Elementary Introduction to Theory of Pseudodifferential Operators (Studies in Advanced Mathematics)

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

Series Editor

Steven G. Krantz Washington University in St. Louis

Editorial Board R. Michael Beals

Gerald B. Folland

Rutgers University

University of Washington

Dennis de Turck

William Helton

University of Pennsylvania

University of California at San Diego

Ronald DeVore

Norberto Salinas

University of South Carolina

University of Kansas

L. Craig Evans

Michael E. Taylor

University of California at Berkeley

University of North Carolina

Volumes in the Series

Real Analysis and Foundations, Steven G. Krantz CR Manifolds and the Tangential Cauchy-Riemann Complex, Albert Boggess Elementary 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 and Ronald 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. cm.

p.

Includes bibliographical references (p. ) and indexes. ISBN 0-8493-7158-9 1. Pseudodifferential operators. I. Title. QA329.7.S25 1991 515'.7242-1c20

91-25184 CIP

This book represents information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Every reasonable effort has been made to give reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. All rights reserved. This book, or any parts thereof, may not be reproduced in any form without 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 0-8493-7158-9

Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Contents

Preface

vii

1

Fourier Transformation and Sobolev Spaces

I

1.1

1.2

Introduction Functions in IR" Fourier transformation and distributions in R"

9

1.3

Sobolev spaces

17

Exercises

23

Notes on Chapter I

27

Pseudodifferential Symbols

28

Introduction to Chapters 2 and 3 Definition and approximation of symbols Oscillatory integrals Operations on symbols Exercises

28

Pseudodifferential Operators Action in S and S'

47

Action in Sobolev spaces Invariance under a change of variables Exercises Notes on Chapters 2 and 3

52

2 2.1

2.2 2.3

3 3.1

3.2 3.3

2

29 32 37

43

47

58 61

67

V

Vi

4 4.1

4.2 4.3

Applications

69

Introduction Local solvability of linear differential operators Wave front sets of solutions of partial differential equations The Cauchy problem for the wave equation Exercises Notes on Chapter 4

69 70

Bibliography

97

76 83

89 94

Index of Notation

103

Index

107

Preface

These notes correspond to about one-third of a one-year graduate course entitled "Introduction to Linear Partial Differential Equations," taught at Purdue University during Fall 1989 and Spring 1990. It is an attempt to present in a very elementary setting the main properties of basic pseudodifferential operators. It is the author's conviction that the development of this theory has reached such a state that the basic results can be considered as a complete whole and should be mastered by all mathematicians, especially those involved in analysis. Unfortunately, the beginning student is immediately faced with a technical difficulty that forms the heart of the theory, namely the extensive use of oscillatory integrals, that is, non-absolutely convergent integrals over ll2". Indeed, all the texts written on these pseudodifferential operators assume explicitly, and even more often implicitly, a good familiarity with such integrals, the theory of which is based on the rather difficult results known as stationary phase formulas, and the authors perform changes of variables, integrations by parts, or interversions of the f exactly as if the integrals were absolutely convergent while the allowed rules are probably not quite clear for the uninitiated reader.

The main originality of these notes, maybe the only one, is to restrict the use of such oscillatory integrals to the case of real quadratic phases for which the theory is both simple and pleasant. Of course, this restriction prevents a full proof of the fundamental result of invariance of pseudodifferential operators under a change of variables. Many other important aspects of the theory are not even 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 the following pages will be reached if this simple setting and the few applications given in the last chapter convince the reader of the fundamental importance of the topic and give sufficient motivations for reading more complete texts. The exposition begins with a chapter devoted to the Fourier transformation and Sobolev spaces in R", which both play a central role in the theory. A sufficient knowledge in classic integration theory (properties of Lebesgue measure and related LP spaces in R) is assumed, and Chapter 1 will provide all the additional

vii

Preface

viii

background needed to take up the next chapters. For the more advanced reader who has encountered these topics before, a quick reading is recommended to get adjusted to the notation used throughout the book. Chapters 2 and 3, respectively devoted to basic symbols and basic operators, form the theory itself. Chapter 4 provides applications to local solvability of linear partial differential equations and to the study of singularities of solutions of such equations . To avoid any ambiguity, it is emphasized that nothing is original in the topics presented here: the text has been based mainly on Hormander [8, Section 18.11 and 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 several exercises can be found in this latter reference). Thus, the specific features of this text lie only in the exposition: it is self-contained with very light prerequisites and all the complements that were not strictly necessary to reach the main results have 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 University for the opportunity I had to teach this course. I also wish to thank Mrs. Judy Mitchell, who with great competence and patience typed the manuscript of this course.

