introduction to the theory of pseudodifferential operators studies in advanced mathematics

The central notion is that of Fourier transformation: for each function a fined on R' with a controlled growth at infinity, one can define its Fouriertransform u., also defined on III, w

I Elementary

Introduction to the Theory of

Pseudodifferential

Operators

STUDIES IN.tDtiANCED MATHEMATICS

Studies in Advanced Mathematics

Elementary Introduction to the Theory of Pseudodifferential Operators

Studies in Advanced Mathematics

University of KansasMichael E TaylorUniversity of North Carolina

Volumes in the Series

Real Analysis and Foundations, Steven G Krantz

CR Manifolds and the Tangential Cauchy-Riemann Complex, Albert BoggessElementary Introduction to the Theory of Pseudodifferential Operators,Xavier Saint Raymond

Fast Fourier Transforms, James S Walker

Measure Theory and Fine Properties of Functions, L Craig Evans andRonald Gariepy

XAVIER SAINT RAYMOND

Universite de Paris-Sud, Departemettt de Mathematiques

Elementary Introduction to the Theory

of Pseudodifferential Operators

CRC PRESS

Boca Raton Ann Arbor Boston London

Library of Congress Cataloging-in-Publication Data

Saint Raymond, Xavier.

Elementary introduction to the theory of pseudodifferential

operators / Xavier Saint Raymond.

All rights reserved This book, or any parts thereof, may not be reproduced in any formwithout written consent from the publisher.

This book was formatted with L TEX by Archetype Publishing Inc., P.O Box 6567,Champaign, IL 61821.

Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida, 33431.

© 1991 by CRC Press, Inc

International Standard Book Number 0-8493-7158-9Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

V

4 Applications

Introduction

4.1 Local solvability of linear differential operators

4.2 Wave front sets of solutions of partial

76 83 89 94

97

103

107

These notes correspond to about one-third of a one-year graduate course entitled

"Introduction to Linear Partial Differential Equations," taught at Purdue sity during Fall 1989 and Spring 1990 It is an attempt to present in a veryelementary setting the main properties of basic pseudodifferential operators

Univer-It is the author's conviction that the development of this theory has reachedsuch a state that the basic results can be considered as a complete whole andshould be mastered by all mathematicians, especially those involved in analysis.Unfortunately, the beginning student is immediately faced with a technical diffi-culty that forms the heart of the theory, namely the extensive use of oscillatoryintegrals, that is, non-absolutely convergent integrals over ll2" Indeed, all thetexts written on these pseudodifferential operators assume explicitly, and evenmore often implicitly, a good familiarity with such integrals, the theory of which

is based on the rather difficult results known as stationary phase formulas, andthe authors perform changes of variables, integrations by parts, or interversions

of the f exactly as if the integrals were absolutely convergent while the allowedrules are probably not quite clear for the uninitiated reader

The main originality of these notes, maybe the only one, is to restrict theuse of such oscillatory integrals to the case of real quadratic phases for whichthe theory is both simple and pleasant Of course, this restriction prevents afull proof of the fundamental result of invariance of pseudodifferential operatorsunder a change of variables Many other important aspects of the theory are noteven mentioned in this course: properties of distribution kernels of the operators;precise description of their local action (properly supported operators); definition

of wider classes of symbols and operators such as in Coifman and Meyer [6),Hbrmander [8), or more recently Bony and Lerner [4] But the goal of thefollowing pages will be reached if this simple setting and the few applicationsgiven in the last chapter convince the reader of the fundamental importance ofthe topic and give sufficient motivations for reading more complete texts.The exposition begins with a chapter devoted to the Fourier transformation andSobolev spaces in R", which both play a central role in the theory A sufficientknowledge in classic integration theory (properties of Lebesgue measure andrelated LP spaces in R) is assumed, and Chapter 1 will provide all the additional

vii

viii Preface

background needed to take up the next chapters For the more advanced readerwho has encountered these topics before, a quick reading is recommended to getadjusted to the notation used throughout the book Chapters 2 and 3, respectivelydevoted to basic symbols and basic operators, form the theory itself Chapter 4provides applications to local solvability of linear partial differential equationsand to the study of singularities of solutions of such equations.

To avoid any ambiguity, it is emphasized that nothing is original in the topicspresented here: the text has been based mainly on Hormander [8, Section 18.11and to some extent on Alinhac and G6rard [ 1, Chap I I (in particular, the origin

of the use of oscillatory integrals as given in Chapter 2 and the origin of severalexercises can be found in this latter reference) Thus, the specific features of thistext lie only in the exposition: it is self-contained with very light prerequisitesand all the complements that were not strictly necessary to reach the main resultshave been avoided, so that it should be considered merely as a first introduction

to the topic

It is my pleasure to thank the Department of Mathematics of Purdue Universityfor the opportunity I had to teach this course I also wish to thank Mrs JudyMitchell, who with great competence and patience typed the manuscript of thiscourse

- X Saint RaymondWest Lafayette, March 1991

The central notion is that of Fourier transformation: for each function a fined on R' (with a controlled growth at infinity), one can define its Fouriertransform u., also defined on III, with the following properties: (i) differentia-tions on u correspond to multiplication by polynomials on u (which is a simpleroperation, particularly with respect to the inversion of such an operation); (ii) onecan recover u from u essentially by achieving the same transformation a secondtime; (iii) the Fourier transform of an L2 function is an L2 function Thus, inorder to study the properties of this transformation, it is more convenient to work

de-in spaces that are closed under operations of differentiation and multiplication

by polynomials, and this leads to the introduction of the Schwartz space S and

of the larger space of temperate distributions S', which contains L2

Since there is this correspondence between differentiation of u and plication of u by a polynomial, there is also a correspondence between thesmoothness of u and the growth of u at infinity (and by symmetry between thegrowth of u at infinity and the smoothness of u) This fact is used to definethe so-called Sobolev spaces, which are much more convenient than the classicclasses Ck of k-times continuously differentiable functions, especially when onedeals with L2 estimates

multi-I

2 Fourier Transformation and Sobolev Spaces

1.1 Functions in !R"

Throughout this course, we are going to study properties of complex-valuedfunctions of n independent real variables and their various derivatives There-fore, we need to develop convenient notation for these variables, functions, andderivatives

The variables will be denoted by x1, , x", or in short by x A function u

of these variables can thus be considered as defined on (a domain of) R", and

we will write u(x) and x E W For any multiindex a = (a 1, , a") E Z"+,

we define its length as the sum j al = al + + an and its factorial as the

product a! = (a1!) (a"!) Moreover, we will write a E Z+ if one has

aj <3, forallj=I, ,n.

