Pseudodifferential Operators and Spectral Theory (Springer Series in Soviet Mathematics)

M. A. Shubin

Pseudodifferential Operators and Spectral Theory

M. A. Shubin

Pseudodifferential Operators and Spectral Theory

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M. A. Shubin

Pseudo differential Operators and Spectral Theory Second Edition Translated from the Russian by Stig I. Andersson

Springer

Mikhail A. Shubin

Department of Mathematics Northeastern University Boston, MA 02115, USA

e-mail: [email protected] Stig 1. Andersson

Institute of Theoretical Physics University of GSteborg 41296 Goteborg, Sweden

Title of the Russian original edition: Psevdodifferentsialnye operatory i spektralnaya teoria Publisher Nauka, Moscow 1978 The first edition was published in 1987 as part of the Springer Series in Soviet Mathematics Advisers: L. D. Faddeev (Leningrad), R. V. Gamkrelidze (Moscow) Mathematics Subject Classification (2000):35S05,35S30,35P20,47G30,58)40

Library of Congers Caulogiag-m-Publicaum Data Shubin. M. A. (Mikhail Aleksardmvich). 1944. rPaevdodiffemrsiarnye operatory i spektral'nais teoriia. English) Pseudodifteremia) operators and spectral theory / M.A. Shubin : translated fio® the Russian by Stig 1. Anderson.- 2nd ad. p. rxn. ISBN 354041195X (softcover : alk. paper)

1. Pseudodiffe emial operators. 2. Spectral theory (Mathematia) QA381 .54813 2001 515'.7242-dc21 2001020695

ISBN 3-54o-41195-X Springer-Verlag Berlin Heidelberg New York ISBN 3-540-13621-5 1st edition Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material

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from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH httpzf/www.springer.de 0 Springer-Verlag Berlin Heidelberg 1987, 2001 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Daten- and Lichtsatz-Service, Wilrzburg Cover design: design 6- production GmbH, Heidelberg 44/3142LK - 5 4 3 21 0 SPIN 10786064 Printed on acid-free paper

Preface to the Second Edition I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to

exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudo-differential operator" but I was unable to check how much the results of these two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews. On the other major topics of the book the results were as follows: Fourier Integral operator: 1022 hits (105 books), Microlocal analysis: 500 hits (82 books), Spectral asymptotic: 367 hits (56 books), Eigenvalue asymptotic: 127 hits (21 books), Pseudodifferential operator AND spectral theory: 142 hits (36 books). Similar results were obtained by searching the Zentralblatt database. And there were only 132 references (total) in the original book. So I decided to quote here additionally only three books which I can not resist quoting (in chronological order): 1. J. Bruning, V. Guillemin (eds.), Mathematics Past and Present. Fourier

Integral Operators. Selected Classical Articles by J.J. Duistermaat, V. W. Guillemin and L. Hormander., Springer-Verlag, 1994. 2. Yu. Safarov, D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Amer. Math. Soc., 1997. 3. V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics. SpringerVerlag, 1998.

Preface to the Second Edition

VI

These books fill what I felt was missing already in the first edition: treatment of more advanced spectral asymptotics by more advanced microlocal analysis (in particular, by Fourier Integral operators). By the reasons quoted above I did not add anything to the old bibliography at the end of the book, but I made the references more precise whenever this was possible. In case of books I added some references to English translations and also switched the references to the newest editions when I was aware of the existence of such editions. I made some clarifying changes to the text in some places where I felt these

changes to be warranted. I am very grateful to the readers of the book who informed me about the places which need clarifying. Unfortunately, I did not make the list of those readers and I beg forgiveness of those whom I do not mention. However, I decided to mention Pablo Ramacher who was among the most recent and most thorough readers. His comments helped me a lot. I am also very grateful to Eugenia Soboleva for her selfless work which she generously put in helping me with the proofreading of the second edition. I hope that my book still has a chance to perform its main function: to teach its readers beautiful and important mathematics. March 21, 2001

Mikhail Shubin

Preface to the Russian Edition The theory of pseudodifferential operators (abbreviated `PDO) is comparatively young; in its modem form it was created in the mid-sixties. The progress

achieved with its help, however, has been so essential that without `PDO it would indeed be difficult to picture modem analysis and mathematical physics. `PDO are of particular importance in the study of elliptic equations. Even the simplest operations on elliptic operators (e.g. taking the inverse or the square root) lead out of the class of differential operators but will, under reasonable assumptions, preserve the class of'FDO. A significant role is played by 'I'DO in the index theory for elliptic operators, where `PDO are needed to extend the class of possible deformations of an operator. Y'DO appear naturally in the reduction to the boundary for any elliptic boundary problem. In this way, `I'DO arise not as an end-in-themselves, but as a powerful and natural tool for the study of partial differential operators (first and foremost elliptic and hypoelliptic ones). In many cases, `PDO allow us not only to establish new theorems but also to have a fresh look at old ones and thereby obtain simpler and more transparent formulations of already known facts. This is, for instance, the case in the theory of Sobolev spaces. A natural generalization of `PDO are the Fourier integral operators (abbreviated FIO), the first version of which was the Maslov canonical operator. The solution operator to the Cauchy problem for a hyperbolic operator provides

an example of a FIO. In this way, FIO play the same role in the theory of hyperbolic equations as'I'DO play in the theory of elliptic equations. One of the most significant areas for applications of `PDO and FIO is the spectral theory of elliptic operators. The possibility of describing the structure of various nontrivial functions of an operator (resolvents, complex powers, exponents, approximate spectral projection) is of importance here. By means of `PDO and FIO one gets the theorem on analytic continuation of the -function of an operator and a number of essential theorems on the asymptotic behaviour of the eigenvalues. This book contains a slightly elaborated and extended version of a course on `PDO and spectral theory which I gave at the Department of Mechanics and Mathematics of Moscow State University. The aim of the course was a complete presentation of the theory of'PDO and FIO in connection with the spectral theory of elliptic and hypoelliptic operators. I have therefore sought to make the presentation accessible to students familiar with the standard Analysis course (including the elementary theory of distributions) and, at the same

VIII

Preface to the Russian Edition

time, tried to lead the reader to the level of modern journal articles. All this has required a fairly restrictive selection of the material, which was naturally influenced by my personal interests. The most essential material of an instructional educational nature is in Chapter I and Appendix 1, which also uses theorems from §17 and §18 of Chapter III (note that § 17 is not based at all on any foregoing material and § 18 is based only on Chapter I). We unite all of this conventionally as the first theme, which constitutes a self-contained introduction to the theory of `I'DO and wave fronts of distributions. In my opinion, this theme is useful to all mathematicians specializing in functional analysis and partial differential equations. Let me emphasize once more that the first theme can be studied independently of the rest. Chapters II and III constitute the second and third themes, respectively. From Chapter II the reader will learn about the theory of complex powers and the -function of an elliptic operator. Apparently the theorem on the poles of the i; -function is one of the most remarkable applications of `PDO. The derivation of a rough form of the asymptotic behaviour of the eigenvalues is also shown in this chapter. In Chapter III there is a more precise form of the theorem on the asymptotic behaviour of the eigenvalues. This theorem makes use of a number of essential facts from the theory of FIO, also presented here. Let us note that it is in exactly this way that further essential progress in spectral theory was achieved, using, however, a more complete theory of FIO which falls outside the framework of this book (see the section "Short Guide to the Literature"). Finally, Chapter IV together with Appendices 2 and 3 constitute the final (fourth) theme. (Appendix 3 contains auxiliary material from functional anal-

ysis which is used in Chapter IV and is singled out in an appendix only for convenience. Advanced readers or those familiar with the material need not look at Appendix 3 or may use it only for reference, whereas it is advisable for a beginner to read it through.) Here we present the theory of `I'DO in IR° which arises in connection with some mathematical questions in quantum mechanics. It is necessary to say a few words about the exercises and problems in this book. The exercises, inserted into the text, are closely connected with it and are an integral part of the text. As a rule the results in these exercises are used in what follows. All these results are readily verified and are not proved in the text only because it is easier to understand them by yourself than to simply read them through. The problems are usually more difficult than the exercises and are not used in the text although they develop the basic material in useful directions. The problems can be used to check your understanding of what you have read and solving them is useful for a better assimilation of concepts and methods. It is, however, hardly worthwhile solving all the problems one after another, since this might strongly slow down the reading of the book. At a first reading the reader should probably solve those problems which seem of most

IX

Preface to the Russian Edition

interest to him. In the problems, as well as in the basic text, apart from the already presented material, we do not use any information falling outside the framework of an ordinary university course. I hope that this book will be useful for beginners as well as for the more experienced mathematicians who wish to familiarize themselves quickly with 'PDO and their important applications and also to all who use or take an interest in spectral theory. In conclusion, I would like to thank V.I. Bezyaev, T.E. Bogorodskaya, T.I. Girya, A.I. Gusev, V.Yu. Kiselev, S.M. Kozlov, M.D. Missarov and A.G. Sergeev who helped to record and perfect the lectures; V.N. Tulovskij who communicated to me his proof of the theorem on propagation of singularities and allowed me to include it in this book; V.L. Roitburd who on my request has written Appendix 2; V.Ya. Ivrii and V.P. Palamodov who have read the manuscript through and made a number of useful comments and also all those who have in any way helped me in the work. M.A. Shubin

Interdependence of the parts of the book Chapter I

I §18

i

Chapter 11

Chapter IV

Chapter III

Appendix 2

I §17

H Appendix I

I

I

Appendix 3

Preface to the English Edition There are so many books on pseudodifferential operators (which was not the case when the Russian edition of this book appeared) that one naturally questions the need for one more. I hope, nevertheless, that this book can be useful because of its selfcontained approach aimed directly at the spectral theory applications. In addition it contains some ideas which have not been described in any other monograph in English. (I should mention, for instance, the approximate spectral projection method which is a universal method of investigating the asymptotic behaviour of the spectrum - see Chapter IV and also a review paper of Levendorskii in Acta Applicandae Mathematicae'.) Certainly many new developments have taken place since the Russian edition of the book appeared. The most important ones can be found in the monographs listed below. September 3, 1985

M. A. Shubin

References

Egorov, Yu. V.: Linear partial differential equations of principal type. Mir Publishers, Moscow 1984 (in Russian). Also: Consultants Bureau, New York, 1986. Hi rmander, L.: The analysis of linear partial differential operators, v. I-IV. Springer-Verlag, Berlin e.a. 1983-1985 Heifer, B.: Theorie spectrale pour des operateurs globalement elliptiques. Societe Math. de France, Asterisque 112, 1984 Ivrii, V.: Precise spectral asymptotics for elliptic operators. Lecture Notes in Math. 1100; Springer-Verlag, Berlin e.a. 1984 Kumano-go,H.: Pseudo-differential operators. MIT Press, Cambridge Mass. 1981 Reed, M., Simon, B.: Methods of modern mathematical physics, v. I-IV. Academic Press, New York e. a. 1972-1979 Rempel, S., Schulze, B.-W.: Index theory of elliptic boundary problems. Akademie-Verlag, Berlin, 1982 Taylor, M.: Pseudodifferential operators. Princeton Univ. Press, Princeton, N.J. 1981 Treves, F.: Introduction to pseudodifferential and Fourier integral operators, v. I, II. Plenum Press, New York e. a. 1980

See also: Levendorskii, S.: Asymptotic distribution of eigenvalues of differential operators. Kluwer Academic Publishers, 1990.

Table of Contents

..... .......... .. ...... ..... .. .. ........ ......... . ...................... ............. . .......... ..... .... .........

Chapter I. Foundations of `PDO Theory §

1.

§ 2. § 3. § 4.

§ 5. § 6. § 7.

§ 8.

I

Oscillatory Integrals . . . . . . . . . . . . . . . . . . . . . . . . Fourier Integral Operators (Preliminaries) 10 The Algebra of Pseudodifferential Operators and Their Symbols . 16 Change of Variables and Pseudodifferential Operators on Manifolds 31 Hypoellipticity and Ellipticity 38 Theorems on Boundedness and Compactness of Pseudodifferential Operators 46 The Sobolev Spaces 52 The Fredholm Property, Index and Spectrum 65

Chapter II. Complex Powers of Elliptic Operators

....... .... ..

§ 9. Pseudodifferential Operators with Parameter. The Resolvent . § 10. Definition and Basic Properties of the Complex Powers of an Elliptic Operator §11. The Structure of the Complex Powers of an Elliptic Operator § 12. Analytic Continuation of the Kernels of Complex Powers . . . . § 13. The C-Function of an Elliptic Operator and Formal Asymptotic Behaviour of the Spectrum § 14. The Tauberian Theorem of Ikehara § 15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem) .

..................... .....

1

77 77 87 94 102

...... . .... ... .. ..... 112 ..

.. ....... ..... .......... .

Chapter III. Asymptotic Behaviour of the Spectral Function

.. .. ..

120 128 133

§ 16. Formulation of the Hormander Theorem and Comments . . . . . 133 § 17. Non-linear First Order Equations 134 § 18. The Action of a Pseudodifferential Operator on an Exponent . . . 141 § 19. Phase Functions Defining the Class of Pseudodifferential Operators 147

...... .. .... ... ..

§20. The Operator exp(-itA)

........ ....... ......... ..

§21. Precise Formulation and Proof of the Hormander Theorem §22. The Laplace Operator on the Sphere

150 156

........... ..... 164

XII

Table of Contents

Chapter IV. Pseudodifferential Operators in IR"

............ 175

§23. An Algebra of Pseudodifferential Operators in IR. §24. The Anti-Wick Symbol. Theorems on Boundedness

and Compactness

........ 175

......................... .

186

§25. Hypoellipticity and Parametrix. Sobolev Spaces. The Fredholm §26. §27. §28. §29. §30.

......... ................... ........193 197 .................... 202 ..................... 215

Property Essential Self-Adjointness. Discreteness of the Spectrum Trace and Trace Class Norm The Approximate Spectral Projection . . . . . . . . . . Operators with Parameter Asymptotic Behaviour of the Eigenvalues . . . . . . . .

Appendix 1. Wave Fronts and Propagation of Singularities

.

.

.

.

.

206

.

.

.

.

.

223

....... 229

........ ........ ..................... 269 ............................... 275 ............................. 285 ..... ..........................

Appendix 2. Quasiclassical Asymptotics of Eigenvalues

240

Appendix 3. Hilbert-Schmidt and Trace Class Operators

257

A Short Guide to the Literature Bibliography

Index of Notation Subject Index

287

Chapter I Foundations of `I 'DO Theory §1. Oscillatory Integrals 1.1 The Fourier transformation. The simplest example of an oscillatory integral is provided by the Fourier transform of a function (or distribution) of tempered growth. Let S (IR") be the Schwartz space of functions u (x) c- C' (R') all derivatives of which decrease faster than any power of I x I as I x

oo, i. e. for

arbitrary a, fi sup I X, (DO U) (x) I < + oo .

(1.1)

XER'

As usual x here stands for (x, , ... , x"); a and fi are multiindices, so for example a= (a...... a") and a , is a non-negative integer; xe = x, - . . . a = (a, , ... , a") a a 181

where a, =

a xj

; as = a; ... a.0 =

axwith If I

M"...

+ $,. The

"

left hand sides of (1.1) define a collection of semi-norms in S(IR") which turn S(IR") into a Frechet space. The Fourier transform of a function u (x) e S (IR") is given by the formula

f e-' u(x)dx,

(1.2)

i = 1/-1 and dx=dx,...dx" is

where HEIR", x-

Lebesgue measure on IR". The integral in (1.2) is taken over the whole of 1K",

which will always be the case unless a domain of integration is explicitly indicated. It is well known that the operator F defines a linear topological isomorphism

F:S(IR")-+S(IR") and that the inverse operator (the inverse Fourier transformation) is given by the inversion formula (F-'4) (x) = u (x) = ! e,x ' e u

where d = (2n)-"dd, ... dd..

d,

(1.3)

Chapter 1. Foundations of I'DO Theory

2

Now we are going to show how to extend the Fourier transformation (1.2) to

continuous functions u(x) satisfying the following condition: there exist constants C > 0 and N > 0 such that I U (x) 15 C<x>',

(1.4)

where (x) stands for (1 + IxI2)112 and Ix12 = x2 + ... + x.. We will define u(l;) E S' (IR"), the dual space of S(IR"), i.e. the space of all continuous linear functionals on S(IR"). So we want to regularize the integral

= If e` "

(1.5)

eS(lR"), an integral which we will also regard as the value of the functional u at the element If u(x)eS(1R") it is obvious that with

since in this case (1.5) converges absolutely. We give two equivalent means of regularizing (1.5) both differing from the well-known method, based on the Parseval identity, and both extendable to considerably more general situations.

First method. Put D; =

a

i1 d xj

, D = (DJ , ... , D") and _ (1 + D; + .. .

+ D,,) "2 (usually we will make use of k with k a non negative even number so

that k becomes a differential operator). The vector D will also be used to indicate differentiation in the variable. To avoid confusion we then denote by D. the just described vector D and by D, the same vector but acting on the variable. We have e-"x.4 = <x>-kke-lx-4.

(1.6)

To begin with, suppose that u (x) e S (IR"). Then inserting this expression for e-4'4 in (1.5) and integrating by parts, we obtain

= Ile

,x*cu(x)<x>-kkV/

(S)dxdd.

(1.7)

This integral is now defined not just for u (x) e S (R") or for absolutely integrable u (x). Indeed, if u (x) satisfies (1.4) and k > N + n, then (1.7) converges absolutely

and we can consider it as the required regularization of (1.5). Exercise 1.1. Verify that formula (1.7) defines a continuous linear functional u e S' (]R") fork > N + n.

Second method. Suppose that qp (x) a Co (IR") (the space of compactly supported infinitely differentiable functions on IR") and that V (0) = 1. Put

§1. Oscillatory Integrals

3

E>0.

1

(1.8)

This integral converges absolutely. It turns out that there is a limit I= lim I, »o

independent of the choice of W (x). Indeed, carrying out in (1.8) the same integration by parts as before, we get

IL= J$e-ix r (ex) u(x)<x>-rt and if k > N + n, by the Lebesgue dominated convergence theorem, the limit as E-0 exist, and equals as defined by formula (1.7).

Exercise 1.2. Verify that for different values of k formula (1.7) leads to the same functional u. 1.2 Definition of the oscillatory integral and its regularization. Now consider an integral more general than (1.5) 1° (au) = J J

e'°(x.6)

a (x, 0) u (x) dx dO.

(1.9)

Here 0 E IR", x E X, where X is an open set in IR" and u (x) a Co (X), i.e. u (x) a Co" (X) and there is a compact set K c X such that u 1 x\ K = 0. To describe a (x, 0) and 4P (x, 0) we introduce a number of definitions.

Definition 1.1. Let m, a and b be real numbers; 0< b 5 1, 0:5 Q 5 1. The class S,',6 (X x IR") consists of functions a (x, 0) a Cm (X x IRN) such that for any multi-indices a, fi and any compact set K c X a constant Ca, a. K exists for which (1.10)

10,'00a(x,0) 1:_5 C..B.K<0)mlal +6161

where xeKand0E1R'. Instead of Si"' o (X x W) we simply write S' (X x IRN). Furthermore, instead of SQ a (X x 1R") we will sometimes simply write SQ e . We also put S-

n s-. M

Definition 1.2. We call 0 (x, 0) a phase Junction if 0 (x, 0) a C' (X x (R^'\ 0)),

O (x, 0) is real valued and positively homogeneous of degree I in 0 (i.e. (x, t 0) = t 0 (x, 0) for any x e X, 0 e IR" and t > 0) and P (x, 0) does not have critical points for 0 4' 0, i. e. PX Q(x, 0) + 0 for x e X and 0 e lR' \ 0 (P a denotes the gradient of 4o (x, 0) with respect to x and 0). Definition 1.3. An integral (1.9) in which a (x, 0) ES'". (X x IR") and 0 (x, 0) is a phase function is called an oscillatory integral.

Exercise 1.3. Verify

that if a (x, 0) a S,'",, (X x IR"`) then

00 "00 a (x, 0)

e SQ a Q Ial + 6181(X x lR"). Verify also that for b (x, 0) a SQ . (X x IR") we have

a(x,0) b(x,0)eSQ

(XxIR").

4

Chapter 1. Foundations of `I'DO Theory

Our immediate goal is the regularization of the oscillatory integral (1.9) which is not, generally speaking, absolutely convergent. The following lemma allows us to write down an equality of the type (1.6) in the general case. Lemma 1.1. There exists on X x 1R', an operator N a a L=E a;(x,0)-+ Y 64(x,0)-+c(x,0), a0j axk

(1.11)

k=,

j=1

such that a; (x, 0) a S° (X x lRN), bk (x, 0) e S -' (X x 1RN), c (x, 0) eS -' (X X 1RN)

and defining the formal adjoint 'L by the formula N

'Lu (x, 0) = -;It

a

a

°

(a; u) - kY-

ao;

sYk

(bku) + cu,

(1.12)

we have

'Le'o = e'o

(1.13)

Exercise 1.4. The operator 'L may also be written in the form (1.11) with other a; , bk and c, still belonging to the same classes as stated in the definition of L.

Exercise 1.5. Show that if M = 'L then L ='M. Proof of Lemma 1.1. We have a

;m

therefore N

;_,

_

am

eio

-'a0;e

a0;e

-i-a

1012

a aoi

ao;

+

100 2

_

Y- 1012

=,

e'm,

axk

k

al --)e a4

^

- i axk ax4/J k=, Y-

+ i 00

ae;I

k=1

I

2

,m

eim

eio w

axk

where w (x, 0) e C" (X x (1R' \ 0)) is positively homogeneous of degree - 2 in 0. Therefore

-iw

JN 10 12

;_,

a"

a

a0; a0;

+

ad;

a

k=, axk axk}

e

i1o = eio

and it remains only to get rid of the singularity at 0 = 0. Let X (0) a Col(1RN), X (O) = 1 for 101 < 'a and X (O) = 0 for 101 > ''z . Let us put

a -=i(1_X)Wa a

N

;_,

ao;

e;

k=,

ax4 ax4

+X

§ 1.

Oscillatory Integrals

5

It is obvious that Mei° = e'° and one can easily verify that all the coefficients of the operator M have the required properties. The same is also true for L = `M (cf. Exercise 1.4). It only remains to note that `L = M in view of Exercise 1.5.

We will achieve the regularization of (1.9) using two different methods. First method. To begin with, let m < - N so that (1.9) converges absolutely. Utilizing (1.13) write in this integral (`L)' ei° instead of e'° and integrate by parts k times. In this way we get I°(au) = f Je1°1",° L'`(a(x,0)u(x))dxd0.

(1.14)

Exercise 1.6. Verify that this operation is well defined.

Putting s = min (p, I - 6), from Exercise 1.3 we deduce Lk(au) a SQ (Xx lR"). If p > 0 and 6 < I (so that s > 0), which will always be assumed in the sequel, then formula (1.14) already allows us to define the integral I° (au) for an

arbitrary m, if we select k so that m - ks < - N. This, of course, makes the integral (1.14) absolutely convergent.

Exercise 1.7. Demonstrate that for fixed a and 0, 1° (au) considered as a functional of u e Co (X), defines a distribution on X, i.e. is an element of 1' (X) (the dual of Co (X)).

Second method. Picking X (0) e Co (lR") such that X (0) = 1 in a neighbourhood of 0 e IR", we put 10,,(au) =

JJx(E0)e'°(=.e)a(x,0) u(x)dxd0,

c > 0.

(1.15)

Integrating by parts as in the first method, we get (1.16)

Note that l ae X (Ee)1 < Cr (9> - irl

(1.17)

where Cr does not depend on e for 0 < e < 1. Exercise 1.8. Verify the estimate (1.17). We now see that using the dominated convergence theorem we may pass to the limit as E - 0 in (1.16). In this way li m I°, E (au) exists and is equal to 4 ° (au) in the sense of formula (1.14). In particular, it follows that the integral (1.14) does not depend on k (provided k is sufficiently large) and also that the limit of (1.15) as t - 0 is independent of the choice of cut-off function X. In what follows we will more or less freely deal with oscillatory integrals, assuming that they could be regularized by one of the above methods.

Chapter I. Foundations of Y'DO Theory

6

Exercise 1.9. Prove that the oscillatory integral Im (au), for fixed 0 and u, represents a continuous linear functional on the Frechet space

SQ a(X x IR"). Here the topology is given by the semi-norms equal to the infima of the constants Cap , in (1.10). Verify that the closure of S- °°(X x IR") in S.',Q a(Xx lR") contains SQ a(X x R") for arbitrary m' < m. In this way we can view regularization as an extension by continuity of a linear functional. 1.3 Smoothness of distributions defined by oscillatory Integrals. Let us intro-

duce the following important notation Cm= {(x,0): xeX, OelR"\O, 45e'(x,9)=0}

(1.18) a4p

(Here 0; denotes the gradient of 0 w. r. t. 0, i. e. the vector 001

60

' .. a8 " )) The set CO is a conic subset of X x (IR"\0), i.e. together with the point (xo, 00) it also contains all points of the form (x0, t0o) with t > 0. Denoting the natural projection by a: X x (R\ 0) - X we set -

So=nCo,

Re=X\Sm.

(1.19)

Consider the distribution A e-9'(X) defined via the oscillatory integral Im (au) by

A, u> = 1m (au).

Theorem 1.1. sing supp A C Sm or, equivalently, A E C°°(R,).

Proof. The assertion of the theorem is equivalent to the existence of A (x) e C°° (Rm) such that if u e Co (R.) then Im (au) = f A (x) u (x) dx,

(1.20)

A(x) = f eimcx.e1a(x,0)d0.

(1.21)

Put

The last integral is itself an oscillatory integral for x e Rm, depending on the parameter x. Differentiating w. r. t this parameter we obtain integrals of the same kind. Essentially here we speak about differentiating w. r. t the parameter of a

convergent integral obtained from (1.21) by the above transformation. Therefore A (x) e C'(R,) and (1.20) is straightforward. O Theorem 1.2. If a E S,'.6 then A e C' (X).

x IR") and a = 0 in a conical neighbourhood of CO,

Proof: Similar to the proof of Theorem 1.1, in that, since 0., (x, 0) * 0 on the

support of a(x, 0), it amounts to a study of the oscillatory integral (1.21). 0

§ I. Oscillatory Integrals

7

Definition 1..4. The phase function 40 (x, 0) is called non-degenerate if the

differentials d (), j = 1, ..., N are all linearly independent on Co or, equivalently rank II ee

Il fee 0ex 11 =

II = N (in detail;

a2.

a2.

ago

a20

a0, a0,

00, a0"

001 ax,

a0, ax"

a24,

a2,

a0"ax,

a0"ax"

). a2 (p a2, (0,a0, "' 00"a0"

Proposition I.I. If 0 is a non-degenerate phase function, then CO is an n-dimensional submanifold in X x (IR"\0).

Proof. A trivial consequence of the implicit function theorem.

The following theorem makes theorem 1.2 more precise in the case of a non-degenerate phase function.

Theorem 1.3. Let 0 be non-degenerate and let a ESQ(X x IR") with the condition:

"either Q > S and Q + 6 = 1 or Q > 6 and 0 is linear in 0"

(1.22).

Then

1) if a has a zero on Co of infinite order then A (x) eC' (X); 2) ifa = 0 on Cm, we canfnd b ESe"(X x IR") such that Im(au) = Im(bu) for arbitrary u e CO (X).

Remark. The latter statement shows, that if a I c. = 0, then the distribution A may also be defined by substituting b (x, 0) for a (x, 0) and keeping the phase function. The function b (x, 0) has a lower degree of growth in 0, meaning higher regularity A (x). To prove theorem 1.3 we need a series of lemmas. The first of these concerns the change of variables in functions of the class S 's . First of all note, that it makes sense to say that a (x, 0) ESQ ,,(U), where U is an arbitrary region in IR" x IR", which is conic with respect to 0. Indeed, we will write that a(x, 0) ESQa(U), if for any compact set Kc (IR" x So U (S" is the unit sphere in IR") and for arbitrary multi-indices a, fi there is a constant C", a. 1e > 0 such that (1.10) is satisfied for (x, 0/101) e K and 101 ? 1. Now assume, that we are given a diffeomorphism from a conical region V C 1R"' X IR"' onto the

conical region Uc IR" x IR", commuting with the natural action of the multiplicative group IR+ of positive numbers, i.e. the diffeomorphism maps a point (y, n) E V to a point (x (y, n), 0 (y, n)) a U, where x (y, n) and 0 (y, n) are positively homogeneous inn of degree 0 and I respectively. Change the variables in a (x, 0):

b(y,n) = a(x(y,17), 0(y,n)).

(1.23)

Chapter I. Foundations of PDO Theory

8

Lemma 1.2. Let a (x, ) E SQ a(U) and assume that one of the following three assumptions hold:

a) Q+B= 1; b) Q+6? 1 and x = x (y) does not depend on q; c) x = x (y), _ (n). Then b(y, q) ESQ6(V).

Proof. Differentiating b (y, q), we obtain from (1.23) ab

=

b(i)

+

b

an,

ant

a = E 0)

ae' +

ab

where boy =

a©;

k

ay,

(x(y, q), 0(y, n)), b(k) =

k

axk

(1.24)

"' ant '

b(k) axk ay,

'

(1.25)

az (x (y, n), 9(y, q)). The functions ant'

axk , a9t and axk are positively homogeneous in q of degrees 0, - 1, 1 and 0 an, ay, ay,

S' and S° respectively respectively. They belong therefore to the classes S°, S (in V). Estimating the derivatives of a, we easily obtain for In 1 >_ 1 ab

5

CK(In1'"-Q+

ant ab

ayr 15

CK(1

q1"'-Q+l +

1q1"'+d),

where K is some compact set in V. If m + b - 1 5 m - Q, i.e. Q + 6 S 1, then from (1.24) we obtain the estimate

ab aqt

I

-

2CKm-°. If x = x(y) then

axk

an,

= 0 and we obtain this estimate from (1.24) without assuming g + 6:!5; 1. Similarly, if m - Q + I <--_ m + b, i.e. Q + b >__ 1, it follows from (1.25) that 1.5

2CK<17>^'+a, and the same estimate is obtained without the extra

Isy assumption Q+ b >- 1 if 0 = 0 (q).

The necessary estimates of the form (1.10) are thus verified when I a + Q 1

-<- 1

for an arbitrary function a (x, 0) E S.I. (U) in all the three cases a), b), c). Now,

inductively, assume that the estimates hold for I a + $1 < k and arbitrary a ESQa(U). In particular, we then obtain that for the derivatives of order <-k of bU) and b0, the estimates of the classes S' (V) and S' (V) respectively hold.

But then we obtain from (1.24), (1.25) by analogous reasoning, that these estimates hold for derivatives of order <-- (k + 1) and for arbitrary aeSQa(U).

§ 1.

Oscillatory Integrals

9

Lemma 1.3 (variant of the Hadamard lemma). Let the functions O, (x, 0), .. . 0&,0) belong to C°° (U), with U a conic region in IR" x (IRN\ 0), and assume that they are positively homogeneous in 0 of degree 0; dP, , ... , dok being linearly independent at points of the set

C={(x,0)eU,Oj(x,0)=0,j=1,...,k}. Let a E Se".j (U), a I c = 0 and Q + 6 = 1. Then there is a representation k

a= Y a, 0j,

(1.26)

i=1

where as (x, 0) e SQ e '(U), j = 1, ... , k. If the function a (x, 0) here has a zero of infinite order on C, then all the functions a; (x, 0), j = 1, ... , k, also have a zero of

infinite order on C.

Proof. Note, that (1.26) is a linear equation in the functions a;. It is therefore enough to be able to find the functions aj locally (for (x, 0/101) close to (xo, 00110,1), (xo, 00) being a fixed point in U). A global solution could then be glued

together in U using a partition of the unity in U consisting of functions homogeneous of degree zero supported in conical regions in which the required functions a; have already been constructed. Thus let (x0 i 00) a U. If (x0, 00) t C, there exists a jo such that P1, (x0, 00) 4 0. For (x, 0/ 101) close to (x0 , 00/ 1001) we can put at, = a/0t, and at = 0 for j $ jo. It remains to verify the existence of a1 locally for (x, 0/101) close to (xo, 00/1001) e C. By the implicit function theorem 0 1, ... , 4)k can be supplemented by functions Ok + 1,

,

01(1= N+ n - 1), homogeneous of degree zero, to form a local

coordinate system 0,,

..., i, on

the manifold {(x,0): 161=1) in a

neighbourhood of (xo, 00/ 100 1). Therefore, the transformation

(x,0)-(1(x,0), ..., 01(x, 0),10I)e1R'x]R+ is a diffeomorphism of a conical neighbourhood of the ray (x0,100) onto the conical set B x IR+ , B a ball in ER', and the image of C has the form {(0,101): 01 = ... = Ok=0). Let us now consider the symbol a(-P, 101)=a(x(O, 101), 0(0, 101)),

obtained from a (x, 0) under this diffeomorphism. It follows from lemma 1.2 that x (10,101) S g'" (B x IR+). But then, by the Newton-Leibnitz formula k

1

a(d',I01)= Y 0, aUl(t01, ..., tok> 0k+1, ..., 0,, 101)dt, j=1

0

where au, = a d eSQ a °(B x 1R+). It remains to carry out the inverse

substitution.

10

Chapter 1. Foundations of `PDO Theory

Proof of Theorem 1.3. Assume that Q + b = 1. Then, if a I c. = 0, by lemma 1.3 with 0; = a8 , we may represent a (x, 0) in the form

a= Y a; ;_,

a

a;eSQ a"(U).

a0;

(1.27)

ao

However, taking into account that e'm= -i a e'm, we obtain, on inteao; ao; grating by parts, that

But a©; e SQ d a - °(U), demonstrating the second statement of the theorem. From

this proof it is obvious, that if a (x, 0) had a zero of infinite order in C, then b (x, 0) could also be chosen to possess this property. So in proving the first statement we can assume a (x, 0) eSQ ' (X x R"), M as large as desired. But then the integral (1.21) converges absolutely and uniformly in x as do the integrals obtained from it by differentiation of degree S 1 (M), where I (M) --+ + oo as

M - + oo, and hence the smoothness of A (x) follows. D Exercise 1.10. Prove theorem 1.3 when the second of the assumptions (1.22) is fulfilled (0 (x, 0) linear in 0).

Hint. It amounts to applying of part c) of Lemma 1.2.

§2. Fourier Integral Operators (Preliminaries) 2.1 Definition of the Fourier Integral operator and its kernel. Let X, Y be open sets in IR'x and IR"r. Consider the expression

Au(x) =

a(x, y,0) u(y) dydO,

(2.1)

where u (y) E Co (Y), x e X, 0 (x, y, 0) is a phase function on X X Y X IR"' and

a(x,y,0)eSm(Xx Yx1R') with Q>O and b <1. Under these conditions the integral = J JJe'.(x.Y.a) a(x,y,0) u(y) v(x) dxdyd0,

veCO (X), (2.2)

is defined and is an ordinary oscillatory integral. It is easily verified, that for fixed u the expression (2.2) viewed as a functional of v, defines a distribution Au E Y'(X). Therefore a linear operator

§2. Fourier Integral Operators (Preliminaries)

11

A: Co (Y)-+-9'(X),

(2.3)

is defined. We will formally write it as the integral (2.1) Definition 2.1. An operator A of the form (2.1) is called a Fourier integral operator (abbreviated FIO) with phase function cp (x, y, 0). Definition 2.2. The distribution K,, c21'(X x Y) defined by the oscillatory integral

w(x,y)dxdydO,

(2.4)

is called the kernel of A.

Proposition 2.1. a)

K,, a Cy (Rm)

where

Rm = {(x, y):

(x, y, 0) + 0,

O c-]RN\0}

b) If a (x, y, 0) = 0 in a conical neighbourhood of the set Co = {(x, Y. 0):.0B (x, y, 0) = 0} , then K,, e C' (X x Y).

Proof. This follows immediately from Theorems 1.1 and 1.2.

O

In view of the obvious formula


ueCo (Y),

veCol" (X),

(2.5)

the kernel K4 is the usual kernel of A in the sense of L. Schwartz. Exercise 2.1. Verify that the kernel K4 is uniquely defined by the map (2.3) given by A and, conversely, uniquely determines this map.

Remark. One can easily construct two different pairs consisting of a phase function and a function a (x, y, 0), both pairs giving rise to the same operator (2.3). Furthermore, as a rule, a (x, y, 0) is not uniquely defined by A, even with the same phase function 0. 2.2 Operator phase functions.

Definition 2.3. A phase function 0 (x, y, 0) is called an operator phase function, if the following two conditions are fulfilled;

y, O) *0 for 0* 0,

xeX,

yeY

(2.6)

s,a(x,y,0)40 for 0 * 0

xeX,

yeY

(2.7)

0 y. a (x,

The role of these two conditions is brought out by the following two propositions.

Chapter 1. Foundations of `PDO Theory

12

Proposition 2.2. Under condition (2.6) the operator (2.1) continuously maps Co (Y) into C°O(X).

Proof. The integral (2.1) is already defined as an oscillatory integral, depending on the parameter x. Its x- derivatives are of the same form. In what follows,.f' (Y) denotes the dual to C°° (Y) (and is the set of compactly

supported distributions in Y). Proposition 2.3. Under condition (2.7) the map (2.3) defined via (2.1) extends by continuity to a continuous map:

A: 6'(Y) -- 2'(X).

(2.8)

Here the continuity is understood in the sense of the weak topologies on X'(Y) and Y'(X). Recall that the weak topology on the space E', consisting of the linear functionals on E, is defined by the family of semi-norms p,o(f) = I (f (p) I, where f E E' and (p is any fixed element of E. Proof. The transposed operator

'Av(y) = J

je'm(X.Y'O) a(x,y,O) v(x)drdO

(2.9)

defines, by proposition 2.2, a map

'A: Co (X) -. C' (Y). Thus, defining A by =

with u e 8'(Y), v e Co (X) we are done.