- X. Saint Raymond West Lafayette, March 1991

1 Fourier Transformation and Sobolev Spaces

Introduction The main purpose of Chapter 1 is to fix the notation used throughout this course; most of the notation is classic, but some is probably unusual, e.g., the notation

P for the space of C' functions with polynomial growths at infinity. This is why a quick reading is recommended, even if the student is already aware of the topics presented here. The central notion is that of Fourier transformation: for each function a defined on R' (with a controlled growth at infinity), one can define its Fourier transform u., also defined on III, with the following properties: (i) differentiations on u correspond to multiplication by polynomials on u (which is a simpler operation, particularly with respect to the inversion of such an operation); (ii) one can recover u from u essentially by achieving the same transformation a second time; (iii) the Fourier transform of an L2 function is an L2 function. Thus, in order to study the properties of this transformation, it is more convenient to work 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 multi-

plication of u by a polynomial, there is also a correspondence between the smoothness of u and the growth of u at infinity (and by symmetry between the growth of u at infinity and the smoothness of u). This fact is used to define the so-called Sobolev spaces, which are much more convenient than the classic classes Ck of k-times continuously differentiable functions, especially when one deals with L2 estimates.

I

Fourier Transformation and Sobolev Spaces

2

1.1

Functions in !R"

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

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"+, + an and its factorial as the we define its length as the sum j al = al + E Z+ if one has product a! = (a1!)...(a"!). Moreover, we will write a

aj n; in particular, we have the precise estimate f(1+IxI2)-ndx 0 and x E Rn

(I +

Ixl)-8

= (1 + 2Ixl + 1x12)-8/2 < (I + 1x12)-8/2 +x2)-s/2n...(I +x2,)-s/2n < (1 +x2i)-s/2n(1

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 will not use the "only if" part of this lemma, which can be proved by writing the integrals in polar coordinates. I

The notation being thus fixed, we now introduce the Schwartz space S of C°° 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 all pairs a, 3 of multiindices. If we denote by lcplo the supremum over R' of a bounded continuous function cp, the implicit topology of S is that defined by the norms' I'P1k =

sup

lx0B1cplo

= sup{Ix°8Qcp(x)I; x E Rn and Ia + i31 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

Functions in R"

7

we divide this function by the constant f cp(x) dx > 0, we get a new function p satisfying

cpECo ,

J'P(x)dx=1 ,

c p>0,

suppVCB1,

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 in the following result which will be used to reduce proofs of global properties to local proofs. LEMMA 1.5 PARTITIONS OF UNITY

Let K C R be a compact set contained in a union of open sets 52;. Then there exist 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. (i) First assume that K C 521. Then for e > 0, let us denote by KE the set of points at distance < e from K, and set 7E(x) = E'"cp(x/E) where cp is a unit test function as above. (Thus, cp, satisfies the same properties as cp but the last one to be replaced with supp TE C Be.) We choose e = one-fourth of the distance from K to the complement of 521, then we set PROOF

*(x) =

'K2,

'E( x - y) dy

which is a CO° function (take derivatives under f) satisfying ?U = I on KE and supp C K3E C 521 as required. (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 are compact. For each j < k, let j E Co (52 j, 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 k

F, cpj = 1 - (I -'+Gl)(1 - 02)...(I -'0k)

I

j=1

We finally end this section by pointing out that the main motivation for the introduction of the Schwartz space S lies in the fact that when dealing with integrals of such functions, all the difficult operations of integration theory (integration by parts, differentiation under f or interversion of f) will be obviously valid thanks to the good decreasing of Schwartz functions at infinity. We re-

mind that for I < p < oo, the Lebesgue space LP is defined as the space of

8

Fourier Transformation and Sobolev Spaces

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

(Jlu(xvdx)

I /p

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 functions

such 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 that can be obtained directly from integration theory. THEOREM 1.6

One has s C nt I on supp (1 - iE ). Thus we get the estimate IMP - S' Ik 5 ne2lwIk+2 +

fork E Z.

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 u, PROOF

u=u=u

e'(x.n)u=T-77)U

Left to the reader as an exercise.

Actually, considering only semi-linear forms on the Schwartz space S, which contains functions with noncompact supports, is equivalent to a certain control of the "growth" at infinity of temperate distributions. To have a good theory of Fourier transformation, we need such a control, since a very wild growth of u at infinity 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 Fourier transformation to keep only the operations of differentiation and multiplication by smooth functions, one can consider much wider classes of distributions, and even distributions defined only locally. Indeed, if ! is any open set in R", one can define a "distribution in f2" as

follows: 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

Fourier Transformation and Sobolev Spaces

16

two constants CK and NK such that

for cp E Co (S2) and supp, C K.

I(u,cp)J < CKIcpINK

(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 has ul,, = vow,. These considerations give meaning to the notion of local behavior of a distribution; the following result shows that the local behavior of a distribution determines it completely. PROPOSITION 1.12

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

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 of PROOF

Lemma 1.5, one can write

(u,VEWJ) = J

J

_

(v, w2 ') = (v,'P

since cpjW has its support in S2, where u = v.

VJ) _ (v, 4%)

I

This property gives clear meaning to the notions of support and singular support 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 sing supp 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 if and only if (u, cp) = 0 for every V E Co' (11) with supp cp fl F = 0.