These multiindices a are used to write polynomials: for x E 1t" and a E Z

one defines x' as the product x' = xi' xn' Similarly, if a; denotes the

operation of taking the partial derivative with respect to x3 (i.e., a; = 9/ax;),one will write

1 "

axalI axon"

Recall that a function u(x) is said to be of class Ck if it has continuous tives up to order k, and these derivatives do not depend on the order used toachieve the differentiation (i.e., 8;ak = aka; when acting on C2 functions).These derivatives are therefore denoted by an u without possible confusion

deriva-To show how this notation can be conveniently used, we prove the followingclassic result known as Taylor's formula

THEOREM 1.1 TAYLOR'S FORMULA

Let u be a Ck function defined on R; then for any x and y E R" one has

PROOF Forall0EZ+, let us write I}={aEZ'+;IaI= 1131-1 and

a < Q}; thus one has

Functions in R' 3

Then, the C' function in t: vk(x, y, t) _ E Q!<k((l - t)!QI y°/a!)8°u(x + ty) satisfies vk(x, y, 1) = u(x + y), vk(x, y,0) _ EI I<k(y"/a!)8°u(x) and

(8vk/8t)(x, y, t) = Ej.j=k k(y°/a!)(1 - t)k-'& u(x + ty) so that the result

simply follows from the fundamental theorem of calculus

is the key to the following result

THEOREM 1.2 BINOMIAL AND LEIBNIZ'S FORMULAS

4 Fourier Transformation and Sobolev Spaces

(ii) (Leibniz's formula) For any C1°1 functions u and v,

e (UV) = E (13) (O u)(e-Av)

PROOF By induction on a The two properties are obvious for a = 0 Then,

assuming they hold for a, we prove they hold for a + b,, where b; is the

multiindex of length 1 with jth component 1, as follows:

according to the fundamental relation given above

It is a classic remark that when one takes x = y = (1,1, ,1) in formula (i),one gets

Although all the notions we introduce below are actually invariant under themost general linear transformations on the variables x, it is convenient to use theeuclidean structure of R" when the system of coordinates is fixed, in particular

to measure the growth of functions at infinity Thus, for any vectors x and

F E R", we will consider the scalar product (x, 0 = X1 l;1 + + x"E and

LEMMA 1.3

The integrals over Rn

are convergent if and only if s > n; in particular, we have the precise estimate

and the result follows from the classic one-dimensional case In particular, for

s = 2n this gives the precise estimate since f c(1 +x2)-l dxt = 2r We willnot use the "only if" part of this lemma, which can be proved by writing theintegrals in polar coordinates I

The notation being thus fixed, we now introduce the Schwartz space S ofC°° functions that are rapidly decreasing at infinity More precisely, the C°°function cp belongs to S if the functions xa89cp(x) are bounded on Rn for allpairs a, 3 of multiindices If we denote by lcplo the supremum over R' of abounded continuous function cp, the implicit topology of S is that defined bythe norms'

I'P1k = sup lx0B1cplo= sup{Ix°8Qcp(x)I; x E Rn and Ia + i31 <_ k}to+aI <k

where k E Z Obviously, S is closed under the operations of differentiationand multiplication by polynomials As a matter of fact, it is even closed undermultiplication by C°° functions with polynomial growths at infinity: if a contin-uous function 1 satisfies an estimate ltli(x)1 < C(1 +Ix12)N for some constants

C and N, we will write r/i E P°, and if rp is a C°° function such that Oat' E P°

'Equipped with these norms, S is then a Frbchet space Throughout the text, we will use some words from functional analysis (such as Frdchet, Banach, or Hilbert spaces) but we will not use any result from this theory The purpose is just to give additional information to the more advanced student Similarly, we will use the words continuous and continuity (e.g in the statement

of Lemma 1.4) only as synonyms of an inequality between norms.

6 Fourier Transformation and Sobolev Spaces

for all a E Z we will write -0 E P (space of CO0 functions with polynomialgrowths at infinity) The following continuity properties will be important

LEMMA 1.4

One has

(i) (Continuity of differentiation) For any a E and cp E S, & E S

with k Ik+I°I for all k E Z.

(ii) (Continuity of multiplication by a V) E P) For any V, E P there existtwo sequences Ck and Nk such that cp E S * z E S with I?GcpIk <

for all k E Z+; in particular if O(x) = x° one has Ix°wIk <2k(a!)1V1k+I°I

PROOF Property (i) is clear, property (ii) follows from Leibniz's formula (cf.Theorem 1.2), which can be written here

and thus implies the estimate if we take Nk then Ck such that

I8' (x)I :5 2k(n -Fkl)Nk (I +IxI2)Nk for I'yl :5k

Similarly, one gets the more precise estimate for Ix°Vlk by substituting 031(x°) _(a!/(a - y)!)x°-'r in Leibniz's formula, then using the estimate

<O \ ! (a

any)!

< 21QI(&)

the proof of which is left to the reader I

The student will benefit by determining himself which of the following

func-tions are in S or P : e-'Ixl for a E C, e'(x,t) for a E Rn, cos IxI Wee arenow going to define an important subspace of S, the space of C°` functionswith compact support, also called test functions

Recall that the support of a continuous function p can be defined as follows:

x V supp cp if and only if cp = 0 in a neighborhood of x The support of 0 isthus always a closed set, and it is compact if and only if it is bounded Thenthe space Co of test functions is defined as the space of C" functions withcompact support If cp E Co and Sl is an open set containing supp gyp, we will

write more precisely p E Co (fi) (thus, Co = Co (R")) It is clear that test

functions are automatically in S, but it is less obvious that Co : 0 The classic

example of a test function is that of cp(x) = f(Ixl2 - 1) where f(t) = ell' if

t < 0, f (t) = 0 if t > 0 Indeed, this function is in CO° since this is true for

f (classic exercise), and its support is B, = {x E Rn; IxI < 11 Moreover, if

and a function with these properties will be called a unit test function.

These unit test functions can be used to construct partitions of unity, as inthe following result which will be used to reduce proofs of global properties tolocal proofs

LEMMA 1.5 PARTITIONS OF UNITY

Let K C R be a compact set contained in a union of open sets 52; Then thereexist a finite number of functions cp,j E Co (52,)(I < j < k) such that cps > 0,

Ej=1

7 < 1 and X:j=1 pj = I in a neighborhood of K

PROOF (i) First assume that K C 521 Then for e > 0, let us denote by KE theset of points at distance < e from K, and set 7E(x) = E'"cp(x/E) where cp is aunit test function as above (Thus, cp, satisfies the same properties as cp but thelast one to be replaced with supp TE C Be.) We choose e = one-fourth of thedistance from K to the complement of 521, then we set

(ii) In the general case, the compact K is actually contained in a finite union

521 U U 52k of open sets 52,, and K = UkI K; for some K3 C 1l that arecompact For each j < k, let j E Co (52j, satisfying iP = I near K as in

part (i) of this proof, then let

'P1 = ,L1, 'P2 ='02(1 -0l), , 'Pk =V)k(I -'01) (I -'bk-I)

These functions solve our problem because they satisfy cpj E Co A)- cpj > 0,and

8 Fourier Transformation and Sobolev Spaces

measurable functions2 u on R!' satisfying NormLP(u) < oc where

I /pNormLP (u) = (Jlu(xvdx) if p < 00,

NormL-(u) = inf{U E 1R lu(x)I < U almost everywhere}

For p = 2 and p = oo we will use the simpler notation

llullo = NormL2(u) and lulo = NorrLs(u)

(note that lulo corresponds to the previous definition when u is continuous).These spaces are Banach spaces; whenever u and v are two measurable functionssuch that uv E L', we will use the notation

(u,v) = fu(x)i(x)dx.