Exercise 2.2. Verify that the operator A, defined in this way, is indeed an extension by continuity of the map (2.3). So an F10 A with operator phase function dy maps Co (Y) into C'(X) and 1'(Y) into 2'(X). We now study the change in the singular support under the action of A. Let us settle a notation. If X and Y are two sets, S a subset of X x Y and K a subset of Y, then S o K is the subset of X consisting of the points x E X, for which there exists a y e K with (x, y) c- S. Theorem 2.1. The following inclusion holds:

sing supp Au c So ° sing supp u

(2.10)

where Sm = (X x Y)\ Rm consists of those pairs (x, y) for which there exists a 0 eIRN\0 with 0; (X, y,©) = 0.

§2. Fourier Integral Operators (Preliminaries)

13

Proof Splitting U E 9'(Y) into a sum of a function in C, '(Y) and a distribution with the support in a neighbourhood of sing supp a we see that it suffices to demonstrate that sing supp (Au) c S. o supp u. Let K = supp u and K' an arbitrary compact set in X not intersecting So o K and so that K' x K C Rm. Since R. is open, there are open neighbourhoods S2 and ST of the compact sets K and K' respectively such that ST x S2 C R,. So it suffices to verify that Au E C°°(Sl'). But this is evident, since KA(x, y) E C°°(R,) and in particular, KA (X, y) E Coo (ST X S2).

Exercise 2.3. Verify the statement used above that if K4 E C (Q' x (2) then A maps d'(Q) into C°°(Q').

2.3 Example 1: The Cauchy problem for the wave equation. Consider the Cauchy problem (2.11)

f I(=o = 0

f' 1,=0 = u(x)

(2.12)

where x e IR", f = f (t, x), d is the Laplacian in x and - to begin with u (x) a Ca (1R"). We solve (2.11)-(2.12) with the help of the Fourier transformation in x, putting je-,,..cf(t,y)dy.

In this way, we have a

are

_-

.JIt=o = 0,

(2.13)

.1j" It=o = uO

(2.14)

where u(i;) is the Fourier transform of u(x). From (2.13) and (2.14) we easily obtain that

f (t,

)=

(,),in t I1

Therefore by the Fourier inversion formula

f(t,x) = f f =ff

sin t14 lu(y)dyd e-`1f1)u(y)dyd(;.

We would like to split the last integral into two parts separating the exponents ei'111 and e-l"41. However this would lead to a singularity at t = 0. To avoid E Co (lit"), such this singularity, let us again use a cut-off function X = x that x (!) = I near 0 and split the integral into three parts:

f (t, x) = f+(t, x) - f_(r, x) + r(t, x)

14

Chapter 1. Foundations of 4'1)0 Theory

f+(t,x) = f- Q. x) = f

r(t,x) =

ffe'lc.,-3'>t+rIt11(1

- x(4))(2i X (.))(2il I)-dyd4, I-1

sintI dyd4.

It is clear e.g. that f+ = Au where A is a FIO with the phase function

) (t, x, y, ) = (x - y) . This is an operator phase function. Since 45{ = x

+ t 1C.

-y+

we have

Cm = (Q. X, y, ) : y -x = tIII), sm=((t,x,y):Ix-y12=t2).

The second term f_(t, x) can be similarly presented as f_ = Au, where A is a FIO with the phase function

<(t,x,y,(x-y) -tll It has the same set Si = Sm. For the third term we clearly have r = Ru where R has a C°° Schwartz kernel KR (t, x, y). In fact, it is easy to see that any such operator R can be also presented as a FlO in the form (2.1) with an arbitrary choice of the phase function and with an amplitude a E S. (See also Exercise 2.4 below.)

So we see that each of the terms ff(t, x), r(t, x) can be presented as a result of the application of a FIO to the initial condition u. In particular, by Proposition 2.3 we can define f (t, x) for any u E e'(lR°). By Theorem 2.1 the singularities of f (t, x) belong to ((t, x) : 3u E sing suppu, Ix - y12 = t2}. This is the classical statement that singularities propagate with the speed of light (which is equal to 1 in this case). In particular, the singularities of the fundamental solution (the case u(y) = 8(y)) belong to the light cone Ix12 = t2.

Note also that if we fix the time to, then 45,,(x, y, ) = ' (to, x, y, 4) remains an operator phase function. Therefore the mapping of u into f (to, ) is a FIO. 2.4 Example 2: Linear differential operators. Let

A= Y a,(x)DR, where aQ (x) e C ' (X), X an open set in IR" and D = i - t

Using the Fourier transformation, we may write

D'u(x) = JJaeitx-r)-Eu(y)dyd,

(2.16)

a

§2. Fourier Integral Operators (Preliminaries)

IS

hence

(2.17)

where a,, (x,

Y_ as (x) ' is called the symbol of the operator A. Since 14I S m

a,, (x, ) a S'(X x lR"), we see from (2.17) that A is an FIO with phase function (x,Y,

2.5 Example 3: Pseudodifferential operators

Definition 2.4. Let n x = n y = N = n and X = Y. Then an FIO with the phase function ds (x, y, l;) _ (x - y) is called a pseudod{ff erential operator (briefly: 'PDO). The class of `PDO, defined by a(x, y, l;) E SQ1(X x X x IR") is denoted oted byn (X) or simply by L'" a. We also put L"' instead of L 'no and L-00

L

.

As demonstrated in the previous example, any linear partial differential operator is a `PDO.

We now display the properties of 'PDO, which follow from the already shown properties of FIO. Proposition 2.4. Let A be a `PDO given by the formula Au(x)=IIetcx-r,-
(2.18)

Let K4 be the kernel of A and d the diagonal in X x X. Then a) K,, a Ca, ((X x X)\ d)); b) A defines continuous linear maps

and

A: Co A: d'(X)- .9'(X)

(2.19)

sing supp Au = sing supp u

(2.21)

(2.20)

for u e d'(X) (this property is called pseudolocality of A); c)

the function a (x, y, ) a S,1.o a(X x X x IR") vanishes for x = y and b <

then we can write A in the form (2.18) with b (x, y, ) e SQ a to -b' (X x X x IR") instead of a (x, y, ). d) (f a (x, y, ) has a zero of infinite order at x = y, then K,, E C'°(X x X) and the operator A transforms 1'(X) into C °° (X).

Proof. Left to the reader as an exercise. Exercise 2.4. Let K(x, y) e C °° (X x X) and A bean operator from C, '(X) to C°°(X) with kernel K(x, y). Prove that A is a `PDO and that in the representation (2.18) we can take a(x, y, ) E S-' (X X X x IR"). Hint. If X

a Co (lR"), X ( ) z 0 and JX

dd =1 we can take

e-ccx-Y'*4K(x,Y)

X()

16

Chapter I. Foundations of `I'DO Theory

Note that linear differential operators enjoy the locality property: supp (Au) c supp u,

u e Co (X)

(2.22)

The following exercise shows that in general this is not the case for IFDO.

Exercise 2.5. Show that an operator whose kernel K(x, y) E C' (X x X) is not identically zero does not obey (2.22) Problem 2.1. Given a linear continuous operator A : Co (X) -+ Co (X) satisfying (2.22), then for any subdomain X' c X, whose closure is compact in X, we get a linear differential operator by restricting A to Co (X'). Hint. Verify that for any fixed x0 E Xthe linear functional given by (A q0) (xo) for cp E Co (X), is supported at xo and thus can be written as (A (p) (xo) _

as (xo) (Da(p) (xo)

Derive from the continuity of A the local finiteness of this sum and the smoothness (in xo) of the coefficients a,,,(xo).

§3. The Algebra of Pseudodifferential Operators and Their Symbols 3.1 Properly supported pseudodifferential operators. Let A be a TDO with kernel KA and let supp KA denote the support of KA (the smallest closed subset

Z C X x X such that KAI(XXX)\z = 0. Consider the canonical projections 171, 172 : supp KA -+X, obtained by restricting the corresponding projections

of the direct product X x X. Recall that a continuous map f : M -> N between topological spaces M, N is called proper if for any compact K C N the inverse image f -I (K) is a compact in M. Definition 3.1. A TDO A is called properly supported if both projections f71,172 : supp KA -+ X are proper maps. Example. Linear differential operators (cf. item 2.4) are properly supported TDO (in this case supp KA = d, the diagonal in X x X).

Proposition 3.1. Let A be a properly supported `PDO. Then A defines a map (3.1) A: Co (X) -+ Co (X) which extends to continuous maps

A:

d'(X)-d'(X)

(3.2)

§3. The Algebra of Pseudodifferential Operators and Their Symbols

A: C'(X)->C'(X) A:

9I'(X) -' 2'(X)

17

(3.3)

(3.4)

Proof. For u (y) E Co (X), we have the inclusion supp (Au) c (supp K4) o (supp u)

(3.5)

Indeed, if v E Co (X) is such that supp v n (supp K4) o (supp u) = 0, then supp K4 n supp [u (y) v (x)] = 0 and therefore = 0 by (2.5). Furthermore, from the obvious formula (supp K4) o (supp u) = 17, (supp K4 nI1 2 ' (supp u))

(3.6)

it follows that the right hand side of (3.5) is compact, so that Au e CO '(X), which establishes (3.1). The continuity is easily verified. Since the kernel Kt,, of the transposed operator' A is obtained from KA by

permuting x and y (more precisely: (KtA, w(x, y)) = (K,,, w(y, x))): then'A also defines a continuous map

'A: Co(X)-iCo(X), which yields (3.4) by duality. Finally, as is easily verified, formula (3.5) also applies for u e t' (X) which gives (3.2). Since this can also be said of'A, by duality we obtain (3.3).

Exercise 3.1. Let X. be a sequence of open subsets of X such that 1)

X,cX2c...

2) Ux,,=x 3)

the closure X. of X. in X is compact in X.

Let X. (x) e Co (X) and X. (x) = I for x e X,,. Finally let A be a properly supported `A'DO in X. Show that if u e 9' (X), then for arbitrary m one can find N = N (m) such that the distribution [A Q. u)] I does not depend on n for n Z N. In this way, we can define Au e 2'(X) by the formula Au = lim

(3.7)

A. M

Show that this definition coincides with the one given above in proving Proposition 3.1.

Exercise 3.2. Show that all the three definitions of a properly supported PDO given above, coincide on C°° (X):

a) by duality from the map'A : 9'(X) -. 8'(X); b) as a restriction to C' (X) of the map A: I'(X) -+ 9'(X), constructed by duality from the map 'A: Co (X) - Co (X), c) by formula (3.7).

Chapter 1. Foundations of PDO Theory

18

Exercise 3.3. Verify that the operator A is properly supported if and only if 'A is.

The importance of properly supported'PDO lies in the fact that they form an algebra where multiplication is the ordinary multiplication (composition) of operators. This statement will be proved below, but presently it is clear from Proposition 3.1, that the composition A oA2 of two properly supported'PDO is defined as a linear continuous operator on the spaces Co (X), -f'(X), C'° (X) or

Note that it is not immediately clear that (2.18) which determines A via a (x, y, ) applies for u E -9'(X) (or even for u e C°°(X)). This is not surprising, since the function a (x, y, ) is not uniquely defined by the operator A. However, utilizing this arbitrariness, we can make a better choice of a (x, y, ). Definition 3.2. We say that a function a (x, y, ) is properly supported if both the projections 111, 172: supp:.r a (x,

are proper maps (by suppz,,, a (x, y, ) we denote the closure of the projection of supp a (x, y, ) in X x K).

In this case the corresponding Y'DO A is obviously properly supported. Proposition 3.2. If A e LQ,,, (X) is properly supported, then A can be put in the form (2.18) with a (x, y, ) e SQ a(X x X x IR") being properly supported.

Proof. Let X (x, y) e C °` (X x X), X (x, y) = I in a neighborhood of supp K4

and let both projections 111, 172: supp X - X be proper (verify that such a function exists!). Then, substituting in (2.18) X (x, y) a (x, y, ) for a (x, y, ), the

kernel K,, and hence the operator are unchanged while X (x, y) a (x, y, ) is properly supported. D Note that if a (x, y, ) is properly supported then (2.18), viewed as an iterated integral, is defined for u e C' (X).

Proposition 3.3. Any 'PDO A can be written in the form A = AO + A, where Ao is a properly supported'PDO and A, has kernel K,,, E C°° (X X X). Proof. Given A in the form (2.18) with function a (x, y, ), we obtain AO and A, by substituting as (x, y, ) = X (x, y) a (x, y, ) and a, (x, y, ) = (1 - X (x, y))

a (x, y, ) respectively instead of a (x, y, ) where X (x, y) equals

1

in a

neighborhood of the diagonal d c X x X and is such that both projections 17, 172: supp X X are proper.

,

Proposition 3.4. Let A be a'PDO. Then A is properly supported if and only if the following two conditions are fuflled ; a) for any compact set K e X we can find a compact set K, e X such that supp u c K implies supp (Au) c K,; b) the same condition with 'A instead of A.

§3. The Algebra of Pseudodifferential Operators and Their Symbols

19

Proof. The necessity of a) and b) is clear from (3.5). So we are left with proving the sufficiency. For instance let us verify that the projection 172: supp K,, -' X is a proper map. Let K be an arbitrary compact set in X. Find a (K) n supp KA c K, x K. If compact set K, as in a) and verify that 172 -'(K) (x0, yo) E (X\K1) x K and if a smooth function w(x, y) = u(y)v(x) is supported in a neighbourhood of this point, then by a) (KA, w) = 0. By linearity and continuity this then also holds for an arbitrary function w E Ca (X X X), supported in a neighbourhood of (xo, yo) from which the desired inclusion

follows. 0 3.2 The symbol of a properly supported pseudodifferentlal operator. We would like to define the symbol of an arbitrary properly supported `PDO A by analogy with example 2 of §2. Let us point out that in this example the following holds Q,, (x, 4) = e _, (x) A e{ (x),

(3.8)

where et (x) = e" t, and that the right hand side of this expression also makes sense for a properly supported `PDO. Definition 3.3. Let A be a properly supported `PDO. Its symbol (or complete symbol) is the function aA(x, ) on X x IR", defined by (3.8). Since e, (x) is an infinitely differentiable function of c with values in C°° (X) and A is a continuous linear operator on C°° (X), it is clear that a,, (x, ) is also an

infinitely differentiable function of

taking values in C°°(X), therefore

aA (x, ) e C °° (X X R").

Writing u (x) E Co (X) as the inverse Fourier transform u (x) = j e4 (x) u (i;) d

where the integral converges in the topology of C`° (X), we see that

Au(x)=!ei" 4CA(x, )4 (4)d

(3.9)

or

Au(x)=

fe`c=-Y' 4aA(x, )u(Y)dyd

(3.10)

(where the integral is viewed as an iterated one), which coincides with the corresponding formulas for differential operators. As is demonstrated by the formulas (3.8) and (3.9), the symbol aA (x, ) defines A and is also defined by A. Below it will be shown that if A E LQ,6(X) and 8 < p, then we will have x IV) implying that (3.10) can also be viewed as an oscillatory a4 (x, ) E Se lm integral ((3.9) is absolutely convergent).

Remark If A is an arbitrary `PDO on X, then its symbol is frequently defined as the symbol aA. (x, !) of a properly supported `PDO A 1 on X such

20

Chapter I. Foundations of'I'DO Theory

that A - AI E L-00. In this case the symbol is not uniquely defined although, as will be seen later, any two such symbols differ by a function r(x, ) E S-°°. 3.3 Asymptotic expansions in S1 a Definition 3.4. Let a, (x, 0) E SQ J8 (X X RN), j=1, 2,

j - +oo, and let a(x,0)eC''(XxWe will write

..., mj - - oo

as

a(x,0) - Y aj(x,0), j=1

if for any integer r ? 2 we have r-1

a(x,0)-

a;(x,0)ESI,,(Xx1R"),

(3.11)

j-1

where in, = max m,. jT,r

From this it follows, in particular, that a ES.1.6 (X x IR").

Proposition 3.5. For a given sequence a, e S; b (X x RN), j = 1, 2, ... , with mj- - oo as j-+ + oo, we can always find a function a (x, 0) such that

If, furthermore, another function a' has the same property a'-

a-a'ES-°°(XxIR").

aj, then j=1

Proof. The second assertion is obvious so let us prove the first one. We can assume that m, > m2 > m3 > .... In fact, if this is not the case, we can always

achieve this situation by a simple rearrangement and gathering the terms of the same order. Let X = U X,, where Xj are open subsets of X and such j=1

that Kj = Xj cc X (i.e. Kj is compact in X). Let 9 (0) E C ' (R') and (P(0) =

for 101 5 1/2

0

11 for 101>1.

Put a (x, 0) _

cp

j=1

(-0) aj (x, 0), tj

(3.12)

where t, approaches + co so quickly as j - + co that I8e8= [q'

(p:) a,(x,0)]2Jm

I=I+610t;-

for x e K1 and I a I + I IJ I + 1 5 j. Let us show that this is always possible.

(3.13)

§3. The Algebra of Pseudodifferential Operators and Their Symbols

21

First observe that 1:5 Ca<0>-lal,

ago

t> 1,

(3.14)

where C. does not depend on t, that is for cp (8/t) and t z I we have uniform in t estimates of class S10.0 . In fact aaq (0/t) _ (aa(p) (©/t) - t-1a1

and

101 5 t5 2 101

for 0 esuppaegp(9/t), from which we obtain (3.14). Further from (3.14) follows that <e>m'-e 1a1 +6161

l aeax [1P (0/t) aj(x, e)J I < Cj

if x e K,

, t? I and 1 a I+ 1#1+1:5j. Let us observe now that <©)m,-e1a1+6161 < g<0>m' .-elal+6161

for (B>'",-' -m' z 1/e. Thus, by the choice of tj we can achieve (3.13) which implies

the convergence of (3.12), together with all its derivatives, uniformly on any compact K c X, and for arbitrary fixed a, Q and 1 we obtain a^ aeaX

L co(©/tj)aj(x,©)

j=r+1

Thus, a -

<2

r<e>m,-ela1+6161

xeK,,IuI+Illl+l<-r.

aj E S,*',, (X x IR") and since ar E S,',,, (X x lR"), we obtain from this j=1

r-1

that a -

aj a S,.,, (X x IR"), as required. j=1

The following proposition facilitates the verification of a -

aj.

Proposition 3.6. Let aj ESQ;6(X x K r), m, -+ - oo as j + oo and let a e C°°(X x IR") so that for each compact Kc X and arbitrary multi-indices a,# there exist constants u = p (a, fi, K) and C = C (a, fi, K) with Iaea,'a(x,0)1 5 C<©>",

xeK.

(3.15)

Furthermore assume that for any compact K c X, there exist numbers p, = p, (K), 1=1, 2, ..., and constants C, = C1(K), such that p, --+ - oo as I - + oo and the following estimate holds

aj(x,0) 5 C,<0>",

a(x,©) i=1

xeK.

(3.16)

22

Chapter 1. Foundations of `FDO Theory w

Then a - Y_ aj. J=1

The point of this proposition is that instead of (3.11) one has to verify the

remainder estimates only for the functions themselves (and not for all the derivatives) provided the fairly weak estimates (3.15) are guaranteed. The proof is based on the following well-known lemma. Lemma 3.1. Let a junction f (t) have continuous derivatives f' (t) and f" (t) for

te[-1,1].PutAj= sup If1J)(t)jj=0,2. Then _1$,51

(3.17)

If (0) 12 5 4Ao(Ao+A2).

Proof. By the intermediate value theorem we have

If'(t) -f'(0)I < A21t1. Therefore I f' (t)1 > 1 /21 f' (0) I for Ai 1 t l 5 1 /2 I f' (0) I, I t 151. Denoting

d = min

1 }we have I f'(t) I >1/21 f'(0) I for t e [- d, A]. We have

2A2

J

2A0? If(A) -f(-A)I ? 2A If'(O)I 2

and consequently, 11(0)1 5 2 e ° = 2Aomax

IA2

IT(O)i,

i}.

or If'(0)I 5 2A,, i.e. either If'(0)12 II (o)I 5 4AOA2 or If, (0) 12 S 4A2 and thus (3.17). O

This implies that either If'(0) 1

54ApA2.

Lemma 3.2. Let K, and K2 be two compact sets in lR° so that K, C Int K2 (the set of interior points in K2). Then there exists a constant C> 0, such that for any smooth function f on a neighborhood of K2, the following estimate holds

(sup

jaj=1

IDaf(x)I

sup If(x)I +ja1=2 sup Z I Daf(x)I(3.18)

Csup If(x)I (xc.Kz xeX2

Proof. Immediate from Lemma 3.1.

0

§3. The Algebra of Pseudodifferential Operators and Their Symbols

23

Pro of of Proposition 3.6. Let b - Y a, (such a function exists by Proposition j=3

3.5). Putting d (x, 0) = a (x, 0) - b (x, 0), we have for every compact set K c X the estimate

xeK,

Iaea,d(x,0)I < C<0>',

(3.19)

where C and p depend on a, fi, K, and additionally

Id(x,0)I 5 C,<0>-', where C,= C,(K). Set do(x, ) = d(x,

xeK,

(3.20)

Then a{a° ds(x, x) 14 =0 = aBax d(x, 0),

and applying Lemma 3.2 with K, = K x 0, K2 = k x { I 151 }, where k is a compact set in X such that Int k =>K, we obtain from (3.20) d(x,0)I\2 ( sup

I0B0P

xQK iai+ipisl

C<0)-, (<0)-'+<0)p) l1

Here r can be choosen arbitrarily, µ depends on a, fl, K and C depends on

a, fi, K and also on r. Thus it follows that for x E K and Ia1 + Illl < 1 0) -+ 0 faster than any power of (0) as 101 -f +oo. By the function induction we obtain the same for arbitrary a, fi which gives d E S-°°(X X IR") as required. 0 3.4 An expression for the symbol of a properly supported `YDO in terms of a (x, y, t). In this and the following subsections we assume that b < Q.

Theorem 3.1. Let A be a properly supported 'PDO given by (2.18), and 04 (x, ) its symbol. Then

a4Dya(x,Y,

(3.21)

where the asymptotic sum runs over all multi-indices a.

Remark. Obviously a sum (3.21) is therefore meaningful.

a

Q-61l

and the asymptotic

Proof of Theorem 3.1. Observing that by Proposition 3.2 we can assume that a (x, y, 4) is properly supported, then (3.8), defining the symbol a4 (x, ), can be rewritten in the form

a4(x, 0 = f

fa(x,y,0)er(x-y)-eei(y-x)

4 dyd0,

(3.22)

Chapter 1. Foundations of `PDO Theory

24

Here the integral is regarded as an iterated one and the integral makes sense since

for each fixed x, the y-integration is over a compact set. In this way, if K is a compact set in X, the oscillatory integral given by (3.22) depends on a parameter x e K. Making the change of variables z = y - x, q = 0 - to simplify the exponent we obtain dzdq.

(3.23)

Expanding a (x, x + z, + q) inn near q = 0, using the Taylor formula, we have:

Y

(3.24)

i:i5N-1

where

N q' a

-'

(3.25)

0

Now observe that

"dzdq=a D,a(x,x+z,U

IS a'4

4

(3.26)

by Fourier inversion formula, this gives the finite terms in formula (3.21). We would now like to use Proposition 3.6. Let us first get a rough estimate for a,, (x, ) of the type (3.15). For this we rewrite (3.23) by integrating by parts


a,,(x,5)= $Je

(3.27)

where v is even and non-negative.

Taking into account the inequality <+'> < 2 <> - , we obtain from (3.27) that la',aX o4 (x, ) I < J°-(1-°)"dq, where p = max (m - p I C (I + 6 1# I, 0), x E K and v is sufficiently large. This gives the desired estimates for the derivatives of Q4 (x, ) of the type (3.15). It remains to estimate the remainder term. Inserting in (3.23) the expression rN (formula (3.25)) for a (x, x+ z, + q) and interchanging the orders of integration over t and over z, q, we see that it is necessary to have a uniform in t c- (0,1 ] and x E K estimate of the integral

where I a I = N. Integrating by parts, we obtain (3.28)

Let us decompose the integral in (3.28) into two parts: Ra.1 = Ra., + R.",,

(3.29)

§3. The Algebra of Pseudodifferential Operators and Their Symbols

25

where in RQ,, the integration is over the set {(z,q): and in R;., it is over the complement to this set. Note that if I q 15 I fl /2 then 11!2 < I + n7 l 5 j I I and moreover in &, the volume of the domain of integration for q doesn't exceed C I I", hence IRa.,(x, )I

(3.30)

C doesn't depend on and t.

Let us next estimate R.",. Integrating by parts and using the formula

' e : It = e

= ''

where v is an even and non-negative number, we see that R.", can be written as a finite sum of terms of the form 11

e-`='q-'D';Da+aa(x,x+z,

+tq)dzdq,

(3.31)

Iql> M12

where 1#1:5v. For I' I ?ICI/2 the expression 84D= +e a (x, x + z, + tq) is m- (Q - b) N+ by ->- 0 and by C for m - (Q - b) N + by < 0 (in both cases C is independent of S, q and t). Taking into account the factor we obtain from (3.31) that for sufficiently estimated in absolute value by Cm -(Q

-8)N+av for

large v I

SC

f

P-(1-a1.

,

Iql>Ifl/2

where p = max { m - (Q - 6)N, 0}. If p - (1- O v + n + 1 < 0, it follows that IRa.O.t(x,S)I S

CP-(1-a)v+n+1 j-n-1 dq S

C doesn't depend on x, and t (for x e K, t e (0,1 ]). Selecting a large enough v we can make the exponent in (3.32) p - (1- b) v + n + I negative and as large as we like in absolute value. Taking (3.29) and (3.30) into account, we obtain for R,,, the estimate C«)'"-(Q-d/N+.,

xeK,

te(0,1],

which ensures the applicability of Proposition 3.6 and so finishes the proof. U Remark. The method of proof of Theorem 3.1 is very typical for the theory of `PDO and the corresponding arguments are to be found in all versions of this theory independently of the mode of presentation. We therefore strongly urge the reader to carefully study the proof of this key theorem.

26

Chapter I. Foundations of'I'DO Theory

3.5 The symbol of the transposed operator and the dual symbol. The transposed operator 'A is defined by =

(3.33)

for any u, v c- Co (X), where
Therefore, if A E Le".,, (X) is given by (2.18), where a(x, y, ) ESQ 6(X x IV), the transpose 'A is given by

'Av(y) = f f e"x-'''C a(x,y,l;) which with the change of variable ry = - gives 'Av(y) = f f

e'"''-X)-4 a(x,

y, -') v(x)dxdrl.

(3.34)

It is therefore obvious that 'A c- V,",, (X).

Theorem 3.2. Let A be a properly supported'I'DO with symbol aA(x, ) and aA(x, ) the symbol of'A, then a'. (x,

0" D,2

(3.35)

Proof. Note that 'A is also properly supported (cf. Exercise 3.3). Also, instead of a (x, y, l;) in the formula (2.18), giving the action of A, we can substitute aA(x, ) (cf. (3.10)). Then (3.34) can be written

'Av(x)= ffe(:-y)-{o'A(Y,- )v(Y)dYd

(3.36)

This is the standard form for a'PDO (cf. (2.18)) where the role of a (x, y, l;) is played by aA(y, -c). It remains only to apply Theorem 3.1. Exercise 3.4. Let A be a properly supported 'I'DO with symbol aA (x, ) and let A* be the "adjoint" operator, defined by (Au, v) = (u, A*v), u, v E Co (X)

where (u, v) = f u(x)v(x) dx. Prove that A* is a properly supported Y'DO whose symbol satisfies

aA* (x,)

a" D' aA(x, &a!

where the bar denotes the complex conjugation.

(3.37)

§3. The Algebra of Pseudodifl'erential Operators and Their Symbols

27

We now introduce the dual symbol by setting aA(x,

O'' (XI -)

(3.38)

Taking into account that '('A) = A, we obtain from (3.36) that A can be expressed via the dual symbol a4 (x, ) in the form Au(x) = J fe"s-YI'c rA(y, ) u(y)dyds or

(Au)(k)= Je -ir

aA(y, )u(y)dv

(3.39)

Theorem 3.3. The dual symbol aA (x, O is connected with the symbol aA (x, via the asymptotic formula aA(x,

)

(3.40)

) ~ Y(-d4)aDxa, (x,0/a! a

Proof. Obvious from (3.38) and (3.35). L 3.6 The composition formula

Theorem 3.4. Let A and B be two properly supported `YDO in a domain X RV and let their symbols be aA (x, l;) and aB (x, ) respectively. The composition

C = B A is then a properly supported`PDO, whose symbol satisfies the relation (3.41)

/a!

QBA(x,

Proof. Using formula (3.39) for A and applying formula (3.9) to B, we obtain Cu (x) =if eitx - ri ' 4 c, (x, ) aA (y, ) u (y) dy dS .

It follows that if A E LQ.6(X) and B E LQ.6(X) then C E

we obtain 'C='A 'BeLQ.a

Analogously

The fact that the `PDO C is properly

supported now follows from Proposition 3.4 and it remains to compute 6BA (x, ) using Theorems 3.1 and 3.3.

We have

8cDy1aB(x,i)eA(y, )]/a!Ir

aBA(x,

= L Ca IQB(x, 0 DxaA(x, )]la! i3{ [a. (x, ) (-at)r Dx+0QA(x, 01/0'! fl!

~ a.0

(3.42)

Chapter I. Foundations of Y'DO Theory

28

Next we state two well-known algebraic lemmas. Lemma 3.3 (Leibniz' rule). Let f(x) and g (x) be two smooth functions in an open set X c lR" and a a multi-index. Then

(f(x) g(x)) _ Y_

r+6=a Yab

[ayf(x)] [a°g(x)]

(3.43)

Lemma 3.4 (Newtons binomial formula). Let x, y e 1R" and a a multi-index, then

(x+y)a _

a .i Y_

xyyd.

(3.44)

,.+,$=a Y.8

Exercise 3.5. Prove Lemmas 3.3 and 3.4. Hint. (3.44) can be shown by induction or by using the Taylor formula for polynomials. (3.43) is obtained from (3.44) by noting that

asU(x) g(x)] = {(ax+3,yV(x) g(y)]} 1r=x. Conclusion of the proof of Theorem 3.4. Rewrite (3.42) using Lemma 3.3. 034 a. 0. Y. b

.+e=a

_ Y_ (-1)ivi [ayae(x, 0.r.a

/

)] 100+6D.011+

.4 (x, )]PP!y!6!

(3.45)

(-1)lel\

= E Ex +\ +a=x E !a! y

J1 [ay 68(x, )] [ac'Dr+Ya, (x, )]lY!

We then obtain from (3.44) with x = -y = e, where e = (1,1, ...,1) and with a = x; x! ee( _ e e)

bjx 0+e=x

atb!

I' = x! E (1

$+s=x

/J18!

here S'x, is the Kronecker symbol, equal to I for x = 0 and 0 for I x 1 > 0. Because

of this relation, in (3.45) there only remain terms with x = 0, which proves (3.41).

Corollary 3.1. Let A E Loa(X), B E LQa(X), 0 S 8 < p< 1, and assume that B is properly supported. Then the operators A B and B A viewed Loa+m,(X). as operators from CO '(X) to C' (X) belong to

Proof. Decompose A into sumA = A, + R, where A, is properly supported and R has a kernel R (x, y) a C°° (X x X). It is easily verified that the operators

§3. The Algebra of Pseudodifferential Operators and Their Symbols

29

BR and RB have smooth kernels equal to B R (x, y) and 'Br R (x, y) respectively, where Bx operates on x keeping y fixed and analogously for `B,,. The assertion of the corollary now follows from Theorem 3.4.

3.7 Classical symbols and pseudodifferential operators. It is sometimes convenient to consider narrower classes of `PDO. Here we describe one of this classes, closed under the majority of the necessary conditions.

Definition 3.5. By a classical symbol we mean a function a (x, 0) E C" (X x IR") such that for some complex m these is an asymptotic expansion 00

a(x,0)

E W(0)

a'"-;(x,0),

j=o

where w E C' (1R"), iy (0) = 0 for 101 S 1/2, v (0) = I for 101 z 1, and a", _j (x, 0) is positive homogeneous of degree m -j in 0, i. e. a, _ j (x, t 0) = t" -j a, _j (x, 0) for all t > 0 and (x, 0) e X x (IR"\0). Denote the class of all symbols fulfilling these

requirements by CSm(X x RN). Furthermore, denote by CL"(X) the class of `PDO which can be written in the form (2.18) with a(x, y.1;) E CS"(X X X X IR"). These operators will be called classical PDO.

If ak(x, 0) is positive homogeneous of degree k in 0, then 8;8fak(x, 0) is positive homogeneous of degree k - I a I in 0. Therefore it is clear that CSTM(X X R"),= SRem(X X 1RN) .

Proposition 3.7. a) If A E CL'" (X) and is properly supported, then a, (x, S) E CSt'(X x IR"). b) If A e CL"* (X) and B e CL!", (X) and both are properly supported then BAECL'",+'"1(X). c) If A e CL'"(X), then 'A E CL!"(X) and A* E CL'"(X).

Proof. Follows immediately from Theorems 3.1-3.4. Thus the class of all classical `PDO is closed under composition, taking the adjoint, and the transpose. In what follows we will show that it is also closed under changing variables, taking the parametrixes (cf. §5) and complex powers of an elliptic operator.

3.8 Exercises and problems

Exercise 3.6. Show the following generalization of Leibniz's rule (3.43): if

p (x, ) = E a, (x) °, and p (x, D) is the corresponding differential operator, then

1219-

p (x, D) (f(x) g(x)) = Y [p'"l(x,D) f(x)1 [D°8(x)]/a!, "

where p("(x, ) = a, 4,p (x, ).

(3.46)

Chapter I. Foundations of `PDO Theory

30

Exercise 3.7. Derive theorem 3.4 for differential operators from the result of Exercise 3.6.

Exercise 3.8. Let x, , xs , ... , xk be n-vectors and a an n-dimensional multiindex. Prove that a.i a

(xl+x2+... +xk)a=

k'

-

a, '

... xkI.

(3.47)

'

Deduce from this that for any smooth functions fl, ... , fk da U (x) ... fk(x)]

ai =a,+. Ya,=a a, !

.

akt (a°1-f, ) (x)

... (aa.f)(x).

(3.48)

Exercise 3.9. Given a function a (x,) E SQ s (X x IR"), X an open set in IR",

show that there exists a properly supported `I'DO A in X, such that a -aA E S-°°(X x IR"). Hint. Consider the operator given by (2.18) with a(x,y, where Z (x, y) is the same as in the proof of Proposition 3.3.

y (x, y) a(x, ),

Exercise 3.10. Derive from Exercises 2.4 and 3.9 that the operation of taking the symbol defines (for b < e) an isomorphism Le.a(X)/L-'°(X)

SQ 8(X x IR")/S-°"(X x IR").

Problem 3.1. Consider the following operator in R"

u(y)dydc,

Au(x)=

(3.49)

where a (x, ) satisfies

laga°a(x,)I

Cae<>"'-Qiai+aiei

(3.50)

Assume that B > 0 and b < 1. Attach a meaning to the integral (3.49) in the following two situations; a) u (x) a S(IR"); b) u (x) E C6 (IR") i. e. I as u (x) I < C. for an arbitrary multi-index a. Show that A defines a continuous transformation of the spaces S(IR") and Cb (IR") into themselves. Show that the symbol a (x, ) is

uniquely defined by the action of A on S(W) or Cb' (W). Problem 3.2. Show that the operators of the form described in Problem 3.1 form an algebra with involution and obtain an asymptotic formula for the symbols of the composition of two operators, of the transpose and of the adjoint operator. Problem 3.3. Let K(x, z) E C" (X x (lR"\0)) be positive homogeneous in z of

degree - n and let S

I=1- I

K(x,z)dS.=O

(3.51)

§4. Change of Variables and Pseudodifferential Operators on Manifolds

31

(integral over the sphere I z I = 1). Show that for u c- Co (X), the following limit exists

Au(x) = lim

J

K(x, x-y) u(y)dy,

(3.52)

i0 Ix-Ylit defining a `I'DO A E CL°(X). The operator A, defined by (3.52) (under condition (3.51)), is called a singular

integral operator. We see that such operators are just special cases of Y'DO. Remark. The solution of Problem 3.2 can be found in one of the works of Kumano-go [1 ]-[3], and the solution of Problem 3.3 can be extracted from the book of Mihlin [1 ]. The solutions of these problems are rather laborious but very useful for understanding 'PDO theory.

§4. Change of Variables and Pseudodifferential Operators on Manifolds 4.1 The action of change of variables on a `PDO. Given a diffeomorphism x: X - X, from one open set X c IR" onto another open set X, C 1R", the induced transformation x': C' taking a function u to the function u o x, is an isomorphism and transforms Co (X,) into Co (X). Let A be a `PDO on X

and define A,: Co

with the help of the commutative diagram

CO (X) A Cs (X) x-J Co_(X,)

A.

.

i-C_(XI)

If xt = x-', then A, u = [A (u o x)] o xl .

(4.1)

Let A be given by (2.18), then

A,u(x) = J

Jei(xI(x)-Y) c a(x,(x),

y = x, (z), we obtain

A,u(x)=

Idetxi(z)Iu(z)dzdd,

(4.2)

where x, is the Jacobi matrix of the transformations xi. It follows from this that A, is a FIO with phase function d' (x, y, ) _ (x, (x) - x1 (y)) 0. We will show, that for 1 - Q < S < q the operator A, is a `I'DO. This fact can be obviously derived from the following more general theorem.

Chapter I. Foundations of `I'DO Theory

32

Theorem 4.1. Let 0 be a phase function in X x X x IR", such that 1) P (x, y, 0) is linear in 0; 2) 0,'(x, y, 0) = 0'e' x = y.

Let A, be a FIO with phase function 4>(x, y, 0) and a(x, y, 0) E Saa (X x X x IR") (cf. formula (2.1)), where

1-QSS
(4.3)

Then A, e LQ, j (X).