17

Sobokv spaces

Of course, these classes D'(1) of distributions contain distributions that are not "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 we mean that distributions with compact supports can be extended to the whole of IR" as temperate distributions. (Actually, any distribution u can be written as a locally 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) for all V E Co (fl) since (u, (1- ')gyp) = 0, and thus one can extend the semi-linear form 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 0, show that u E B. More generally, show that if F(z, Z) = F,,3 (z" /a!) (Z1113!) is convergent 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. then the 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 real valued if and only if (u, cp) E R for all real-valued V E S. Show that iz is real valued 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 that supp 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, depends holomorphically on c). first study the case c E R. then prove that

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) by the 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 that G(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 the limit for Re c - 0* in the formula (h, rp) = (h, gyp), use the results of Exercise 1.4 to find the expression of h also when Re c = 0 (but c # 0).

25

Exercises

(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 transform is given by the formula

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

s E R for which H E H. 1.7 The Paley-Wiener theorem for smooth functions. In questions (b) and (c), the number A > 0 is fixed. (a) Show by direct calculation that the Fourier transform of e-1=12/2 can be extended to C" as an entire function, but that it does not satisfy any estimate of the types given in Theorem 1.13. (b) Let u E C'° such that supp u C BA. Set U(() = f dx and show 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) _

A. Show that u(() = U(() and u E H°°. Show that for any e > 0, (2r)-" f

u(x) = (21r)-" f e'(=

i(x/e)) d{, then prove that u(x) = 0

for Ixi > A and finally give the conclusion. 1.8 The Paley-Wiener-Schwartz theorem for distributions. In this exercise, use the results of Exercise 1.7. In particular, if tG E Co , denote by r1'(() the entire extension 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 transform of a function r' , E S (resp. of a distribution u, E S'), that lim,.o

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,pandyE B,}, 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 of u. Let 0 be a C°C function of one variable t satisfying fi(t) = 0 for t > I

and 0(t) = I for t < 1/2, and set *((x) = 0(I(I(IxI A))e'(=. 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 ; E H' for all cp E Coo), Hi° for all j < N and some s > n/2 (i.e., and set Fu (x) = F(x, u(x)). Choose a unit test function cp, and for any locally 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 compact set 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 the generalized 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 implies

F°EHH.

Notes on Chapter 1

27

Notes on Chapter 1 During the eighteenth century, trigonometric series were introduced in the problems of interpolation (Euler), astronomy (Clairaut), and sound (Lagrange). By the end of the century, they played a central role in the famous controversy over

the vibrating string problem, which would lead eventually to the revision of the bases of analysis initiated by Cauchy [28]. The integral transformation also is introduced in Fourier's me moire [35], considered a fundamental contribution to the theory of trigonometric series despite its lack of rigor. (Actually, the same results were obtained concurrently by Cauchy and Poisson.) The Fourier transformation was then extended, thanks especially to the Lebesgue integration theory, but it is the introduction of distributions by Schwartz [61] that simplified and unified the theory. The best account on the origins of distribution theory is to be found in the introduction of Schwartz [61]: this theory has roots in the symbolic calculus of engineers initiated by Heaviside [39] and continued by the physicist Dirac [29], in the turbulent solutions of Leray [51], in the derivatives of Sobolev [65], in the finite parts of Hadamard [38], in the Fourier transformation as extended by Bochner [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 also Hormander [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 of these (and related) spaces in Adams [14].

2 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 operator a(x, D) a°(x)D°, these formulas thus lead to the following expression:

a(x, D)V(x) =

(27r)_n

J e'(x,Oa(x,

for cp E S

where the "symbol" a(x, ) of the operator a(x, D) is simply the polynomial et a(x, S) = L{a{<m a.(x)S°.

On the other hand, we can remark that the operators a'(D) introduced in Section 1.3 can actually be defined by a similar formula where the symbol a(x,e) is replaced with the symbol (1 + ICf2)'/2, which is no longer a polynomial (see Proposition 1.20). Moreover, the operator A-2(D) is a twosided inverse of A2(D) = 1+Ej Dj' = 1-A where A = Fj OJ2 is the euclidean Laplacian operator. This simple remark shows that the operator A-2(D) can be used to study the equation (1 - O)u = f. Indeed, for any f E S', this equation

has 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 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 algebraic calculations on their symbols. If we consider sufficiently large classes of symbols (which will contain functions that are not polynomials, corresponding to operators 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 variable coefficients 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 is not an exact inverse of the we would get by using the symbol operator 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 and hypoellipticity of these operators. To construct a good theory of pseudodifferential operators, one must restrict the 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 further developments of the theory has been to extend the fundamental properties of these basic pseudodifferential operators to larger classes of symbols, which can 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 Coifman 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, we close Chapter 2 by defining two fundamental operations on symbols, the use of which will be essential in the next chapter. The only motivations for the results presented in this chapter lie in the description given in Chapter 3 of the essential features of the theory. Chapter 3 will thus provide the definition of the operators and the proof of their continuity in Sobolev spaces, a description of the symbolic calculus (i.e., the correspondence between operations or estimates on symbols and operations or estimates on operators), and a sketch of the invariance property under a change of variables (which allows us to define the corresponding operators on a manifold). This set of results can be considered as the most basic properties of pseudodifferential operators, and we will see in Chapter 4 that it is already sufficient to be conveniently used in the solution of difficult problems of partial differential equations.