This product is linear in u and semi-linear in v (i.e one has the relation (u,

v +µw) = (u, v) + p(u, w)), and since llullo = (u, u) for u E L2, (u, v) is a

scalar product that defines a Hilbert space structure on L2

The following statement gives the properties of the Schwartz space S thatcan be obtained directly from integration theory

(ii) For any measurable u such that up E L' for all ep E S,

(u, <p) = 0 for all W E S

Fourier transformation and distributions in R" 9

PROOF For p = oo and p E S, it is clear that NormL- (<p) = Io oc Forany 1 < p < oo and cp E S, one can write

kP(x)1 n <_ sup

xER" O(x)11(1 + xi'

J=1 n

Thus by integration (cf Lemma 1.3) we get NormLP (gyp) < irn/p2n IWI2n

Prop-erty (i) then follows by using Holder's inequality I(u,cp)I < NormLP(u)

NormLQ (gyp) where q = p/(p - 1) is the conjugate exponent of p

If a measurable u satisfies (u, gyp) = 0 for all cp E S, we will prove that onehas (u, XE) = 0 for any bounded measurable set E with characteristic function

XE, for this classically implies that u = 0 almost everywhere First, if E is anopen set, the sets Kj C E of points at distance larger than or equal to 1 I j fromthe complement of E are compact, and the sequence W E Co (E) of functionssatisfying cpj = 1 on Kj as in Lemma 1.5 satisfies also V = XE point-wise, so that (u, XE) = limt , (u, V j) = 0 by dominated convergence Now, ageneral bounded measurable E is, up to a negligible error, the limit of a sequence

of open sets with characteristic functions Xj Therefore, XE = limy-"" Xj

al-most everywhere, then (u, XE) = limj (u, X j) = 0 again by dominatedconvergence Thus, we get (ii)

Finally, given a semi-linear form U as in (iii), the existence of a u E L2 suchthat U(cp) = (u, cp) and Ilullo < C follows from Riesz's representation theorem,while the uniqueness comes from property (ii) I

1.2 Fourier transformation and distributions in R"

For any u E L', the Fourier transform u of u, defined by the formula

u(0 = Jc_i()u(x)dx,

is a bounded continuous function since obviously NormL, (u) for all i; ER", while the continuity follows from dominated convergence The purpose ofthis section is to extend this transformation to a large class of objects (containing,

in particular, L2 functions) called temperate distributions and to establish itsbasic properties Let us begin with an example

10 Fourier Transformation and Sobokv Spaces

Example 1.7

The Fourier transform of cp(x) = e-IS12/2 (21r)f12e-IEI2/2 0

PROOF By definition, one can write

dx

<

which yields Ell e-(x+`f)2/2 dx = f e-x2/2 dx by taking the limit for A -a

oo Thus the result comes from the identity f !: e-x2/2dx = (2Tr)h/2, whichcan be proved by computing its square with polar coordinates as follows:

Jo d9 /JJJo e-r /2r dr = 2ir.

As a first step, we establish properties of Fourier transformation in theSchwartz space S To simplify the formulas, we introduce the operators Dj =

Fourier transformation and distributions in R" 11

-i8;, their powers D° = and the notation u(x) = u(-x) Then,

we can state the following theorem

THEOREM 1.8

For any 0 E S, one has cp E S with IWIk < (8ir)n(k + 1)!IWI2n+k (continuity

of Fourier transformation) Moreover, the Fourier transform cp of a cp E Ssatisfies

(i) For any a E Z+, Dx cp(s) and

(ii)

(iii)

For any u E L', (u,

(Inversion formula) P = (2ir)"cp or in other words, cp(x) = (21r)-n

(iv) (Parseval's formula) For any '0 E S, (21r) n

PROOF Since W ELI, is bounded and continuous with NormLl(cp) <(27r)nIWI2n (cf Theorem 1.6) Since the integrand is in S, we candifferentiate under f or integrate by parts, and these operations give for a E Zn+

DF JD(e())W(x)dx = fe_i()(_x)W(x)cix= (-x

L5 fe2D(x)dx = f p(x)(-Dx)'(e-i(x,f))dx

which are formulas (i) In addition, these formulas prove that aO is theFourier transform of some function in S; therefore, it is bounded and continuouswith the estimate

ICav3;3Io = IOxOcoIo <

(cf Lemma 1.4), and this gives E S with I Ik S (8ir)' (k+

If u E L' and cP E S, then u(x)cp(4) E L' (R2n) and by Fubini's theorem andthe change of variables y = -x one gets

(fl, W) = f (fe'>u(x)dx) P(0 4

= Ju(_)(Je1(YJw()de) dy = (u, 0)

For the inversion formula, the proof is more difficult because e`(x-v,E>cp(y)L' (R2') as a function of y and C To overcome this difficulty, we introduce afactor e-l0/2 to obtain absolute convergence, and by Fubini's theorem

12 Fourier Transformation and Sobokv Spaces

and the change of variables y = x + ez, _ (/e we get

and this is the result according to the formula given in Example 1.7

Finally, Parseval's formula easily follows from (ii) and (iii) since these erties imply

Property (iii) in Theorem 1.8 will allow us to extend the Fourier mation as announced Indeed, on one hand we remarked in Theorem 1.6(i)and (ii) that each LP space can be considered as a subspace of the space ofsemi-linear forms on S On the other hand, the relations (u, cp) = (u, gyp) (which

transfor-follows from the change of variables y = -x when u is any function) and

(u, cp) = (u, c) (which holds for any integrable u according to Theorem 1.8(ii))still make sense for general semi-linear forms cp '- (u, gyp) on S, even not de-fined by a function u, and can be taken as definitions of new semi-linear forms

u and u. Actually, to get a good theory where we can also take limits, wemust restrict ourselves to continuous semi-linear forms, and this leads to thefollowing definition: we say that u is a temperate distribution, and we write

u E S', if u is a semi-linear form cp f-, (u, cp) on S (not necessarily defined by

a function u, even if we keep the same notation) with two constants C E R and

N E Z+ such that

I(uMI < CkOI N for cp E S

It follows from Theorem 1.6(i) and (ii) that every Lebesgue space LP is asubspace of S Thus, the extension of the Fourier transformation to S' willgive a meaning to u for all u in any LP (but this it will merely be a continuoussemi-linear form, not always a function) The first properties of the Fouriertransformation in S' can be stated as follows

THEOREM 1.9

Let U E S'; then the formulas

(u, w) = (u, 0) and (u, p) for cpES

Fourier transformation and distributions in R" 13

define distributions u and u E S Moreover, one has u = (2a)"u (inversionformula), and u E L2 implies u E L2 with Parseval's formula:

(u, v) = (2n)"(u,v) for u,v E L2.

(The similar formula (u, cp) = (27r)" (u, gyp) for u E S' and cp E S followsdirectly from the inversion formula.)