For the proof we need Lemma 4.1. Let the phase function 'P satisfy conditions 1) and 2) of theorem 4.1. Then there exists a neighbourhood S2 of the diagonal d C X x X and a COO-map i/r : S2 -+ GL(n, IR) (non-degenerate matrix-function 1`r(x, y)), such that (4.4) (x, y) E S2, 'P(x,Y, W (x,Y)c) _ (x-y) - , where

det'P,(x,y,0)1,,=x= 1.

detyr(x,x)

(4.5)

Proof. We have

0(x,Y,0)

.0j(x,Y)0;,

(4.6)

where 0; (x, x) = 0 and if 'P (x, y) = 0, j = 1 , 2, ... , n, then x = y. Further "

ox.e =

;i ax,

0;,

..., fi

a4s;

ax"

0,, 4s, ,

... 10

Note that differentiation of the relation 45 (x, x, 0) = 0 with respect to x shows that 'P;, (x =,, = - Px 1 x s,. Now, by definition of the phase function 4sx , e * 0 for 0 * 0, so in order that 0#'(x, x, 0) = 0, it is necessary that

'P (x, x, 0) $ 0, i.e. for arbitrary 0 + 0 there exists k, 16 k S n, such that "

a0,

l

axk x=y

0 * 0.

It follows that det

(a'; (x, Y) (4.7)

``

axk

J

By the Hadamard Lemma we have O; (x, Y)

4sk; (x, Y) (xk - Y0, k=1

§4. Change of Variables and Pseudodifferential Operators on Manifolds

33

for close x and y, Pkj E C°° (ST), ST some neighbourhood of the diagonal in

X x X and (x, x) _

aoj (x, Y) (4.8)

(0xk

1.=y

Denoting by 0(x, y) the matrix (Pkj(x, y))_ , we see from (4.7) and (4.8) that there exists a neighbourhood 0 of the diagonal in Ix X such that det 41(x, y) 4 0 for (x, y) E Q. Put W (x, Y) = 0(x,Y)-1.

(4.9)

Since 45 (x,Y,0) _

j.k=1

-Oki(x,Y) ©j(xk-Yk) = (x-Y) - (4 (x,Y)0),

and putting t1 (x, y) 9 = , we clearly obtain (4.4). The formula (4.5) follows from (4.8) and (4.9). Proof. of Theorem 4.1. In view of Proposition 2.1 and Exercise 2.4 we may assume that A is given by (2.1), where a (x, y, 0) = 0 for (x, y) t 0' with Q' any neighbourhood of the diagonal. Making the change of variables 0 = W (x, y) in the integral (2.1), we obtain

A1u(x)=

IdetW(x,Y)I u(Y)dydl.

(4.10)

It remains only to remark that for a, (x, y, ) = a (x, y, W (x, y) ) condition (4.3) guarantees that a1 (x, y, ) E SQ a (X x X x lR"), in view of Lemma 1.2. El

4.2 Formulae for transformations of symbols

Theorem 4.2. Given a diffeomorphism x: I-' X1 and a properly supported `PDO A E Lo 8 (X) with I - Q 5 8 < g, let A l be determined by (4.1). Then Q.1,(Y, 1)ly=x(x)^ L

a

a

(x, ) = 8'aA(x, ) and xx (z) is given by

xx (z) = x (z) - x (x) - x'(x) (z - X).

(4.12)

Proof. Note first of all that the function x, (z) has a zero of second order for z = x. Therefore, denoting

0.(x,n)=D=elx;(=) "I:=x,

(4.13)

Chapter 1. Foundations of'VDO Theory

34

we have that 4>a (x, q) is a polynomial in q of degree no higher than I a 1/2. Taking

Lemma 1.2 into consideration, we see that vI

=

S' -(e-1/2)121 (XxIR").

But from the condition I - p < b < p it follows that p > 1/2, so the asymptotic sum (4.11) is well defined. To prove formula (4.11), we utilize formula (4.2) with a (x, y, 0) = a,, (x, 0). Using the transformation described in the proof of Theorem 4.1, we get A, u (x) = J J e' -r) " aA (xi (x), W (x, Y) q) D (x, Y) u (y) dy dq ,

(4.14)

where D (x,),) = I det x, (x) I I det' (x, y) 1. By Theorem 3.1, we have QA,(Xl q) - Y- aMDy [aA(xi (x), W (x,Y)q)D(x,Y)]/

!

I

r=..

(4.15)

a

From the terms with multi-index a, we obtain (before substituting x = y) a sum of terms of the form c(x,Y) 01'0'A

W (x,Y)q),

(4.16)

where c (x, y) depends only on the diffeomorphism (but not on A). For the multiindices y and Q in (4.16) we have the estimates I13I < 21a1,

(4.17)

I'I+IalsIll.

(4.18)

Here, (4.17) is obvious and (4.18) follows from the fact that applying Dy to expressions of the type (4.16) does not change I Q I - I Y I and 0, increases I /31 - I Y I

by 1.

From (4.17) and (4.18) we have

II 11l-lal: 11l-1/31/2=1/31/2.

(4.19)

Note now, that applying (4.5) to 0 (x, y, 0) = (x, (x) - x, (y)) 0 gives w (x, x) = (`x, (x))-'. Also, rearranging in (4.15) the terms of the form (4.16), collecting together all the terms with the same /3, we obtain

aT'(xi(x), (`xiWY `q) Pp(x,q)//3!,

aA,(x,1)

(4.20)

9

where'P0(x, q) is a polynomial in q of degree no higher than 1131/2 (with C°°(X,)-

coefficients) and independent of A. Here To

1.

§4. Change of Variables and Pseudodifferential Operators on Manifolds

35

Replacing x by x(x) in (4.20) we easily obtain the equivalent formula aA,(X(x),n)

Q°'(x> 5'(x),)

(4.21)

A

where 'P (x, n) is a polynomial inn of degree no higher than 1#1/2 (with C°° (X)coefficients) independent of A and where 4P0 - 1. It remains to show that these polynomials are given by (4.13).

We will compute the polynomials P (x, q) with the help of differential operators. For the differential operator A we have iy' q

ly=x(x) CA,(Y,q)ly=x(x)-e = e-ix(=)'" aA(z D:) eix(x) "I 1:=x A)ely."

(4.22)

(here a, (z, D:) denotes the operator A, acting on the variable z). We write now x (Z) = X (x) + X' (x) (z - x) + Xi (Z),

from which

(z) q-x'`X'(x)n.

X(z)

Putting this into formula (4.22), we obtain 0 A, (Y, i) l y=x(x) = e-' ',(x), {aA (Z, Dz) eis 'x'(x)q eix;(Z) q,l I. -x

(4.23)

Now use the Leibniz rule (3.46) (Exercise 3.6) to differentiate the product of two exponents in (4.23). We then obtain clearly

aA,(X(x),n)=Y-

1

aA(a)

(x,fX'(x)n)'Da

a)

:eix:(:)

I==x

(4.24)

(we have used here yet another obvious formula for differentiating a linear exponent : aA (z, D:) a i= - c =

e11-4

aA (z, c))

Formula (4.24) signifies the validity for differential operators of (4.13) for

the polynomials P] (x, q) in (4.21). But in view of the universality of the polynomials 0a(x,n), then (4.13) is valid also in the general case. Examples. 00 = 1, 4i,=O for I P 1= 1, bg (x, )7) = Dr (i x (x) n) for I Ii I = 2. Corollary 4.1.

0A,(Y,n) - aA(XI(Y), (N

i(Y))-ln) ESQ a

z(o-)lz)(X1 xlR").

(4.25)

This statement shows, that modulo symbols of lower order, the symbols of all operators obtained from A by a change of variables form a well-defined function on the cotangent bundle TX.

36

Chapter I. Foundations of `l'DO Theory

Corollary 4.2. If A E CL"(X ), then A, e CL'" (X, ).

Proof. Obvious from formula (4.11).

4.3 Pseudodifferential operators on a manifold. Let M be a smooth ndimensional manifold (of class C"'). We will denote by C' (M) and Co (M) the space of all smooth complex-valued functions on M and the subspace of all functions with compact support respectively. Assume that we are given a linear operator

A: C(M)-C(M). If X is some chart in M (not necessarily connected) and x: X -. X, its diffeomorphism onto an open set X, c 1R", then let A 1 be defined by the diagram

C( (X)

'' i C°°(X)

CC (X,)

"-' -C"(X1)

(note, in the upper row is the operator ra.o A o ia., where ix. is the natural embedding iX: Co (X) - Co (M) and rx is the natural restriction rX: C"(M) - C' (X); for brevity we denote this operator by the same letter A as the original operator).

Definition 4.1. An operator A: C o (M) -. C°` (M) is called a pseudodifferential operator on M if for any chart diffeomorphism x: X - X, the operator A I defined above is a `'DO on X 1.

Theorem 4.1 shows that the `PDO on an open set X C IR" for 1-Q 3 < Q are `PDO on the manifold X. Furthermore, from Lemma 1.2, we see that the class of symbols SQ a(T*M), as well as the class of operators LQ,,,(M), are well-defined for I - o <_ 6 < o, and Lemma 4.1 shows that the principal symbol is well-defined as an element of the S"'.,(T*M)/S'"-2,Q-1/2)(T*M). quotient space Q. Q. Also, in view of Theorem 4.2 the class of classical `PDO CL'"(M) is welldefined on M. If A E CL"(M), then the principal symbol of A can be considered as a homogenous function a,, (x, c) on T*M with degree of homogeneity equal to m, since two functions a, (x, ) and a2 (x,1;), positively homogeneous in for I I z I of degree m, which define the same equivalence class in Sm(X x 1R") modulo S' -1(X x 1R"), must coincide for I I >_ 1. In conclusion note, that it is essential to allow the use of non-connected charts in definition 4.1, since otherwise we would have to consider the reflection f (x) -. f(- x) in C'0 (IRS {0}) as a pseudo-differential operator.

§4. Change of Variables and Pseudodifferential Operators on Manifolds

37

Exercise 4.1. Show that a `PDO A on a manifold M can be extended by continuity to a mapping

A: 4f'(M) - 9'(M) where t' (M) and 9' (M) denote the spaces dual to the spaces of smooth sections and smooth sections with compact support respectively of the line bundle of densities I A"(T*M) I. (This bundle can be defined for instance, by choosing a covering of M by charts, regarding the bundle as trivial on each chart and setting

the transition functions equal to the absolute values of the Jacobians of the coordinate transformations. The sections of the density bundle can be integrated on the manifold, which cannot be said of exterior n-forms, where one needs an orientation, i.e. essentially an isomorphism between A"(T*M) and I A"(T*M) 1. If we fixed a smooth positive density on M, then this gives us an isomorphism of the bundle I A"(T*M) I and M x 1R', which allows us to consider functions as densities and therefore to view the elements in if'(M) and 9'(M) as functionals on functions.)

The inclusion C'(M)c- 9'(M), inducing the inclusion Ca (M)- or'(M), is defined in a natural way by the formula
(4.26)

M

where u e C'°(M) and (p is a smooth density with compact support (so that u

(P

is also a smooth density with compact support). Verify the property of pseudolocality for the operator A. it:

Exercise 4.2. Let E and F be smooth vector bundles on the manifold M; let be the natural projection; n*Eand n*Fthe induced vector bundles

over T'M. Define a `PDO A : Ca (M, E) -+ C°°(M, F) (Co (M, E) the space of smooth compactly supported sections of E and C°°(M, F) that of the smooth sections of F) and show that its principal symbol is a well-defined element in the space SQ e(Hom(n*E, 1t*F))/SQ a

2 (a-112) (Hom(n*E,n*F)).

Problem 4.1. Let A be a differential operator of order m on a manifold M (an operator A: C o (M) -+ Co (M), such that any operator A 1: Co (X1) -' Co' (X1), as defined before, is a differential operator of order m). Give an invariant definition of the principal symbol a function on T*M which is a homogeneous polynomial of order m in (i.e. along the fibres).

Hint. Use the formula lim

2q. eC°°(M).

(4.27)

Chapter I. Foundations of'I'DO Theory

38

Problem 4.2. Compute the principal symbol of the (de Rham) exterior differentiation operator: d: AD (M) - i1v+' (M) ,

where A"(M) denotes the space of smooth exterior k-forms on M (k = 0, .. , n).

Problem 4.3. Prove that the one-dimensional singular integral operator on

a smooth closed curve r c c Au(t) = a(t) u(t) + lim

!

`.,p It-alas

K(t, T)

t-T

u(T)dr,

t, T er,

for a(t) E C110 (F), K(t, T) E Co (Jr x r), is a classical `PDO and belongs to

the class CL°(r).

§5. Hypoellipticity and Ellipticity 5.1 Definition of hypoelliptic symbols, operators and examples

Definition 5.1. A function a (x, ) e C Z (X x IR"), where X is an open set in IR", is called a hypoelliptic symbol if the following conditions are fulfilled: a) there exist real numbers m(, and m, such that for an arbitrary compact set

Kc X one can find positive constants R, C, and C2 such that C2

IfI ? R, xeK;

(5.1)

b) there exist numbers P and 6, with 0:5;6 < Q 51, and for each compact set K c X a constant R such that for any multi-indices a and fi

IfI ? R, xeK,

(5.2)

with some constant C,,B, K . Denote by HSQ a °(X x IR") the class of symbols satisfying (5.1) and (5.2) for

fixed m, m0, p and 6. Sometimes we will denote this space simply by HS,',"*, if the domain X is obvious (or irrelevant). From (5.1) and (5.2) it obviously follows that

HSQ',°(XxJR")cSQ, (XxIR").

We will denote by H La a ° (X ), X open, the class of properly supported `PDOA for which CA(x,t;) E HSQa °(X x IR").

§5. Hypoellipticity and Ellipticity

39

Definition 5.2. A 'PDO A is called hypoelliptic if there exists a prop-

erly supported 'PDO Al E HL" °(X) such that A = A, + R,, where R, E L-°°(X), i.e. R, is an operator with infinitely differentiable kernel.

Note, for any representation of the hypoelliptic operator A in the form A = A, + R, , where A, is a properly supported PDO and R, is an operator with smooth kernel, it is true that A, e HLQ; a '(X).

Example 5.1. Let A be a differential operator, i.e. A = aa(X)D°, with lalsm a" e C°°(X). Denote by am(x, ) the principal symbol

am (x, ) _ Y a(x).

(5.3)

Iat=m

Definition 5.3. A differential operator A is called elliptic, if

am(x, ) + 0

(x, ) E X x (IR"\0) .

for

(5.4)

Proposition 5.1. The following conditions are equivalent for a differential operator A: a) A is elliptic;

b) AEHUI;o(X). Proof. The implication b) a) is obvious. To show the converse implication, we introduce the complete symbol of the operator A

a (x, ) = Y_ a(x) `

(5.5)

IsI sm

and notice that, if a) is fulfilled, then a(x, ) = 1

am (x, )

where the functions b _j (x, ) a C°° (X x (R"\ 0)) are homogeneous in 4 of degree

-j. From this (5.1) follows with m = mo and (5.2) is obtained similarly. O Examples of elliptic operators:

the Laplace operator

d=

a2

a2

in iI";

ax + ... + ax^

the Cauchy-Riemann operator

a a2

=

a

ax,

+i

a axe

Example 5.2. The heat operator a

a

at-d=at-a

a2

a2

-

-ax2

is hypoelliptic in 1R"+', although it is not elliptic.

in 1R2.

40

Chapter 1. Foundations of `l'DO Theory

Exercise 5.1. Verify the hypoellipticity of the heat operator and find the corresponding in, mo, p, S. Example 5.3. Let A be a classical `l'DO with principal symbol a," (x, ). Then the following definition makes sense.

Definition 53'. An operator A E CL'"(X) is called elliptic, if its principal symbol a. (x, ,) satisfies condition (5.4).

As in the proof of Proposition 5.1 it is easy to verify that if A E CL"(X) is properly supported then its ellipticity is equivalent to the inclusion A E HL" '(X). (X). Generally, A E CLm(X) is elliptic if and only if A = At + R where AI E HL" L",0" and R E L'°°(X). Examples 5.1 and 5.3 motivate the following

Definition 5.3". An operator A E Lma(X) is called elliptic if A = AI + R where Al E HLQa (X) and R E L

Proposition 5.1'. For a properly supported A E L',,,(X) (X) to be elliptic it is necessary and sufficient that the condition a) in Definition 5.1 is satisfied for

its symbol with mo = m. Generally A E L(X) is elliptic if and only if A = A I + R where A 1 is properly supported and the condition a) in Definition 5.1 is satisfied for the symbol of A 1 with mo = m. In this case this is true for any presentation A = AI + R as above.

Proof. The proof is left to the reader as an excercise.

0

5.2 Basic properties of hypoelliptic symbols. First of all, say that a (x, ) E SQ a for large , if for any compact set K c X there is an R = R (K), such that v (x, ) is defined for x e K, I I >_ R (K) and for these (x, c) all the necessary estimates of type (1.10) are fulfilled. If, in addition, the estimates (5.1) and (5.2) are fulfilled, we say that a (x, ) E HSQ "0 for large 4. Note that if a (x, ) belongs to SQa or HS','- for large then, multiplying by a smooth cut-off function Vi (x, ), equal to 1 "for large " (e. g. for x e K, I I z R(K) + 2 for any compact set K) and equal to 0 in a neighbourhood of the set where the symbol a is not defined (e.g. for x e K, I t I < R (K) + 1), we obtain a symbol a1(x, ) E SQa(X x IR°) or HSQa °(X x IR") respectively, which coincides with a(x, ) "for large ". Lemma 5.1. If a (x, ) E HSQ o jor large 4, then a -' (x, c) E HS,-'-' a -'for a multi-indices a, ft we have for large large . Furthermore, for arbitrary , that a240!a (x, )/a (x, ) ESQ.8I"1 +aIBI

Proof. Let y = (x, ) and y, S be 2n-dimensional multi-indices. By induction in 16 1 one verifies that

§5. Hypoellipticity and Ellipticity

_

E1aYa(Y)

a(Y)

161

ca,...a, k

41

as°Q(Y) a(Y)

k

a6a(Y)

(5.6)

rn a(y)

Obviously, setting y = (J1, a) we obtain from the definitions all the necessary estimates for the proof of the lemma. El Lemma 5.2. If or' e HSQ ,

and a" E HS, 7j "; then a' o a" a HSQ .g

Proof. Direct from the Leibniz rule.

+

;l

Lemma 5.3. If a (x, ) E HS,'-'"- and r (x, ) e SQ -j, where m, < mo , then

a + r a (1 + r/a) and using Lemmas 5.1 and 5.2, we see that it suffices to consider the case or = I and mo = m = 0, i.e. m, < 0. But then the assertion of the lemma is trivial. Lemma 5.4. Let a (x, 4) E HS,,- e ' for large and let a, (y, q) = a (x (y), (y, q)), where the map (y, rl) --+ (x(y), (x, q)) is a Coo map from X I x (NV \ 0) into X x (1W' \ 0) and where 4(y, rl) is positive homogeneous of degree 1 in rl. Assume that 1-Q < S < Q. Then a, (y, rl) E HS."" ° for large >l.

Proof. Completely analogous to the proof of Lemma 1.2 and is left to the reader. 5.3 Basic properties of hypoelliptic operators Proposition 5.2.

If

A' c- HLQ a°'a (X)

and

A" e HL-,,. "y"; (X),

then

A' o A" E HLQ. j mm"-;+,m°(X)

Proof. By theorem 3.4 aA'A-(x, 0 - aA'(x,

1 + Y

)

L

12121

ark

aA'

and from Lemmas 5.1 and 5.3, we see that the series in square brackets is an asymptotic sum which (in the sense of Proposition 3.4) belongs to HSQ a° . It remains to use Lemma 5.2. Proposition 53. If A E HLQ:6'(X), then both 'A and A* belong to HE,", e o(X).

Proof. Similar to the proof of the preceeding proposition. Proposition 5.4. If A E HLQ;; ^(r) and R E V," ,-,(X) with m, < ma, where R is properly supported, then A+ R EHLQ:a o(X).

Proof. The statement follows immediately from Lemma 5.3. Proposition 5.5. If I - L q< S < e, then HL!,":'-(X) is invariant with respect to

changes of variables i.e. if we are given a diffeomorphism x: X-+ X, and an operator A 1 is defined as in §4 (formula (4.1)), then A I E H L'"'"'0 (X ).

42

Chapter 1. Foundations of TDO Theory

Proof. By Lemma 5.4 0' .4 (X I (y), ('X; (y))-

-" (X)

(here x, = x-1). But then, by Theorem 4.2 and Lemma 5.1, it is obvious that QA, (y, q) = QA (x1 (y), (`x'1 (y))-' q) (1 +r(y, 1)), where r (y, q) e SQ. 2(e -' n) for large q. The assertion of the proposition now a follows from Lemmas 5.2 and 5.3.

O

Proposition 5.5 allows us to define the class HL" °(M) of hypo-elliptic TDO to an arbitrary manifold M provided I - Q = S < Q. 5.4 The parametrix and the rough regularity theorem Theorem 5.1. Let A e HLQ; a "(M), with either 1 - Q = S < Q or b < Q and M a

such that

domain in W. Then there exists an operator BEHLQ

BA = I + R1,

AB = I + R2,

(5.7)

where Rt e L- ' (M), j = 1, 2, and I is the identity operator. If, furthermore, B' is which either B'A = I + R; or AB' = I + Rz (where

another TDO for

R,'E L- (M)), then B'- BE L-'O (M). Corollary 5.1. If A is a hypoelliptic 1PDO on M (not necessarily properly supported), then there exists a properly supported TDO B, such that (5.7) holds.

Proof of Theorem 5.1. First let M be a domain X in IR" and aA the symbol of

A. Consider a function bo (x,) E HSQ e -'"(X x IR") such that bo (,r, )

= a,' (x, ) for large . Next choose a properly supported operator B E HLQ - -'"(X) such that o80 - bo c- S -'° (X x IR"). Let us verify that BOA = 1+ RO,

(5.8)

with RO a LQ tg-°'(X).

In fact, by Theorem 3.4 we have for large

a,A(x, )^ 1+I2IY l al o

1

i

a 'DxaA=1-1

D.QA

1

1x121 at

a;'

6A

and it remains to use Lemma 5.1. Now let Co be a properly supported PDO such that CO

CO

Y J=O

i. e.

§5. Hypoellipticity and Ellipticity

43

X

oc.

Y (-1)j j=0

From (5.9) we clearly have

C0(I+Ro) - IEL', so that putting B, = C0B0, we have

B,A=1+R,

(5.10)

where R, eL-°°(X). From the construction it is clear that B, E HLQ sy -"'(X). Further, we can similarly construct an operator B2 EHLQ b^ -"'(X), such that A B2 = I+ R2

(5.11)

where R, a L- °° M. Let us now verify, that if B, and B2 are two arbitrary `FDO, for which (5.10)

and (5.11) hold, then B, - B2 E L-°°(X). This will then also demonstrate the existence of the required B (for which we may take either of the operators B, and B2) and its uniqueness (modulo L- `° (X)). Note, that B, and B. can be taken to be properly supported. Multiplying (5.10) on the right by B2 and using (5.11), we obtain B, - B2 = R, B2 - B, R2 and it only remians to note that B, R2 and R, B2 both belong to L- `° (X). Now let M = U XY be an arbitrary manifold with a covering by charts X7. Then, (by the results just shown) there is a properly supported operator BY in X",

such that

B'-A=I+R",

A- B''=I+R2,

where R; and RI are operators with smooth kernels. Now let (pj, j = 1, 2, ..., be a partition of unity subordinate to the covering of M by the X. This means that the following conditions are fulfilled: 1) (pi E Co (M), Qj > 0, suppcpj C XY for some y = y(j); 2) for any x E M, there exists a neighbourhood IM, of x in M, such that W, intersects only a finite number of sets supp Vj;

3)(pi=1.

(See e.g. Theorem 6.20 in Rudin [1].) Now let us construct functions tfj E Co (M) such that they still satisfy the conditions 1) and 2) above (with the same y(j)) and in addition if/j = 1 in a neighbourhood of supp pj. Denote by Oj and Wj the multiplication operators by fpj and >lij respectively. We set then

B = E OjBY(j),Iii

Chapter I. Foundations of `PDO Theory

44

where necessary operations of restriction and extension by zero are understood. We claim that then B satisfies all the required conditions. Clearly B is properly supported. Note also that on the intersection X Y tl X Y' the operators BY and BY' differ by an operator with a smooth kernel, so modulo operators from L-00 they may be given by the same symbol. This allows us to

calculate the compositions BA and AB modulo L-°° using the composition formula (3.41). For example, the symbol of BA will locally have the form aBA(x, )

a O;(x)(a' ae(x, ))D«(aA(x, )P;(x))

If we apply a derivative in x to P;, then the resulting term will vanish because 45; D; WL; - 0 for any a 0 0. Therefore we conclude that aBA(x,

'Pi(x)y;(x)

which is equivalent to the first relation in (5.7). The proof of the second relation is not different. O

Remark. The formula for the parametrix B above can be simplified if we add small neighbourhoods of supp rp; to the set of all XY. In this case we can simply write

B=E Y

where (py satisfy the same properties as (pj above and it is understood that some of the functions co can be identically 0. If M is a closed manifold then we can also assume the covering (XY) to be finite, so the formula for B will be a finite sum.

Definition 5.4. An operator B satisfying the condition (5.7) is called a parameirix of the operator A. Corollary 5.2. Any elliptic operator AeLQ,a(M) has a parametrix BeHLQ

a hypoelliptic 1'DO, then sing supp Au = sing supp u,

u e e' (M) .

(5.12)

In other words, if S2 is an open submanifold of M, then Au JQ e C'° (Q) if and only if u JOEC'(S2).

If A is a properly supported hypoelliptic `YDO, then (5.12) is true for an arbitrary distribution u E 2'(M).

§5. Hypoellipticity and Ellipticity

45

Proof. It obviously suffices to prove the first part of the theorem. For this, the inclusion sing supp Au c sing supp u follows from the pseudolocality of A (cf.

Proposition 2.4) and it only remains to show the inclusion sing supp u c sing supp Au

(5.13)

Let B be a properly supported parametrix of A. Then, from the formula u = B(Au) - R,u and the pseudolocality of B, it follows that sing supp u c sing supp (Au) using supp R, u, and since R, u e C°° (M), we have that sing supp R, u = O proving (5.13).

Theorem 5.2 is a rough regularity theorem for solutions of hypoelliptic equations of the form Au = f More precise theorems will be proved after we have introduced exact regularity classes of functions, i.e. the scale of Sobolev spaces.

5.5 A parametrix for classical elliptic pseudo-differential operators. In this case a parametrix can be constructed in a much more explicit way. Let A be a classical'PDO in an open set X C IR", whose symbol for large r; admits the asymptotic expansion z x

a(x,S)

j=o

(5.14)

a. (x, S),

(x, ) E C°° (X x (IR" \0)), a,"_; is positive homogeneous of degree where m - j in l and also satisfies the ellipticity condition (5.4). Let B be a parametrix of A. We will show that B is a classical'PDO, whose symbol b(x, ) for large admits an asymptotic expansion

b-m-i(x,5),

b(x,

(5.15)

J=O

where b

(x, ) e C`° (X x (R"\0)) and b

_j (x, ) is positive homogeneous

of degree - m -j in g. The composition formula shows that the symbol b (x, ) must satisfy the condition 1, or 1 .

k.i

(5.16)

Chapter 1. Foundations of'PDO Theory

46

Clearly, regrouping the terms in (5.16) by their homogeneity degree we make (5.16) into just an equality of the corresponding homogeneous components, i.e.

E

tam-k(x,) Dxb-m-i(x,)l«! = So,

p = 0, 1, ....

(5.17)

k+j+lal-D

or, more explicitly am a,,,

(a*gam-k)

(DXb_._1)/a! = 0,

j = 1, 2, ... (5.17")

k+l+lal=j '<j

Clearly, from (5.17) the functions b_.-;(x, 4), positive homogeneous of de-

grees -m - j (j = 0, 1, ...), are uniquely defined. If we now define b(x, ) by (5.15) and find a properly supported'PDO B such that aB (x, b(x, ) E S-°°(X x IR"), then this operator B is a parametrix of A. Formula (5.17) defines a parametrix of A also in the case when A is a matrix PDO: in this case are square matrix functions and the ellipticity condition takes the form (x, ) E (X x (IR" \ 0)

det am (x, ) $ 0,

(5.18)

Problem 5.1. Show that the terms b _ m _, (x, ) (j > 0) in the parametrix of the classical elliptic operator A in the scalar case can be expressed via am -,t (x, 4) by 2j+1

,

b_m-j(x, ) = Y c,(x, ) (am(x,

(5.19)

1=2

where c, (x, ) is a function positive homogenous of degree m (1-1) -j in , polynomial in the functions am , a.-,, ..., am _; and their derivatives of order <j. The analogous formula in the matrix case is of the form 2j+1

b- m -; (x, ) =

(x, )

1

fl (ck., (x, S) a;' (x, )]

1=2 k=1

.

(5.20)

§6. Theorems on Boundedness and Compactness of Pseudodifferential Operators 6.1 Formulation of the basic boundedness theorem. Let A be a'PDO in IR". Consider A as a map A : Co (lR") - C'° (lR") .

§6. Theorems on Boundedness and Compactness of Pseudodifferential Operators 47

Let K,, be the kernel of A in the sense of L. Schwartz. If supp IC4 , is compact in

IR" x IIt", then A defines a map A : Co (IR") - Co (R") .

Is it possible to extend the operator A to a continuous linear operator A : L2 (IR")

L2 (1R") ?

Clearly, this is so if and only if the following estimate holds IIAuII

CIlull,

uECo (IR"),

(6.1)

where C> 0 does not depend on u and II ' II denotes the norm in L2(IR") Theorem 6.1. Let A E LO O's (IR" ), 0

3 < Q < 1. and let supp KA be

compact in IR" x HR". Then (6.1) holds and A can be extended to a linear continuous operator on L' (UV). 6.2 Auxiliary results and proof of Theorem 6.1. In the sequel we will use the notation lim I a(x, ) I = lim sup I a (x, ) 1.

- Ici?t xeK

i
Theorem 6.2. Let A be a properly supported `YDO in LQ a (X), with 0 E < Q < 1 and X an open set in R". Suppose there exists a constant M such that lim

M

(6.2)

ui-W xeK

for any compact set K C X. Then there exists a properly supported integral operator R with hermitean kernel R E C°° (X x X), such that

(Au,Au)<M2(u,u)+(Ru,u),

ueC0 (X).

(6.3)

If, in addition, supp K,, is compact in X x X, then supp R is also compact in X x X.

Proof that Theorem 6.2 . Theorem 6.1. It suffices to show the boundedness

in L2(IR") of an operator R with smooth compactly supported kernel. This, however is well known (one can show for instance that 11R112 < f I KR (x, y)12

dxdy, where IIRII is the operator norm of R in L2(IR") and KR (x, y) is the kernel of the operator R). 0

To prove Theorem 6.2 it suffices, in view of the relation (Au, Au) _ (A" Au, u), to construct a properly supported operator B E LO a such that

A*A + B*B - M2 = R

(6.4)

48

Chapter I. Foundations of `1'DO Theory

where R has a smooth kernel (in which case R is properly supported since the left hand side of (6.4) is a properly supported operator). Rewriting (6.4) in the form B*B = M2 - A*A + R, we note that the symbol of M2 - A*A is equal to M2 - Ia., (x,4)12 modulo symbols of class SQ,60-8 (X), from which we infer lim_ Re [a(,a,_4.A) (x, 0] > 0

(6.5)

iti- W xeK

for an arbitrary compact set K c X. Therefore we derive Theorem 6.2 from the following proposition. Proposition 6.1. Let C e LQ, a (X) and be properly supported, 0 5 6< Q 5 I and let C* = C and assume lim Re ac(x, ) > 0 -fEK

for arbitrary compact sets K c X. Then there exists a properly supported operator

B E LQ4(X) such that R = B*B - C has a C°° kernel. Lemma 6.1. Let a (x, ) a SQ a (X x IR") and let a (x, c), for arbitrary (x, ) e X x IR", take values in a compact set K c C. Let a complex-valued junction

f(z) be defined on a neighbourhood of K and be infinitely differentiable as a function of two real variables Re z and Im z. Then

f (a (x, )) E SQS(X X M')

(6.7)

Proof.a Denote u = Rez and v = Im z. Then we evidently have vyJ (a(y)) =

Y_

x Op (Rea) ... ay.(Rea) from which (6.7) follows, since I (a;pvg1 f) (a (y))1:5 Cpq .

(a(y)) x

a;.(Ima),

(6.8)

1-j

Proof of Proposition 6.1. It follows from Lemma 6.1 that j/Re ac (x, ) belongs to SQ,,, for large . Therefore there exists a properly supported WM Bo E L°q a(X), such that if bo (x, ) is its symbol then

From this it follows that C - Bo B0 a L; (61? -6)(X).

(6.9)

The operator Bo serves as the "zero order approximation" to B. We will seek a first order approximation in the form Bo + B, , where B, e L.1 °"(X).

§6. Theorems on Boundedness and Compactness of Pseudodifferential Operators 49

We have

C- (Bo + Bi) (Bo + B,) = (C- Ba Bo) - (Bo B, + B* Bo) - Bi B,

.

(6.10)

The point is to reduce the order of the operator on the left-hand side, taking B1 to be properly supported with symbol b1 (x, ), such that for large

2b, (x, ) bo(x,) = ac_(x,),

(6.11)

which is obviously possible, since by Lemma 6.1 bo '(x, ) E S.O. a for large . It follows from (6.10) and (6.11), that

C-(Bo+B,)* (Bo+B,)EL..(6.12) Arguing by induction, we may in exactly the same way construct properly

supported `PDO B; E L(X), j = 0, 1, 2, ..., such that C - (Bo+ ... +B,)* (Bo+ ... +B;) ELQ aQ-s'(X).

(6.13)

Now let b1(x, ) be the symbol of B3 . It only remains to construct a properly

supported operator B, such that Y b, (x, ) i=o

It follows easily from (6.13), that this operator will be the one we are looking for.

Thus Proposition 6.1 is proved and together with it Theorems 6.1 and 6.2. D

6.3 The compactness theorem. We will derive the compactness theorem from the following much more general statement. Theorem 6.3. Let A E LQ,a(IR"), 0:!9 6 < q:5 1, let the kernel K,, have compact

support in IV x IR" and let the symbol a,, (x, ) satisfy I'M I a,, (x, c) I < M'

(6.14)

Then there exists an operator A, such that A - A, E L- (IR"), the kernel K,, has compact support and

IIA,uII S MIIuII,

ueCO (R").

(6.15)

P r o o f. Let X e C o (IR) be such that X (x) ? 0, f X (x) dx = 1, 0 < X 1. Such a function can be found. Indeed, to begin with let the function Xo (x) be such that Xo (x) a C o (IR"), Xo (x) z 0 and J Xo (x) d x = 1. Then obviously I Xo 1. Put now X (x) = f Xo (x+ y) Xo (y) dy. In view of the fact that j (e) = 110 ( ) I Z the I

function X (x) fulfills all the requirements.

50

Chapter I. Foundations of `YDO Theory

Now put xL(x) = E-"x(x/E) and define the operator AL by (6.16)

AL u = Au - A (xL * u),

where (xL * u) (x), the convolution of xL and u, is defined by

(XL*u)(x)= 1x.(x-y) u(y)dy= ju(x-y) X.(y)dy Now, in view of Theorem 6.2

IIALuII2 s M2IIu-xL*u112+ (R(u-xL*u), u-xL*u),

(6.17)

where R is an operator with kernel R (x, y) e Co (IR" x Ilk"). Note that the Fourier transform of u - xL * u is (1- 2 (En)) u ( ) and from the condition 0 < X < 1, it follows that

IIu-xL*ull < (lull.

(6.18)

Further, denote by RL the operator which maps u into R (u - xL * u), then its kernel is given by the formula

RL(x,y)= R(x,y) - IR(x,z)

E)

or

RL (x, y) = R (x, y) - I R (x, y + t z) x (z) dz,

from which it is obvious, that supp RL (x, y) lies in some fixed compact set K (independent of z for 0 < &:5 1) and, in addition, for

sup X. Y

It follows that II RL II -i 0 for a - 0. We now obtain from (6.17) and (6.18) that IIALu112 < M2IIu112+

IIRLUII (lull.

(6.19)

From the conditions of the theorem it is evident that we may replace M by M - 6, where b is sufficiently small. But then it follows from (6.19), that for sufficiently small z > 0 11ALuII2 s M211u112

Put A, = AL . Since the symbol of the convolution operator with xL is (Ed) e S - (1R" x IR"), it is evident that A - A, a L- W (IR"). It is also easily

verified, that the kernel KA of A, has compact support.

§6. Theorems on Boundedness and Compactness of Pseudodifferential Operators 51

Theorem 6.4. Let A e LQ (IR"), 0 <_ b < Q < 1, let the kernel KA have compact support and

sup la4(x,c)I-0

as

ICI -s +oo.

(6.20)

X

Then A extends to a compact operator in L2(IR").

Proof. By Theorem 6.3 there exists a decomposition A = A, + R, for arbitrary e > 0, where II Ar II < e and Re has a smooth compactly supported kernel

(and is thus compact). Therefore lim II A - R, II = 0 and the compactness of A follows.

r"0

Corollary 6.1. Let A E L,", j (IR"), 0 5 b < e5 1, m < 0 and K4 have compact support. Then A extends to a compact operator on L2 (IV).

6.4 The case of operators on a manifold. Consider first the case of a closed manifold M (a compact manifold without boundary). Using a partition of unity on M, it is easy to introduce a measure, having a smooth positive density with

respect to the Lebesgue measure in any local coordinates. If du is any such measure, the Hilbert space L2 (M, du) is defined. Note, that the elements and the

topology in L2 (M, dµ) do not depend on the choice of dµ. It is therefore meaningful to talk about the space I.? (df) as a topological vector space in which

the topology can be defined using some non-uniquely defined Hilbert scalar product. Theorems 6.1 and 6.4 obviously imply Theorem 6.5. Let M be a closed manifold, A e LQ (M), I - Q S 6 < Q. Then 1) operator A extends to a linear continuous operator

A: L2(M)-+L2(M); 2) if the principal symbol a,,(x, ) ESQ 8(T*M)ISQ,s a+' (T*M) satisfies cona?ticri (t .2-0) (k is the same for all representatives of u"ir r"yui'vaknt rC class in

S8(TM)/S'-'(TM)), then the operator so obtained is a compact operator L2(M) -4 L2(M).