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

30

Pseudodiferential Symbols

A1131-`&Oa

are bounded on R" x Rn for all multiindices a, >3 E Z. If one prefers to write estimates, this means that there exist constants Cap such that C

(1 +

for

(X, C) E Rn x R",

a,0 E Z.

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 a lower f. Thus, besides an exact calculus (without rests), we will also develop an approximate calculus (modulo terms of lower order than principal terms). When looking for progressively more precise approximations, this point of view will lead 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 to study local properties, we can reduce the problem to the previous case where the symbol is in a class S' merely by multiplying the coefficients by a cut-off function 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 satisfy

A' E S'". Our third example will be used in Section 4.1 to study the local solvability 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 is also C°° at = 0, it is a polynomial), then there is a symbol b E S"' (uniquely

determined modulo S-OC)t such that b(x, t;) = a(x, ) for ICI > 1. Indeed, to transform a into an actual symbol satisfying the definition, it is sufficient to take

b(x, i;) = (1 near

where V E Co (R") with supp V C B, and V = I

= 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 usually use the same letter a to denote the modified symbol b = (I - yo)a, since this will 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, to study 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 remarked that S° is closed under multiplication (it is an algebra); one can even prove the following 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°. 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) PROOF

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 an exercise. r

In setting up an asymptotic calculus as announced above, we will use the following lemma as a substitute for the summation of a series. LEMMA 2.2

Let aj E S'"-' for j E Z+; then there exists a symbol a E S' (unique modulo S-'°) such that for any k E Z+

a - E a, E

Sm-k.

I,v, we have AI$I-(m-k)000$

r I'\l,9l-,n+kO

b

31ibj

j> N

j>N
0,

J

e`Q(=)a(x)V(ex) dx = lim

J

e'q(x)a(x)'(ex)0(2-3x) dx

by dominated convergence, we will just set

li(e) = f e+q(z)a(x)(I

-

A(Ex))V,(2-3x)dx

and show that lim3.o,, Ii exists (resp. that limi_m IJ(e) = 0(e)). To get these results, we use the change of variables y = 2-3x:

li - ii_, =

f

e'Q(x)a(x)(,(2-ix)

- V,(2'-Jx)) dx

= r ei2''9(y)a(21 y)(,0(y) - 0(2y))23' dy

(resp. li(e)-I,_,(e) =1 et22jq(y)a(2iy)(I-'P(e23y))(il'(y)-rl'(2y))2'"dy

.

34

Pseudodiferential Symbols

The function X(y) = i(y) - '(2y) satisfies X E Co and supp X C {y; 1/2 < IyI < 2}. Moreover, for y E supp x, I(&a)(23y)I S

IIIaiIIIQi(1+22jlY12)m/2
0 such that Ia(x, C)I > eA"'(l;) for ICI >

/e.

Moreover, when these conditions are fulfilled, a is said to be elliptic, and there

exists an ae E S-' such that b solves (i) PROOF

a

b solves (ii)

a

b - a0 E S-°°.

If a E S"' and b E S-'n, then a#b = b#a = ab modulo S-1 in

view of the asymptotic expansion given in Theorem 2.7. Thus each of the statements (i) and (ii) implies (iii); then (iii) implies the existence of an e > 0 such that 11 < 1/2 for ICI > 1/e, so that we have for such : 1/2, then since bo E S-'n Ambo bounded, and this is (iv). Conversely, if (iv) is satisfied, then c = A-'"a E So and satisfies Ic(x, ) I > e for Il;') > 1/e. If F(z) is a C°° function defined on C and equal to 1/z for

Izi > e, F(c) E So thanks to Lemma 2.1, and then bo = A-mF(c) E S-' satisfies abo(x,l;) = 1 for ICI > 1/e, which implies (iii). Now, if (iii) is satisfied, then a#bo = I - rl and bo#a = 1 - sl with rl and sl E S-1. Let

us set rj = rl#rj_l E S-j, sj = S,_l#sl E S-J, b, = bo#rj E S-"'-I, cj =sj#b0ES-'"-j, and finally bNF_12!ob, ES-' and c,,bo+>,2,lcj E S-' by using the construction given in Lemma 2.2. One then has for any fixed k E Z the following equalities modulo S-k:

a#b=a#l:bj=(1-rl)#(1+ > rj I =1-rk=1 j 1 nor on b E C°°. (b)

Let k and i E lR be such that k + i > 1, and assume that p is also homogeneous of degree k. Let a be any amplitude taken from the space

AL = {a E C°`; At* -m(x)O"a(x) is bounded on 1R" for all a E Z. }. Show that for any t' E S such that t;,(0) = 1. the limit

J

lim I e"0(=)a(x)y)(ex) dx o

exists, is independent of io (as long as t!)(0) = 1) and is equal to

J e"°t )a(x) dx when a E L) (this limit will also be denoted by f e"')=)a(x) dx when (c)

Give an estimate of this integral, as in Theorem 2.3. If 96 0, show that the function W4 (x) (x, ) satisfies the assumptions of question (b).