PROOF To prove that u and u E S', we just have to check that cp and cp

depend continuously on cp E S This is obvious for cp, and for cp this followsfrom Theorem 1.8

The inversion formula also comes from results of Theorem 1.8, since for all

(u, (u, ) = (u, i) = (27r)" (u, +b) = (27r)" (u, +Il) = (27r)"(u, cp)According to Theorem 1.8(iv) we have for u E L2

I (u,'p)I = I (u, v)I <- IIullollcvllo = (2ir)n'2IIulloIIcpIIo

so that the semi-linear form U(W) = (u, cp) satisfies the assumptions in rem 1.6(iii) with the constant C = (27r)"'2IIullo Therefore the distribution u isequal to a square integrable function with IIuIIo < (27r)"/2IIulIo; thus if we usethe inversion formula we get

Theo-(27r )"/2IIuIIO = (2r)"/2IIuIlo = Theo-(27r) ""IIuIIo 5 IIuIIo <- (2ir)""2IIulI0with the equality all along, which gives IIuIIo = Finally, Parseval'sformula then follows since any pair u, v of square integrable functions satisfiesthe elementary identity

(u, v) = I (Ilu + vllo- Ilu -vllo + i11 U + ivllo - illu -ivllo) I

As a matter of fact, we can also extend to S' several other simple operations

on functions

First, the operation of differentiation: indeed, as soon as a function u is

smooth enough to define Dau without ambiguity, we can integrate by parts in(Dau, cp), and this gives

(Dau, cp) = (u, Da p) for cp E Cosince the integrated terms vanish If, moreover, the functions u and Dau definecontinuous semi-linear forms on S, the semi-linear form cp ' , (u, Dace) is also

in S' (cf Lemma 1.4(i)) and agrees with cp H (Dau, cp) on Co and then even

on S thanks to the following result

14 Fourier Transformation and Sobokv Spaces

LEMMA 1.10

If u and v E S' satisfy (u, gyp) = (v, cp) for all c E C01, then u = v (i.e., (u,cp)=(v,cp) forV ES).

PROOF Choose a * E Co such that io = I on B1, then for 0 < e < 1 set

zji,(x) = '(ex), and also for cp E S set c', = -0fcp E Co One can estimate thenorms of cp - ct by writing

x°e(sv - 0E) = (I - 00x°O + E ( a )O

7#0For -y # 0, 6P7 P, = 0(e) so that the sum is bounded by As for thefirst term, one has 11 - AEI < e21xI2 since 0 < I - ipE < I and lexl > I onsupp (1 - iE ) Thus we get the estimate

Now, u - v E S' satisfies an estimate I(u - v, p)I < CI API N for some constants

C and N, and since (u - v, wE) = 0 for <p, E Co, one has

I(u - v, v)I = I(u - v, V - We)I <- COeIIPIN+2

It follows that (u, gyp) = (v, <p) for all W E S by taking the limit for e -i 0 U

Thus, if u E S', it follows from Lemma 1.4(i) that the formula

(D* u, gyp) = (u, D°cp) for p E S

defines a distribution D°u E S' for any a E Z+, and from the discussion

given above it is clear that this operation extends the usual differentiation offunctions The student will remark that differentiation is always possible inthe space of distributions, and this is an important improvement of the classictheory of functions: we can now always differentiate a function, even when it

is not "classically" differentiable (but in that case, of course, the result will not

be a function, but merely a continuous semi-linear form) Also notice that wealways have D2 Dk = Dk Dj, since this is true for C°° functions

These wonderful properties, however, are not compatible with a good tiplication theory: indeed, it has been proved that it is impossible to define ingeneral the product of two distributions with the usual properties of products offunctions (e.g., see Exercise 4.5(a)) Here, we start from the formula we canwrite in the case of two functions and u,

mul-(tu, 0) = f V)(x)u(x)cc(x)dx = (u, V),

so that we see such a formula will define a distribution Ou E S' for any u E S'only if ,p E S for all cp E S. Thus, the operation of multiplication will

Fourier and distributions in R" 15

be restricted to the following two situations: (i) when & and u are functions,,ou is defined in the usual way; (ii) when E P and u E S', the formula

(mu, cp) = (u, cp) for cp E S defines a distribution iiu E S' according to

Lemma 1.4(ii), and this agrees with the usual definition when u is also a function

We complete this list of elementary operations on distributions by giving thefollowing two: if u is a function and p E S then

(u, gyp) = fu(x)cp(x) dx = fu(x)V(x) dx =

and if ru denotes the function ryu(x) = u(x + y),

/(ryu, cp) = f u(x + y)(x) dx = J u(z)c3(z - y) dz = (u,T_yW)

so that we can define distributions u and ryu by these formulas for a general

uES' Relations such as fi=u=u,u=u, rru=rytZuandsoforth are

completely obvious; however, the student is strongly encouraged to prove thefollowing collection of less obvious, but still easy, useful formulas

ryu=&('>u, u=u=u e'(x.n)u=T-77)U

PROOF Left to the reader as an exercise

Actually, considering only semi-linear forms on the Schwartz space S, whichcontains functions with noncompact supports, is equivalent to a certain control

of the "growth" at infinity of temperate distributions To have a good theory ofFourier transformation, we need such a control, since a very wild growth of u atinfinity would correspond to a very singular local behavior of u, and too singular

an object cannot even be a distribution However, if one gives up the Fouriertransformation to keep only the operations of differentiation and multiplication

by smooth functions, one can consider much wider classes of distributions, andeven distributions defined only locally

Indeed, if ! is any open set in R", one can define a "distribution in f2" asfollows: u E D'(il) (the space of distributions in 1) if u is a semilinear form

on CI (Q) continuous in the sense that for each compact set K C 9, there exist

16 Fourier Transformation and Sobolev Spaces

two constants CK and NK such that

I(u,cp)J < CKIcpINK for cp E Co (S2) and supp, C K

(We will write just D' for D'(R" ).) The same formula (4'u, (p) = (u, W) as

above allows us to define the product 4'u E D'(S2) of any u E V(Q) and

1P E Coo (Q).

It is clear from Lemma 1.10 that S' can be considered as a subspace of V.Moreover, given a distribution u in S2 C R, we can define its restriction u,

to a smaller open set w C 1 simply by restricting the semi-linear form u to

Co (w) One then says that u and v E D'(tl) satisfy u = v in w C 9 if one hasul,, = vow, These considerations give meaning to the notion of local behavior of

a distribution; the following result shows that the local behavior of a distributiondetermines it completely

PROPOSITION 1.12

Let 0 be an open set in R"; if it and v are two distributions in SZ such thatevery point x E 0 has a neighborhood where u = v, then u = v in Q

PROOF For any cp E Co (S2), K = supp p is covered by open sets Q., C

SI where u = v by assumption Then, using the partition of unity E VJ ofLemma 1.5, one can write