Corollary 6.2. If A e Lr, d (M), i - Q 5 6 < Q and m < 0, then A extends to a compact operator

A: L2(M) - L2(M). We now formulate a version of the boundedness theorem, adequate for non-

compact M. For this we introduce the spaces LA(M) and LP(M). Let f be a complex-valued function on M, defined everywhere except, perhaps on a set of measure 0. In the sequel, we consider functions f, and f2 as equivalent if they coincide outside some set of measure 0. Indeed, the elements of the spaces LL (M) and

52

Chapter I. Foundations of Y'DO Theory

LL (M) are not functions but equivalence classes, although we will write f e LA(M) for a function f, by abuse of language. We will write that f e L L(M) if for an arbitrary diffeomorphism x: X- X, of an open set X c lR" into an open set X, c M and an arbitrary open subset X° C- X, such that Xo is compact in X, we have [x* (f I x)] Ix,e LZ(X°, dx), where dx is the Lebesgue measure on X0 induced by the Lebesgue measure on W. The topology of L? ,,,,(M) is given by a family of seminorms

III IL, x,= II[x'Ulx,)]Ix,liL2cx,.ax)'

If M has a countable basis, then L ',,,,(M) is a Frechet space (a complete metrizable and locally convex space or, what is the same thing, a complete countably normed space). the linear subset of LA(M), consisting Further, we will denote by of those elements f e LL(M) for which suppf is compact in M. Given x: X - X, and X° c X as described above, define the inclusion i,, xo: LZ (X°, dx) - Lo.P(M), mapping a function f ° e LZ (X°, dx) into the function J(y) on M, equal to f (x) at

the point x (x) of M and to 0 at y e M\ x (X°). The topology of ...p(M) is defined as the inductive topology, i.e. the strongest locally convex topology for which all the inclusions i, , are continuous. From this it follows that the linear operator A: L,mp(M) -+ E, E any locally convex space, is continuous if and only if all the compositions A o x, are continuous. This circumstance being taken into account, we clearly get the following Theorem 6.6. If A e LQ, a (M), where 1 - Q < b < p, then A extends to a linear continuous operator

A: Loomp(M) - 9.(M) Exercise 6.1. Prove that if A e L°Q.a(M), I - Q S b < g, and if A is properly supported, then it extends to a linear continuous operator A: L2.mP(M) -. LICOMP (M)

and also to a linear continuous operator

A: L« (M) -* 1.(M)

.

§7. The Sobolev Spaces 7.1 Definition of the Sobolev spaces

Lemma 7.1. Let M be an arbitrary manifold. Then for any real s there exists on M a properly supported, classical elliptic 'I'DO A, of order s with positive principal symbol (for 1

0).

§7. The Sobolev Spaces

53

Proof. To begin with let M = X, a domain in W. Then we may take as A, an arbitrary properlysupported'YDO A, e CL'(X) with principal symbol I Is. Next let M be arbitrary and let there be given a covering of M by charts Al = U X17. Y

We will denote by x' any coordinate diffeomorphism x' : X' -+ X; , where X' is

an open set in R". We construct on X1 operators AL, having the required properties and transport these to Xl by the standard procedure (cf. §4) using the diffeomorphisms x', producing a properly supported, classical TDOA'.I on X,' with positive principal symbol. The operator A. on M can be glued together from the operators AL I by the process used to construct B from By in the proof of theorem 5.1.

Definition 7.1. We write u e HA(M), if u E.Q'(M) and A,u E LA(M). Further set HA(M) = H;a(M) n f'(M). (Concerning -Q'(M) and 1'(M), see Exercise 4.1.) If M is a closed manifold, we denote HHa(M) = HHmp(M) simply

by H'(M). If K is a compact in M, we denote by H'(K) the set of all u E for which supp u C K. and There is a well-defined topology in the spaces H'(K), but for the time being only the set of elements in these spaces is essential to us. Below we will show that these spaces do not depend on the choice of the operator As.

7.2 The action of'YDO on Sobolev spaces. The precise regularity theorem. Theorem 7.1. If A e LQ, a(M), 1 - Q 5 S < p, or S < Q and M = X, an open set in W, then A defines a mapH,,m,(M) -. HH, "'(M). If A, in addition, is properly supported, then A defines maps

A: HL,mp(M) -' H o (M) ,

A: HA(M)-+Hiam(M). Proof. Without loss of generality, we may assume that the operator A _, is a parametrix of A, for arbitrary s e 1R, i.e.

A-,oA,=I+R,,

(7.1)

where R, is a properly supported operator with smooth kernel and therefore transforming 0'(M) into Ca (M) and 9'(M) into Cm(M).

If ueH(M), then, setting Au = ua, we will obtain from (7.1), that u = A-,ua + v, where uo a LL,mp(M), v e Co (M). Therefore

A,_",Au = A,_.A(A_,uo+v) = A,_mAA_,u0+ A'Av,

Chapter 1. Foundations of `PDO Theory

54

and in view of the fact that A,_mAA_,ELQ.6(M), we obtain from Theorem 6.6, from which A,_mAueLL(M), that A,_mAA_,u0EL4.,(M), i.e. Au E H;a '(M). Thereby we have proved the first assertion of Theorem 7.1.

The remaining ones follow from this one or are shown similarly. O

We are now in a position to give a definition of Sobolev spaces, not depending on the choice of A,. It clearly suffices to define HA(M). Definition 7.1'. We will write u e HHa(M), if u e 9'(M) and Au e L «(M) for any properly supported A E L1.0(M).

The equivalence of Definitions 7.1 and 7.1' follows in an obvious manner from Theorem 7.1. Theorem 7.2. Let A e HLQ: a '(M), where I - Q < 6 < Q, orb < Q and M = X, an open set in IV. Then, if u e -9'(M) and Au E HA(M) we have u e 3M+"`-(M).

Proof Let B be a parametrix for A, BeHLQ '"-,-"(M). Then, by Theorem 7.1 we have BAu E H',+"'-(M). But BAu = u + v, where v e C°°(M) so

u E H "- (M). O Corollary 7.1. a) If A is elliptic of order m, u e if' (M) and Au E Hj«(M ), then

b) If A is properly supported (in particular if A is a differential operator) and elliptic of order m, u c -!?'(M) and Au e HL(M), then u e H,« '(M).

7.3 Localization. Theorem 7.1 clearly implies that if u e H;«(M) and a(x) e C' (M), then au a HHa(M).

If, in particular, a (x) E Co (M), then au E H ,(M). The following is a precise version of the converse.

Proposition 7.1. Let the distribution u e 2'(M) be such that for any point e Co (M), such that cpxe(xo) + 0 and

xo e M one can find a function e H ,(M). Then u e H;«(M).

Proof. We may select from the functions a system of functions { qy}, such that some neighbourhoods of supp cp, form a locally finite covering of M and for any point xo e M one can find y such that cp., (x0) * 0. Put now

_ (Pr. (r

.

Y- Ic'yI2 Y

Then obviously Vi, has the same properties as cp., and, in addition, Wy Z 0 and = 1. We now have

(M) because, by the fact that A, is properly supported, the sum finite.

A, (s1 u) is locally

§7. The Sobolev Spaces

55

Corollary 7.2. Let M = U X, be an open covering of M, u e 9'(M). Then the r c o n d i t i o n u e HA(M) i s e q u i v a l e n t to

u I x, e H (X,) for any y.

Proposition 7.1 shows that it is essentially sufficient to study H'(K) for K compact in IR". 7.4 The space H'(IR").

Definition 7.2. Let s E IR, U E S' (R'). We will write u e H'(IR"), if E L1Z,C(IR") and 11U11.2

= J I u (x)12 <>2s d <

(this also serves as a definition of the norm 11 '

+00

(7.2)

II,).

Exercise. Show the completeness of H'(IR") (with the norm II ' IIs) The following Hilbert scalar product can be introduced in H'(IR") (7.3)

(u, v). = Ju(g) 6(e)

and the map (D)', mapping u E S'(IR") into

(F is the Fourier

transformation), provides an isometric isomorphism

': H'(W)- L2(IR").

(7.4)

Lemma 7.2. Let K be compact in IR". Then

H'(K) = 9'(K)nH'(IR")

(7.5)

d'(K) denotes the set of all u E d'(IR"), such that supp u c K).

Proof. 1. Since 'EL1j1.0(IR") and is elliptic, then by Corollary7.1 it is clear that u E of' (K) and ' u e L2 (IR") implies u e H'(K), i.e. 9'(K) n H'(IR") c Hs (K). 2. Now let u e H'(K). We must verify that 'u a L2(IR") if it is known that 'u e LL(IR"). Indeed, a stronger result is valid: if (p (x) e Co (II."), q = I in a neighbourhood of K. then (1- (p) 'u a S(I V). Let us prove this. Let W e Co (IR") be such that rp = I in a neighbourhood of supp ' (in

particular (1- (p) w =- 0) and moreover w = I in a neighbourhood of K. Then +y u = u and (1- (p) 'u = (1- cp) 'yi u. Let us study the operator (1-(p)'w. Clearly its kernel K(x, y) belongs to C°°(IR" x IR"). It suffices

to verify that K(x,y) ES(IR" x IR"). But K(x,y) is given by an oscillatory integral:

K(x,y)=

Je'(:-ri

e(1-(,(x))<>'W(y)dd,

56

Chapter I. Foundations of 'I'DO Theory

which can be rewritten as

Ix - yl-2N

K(x,Y) =

(p (X)) W(Y) (-d4)N<)sd

since Ix-yI _ a>O forxesupp(1-(p),yesuppqi. For2N>s+n weobtaina convergent integral, which can be estimated by The C(1+IxI+IYI)-2N

derivatives of the kernel K(x, y) are estimated in a similar way. 1]

Noting that H'(K) is a closed subspace of H'(1R"), we see that the scalar product (u, v)s induces a Hilbert space structure on H'(K).

Lemma 7.3. Let k be a compact set in lR" such that Kc Int k. Then for u E H' (K), there exists a sequence u" a Co (lR"), supp u" a k, such that Ilu" - uI1,--* 0 for n-' +oo. Proof. Let (p (x) E Co (lR"), J cp (x) dx = I and gyp, (x) = s -"gyp (x/s). We put uL (x) _ ((P, * u) (x) _

r>0.

,

It is clear that u, (x) e Co (lR") so let us prove that lim II u: - u Its = 0. Since a»o+ apt ( ) = ip and rp (0) = 1, the question reduces to establishing the relation lim t»+o

0,

I21

which is evident from the dominated convergence theorem. L 7.5 Topology in the Sobolev spaces on a manifold. Let M = U X; be a locally

finite covering of a manifold M by relatively compact coordinate neighbourhoods X', xY : X Y -* X Y the coordinate diffeomorphisms (X Y an open set in IR") and co' a partition of unity on M subordinate to (X',). Let K be a compact set in M. Introduce a scalar product in H'(K), setting (u, v), _ Y_ ((x')* ((,YU), (x'')* ((o'v))s ,

u, v E H'

(K) .

(7.6)

Y

Proposition 7.2. The scalar product (7.6) induces a Hilbert space structure on

H'(K). Proof. Clearly we only need to verify the completeness of H'(K) in the norm II I. defined by the scalar product (7.6). Let the sequence of elements in H'(K) be a Cauchy sequence with respect to 11 II,. Then (x')*VYU", -+ ui E H'(xy(suppSpY))

(7.7)

in the H'(lR") norm (here x;=(x')-'). We may take v'=(x;)*vi, thus v; = (xl)* v1. Then vY a H' (supp q ) and we put u =

vY (which is obviously a Y

§7. The Sobolev Spaces

57

finite sum). It is clear that u E H'(K) since vY E H'(K). It remains to prove that lim Plum - ull., = 0. This means in view of (7.6), that M-oo

lim II ('Y)'9p'(um - u)II., = 0

M-M

for any y. But it is evident from (7.7) that it suffices to verify vY = W'u, which

is clear since ur -- u in 1'(M), hence (PYUm (p'U in 1'(M), and (7.7) implies V u -+ vY in 1'(M) (convergence in 1'(M) in the weak sense). U

We now want to prove that the topology, induced on H'(K) by the norm II

'

II, , is independent of the choice of arbitrary elements in (7.6) (the covering, the

partition of unity and the coordinate diffeomorphisms). For this

it is

appropriate to give another definition of this topology. Let A, be as in Lemma 7.1 so that A _, is a parametrix for A, (relation (7.1) is satisfied). Then we have

u= A_,A,u-R,u.

(7.8)

Let p > s be a positive integer and Q 1, ... , QN differential operators, generating the left C°°(M)-module of all differential operators of order not greater than p

on M. We then set N

(u, v): = (A,u, A,v) + Y (QkR,u, Qk R,v),

(7.9)

k=1

where u, v eH'(K), (, ) the scalar product in L'C,,,,(M) induced by any smooth positive density on M. From (u, u)s = 0 it follows that A,u = 0 and R,u = 0 and

then, in view of (7.8), we have u = 0. Therefore the scalar product (7.9) is well defined and we will denote the corresponding Hilbert norm by II

H.

Proposition 7.2'. The scalar product (7.9) induces a Hilbert space structure on

H'(K). Proof. Once again, we need only verify the completeness. For the beginning note, that from the convergence of a sequence u, E H' (K) with respect to the norm II I1; the weak convergence in J'(M) follows. If a sequence U. E H' (K) is a Cauchy sequence with respect to 11 then this means, that the following limits exist in the L2(K)-topology (K some compact set in M)

lim A,um = v,

M-00

lim QR,u,,, = wQ, Trot

(7.10)

where Q is any differential operator of order <-- p. In particular A,u,,, and R,u,, converge in the topology of L (M), so that weak convergence of u, in 1'(M) results from (7.8). Denoting the limit of u,,, (in the weak topology of 1'(M)) by u, we obviously have

58

Chapter I. Foundations of `FDO Theory

u=A_,v-w,,

(7.11)

where wl = lim R,ur. If as Q in (7.10) we take an elliptic operator of order T-00 p, we have

Qwl = lim QR,um = WQ E L (M), M-00

from which w1 E Hi (M), therefore w1 E but then w1 E HIP(K) for some compact set K since the operators A, and R, are properly supported. Therefore U E H'(K). It remains to verify that lim IIum - u111 = 0.

We have

um - U = u, - A_,v + wl = A_,A,Um - R,u,,, - A_,v + wl = A-,(A.su, - v) - (R,um - wl), hence N

Ilum - uil, < IIA,A-,(A,um - v)II + E II Q&R,A_,(A,Um - v)II k=1

N

+ IIA,(R,um - wl)II +

II QkR,(R,ur - w1)II

(7.12)

k=I

(here the norm II (7.9)).

II

is induced by the same scalar product

as in formula

The convergence to 0 of the first, second and last terms in (7.12) follows from the boundedness of the operators A, A_ Qk R, A_,. and Qk R, as operators from L2(K) into L2(K), and in the case of the third term from the following argument. Take an elliptic differential operator Q of order p and its properly supported parametrix Q1. Then

-

As(R,um - w1) = A,QI Q(R,u,,, wl) + A,R(R,u,,, - wl), The desired result now follows from the fact that A, Q, is a where R e L properly supported PDO of order s - p ;S 0 continuously mapping L2 (K) into L2 (RC).

There also exists a third way to introduce a topology on H'(K), via the seminorms IIuIIA,k= IIAuIRII,

AeL;,0(M).

(7.13)

Proposition 7.3. The three topologies introduced on H'(K) above (i.e. via II

' II,,

11

'

Its and the seminorms (7.13)) coincide.

§7. The Sobolev Spaces

59

Proof. The equivalence of the topologies determined by the norms II ' Its and 1I

' II;

is clear from the closed graph theorem. The topology given by the

seminorms (7.13) is obviously stronger than the one given by the norm II ' II;, since the latter can be estimated by a sum of N + I semi norms of the type (7.13), where the corresponding operator A is equal to A, and Qk R, . To verify the

equivalence of the two topologies, it therefore sufficies to establish that if A e Lj.0(M) and K, k are compact in M, then the following estimate holds IIAuIRII < C11ulls,

uEH'(K).

(7.14)

Now, writing u in the form (7.8) we obtain

Au = (AA-,)A,u - AR,u, and, since AA _, a L°, o(M), AR, a L- (M), (7.14) follows from Theorem

6.6. O Corollary 7.3. The topology defined in H'(K) by the norms II ' II, and II ' II; does not depend on arbitrary elements entering the definition of these norms.

Proposition 7.4. Let k be a compact set in M, such that K CZ Int K. Then if u e H'(K), there is a sequence w" E Co (R), such that cp" -. u as n - + oo in the topology of H'(k).

Proof. Follows from Lemma 7.3 taking into account the definition of the norm II

.

II,

Proposition 7.5. Let K be a compact set in M and let A e LQ 6(M), where either 1 - Q S 6 < Q or b < Q and M = X an open set in W. Then, provided A is properly supported, it defines a linear continuous operator

A: H3(K) -. H,-,(k), where k is a compact set in M depending on K. Without assuming that A is properly

supported, the same holds for the operator 4pA, * a Co (M).

Proof. It is most convenient to use the norm 11 ' 11;. Then, to verify the continuity of A acting from H' (K) into H' -"(k), we have to estimate II A, _ ", Au II

and IIQR,_,"Aull by CIIA,ull and CIIQ'R,uII (here Q and Q' are differential operators). But from (7.8) we have

Au = (AA_,) (it,u) - AR,u, from which

(A,-.AA_,) (A,u) - (A,-.AR,) u, QR,_",Au = QR,-",A-,A,u and the required estimate follows from Theorems 6.1 and 6.6, if we take into

Chapter I. Foundations of PDO Theory

60

consideration that As_mAA_seLe°.a(M) and that the operators AS_JARC, QRs _ m A _ s and QRs _,,,

belong to L(M). 0

We now introduce topologies in HHa(M) and H;mp(M).

The topology of H;a(M) is defined as the weakest locally convex topology making all the mappings M.: HA(M) - Hs(supp (p), (p a Co (M) and M,u = cpu, continuous. In other words, this topology is given by the system of seminorms IIuIL.m= Ogvuf"

cpeCC (M).

(7.15)

The topology on is defined as the strongest locally convex topology, making all the embeddings iK: H5(K) - H'P(M) continuous. The most important characteristic of this topology (called the inductive topology) is that a linear map f- H.mp(M) -' E to any locally convex space E is continuous if

and only if all the compositions f o iK: H(K) - E are continuous. These definitions and Proposition 7.5 imply the following Theorem 7.3. Let A E L' a (M) with either I - Q 5 b < Q or b < p and M = X, an open set in W. Then A is a linear continuous operator for any s e IR

A: HHmp(M) - H M(M) . If A is properly supported it extends to linear continuous operators

A: H(M)-+H(M) and

A: HL(M) - Hi«"'(M). 7.6 Embedding theorems. First, note the completely trivial (and already used) fact, that for s > s', we have the embeddings

H'(K) c H''(K),

H;a(M) c HL(M),

H',(M) c H p(M), which are continuous. Less trivial is the following

Theorem 7.4. Let s > s' and K a compact set in M. Then the embedding operator

is": Hs(K) -Hs'(K) is a compact operator.

Proof. By the equality (7.8), we obtain

§7. The Sobolev Spaces

61

(A: A_,) (A. u) - (A: R,) u _ (As A-,) (A. u) - (A.,, A

(A, R. u) + (As R,) (R,u)

(3-'1(M) (hence from Corollary 6.1 for any compact set K, Since A; A_,eL-1.0 one can find a compact set K2, such that A,. A _, is a compact operator from L2(K1) to L2(K2)) it is clear that if u runs through a bounded set in H'(K) (and, consequently A,u and R,u run through a bounded set in L2(K,)) then As u runs through a precompact set in L2(K2). Similarly one shows that in this case QR,,,u runs through a precompact set in L2(K2) for any differential operator Q. But this, in view of the equality (7.8) and the definition of the norm, implies the compactness of the corresponding set in H' (K2), hence in H"(K), since, actually, it belongs to H" (K) and the topology in H"(K) is induced by the

one in H(K2) provided K C K2. A generalization of Theorem 7.4 is Theorem 7.5. Let A e L"Q, a (M), A properly supported, either I - Q < b < Q or b < Q and M = X, an open set in IR'. Let the numbers s, s' e IR be such that s' < s - m. Let K be a compact set in M and k a compact set in M (depending on K) such

that A 4f '(K) a r'(R). Then the operator A: H5(K) --- Hs'(k) is compact.

Proof. Theorem 7.5 is a consequence of Proposition 7.5 and Theorem 7.4, since

the operator A: H'(K) -, H" (k) can be viewed as a composition

H'(K)

A H'-M(R)

H''(R).

Cl

Denote by CP(M) the space of functions on M having continuous derivatives of order Sp in any local coordinates. The topology in CP(M) is defined by the seminorms IIu1IA,K= sup I Au(x)

(7.16)

xeX

where A is any differential operator of order Sp. We denote by Cg(K) the subspace of the functions u e CP(M) with supp u c K. It is clear that the topology of CP(M) induces a topology on Cg(K), which can be given by a Banach norm.

Theorem 7.6. If s > n/2 + p, then H;a(M) c CP(M) is a continuous embedding. If K is a compact set in M, then the embedding H'(K) c Cg (K) is a compact operator under the same assumption s > n/2 + p.

Proof. Since differential operators of order p are continuous maps H'(K) - H'-P(K), it is obvious that it suffices to consider the case p = 0. Further, it suffices to verify that for s > n/2, we have a continuous embedding

Chapter I. Foundations of PDO Theory

62

Hs(K) c Co(K), since the compactness of this embedding is obtained by writing it as a composition

H'(K) c H'-`(K) c Co (K) (e > 0 such that s - e > n/2) and using Theorem 7.4. Finally, it is clear that it suffices to consider the case of K lying inside a chart, i.e. the question reduces to the case M = IR". Thus, let K be a compact set in IR" and s > n12. It follows from Lemma 7.3,

that it suffices to prove the estimate sup Iu(x)I < C11u115,

u c- Co (K),

(7.17)

xCR'

where C does not depend on u. We will prove this estimate with C even independent of K. We have

lu(x)I= I je1z fu( dal [jlu( )12< >z'd where C =

+oo, as required.

Corollary 7.4. f Hja(M) = C-(M). This corollary is obvious. Let us also note the dual fact: U Hs(K) = f'(K) for any compact set K c M. This fact follows from the well-known statement of distribution theory, that if u e d'(K), then u can be written as u = Y_ Q;v;,

where v; E L2 (k), k is compact and Q J are differential operators. if If m is the

greatest order of the Q;, then uEH-'(K). 7.7. Duality. Let there be given a smooth positive density du on M. This defines a bilinear form = j u (x) v (x) du (x),

(7.18)

for instance, if u r= Co (M) and v E C°° (M).

Theorem 7.7. The bilinear form (7.18) extends for any s e IR to a pairing (separately continuous bilinear mapping)

H.'.P(M) x H,; (M) - C

(7.19)

which we will denote as before by < -, >. The spaces Ho,,,P and HL are dual to each

other with respect to this pairing, i.e. any continuous linear functional 1(u) on can be written in the form for some v e H1« '(M), and any

§7. The Sobolev Spaces

63

continuous linear functional 1(v) on H,« (M) can be written as , where If the manifold M is closed, then the transformation which attaches

u

to any v eH-'(M) the linear functional is an invertible linear continuous operator from H-'(M) into (H'(M))* (where the latter space is endowed with the natural Banach space topology).

Proof. 1. First let us verify that the form (7.18) extends to the pairing (7.19).

Note that the operator A appearing in the definition of the Sobolev space can be chosen symmetric with respect to the given density, i.e. such that =
construct all the A s >_ 0, as symmetric operators, and then consider their parametrices A, and take for A-, the symmetrization of A',: A-s

1/2(A,

,+'A' ,)).

From the definition of the topology on extend (7.18) to a pairing

it follows, that it suffices to

H'(K) x H,a (M) -, C

(7.20)

where K is any compact set in M. Clearly this is possible for s = 0. For s + 0, we take u e Co (K), v c- C°° (M) and write u in the form (7.8). Then = = - ,

(7.21)

from which the extendability of < , > to the pairing (7.20) follows since A. and

A, are continuous linear mappings A,:

A-,: Hj«(M)-.L,«(M)

2. Now let be a linear continuous functional on H,O'(M). We will show that it can be written in the form I (v) = , with u e 11,11,,,(M). First of all, since C' (M) c H,; !(M) is a continuous embedding, the restriction of 1(v) to C°`(M) can be written in the form , u E 9'(K) for some compact K in M. The distribution u is thereby uniquely defined and it only remains to verify that uEH3(K), i.e. A,ueL2(k). But =
and therefore the desired statement can be derived from the fact that A. is a continuous mapping LL(M) -+ H,a (M) and also from the Riesz theorem, which guarantees the assertion for s = 0.

64

Chapter 1. Foundations of `PDO Theory

on HomP(M) Similarly one shows the representability of the functional in the form l (u) = , where v E H,a (M). 3. Now let M be a closed manifold and let us verify that the map v -+ 1"(-)

=<-,v> is a topological isomorphism between H-'(M) and Obviously this is true for s = 0 by the Riesz theorem. Consider the case of an has already been arbitrary s E IR. Since the bijectivity of the map v - i established in 2., it suffices to verify its continuity. But this follows at once from (7.21).

7.8 Exercises and problems

Exercise 7.1. Verify that 6 (x) a H'(1R") for s < - n/2.

Exercise 7.2. Show that the embedding operator H'(IR") c H" (W) is not compact for any s, s'(s>s').

Exercise 7.3. Let T' = R"/2,r71" be the n-dimensional torus (7Z" is the lattice of points with integer coordinates in ]R"). If f E C°°(lr) (= Co (1r)), then f decomposes into a Fourier series fk e",

f(x) =

(7.22)

keZ*

where fk are the Fourier coefficients, given by the formula

f = (2n)-" $ f(x) e"dx.

(7.23)

The same formula also applies to f e 9'(1'") if as the integral in (7.23) we take the value of the functional f at the function e-jk (in this case the series (7.22) converges in the weak topology of then the condition f c- H'(Y") is equivalent to Show that if f keZ'

1112(1+Ik12)'<+00,

(7.24)

where the left-hand side of (7.24) defines the square of a norm in H'(T"), equivalent to any of the previous norms. Exercise 7.4. Show that if A satisfies the conditions of Theorem 6.4, then A:

H'(K) - H" (k) is a compact operator for any s e IR. Exercise 7.5. Show that the spaces H'(IR") and H--(R") are dual to each other with respect to the bilinear form = r f (x) g (x) dx. Problem 7.1. Verify that if N is a submanifold of M of codimension d, then the restriction map f -f IN (defined a priori for f e C°° (M)) extends for s > d/2 to a linear continuous map

§8. The Fredholm Property, Index and Spectrum

65

Hi «(M) -' Hi« d12 (N).

Problem 7.2. Show that the map defined in Problem 7.1 is surjective.

§8. The Fredholm Property, Index and Spectrum 8.1 The basic properties of Fredholm operators

Definition 8.1. Let E, and E2 be Banach spaces and A : E, -. E2 a linear continuous operator. It is said to be Fredholm if dim Ker A < + oo and dim Coker A < + oo (recall that Ker A = {x c- E,: Ax = 0}, Coker A = E2/Im A, where Im A = AE, and the quotient space is meant in the algebraic sense, i.e. regardless of a topology). The index of a Fredholm operator is the number index A = dim Ker A - dim Coker A.

(8.1)

We will denote by V (E, , E2) the set of all linear continuous operators A : E, -' E2 and the set of all Fredholm operators A E 2 (E, , E2) will be denoted by Fred (El, E2).

Lemma 8.1. Let A e 2 (El, E2) and let dim Coker A < + oo. Then Im A is a closed subspace in E2.

Proof. Clearly, Ker A is a closed subspace of E, and therefore the quotient space E, /Ker A has a natural Banach space structure. The operator A induces a continuous map A, : E, /Ker A - E2 with Im A, = Im A and Ker A, = 0. Now let C denote any finite-dimensional subspace of E2 for which E2 = ]m A ® C (direct sum in the algebraic sense). Define the operator

A: E,/KerAOC-'E2

(8.2)

mapping a pair {x, c} into A, x + c E E2. Obviously A is bijective and continuous, if the space on the left hand side is considered as a Banach direct sum (e. g. with the norm II {x, c} II = II x II + II c 11, where If c 11 is defined by any norm on

Q. By the Banach inverse operator theorem, A is a topological isomorphism implying that Im A is closed in E2 since A-' (Im A) = E, /Ker A ® 0 is closed

in E, /Ker A ®C. 0 Corollary 8.1. If dim Coker A < + oo and L, is a closed subspace in E, such that E, = L, e Ker A, then A defines a topological isomorphism A: L, - Im A.

Note that in the case A e Fred (E,, E2) there always exists a subspace L, of this type, since by the Hahn-Banach theorem, we may extend the identity map Ker A -. Ker A to a continuous linear operator PI: E, --+ Ker A and then put L,

=KerP,.

66

Chapter I. Foundations of `YDO Theory

Corollary 8.2. If A e 2 (Et, E2) and dim Coker A < + oo, then dim Coker A

= dim KerA*, where A* is the adjoint operator A*: Ez -. E. Proof. It is known (and trivial) that

KerA*= {feE2: =O}. From the closedness of Im A and the Hahn-Banach theorem it follows that

ImA = {xeE2:
C!

Lemma 8.2. Let E be a Banach space and let T e

(E, E) be of finite rank, i.e.

dim Im T < + oo. Then the operator I + T is Fredholm and index (1+ T) = 0.

Proof. It is easily seen that there exists a decomposition E = Lo ® L1, where Lo is a closed subspace, La a Ker T, L1 Im T, dim L1 < + oo. Then (1+ T)14 = I I L., (1+T) L1 c L1, because TL1 c Im Tc L1. Therefore Lo and L1 are invariant subspaces for (1+ T) with Im (1+ T) Lo, Ker (I+ T) C L1. Therefore 1+ T is Fredholm and index (1+ T) equals the index of (I+ 7), viewed as an operator from L1 into L1, which means that the whole matter reduces to a trivial statement from linear algebra. O Lemma 8.3. If A E Fred (El, E2), then there exists an operator B E Fred (E2, E1) such that

BA=1-P1,

AB=I-P2

(8.3)

where P1 is a projection onto Ker A and (I- P2) a projection onto Im A (so that P1 and P2 are of finite rank).

Proof. Let L, be a closed complement to KerA in E1 and L2 any complement to Im A. Define the operator B such that

BAIL,=11L,,

BIL,=O.

From Corollary 8.1 it is clear that Be 2 (E2 , El) and Ker B = L2 , Im B = L1,

from which the Fredholm property of B follows. The relation (8.3) immediately verified.

is

El

Lemma 8.4. Let A E Y (E1, E2) and let B1, B2 a 2' (E2, El) be such that

B1A=I+T1,

AB2=1+T2,

where T, , T2 have finite rank. Then A is Fredholm.

(8.4)

§8. The Fredholm Property, Index and Spectrum

67

Proof. The statement follows from the obvious inclusions Ker A c Ker (B, A) = Ker (I+ T,), Im A : Im (AB2) = Im (I+ T2)

and from Lemma 8.2.

17

Lemma 8.5. Let A e Fred (E, , E2), B e Fred (E2, E3). Then BA e Fred (E,, E3)

and index BA = index A + index B

(8.5)

Proof. We show first of all, that there exist closed subspaces LC Ej, j = 1, 2, 3, such that Ker A I L = 0, AL, = L2 , Ker B I L2 = 0, BL2 = L3, where codim Lj < + oo, j = 1, 2, 3. (The codimension of a subspace L of E is codim L = dim (E/L). Here, in particular we have codim Lj = dim (Ej/Lj), j= 1, 2, 3.) Indeed, if L, is a closed complement of Ker A in E1, LZ a closed complement of Ker B in E2, we may put L2 = Li n Im A,

L' = (AIL.) -' (L2),

L3 = BL2 .

Let us now note the following fact: Let L1, L2 be closed subspaces in El, E2 respectively, A e Fred (E, , E2), Ker A IL, = 0 and AL, = L2. Then, denoting by A: E1/L, -+ E2/L2 the map induced by A, we have A E Fred (E,/L1, E2/L2) and index A = index A. Therefore, using the above constructed subspaces Ljc Ej reduces the proof of (8.5) to the case dim EE < + oo, j =1, 2, 3, which is evident, since if A e2 (E,, E2) and dim Ej < + oo, j = 1, 2, then index A = dim E, - dim E2 . L Proposition 8.1. Fred (El, E2) is an open subset of .' (El, E2) (in the uniform operator topology, i.e. the topology defined by the operator norm) and the function

index: Fred (E,, E2) - Z is continuous (i.e. constant on each connected component of Fred (E, , E2)). In particular, if A, is a continuous (in the norm) operator-valued function, oft a [0,1 ], with values in Fred (E, , E.) then index Ao = index A,.

Proof. Let A E Fred (E, , E2). We have to prove the existence of E > 0 such that if D e 2' (E, , E2) and JID II< E then A + D e Fred (E, , E2) and index (A + D) = index A.

Let B e 2 (E2, E,) be an operator such that

BA =I+T1,

AB =I+T2

(8.6)

with T,, TT of finite rank (which is always possible by Lemma 8.3). We will verify

68

Chapter 1. Foundations of 1I'DO Theory

that one may take e = II B II '. Indeed, let 11D II
+ T, and if we put B, =(I+ BD) -' B, then B, (A+ D) = 1+ T,, where T, is of finite rank. Note that index B = index B, . Analogously, there is an operator B2 such that (A+D)B2 = I+ T2, with T2 of finite rank. By Lemma 8.4, the operator A + D is Fredholm and by Lemmas 8.5 and 8.2 we have

index (A + D) = - index B, = - index B = index A. D In what follows we will denote by C(E,,EZ) the set of all compact linear operators from E, into E2.

Lemma 8.6. Let E be a Banach space and let R e C (E, E).

Then

I + R e Fred (E, E) and index (1+ R) = 0. Proof. Since I I Ke,(1+R) = - R I Ker(I+R), the unit ball in Ker (1+ R) is compact and therefore dim Ker (1 + R) < + oo. Further, since R* is also compact, dim Ker (1+ R)* < + oo and to show the Fredholm property of (I+ R) it only remains to verify the closedness of Im (1+ R) (since then dim Coker (1+R) = dim Ker (/+ R)*). n-++oo. We need to verify Let x.eE,n=1,2,...,and the existence of an x e E, such that (1+ R)x = y. Let L be any closed subspace complementary to Ker (1+ R) in E. Adding to xn vectors from Ker (I+ R) (which

does not change yn), we may assume that x e L for all n.

Let us show that the sequence x is bounded. Indeed, if this is not as the case, taking a subsequence of {xn), we may assume that n--++oo. But then, putting x;, = xn/I)x11, y.' = (I+R)x;, we have y;,-'0 as n - + oc, and since II x;, II = I we may assume that lim Rx;, exists. But then

+.

also lim xn = - lim Rx;, = x and clearly II x 11= 1, x E L, (I+ R) x = 0, contra-

n+ac n dicting the choice of L. Thus the sequence {xn} is bounded and we may assume that lim Rx exists n»+x and, consequently so does lim xn =Y - lim Rxn . Denoting x = lim x., n»+m n»+z. n»+x we clearly have (1+ R) x = y, proving the closedness of Im (I+ R), i. e. the Fredholm property of (1+ R). By Proposition 8.1, we have index (1+ tR) = const for t e [0,11 implying index (1+ R) = Index I= 0.

Proposition 8.2. Let A e . (El, E2) and let there exist B, and B2 such that

B,A=I+R,,

AB2=I+R2,

(8.7)

where R,eC(E;.EE), j=1,2. Then A eFred (E,,E2). Proof. It immediately follows from Lemma 8.6 in a similar way to the proof of Lemma 8.4. LJ

Proposition 8.3. Let A e Fred (E,, E2), Fred (E,, E2) and index (A + R) = index A.

R e C (E,, E2).

Then

A+Re

§8. The Fredholm Property, Index and Spectrum

69

Proof. Obvious from Proposition 8.2 and Lemmas 8.5 and 8.6. 8.2 The Fredholm property and the index of elliptic operators on a closed manifold Theorem 8.1. Let M be a closed manifold and A E HLQ .a (M), 1 - Q =< S < Q. For any s e IR construct the operator A, E 2' (H' (M ), H' - '(M)) the extension of A by continuity. Then,

a) A,EFred(H'(M),

H'_T(M));

b) Ker A, c C' (M), therefore Ker A, does not depend on sand will be denoted simply by Ker A; c) index A, does not depend on s (so we will denote it simply by index A) and is expressed by the formula

index A = dim Ker A - dim Ker A*,

(8.8)

where AS is the formal adjoint PDO (cf. §3) in the sense of a scalar product determined by any smooth density. d) if D e LQ 'd(M), where m' < m, then index (A + D) = index A. Proof. By Theorem 5.1 we may construct a parametrix B E HL-

'(M) of

the operator A. In view of Theorem 7.5, the operators Rt E L- ' (M) can be e C(H'(M), H'(M)) for arbitrary s EIR. But now, extended to operators the Fredholm property for all the operators A, follows from proposition 8.2. Further, from Theorem 5.2 statement b) of the theorem follows and since A* a HL'- e , we also obtain c). Finally, d) follows from Proposition 8.3 since if D E LQ;b(M), m' < m, then

D e C(H'(M), H'-'"(M)) by Theorem 7.5. Remark 8.1. The assertion of this theorem is clearly true not only for scalar operators, but also for operators acting on the sections of vector bundles. Remark 8.2. For classical elliptic 'PDO, d) says that the index depends only on the principal symbol. It is easy to deduce from Theorem 6.2 that the index does not change with arbitrary continuous deformations of the principal symbol within the class of homogenous elliptic symbols. This is important in the index theory of elliptic operators. 8.3 The spectrum (basic facts). Let M be a closed manifold, A E HLQ, a (M),

1 - Q S 6 < Q, m > 0. In the space L2 (M) consider the unbounded linear operator defined by A by taking as domain the space H'(M). We will denote this unbounded operator by A0, or sometimes just by A if there can be no confusion. So the domain of Ao is DAa = H'(M). Proposition 8.4. The operator A0 is closed, i.e. if for u" E H'" (M), n =1, 2, ... ,

the limits u = lim u" and f = lim Au" exist in L2 (M) then u e H"' (M) R. + "" + °` and Au =f

Chapter 1. Foundations of `PDO Theory

70

Proof. Since convergence in L2(M) implies convergence in 9'(M) and since A is continuous in 9'(M) (in the sense of, e.g. the weak topology), we obtain Au =f, and then u E HM(M) in view of Theorem 7.2. From the fact that Cm(M) is dense in Hm(M) (in the Hm(M)-topology), we have Corollary 8.3. The operator Ao is the closure (in L2(M)) of the operator A 1 c=(M)

.