If a E A' for some m and i > 0, show that the function f e" °f (=)a(x) dx satisfies A E P. Show that if t' E S satisfies ib = 0 near 2.5

= 0. then (A,ty)

conclusion? Let a E S'" and b E St. Rewrite the asymptotic expansions of Theorem 2.7 to get

simple expressions of a' and a#b modulo S` and Sm+r-2, respectively. Then, relate the symbol a#b - b#a to the Poisson bracket of a and b, which is defined as (a, b} = (efa, 8=b) - (8=a, O (b). 2.6

Let a and b be two elliptic symbols. Prove that a' and a#b are elliptic and express

their inverses (a')' and (a#b)' in terms of the inverses a' and V.

2.7

Conversely, assume that a and b E S°° are such that a#b is elliptic. Show that a and b are elliptic. What conclusion could you give when you assume that a#b#c is elliptic? Let a E S' satisfying eX"'({) for all I/e and some e > 0,

and let k E Z.+ \ {0}. Show how to construct a symbol b E Sm/k such that

a = b#b#...#b (k terms) modulo S. 2.8

This problem is made up of two exercises: in questions (a) and (b), one proves that if a symbol a of unknown order satisfies a single estimate Ial < Cat, it is automatically in St+ = n>IS,; in questions (c) and (d), one uses this property in the study of nilpotent and idempotent symbols.

45

Exercises

(a)

Let k and q be two positive integers. Using Holder inequality, show that IIaIIk I and loO a=,,,Io < C°Qµp-". such that Ia:,,Io 5 Cooµt Use question (a) to show that there exists a sequence Ck independent of

x E R" and µ > I such that IIa,,,,Ilk 5 Ck for all k E Z+. As in Exercise 2.2, conclude that a E S'". (c) A symbol a E S°° is said to be nilpotent if there exists a k E Z+ such that a#a# ... #a = 0 (k terms). Show that if this relation holds for an a E S', then a E S"'-('/2k), and conclude that nilpotent symbols belong to S-°`. (d) A symbol a E S°` is said to be idempotent if a#a = a. Show that a is idempotent if and only if I - a is idempotent. Show that if a is idempotent, then a E S' for some m > 0 implies that a E S2m/3 U S"'-(1/3), and a E Sm for some m < 0 implies that a E S2m. Show that an idempotent symbol a satisfies a E Sol = fl,">OSm and that a2 - a = b for some b E S-213. Then, show that there exists an e > 0 such 1/4,Re(1 +4b(x,e)) > 0 (so that that for ICI > 1/e one has you can take the usual definition of (1 + 4b)' /2), 11 - (1 + 4b(x, ) )' /2I < Finally, conclude that 4Ib(x,t)i and I1+(1+4b(x,C))'/2I >

a E S-'13 or 1- a E S-'/3 (one assumes n> 1). 2.9

Prove that an idempotent symbol a satisfies a E S-°° or 1- a E S-O°. This long exercise will be continued in Exercise 3.6. (a)

Quasi-elliptic operators. For any p E Z+ such that µ, > 0 for all j < n, one sets Ia : µI = (a,/µ1)+...+(a"lµ") and Ii;(N)I = (Ej 2"' )'l2. The differential operator a(x, D) is said to be quasi-elliptic at xo if there exists a p as above such that a(x, D) = >I°:,AI 1/e for some m E R and c > 0, and all a, 8 E Z. (cf. question (a)). Show that there exists a E Co (R") such that the function bo(x, = co(x,l;) _ (1 -,(l;))/a(x,satisfies bo = co E SP0 and .1P10+61(0,0 a) (&'Obo) bounded for all a, )3, y, b E Z. Show that the symbols b, and c, defined by induction as

(a)(Dzbk)

bj = -bo Inl+k=j k<j and

(Mt ck)(Dx a)

c' = -co

1a1+k= <j

satisfy b, and c, E Sp o P2 and and

AP(J+Ip+61)(ooc,)(8

a)

b E Z+ . bounded for all a, Show how to construct symbols b and c E S" no such that a#b - I and c#a - 1 are in S_0°1

3 Pseudodifferential Operators

3.1

Action in S and S'

For a E SO° and V E S, the integral defining a(x, D)V(x) in the introduction to Chapters 2 and 3 (see also the statement of Theorem 3.1 below) is absolutely convergent, and we are going to show that it even defines a function in S. THEOREM 3.1

If a E SO` and cp E S, the formula a(x, D),p(x) = (27r)

n

r el(=.f)a(x,

)(P(C) d

defines a function a(x, D)So E S. and there exist constants N E Z+ and Ck for k E Z+ depending on a such that Ia(x, D)wlk < CkIwik+N (continuity property). PROOF

Since St C S'n for £ < m, one can assume that a E S2,n for some

m E Z+. Then wp E S implies cp E S and one can write l a(x, D)w(x)I

(fei'

We thus have to prove IS = I' and 100 = it.

b(z,

4) dz) do) f(x) dx.