(u,VEWJ) =

_ (v, w2 ') = (v,'P VJ) _ (v, 4%)since cpjW has its support in S2, where u = v I

This property gives clear meaning to the notions of support and singularsupport of a distribution, which we now introduce Indeed, if u E V(Q) and

x E Q, we say that x V supp u if x has a neighborhood where u = 0 (i.e.,

the same definition as in the case of a function u) and we say that x V singsupp u if x has a neighborhood where u is a smooth function (i.e., if there exist

an w C St with x E w and a 4 E C°° (w) such that (u, cp) = (4', cp) for all

cp E C0 (w)) It is clear that supp u and sing supp u are closed subsets of 0,and

supp (4u) C (supp 4/i) fl (supp u) and sing supp (4'u) C sing supp it

if 4' E C°°(1) and u E D'(Il) The following characterizations are also useful,but we leave their easy proofs to the student as exercises: x i supp u (resp

x 0 sing supp u) if and only if x has a neighborhood w such that cpu = 0 (resp.cpu E Co) for every cp E Co (w); if F is a closed subset of Q, supp u C F ifand only if (u, cp) = 0 for every V E Co' (11) with supp cp fl F = 0

Sobokv spaces 17

Of course, these classes D'(1) of distributions contain distributions that arenot "temperate" for two reasons: because they are not defined on the whole

of R", nor even when 1 = lR", because there is no control of their "growth"

at infinity Locally, however, these distributions are "temperate": by that wemean that distributions with compact supports can be extended to the whole ofIR" as temperate distributions (Actually, any distribution u can be written as alocally finite sum of distributions with compact supports if one multiplies u by

a partition of unity slightly more general than that from Lemma 1.5.)

Indeed, if u E D'(Sl) has a compact support K and if we choose a 0 E

Co (Sl) such that 1' = 1 near K (cf Lemma 1.5), one has (u, gyp) = (u, iV) forall V E Co (fl) since (u, (1- ')gyp) = 0, and thus one can extend the semi-linearform u to S (and even to C°° (IY" )) by setting (u, gyp) = (u, ?Pp) for cp E S.Moreover, this extended form satisfies

I(u,w)I <CKI'01PINA <CVCKICINK

if K = supp b without any constraint on supp cp, so that u E S' (This extensionconsists essentially in setting u = 0 outside Q.) The subspace of D'(1) formed

by distributions with compact supports is denoted by £'(f) (just £' for £'(R" ));

if wCfl,one has £'(w)C£'(Q)CS'.

Finally, we end this section by stating the classic Paley-Wiener-Schwartztheorem, which shows that distributions (resp smooth functions) with compactsupport can be recognized on their Fourier transforms Since we will use thisresult only in the very last application (in Section 4.3), we provide the proof inExercises 1.7 and 1.8 rather than here

THEOREM 1.13 PALEY-WIENER-SCHWARTZ THEOREM

Let U(() be a function defined on R' Then, it is the Fourier transform of

a distribution (resp a C°° function) with support contained in BA = {x ER"; Ixl 5 A} if and only if U can be extended as an entire function U(() onC" satisfying an estimate

18 Fourier Transformation and Sobokv Spaces

correspond to multiplication by polynomials on u, it is clear that the smoothness

of u can be measured by the growth of u at infinity The definition of Sobolevspaces is based on these properties, and this way of measuring smoothness will

be convenient especially when we will deal with L2 estimates

From now on, the Greek letter A will denote the function A(e) = (1 + 112)1/2defined on R", and more generally we will write A" (t) = (1 + 1e12) 8/2 for s E Rand C E R For any s E JR, we say that u E H8 (the Sobolev space of exponents) if u E S' and A'u E L2 In other words, u E H' if u is a function satisfying

IIuI12 = (27r)"f(I + 1C12)81u(C)I2 d < 00(the factor (27r)-" is introduced here to keep the convenient relation 11ullo =NormL: (u), cf Parseval's formula in Theorem 1.9) or more shortly satisfying11u11, = (21r)-n/211Aeu11o < oo Since H' C Ht if s > t, we will also use

the notation H-OO = U8H' aqd H°° = fl,H' The inclusions S C HO° C

H-°° C S' are immediate

The first properties of these spaces that we prove show that they measureessentially the same smoothness as the classic classes Ck up to a fixed shift ofexponents Indeed, they clearly imply that u E Ck = u E HIM (which means

Vu E Hk for all So E C01), and, for example, that u E H"+k = u E Ck

PROPOSITION 1.14

For all s E R one has

uEH'+' q u,Diu, , andD"uEH8

with the equality 11u118+i = 1ju118 + >, 11 Diu118 Moreover, for any k E

Z+ U {oo},

(i) uEHkt* D"uEL2forall lal <k.

(ii) s > (n/2) + k and u E H' D' u are bounded continuous

func-tions for la1 < k (with 1D'ulo < C,.k1Mu)i,: for example, one has

lulo S 2-"'211ul1" for u E H")

PROOF Since A2 (l;) = 1 + (1;12 = 1 + E, ,2, one has for any function uIAe+l,u12

=A21aeu12 = 1A'u12 + lA'Ciu12 = 1ABu12 + E 1A'D,j"u12

from which the equivalence and the equality of norms follow Properties (i)and (ii) can then be proved by induction Indeed, (i) is true for k = 0 in view

of Theorem 1.9 As for (ii) for k = 0, s > n/2 implies \-" E L2 thanks to

Lemma 1.3, so that u E H' implies u = (A_')(a'u) E L' as a product of twoL2 functions, therefore u E co fl L°° Finally, the estimate lulo < C8,o11u118

Sobolev spaces 19

follows from the Cauchy-Schwarz inequality: in the case s = n, for example,

Iulo <- (21r)-nNorMLI(u) < (27r)-nlllla-n1Io(21r)-n,2ll,\null0

<- 2-n/2llulin

since III-nllo < irn/2 as proved in Lemma 1.3 1

REMARK Actually, property (ii) can be improved as follows: one has u EH' = u E CE (the Holder space of exponent e, i.e., the space of functions u

such that Iu(x) - u(y)I < CIx - yl' - it is assumed implicitly that e < 1) as

long as s - e > n/2 (cf Exercise 1.9); one can also prove that for any s > n/2functions in H' tend to 0 at infinity (cf Exercise 1.10) 1

Through Riesz's representation theorem, the H' distributions can also becharacterized as the continuous semi-linear forms on H' This property can

be stated more precisely as follows

PROPOSITION 1.1 S

If u E H' and V E S, then I (u, W)I < Conversely, if u E S'satisfies I (u, W) I < CII VII-, for some constant C and a!1 W E S, then u E H'with IIull, < C

PROOF We get the first conclusion from Parseval's formula and the Schwarz inequality:

Cauchy-I(u, O)I = (2 r)-nl(uov)I = (21r)-nl(A'u, A-',v)I < Ilull,llpll-,.Conversely, if one has the estimate R u, co)l < for all cp E S, then

(Au, p)I = I(u, ))l = I(u, A'V)I

C(27r)-n/2lla-ea'wllo = C(2a)n/211 wvllo.