Definition 8.2. The spectrum of A is the subset a (A) of the complex plane,

defined as follows: for A E C, d 4 a (A) is equivalent to (A0 - Al) having a bounded everywhere defined inverse (A0-Al)-' in L2(M). It is easy to verify that a (A) is a closed subset of C and that (A0 - A1)

is a

holomorphic operator-valued function of A on C\a(A) with values in & (L2(M), L2(M)). The function R, = (A0 - 21)-' is called the resolvent of A.

Proposition 8.5. Let a fixed positive smooth density dµ(x) bejixed on M. Then the conditions A ' a (A) and Ker (A - ,Xl) = Ker (A' -.k1) = O are equivalent. Proof. The statement follows from Theorem 8.1, since (A -Al) E HLQ. a (M)

because m > 0. O Theorem 8.2. (the inverse operator theorem) Let A e HLQ a (M), I - e S 6

<e, m > 0 and let AO be constructed as before. Let also ,1 t a (A). Then

(A,-Al)-' is an extension by continuity (from C(M)) or restriction (from 9'(M)) of an operator from HLQ a -'"(M) (we denote it by (A-AI) I n particular, (A0 - AI) -' is compact in L2 (M).

Proof. It suffices to consider the case A=O. Let B E HLo a-'(M) be a parametrix for A. More precisely

AB =1- R

(8.9)

where R is an operator with smooth kernel R (x, y) (for simplicity we assume that

a smooth, positive density on M is fixed, so the kernel R (x, y) is an ordinary function on M x M). It follows from (8.9) that

A-'=B+A-'R,

(8.10)

and it remains to verify that A -'R is an operator with smooth kernel. But by (M) for any s e IR, and, moreover, is Theorem 7.2, A -' maps H' (M) into H" (M) continuous by the closed graph theorem. By the embedding Theorem 7.6, A -'

maps C°°(M) into C°°(M). But then A-'R is given by the smooth kernel R, (x, y) = [A -' R (- , y)](x).

§8. The Fredholm Property, Index and Spectrum

71

Theorem 8.3. Let a fixed smooth positive density dµ(x) on a closed man-

ifold M be fixed. Let A' = A E HL% (M), 1 - Q < 8 < Q, m > 0. Then Ao is a self-adjoins operator in L2(M) and there exists in this space a complete orthonormal system {tp; }, j = 1, 2, ... of eigenfunctions of Ao. Here (pi E CO0(M), Acp; = Atp; and the eigenvalues A; are real, with IA; I -+ +oo as j - +oo. The spectrum or (A) coincides with the set of all eigenvalues. Proof. Note first of all that a (A) c IR in view of Proposition 8.5, since A is symmetric on C' (M) and can thus have no non-real eigenvalues. Next, we want to show that a (A) * IR. Assuming a (A) = IR then we could for any A e IR find a function cpx e C°° (M), such that Aqpj = Aqp,, and II tpx II = 1

But then ((p,, cp,) = 0 for A * a by the symmetry of A, contradicting the separability of LZ(M). Now take A. a IRS a (A). By Theorem 8.2, R4 = (A - AO 1) -' is a compact self-adjoint operator in LZ (M). By a known theorem from functional analysis there is an orthonormal basis { v;)- , of eigenfunctions, where the eigenvalues r,

tend to 0 as j- + oo. Now note that r; 4 0 (since Ker (A - ).o 1) -' = 0). The condition R,,otp; = r; q7;

can therefore be rewritten in the form (A-AoI)(p; = r1 'cp;

or

A(Q;=(r,-i +Ao)rp;.

(8.11)

It is obvious from (8.11) that tp; E C °° (M) and the V, are eigenfunctions of A with

eigenvalues A, = r; ' + A0. It is also clear that I A; I -. + oo as j-' + oo. The remaining assertions of Theorem 8.3 are obvious. The fact that the spectrum a (A) coincides with the set of all eigenvalues {A;} follows from Proposition 8.5 and the self-adjointness from the representation A = Rxo + Aol.

The following theorem extends one of the statements of Theorem 8.3 to the non-selfadjoint case.

Theorem 8.4. Let A e HL'" -h (M), 1 - e <= 8 < e and m > 0. Then for the spectrum or (A), there are two possibilities:

a) a (A) = C (which, in particular, is the case if index A 4 0); b) a (A) is a discrete (maybe empty) subset of C (subset without limit points). If b) holds and AO e a (A) then there is a decomposition L2 (M) = E,, (D EL such that the following conditions are satisfied: 1) E,1, c C I (M), dim E,. < + oo, and E4 is an invariant subspace of A such that there exists a positive integer N > 0 with (A - A0 I)" E,. = 0 (in other words, the operator A I s,, has only the eigenvalue AO and is equal to the direct sum of Jordan cells of degree S N);

2) E,, is a closed subspace of L2(M), invariant with respect to AO (i.e. A (D,,.nE,) e and if we denote by A, the restriction AO I Ey, (understood as

Chapter 1. Foundations of'I'DO Theory

72

an unbounded operator in E;0 with domain DA° fl E;.O), then A' - AoI has a bounded inverse ( o r , in other w o r d s, A o ¢ a (A1Q)-

Proof. 1. Let or (A) $ C. Let us prove that a (A) is a discrete subset in C. There is a point AOe C\ a (A) and we may, without loss of generality, assume that AO = 0, so that by Theorem 8.2 A0 has a compact inverse Ao '. Then since

A0 - Al = (I-).A0 ')A0 the inclusion A e a (A) is equivalent to A + 0 and A -' e a (Ao ' ). Discreteness of a (A) follows from the fact that a (A. ') may have

only 0 as an accumulation point. 2. Let a (A) * C, AO e a (A). Once again, without loss of generality, we may assume that A0 = 0. Let /'0 be a contour in the complex plane, encircling 0 and not containing any other points of a (A) (e. g. a circle, sufficiently small and with centre at the origin). Consider the operator 1

PO =

2I

I

(8.12)

Standard arguments (cf. Riesz, Sz.-Nagy [1), ChapterXI) show that PO is a projection, of finite rank in view of the compactness of R, commuting with all the operators RA (and with AO in the sense that POAO c AOPO) and such that

if EEo = PO (L2 (M)), E2= (I - PO) (LZ (M)), then conclusions 1) and 2) of Theorem 8.4 hold. We leave it as exercise for the reader to take care about the details. We note only that the inclusion Eao C C°°(M) follows from Ao Ea° = 0 if we take into account the ellipticity of A and utilize the regularity Theorem 5.2. 8.4 Problems

Problem 8.1. Let E be a separable Hilbert space, nO (Fred (E, E)) the set of connected components of Fred (E, E) provided with the semigroup structure induced by the multiplication. Show that taking the index gives an isomorphism index: nO (Fred (E, E)) =- Z.

Hint. An operator A of index 0 can be written in the form A = A0 + T, where A0 is invertible and T has finite rank. Show (by use of the polar decomposition) the connectedness of the group of all invertible operators in E.

In all the following problems M is a closed manifold and 1 - g S S < g, m > 0.

Problem 8.2. Let A e HL'"-,1'-(M). Prove that A is a Fredholm operator in

C(M), i.e. that dim Ker A < + oo, ACW (M) is closed in C(M) and dim CokerA < +oo, where Coker A= C°°(M)/AC°°(M). Show that AC`°(M) consists of all f c C`° (M) for which (f, g) = 0 for any g e Ker A' (here ( , ) is a scalar product determined by some smooth positive density and A* is the adjoint 'PDO with respect to this scalar product).

§8. The Fredholm Property, Index and Spectrum

73

Problem 8.3. Let A e HL!,":--(M) and mo > 0. Let there be given on M a smooth positive density defining the scalar product(-, -)and the formal adjoint

'DO A. Assume A = A'. Let AO be the closure of the operator AIc.(M). Then Ao is self-adjoint in the Hilbert sense in the space L2(M). Problem 8.4. Let A e HLQ: s o(M), let A* be the formal adjoint operator and A0, Ao the closures of A I c. (M) and A I c.(M) in L2 (M) respectively. Show that AO

and Ao are adjoint to each other in the sense of the Hilbert space P(M). Hint. Consider the matrix of 'PDO'2I =

0 A

AO

'

Problem 8.5. Find an example of an operator A e HL ' (M) for which a (A) = C.

Problem 8.6. A sequence of Hilbert spaces E, and linear continuous operators d,: 0

°-`1E0 d o E1

-EN -1

°-'

) EN

°" 60 (8.13)

is called a complex if d 1+

1

dj = 0 for all j = 0, 1, ..., N-2. Put

Z'=Kerdt, B'=lmd,-1, H'=Z'/B', j=0, 1, ..., N. (if (8.13) is a complex, BicZ'). The spaces H' are called the cohomology of the complex (8.13). The complex is called Fredholm if dim H' < oo for all

j=0, 1, ..., N. a) show that if the complex (8.13) is Fredholm, then the B' are closed subspaces of Z'. b) Let d; = b;df + dj-1 b;-1, where bj = d,*. The operators d; are called the Laplacians of the complex (8.13) (or the Laplace-Hodge operators). Put I'' = Ker A, . Show that for the complex (8.13) to be Fredholm it is necessary and sufficient that all A, are Fredholm operators in Ej, j = 0, 1, 2, ... , N. In this case

dim H'= dim T'.

More precisely, I'' c Z' and the map T' - H' induced by the canonical projection

is an isomorphism (in the case of a Fredholm complex).

c) Put now N

X(E) _ E (-1)'dimH' J=0

(the Euler characteristic of the Fredholm complex E). Prove that if N = 1, then the Euler characteristic of the complex

74

Chapter I. Foundations of `YDO Theory

0 -. Eo °6 E1 - 0 is simply the index of do.

Prove that if dim Ej < +oo, j = 1, 2, ... , N, then N

X (E) _ E (-1)j dim Ej. J=o

d) Show that X (E) does not change under a uniform deformation of all the operators d, if under this deformation the sequence (8.13) remains a Fredholm complex. Problem 8.7. Let V ; (j = 0, 1, ... , N) be vector bundles on a closed manifold M and H'(M, Vj) the Sobolev spaces of sections. Let d,: C°°(M, Vj) - C'

(M, V.1) be classical PDO of the same order m. Let To (M) be the cotangent

bundle over M without the zero section and no: To (M) --* M the natural projection. Assume that the operators ...

0 ° _. C°°(M, Vo) a ' C_ (M' V1)

16

C°°(M, VN) , 0 (8.14)

form a complex. Let ad'- no V -+ no Vj+, be the principal symbols of the operators d, (homogenous functions in of order m). The complex (8.14) is called elliptic if the sequence of vector bundles ,

o

0 -+ no Vo

o

no VN --* 0

nO VI

is exact (i. e. an exact sequence of vector spaces at every point (x, ) a To (M)). a) Show that ellipticity of the complex (8.14) is equivalent to ellipticity

of all the Laplacians Aj = bjdj + dj_,8,_1, where 8j is the'PDO adjoint to d, with respect to some density on M and a Hermitean scalar product on the vector bundles Vj. b) Show that if (8.14) is an elliptic complex, then for any s e IR, the complex

H'--(M, V1)....

0 d - HS(M, V0) -d'

dam - Ha-N",(M, VN) °-" . 0

is Fredholm and the dimension of its cohomology (and thus the Euler characteristic) does not depend on s. The cohomology itself can be defined also as the cohomology of the complex (8.14), i.e. putting Hi= Ker(dJIc-i,M.v,))/dj-1(C00(M,Vj-1))

Problem 8.8. Show that the de Rham complex on a real n-manifold M

0 - A0(M)

? A' (M) - A2 (M) d ° i

_, ....

0 A"(M) --; 0

§8. The Fredholm Property, Index and Spectrum

75

(AJ (M) is the space of smooth exterior j-forms on M, d is the exterior differential)

and the Dolbeault complex on a complex manifold M, dim, M = n,

0 -, Aa. o(M)

. AP-'(M)

-L ...

, AP-"(M) - 0

(AD.9(M) is the space of smooth forms of type (p, q) on M and is the CauchyRiemann-Dolbeault operator) are elliptic complexes. Derive from this the finite-dimensionality of the de Rham and Dolbeault cohomology in case of a closed M.

Chapter II Complex Powers of Elliptic Operators §9. Pseudodifferential Operators with Parameter. The Resolvent 9.1 Preliminaries. Let A be a subset of the complex plane (in the applications this will, as a rule, be an angle with the vertex at the origin). In spectral theory it is useful to consider operators depending on a parameter A E A

(an example of such an operator is the resolvent (A -).I) To begin with, we introduce some symbol classes. Let X be an open set in IR" and let a (x, 0, A) be a function on X x lR" x A,

xeX, OEIR, AEA. Definition 9.1. Let m, Q, b, d be real numbers with 0 5 S < g ;S 1, 0 < d < + oo. The class So o. d(X x R, A) consists of the functions a (x, 0, A) such that 1) a (x, 0, AO) E C°° (X x JR') for every fixed A0 E A;

2) For arbitrary multi-indices a and f and for any compact set K C X there exist constants

such that

laeafa(x,9,A)j S

(9.1)

for x e K, 0 E lR1, A E A. As usual we put

S-°°(XxIR",A) = n S'",Q a:a(XxIR",A) MCR

(the right-hand side does not depend on g, b and d). If a(x, y, 4, A) E SQ J;d(X x X x IRA, A), we may construct a `PDO A,,, depending on the parameter). E A: (Ax u) (x) = l l e1 , - y t a (x, y, , A) u (y) dy dd ,

(9.2)

for u E Co (X). In this case we will write A,ELQ a:d(X,A)

Note that A, a L- Ic (X, A) if and only if the operator A, has a smooth kernel K, (x, y) for any fixed A E A and there exist constants C; s!K (K a compact set in X,

Chapter H. Complex Powers of Elliptic Operators

78

a and fi multi-indices and N a positive integer) such that the following estimate holds

Iaxaf K2(x,y)I 5

Cvp'x(I+IA1)-",

x, yEK.

(9.3)

Many of the statements about `PDO without a parameter (cf. §§3-7) can also be proved for the case with a parameter A. We indicate now some of these statements, which are necessary in what follows. First, note that the whole theory of asymptotic summation (Definition 3.4 and Propositions 3.5 and 3.6) carries over to symbols depending on a parameter. The corresponding formulations are obtained by changing SQ a (X x IR") to SQ b;d (X

X IR", A) and the proofs are almost verbatim repetitions of the arguments in 3.3 and are left for the reader as an exercise. We only state that the role of <0> in these proofs (as in the following) is now played by (1+I©I2+ 1A121d),12

Further we will call an operator A,1 E LQ d;d(X, A) properly supported if it is uniformly properly supported in A, i. e. there exists a closed set L C X x X, having proper projections on each factor in X x X, such that supp KA, c L for all A E A.

Note that any operator A ELQ 6;d(X, A) can be decomposed into a sum A = A, + R1, where A 1 (depending on a parameter) is properly supported in the sense described and R, a Lo s d(X, A). For properly supported `PDO AA depending on a parameter, the symbol aA.(x, S) = aA (x, , A) is defined and a theorem of type 3.1 is valid. Naturally, we have to interpret formula (3.21) taking the parameter into account, i.e.

aA(x, S, A) - L « G A S Dy a(x, y, S,A)Iy_xe 121S"-1

ESQ d

d-d!"(XxIR", A).

In an analogous way Theorems 3.2-3.4 on the transpose and adjoint operato and composition can be generalized.

Exercise 9.1. Prove all the statements in sec. 3 in the case of operators an symbols depending on a parameter. Further, repeating the arguments of §4, for 1 - Q < 6 < Q, we may introduce the classes LQ a:d (M, A) on a manifold M. Let us now pass to considering hypoellipticity and ellipticity. We introduce the class HS,',-.' , (X x W, A) of symbols a (x, , A) (we will call them hypoelliptic with parameter), belonging to S',,,,, (X x 1R", A) and satisfying the estimates C1 (ICI+IAI"d)m, < l a(x,

C2(Ifl+IAi"d)

(9.4)

for x e K (K compact in X), I I + I A I ? R, C1 > 0 and R, C, , C2 may depend on K;

§9. Operators with Parameters

s

I [asaXa(x,

79

(9.5)

+ IA I z R (here, as above, R may depend on K). We will denote by HL"" a a (X, A) the class of properly supported `'DO (depending on the parameter I E A), whose symbols belong to HS0' a (X X for x e K, I

IRA, A). We have an analogue of Theorem 5.1: If A e HL Q (X, A), then there exists an operator B, a HLQ called the parametrix of the operator A,, such that

B,A,=!+Rx,

A,BA =I+Ri

(X, A)

(9.6)

where R;, Rx E L-w(X, A). The same statement is also true when X is a manifold.

Exercise 9.2. Prove this analogue of Theorem 5.1. It is natural also to consider classical `I'DO depending on a parameter. In this case A is assumed to be an angle with the vertex at 0. The corresponding symbols a(x, ! , A) admit asymptotic expansions (for I I + III' ? 1) of the form Tco

Y am -i(x, ,2),

(9.7)

f=o

where am _, (x, , A) is positive homogeneous in

A"') of degree m -j, i. e.

am-1(x, t > 0, A e A and t') e A. Here m can be any complex number. This class of symbols will be denoted by CS, (X x IV, A) and the corresponding class of operators by CLd (X, A). This class is stable under composition, taking the transpose and taking the adjoint.

We will say that the operator A, e CLa (X, A) is elliptic with parameter if it is

properly supported and

a,, (x, , 2) 4 0

if

x e X and I f I + 12I"' * 0 .

(9.9)

It clearly follows that A, a HL; ; '" d (X, A). There exists a parametrix B, a CL;- m(X, A) of a classical elliptic operator with parameter, which is also an elliptic operator with parameter. Example 9.1. Let A be a differential operator in X of degree m and I the unit

operator. Then A - Al a CL (X, C) and the principal symbol is given by the formula am (x, ) is the principal symbol of A. If A is a closed angle in the complex plane with vertex at the origin such that am (x, ) for I I = I does not take values

Chapter If. Complex Powers of Elliptic Operators

80

in A, then the operator A - 2I is elliptic with parameter (and, in particular, belongs to HL-,-

m

(X, A)).

9.2 Norms of operators with parameter. In this subsection we will consider operators with parameter of two kinds: 1) operators Ax in IR", such that supp KAA lies in a fixed compact set k c IR2rt

(where 056
m ands, / e JR. Let one of the numbers

6, s, s - I be equal to 0. Then 11A,, II,.,-, C, (1+IA11'd)m,

IIAx11s,s-l

if if

Cs.l(1+jAj11d)-p-m),

Corollary 9.1. If AA E LQ, b d (X, A), where m

1;_> 0,

/<_0.

(9.11)

(9.12)

0, then

IIA,II < C(1+I2l a)m,

(9.13)

where II AA II denotes the operator norm of A. in L2 (X).

We will need the following useful Lemma 9.1 (Schur Lemma). If A is an operator with the Schwartz kernel KA such that

Sup f I KA(X, y)ldy < C and x

sup f I KA(X, y)ldx < C, t.

then IIA:L2--), L211
(This lemma holds for integral operators in L2 on any measure space, or for L2(X) for any measure spaces X, Y.) integral operators L2(Y) Proof. By the Cauchy-Schwarz inequality we have

IAu(x)I2 < (f I KA(x, y)IIu(y)Idy)2 _ (f IK,+(x, y)1112IKA(x, y)I1/2Iu(y)Idy)2

f I KA(x, y) Idy f I KA(x, y)IIu(y)I2dy

< C f IKA(x, y)IIu(y)l2dy.

§9. Operators with Parameters

81

Now integrating with respect to x and changing the order of integration we obtain the desired norm estimate. Proof of Theorem 9.1. 1. Note first of all that a partition of unity reduces the case X = M to the case X = IR", which we will now study.

2. Consider first the case s = I = m = 0. The statement of the theorem reduces to the estimate (9.14)

IIkII:5 C,

which is proved by repeating verbatim the argument in §6 (we recommend the reader to work this through as an exercise). Note now that the estimate (9.14) could be proved directly utilizing the results from §6 if the constants Ca. a (A) = sup l as as a (x, , 2) I « ° lal alai

(9.15)

X.(

where bounded as 12 I -+ + oc. This evidently follows from (9.1) for S = 0, but for b > 0 some of the constants Ca, a (A) can grow as I A I -+ + oo. Therefore, for 6 > 0

it is indeed necessary to repeat the argument from §6. 3. Now consider in IR" the standard operator-valued 0m()) E L m ,. o; a (R",

C) with the symbol f p m (x, , 2) = (1 + I

I2 + 1 A

function 121a)"n. This

1PDO with parameter will be useful to us, although it does not satisfy condition 1). Let us estimate the norm of the operator 4',,,(A). The operator 0. (0) induces an isometric isomorphism of H"(IR") onto L2(IR"). Therefore II0"(A)Ils.s-I = Il0:-1(0) 0. (A) It -s(0)II

But 0.-1(0) 0.(A) 4P _,(0) =0.(A) 0_1(0) is simply the multiplication operator in V(W) of the Fourier transform by and therefore its norm is equal to 21a)m/2

W,",(A) = sup (1+1

12+IA121d)"r2

4CW

We obviously have

Y/., (A) 5 Csup (1+Ici+IAI1jd)"0 4eR

where C only depends on m and / (but not on A). Now, from the easily verified relation

sup(1+x+1)"(1+x)-'= Xao

5(1+if /?0, C", t" ', if

(we assume m:51 everywhere) it follows that

1_< 0

and

i z R",

Chapter II. Complex Powers of Elliptic Operators

82

Wm (2) 5 Cm (1 + W.1 (A)

I2rd)m,

if 1 z 0,

if 150,

Cm1

i.e. for 0m (2) one of the norm-estimates (9.11), (9.12) holds as asserted in the theorem. 4. Consider now the general case AA E LQ a, d (1R", A). We obviously have IIAAIIa.a- =

(0-m(2) AA)II,s(9.16)

III-m(2) AAlls.a

and analogously IIAAII3.a- = II(AA-0-m(2))0.(2)I1a.a5 II II'm(2)Ils.a-

(9.17)

Using the already proven norm-estimate for II0m(A)I1a.a- we see that in order to complete the proof of the theorem when s = 0 or s -I= 0 it suffices to verify that II'-m(A)AAII 5 C,

(9.18)

II AA- 0-m(2)II S C.

5. Let us define -m (A) as an operator with the Schwartz kernel which is obtained by multiplying the Schwartz kernel of 45..m(A) by tp(x - y), where

(p E Co (IR"), (P = I in a neighbourhood of 0 E IR". Then &_m(A) is a uniformly properly supported `PDO in L" o;d(1R", (r). Let us write

(9.19)

d5_m(A) + R-m(A)

and investigate the remainder operator R_m (A) which has a Schwartz kernel KR_", vanishing in a s-neighbourhood of the diagonal. In fact it is a convolution operator, so KR_. depends on x - y and A only. It is easy to see from the construction of R_m (A) that KR_", (x, y; A) = r-m (x - y, ),) with

r-m(z, A) = f e'-{(I - V (Z)) (I +

IAI2/d)-m/2 a,

Ia"

where W E Co (IV), (p = 1 is a neighbourhood of 0.

Now it is easy to prove that

I) E S(lR") for every fixed A E C.

A) in S(IR") decay as III -' oo faster than Moreover, all seminorms of any power of IA I. Indeed, we can apply the standard integration by parts to get

r-m(z, A) = f e''{IzI-2N(l _'p(z)) [(-at)"(1 + IRR

hence

IAI21d)-m/2]

§9. Operators with Parameters

83

z'Dlr-.(z,),) = f zaDq [eiz'EIZI-2N(1 - p(Z))} L

(R"

X [(-e{)N(1 + 112 + IAI2/d)-m/21 dt,

for an arbitrary integer N > 0 and any fixed multiindices of, P. The integrand can be estimated by Izl)iai-2N(1

C(1 +

+ 141)1ai(t + 141 +

IAI11d)-m-2N

with a constant C > 0. We can assume that m + 2N > 0 and use the obvious estimate IAI11d)_m-2N

(I + I4I +

< (1 +

IAI11d)-m12-N

to arrive to the estimate lzaDar-m(z, ),)I < Ca0N(1 +

IAI1/d)-m/2-",

which holds for sufficiently large N and implies the desired result. 6. Now we will sketch two possible proofs of estimates of type (9.18) for R_," (A) A, and AAR_m (A).

Let us recall that it is assumed that the Schwartz kernels of A, are supported in a fixed (independent of A) compact subset of IR" x IR".

(a) Note that the proof of the boundedness result (e.g. Proposition 7.5) implies that IIAaII,;-m <_ C(I + ICI)"',

where M = M(s, A) is independent of A. This is a rough estimate and it is easy to obtain by following the steps of the proof of Proposition 7.5 and of the necessary results from Sect. 6. Now note that R_,,(A) is infinitely smoothing in the Sobolev scale Hr(IR"), and, more precisely, IIR-m(A)II,. < C.r.t.N(I +

IAI)-N

for all s, t E IR and N > 0. This holds because the convolution operator R_m(),) can be presented as the multiplication of the Fourier transform by a function r_m(i;, A) = F2.4r(z, A) which is in S(IR") with respect to t with seminorms which decay as IAI --+ oo faster than any power of IAI due to the estimates obtained in the first part of my comments to this question above. Combining the two inequalities above we immediately obtain the desired estimates of the type (9.18) for R_m ().)A, and A, R_m (A).

(b) Another way to establish these estimates is to study first the structure of the operators R_m(A)A, and A,R_m(A). It suffices to consider the operator A, R_. (A) and then use the relation

84

Chapter II. Complex Powers of Elliptic Operators

(R-m(),)Ai)* = AxR-.,(X) to establish the same estimates for the operators of the type R_," (1) Ax. Clearly Ax R_", (1) can be written in the standard form (3.9) with the symbol ax.. (x, 4) = aA, (x, 4) L. (t, )A), where the function F was defined above. Note that o,,, (x, 4) has a compact support with respect to x, uniformly in 4. Taking into account the behavior of r` we see that ax,m (x, 4) satisfies the estimates Ia4 D ax.m(x, 4)1 < Ca.Pm.M.x(l + 151)-'"(l +

IAI)_N

for any M, N > 0 and any multiindices a, P. The Schwartz kernel of the operator A,, R_", (1) has the form

KA(x, y) = f

4) d4.

IR"

The estimates for ax immediately imply that

IKx(x, y)l < CN(1 + Ix - yl)-N, and the required norm estimates follow from the Schur lemma. 7. Now let 8 = 0. It is clear from (9.16) that we need to show that (9.20)

II0-m(A)AAIL.. _< C,

where C does not depend on ,l (but may depend on s). Clearly (9.21)

III-,(A)AxIIF..T =

Acting as in the part 5 of this proof, we may replace 'P, (A) by a properly supported `YDO 0, (A) (for any t E IR) and instead of (9.20) prove the estimate

C.

(9.22)

Denote the symbol of 4;_m(A)A, by b(x, 4, A). We claim that b(x, 4, x) E S°O.d(IR"; A)

in the sense of the uniform classes in IR" (see problems 3.1 and 3.2), and in particular b(x, i . ,l) E SO °(1R2') uniformly in A, i.e. Q

sup[Ia,dfb(x,

+oo

x.E.)L

Using an appropriate composition formula (e.g. in the uniform classes discussed in Problems 3.1 and 3.2) we see that the same estimates hold for the symbol a(x, 4. X) of the operator ds,(0)dy_m(A)Ax _,(0). Now applying the boundedness theorem (again extended to uniform operators in lR") we get the desired estimate (9.22) for the operator norm. Another possible way of arguing (to avoid using uniform operators in IR"): introduce appropriate cut-off functions to reduce everything to functions sup-

§9. Operators with Parameters

85

ported in a fixed compact set, and then investigate remainders, using the Schur

Lemma. 0 9.3 The inverse of operators with parameter. In this part we will consider only operators on a closed manifold M. Theorem 9.2. Let A .x eHLQa5(M, A). Then there exists R > 0, such that for IzI z R, the operator A, is invertible with AA-' EHLQ.e°e '`(M,i1R).

(9.23)

where AR = A o (.111) I z R}. More precisely, if B, is a parametrix for the operator with parameter A, , i.e. condition (9.6) is fulfilled, then AA-' - B,1 E L- x (M, AR).

(9.24)

Proof. Let B, be a parametrix for the operator A,. Then it is obvious from (9.6) that it suffices to prove that I + R, with R, e L- ac (M, AR) is invertible for small A and

(I+R,,-' - 1EL-°O(M, AR).

(9.25)

Note that for arbitrary N > 0 and s, t E 1R

IIR1II, < C;"'(1+IAI)-".

(9.26)

From this it follows, in particular, that there exists R > 0 such that II R, II R is guaranteed in each of the spaces H' (M). To prove (9.25) it is convenient to use

(1+R,)-' - 1= -R,(I+R,)-'

(9.27)

and (9.26). Denoting the left hand side of (9.27) by Q, we see that estimates of the

form (9.26) hold for Q, . The kernel Q,(x, y) of Q, can be expressed by the formula Q,I (x, Y) _ [Qab( - -Y)l(x),

(9.28)

where b (z - y) is the b-function (in z) at a point y e M on which it depends as a parameter. The operator Q, in (9.28) acts on the variable z and the result is taken at the point x. Note that if s < - n/2, then b (- - y) a H'(M). Further, b (- - y) is

Chapter II. Complex Powers of Elliptic Operators

86

a differentiable function of y with values in H' -' (M) and is more generally a k times differentiable function of y with values in H-' -'(M). Therefore from (9.26) it follows that estimates of the form (9.3) hold for the kernel Q j(x, y), which also demonstrates that Q, a L- °° (M, AR). The inclusions (9.23) and (9.24) then also readily follow. 9.4 The resolvent of an elliptic operator. Returning to example 9.1 in the case

of an operator on a manifold and applying the results obtained we get Theorem 93. Let A be a differential operator on a closed manifold M with principal symbol a. (x, ) and A a closed angle in the complex plane C with vertex 0 e C. Let A be elliptic with parameter relative to A, i. e. for $ 0, a.(x, ) does not take values in A. Then a) there exists R > 0 such that A - AI is invertible for A EAR with

(A-AI)-`eCL.-"(M,AR);

(9.29)

b) the following norm estimate holds

II(A-AI)-'II,.,rj:5 C,.r/I2I'-um,

0515m,

AFAR,

(9.30)

where s is any real number.

Proof a) follows from Theorem 9.2 and b) from a) and Theorem 9.1. Corollary 9.2. Under the conditions of Theorem 9.3 we have II(A-AI)-

II 5 C/IAI,

AEAR.

(9.31)

Corollary 9.3. Let A be an elliptic self-adjoint differential operator on a closed manifold M with principal symbol a (x, l;). Assume that a. (x, i;) > O for all (x, g), 4 0. Then A is semi-bounded from below, i. e. there is a constant C > 0 such that

A>- -CI or (Au,u) >_ -C(u,u),

(9.32)

We mention here another important fact, namely that under the condition of ellipticity with parameter, the resolvent (A -AI)-' differs from the parametrix only by an operator of L- °° (M, A). This is used in the theory of complex powers along with the explicit construction of a parametrix as given in the elliptic theory (cf. section 5.5).

Problem 9.1. Extend the theory of elliptic operators with parameter to the case of matrix operators, i.e. to systems.

§ 10. Definition and Basic Properties

87

Hint. The condition of ellipticity with parameter for a matrix function a, (x, ), means that det (a. (x, ) - A) $ 0 for * 0 and A e A or, equivalently, that the eigenvalues of a. (x, ) do not belong to A for I # 0. Problem 9.2. Let A be an elliptic differential operator on a closed manifold

M, and suppose that for some angle A the operator A - Al is elliptic with parameter for A e A. Show that index A = 0.

§10. Definition and Basic Properties of the Complex Powers of an Elliptic Operator 10.1 Definition of the holomorphic semigroup A= . Let A be an elliptic differential operator of order m on a closed n-dimensional manifold M and a,,, (x, ) the principal symbol of A. Assume that a,,, (x, c) does not take values in a

dosed angle A of the complex plane C for * 0 (here the vertex of A is assumed to be at 0 e C). In other words, in the notation of §9, A - AI e CLT (M, A) and satisfies the condition of ellipticity with parameter. It follows from Theorem 9.3 that the resolvent R,, = (A - A 1) -' is defined for ? R, A e A i. e. for A e AR. Now, in view of Theorem 8.4, we see that the spectrum a (A) of A is a discrete subset of C. Hence, in the angle there can be only a finite number of points of a (A). We may therefore draw a ray Lo, starting at 0 and running inside A such that a (A) n Lo is either empty or consists of the point 0 only. In what follows, we assume for convenience firstly that 0 * a (A), i. e. A -' exists as an operator (cf. §8), and secondly that Lo = (- oo, 0]. Neither of these assumptions is very essential; we may get rid of the first one by replacing A with A + e1 and the second by studying e'"A instead of A. So, finally, our assumptions are as follows: 1)

for +0 and 1.E(-oo,01;

2) or (A) n (- oo, 0] = 0.

(10.1) (10.2)

It follows from conditions 1) and 2) that for some angle A of the form (n - a:5 arg2 <- it + e), with e > 0, the following hold:

1')a,.(x,I;)-A#0for 4*0and XEA;

(10.1')

2') a(A)nA=0.

(10.2')

In what follows we shall assume that A has been chosen in this way. Since 0 ¢ a(A), we see that a(A) does not intersect a disk I X I < 2q in the

complex A-plane. Now select in this plane a contour r = rQ of the form

r=r, ur2vr,, where (Fig.1)

Chapter 11. Complex Powers of Elliptic Operators

88

lal

or9 a = a 0

r

Fig. I

A= rei" (+ oc > r > Q)

on r,,

.1=Qe'' (n>qp>-n) A.=re-'" (Q
on r2 , on F3.

Consider the integral

Az=Zn fAx(A-AI)-'dA,

(10.3)

r

where z e C, Az is defined as a holomorphic function of A for A e C\ (- oo, 0], equal to ei1nz fort > 0 (here it is assumed that In ,1 a IR for A. > 0). In other words,

on 1' we set A: = er'n1 = ezlnlll+'zar9A =I A IzeizarsA,

(10.4)

where - it < arg A 5 it (arg A is described more precisely in the definition of the

contour r). Note that in view of the estimate (9.31) (Corollary 9.2) the integral (10.3) converges in the operator norm on L2 (M) for Re z < 0 and also A2 is a bounded

operator on L2(M). In the same ray, by Theorem 9.3, the integral (10.3) converges in the operator norm on H5 (M) for arbitrary s E IR and also, for Rez <0, AQ maps H2(M) into H5(M) hence also maps C°`(M) into C' (M) as

well as I'(M) into 9'(M), since

C'°(M) _ fHs(M)

and

W(M) = U Hs(M).

J

S

Proposition 10.1. a) For Re z < 0 and Re w < 0 we have the semigroup property A=Aw= Asrw

(10.5)

A -,= (A-')'`.

(10.6)

b) If k E Z and k> 0 then

c) For arbitrary s E R, A= is a holomorphic operator function of z (for Re z < 0) with values in the algebra of bounded operators on the Hilbert space H2(M).

Proof. a) Construct a contour r' (Fig. 2) of the form r' = r; u r2 u r; , where

89

§ 10. Definition and Basic Properties

Fig. 2

A= reA''-O (+oo>r>i Q) A = z Qe' (n-e>(p> -n+e)

on I'1

on r; on r;.

.l = re'(-x+e) (Z Q
The number a> 0 is chosen so that (10.1') and (10.2') are satisfied. The contour F

is contained "within" I", and in view of (9.31) and the condition on I" it is obvious that the integral (10.3) does not change if we replace F by V. Utilizing this fact, we obtain A=Aw

4I

I

(A-µl)-' Azpw dpd1

J(A-2l)-'

r, r

Azµ"'

4rt rJ rJ x_ p [(A-AI)- -

(A-,uI)-'] dµdA

= L J Aw+z(A-AI)-`dA+..± J J (A -µl)-` r r r,

: w

p

dldp

= Az+w+ 0 = Az+w

In this computation we have used the Cauchy formula and the so-called Hilbert identity (A

(A

[(A

-(A

(10.7)

which is clear if we, for instance, multiply both sides by (A-Al) (A-uI).

b) Note that if z = -1, -2, ..., then (re' )' = (re-ix)' and the integrals along the straight line parts of r in (10.3) cancel. Therefore A-k __

J -' (A-.ll) 2s r,

' d2,

90

Chapter H. Complex Powers of Elliptic Operators

where 12 = {IAI = e}, traversed clockwise. Now make a change of variables, putting A = 1/p which gives r x Jpk(A_p-'I)-ip-Zdp, A_k= -2

r;

where I'2 = fl p I =1 /e}, also traversed clockwise. Taking into account that (A

-p`I)-' =,uA-1(p1-A-')-', we may now write lA-' A-k = - 2n Jp k-1 (pl-A- )-'du = A- (A r; 1

1

1)k

1

-= (A

1)k,

since the entire spectrum of the bounded operator A-' is situated inside the contour f and we may use the Cauchy formula. c) Differentiating the integral (10.3) with respect to z we obtain the integral

J2Z (ln.l) (A-AI)-1 d.1, 2n r

(10.8)

converging in operator norm (in H'(M)) uniformly for Re z:5 - s < 0. We conclude, that the operator function A. is holomorphic in z and the derivative dz

As is equal to the integral (10.8).