Action in S and S'

49

Here we will provide the proof only for * since it is a good exercise for the student to write the details for #. First, we write that Io is the limit for e 0 of the integral IE _ (227r)-2n

t J where x E S can be chosen so that x = I in B1. Then 1' - I. = IE + IE + IE with

IE =

(21r)-n

fe

(l

- X(e)X(Ez))a(z, D)O(z) d. dz,

x(E())0(()d4 dzd(,

1, = (27r) -2n IE = (21r) -2n

ei((=,()+(_,()-(z,())g()a(z,

J

()x(e )X(Ez)x(E()

(1 - x(ex))(x) Adz d(dx. The integral IE tends to 0 with a by dominated convergence. The integrals IE and IE also tend to 0 with e, thanks to the following result. I

LEMMA 3.3

Let a(x, y) E A'n(Rn x RP), cp a real-valued function, and x, -0, and w E S with x = 1 in B1. Then dx dy = 0.

y)w(Ex)(1 -

lim I E---o

PROOF The change of variables z = ex gives

f

ei'P(=/(,Y)a(z/E,

y)w(z)(1 - X(Ey)) (y)e-n dz dy,

then a(z/e, y) is estimated by IIIaIIIo(1 + Iz/EI2 + Iy12)m/2
1/c, and this gives 11 + IyI2 1

1 +p

m-n-p

)


EVrit everywhere for some large constant Co, and conversely if this latter estimate holds, Re a can be proved to satisfy an elliptic type estimate with a smaller f > 0. In our statement we will therefore use this more convenient form of the assumption. THEOREM 3.9 GARDING'S INEQUALITY

Let a E S2m and assume that for some Co and c > 0 one has Re E)I2m (assumption satisfied in particular when Re a(x, C) > I /E). Then for any N > 0 there exists a constant CN such that 2Re (a(x, D)V, gyp) > EIIV1122 - CNIIc I12 _N

a+CoA2m- i >

for 1 I >

for all cp E S.

Action in Sobolev spaces

57

Let us set b = \-m#a#a-m E So. Since b = A-2ma modulo S-1, the assumption on a implies that Re b + (Co + CI )A- I > for some constant C1, so that b itself satisfies the assumption in the theorem with m = 0. If we assume momentarily that the result is proved when m = 0, we can write for ,p E S PROOF

2Re (a(x, D),p, ,p) = 2Re (b(x, D)Am (D),,, A' (D),p) ?EIIA"`(D)WIIO-CNIIA'(D)wII? N=EIIwIIm-CNIIwII,,,-N (cf. Proposition 1.20). Therefore, it is sufficient to prove the theorem in the case

m = 0. Thus assume a E So with Re a + CoA-' > E. We can choose a function F E C°°(C) such that F(z) = z)1/2 for z E lll, and since 2(Rea + CoA-' - ) E So is nonnegative, it follows from Lemma 2.1 that b = (2Re a + F(2(Rea + CoA-' - E)) E So. We can write modulo S-1: b*#b = 2Rea - (3/2) = a + a* - (3/2)E, which implies 2CoA-1

a+ a*

for some cE S-1.

Then for w E S,

2Re (a(x, D),p,,p) = (a(x, D),p,,p) + (,p, a(x, D),p) = ((a + a')(x, D)w, w) = (b"#b(x, D)w, V) +

(23

'Y' 1P

+ (c(x, D)w, V)

> Ilb(x, D)wll0 + 2EIlwll0 - II c(x, D)wlII/211wI1-1/2 >_ Ellwll0 + (211w110 - C112114' 1/2)

for some constant CI/2 since c E S'. Thus the result will finally come from the estimate CI/2IIwIL1/2 !5 211wII0 + CNI1wII2-N

with CN =

(2C)2N

which can be proved as follows: when CI/2A-V) > /2, one has then

C1/2A-V) = so that CI/2A-1
0, and a constant CO such that

IIt1IIm-1+6 C Co(IIa(x, D)'Ilo + ft1IIm-1) for t' E Co (Q). Here, a(x, D) is assumed to be subelliptic at xo. Show that if w has a compact closure in fl, there exist constants (C,),ER such that for any s E R for zV E C0 00(w). IIYIla+m-1+6 !5 C,(IIa(x, D) 'II, + Use the functions 7P, = p,(D)(Xu), where 0,(D) is a Friedrichs mollifier as in part (a) and X is a cut-off function, to show that if it E H'+'4-' and

a(x, D)u E H' in some ball centered at xo, then it E

H'+--1+6 in

any

smaller ball.2

Show that if it E S' satisfies a(x, D)u E C° near xo, then u E C°° near xo (property of hypoellipticity). (c)

A subelliptic operator. For k E Z4., let a2k(x,C) = 1 + ix2 kl;2. For x1 (xik+' - y2k+1)/(2k + 1), then for t2 E R, and y E R define B(xi, y) = eB(x1,v)(2 if 52(x1 - y) < 0 K2(x1,y) _ 0'(z if {2(x l - y) ? 0,

2We say that u E H' in a ball B if there exists a v E H' such that it = v in B.

66

Pseudod(ferential Operators

and finally for 0 E S = S(W), set

(y,l;z) = J e-"={=,L(y,z)dz,

/ ?0(xi,t2) = J

and

K,O(xi,x2) = (27)-i / eix2fz,I0(xl,6)42.