Thus 1'ui is a semilinear form on S satisfying the assumption in Theorem 1.6(iii),

so that \'u E L2 with IPki llo < C(21r)n/2, i.e., u E H' with llull, < C I

We already mentioned the inclusions S C H°° C H-°° C S'; they are allstrict It is not true that S = H°° (take u(x) = (1 + Ix12)-n, which satisfies

u E H°'D but u S), nor is it true that H-OD = S' For the latter, the reason isthat the control of the "growth" of u at infinity is not sufficient, or on the Fourierside the smoothness of u is not sufficient, i.e., u is not a function Indeed, anytemperate distribution can be thought as an H' distribution for some s E R if

we force the control at infinity, as in the following lemma

20 Fourier Transformation and Sobokv Spaces

LEMMA 1.16

For any u E S' there exists an N E Z such that ? u E H-N if 1' E S orV)(x) = (1 +IzI2)-N

PROOF Since U E S', one has an estimate I (u, w)I < Clwlk for some C and

k, and all o E S; one can then write

C2kl,lk

101:5k

by using Leibniz's formula (cf Theorem 1.2) for expressing derivatives of

Furthermore, 11ilk is a finite constant if '0 E S, and also if O(x) = (1 +

lxl2)-N with N > k/2 On the other hand, I1,PIIn.+I81 in view

of Proposition 1.14(ii), so that the estimate becomes l('u,V)l <

Hence, thanks to Proposition 1.15, we get z'u E H-N with N = n + k I

When using H' distributions in the study of linear partial differential tions with variable coefficients, we will have to consider products of such dis-tributions with the coefficients of the equation To measure the smoothness ofthese products, we first compute their Fourier transforms

holds for any u and v E H'°I (and also for u E H°° and v E H-°°, but this

already follows from Proposition 1.11 since H°° C 1'; cf Proposition 1.14(11)).PROOF Since u and v E L2 uv E L', we have the formula

Sobokv spaces 21

as required If U E H°° and ' E S, they are both in L2 so that we can write asabove

uVG(n) = (2ir) " f u(77 (27r)-' f UW - 76(0 A.

Thus, for u E H°°, v E H'°°, and cp E S, setting (21r)-n (-C) (so

that V = 1') and writing

(uv, gyp) = (uv, (27r)"(uv, z/)) _ (27r)' (v, iiV,)

(27r)-" f v(rl)u( - WOWA drl

=

J (_n f u( - n)v(r1) dill dCagain give the same formula for The other equality with the integral in((E R") comes from the change of variables n+(' = t, while Leibniz's formulafollows from multiplication of v(t) by

(cf Theorem 1.2(i)) I

To get estimates on integrals as in Lemma 1.17, we will need the followingelementary result, known as Peetre's inequality

LEMMA 1.18 PEETRE'S INEQUALITY

For any aERandall{,77 ER", one has

7A8(e) 2181,\I81(C - n)A8(n).

PROOF The triangular inequality for the euclidean norm in R" gives

(I + ICI) <- (1 + IC -101 + 1771) <- (1 + IC - nl)(I + Inl)

22 Fourier Transformation and Sobolev Spaces

see that

which can be rewritten

From these lemmas we get the following continuity properties

From Proposition 1.14, it is clear that each D° maps continuously H8 intoH°-1Q1 for any a E Z and s E R Therefore, the result for a(x, D) followsfrom the first part of this corollary I

Finally, we will close this introductory chapter by giving our first examples ofpseudodifferential operators, which here provide merely another way of writing

the H8 norm Indeed, if we denote by )8(D) the operator from S' into S'

defined by D)u = au for u E S', we will have Ilull8 = (2n)-ni211A°ullo =(2ir)-"'2IIA8(D)ullo = Ilae(D)ullo, thanks to Parseval's formula The identityAs(D)At(D)u = `+t(D)u

is immediate, and the following properties are notmore difficult

PROPOSITION 120

ForanysER, coES,anduES',

()"(D)u,V) = (u, A'(D)W)

Moreover, for any t E R, is E H' if and only if At(D)u E H°-t, with 11ullr =IPPt(D)uIIs-t

PROOF The formula for )t'(D)co follows from the inversion formula since

0 E S A-4(3 E S, while the formula for A'(D)u comes from

()R(D)u,V)

Finally, the identity A8-tat(D)u = a'-'A'u = A'ti shows the equivalence

between u E H' and At(D)u E H'-t, and also the equality of norms I

Exercises

1.1 A Banach algebra of holomorphic functions The goal of this exercise is to proveproperties of multiindices a E Z and to use them to construct a Banach algebra ofholomorphic functions (i.e., a Banach space with a continuous product) Such analgebra is useful when studying nonlinear problems with holomorphic functions.(a) Use an induction on the dimension n to show that for any z = (z1, , z") EC",aEZn, andNEZ+,

(c) One considers the space of formal power series u(z) u(z°/a!)

(i.e., the space of sequences uo of complex numbers indexed by Z andequipped with the formal product

and one defines

NormB(u) = sup IuaI(IciI + l)2

24 Fourier Transformation and Sobokv Spaces

Compute the norm of u(z) = (zO//3!) (i.e., uo = 0 if a 36 /3, up = 1).Show that NormB (u) is a norm on B = {u; Norm3 (u) < oo} and that B

is a Banach space for this norm Show that for any u and v E B, uv E Bwith NormB (uv) < 16Nonn8 (u)NormB (v).

(d) Show that u E B implies that E u,,(z°/a!) is absolutely convergent

in 1z, I + + 1. Conversely, if F u0(z' /a!) is convergent inmax{lz,l, j < n} < I + f for some f > 0, show that u E B

More generally, show that if F(z, Z) = F,,3 (z" /a!) (Z1113!) isconvergent in max{lz,1, j < n; JZkl, k < N} < 1 + f for some f > 0 and

if u = (u1, , UN) E BN with NormB (uk) < 1/16 for all k < N thenthe function f (z) = F(z, u(z)) satisfies f E B

1.2 A distribution u E S' is said to be "real valued" if ii = u Show that u is realvalued if and only if (u, cp) E R for all real-valued V E S Show that iz is realvalued if and only if u = u

1.3 Let b be the semilinear form S 3 '- (6, p) = (P(0) Show that 6 E S' and thatsupp 6 = {0} Compute b and determine all the s E R such that 6 E H'.1.4 For a < b E R and c E C, one defines on R(n = 1) the function

f (x) = e`= if x E [a, b]; f (x) = 0 if x [a, b].

If, moreover, Re c > 0, one also defines on R

g(x) = e-'Izl and h(x) = e-"'/2.

Show that these functions are in L' and compute their Fourier transforms (for h,first study the case c E R then prove that depends holomorphically on c).1.5 Forc E C, Rec > 0and E R(n = 1) one sets G(4) =

Show that G E L' and compute C (without using the results of Exercise 1.4) bythe following method.

For x < 0, compute C(x) by using Cauchy's integral formula with the path

for large R

For x = 0, compute d(O) by a direct integration if c is real, then remark thatG(0) depends holomorphically on c

For x > 0, use the same kind of device as for x < 0

Finally, compare your results with those of Exercise 1.4.

