribution solution, 111
elliptic, 69, 73
Erhling’s inequality, 93
extension, 95
first resolvent formula, 141
formal adjoint, 61
Fourier inversion formula, 22, 29
Fourier transform, 17, 28
Fredholm operator, 143
fundamental solution, 93
Garding’s inequality, 113
global regularity, 106
Hormander’s multiplier theorem, 85
heat equation, 129
Hermite function of order n, 25
Hilbert-Schmidt operator, 148
Hille-Yosida-Phillips theorem, 132
index, 143
infinitely smoothing, 69, 169
infinitesimal generator, 130
initial value problem, 129
kernel, 149
Lax-Milgram lemma, 122
left parametrix, 72
Leibniz’s formula, 3
length, 2
maximal operator, 101
minimal operator, 96, 99
Minkowski’s inequality in integral
form, 3
multi-index, 2
one-parameter contraction semigroup,
142
one-parameter semigroup, 129
183

184
An Introduction to Pseudo-Differential Operators, 3rd Edition
order, 167
parametrix, 69
partial differential operator, 1
partition of unity, 43
Peetre’s inequality, 157
Plancherel’s theorem, 23, 24
positive operator, 148
principal symbol, 73
product, 55
pseudo-differential operator, 32
resolvent, 132
resolvent set, 132, 180
Riemann-Lebesgue lemma, 19
right parametrix, 72
Schwartz space, 13
SG elliptic, 168
SG parametrix, 169
SG pseudo-differential operator, 167
SG symbol, 167
singular integral operator, 79
Sobolev embedding theorem, 88, 170
Sobolev space, 88, 169
spectral invariance, 177
spectrum, 180
strong solution, 121
strongly elliptic, 113
support, 11
symbol, 2, 32
Taylor’s formula with integral
remainder, 49
tempered distribution, 27
tempered function, 27
total set, 98
trace, 149
trace class operator, 148
weak solution, 109
Young’s inequality, 9