CJ

10.2 Definition of the complex powers of an operator Definition 10.1. Let z e C and k e Z be such that Re z < k. Put, on C °° (M) or -9, (M) AZ = AkAZ k -

(10.9)

We need to verify that this is well-defined and that is the content of the first part of the following theorem.

Theorem 10.1. a) The operator A' as defined by (10.9) is independent of the choice of integer k, provided Re z < k. b) If Re z < 0, then A' = AZ . c) The group property holds

AzA'"=AZT",

z, weC

(10.10)

d) If k c- Z, then (10.9) with z = k gives the usual k-th power of the operator A

(in particular A° = I, A'= A and A - ' is the inverse to A).

§ 10. Definition and Basic Properties

91

e) For arbitrary k e Z and s e IR, the function A= is a holomorphic operator function of z in the half-plane Re z < k with values in the Banach space .V (H'(M),

H'-Mk(M)) of bounded linear operators from H'(M) to H' -'k (M). Proof. a) Let z e C, k, I e Z be such that Rez < k, Re z < 1. We need to verify

that A&A=_k = A'A=_1

(10.11)

Assume that k > 1 and put k -1= p and z - k = w. Then (10.11) reduces to the

equality Aw=A-PA,,.,, with p a positive integer and Re(w+p)<0. This however, follows at once from Proposition 10.1 since A -P =A -P by (10.6) and it only remains to use the semi-group property (10.5).

b)-d) These properties follow in an obvious manner from a) and Proposition 10.1.

e) This statement follows straightforwardly from a) and c) in Proposition 10.1, if we remember that Ak, for k an integer, maps H'(M) continuously into H'-"(M) (this follows from Theorem 7.3 (on boundedness)

and the fact that A-'eCL-M(M), as is clear from Theorem 8.2 (about the inverse operator)).

10.3 The self adjoint case. Let a smooth positive density on M be given defining a scalar product on L2(M) and A a self-adjoint elliptic differential operator of order m on M. Then its principal symbol a. (x, 1;) is real-valued. In this case, conditions (10.1) and (10.2), which we assume to hold, mean that

a.(x,4)>0,

A?bI,

4#0,

6>0,

(10.12) (10.13)

Let {(p}' 1 be a complete orthonorRemember that mal system of eigenfunctions for A with eigenvalues Aj-. + oo as j - oo (cf. Theorem 8.3). It follows from (10.13) that A,? 6 > 0 for all j = 1, 2, .... Now, any distribution f e 9' (M) may be represented as a Fourier series r

i.e. (Au,u) z 6(u,u) for any

f- > fapt(x),

where

xeM,

(10.14)

1=J

f=(f,w;)

(10.15)

Here, when f e L2 (M) we of course have the usual scalar product in LZ (M).

If, however f e 9' (M), then (f, 9j) denotes U; ipjdp>, where du is the fixed density on M (recall that the distributions are linear continuous functionals on the space of smooth densities). We now describe the properties of the Fourier series of smooth functions and distributions.

92

Chapter 11. Complex Powers of Elliptic Operators

Proposition 10.2. For a series OD

(10.16)

Y- ci(pj(x)

j=1

with complex coefficients cj the following properties are equivalent: a) the series (10.16) converges in the C00(M)-topology; b) the series (10.16) is the Fourier series of some function feC°°(M);

c) for any integer N Icj12AJN<+oo.

(10.17)

j=1

Furthermore, conditions d), e) and f) are also equivalent: d) the series (10.16) converges in the weak topology of 9'(M);

e) the series (10.16) is the Fourier series of some distribution f e 2'(M); f) the exists an integer N (perhaps negative), such that (10.17) is fulfilled.

Proof. The basic idea of the proof is to use the relations

C°°(M) = nHs(M),

9'(M) _ y Hs(M), a

s

and the fact that A' is a topological isomorphism between the spaces H'"N(M) and LZ(M). Now, LZ(M) is easily characterized in terms of the coefficients of Fourier series by the Parceval equality. Therefore the topology of C°`(M) may be determined via the seminorms II f II A; N where IIJ IIA:N = j=1

'ii

'J",

because ANf has the Fourier coefficients ,tN J. From this the equivalence of conditions a), b) and c) is obvious. Let us verify the equivalence of d), e) and f). If d) is satisfied denote by f the sum of the series (10.16), so that f e 9' (M) and W

(f (P) _ Y- cj((Pj,w) J=1

for any p e C'° (M). In particular, taking (p = (pj we obtain cj = j i.e. precisely e). Further, if e) is satisfied, i. e. cj = (f, qpj), with f c- 9' (M), then selecting an integer N such that AN f e LZ (M) we also see that f) is satisfied. Finally, if f) is satisfied,

then the series (10.16) converges in the norm of HN(M), thus giving d).

§ 10. Definition and Basic Properties

93

We now introduce a "spectral" characterization of the complex powers of a

self-adjoint operator A in terms of the coefficients of the eigenfunction expansion associated with the eigenfunctions rpt. x

Proposition 10.3. Let f e 21'(M) and let f (x) = Y_ fjgpj (x) be the Fourier I=I

series expansion off in the eigenfunctions of the operator A. Then

Azf =

A; f gyp; (x)

(10.18)

.

l=1

In particular, q (x) are the eigenfunctions of the operator As with eigenvalucs Aj .

Proof. The operator AZ maps C'(M) continuously into itself. In view of the easily verified relation (AZ)* = A4 we see that AZ being the adjoint of AZ

continuously maps 2'(M) into 1'(M) provided I'(M) is endowed with its weak topology. Since the series on the right hand side of (10.18) converges weakly in -9'(M) by Proposition 10.2, in order to verify (10.18) in the general case it suffices to do so for f = V,, Re z < 0. But, f = gyp; and Re z < 0 imply JA'(A-AI)-I

AZpt = 2n r

(ppd2 = 2r r

Az(Aj-A)

d) = )Jrpt

r

by the Cauchy formula. But this is exactly (10.18) for f = (Pp

0

Exercise 10.1. Let A satisfy (10.1) but instead of (10.2) require the weaker condition

a(A)n(-oo,0) _0, so that KerA may be a non-empty finite-dimensional subspace in C'(M). Define A. via the contour integral (10.3) and let E0, EE be the invariant subspaces of the operator A introduced in Theorem 8.4. a) Show that A2 E° = 0 and EE is an invariant subspace of A. for Re z < 0. b) Show that for the operators Az the semigroup property AZ A.= Re z < 0, Re w < 0 holds, allowing Az to be well-defined for all z by formula (10.9), so that the group property (10.10) holds for the operators A. c) Verify that Az for a sufficiently large positive integer z is the usual power of A whereas

A°=I-P° where P° is the projection onto the subspace E° parallel to Ea and that, analogously, A _k for a sufficiently large positive integer k is the inverse of the operator Ak on EE and equals 0 on E°.

94

Chapter Il. Complex Powers of Elliptic Operators

Problem 10.1. Let A be an elliptic differential operator with principal symbol a, (x, ) on a closed manifold M. Assume that Re aT (x, ) < 0 for # 0. Show that the Cauchy problem au

81=Au,

t>0;

U I'=0 = (P W;

(10.19)

has a unique solution in C°°(M) and .9'(M). Hint. The solution u (1, x) will necessarily be of the form u = e`Aco ,

(10.20)

where the operator e" is determined via the contour integral e'" = 21

r

Se-t'(A-2I)-1 dA.

(10.21)

Assuming that the spectrum a (A) is situated in the half-plane Re A < 0 (which can be achieved by changing A into A - CI or substituting u = ve' in (10.19)), it suffices to take r =1', u T2 , where r, and r2 are the following two rays:

A=re'lz-` (+oo>r>0) on I',, A = re`

Z

(0 < r < + oo) on F2 .

The uniqueness of the generalized solution of the problem (10.19) is demonstrated using the Holmgren principle (cf. e. g. Gel'fand 1. M., gilov G.E. [1], vol. 3).

§ 11. The Structure of the Complex Powers of an Elliptic Operator 11.1 The symbol of the resolvent. Let A be an elliptic differential operator on a closed manifold M. We shall next construct in local coordinates the symbol of a special parametrix of the operator with parameter A - Al (which we view here as an operator in CL.(M, A), A a closed angle in C with vertex at 0). We assume that A satisfies the conditions for ellipticity with parameter relative to A, where

the angle A is as described in §10 (i.e. it satisfies (10.1') and (10.2'), and A contains the semi-axis (- oo, 01). The parametrix will be constructed in a chart XcM and we will identify X with an open set in 1R" using a coordinate system on X. Let the operator A on X be of the form

§ 11. The Structure of the Complex Powers

95

A = E aa(x)D*.

(11.1)

IaIS m

Its total symbol Y_ a. (x)

(11.2)

I*IS on

may be decomposed into homogeneous components

a; (x, ) _ Y as (x)

j = 0, 1.... , m .

(11.3)

I=I=;

The total symbol a (x, , A) = a (x, ) - A of A - AI may be decomposed into components homogeneous in (, A"'") given by the formulas

a. (x, S, A) = a,, (x,0-), a, (x,

(11.4)

j=0, 1, ..., m-1.

(11.5)

The condition of ellipticity with parameter means that

for xEX, J.EA,

it 1+1A1'tm

0

(11.6)

It is natural to look for the symbol of the parametrix of A - ).1 in the form of

Denote these an asymptotic sum of functions homogeneous in functions by b°-m_j (x, , A), j = 0, 1, 2, ... , where the lower index indicates the degree of homogeneity:

t>0, IfI+IAI'1'"*0.

(11.7)

These functions are recursively defined by the relations

am(x,

0,

(11.9)

k+I+IaI=J

I<j

j=1,2,...

(compare the construction of a parametrix for the classical elliptic PDO in §5, formulas (5.17') and (5.17")). To obtain a real parametrix from these functions b°- m -t (x, , A) it is first of all necessary to eliminate their singularities for I I + 1). 1 = 0 by multiplication

with a cut-off function and, secondly, to glue together the various local parametrices using a partition of unity (cf. §5).

Chapter II. Complex Powers of Elliptic Operators

96

11.2 The Symbols of complex powers. We shall construct the homogeneous components of the symbol of Az using the homogeneous components of the parametrix constructed earlier, in exactly the same way as the powers AZ were constructed using the resolvent (A - Al) - `. Indeed, from the condition of ellipticity with a parameter there follows the existence of Q = Q (x, ) such that am (x, , A) * 0 for I A I < 2 Q, r; 0 and A e C. From (11.8) and (11.9) it is clear that b-m -i (x, r;, A) is holomorphic in A in the disc I A I < 2,q. Forming the contour F as in §10, we may define the functions b(ZZ. ) for Rez < 0 by the formulas 1(x,

b(=z.0

(x, ) =

j=0, 1, 2, ...,

fA

27r

r

(11.10)

where the branch AZ is defined as in §10.

In particular, for j = 0, we obtain by the Cauchy formula that bTZ=.°(x,

) =27rt

)-A)-'dA

I AZ(a,(x,

r

= a, ,(x, ).

(11.11)

Let us note that for a sufficiently small Q the integral (11.10) is independent of the

choice of g (in the disc I A I< 2 Q there are no singularities of the functions (x, r;) is positively homogeneous in of b _ m _i (x, , A)). The function degree mz - j, i.e.

t>0,

r;$ 0.

(11.12)

To prove this, it is necessary to perform a change of variables in the integral (11.10) and use the homogeneity of bom-i (x, 4, A): 2n

r = 2 f (tmp)zb-M-i(x, tm1) . tmdp r'

=

tmz-i

-t

mz-I

i

_

2n 1µ t

2n r

mz-j (z).0 fµzb-m-i(x,,p)dp=t 0 bmz-i(x,

Here F` is the contour t-"F (it has the same shape as F but the radius of the curved part is t-mQ instead of Q). (x, ) to all z E C. This is Now it is necessary to extend the definition of done in the same way as the construction of AZ for Z E C in §10. The following analogue of Proposition 10.1 holds.

Proposition 11.1. a) For Re z < 0 and Re w < 0 we have the semigroup property

97

§ 11. The Structure of the Complex Powers as 1a1 +p+q = j

o b(2). '° (x ) xD41 b"1- o, (x )/I = b(2 { n,z-P mw-q ,n(2+w)-j (x,

j=0,1,2,... j (x, l; ), j = 0, 1, 2, ... , is the set of b) If k e Z and k > 0, then the set b! homogeneous components of the parametrix of Ak. c) For any multi-indices a, fi the derivative 64 '00 b'21-_°j (x, c) is a holomorphic

function of z for Re z < 0 and t + 0.

Proof. This is achieved by repeating on the symbol level the proof of Proposition 10.1; recommended to the reader as a useful exercise.

L

In what follows it is convenient to denote by a;k) (x, ), k > 0 and integer, the homogeneous components (of degree j) of the symbol a(') (x, S) of the operator Ak, so that a(k)(x,

S) _ Mk > aak)(x, j=° FF

xx

If k is an integer and k < 0, then by a,") (x, ) we denote the homogeneous components of the symbol of the parametrix to the operator Ak, or, what is the

same thing, the homogeneous components of the symbol of A-;. They are defined recursively by the relations a! ,'A(x, a(-nik (x, S) ' a4akA -j (x,

S) +

a a(mA(x,)

+

(11.14)

akA(x,

Dsa'"kA-4

(x, )la! = 0,

j = ], 2. ...

(11.15)

P+q+ 1411 =j q<j

Definition 11.1. Let z e C and k e Z be so chosen that Re z < k. Put b,(z=,o

(x,) _

a4a(kA-P(x,i)

Dxbrnz(:!c)k)-9(x,)/«!,

P+4+1411 -j

j=0, 1,...

(11.16)

Theorem 11.1. a) The function b _ , j (x, ) as defined via formula (11. 16) is independent of the choice of the integer k as long as Re z < k. b) If Re z < 0, then the functions b.a. ° (x, ) obtained by formula (11.16) coincide with the functions, denoted by the same symbol, obtained via the contour integral (11.10). c) The group property (11.13) holds for any z, w e C. d) If k e Z, then b (kA'-jo (x,

a(.k) o -j

(x,

)

e) For any multi-indices a, fi, any x, l; ( * 0) and any j = 0, 04"6$ x Ma-j(x, ) is an entire function in z.

1,

... ,

Chapter If. Complex Powers of Elliptic Operators

98

Proof. Statements a)-d) are proved in exactly the same way as the corresponding statements in Theorem 10.1 (it is only necessary to pass from the

operator algebra to the symbol algebra consisting of formal series of homogeneous functions and with multiplication given by the composition formula for symbols). The proof of e) is obtained immediately by looking at the formulae defining the functions b.(=Z, ° (x, 0. 11.3 Smoothed resolvent symbols. Let w (t) E C °° (R'), so that co (t) = 0 for

t- 1. Put (11.17)

IZIZ'm)

and let us define (11.18)

With the help of the function b_m -j (x, , A) we construct on the manifold M

a parametrix of the operator with parameter A - Al in a way similar to the second part of the proof of Theorem 5.1. Now let M = U Xr be a finite covering r

of M by charts, qv a subordinated partition of unity and the functions wr e Co (Xr) be such that wr =_ I in a neighbourhood of supp Apr. Further let 01, Tr be the multiplication operators by q and wr and By m-j(Z) a pseudo-

differential operator on Xr with the symbol by m_ j(x, , Z) constructed by formula (11.18) in the coordinate neighbourhood Xr. Now put B-m_j(Z) _ Y_ V'Bym-j(A) V7

(11.19)

r

and

N-1

B(N)(A) _ Y

B-m-j(Z).

(11.20)

j=o

Proposition 11.2.

(A-J.I)-' - B(N)(Z)ECL;m-N(M,A).

(11.21)

Proof. First we construct an exact parametrix B (Z) of the operator A - Al, putting 00

B(2) - Y B-.-j(Z)

(11.22)

j=o

The precise meaning of this formula is: construct in each Xr an asymptotic sum

Y

j=o

(11.23)

then take the operator By (Z) in Xr with the symbol by (x, , Z) and finally put

§ 11. The Structure of the Complex Powers

B(,) = EOYBYVIY,

99

(11.24)

Y

where 0Y, TI are as in formula (11.19). Then we obviously have

B(A)-B(N)(A)ECL;"'_N(M,A).

(11.25)

But from Theorem 9.2 it follows that

(A-..n-' - B(2)EL-°°(M,A).

(11.26)

Equating (11.25) and (11.26) we arrive at (11.21). C 11.4 Smoothed symbols of complex powers and the structure theorem. Let w (r) be the same function on IR' as at the beginning of 11.3. Put 6(e) =

(11.27)

and define

b"': _

0 (x,

(x,

(11.28)

The construction of (x, and bMz _ j (x, ) can be carried out in any coordinate neighbourhood XY (the corresponding functions will be denoted by

b? Y (x, ) and

Y

j (x, ) if we need to know exactly which coordinate

",-

neighbourhood). Denoting by B( .Z=_7. the operator in XY with symbol b(')-Y (x, ),

we once again put

Bz) - j _ Y 45 7 B(,nzz). Yj T Y

(11.29)

V

and

-

N-1

B(Nlz)) -

Bmz(z)

j=o

(11.30)

-

For the statement of the basic structure theorem, we shall also need the definition of holomorphic families of `PDO, which constitute the set d (G, LQ a (M)), where G is a domain in the complex plane C.

Definition 11.2. Let X be an open set in IR", G a domain in C, a(x, S, z) E C°° (X x IR" x G) where a (x, , z) is holomorphic in z. We shall write a (x, , z) E d (G. SQ a (X)) if for any multi-indices a, fi any k e Z + and any compacts K1 c X, K2 c G there exists a constant C = C (a, fl, k, K, , K2) such that l

IPl,

for (x, , z) E K, x IR" x K2. If A (z) is a `PDO in X, depending on the parameter z, then we shall write that A (z) E d (G, LQ,,, (X)) if A (z) = A, (z) + R (z), where A, (z) is a properly supported TDO on X with symbol a (x, , z) e 0 (G, SQ d (X))

100

Chapter 11. Complex Powers of Elliptic Operators

and R (z) has kernel R (x, y, z) e C°° (X x X x G), holomorphic in z. Finally, we shall write that A (z) e d (G, L. a (M)), if A (z) is a PDO on M, depending on the

parameter z c- G such that for any coordinate neighbourhood Xc M and any local coordinates x: X + X, , X, an open set in IR", the family of operators AK(z) on X, induced by the operators A (z) on M via the diffeomorphism x is such that A"(z) e 0 (G, LQ a (Xi ))

Repeating the arguments of §4 we see that to verify that A (z) e 4 (G. L, a(M)) for 1 - g:5- 6 < p it suffices to do so for a fixed coordinate covering of M and fixed coordinate diffeomorphisms. Also, repeating the argument in §3 and §5, we see that composition, taking the adjoint and forming the parametrix of a hypoelliptic operator (we assume that the condition for hypoellipticity is fulfilled uniformly in z) does not take us outside the holomorphic families, specified in Definition 11.2. We shall now formulate the basic structure theorem.

Theorem 11.2. For any z E 4 A= E CL" (M),

(11.32)

moreover for any integer N;-> 0 and t E lR

A= - B((N) e0 (Rez < 1, L1"0'(M)) .

(11.33)

Proof. 1. First let us show that (11.32) follows from (11.33). For fixed z e 4 set

(11.34) i=o

(this formula can be decoded in the same way as (11.22)). Then it is obvious that we may assume B(=) e CL"'=(M).

(11.35)

Now, fixing in (11.33) t e IR such that t > Rez and then letting N tend to + oo, we

obtain

A=- B{=)eL-'(M),

(11.36)

from which (11.32) follows. 2. Let us prove (11.33). Set RN' = A= - BAN), .

(11.37)

§ 11. The Structure of the Complex Powers

101

First of all note, that the group property of the complex powers A= and its, for the time being hypothetical, symbols b.==_0 (Theorem 10.1 and 11.1) together

with the fact that the composition of holomorphic families yields again a holomorphic family allow us to reduce the proof to the case t = 0 (N may be arbitrary). In other words, it suffices to verify that

R((N)e0(Rez<0,Li

(11.38)

o(M)).

In order to make use of Proposition 11.2 it is convenient to consider for Re z < 0 and together with the operator Ba)), the operator (11.39)

SA B)N)(A)d? .

2n r

'B((") =

It is easily verified that 'Bj(N)) -

B(()))e0(Rez<0,L-x(M)).

(11.40)

Indeed, we obviously have N-1

'Bl INz) )

= Y- ,B(z) mz-;+ i=0

where 2n

JA

(11.42)

(x, ) of the operator 'Bm(z=_; is expressed in and this is why the symbol some local coordinate system by the formula 'bmz:-;(x, 4) = 2n

(11.43)

r

From this it obviously follows that b;.z. -;

(x, ) =

=1(x, )

for

1

I>1

(11.44)

But then the same holds for the symbols j(x, ) of the operators B(z)-,, the sum of which gives the operator B((N) (cf. the formulas (11.28)-(11.30)). Taking the obvious estimates for the derivatives with respect to z into account, (11.40) is

obtained at once. Now, in view of (11.40), it is clear that it suffices to verify a memberthe symbol ship of the type (11.38) for 'R((N) = Az -'B(N). Denote by of 'R(()1 in some local coordinate system and by r(N)(x, , )) the symbol of R1N)(A)=(A-AI)-1_B(N#). Then

Chapter II. Complex Powers of Elliptic Operators

102

(11.45)

JA

and we have

2n r

J A (in A)' 01,V rN)(x,

(11.46)

But in view of Proposition 11.2, we have the estimate Ia

s

xeK,

(11.47)

where K is some compact set in the coordinate neighbourhood under consideration. From this it follows that

xeK,

I64aBr(N)(x, ,A)I <

(11.48)

and via (11.46) we get (11.38) for

Exercise 11.1. Extend Theorem 11.2 to the situation described in Exercise 10.1.

§ 12. Analytic Continuation of the Kernels of Complex Powers 12.1 Statement of the problem. Expressing the kernel in terms of the symbol. Let M be a closed manifold, A an elliptic operator on M, satisfying (10.1) and (10.2), which makes it possible to construct the complex powers. For Re z < - n/m we denote by A, (x, y) dy the kernel of A2 (this then depends on the parameter x e M and is a density on M and may, in local coordinates defined for y E Y, be expressed as A. (x, y) dy, where dy is the Lebesgue measure defined by

the local coordinates and A. (x, y) is a continuous function on M x Y). By a abuse of language, this function A, (x, y), which depends on z and on the local coordinates in a neighbourhood of y, is called the kernel. Our immediate goal is to construct an analytic continuation (in z) of the kernel A. (x, y) to the entire complex z-plane C. Note, that if X, Y are open subsets of M, then A. (x, y) is uniquely defined for x e X, y e Y by the values (A=u, v) for u e Co (X) and v c- Co (Y).

Now let X be a coordinate neighbourhood (not necessarily connected), which we identify with an open subset of IR". If we write the `NDO B E Lm (X) in the form

Bu(x)=

-Y4

u(y)dydt,

§ 12. Analytic Continuation of the Kernels

103

where b e Sr(X x R), then for I < - n the kernel A (x, y) is continuous and is of the form

A(x,y)= The restriction of the `FDO A= to X (cf. 4.3) can be represented in the form

A' = A,,,+ R,,,, where A,,, is a properly supported `I'DO and R,,, is an operator with kernel R, (x, y, z) e C °° (X x X x (), holomorphic in z and equal to zero for x and y close to each other. Denoting by a (x, , z) the symbol of A,,, we

will also by abuse of language, call this symbol the symbol of A'. The kernel A,(x, y), for x, y e X close to each other, may be represented in the form

.4,(x,),)-

(171)

The kernel A, (x, y), for Rez < - n/m, is continuous and holomorphic in z. For x = y we obtain A, (x, x) = I a (x, , z) d; .

(12.2)

Note that the result of the integration in (12.2) (and in (12.1) for x and y close to

each other) does not depend on the choice of the "symbol" a (x, , z). 12.2 Statement of the result. In the statement of the result we will make use °,(x, ) of the symbol of A', which were of the homogeneous components

constructed in §11. Note here that for Rez < j/m, these homogeneous components are given by the formulas b

;'-°t(x, )=2n

A 1*

where bo_",_;(x, , A) are the homogeneous components of the symbol of the parametrix for the operator A - Al, also constructed in §11.

Earlier (12.3) was applied only for Rez < 0, but the integral in (12.3) converges for Rez <J/m, hence both parts of (12.3) are holomorphic in z for Re z < J/m demonstrating their equality for these Z. We shall also need the functions

jr'bo-(x,i,-r)dr,

j=0, 1, 2. ...,

(12.4)

o

defined for - I < Re.- < J/m and positively homogeneous in .; of degree mz - j.

Theorem 12.1. Let X be a fixed arbitrary coordinate neighbourhood on M, A, (x, y) the kernels of the complex powers A' of the elliptic operator A. defined for

Rez < - n/m and for x, y e X. Then y the function A,(x, y) can be extended to an entire function 1) for x of z, equal to O for z = 0, 1, 2, ...;

Chapter II. Complex Powers of Elliptic Operators

104

2) AZ(x, x) can be extended to a meromorphic function in the whole com-

plex z-plane with at most simple poles, which may be situated only at the points of the arithmetic progression z; = I n, j = 0, 1, ..., and the residue of A, (x, x) at z, is equal to

Y,(x) _ - I

J

(12.5)

m 141=1

where

(2n)'"dd' and

is the surface element of the sphere I fl l = 1;

3) if z , = l is a non-negative integer then 'y1(x) = 0 and the value x, (x) = A, (x, x) of the analytically extended kernel at z =1 is given by the formula

x,(x) _ (-1), 1

(12.6)

J d()- °(x, )dd'.

m 1t1=

Statement 1) is valid uniformly in x E K, , y e K2; K, and K2 being disjoint compact sets in X, i.e. K= (x, y) can be continued to an entire function of z with values in C (K, x K2). Similarly, statements 2) and 3) are uniform in x E K, where K is a compact set, i. e. the map z -+ A= (x, x) viewed as a function of z with values in C (K), can be extended meromorphically to the whole complex z -plane with poles at the points z, f = 0, 1, ... , with residues y, (x)1 K at these poles and values x, (x) I

at 1=0, 1,2,....

Remarks. 1) In formula (12.5) the function O%; ° (x, ) appears, which may, according to the notation in § 11, be written also b'=;l. ° (x,

)=

b':d. o (x,

(12.7)

MzJ-J

since mzj -j = -n. 2) Since z;
,-"

i

m- 2n

J

d1' J k - b°_,,-1(x, , A) d7..

r

ICI=

(12.8)

A similar expression can be written also for x&):

x1(x) _ (-1)

1

J

m 141=1

d

-r)dr,

J

(12.9)

0

where I is a non-negative integer (the subscript -m (1 + 1) - n in (12.9) is In obtained by expressing j in terms of l by J - = l ). n

3) Note the special formula for the residue of the left-most pole z°

n/m:

§ 12. Analytic Continuation of the Kernels

yo(x)= --

105

(12.10)

J

m itt =

In the important special case, when a. (x, ) > 0 for * 0, it follows from this formula that yo (x) + 0. In what follows in this case we transform (12.10) to a form more convenient for applications.

Proof of Theorem 12.1 1. We will make use of the structure theorem 11.2 and the notation used there, viz. B,,,--) and R(N) = Aa - Ba)). Let us denote by RAN) (x, y) the kernel of RAN) and by r((N) (x, ) its symbol in some chart X. Then, if

x, y e X we have

Ri))(x,y)= This integral converges for Re z <

dc.

N- n m

(12.11)

and defines for these z a holomorphic

function of z with values in C(K1 x K2), where K,, K2 are arbitrary compact sets in X. Therefore the statements about holomorphy, poles and residues reduce to the corresponding statements about the kernels of the operators B((N) .

The kernel of B(()), in its turn, is a sum of terms of the form BMl.. (x, y) =

Je'(x-r

(12.12)

therefore in what follows we will study integrals of the form (12.12). 2. Let K, and KZ be disjoint compact sets in X. We will show that

L, (x, y)

is an entire function of z with values in C(K1 x K2). Integrating by parts in (12.12) for x # y, we obtain

Bl')-i(x,y)=I e'(x-Y)'4 IX-yI_2M4
(12.13)

where M is a positive integer. This integral converges already for Re z <

2M+j-n m

and defines a holomorphic function of these z with range in

C (K, x K2), since l x - y I > e > 0 for x e K, , y E K2 . Since M is arbitrary, it is clear that B'";,)_, (x, y) is an entire function of z with values in C (KI x K2). 3. We have for x = y

or

B(z) -i(x,x) = B(ai _; (x, x) = J 0

b,MZ_1(x, )

(12.14)

co (IgI),w(T)eC°`(IR1),w(t)=0 for t5i,w(T)=1 for t where 0( Passing to polar coordinates in the integral (12.14), we obtain

Chapter II. Complex Powers of Elliptic Operators

106

BM=-i (x, x) =

f cu (r) r"-J+4-1 dr

bmz= 0 (x, ) dd'

.

(12.15)

iSlJ

Obviously, the second factor is an entire function of z with values in C(K) (K any

compact set in X). The first factor may be decomposed into the sum 00

1

5 w(r) rmz-j+ n- 1 dr = I w (r) r'"t-j+ e- 1 dr + I rm:-1+n- 1 dr. 0

0

(12.16)

1

The first integral in (12.16) is an entire function of z and the second one can be computed

f rmz-t+n-1dr= -

where z, =i mn. Therefore

1

mz-j+n

=_

1

1

m z-zz

(12.17)

j (x, x) has one pole with the residue y,(x),

given by the formula (12.5). Let us verify that y, (x) = 0 if z, =1, a non-negative integer. This is clear from

Theorem 11.1, part d) (the functions b° (x, ) are the homogenous components of the symbol of the differential operator A' and therefore vanish for

ml -j < 0 and, in particular, for ml -j = - n). 4. To conclude the proof of 1) and 2) in Theorem 12.1 it remains to show that A, (x, y) = 0 for x + y and 1 e Z + . For this, it suffices to show that if u, v E Co (X) and supp u n supp v = 0, then f A, (x, y) u (y) v (x) dy dx = 0.

(12.18)

This however is equivalent to the fact that = 0, since in view of Theorem 10.1 part e), the function is an entire function of z. Thus we have shown 1) and 2) of Theorem 12.1 and the absence of poles for z e Z + . In what follows we will in fact give a new independent (although also more intricate) proof, allowing us to compute A, (x, x) for 1 e Z + . The reader who is not interested in this computation may omit the remainder of the proof. without any loss of understanding of the later parts of the book. 5. To investigate the values of A= (x, x) for z e Z + , it is convenient to use

another approximation of A= (x, y) which is obtained if we smooth off the symbols of the parametrix b°_-m _; (x, , A) instead of the symbols M' (x, ) of the complex powers. This was essentially done in 11.3, where we introduced the J, 'BiN) and 'R,N) and their symbols, operators B_m_ j(A), B(N)(A), R(N)(A), defined in any coordinate neighbourhood X, b _ m _; (x, , A), b(N) (x, , A), r(N) (x, , (x, ), 'r(J) (x, f) (cf. formulae (11.17)-(11.48)). As

),

107

§ 12. Analytic Continuation of the Kernels

an approximation of the kernel A. (x, y) we will use here the kernels 'B((H) (x, y) of

the operators

:

N-1

'Bi )) (x, y)

'B! (x, y)

j, expressed by the formula

where 'B',,;= _ j (x, y) is the kernel of the operator 'B(z)

mz-;

,

(X,

(12.19)

=o

Y) _ (eicx -r

'b(:)

111:-J

(x )d

(12.20)

or

b_m_j(x,c,A)dl.

'B(z=-1(x,y) = 2x Jd J'1 el(x-Y

r

(12.21)

Here an important role is played by the choice of contour F, since, generally speaking, the integral (12.21) depends on the radius of the curved part of the A) entering the definition of b _ m _1(x, 4, A) is contour (the cut-off function not holomorphic in A). We shall denote by em, the radius of the curved part of the A) in contour F, Q > 0. In view of the homogeneity of bo_m_ _j A 'I-), it is

clear that there is a constant L > 0 such that the function bo_ m _ j (x, ,

)) is

holomorphic in A for I A I < Lm I ' Im. We take the radius of the curved part of F

equal to Qm, Q > 0 and such that

e < L/(2 11L? + 1).

(12.22)

Then, if I I + I I2/m = 1 /4 and A e T, we have either I A I > Qm (i. e. A belongs to the straight line part of F) or I A I = Qm and Lm Lm

14

(1 _ QZ)mr2> Qm= IAI

l4

(the last inequality is equivalent to (12.22)). In this way, and in view of the fact that 0 A) = 0 for I2 + 1212im S 1 /2, we may always assume that bo_m_;(x, , d) is holomorphic inside the curved parts of T. The kernels 'B.I=z _, (x, y) are defined and holomorphic for Re z < j/m (for these z-values, the integral in (12.21) converges absolutely). Our present aim is to analytically continue these kernels to the entire complex z-plane. Let us show first of all, that for x $ y, the kernel 'B,4== _; (x, y) may be continued to an entire function of z (and furthermore, if K, and K2 are disjoint compact subsets of X, then the kernel 'Bm= _; (x, y) may be continued to an entire function of z with values in C (K, x K2)). Indeed by the standard integrating by parts I

108

Chapter 11. Complex Powers of Elliptic Operators

B- (x, y) =2n fdd

Ix-y1-2MA

r

(12.23)

where M is a positive integer, and from this expression the holomorphy of j (x, y) for z such that Re z < (j+ 2 M)/m is obvious. 6. We now demonstrate that ' B m z _ j (x, y) = 0 f o r x 4 y and z E Z + . If z e Z + , then A' is a single-valued function and the integrals along the straight line

parts of r in (12.23) cancel. But also the integral along the curved part of r equals 0 by the Cauchy theorem, since dE (b-m-;

[ax

Y_

IaI+IBI- 2M

the function as b°_,, - j (x, , A) is holomorphic in A inside the curved part of r and any derivative Of 0 A) is constant in A for I A I = const.

7. Let us now study the analytic continuation of the integral 'B(z)_j(x,x) =

2n !dt

b°-

;(x, ,.l)d1..

Let r' be the part of r, in I 12 + IA > I and r 2 the part where I < 1 (for I I > I this set is empty). We then clearly have

(12.24)

12/m

f AzObO_m_jdA

r

= f Azb°- m -jdA+ r'

12 + I A

12p

f A"Ob°-._jdA. r=

It is obvious that the integral

fdf

2n

b°-

r

is an entire function of z. This entire function equals 0 for z = 0, 1, 2, ... , since then the integrals along the straight line parts of r2 cancel due to the singlevaluedness of Az, and the integral along the curved part is 0 by Cauchy's theorem is constant for I f I = const and I A I = const and b°_._j (x, 4, A) is holomorphic in A inside the curved part of F'). We may therefore prove all the statements on continuation for the integral

1(z)= f- fd4 Ff It is obvious that

two rays for

1

1<

r' = r for

1 - Q2 .

(12.25)

I

I>

1 - e2

and r'

Let us put F,=F' for

1

1<

consists of 1 - Q2 and

109

§ 12. Analytic Continuation of the Kernels

!'r = I I"(1 _ Q2)-m12 F for I I Z

1 - Q2. Since it follows from (12.22) that

e1 VI - Q2
I/

1 - Q2

and by the Cauchy theorem f Azb°-

f )=bo-

1(x,.,A)dl.

(12.26)

Let us now make a change of coordinates, putting

)1= et l t' on the straight line parts of F,;

A = a' 1 I'a", where a =

, - it 9;5 it on the curved part of f,' .

Q 1

- Qz

The purpose of this change is to derive from (12.25) an integral of a t) and to proceed as in part 3. of this proof. After

homogeneous function in the change, we obtain

if

Azbo-.-jdA = 11

2n r,-I -I

2n r,1 Alb°-

jdA=12+13

1 - Q2,

ICI <

if

1 -Q2,

ICI Z

where 11 = m sin nz n

+0 J

t'":+",-1

b_,"-j(x,

,

-t")dt,

1-141'

t

=

P191

7p2

t Fig. 3

12=m

sinnz It

13=f n

"

-"

+

t"':+,"-1bo 2141

(x, ,-t"')dt,

Chapter II. Complex Powers of Elliptic Operators

110

Let us remark that the sum

I, d +

f

f

12 d

(12.27)

is an integral of a function in t), homogenous of order mz + m - I - m - j in the domain consisting of the intersection of a conic domain and the complement of a ball (Fig. 3). Introducing spherical coordinates in the space t), we see that the integral

(12.27) converges absolutely for Re(mz-j-1) < -n - I and that, in view of (12.17), it can be written in the form

msinnz n(mz I-I-n)

f

iei +t t-

t" 1 bo_j(x,

(12.28)

t>Q

where

t)' is the surface element on the unit sphere S" in the

t)-space. Since

p > 0 and t > p, the integral (12.28) is well-defined as an entire function of z. Further this whole expression can have only one pole for z = z, =1

Mn

; if zj * 0,

1, 2, ... , it vanishes for all z e Z + ; if z j = l E Z + it vanishes for all integer z $l and there is no longer a pole at z = 1. Here one can, of course, write down the value of (12.28) for z = z j =1 E Z + , but it will be more convenient to do this later for the whole integral (12.25).