1.6 (a) For c E C\{0} and Rec > 0, the functions defined on R(n = 1)h(x) _

e-`Z2/2 are all bounded by I (they are uniformly in L°°) By taking thelimit for Re c - 0* in the formula (h, rp) = (h, gyp), use the results ofExercise 1.4 to find the expression of h also when Re c = 0 (but c # 0)

Exercises 25

(b) If A is a real symmetric nonsingular n x n matrix, the function H(x) =e`(A=,=)/2 defined on R" is obviously bounded Show that its Fourier trans-form is given by the formula

k(f) = (2a)"J2IdetAI-1/2eIf (spA)e-:(Awhere sgn A is the signature of A, that is, the number of positive eigenval-ues minus the number of negative eigenvalues Finally, determine all the

of the types given in Theorem 1.13

(b) Let u E C'° such that supp u C BA Set U(() = f dx andshow that U is an entire function on C' extending the Fourier transform of

u and satisfying estimates IU(()I < CN(1+I(I2)-NeAltm(l for all N E Z+and some sequence CN.

(c) Let U be an entire function on C" satisfying estimates iU(()I < CN(I + I(I2)-NeAllmll for all N E Z+ and some sequence CN Set u(x) _

Show that u(() = U(() and u E H°° Show that for any e > 0,u(x) = (21r)-" f e'(= i(x/e)) d{, then prove that u(x) = 0for Ixi > A and finally give the conclusion

1.8 The Paley-Wiener-Schwartz theorem for distributions In this exercise, use theresults of Exercise 1.7 In particular, if tG E Co , denote by r1'(() the entireextension of ' In questions (b) and (c), the number A > 0 is fixed

(a) Let ', E S, u E S', and So be a unit test function

Show that for any e > 0, gyp(-e()t/, (resp c'(e()u) is the Fourier form of a function r' , E S (resp of a distribution u, E S'), that lim,.o

trans-10 - rb-, Ik = 0 for all k E Z+, and that lim, o(u,, iP) = (u, 0)

Show that supp V), Csuppr(i+B,={x+yER";xEsupp,pandyEB,}, then that supp u, C supp u + B,

(b) Let u E S' such that supp u C BA

Using Lemma 1.16, show that the distribution u, defined in (a) satisfies

u, E H'C and supp u, C BA+, Then show that U(() = lim,-ou,(()

is an entire function and that its restriction to R" is the Fourier transform

(c) Let U be an entire function on Cl satisfying an estimate IU(()I < C(I + I(I2)NeAUm(I

Show that there exists a u E S' such that it = UIR". Show that thedistribution u, defined in (a) satisfies u, E C°° with supp u, C BA+E, thenthat supp u C BA, and give the conclusion

1.9 For 0 < e < 1, one defines the HSlder space C' as the space of functions u

26 Fourier Transformation and Sobolev Spaces

defined on R" such that lu(x) - u(y)l < Clx - yl' for some constant C and anyxandyER".

Show that for any 0 < e < I and x, y,£ E R",le`(',() - e'(11,0 I < 2'

lx - y1`a`({) Show that if it E H' with s > (n/2) + e, then it E C

1.10 Let s > n/2, u E H', and cp E C°° such that ,p(0) = I and supp cp C B1 Forany U E Lz, one sets U,(C) = (21r) -" f U({ - en)0(77) dj7

Show that (fi), is the Fourier transform of a bounded continuous function u` (x)and that supsx,,18 1U(x)1 < lu - 00

Show that for any , rt E R" one has

A'(n)l <- s2'-' If - nja'-' (,7)A'-' ( -'1)

Show that for any U E LZ one has )-'U E fL2 and that one can write

UU(f)

-akernel K,(t,'i) satisfying for some constant Co

f IKf(f,71)ldC < Coe and JlK('i)ldii Coe.

Use Schur's lemma (i.e., Lemma 3.7) to prove that for any U E Lz, lim,.,oIIUe - A'0`17)(110 = 0

Using the L2 function U = a'u, prove that u(x) = 0

1.11 The algebra H' for s > n/2 The following properties of Sobolev spaces lead totheir use in the study of nonlinear problems as well

(a) Let r, s, t E R+ be such that r < s, r < t, and r < s + t - (n/2).Show that if l -'ii <_ InI, one has A2r(4) < 5rA2t('i)Azr-zt( - ii).Similarly, prove \2"(e) < 5ra2'(e - p)A2r-2a(r!) for l - r71 ? Inl.Let it E H' and V E Ht Write the integral of Lemma 1.17 as a sum oftwo integrals on the domains 1 - rll S Inl and l - 7I ? I'l respectively,then show that uv E Hr With Iluvjlr < CIIuIIa1IvIIt where the constant Cdepends only on r, s, and t

Let it and v E H' for some s > n/2: show that uv E H' with Iluvlla <Callullallulla-

(b) More generally, let F(x, X) be a C' function defined on R" x RN, u =(it,, , UN) a function defined on R" and valued in RN satisfying u, E

Hi° for all j < N and some s > n/2 (i.e., ; E H' for all cp E Coo),and set Fu (x) = F(x, u(x)) Choose a unit test function cp, and for anylocally integrable v set v, (x) = f v(x - ey)cp(y)dy fore > 0 Prove that

va E C°°

If v is continuous, show that v = lime_o v, uniformly on every compactset of R" If v is square integrable, compute the Fourier transform of v,(see Lemma 1.17) and show that limf_o 11va - v110 = 0.

Assuming s > I and using similar arguments, show the validity of thegeneralized chain rule

(F", Dkib) = ((DkF)u+>(D,,u,) (F)

+G)

for all (,EC0

Prove by induction that for any t E Z+ with t < s, F E Ct impliesF°EHH.

Notes on Chapter 1 27

Notes on Chapter 1

During the eighteenth century, trigonometric series were introduced in the lems of interpolation (Euler), astronomy (Clairaut), and sound (Lagrange) Bythe end of the century, they played a central role in the famous controversy overthe vibrating string problem, which would lead eventually to the revision ofthe bases of analysis initiated by Cauchy [28] The integral transformation also

prob-is introduced in Fourier's me moire [35], considered a fundamental contribution

to the theory of trigonometric series despite its lack of rigor (Actually, thesame results were obtained concurrently by Cauchy and Poisson.) The Fouriertransformation was then extended, thanks especially to the Lebesgue integrationtheory, but it is the introduction of distributions by Schwartz [61] that simplifiedand unified the theory

The best account on the origins of distribution theory is to be found in theintroduction of Schwartz [61]: this theory has roots in the symbolic calculus ofengineers initiated by Heaviside [39] and continued by the physicist Dirac [29],

in the turbulent solutions of Leray [51], in the derivatives of Sobolev [65], inthe finite parts of Hadamard [38], in the Fourier transformation as extended byBochner [18], etc Expository texts on distribution theory are Schwartz [61],Treves [67, Part II], and Gelfand and Silov [37] Extensions of these ideas can

be found in Beurling [17] and in Sato's theory of hyperfunctions [60] (see alsoHormander [8, Chap 9])

Finally, Sobolev spaces were first introduced for positive integral exponents

by Sobolev [64,65]; they now play an increasingly important role in the theory

of partial differential equations The student will find a systematic study ofthese (and related) spaces in Adams [14]