J

Show that for any ?1' E S, Kii is aCOO function and -0 = Ka2k(x, D) V,. Show that there exists a constant C such that

f IKe2(xj,y)I dx1
ojq, satisfies {p, p} = 2iRe (q0p) in a neighborhood of K, which finally gives {p, p} = 2iRe (qp) for f E ' \ 0 and x close to xo by homogeneity if we set q(x, ) = I I m_ I qo(x, /ICI). Principally normal operators obviously satisfy Hormander's condition {p, p} = 0 on p = 0. The converse will be discussed at the end of this section (see Corollary 4.4 and its comments). I The main result of this section is the following. THEOREM 4.1

Let a(x, D) be an mth order principally normal operator of principal type at xo. Then there exists a neighborhood S2 of xo such that the equation a(x, D)u = f

(in S2) has a solution u E L2 (Q) for any f E H'-'.

Example Operators with real or constant coefficients in the principal part satisfy {p, p} 0, and therefore they are principally normal (take q = 0). Elliptic operators have nonvanishing principal symbols (except at = 0), so that - thanks to the

previous remark - they satisfy the assumptions of Theorem 4.1. The student will find other operators satisfying these assumptions by solving the following questions. If a is of principal type, is a' also of principal type? If a is principally normal, is a* also principally normal? Is a#b of principal type when a and b are of principal type? What about the converse? Assuming that a#b is of principal type, that a and b are principally normal, and that the order of c is smaller than the order of a#b, does a#b + c satisfy the assumptions of Theorem 4.1? 0 In the proof of Theorem 4.1, our first step will be to show that the principal type and the normality correspond to some a priori estimates, but before stating and proving this result, we give a lemma that provides estimates for functions

with small support. In the following, we denote by S26 = {x E R"; Ixi < b} the open ball of radius b > 0. LEMMA 4.2

For allb>0andmEZ

,

for all p E Co (SZ6).

II'PIIm 2E1e12m-2 for some e > 0 and all x in some S22N. Therefore, the symbol b + EA2m-2#(1 - V)) satisfies the assumption of Theorem 3.9 (GArding's inequality) if 0 E Co (126o)

and 0 < 1. Moreover, if 0 = 1 in !

and 6 < bo, (1 - V,)Ip = 0 for all

Local solvability of linear differential operators

73

W E Co (f26) then (b(x, D) + EA2m-2(D)(1 - 0))y7 = Bye. It follows that we have n

2Re (By7, p) ? EIIsvIIm-i - C'IIVIIm-2

for some constant C. On the other hand, for each operator Qj one can write

= (A(ix.iV) - ix3(Ap),Q3p) (ix3sc, A`Q, ) - (ixj (Afw), Qif')

(ixi'p, [A',Qi] 'p) + (Q(ix.,'p), For V E Co (06), this can be estimated by using Lemma 4.2: IIQ2wII
0 as in Proposition 4.3(iii). Then, the operator a*(x, D) is injective on Co (S16), and we can consider its inverse (A')-1 which is well defined on the space

E = {zL' E Co (Slb); 3cp E Co (16) with iP = a'(x, D)y'}.

For f E H1-" we define on E the semi-linear form U(ii) = (f, (A')-1 ') which satisfies

IU(0)I =1(f, p)l 0). (Using the asymptotic expansion for the operation # and Lemma 2.2, one gets that if a E and b E St, then a#b and b # a can be written as sums of a term in ScornP (F) and a term in S-OD.) a E S°° is said to satisfy a E Sa(F) if ab E S'" for all b E S°°,,,p(r) (this also implies that ab, a#b, and b#a are in S°'+t for all b E S ...P(17)). The symbol a E S"' is said to be elliptic at (xo, Co) E T*R" \0, or (xo, o) is said to be noncharacteristic for a, if there exist a b E S-'" and a conic neighborhood I' of (xo, o) such that ab - I E S;.' (t). The student will check that the proof of Theorem 2.10 can then be adapted to construct a b E S-' such that a#b- 1 and b#a -1 are in Sj;°°(F), maybe for a smaller F. Finally, the set of characteristic

said to satisfy a E S

Wave front sets of solutions of partial differential equations

77

points for a will be denoted by Char a, and from its definition it is thus a conic closed subset of T'R1 \ 0.

Example If a(x, D)

p(x,.) =

a,, (x)D' is a differential operator with principal symbol

a,,(x)t , the characteristic set of a is simply Char a = { (x, l;) E T* R' \ 0; p(x, t;) = 0}. Indeed, if p(xo, to) # 0, one can define EJa1_,,,

b(x, t;) = 1 /p(x, f) in a conic neighborhood r of (xo, CO) since p is homogenous,

and define b E S-"` anyhow out of r, and it is then clear that ab - I E Si-.' (I'). Conversely, if p(xo, to) = 0 and b E S-'°, a(xo, µl o)b(xo, 14o) = O(µ-1) and this shows that ab - 1 ¢ Sj;' (F) for any conic neighborhood r of (xo, co). 0 The wave front set of a distribution is then defined as follows.