Let us consider the remaining term

Here it is convenient to go over to spherical coordinates in the c-space. We then obtain R

V1 -a

mz-j+n

eiv(:+n f

(12.29)

141-1

where C= const. From this the fact that the integral is meromorphic is obvious and the only possible pole is at z = zj. If z =1 e Z + we have

f _R

/

f IWI-I

since the function b°-_ _ j (x, , A) is holomorphic in A for JAI < a'. From this it is also clear that for z = I E Z + the integral (12.29) vanishes, except maybe at z = zj

(if zjeZ), where in this case there is no pole.

8. In this way we have demonstrated that the integral (12.24)

is

meromorphically extendable to the whole complex plane, having no more than

§12. Analytic Continuation of the Kernels

11 I

n

one simple pole, which is only possible for z = zj = - . Further, the value of m

this integral for z =I e Z + is zero, with the possible exception of the point z =1= zj, but then we have no pole at z,. Let us consider now just this case, z =1= z j E Z + , and compute the value of the analytic continuation (12.24) at

z=I=zj.

Note that the integral with respect to

in (12.24) is convergent for z = zj,

since zj = J - n < j/m. Decompose the c-integral into a sum of the integrals over

m the ball I < I and over its complement I I > 1. Standard arguments, already used before, show that the integral over the ball I <1 is 0 for z = zj. For I I ? I we have 0 A) = 1 and instead of (12.24) it is enough to consider the integral

I =2n1iflel J d rz

(x,, ) d.I .

(12.30)

Changing to spherical coordinates in the -space, we obtain

I

mz

'

+n

b°-m-, (x ,

r R,-, 2n JA

) ) dAd ' .

, .

(12 . 31)

Using the Cauchy theorem for Re z > -1 we may contract the curved part of I' to 0. Then we obtain for (12.31)

sinnz

n(mz-j+n)

Srzbo_(x, I4I=1

o

The value of this expression for z = z j =1 e Z + equals exactly x, (x), where x, (x)

is given by the formula (12.6) or (12.9). Therefore 'B'';, (x, x) = x, (x).

9. Let us now note that the difference 'R( 'N') (x, y) = A. (x, y)

=- Ids 2

'BiN'i (x, y)

fAze'(X-r)

(12.32)

1r

can be extended to a holomorphic function of z for Re z <

N -n m

with values in

C(K x K), where K is any compact set in X. From this one can see that in the half-

plane Re z <

N- n the functions Az (x, y) and 'BAN) (x, y) have the same poles m

with identical residues. Further, if z = l e Z + and x * y, then A, (x, y) _ 'BAN) (x, y) = 0 (cf. parts 4 and 6 of this proof). It is therefore clear, that by

Chapter 11. Complex Powers of Elliptic Operators

112

continuity A, (x, y) = 'BAN, (x, y) for all x, y (if I< Nm n). In particular, A, (x, x) = 'B()1 (x, x), which together with the result of part 8. completes the proof of Theorem 12.1. O

§ 13. The c-function of an Elliptic Operator and Formal Asymptotic Behaviour of the Spectrum 13.1 Definition and the continuation theorem. Let A be an elliptic operator

on a closed manifold M, satisfying the same conditions as in the foregoing section. Let A2 (x, y) dy be the kernel of A2. For x = y we obtain from this kernel the density A2 (x, x) dx which is well-defined on the whole manifold M and which

can be integrated over M. Definition 13.1. The function

CA(z)_ JA.(x,x)dx

(13.1)

N

is called the S function of the elliptic operator A. In the next section we show that (A (z) can be expressed via the eigenvalues of

A allowing us in the self-adjoint case to obtain the simplest theorem on the asymptotic behaviour of the eigenvalues. For now, we shall be content with the

formal Definition 13.1 and will formulate a theorem on the analytic continuation of the t-function. Theorem 13.1. The function CA(z) defined by the formula (13.1) for Re z < - n/m can be continued to a meromorphic function in the entire complex z -plane with at most simple poles, which can be situated only in the points of

the arithmetic progression z, = j

n j = 0, ], 2,

m

.., except for the points

z; = I = 0, 1, 2, ... , and where the residue y; at zj and the value x, = C (I), in the notations of Theorem 12.1, are given by the formulae y, = J y; (x) dx = -

1 J dx f m At

At

ia

i

f tam Alfdx 1t1=1

b"A o (x, t) d '

141=t

r

(13.2)

1 Jdx j xr= Jx,(x)dr=(-I)' mM It1=1 w _ (-1)' 1 Jdx f m Al

Kl=1

(13.3) 0

§ 13. The C-Function and Formal Asymptotic Behaviour

113

Proof. Follows from Theorem 12.1 by integrating (12.5) and (12.6) with respect to x, taking into account the remarks following Theorem 12.1. i 13.2 The spectral meaning of the C-function. In this section we shall assume that on M there is given a smooth positive density, which by abuse of notation is denoted dx. Then the kernel of an operator may be identified with an ordinary function on M x M. In addition, the self-adjointness of an elliptic operator on M is a meaningful concept. Theorem 13.2. Let A be a self-adjoint positive elliptic differential operator on

M and let 2j (j =1, 2,...) be its eigenvalues. Then C,, (z) _

Rez < - n/m ,

Aj ,

(13.4)

1=1

where the right-hand side converges absolutely for the indicated z-values. This convergence is uniform in z in the half-plane Re z < - nlm - e for arbitrary e > 0.

Proof. Let Rez < - n/m and let A, (x, y) be the kernel of AZ, which is a continuous function on M x M. Let { gp1(x)},'. , be complete orthonormal system

of eigenfunctions for A. Decomposing AZ (x, y) into a Fourier series in the we obtain complete orthonormal system of functions (co A. (x, y) _

J=1

2 j (pi (x) (pi (Y) .

(13.5)

where the series converges in L2 (M x M). If z is real, then by the Mercer theorem

(cf. Riesz and Sz.-Nagy [1), §98) the series (13.5) converges absolutely and uniformly. Putting x = y in (13.5) and integrating over x, we obtain the identity (13.4). In the case of a non-real z it is only necessary to note that I A z I = 2i °Z, from

which it follows that the series in (13.4) and (13.5) converge absolutely and uniformly. The last statement of the theorem follows from the fact that if so E

IR, so < -n/m, then the series (13.5) for Rez < so is majorized in absolute value by the sums of the series Co

2 Y A°I(P;(x)12+ j=1

1

'O

j

-1'lj°Iq';(Y)I2,

which are themselves absolutely and uniformly converging series with positive

terms. O Remark 1. One may also prove Theorem 13.2 without using the Mercer

theorem, noting that for Rez < -n/(2m) the operator AZ has a kernel AZ (x, y) a L2 (M x M) (i. e. A' is a Hilbert-Schmidt operator) and in view of the Parceval identity we have 00

Y Ajs= f= 1

A,(x,Y)12dxdy, MxM

s< -n/(2m).

(13.6)

Chapter 11. Complex Powers of Elliptic Operators

114

From the group property it follows that for s < - n A, (x, y) = f A,, (x, z) A512 (z, Y) dz, or

A, (x, y) = f Aij2 (x, z) A:12 (Y, z) dz

using the fact that the kernel A512 (x, y) is hermitean. Putting now x = y and integrating in x we obtain from (13.6) that (13.4) holds for real z < -n/m. The transition to complex z is accomplished in the same way as above or by using analytic continuation. Let us remark however that from this proof it is hard to get exact information on the decomposition (13.5) (in particular about the uniform convergence of the series there).

Remark 2. The equality (13.4) is valid also without the assumption on selfadjointness of A. The proof is easily obtained from the theorem of V. B. Lidskii

(cf. Gohberg I.C. and Krein M.G. [I ], Theorem 8.4). However, in the nonselfadjoint case it is not possible to extract any kind of interesting information about the eigenvalues from (13.4). The only exception is the case of a normal operator, where in fact the results may be deduced from the corresponding results in the self-adjoint case. 13.3 Formal asymptotic behaviour of the function NQ) in the self-adjoint case. The function V(I). Let A be as in Theorem 13.2. Set

N(t) = Y 1

(13.7)

I's, for arbitrary t EIR, i.e. N(t) is the number of eigenvalues of A not exceeding t (counting multiplicity). It is clear that N(t) is a non-decreasing function of t which equals 0 for t < A1. We assume here for convenience that the eigenvalues have been arranged in increasing order:

0<,11 ;9 .125.135...

(13.8)

We then have an obvious formula, expressing S,1 (z) in terms of N(t) in the form of a Stieltjes integral: a-,

CA(z) = f t'dN(t).

(13.9)

0

Assume now that N(t) admits the following asymptotic expansion as

N(t)=c1t +C2t

(13.10)

§ 13. The -Function and Formal Asymptotic Behaviour

where Re a, > Re a 2 > ... > Reak > Re ak + 1

then

x

k

CA (Z) _

,

115

t=d(t'') +l (z),

c1

1=1

1

where f (z) is holomorphic for Re z < - Reak + I. Since W

J t=d(t-,) = f a, t=+- 1 dt = -

a,

z+a,

,

it follows that (13.11) may be rewritten as (13.12)

Hence for Re z < - Re ak+ 1 the function CA (z) has simple poles at - a, with

residues -c,a1, 1=1, ... , k. Therefore knowing the poles of CA(s) and the residues at these poles allows us to write down a formal asymptotic expansion

for N(t). In reality however, only the computation of the first term of the asymptotic expansion works out well. This term, dictated by comparison of rr0

(13.10) and (13.12), must have the form - t5' if CA (s) has its left-most pole at - so < 0 with residue ro . SO The formula r-0

N(t)

r-

(13.13)

SO

will be rigorously proved in the following section using the Tauberian theorem of Ikehara, while for the present we shall occupy ourselves with the specification of its coefficients. By Theorem 13.1 we have s0 = n/m and

r0 = -

1

M

j

I dx M

a,- "* (x, ) d '

(13.14)

141=1

Since in the situation under consideration a. (x, ) > 0 for + 0, then r0 + 0, so that the pole at -n/m really exists. Formula (13.13) may now be written

N(t) ^'

1

J a(x,)dc' t(13.15)

j dx nkr Ic1=1

Now rewrite the right-hand side of (13.15) in a more natural form. For this, introduce the function

116

Chapter If. Complex Powers of Elliptic Operators

V(t) = (2n)-"

(13.16)

J a.(x.f)<e

Note that this function has an invariant meaning: it is the volume in T*M of all points (x,4) such that a. (x, ) < t, multiplied by (2n)-". Here the volume is given in T*M by the measure, induced by the canonical symplectic structure (cf. Arnol'd [11). Lemma 13.1. We have the following formula

V(t)= 1 Jdx f nM

(13.17)

itl=i

and may, therefore, instead of (13.15) write

N(t) - V(t).

(13.18)

Proof. Let us remark to begin with, that the condition a", (x, ) < t, in view of the homogeneity of is equivalent to a", (x, t< 1. Thus, changing variables in (13.16), q = t-'1' , we obtain

V(t)

dxdrl

t"/1".

The later arguments will take place for a fixed value of x, and we shall write a",

instead of a. (x, 5).

Fig. 4

We have to show that J

d = 1 J a,

d'

.

(13.19)

§ 13. The {' -Function and Formal Asymptotic Behaviour

Let 14'1= land d

=sup { t : am (t ') < 1 } (Fig. 4), so that d

117

is the

i>0

distance of 0 from the surface am =I in the direction '. Let us consider the infinitesimal cone with vertex in 0 and height d(4') - ', cutting out the area dc'. on the surface =1. The volume of this cone equals j (d Therefore dd

But am

am (

am(t:); hence if

I

I

'

I

I

j

=1

".ct>
J = I f I m am I 1

then

(13.20)

n u'+-I I

1

),

in view of the homogeneity of

a.( ' )=Iw-, from which d(')= l

I

i

Substituting this expression for

in (13.20) we obtain

(13.19).

13.4 Asymptotic behaviour of the eigenvalues. We shall now state an asymptotic formula for Ak ask -' + oo, equivalent to the asymptotic formula for N(t) ((13.13), (13.15) or (13.18)), and infer from this that, essentially, N(t) as a function of t, and .k as a function of k, are mutually inverse functions. Denote by V, the coefficient of the term t"'' in (13.17), so that V, = V(1). Then the desired formula is of the form Ak- V,- m'"

k"'"

as

k-*+oo.

(13.21)

Proposition 13.1. The asymptotic formulae (13.21) and (13.18) are equivalent (i.e. each implies the other).

Proof. 1. Suppose (13.18) holds, i.e. for any c > 0 there exists to > 0 such that

1-c
(13.22)

for t > to. Choose an integer ko > 0, such that ) ku > to and Ak,+ I > 2k, Then show that

-c
(13.23)

fork z ko. Indeed, for any k ko there exist integers k, and k2 such that ko <= k, < k < k2 and 2k, < Ak,+I ='lk, < Ak,+, . In particular, we have N(2k) = k, and N(Ak) = k2, so that from (13.22) it follows that (13.24)

1-e:5 k2V,'.lk="'"S1+E.

(13.25)

118

Chapter H. Complex Powers of Elliptic Operators

Further, N (t) = k, for Ak S t < Ak, so that for these t 1

E:5 k, V,-'t

5I+s

and by continuity

1eSk,Vj-'A-Rlm51+e.

(13.26)

It follows from (13.25) and (13.26) that

1- e 5 kV,-' 2;"/m 5 I + e, and it remains to note that Ak = follows that (1

Ak,.

(13.27)

Hence (13.23) is proven. But from this it

+e)-m/" V1-ml"kmI" 5 A.k S (1

-e)-m/" Vim/"k"m/",

(13.28)

and this implies (13.21).

2. Now let (13.21) hold. Choose e > 0 and let ko be an integer such that for any integer k > ko, the inequality (13.28) holds. Let Ak 5 t < Ak,, where k, and

k2 are the same as in part 1. of this proof. It follows from (13.28) that, in particular,

1-eSk,V1'1k"/"'5I+e,

(13.29)

1-e5(k,+1)Vj- 'Ak"/m51+e,

(13.30)

Ak, . Choose now a number ko so large that V,-' .1k "Im 5 e for k>ko. We then obtain from (13.30) that since 2k,

I-

(13.31)

But N(t) = k, and therefore from (13.29) and (13.31) we get

1 -2e5N(t) V '.lk

1 +2e,

1 - 2e 5 N(t) V1 '.1k,"/m < 1 + 2e,

from which it follows that

I -2c5N(t) V;'t-"/'"51+2e, in view of the fact that '1k < t < Akj, as required.

§ 13. The c-Function and Formal Asymptotic Behaviour

119

13.5 Problems

Problem 13.1. Find the poles, residues and values at the non-negative z

integer points of the C-function of the operator A = - dz2 on the circle IR/2 x Z. Express the classical Riemann C-function C (z) =

(13.32)

in terms of S,, (z) and find all the poles, residues and values of C (z) at z = 0, - 2,

-4,....

Problem 13.2. Consider the Schrodinger operator A = - A + q (x) on the torus 12= R2/2n V. Here q (x) e C' (12). Express C., (0) in terms of q(x). Problem 13.3. Let A be an elliptic differential operator, mapping Cm (M, E) into C°° (M, fl, where E and F are smooth vector bundles on M. Let there be given a smooth density on M and hermitean structures on E and F (a hermitean metric on each fiber). Show that index A = Cl+AA (z)

(13.33)

where the right-hand side does not depend on z. (This formula allows us, in principle to write down the index A in terms of the symbol of A, using Theorem 13.1 which gives the possibility of computing (e(0)

for B= I+A*A or B= I + AA*.) Hint: All non-zero eigenvalues of AA* and A*A have the same multiplicities, since A maps an eigensubspace of A *A into an eigensubspace of AA* and A* acts in the opposite direction.

Problem 13.4. Show that the kernel K(t, x, y) of the operator eAt from Problem 10.1 is infinitely differentiable in t, x and y for t > 0 and for all x e M, y e M. As t -+ + 0, we have the following asymptotic properties of K(t, x, y):

a) If x + y, then K (t, x, y) = 0 (t") for any N > 0. b) K(t, x, x) has the following asymptotic expansion as t-. +0: K(t, x, x)

E aj (x) t

j=o

(13.34)

where a j (x) e C °° (M). Express aj (x) in terms of yj (x) and x, (x) (cf. Theorem 12.1) and write down an expression for aj (x) in terms of the symbol of A.

120

Chapter 11. Complex Powers of Elliptic Operators

Verify that if A* = A, then

K(t,x,y)_

e- 1,1(p;(x),pt(y)

(13.35)

0(t)= Y, e-''= JK(t,x,x)dx

(13.36)

and the 6-function

j=o

u

J=O

has the asymptotic expansion 0 (t) - 2 aJ t M

.

(13.37)

i=o

Express index A in terms of the 0-functions of the operators A*A and AA*.

Problem 13.5. Let E be a hermitean vector bundle on a closed manifold M with smooth positive density and let A be an elliptic self-adjoint differential operator mapping C a (M, E) into C °° (M, E) (not necessarily semibounded). Consider the function U

_

(sign A)

t,.Iz,

(13.38)

where the sum runs over all the eigenvalues of A. Show that the series (13.38) converges absolutely for Re z < - n/m and the function defined by it, >,4 (z), may be continued to the whole complex z-plane as a meromorphic function with simple poles at z, = J - n , j = 0, 1, 2, .... Express the residues at these poles via m the symbol of A. Hint. Express 17A(z) in terms of (; (z) and C" (z) where C. (z) and C;; (z) are two C-functions of A, obtained by different choices of the branch for Az with cuts along the upper and lower semi axes of the imaginary axis.

§ 14. The Tauberian Theorem of Ikehara 14.1 Formulation. The Tauberian theorem of Ikehara allows us to deduce from the fact that the C-function is meromorphic asymptotic formulae for N (1) as t-. + co or for A,% as k -+ + oo (cf. §13). Let us give its exact formulation. Theorem 14.1. Let N (t) be a non-decreasing junction equal to 0 jor t:5 1 and such that the integral

(z) = J tzdN(t)

(14.1)

§ 14. The Tauberian Theorem of Ikehara

121

converges for Re z < -ko, where ko > 0 and the function

(z)+z+A

k0

can be extended by continuity to the closed half-plane Re z < -ko. We will assume that A 0 0. Then, as t -+ +oo we have

N(t) - k t*-

(14.2)

0

(recall that f, (t) - f2 (t) as t - + oo means that lim f, (t)/f2 (t) =1). ,.+,0 (The convergence of the integral in (14.1) for Re z < - k0 easily follows from a weaker condition. Namely, it suffices to suppose that the integral converges for

Re z < - k, for some k, and the function C (z) thus expressed can be holomorphically continued to the half-plane Re z < -ko). Corollary 14.1. Suppose that the function C (z), defined for Re z < - ko by (14.1), can be meromorphically continued into the larger half-plane Re z < - ko + e, where e > 0, so that on the line Re z = - ko there is a single and moreover simple pole at - ko with residue - A. Then the asymptotic formula (14.2) holds. 14.2 Beginning of the proof of Theorem 14.1: The reductions.

1st reduction. It is convenient to consider instead of C (z) the function f (z) = C (- z). We then obtain

f(z) = J t-=dN(t),

(14.3)

where the integral converges for Re z > ko and the function f (z) continuous for Re z ? ko.

- z -Ako

is

2nd reduction: Reduction to the case ko = 1. By introducing the function f, (z) = f (ko z), we obtain

f,(z)= Jt-k":dN(t)= TdN,(T), where N , (T) = N (T' /k"). Since

.f(koz)-k

A

0z

=f(z)-kA z-1 0 1

-k0

1

and since N(t) , A tr- is equivalent to N, (T) - A T, then Theorem 14.1 reduces TO

to the following statement:

ko

122

Chapter 11, Complex Powers of Elliptic Operators

Let N(t) be a non-decreasing function and let the integral 00

f(z) = $ t-zdN(t) be convergent for Re z > 1, where f(z)

N(t) - At

A

(14.4)

is continuous for Re z 2! 1. Then

-z-1

t-'+oo.

as

Note that from the continuity of f(z) -

z

A

1

(14.5)

for Re z z 1 and the fact that

f(z) z 0 for real z;-> 1, it follows that A > 0. Changing N(i) for A - 1 N(t), which results in changing f (z) for A -1 f (z), we see that it suffices to show the statement for A = 1.

3rd reduction. Let us pass from the Melin transformation to the Laplace transformation, i. e. make a change of variables t = ex. Put N (ex) = 9 (x). We then see that rp (x) is a non-decreasing function, equal to zero for x < 0 and that the integral

f(z) = ! e-zxd(p(x)

(14.6)

0

converges for Re z > 1 and f (z) z-1 show that

is continuous for Re z z 1. We must

lim e-xgp(x)=1.

(14.7)

x.+w

4th reduction. Denote H(x) = e-x(p(x). The So(x) is non-decreasing if and only if H(y)zH(x)ex-y

y> x.

for

(14.8)

Integrating by parts in (14.6) gives, for Re z > I

f(z) = zfe-'x(p(x)dx =

ze-(z-1)x H(x)dx 0

0

Now put z =1 + s + it, where z > 0 and t is real. Note that °D

0

therefore, (14.9) implies

e-(z- ,)xdx = 1 z-1

(14.9)

§ 14. The Tauberian Theorem of Ikehara

f(z) _ z

= $e

1

z-

c=

123

.

0

Since

fZZ)- Z

1= 1(f(z)- Z

1

I-

1

then, putting

h,(t)=Z(f(z)- z

1

1

-1)

(14.10) =+e+lf

we obtain 00 h.(t) = J e-`x_"x(H(x)- 1)dx.

(14.11)

0

We may now give the following reformulation of Theorem 14.1.

Theorem 14.1'. Let H(x) be a function, equal to 0 for x < 0 and satisfying (14.8) for all real x and y. Assume that the integral (14.11) converges absolutely for

any e > 0 and the function h, (t) defined by it, is such that the limit lim h, (t) = h (t)

(14.12)

exists and is uniform on any finite segment I t 15 22. Then

lim H(x) = 1.

(14.13)

Remark. If H(x) tends to I sufficiently quickly (if e.g. H(x) -1 EL' ([0, + oo))), then we obtain (14.12) from (14.13) by passing to the limit under the integral sign, which one may do in view of the dominated convergence theorem (the function h (t) then equals the Fourier transform of O (x) (H(x) - 1), 6(x) the Heaviside function). In some sense, the Tauberian condition (14.8) allows one to invert this statement.

143 The basic lemma. It is clear that in order to prove Theorem 14.1' we have to somehow express H(x) - 1 in terms of h (t) which, formally, is possible by the inverse Fourier transformation. However, we know nothing about the behaviour of h (t) as t -. + oo or about the nature of the convergence of h. (t) to h (t) on the whole line and it is therefore necessary, to begin with, to multiply the limit equality (14.12) with a finite cut-off function j (t). These considerations, linked to the convenience of having transformations with positive kernels (of the Fejer type), demonstrate that it is convenient to consider p (t) to be the Fourier transform of a non-negative function g (v) c- L' (R):

Q(t)= Je ""go (v)dv.

(14.14)

Chapter 11. Complex Powers of Elliptic Operators

124

We shall assume that Q (t) is a continuous function with compact support such that a (0) = 1, p (v) >_ 0 and p (v) a Ll (IR). From this it follows that

f p(v)dv=1.

(14.15)

-OD

The existence of a function B (t) of the type described may be shown in the same way as in 6.3 (at the beginning of the proof of Theorem 6.3 a function fi(t) E Co (IRl) is constructed which satisfies all these requirements). We can also explicitly define Q(t) , putting

1- 121;

(t) =

A

0

f

111>2.

;

Then indeed, for a fixed v * 0 we have

p()v =

I2tll

J e

dt=-

2 ertV

;,

/

d l-

I 2t11

eu°

2

+2rziv

2

e"'

2

e"° sign =-2J2iv - t dt

4nv

2

2

4av

10

2

I2) tII

""v

0

2

1

- 1 -cos2v 2nv 2

I since n

v

2

,

from which all the necessary properties of p (v) are obvious. Lemma 14.1. For any fixed A >/0 lim Y.

\

! H ` y - Al p (v) dv = 1

.

(14.16)

OD -x

Proof. 1. Put e, (t) = B (t/A) and Q, (v) = Ap (A v) so that p, (t) is the Fourier

transform of p, (v). It is clear that + 00

I Hfp(v)dv= $ H(y-v) p,(v)dv,

-x

(14.17)

-ac

a nd since p, (v) possesses the same properties as p (v), it suffices to prove (14.16)

forA=1. 2. Putting F(t) = e(t)ht(t), we compute the inverse Fourier transform of the function F (t) with compact support, taking into account that a (t) and ht (I)

are the Fourier transforms of the absolutely integrable functions p (v) and

0(v) (H(v)-1)e-t 00

+

-m

+ cc

ettYF.(t)dt = J eitvp(t) m

[1wx_ 1)eidx]dt 0

§ 14. The Tauberian Theorem of Ikehara

125

J Q(t) a""y-x'dt dx

(H(x)-1)e-`x

J

(14.18)

-00

o

J (H(x)_ 1) e-`xQ(y-x)dx. 0

As a result, as one might have anticipated, we obtain a convolution and we have made sure that (14.18) holds everywhere and in the usual sense (the change

in the order of integration is permitted by the Fubini's theorem). Let us now rewrite (14.18) in the form

J e"''F(t)dt+ Je-`xQ(y-x)dx = JH(x) e-" Lg(y-x)dx -00

(14.19)

0

0

and take the limit as a- + 0. Since supp F c supp Q and F (t) - F(t) uniformly in t e supp j (here F(t) = Q (t) h (t)), then the first integral on the left-hand

side has a limit as s-s +0 for any y. The same also holds for the second integral (e.g. by the dominated convergence theorem). Therefore, the integral on the right-hand side of (14.19) has for any y a limit as a- +0. Since

H(x)e-"e(y-x) converges monotonely as a-++0 to H(x) e(y-x), we get + ac

00

OD

J e"yF(t)dt+ f Q(y-x)dx= JH(x)Q(y-x)dx. 0

(14.20)

0

Now let y tend to + oo. By the Riemann lemma +m

lim

J e"y F(t) dt = 0.

In addition, it is clear that lim J Q (y - x) dx = 1. Therefore, it follows from .D o (14.20) that lim JH(x) y..+x; 0

Q(y-x)dx= 1.

(14.21)

But ac

+W

+co

JH(x)Q(y-x)dx= J H(x) Q(y-x)dx= J H(y-v) Q(v)dv, - CO

so that (14.21) implies the statement of the lemma.

126

Chapter 11. Complex Powers of Elliptic Operators

14.4 Proof of Theorem 14.1'. 1. First, we show that lim H (y) <_ 1 y.+m

(14.22)

.

Since

+,o

a

J H y-

Q(v)dv S J H y-

-a

(

-00

J Q(v)dv

for any a > 0, it follows from Lemma 14.1 that

J H(y-V) Q(v)dv 5 1

slim

.

(14.23)

Now, in view of the Tauberian condition (14.8) we have

H(y-v)>H(y-ale 2i

for

ve[-a,a].

Now, it follows from (14.23) that

/

2 _a a

lim H l y- )/ e y.+m

A J Q (v) dv 5 1,

-a

or 2a

lim H (y) < e z

a

(J

1

Q (v) dv)

(14.24)

-a

Inequality (14.24) holds for any a > 0 and A > 0. Let a - + oo and A - + co in this inequality in such a way that a/A--'0. Then we obtain the required estimate (14.22) from (14.24). 2. We will now verify that

lim H(y) z 1.

y.+m

(14.25)

To begin with, note that (14.22) implies the boundedness of H(y):

IH(y)I <M,

yeIR ,

(14.26)

in view of which //

J HI y-- I Q(v)dv < (b), roiab

`

I

(14.27)

§ 14. The Tauberian Theorem of Ikehara

127

where a (b) - 0 as b - + oo (this means, in particular that the integral on the lefthand side of (14.27) approaches 0 as b - + oo, uniformly in y and A). Since

)v 0 (V) dV

+m

-m

H

b

-b

IvI?b

then, by (14.27) and Lemma 14.1 we obtain for arbitrary b > 0 lim j H (Y - v) Q (v) dv >-- 1 - a (b). y.+m IvISb

\

(14.28)

Let us again use condition (14.8). We have

b\

2b

U

ve[-b,b],

from which, in view of (14.28), it follows that

mH Cy+ b/ e'

y li

J Q(v)duz I -a(b),

2b -b b

or zz

lien H(y) z (1-a(b)) e 7++m

b

(J Q (v) du)

(14.29)

b

Now let b -+ + oo and A -+ + oo, so that b/2 - 0. Then from (14.29) we obtain the

desired inequality (14.25). D

Problem 14.1. Let N(t) be a non-decreasing function, equal to 0 for t5 I and let the integral (14.1) be convergent for Re z < - ko, some ko > 0. Assume furthermore, that the function a (z), defined by (14.1), can be meromorphically continued to larger half-space Re z < - ko + e, where e > 0 so that on the line Re z = - ko there is a single pole at - ko with principal part A (z + ko) -' in the Laurent expansion (here I is a positive integer, equal to the order of the pole at

-ko). Show that

N(t) ' (-1)'''A t,o(lnt)r-t

(I-1)!

as

Problem 14.2. Prove the Karamata Tauberian theorem: Let N(t) be a non-decreasing function oft E IR', equal to 0 for t < 1 and such that the integral

128

Chapter II. Complex Powers of Elliptic Operators

0(z) = f e-s`dN(t)

(14.30)

0

converges for all z > 0 and

0(z) - Az

as

z-.+0

(14.31)

(here A > 0 and a > 0 are constants). Then

N(t)

r( +1)

to

as

t-++oo.

(14.32)

§15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem) 15.1 The spectral function and its asymptotic behaviour on the diagonal. Let M be a closed n-dimensional manifold on which there is given a smooth positive

density and let A be a self-adjoint, elliptic operator on M such that (15.1)

Then A is semibounded. Denote by A its eigenvalues, enumerated in increasing order (counting multiplicities):

A,5A2;9A35..., By 47j(x) we denote the corresponding eigenfunctions, which constitute an orthonormal system. Let E, be the spectral projection of A (the orthogonal projection onto the linear hull of all eigenvectors with eigenvalues not exceeding t). It is clear that E,u = Y_ (u, apt) p,

(15.2)

x,St

Definition 15.1. The spectral junction of A is the kernel (in the sense of L. Schwartz) of the operator E, . Taking into account that on M there is a correspondence between functions and densities, we may assume that the spectral function is a function, not a density. From (15.2) it is obvious that this function, e(x, y, t), is given by the formula e (x, y, t) _ > (Pt (x) *P; (y) A's,

(15.3)

§ 15. Asymptotic Behaviour of the Spectral Function and Eigenvalues

129

and, in particular, belongs to C°°(M x M) for every fixed t. Let us note immediately the following properties of e (x, y, t): 1) e (x, x, t) is a non-decreasing function of t for any fixed x c- M; 2) the function N(t) introduced in section 13.3, can be expressed in terms of e (x, x, t) by the formula

N(t)= $e(x,x,t)dx,

(15.4)

where dx is a fixed density on M. Now assume that local coordinates in a neighbourhood of x are so chosen, that the density coincides with the Lebesgue measure in these coordinates and put

I

V. (1) _

d

(15.5)

.

Theorem 15.1. For any x e M the following holds:

e(x,x,1) - V. (t)

as

(15.6)

Proof. 1.To begin with, note that without loss of generality we may assume Al ? 1. Indeed this is satisfied by the operator A I = A + MI for sufficiently large M. Now, if eI (x, y, 1) is the spectral function of AI , we have e (x, y, 1) = e 1(x, y, t + M). Therefore, the asymptotic formulae 1(x, x, t) - Vx (1)

implies e(x, x, t) - Vx(t+M). But Vx(t+M)=Vx(1) (t+M)"lm=Vx(1) t°Im(1+O(t-I))_. Vx(1) t"""'=V.0), which implies (15.6).

2. Thus let A,;-> 1. We may then define complex powers Az of A in accordance with the scheme of §10. Using (13.5), we may for x = y express the kernel Az (x, y) of A= in terms of the spectral function as follows 00

A, (x, x) = J t z de (x, x, t),

(15.7)

0

where d signifies the differentiation with respect to t (for a fixed x this is simply a Stieltjes integral). In view of Theorems 12.1 and 14.1 we obtain now fore (x, x, t)

the asymptotic formula

e(x,,x,t)- I 11,

I 141 =1

a-film (x,

,)A,l J

.

t°/"g.

(15.8)

130

Chapter II. Complex Powers of Elliptic Operators

An elementary transformation of the right-hand side of this formula, carried out in the proof of Lemma 13.1, shows that it equals V,,(t), implying (15.6). 15.2 Asymptotic behaviour of the Eigenvalues Theorem 15.2. Let A satisfy the conditions described at the beginning of this section. Then one has the following asymptotic relations

N(t) - V(t),

(15.9)

Ak - V(1)-""k"",

(15.10)

where V(t) is defined by the formula (13.16).

Proof. In §13 we showed the equivalence of (15.9) and (15.10). (Proposition 13.1). Let us prove (15.9). This is done on the basis of the Tauberian theorem of Ikehara, by analogy with the proof of Theorem 15.1. Indeed, again we may assume that A, > 1. Then for Re a < - n/m, we clearly have the formula

S,(z)= f tzdN(t). It remains to use Theorems 13.1 and 14.1 and Lemma 13.1. Remark. One can derive (15.9) from (15.6) by integration over x. To justify this integration, it is necessary, however, to prove the uniformity in x of (15.6), which requires in several places (in particular, in the proof of the Ikehara theorem) the verification of uniformity in the parameter. To avoid this cumbersome verification, we have preferred to give an independent proof.

15.3 Problems

Problem 15.1. In the situation of this section prove the estimate

Ie(x,y,t)I SCt"1T, where x, y e M and the constant C > 0 does not depend on x, y and t (t z 1). Problem 15.2. Let A be an elliptic differential operator, on closed manifold M with smooth positive density, which is normal, i.e. A*A = AA*.

(15.12)

a) Show that A has an orthonormal basis of smooth eigenfunctions (p; (x),

j = 1, 2, . . ., with eigenvalues A; E C, such that IA;I

+oo

as

(15.13)

b) Show that if N (t) denotes the number of A,, such that I.ij I S t, and if V (t) is defined by the formula

§ 15. Asymptotic Behaviour of the Spectral Function and Eigenvalues

V(t) = (2n)""

f

131

(15.14)

la.(x, 0 1
then

N(t)-V([)

(15.15)

as

Problem 15.3. Deduce Theorems 15.1 and 15.2 from the results of Problem 13.4 and the Tauberian theorem of Karamata (Problem 14.2). Problem 15.4. Let M be a closed manifold, A an elliptic differential operator on M, such that a, (x, ) > 0 for c + 0. Let A j be its eigenvalues and N, (1) the number of eigenvalues with Re.lj < t (here we take for the multiplicity of an eigenvalue Ao the dimension of the root subspace EA., cf. Theorem 8.4), N2 (t) the

number of eigenvalues with IA; < t. Show that

N,(t)-N2(t)-V(t)

as t-++oo,

(15.16)

k -++oo

(15.17)

where V(t) is defined as before. Show that

2k _ V(1)-nf"krl"

(this means, in particular, that Im Ak has a lower degree of growth than Re Ak).

Chapter III Asymptotic Behaviour of the Spectral Function §16. Formulation of the Hormander Theorem and Comments 16.1 Formulation and an example. Let M be a closed n-dimensional manifold on which there is given a smooth positive density dx and let A be an elliptic, self-adjoint operator of degree m on M such that aM (x, ) > 0 for + 0. We will use the notations e (x, y, A), N (A), V. (A) and V (A) introduced in § 15. The

following theorem refines Theorems 15.1 and 15.2.

Theorem 16.1 (L. Hormander). The following estimate holds I e (x, x, A) - Vx (A) 15 CA(" -')/m,

A>_ 1,

xeM,

(16.1)

where the constant C > 0 is independent of x and A.

Corollary 16.1. The following asymptotic formula holds

N(A)=V(A)(1+O(A-'/°'))

A --+oo

as

(16.2)

Remark 16.1. In general the estimate of the remainder in (16.1), (16.2) cannot be improved. This can be seen by looking, for instance, at the operator

A=-

on the circle S' =1R/2,t Z. The corresponding eigenfunctions are

of the form yrk (x) =

1

e'kx,

k = 0,

±1,

and the eigenvalues are Ak = k2, k = 0, ± 1, ± 2,

± 2, ... ,

... .

Further, since tWk(x)12= (2n)-', then clearly e(x,x,A) =(21r)-N(A). Since V(A) _ (2x) -' V (A) then (16.1) and (16.2) are equivalent. So it suffices to show that the estimate of remainder in (16.2) can not be improved.

But in this example (16.2) has the form N(A) = V(A) (1 +O(A-'l2)) or N (A) = 2 i/ + O (1). The estimate O (1) can not be improved because N (A) has only integer values.

Chapter 111. Asymptotic Behaviour of the Spectral Function

134

Later, in §22, we will study a more interesting example, which is a generalization of the present one (the Laplace operator on the sphere) and shows

that (16.1) cannot be improved in the case of arbitrary n and m.

16.2 Sketch of the proof. First of all, the theory of complex powers of operators, allows a reduction to the case when A is a'f'DO of order 1. In this situation we will show, that for small t, e«4 is itself an FIO, with a phase function

which is a solution of a certain first order non-linear equation. Let us now remark, that the kernel of eu4 is the Fourier transform (in A) of the spectral function of A. From this the asymptotic (16.1) is obtained, by invoking Tauberian type arguments for the Fourier transformation. The remainder of this chapter is as follows: § 17 contains some indispensible information on first order non-linear equations; in § 18 an important theorem on

the action of `PDO on exponents is proved, from which, in particular, the composition formula for a `IMO with an FIO follows; in § 19 the class of phase functions corresponding to'I'DO is studied; in §20 we construct the operator e"4 in the form of an FI O for a first order operator A; in §21 Theorem 16.1 is proved in the general case (there is also information about e(x, y, A) for x * y); finally, §22 contains the definition of the Laplace operator on a Riemannian manifold and the computation of its spectral function in the case of a sphere.