Pseudodifferential Symbols

Introduction to Chapters 2 and 3

Elementary properties of Fourier transformation allow us to write for (p E S

then

D°co(x) = (27r)-n

by the Fourier inversion formula For a linear partial differential operatora(x, D) a°(x)D°, these formulas thus lead to the following ex-pression:

a polynomial (see Proposition 1.20) Moreover, the operator A-2(D) is a sided inverse of A2(D) = 1+Ej Dj' = 1-A where A = Fj OJ2 is the euclideanLaplacian operator This simple remark shows that the operator A-2(D) can beused to study the equation (1 - O)u = f Indeed, for any f E S', this equationhas at least the solution u = A-2(D) f (property of solvability), and if f E S,then any solution u E S' is actually in S since u = A-2(D)(1 - A)u =A-2(D) f E S (property of hypoellipticity, i.e., the solutions are smooth as soon

two-as the right side is)

The purpose of the theory of pseudodifferential operators is to extend this kind

of proof to more general linear partial differential equations than (I - O)u = f

28

Definition and approximation of symbols 29

The key idea is to replace all the computations on the operators with algebraiccalculations on their symbols If we consider sufficiently large classes of sym-bols (which will contain functions that are not polynomials, corresponding tooperators that are not differential, therefore called pseudodifferential operators),

we will be able to find inverses of these operators, at least for the best of them,called elliptic operators However, one can notice that the problem with variablecoefficients is much harder than with constant coefficients, since a(x, )c (f) is

no longer the Fourier transform of a(x, D)p, and it turns out that the operator

we would get by using the symbol is not an exact inverse of theoperator a(x, D) Thus, in general, we will construct only approximate inverses

of elliptic operators, but this will be sufficient for the study of solvability andhypoellipticity of these operators

To construct a good theory of pseudodifferential operators, one must restrictthe class of allowed symbols, and this is the main topic treated in Chapter 2.Here we present the basic classes S', also known as S o One of the main fur-ther developments of the theory has been to extend the fundamental properties

of these basic pseudodifferential operators to larger classes of symbols, whichcan be adapted to the study of various problems in partial differential equations,but this is definitely beyond the scope of this course (We simply refer to Coif-man and Meyer [6], Hormander 18, Chaps 18.5-18.6], and Bony and Lerner [4]for such extensions.) After a section devoted to simple oscillatory integrals, weclose Chapter 2 by defining two fundamental operations on symbols, the use ofwhich will be essential in the next chapter

The only motivations for the results presented in this chapter lie in the scription given in Chapter 3 of the essential features of the theory Chapter 3will thus provide the definition of the operators and the proof of their continuity

de-in Sobolev spaces, a description of the symbolic calculus (i.e., the dence between operations or estimates on symbols and operations or estimates

correspon-on operators), and a sketch of the invariance property under a change of ables (which allows us to define the corresponding operators on a manifold).This set of results can be considered as the most basic properties of pseudodif-ferential operators, and we will see in Chapter 4 that it is already sufficient to

vari-be conveniently used in the solution of difficult problems of partial differentialequations

2.1 Definition and approximation of symbols

As in Section 1.3, we will keep the notation a" for the function )a(£) _

(1+ICI2)8/2 where CElR and sE R

Let m E R and a(x, C) be a C°° complex-valued function defined on 1[2" x RT'Then we say that a is a symbol of order m, and we write a E S', if the functions

Since St C S"' when f < m, we will also use Sc °= U,"S" and S =

f1,nS' The student will prove that if a E S', b E St, and a,,3 E Zn+, then OOa E S 1,31 and ab E S As a consequence of the inclusions

St C S'" for £ < m, we can consider computations in S" modulo Sf with alower f Thus, besides an exact calculus (without rests), we will also develop anapproximate calculus (modulo terms of lower order than principal terms) Whenlooking for progressively more precise approximations, this point of view willlead to the development of an asymptotic calculus (modulo S-°°)

Our first example is that of symbols of differential operators If a(x,

a polynomial with coefficients aQ E H'°, then a E S'n (cf.Proposition 1.14(ii)) If the coefficients are "only" of class C°° and we want tostudy local properties, we can reduce the problem to the previous case wherethe symbol is in a class S' merely by multiplying the coefficients by a cut-offfunction locally equal to I since the modified operator thus has its coefficients in

Co C H°° A second example is that of the functions a', which clearly satisfyA' E S'" Our third example will be used in Section 4.1 to study the localsolvability of differential operators: if a(x, %) is a function with compact support

in x, (positively) homogeneous of degree m in l; and C°° outside = 0 (if a isalso C°° at = 0, it is a polynomial), then there is a symbol b E S"' (uniquelydetermined modulo S-OC)t such that b(x, t;) = a(x, ) for ICI > 1 Indeed, totransform a into an actual symbol satisfying the definition, it is sufficient to takeb(x, i;) = (1 - where V E Co (R") with supp V C B, and V = Inear = 0, and it is clear that if b(x, f) = a(x,.) = c(x, ) for ICI > I, the

difference b - c is in Co (R" x R") C S-°O In this situation, we will usuallyuse the same letter a to denote the modified symbol b = (I - yo)a, since thiswill not bring too much confusion

Classes S'n can be characterized by the following equivalence: a E S' '"a

E S°, so that it would be sufficient, from a theoretical point of view, tostudy only the class S° of zero-order symbols However, from a practical point

of view, it is better to have at hand all the orders m E R We already remarkedthat S° is closed under multiplication (it is an algebra); one can even prove thefollowing result

'We say that a symbol possessing certain properties is uniquely determined modulo S-°° if, given two symbols with these properties, their difference is always in S.

Definition and approximation of symbols 31

LEMMA 2.1

If a E S° and F E C' (C), then F(a) E S°.

PROOF Let us write a = b + is where b and c are real valued Since a E S°,

we have b and c E S° C C° fl L°°, and therefore the function F(a) = F(b, c)

satisfies 1((9'rF)(b,c)j < C., for all ry E Z2+ The estimates on rO[F(a)]

can then be proved through an easy induction, which is left to the reader as anexercise r

In setting up an asymptotic calculus as announced above, we will use thefollowing lemma as a substitute for the summation of a series

-Since bj = a, for C outside a compact set in R" we have bj - a, E S-°° so

that b, E S'"-J However, we will need some more precise estimates beforetaking the sum of the series

For JCJ < 2/f1, one has Ac, < v" whence

IO&'b,l < C' A ,-101

for some constants CQ0, and we have the same kind of estimates for JCJ > 2/e,since b, = a there Moreover, since 1 < E., ICI in supp (1 - gyp) D supp b., wecan refine this estimate as follows:

It Obit < 3AI8 ebjI < e3CJ A"-j-01 - 00Thus, if we choose Ej < min{l/CQO; Ia + 31 < j} we will have

IA101-moitb3 I < Al-j for 1a +,Q1 :5j

Since Ej 0, the sum a(x, ) = Ej>° b,(x, ) is a finite sum near any fixed 1;o,and therefore this formula defines a function a E C°° If k E Z+ and a,,3 E Z"+