DEFINITION 4S Let u E S'. One says that the point (xo, to) E T'R" \ 0 is not in the wave front set of u, or (xo, to) 0 WFu, if there exists a conic neighborhood r of (xo, to) such that a(x, D)u E C°° for all a E S P(r). From its definition, WFu is thus a conic closed subset of T*R \ 0. The wave front set is related to the singular support through the following result.

THEOREM 4.6 PROJECTION THEOREM

Let u E S'; then one has sing supp u = {x E lR

;

there exists a C # 0 with (x, f) E WFu}.

PROOF Let xo ¢ sing supp u, V E Co such that Vu E C°` and cp = 1 in a neighborhood f of xo, and r = St x (Rn \ 0) which is a conic neighborhood of (xo,l;) for every t 0. Then if a E S(r) one writes

a(x, D)u = a(x, D%pu) + a(x, D)((l - V)u). Since Vu E Co, one has a(x, D)(Vu) E S. Moreover, a(x, D)((1 - cp)u) _ b(x, D)u where b = a#(1 - c') E S-OC thanks to the asymptotic expansion of the operation # since the supports of a and (1 - cp) do not meet. It follows that

a(x, D)((l - V)u) E P (cf. Corollary 3.8) and a(x, D)u E C°°. Conversely assume that (xo,1;) it WFu for all t E R'a \ 0. For every such £, there exists a conic neighborhood r(t) of (xo,t;) as in Definition 4.5. The compact set { (xo,1; ); It I = 1 } is covered by these neighborhoods and one can find a finite number of them F1,..., rk and some functions cps E Co (F3) such that Vi (x,.) = 1 in KE = {(x, t;); Ix - xol + IIt;I - 1I < e} for some e > 0 (cf. Lemma 1.5). Choosing also a function -0 E Co(k') satisfying 'o = 1 near t = 0, one sets aj(x,l) = (1 -''(t;))cp?(x,t;/I1;I) E S.' P(FD). Therefore

Applications

78

F,, aj 1 for Ix - xoI < e and E R1. We then take S2={x;Ix-xoI <e}; for any WECo (Q),cp=v7P+J:,ypajand one has vp

-s}}, the Cauchy problem

(t - ib' (x, D))z'(t) = cp(t) ,0(-s) = 0 has a solution E C°(I; S), namely the restriction to I of the Vi given in the definition of E. This solution is unique according to step (ii) and it satisfies the energy estimate SUP El II (t)II-k < Ck f j

dt for any k E Z. Then,

for g E C°(I; S) and X E S, one defines a semilinear form W on E by

W(A _ (X, -005)) -

J

(9(t), 0(t)) dt

where zji is the unique solution of the previous Cauchy problem. One has

IWMI 0 and a function E Co (f26) with o = I near 0 and such that the operator c(x, D) = p(x) (I - a#b(x, D)) satisfies

IIc(x, D)iPIj-, 5 1 M-

for all 1P E Ca (S26).

Let s E IR and f E H"; explain why the formulas W'o = c(x, D) f,

ip,+i = c(x, D)>yr,

and ?G., _

TV

3>o

define functions ?, = Co (f26) (for j = oo, first prove that ?Pa E H-1, then observe that tj°° = o + c(x, D)?Pa). Finally, show that the formula

u = b(x,D)(f +tV) defines an H"-' distribution solution of a(x, D)u = f in a neighborhood of 0 (here, m denotes the order of b).

4.2 Let b : 1(P -. R be a C°° function, and let a(x, ) = t;i + ib(xi )l;2. We want to study the local solvability of a(x, D) at the origin of R. (a) Determine conditions on b equivalent to the following properties:

- a(x, D) is of principal type. - a(x, D) is principally normal. - a(x, D) satisfies Hdrmander's condition p = 0 = {p, p} = 0, where p is the principal symbol. Then describe for what functions b the operator a(x, D) is locally solvable (resp. is not locally solvable) at the origin thanks to the results given in

Applications

90

Section 4.1, but exclude the case where b would have a zero of infinite order at x, = 0. (b) Assume that b does not change sign for lx, I < E < 1, and set 9 = (-E, E) X (-E, E).

Take B(x3) = fo' tlb(t)I dt and prove that the operator c(x, D) defined by c(x, D)t,(x) = eB(x D)(e-B(nl)?i)(x) satisfies the a priori estimate II

for all

'IIo

E CO '(Q).

(Hint: Integrate by parts the scalar product Im (c(x, D)tli, (x, ± D2)tP).) Show that there exists a constant C such that IkcIIo rll a' (x, D)Vll r for some

p E CI(K) I is open in CO(K). Assume that a(x, D)u = f has a solution u E 1Y (Q) for all f E Co (K) where K C S1 are two neighborhoods of the origin (K is compact). Use Baire's theorem to prove that there exist fo r= CI (K), c > 0, and r and s E Z+ such that

Ilf - foil, < 2E =* f V flr. Finally, prove that if a(x, D) is locally solvable at the origin, there exist a compact neighborhood K of the origin, a constant C. and two integers r, s E 7L+ such that

for all cp,fECo (K). 1(p,f)l