Problem 16.1. Compute N (2) and e (x, x, A) for the operator

A=-d=- C a2

a2

02 ax-Y+ax2+...+ax

on the torus T" = lR"/2nZ" and verify that the asymptotic formulae (16.1) and (16.2) hold.

§17. Non-linear First Order Equations 17.1 Bicharacteristics. Let M be an n-dimensional manifold and a (x, ) a smooth real-valued function, defined on an open subset of T M. Consider the Hamiltonian system on T*M, generated by as Hamiltonian: rx =

where a4 = S

as

as

a_

' ... , a ")' I

a{

-a,,

(17.1)

as as and (z, )are the coordinates axl ' ... , ax"

on T' M, induced by a local coordinate system on M. It is well-kown, that the vector field on T' M, defined by the right-hand side of (17.1), is independent of the choice of local coordinates on M (cf. e.g. V.I. Arnol'd [1]).

§ 17. Non-linear First Order Equations

135

Definition 17.1. A solution curve (x(t), fi(t)) of (17.1) bicharacteristic of the function a (x, ).

is

called a

A bicharacteristic is not necessarily defined for all t E R. In this case we assume that it is defined on the maximal possible interval (concerning this consult also Problems 17.1 and 17.2).

Proposition 17.1. The function a (x, ) is a first integral of the system (17.1), i.e. if (x(t), l; (t)) is a bicharacteristic of the function a(x, ), then a(x(t), fi(t)) = const Proof. We have

d a(x(t),

(t)) = a,,.z + a, = a., a, - a. a, = 0. 1]

Proposition 17.1 makes sense of the following definition:

Definition 17.2. A bicharacteristic (x(t),1; (t)) of the function a(x, ) is called a null-bicharacteristic if a(x(t),: (t)) = 0. 17.2 The Hamilton-Jacobi equation. Consider the first order partial differential equation

a(x,(px(x)) = 0,

(17.2)

where (p is a smooth, real-valued function, defined on an open subset of M and gyp,, its gradient. Such an equation is called a Hamilton-Jacobi equation. For its treatment, it is convenient to introduce the graph of qpx, i.e. the set

F, = {(x, (px (x)), x e M} c T"M.

(17.3)

Proposition 17.2. If cp is a solution of (17.2), then the manifold F. is invariant under the phase flow of the system (17.1), i. e. if (x (t), c (t)) is a bicharacteristic of a, x (t) for t e [0, b] belongs to the domain of (p and (x (0), (0)) a f, then (x(t), fi(t)) ef, for all t c- [0, b].

Proof. In view of the uniqueness theorem, it sufficies to verify that the Hamiltonian vector field (a,, -a.,) is tangent to r. at all its points. This is equivalent to the following: if (x(t), (t)) is a bicharalcteristic and (x(0), (0)) e r, (i. e.

dt t) - 4 (xId11=0 (t ))1 } = 0. But this follows (0) = cpx (x (0))), [fi (then

from the computation :

111

(dt R(t)-(Px(x(t))11Ii3o=(4-Px:

t = - ax (x (0),

(0)) - coxx (x (0)) at (x (0),

a

= - ax

oI,-o

[a (X, (P.(x))],, =X(0) = 0. C

(0))

Chapter III. Asymptotic Behaviour of the Spectral Function

136

In what follows the only important case for us is when a(x, ) is positively homogeneous with respect to of degree m, i.e. a (x,

tm a (x, ),

t>0, +0,

(17.4)

where m is any real number. Such functions are characterized by the Euler theorem:

-a,=ma.

(17.5)

Proposition 17.3. Let a (x, ) be homogeneous of degree m and (P (x) a solution of (17.2). Then (p (x) is constant along the projections of the null-bicharacteristics of the function a (x, l;) belonging to I',, i.e. if (x (t), l; (t)) is a null-bicharacteristic and 4 (0) = V. (x (0)), then tp(x (t)) = const.

Proof. We have

dt

q,(x(t)) = cxx = (pxas = (px(x(t)) ag(x(t), fi(t))

= (px(x(t)) a,(x(t),cpx(x(t))) = ma(x(t),cpx(x(t))) = 0.

173 The Cauchy problem. The Cauchy problem for the Hamilton-Jacobi equation (17.2) consists in finding a solution rp (x) of this equation, subject to the condition

was=W,

(17.6)

where S is a hypersurface (submanifold of codimension 1) in M and W e C m (S). Locally, we may consider the hypersurface as a hyperplane, i.e. by choosing the local coordinate system in a neighbourhood of a point xo e S, we may achieve

that

S= {x:

(17.7)

so that yi = W (x'), where x' = (x, , . . . , x. _ ). In this coordinate system, it is convenient to formulate the condition of being non-characteristic, guaranteeing local solvability of the Cauchy problem in a neighbourhood of the point x' eS: the equation a (x', 0, Wx (x'), A) = 0

(17.8)

has a simple root A, i. e. a root A a IR, which in addition to (17.8) satisfies as (x', 0, yrx (x'), A) 4 0.

(17.9)

§ 17. Non-linear First Order Equations

137

Let a point 0 eS be fixed. Then by the implicit function theorem, the equation (17.10)

0

for I x I < s and I ' - Wx (0) I < e has a solution A = a' (x, '), which is a smooth function of x and '. It is easy to verify that a' (x, ') is homogeneous of the first

order in i;', so we may assume that it is defined for Ix I < e and for all i' 0 0 in a conical neighbourhood of *,,(0). Equation (17.10) for Ix I < e and for a vector (4', A) close to the direction of (*j,(0), a'(0, *,(0))), may be represented in the form

A - a'(x, ') = 0.

(17.11)

Therefore the local Cauchy problem takes the following form: find a solution (p = cp(x) of (17.2), which satisfies (17.6) and, additionally, satisfies O(P

ax

(0, 0) =a ' ( 0 ,( 0 ) ) .

(17.12)

Since in this situation it is possible to pass from (17.10) to (17.11), our problem may be written in the following form

ax -

a

(a (P

= 0,

(17.13)

(17.14)

(P lx.-o=W(x'), i.e. the matter reduces to the case

(17.]5)

Let us consider the bicharacteristics of a (x, ) of the form (17.15). Their equations are (17.16)

=a.' (x, 7 Consider a null-bicharacteristics (x(t), (t)) belonging to 1', and starting in S, i.e. such that 0. Then it is obvious from (17.16) that x. (t) = t. Fix another point x' = x' (0) a S. It is clear that the condition (x (0), (0)) e f, means the following

' (0) = W x (x'), , (0) _ W (x', 0) , X.

(17.17)

138

Chapter 111. Asymptotic Behaviour of the Spectral Function

and the condition a (x (0), (0)) = 0 gives

(0) = a'(x',0,Ws.(x')).

(17.18)

Therefore, the null-bicharacteristic belonging to r,, and such that x (0) = 0

and x'(0) = x', is uniquely defined. From (17.17) and (17.18) the smooth dependence on x' is clear. In addition, if we consider the transformation

g:

(17.19)

defined for I x I < c, then from the initial condition x'(0) = x', it follows that its Jacobian is I for x = 0, so that g is a local diffeomorphism. Now, from Proposition 17.3, it necessarily follows that

w(x)=W([g-`(x)]'),

(17.20)

where [g-' (x)]' is the vector, obtained from g- `(x) by neglecting the last component (corresponding to the notation x' for x = (x', x.)). Therefore, we have shown the uniqueness of the solution of the local Cauchy problem and obtained a formula, (17.20), for this solution. The existence of this solution. is a simple verification. We recommend the reader to do the following exercise.

Exercise 17.1. Show that formula (17.20) actually gives a solution of the local Cauchy problem as described above. 17.4 Global formulation. We would like to formulate sufficient conditions for the existence of a solution of the Cauchy problem in a neighbourhood of S

without restricting to a small neighbourhood of a point on S (although the neighbourhood of the hypersurface S may be very small, in the sense of, for example, some distance from S). First, these conditions must of course, guarantee the existence of solutions of the local problem at any point x e S and secondly, roughly speaking, provide continuous dependence of the root d of

equation (17.8) on x. This means, that on S we may define a covector field _ 4 (x') E Ts M, continuously depending on x' c- S and such that 1) i a (x') = Wx. (x'), where is S-* M is the natural inclusion map and W.., (x') is the gradient of W (x') at x' c- S, viewed as a covector on S (an element of T. S);

2) Introduce local coordinates as described in 17.3 in a neighbourhood of any point x' a S. Then

(x') = (x') is a root of (17.8), satisfying (17.9), i. e. satisfying all the conditions for the local solvability of the Cauchy problem. Let us note that (17.12) may be written, here without local coordinates as

cpx(x') = (x'), x'ES

(17.21)

§ 17. Non-linear First Order Equations

139

Therefore, the final statement of the Cauchy problem goes as follows: find a solution of (17.2), defined on a connected neighbourhood of the hypersurface S, satisfying the initial condition (17.6) and the additional condition (17.21). In this form, the problem has a unique solution, depending smoothly on the parameters (if any), provided that the given quantities a, S, i, and also depend smoothly on these parameters.

Remark 17.1. Condition 1) is obviously necessary (assuming the rest is also fulfilled) for the solvability of the Cauchy problem and signifies simply the absence of topological obstructions to the global existence of a field (x'), the local existence and smoothness of which is ensured by solvability conditions of the local problems at the points x' a S. 17.5 Linear homogeneous equations. Equation (17.2) is called linear homo-

genous if a (x, ) is linear in , i.e.

a (x, ) = V (x) ,

(17.22)

where V (x) is a vector field on M. The projections on M of the bicharacteristics, are in this case the solutions of the system x = V (x),

(17.23)

and the solutions of (17.2) are simply the first integrals of the system (17.23). The

same system (17.1) contains also, along with (17.23), the equations

4 = - V.(x) which are linear in . A standard growth estimate for

(17.24) I

(t) I shows that if

x (t) E K, where K is a compact set in M, then I (t) I is bounded on any finite interval on the t-axis. Therefore a bicharacteristic is either defined for all t or its projection x(t) will leave any compact set K C M. The condition that S is noncharacteristic means, that V(x) is everywhere transversal to S. Let us consider the mapg mapping (x', t) into x (t) with x (t) a solution of the system (17.23) with the initial value x (0) = x'. If there exists e > 0 such that x(t) is defined for any x' for all I I I < e, then g determines a map

g: S x ( - e, e) - M.

(17.25)

If g is a diffeomorphism, then the solution of the Cauchy problem with initial data on S is defined on the image of g. It is therefore important to be able to

estimate from below the number e > 0, for which the map (17.25) is a diffeomorphism. One important case, where such an estimate is possible will be shown below.

Chapter III. Asymptotic Behaviour of the Spectral Function

140

17.6 Non-homogeneous linear equations. These are equations of the form V (x) (px (x) + b (x) (p (x) = f (x) ,

(17.26)

where b (x), f (x) E C °° (M), V (x) is a vector field on M, (p (x) is an unknown

function and (px its gradient. If x (t) is a solution of the system (17.23) then obviously dt (p(x(r)) +

b(x(t)) (p(x(t)) = f(x(t)),

from which (p (x(t)) can be found as a solution of an ordinary first order linear

differential equation, provided that (p (x (0)) is known. The basic feature following from this is that the domain, on which a solution of the Cauchy problem exists, depends only on V (x) and Sand is independent of the right-hand side f(x) and the initial value V/ e C' (S). In particular, in what follows, we will need an equation of the special form

OX.

-

amp

a; (x) ;_1

ax,

+ b (x) 9 = J(x),

(17.27)

where x = (x', xp), x' c- M' for some (n -1)-dimensional closed manifold M' and x E (- a, a) with a > 0. The system (17.23) (for the corresponding homogeneous equation) is of the form

x' = V'(x),

(17.28)

The solutions x(1) of this system which start at x = 0, are defined for t e (- a, a) and if we put S = M'= {x: x = 0}, then the map g of the preceding section becomes a diffeomorphism g: M-. M, where M = M' x (- a, a) and where it is clear from (17.28) that the "fiber" M' x xo is mapped onto itself diffeomorphically. Because of this, the Cauchy problem for (17.27) with initial condition W lx.=o = W W),

XE

,

(17.29)

has a solution (p e C°` (M).

In a number of cases one can carry out similar arguments also for noncompact M'. Problem 17.1. Let a (x , 4) be defined for x E M, $ 0 with degree of homogeneity I in . Show that if (x(t), fi(t)) is a bicharacteristic, then it is either defined for all t or x (t) will leave any compact set K C M. In particular, if M is compact, then all bicharacteristics are defined for all t E IR.

§ 18. The Action of an Operator on an Exponent

141

Problem 17.2. Show that the same holds for an arbitrary degree of homogeneity of a (x, 4), if condition of ellipticity holds:

for X40, xEM.

§18. The Action of a Pseudodifferential Operator on an Exponent 18.1 Formulation of the result. Here we describe the asymptotic behaviour as A -- +oo of the expression A (e"*(-,)), with A a 'PDO and t a smooth function without critical points. Theorem 18.1. Let X be an o p e n set i n 1R", A E LQ a(X), 1 - Q S 8 < Q, A properly supported and with symbol a (x, ). Let Vi (x) e C' (X) and V,, (x) $ O for x e X (here Wx denotes the gradient of W). Then for any function f e C" (X) and

arbitrary integer N ? 0, for A > 1 we have

A(fe«.) = e14v

I

Y-

a(a,(X,2W:(x)) D: (f(z) a!

(a!
+Am-(Q-112)NR

N(X,

13-X+ (18.1)

) ,

where Qx(Y) = W (y) - W (x) - (y - x) - Wx (x), a(a'(x, ) = c's a (x, ), and for RN (x, A) the following estimate holds

a'RN(x,A)15 Cy.N.KAoIII,

XEK,

(18.2)

where K is compact in X and the constants C7, N. K do not depend on A. If there are families of functions f (x), Vi (x), bounded in C °° (X) (i.e. the derivatives 8' f (x), a' w (x) are uniformly bounded on any compact set for arbitrary fixed y) and if the gradients op; are uniformly separated (in absolute value) from 0 on any compact set, then the constants C,,N.K in the estimates (18.2) are independent of the choice of functions f, W of these families.

Remark 18.1. The statement of this theorem is similar to that of Theorem 4.2 on the transformation of the symbol under diffeomorphisms and, as will become clear in what follows, the theorems are actually equivalent. However, in view of the importance of Theorem 18.1, we shall give two proofs for it: one derived from Theorem 4.2 and an independent one, essentially based

on the stationary phase method. The latter allows us to deduce from Theorem 18.1 the invariance of the class of'PDO under diffeomorphisms and the formulae for change of coordinates. The reader is recommended to pursue this in the form of an exercise (historically this was the first method by which the invariance of the class of Y'DO under diffeomorphisms was demonstrated).

Chapter Ill. Asymptotic Behaviour of the Spectral Function

142

Remark 18.2. It is useful to note right from the beginning, in what sense (18.1) is asymptotic as 2-*+oo. It is clear that the condition Vt.' (x) + 0 will ensure the estimate Ca,xAm - olal,

I a(')(x,Aw (x))I5

x eK.

(18.3)

is a polynomial of degree not

Let us now verify that higher than I a 1/2. Indeed we have Dz

(18.4)

(f(z)e1AQ.(z))

c1,...r, (DY°f(z)) . ).k (D:' Qx(z)) ... (D=' Q. (z)) er) x(z),

G

ro+r.+...+r.=a

where in this sum I y j I z 1 for j =1, 2, . . ., k. Since ax (z) has a zero of second order for x = z, then in (18.4) for x = z only terms in which I y j I ? 2, j =1, ... k,

remain. But i Iyj I S I a i, so we obtain 2k:5 Ia I as required. j=1

Now, taking (18.3) into account, we see that a(a)( x

Wx ( x))

D za (f(z) ei1Q,(z)) aI

SC

2m-(Q-1I2)lal

XEK,

:=x

giving the required decrease in the degrees of growth of the finite terms in formula (18.4), since g > 1/2.

18.2 First proof of Theorem 18.1. Let us make a change of coordinates y = x(x) = (x,, ... , x" w (x)). This change has Jacobian y,X, and may be made, therefore, only where yi;,, * 0. The general case however can be easily reduced to this case using a partition of unity and a rearrangement of the coordinate axes. Indeed A is properly supported and consequently for any compact set K, there

exists a compact set K2 such that Au I x, only depends on u I K,. Therefore A (fel)w) I x, may be written as a finite sum of terms of the form A (fj e12r'), with

j E Co (X) and where to any j there exists an integer k, 15 k 5 n, such that yrX,(x) $ 0 for x e suppJ . Thus, let Wz, + 0, x E Xand let A 1 be the operator A written in the coordinates

y= x(x) (cf. §4). Let a, (y,,) be the symbol of A1. By Theorem 4.2 we have a,(Y,11)I

x

e-rx(x) aA IaI
a(a) (x,'x'(x),) (D.* e,x;()''1)1.=x+ rN(X,

where

x" (z) = x (z) - x (x) - x' (x) (z - x),

rN E SQ 6 (Q -1

/2) N (X

x lR") .

§ 18. The Action of an Operator on an Exponent

143

In particular, putting q = (0, . . . , 0, A), we obtain precisely formula (18.1) for

j- 1 and where RN can be estimated as in (18.2). The statements about uniformity and the case of arbitrary f are obtained by repeating the proof of Theorem 4.2; we leave this to the reader as an exercise. 18.3 Second proof of Theorem 18.1. For simplicity, we shall assume in this proof that 6 = 0 and Q =1. Note, once again, that since A is properly supported, we may assume that (x) (for a fixed x). We have

f e Co (X). Put, for brevity, 1(A) = e -iAc(x) A

(18.5)

C dy

I (A) = J a (x, C) f(y)

Let us now make the change of coordinates 4 = AC: I(A) = A" Ja(x,AC)f(y) eiAIP(Y)_ (x)-(Y-x) C1dydC.

(18.6)

We want to find the asymptotic behaviour of this integral as A - + oo. We shall see that a major role (and this is the point of the stationary phase method) is played by neighbourhoods of the critical points of the function

g(y,C)=w(y)-W(x)-(y-x) C.

(18.7)

Now, since gC' = x - y, gy = w; (y) - C, there exists exactly one critical point of this function, the point y = x, C = yi,, (x). For short we put Cx = yi;,(x). Introduce the cut-off function X e Co (1R"), X (z) =1 for I z e, where

e > 0 and consider the integral I(A) = A" J a (x, AC) x (y - x) x (C - fix) fly) eiA (Y-O dy dC .

(18.8)

Then for any N > 0 II(A)-7(A)1 5 CNA-N.

(18.9)

Indeed, putting

rL=A-1I(w'(Y)-C)2+(x-y)2]-1 [(w'(y)-C) -

Dy+(x-y).DC],

we see that `Le'2:1 O= Integrating by parts in the oscillatory integral 1(A) - I(A), and considering that

I(v"(Y)-572+(x-y)21 > e, > 0

on

supp(1 -X(Y-x) X(C-Cx)],

we see that I (A) - 7(A) =

A" J e'A9cy.0 LN [a (x, AC) (1- X (y -

x) X (C - C3))f(y)] dy dd

,

Chapter 111. Asymptotic Behaviour of the Spectral Function

144

and transforming this oscillatory integral into an absolutely convergent one

(cf. §1) we easily obtain the estimate (18.9) due to the factor A', in the expression for L. Analogously, one also obtains estimates for the x-derivatives

from the difference I().) -1(2). However, note that they follow from the estimates of I(2) - 1(X) with arguments similar to the proof of Proposition 3.6. In the sequel we shall omit estimates of the derivatives, leaving them to the reader. Thus, instead of I(2), we may consider 1(2). Making yet another change of coordinates { _ x + 2 ', . we obtain 1(2) _ !e'(x-Y) a

a (x, .lax + ry) in a Taylor series at j7 = 0: a (x, AS. + q) = Y a(a) (x, A4x) ! + rN (x 1, A) Ial
V

,

where )

rN(x,n,2) = E c, !(1 - t)N - I lal=N

0

Multiplying this expansion with the cut-off function X (ryl1) X (y - x) and substituting the result into (18.10) we obtain 1(2) = Y ! ei(x-Y). 'i a(a) (x, lal
!N'7 Xt'7 X(y-x)f(Y) e' Q') dydp + RN(x,1),

(18.11)

where

RN (x,))=

Caldydry J lal=N

ir a(a)(x,LLx+tq)

01

X X(Y-x) X11) e1(x-A-gfly) e'xQ.(y)dt.

(18.12)

As follows from the arguments above, the asymptotic behaviour of the finite parts in formula (18.11) does not change if we remove the cut-off function X (, /2). But then these terms can be easily transformed, by the Fourier inversion formula: e'(x-y)-7

=

a(a)

X(y-x)fly) euQ.(Y)dyd7

(x, AS.) Dy (f(y) euQ.(y)) I Y =x,

§ 18. The Action of an Operator on an Exponent

145

so that it only remains to estimate the remainder RN or to estimate, uniformly in t

(0<1;51), the integral

r,(x,.,t) =

x+tn) X(y-x)

e'(.-r)-"

a(=)(x,4:+ 11) X(',)(Y-x) X

et",

a +r':

U(Y)eiaa.(r)) dyd7

() D' (f(Y) e'4.(r)) dydn (18.13)

where I a I = N.

Let us introduce the notation a(a)

as (x,1,2,1) = XI )

If the numbers in the definition of X (z) is chosen so that a

1/2, then for a, one

has the estimates IaR

2(x,ti,Z,t)I s

x Irl

(18.14)

Now let us use (18.4) and substitute into (18.13) the expression obtained from this for (f (y)e'AQ'(Y)). In this expression all the terms contain products D,"'

(18.15)

Ak (D7, ex (Y)) ... (Dy- a,, (y)).

in which 1y1 I + ... + I yk I S N. If k::5; N/2, we do not transform this product. If k

> N/2, then by the Dirichlet principle, in (18.15) there are no less than k - N/2 indices y, such that I y1 I = 1. But then, by the Hadamard lemma

(Dy'Q:(Y)) ... (Dy'e:(y)) _

gr(Y,x) (x-Y)r, bl2k-N/2

wheregr (y, x) is a smooth function (in x and y), defined for y sufficiently close to

x. Inserting this expression into (18.13) and integrating by parts (utilizing the exponent allowing us to change (x-y)r into (-D")r), we see that r, (x, A, t) is a linear combination of terms of the form I2 ()) = A" j aar (x, n, A, t) e;(=-P) - "

x) dy d11,

(18.16)

where ar = 87 as and AY, x) is smooth (in x and y) and supported in I y - x I S E. The indices k and y are related by I y I z k - N12. Taking into account that the volume of the domain of integration inn in (18.16) does not exceed C).", and using (18.14) we obtain for I1(A) the estimate II1(A)I < CA*+M-(N+IYI) +"5 CA.+"-N12,

146

Chapter III. Asymptotic Behaviour of the Spectral Function

which allows us to conclude the proof by applying the type of arguments used in the proof of Proposition 3.6.

18A The product of a pseudodifereatial operator and a Fourier integral operator. Let X, Y be open sets in 1R"x and IR"r and let P be an FIO of the form

Pu(x) = jp(x,y,0) e`'(-.r-O) u(y) dyd0,

(18.17)

where p (x, y, 9) a S" (X x Y x IR") and cp (x, y, 9) is an operator phase function

(cf. §2, Definition 2.3). Let there also be on X a properly supported PDO A e L',§ (X) with symbol a (x, ). Since P maps Co (Y) into CI (X) and d' (Y) into 9'(X) and A maps the spaces C°° (X) and 9'(X) into themselves, then the composition A P is defined as an operator, mapping COI(Y) into C°°(X) and

9'(Y) into 9'(X). Theorem 18.2. Let 1 - Q 5 6 < Q 5 1. Then the composition Q = A P is also of the form (18.17) with the same phase function (p (x, y, 0) as P and with an amplitude of the form 9(x,y,9) = e-io(x.r.9)a(x,Dx) (p(x,y,0) e+-(X-r.8)],

(18.18)

with the asymptotic formula q (x, y, 0) ^.

a'°) (x, Px (x, y, 0))

D= (p(z Y 0)

a

as

e'ecx.x.r.ei)

a.

I91-++oo

x=x

(18.19)

where g (z, x, y, 0) = q, (z, y, 0) - N (x, y, 0) - (z - x) - cx (x, y, 0).

Remark 18.3. Since ap (x, y, 0) is not smooth for 9 = 0, it is not immediately clear from (18.18) that Q is an F1 0. This is the case however, since adding an operator with smooth kernel to P we may assume that p (x, y, 0) = 0 for 101 < 1. Then (18.18) defines a smooth function in all the variables and the same holds for

all terms in the expansion (18.19), which has the usual meaning (cf. Definition 3.4). However, an operator with smooth kernel may always be written in the form (18.17) with an amplitude p (x, y, 9) which has compact support in 0 and equal 0 for 10 1 < I (cf. the hint to Exercise 2.4). Therefore Q is an FIO with phase function cp.

Proof of Theorem 18.2. Let us introduce the set

C. _ ((x,y,0): (Pa(x,y,0)=0) . used in §1 and §2. Note that cpx (x, y, 0) * 0 for (x, y, 0) a C, by the definition of an operator phase function. Changing P by adding an operator with a smooth

§19. Phase Functions

147

kernel, we may assume that suppp (x, y, 0) lies in an arbitrarily small conical neighbourhood of the set C, (cf. Proposition 2.1) and, in particular that qi * 0

on suppp. In addition and in accordance with Remark 18.3, assume that p(x,y,0)=0 for 101<1. Now note, that since A continuously maps C °° (X) into C°° (X), we may apply

it under the integral sign in (18.17) (for this it is necessary to begin by transforming the integral into an absolutely convergent one, as in §1; note that the variable x is not involved in this change, being just a parameter). Now it is only necessary to verify (18.19), understood in the sense of Definition 3.4 (cf. also Remark 18.3). But this is a trivial consequence of Theorem 18.1, putting A = 10 and viewing y and 0 as parameters. Indeed, we have

4(x,Y,0) =

2m.

e-+'14.1.91 a(x,D.) [A

p(x, y,0) ell'(--Y.9')] .

where 0' = 0/101. Noting now that by varying the parameters y, 0 the functions q, (x, y, 0'), A - p (x, y, 0) belong to a bounded subset of C1 (X), we see that

Theorem 18.1 applies. J Exercise 18.1. Obtain from Theorem 18.2 the composition formula for two properly supported'PDO of the type Lm(X) (cf. Theorem 3.4). Exercise 18.2. Let A and P be as in Theorem 18.2. Prove the result, similar to

Theorem 18.2, for the operator Q1 = P A. Hint. Use transposition. Problem 18.1. Obtain from Theorem 18.1 the change of variable formula for PDO (cf. Theorem 4.2).

§ 19. Phase Functions Defining the Class of Pseudodifferential Operators 19.1 In the formulation of Theorem 4.1 there is an example of a class of phase functions, for which the corresponding class of FIO coincides with the class of'PDO. In the sequel, we shall need the following variant of this theorem for non-linear phase functions. Theorem 19.1. Let X be an open set in 1R" and q, (x, y, ) a phase function on

XxXxR such that 1) tp4'(x,Y, )=0px=Y; 2) qi (x, x, ) = . Then, if 1 - Q < b < Q, the class of FIO with phase function (p and amplitude p (x, y, *) a SQ a (X x X x 1R') coincides with the class LQ a (X). The class of FIO with phase function qi and with an amplitude a (x, y, ) which is classical, coincides

with the class of classical 'YDO.

148

Chapter III. Asymptotic Behaviour of the Spectral Function

Remark 19.1. Conditions 1) and 2) for nearby x and y, may be expressed in one single condition I).

(19.1)

19.2 Proof of Theorem 19.1. 1. Since the kernels of FIO with phase functions satisfying condition 1) of Theorem 19.1 are smooth for x * y by Proposition 2.1, then we may assume that all amplitudes are supported on an arbitrarily small neighbourhood of the diagonal x = y. 2. Let (p, gyp, be two phase functions, satisfying the conditions of Theorem 19.1. Denoting by LQ 8 (X, q) the class of FIO with phase function rp and with amplitudes in the class S.'.6 (X x X x 1R"), we see that it suffices to verify the inclusion LQ a (X, (p)

L7,,, (X, 91) ,

(19.2)

because it clearly implies that all the classes LQe (X, (p) coincide and, in particular, that they coincide with LQ a (X). 3. Denote by JE°k the class of all functions w (x, y, ), which are positively homogenous of degree k in , smooth for + 0 and defined for nearby x and y. Note that JE°o is an algebra, containing the smooth functions of x and y as a subalgebra, and that Jr, is a Jro-module. For us it is essential that for nearby x and y, the difference gyp, - cp can be written in the form "

(P - P =;. L. °Jk

aq a(P ask'

b,,k c -.W, .

(19.3)

Let us verify this. From the Taylor formula it follows that aP

"

(x;-y;) +kj a,k(xk-yk),

where a,k E.Wo, a,k (x, x, ) = 0. This can also be written in the form

,p' = (I+A)(x-y), where I is the unit matrix and A is a matrix with elements in Jl°o, equal to 0 for x = Y. But then, for nearby x and y the matrix (1+ A) -' exists and has elements from A.. This means that we may write X, - y,

a

= k1 ajk a k'

a-,, eJ°o .

(19.4)

Now, using (19.1) for qp and (p,, we see that on the diagonal (as x = y) !p, - tp has a zero of order two and by the Taylor formula

§19. Phase Functions

149

(pl - V = L, 6;k(xj-Y;) (xk-Yk), j.k=1

'Fit, E.*

.

Inserting here the expression for x, - yj from (19.4), we obtain (19.3).

4. Consider now the homotopy

qp,=(P+t(rp1-q'),

0--<_ts1.

(19.5)

Each of the functions (p, satisfies (19.1). A trivial repetition of the above argument shows that instead of (19.3) we may write

wl - (P= I b,k

aSj rc

(19.6)

xx

j.k=1

aSk

where bik E.t'1 and depends smoothly on t. Now let P, be the FIO given by the formula p,u(x) =I ero.(x.r.t)

u(y) dydc .

Then r

dt' (P,u) = !!p(x,Y, ) i'(p1-w)'

u(Y) dyd

a(p,

_

!! d;..... ,,.

e'", pu dy dd ,

(19.7)

where dj...... j,, E.z°r . Without loss of generality, we may assume that p (x, y, ) = 0 for I I < 1, so that dj..... ..p c- S"` +' (X x X x IV). We now integrate by parts in (19.7), using the formula

This integration demonstrates that

d' p,ELQ ar(1-2e)(X,(Pr), where all estimates are uniform in t. But if we now put Qj

(-1)j dP

)

then, by the Taylor formula,

dtjrELQd;u-2Q)(X,gg1),

(19.8)

150

Chapter M. Asymptotic Behaviour of the Spectral Function k-I

Po= i=o

tk-1

I

Qj+(-1)k! (k-1)l

dkP

0

dtkt dt.

(19.9)

The remainder in (19.9) has a kernel with increasing smoothness as k -+ + oo. It is

therefore clear that if Q - E Qt (adding the amplitudes asymptotically), then i=o

QELQ a(X,(p1)

Po-QeL-°°(X),

and

which shows (19.2). The fact that classical amplitudes remain classical under this procedure, is clear from the construction.

Problem 19.1. Let V1, W2 be two phase functions such that

471-92

" '092 42 J1Ll bik ab, abk'

where bA e C °° (X x X x (1R"\ 0)) and b;,; homogenous in of degree 1. Show that L'.a(X (P 2) c LQ a(X,(pI).

§20. The Operator exp (- itA) 20.1 Definition and formulation of results. Let M be a closed n-dimensional manifold with a smooth density and let A be a self-adjoint, classical `I'DO of degree 1 on M with principal symbol a1(x, i;) satisfying the condition a1

0

for

* 0

(20.1)

(in particular, A is elliptic). Let {q7k}k_ 1. 2.... be a complete orthonormal system of eigenvectors for A and Ak the corresponding eigenvalues. If u (x) E C", (M) we

denote by uk the Fourier coefficients of u(x) with respect to the system {(Pk(x)): Uk = (u,tPk).

(20.2)

Proposition 20.1. If u (x) e C °° (M), then 00

U (x) = F, uk(Pk(x), k=1

where the series converges in the topology of C°`(M).

(20.3)

§20. The Operator exp(-itA)

151

Proof. We clearly have uk,Ik = (u, A"(Pk) = (A"u, (pk),

and since A"u E C°°(M), we may apply A" termwise to the series (20.3) for any N E Z,. , obtaining each time a series converging in L2 (M). But from this it immediately follows that for any s E R, the series (20.3) converges in norm in

H'(M), since for s = NEZ 4 this norm is equivalent to II u II + II A"u II, With II

.

II

the L2(M)-norm. From this and the embedding theorem 7.6, the required statement follows. This proposition allows us to make the following Definition 20.1. The operator exp (- itA) for t E 1R is defined by the formula

exp (- itA) u (x) _ Z 00exp (- it A) uk tpk (x) .

(20.4)

k=1

Clearly the series (20.4) for u c- C 0D (M) converges in the topology of C °° (M).

Further, if u e H'(M), s an integer, then this series converges in norm in H'(M) (cf. the proof of Proposition 10.2). The operator exp (- it A) is for integer s a bounded operator on H'(M). Note, that it is also a unitary operator on L2(M). Another definition consists in considering the Cauchy problem

r + iAu = 0,

(20.5)

u l,.o = uo,

(20.6)

where u = u(t, x) E C°°(lR x M), uo e C°°(M) and the operator A in (20.5) acts on x for any fixed t. Solving this problem by the "Fourier method", we see that the solution is given by (20.4), with u replaced by uo, i.e.

u (t, x) = exp (- itA) uo (x).

(20.7)

The solution of (20.5)-(20.6) is also unique. This can be seen, for instance by writing for a solution u (t, x) the expansion

u(t,x) = E Ck(t) (k (x), k=1

which, being inserted into (20.5)-(20.6) gives the equations ck (t) + iAkck (t) = 0, with the initial values ck(0) = (uo, (pk), from which one can uniquely recover ck (t ).

Theorem 20.1. I f e > 0 i s s u f f i c i e n t l y s m a ll, then f o r I t I
tP (t, x, y, ) = W (x, y, ) - ta1(y, )

(20.8)

Chapter III. Asymptotic Behaviour of the Spectral Function

152

linear in t, and by an amplitude p (t, x, y, ) which is a classical symbol of order 0, smooth in t and such that the following estimate is fulfilled

Ia'a°=..,P(1,X.Y,o I5 Cav«>-1°1.

(20.9)

Example 20.1. Consider in IR" the operator A = V---A with symbol ICI. The Cauchy problem (20.5)-(20.6) for functions decreasing as I x I - + co can be solved using the Fourier transformation in x. Indeed, since Au (x)

=

then

u(t,x) = exp(-itA) u0 (x) = J u(Y)dydd. We see that in this case (formally this does not follow from the theorem however) the operator exp(-i t A) is an FIO with phase function (x -y).4 - t I I

20.2 Proof of Theorem 20.1. 1. Let us construct the operator Q(t), which approximates U(t) and is an FIO of the form ei®u.x.y.af(Y)dydg.

Q(t)f(x)=

(20.10)

The operator U(t) satisfies the conditions

( (D,+A) U(t) = 0,

(20.11)

1 U (O) = 1.

(20.12)

We will try to choose Q(t) satisfying the conditions

(D,+A) Q(1) eL'(M), 1 Q(0)- IeL-°°(M).

(20.13) (20.14)

More precisely, the left side of (20.13) will also be a smooth function of t with

values in L- (M). The linearity of the problem, allows a reduction, using a partition of unity, to the case of constructing q and V in local coordinates. In view of Theorem 18.2, (20.13) will be satisfied if e-io [D,+A] (ge1o) eS-`°,

(20.15)

where A operates on x.

Writing down an asymptotic expansion for the symbol a (x, ) and for q (1, x, y, ) in terms of homogeneous functions, we have

a-a,+ao+a-,+..., q-qo+q-t+q-2+...

§20. The Operator exp(-itA)

153

Then by Theorem 18.2

e-`o[D,+A]

(q,,+a,(x,(px)) q0+ ro+ r_, + ... ,

where r, is the sum of terms of degree of homogeneity j. We require ,p, + a, (x,

0.

(20.16)

The initial values W 1, = o = W (x, y,)

(20.17)

must be chosen, so that for t = 0 we may guarantee (20.14). But for this we have to require that W be a phase function, corresponding to the class of `I'DO (cf. §19), i.e. for nearby x and y

W(x,Y, )=(x-Y)' +D(Ix-Y1'ItD.

(20.18)

We will look for this function in a neighbourhood of the diagonal x = y. The term O (1 x - y I I I 1) in (20.18) is necessary in order to achieve linearity in t for

the function (p (t, x, y, ), which - in its turn - is useful since later on we will take the Fourier transformation in t. Thus, we look for rp in the form co (1, x, Y,

) = W (x, y, ) - ta' (Y, 4).

This is yet another requirement on (p. We will see that it can be satisfied. Putting this expression for V into (20.16) yields:

-a'(Y,4)+aI(x,Wx(x,Y,4))=0. Setting x = y, we obtain using (20.18) that a' (y, ) = a, (y, ). Hence

co(t,x,Y,4)=W(x,Y, )- taI(Y,4).

(20.19)

where W (x, y, 4) satisfies the equation ''r: (x, Y,

)) = ai (y, )

(20.20)

Instead of (20.18) we require (20.21)

W. (x, Y, 4)1.=, _ 4,

(20.22)

from which (20.18) immediately follows. The relations (20.20)-(20.22) define a Cauchy problem for yr, in which y and are parameters. In this, we may assume

Chapter Ill. Asymptotic Behaviour of the Spectral Function

154

that I I = 1, since solving this problem for I I =1 we can afterwards extend W by homogeneity (of degree 1) to all values of .

Let us note, that the plane (x -y) = 0 is non-characteristic, because 64a, (x, ) r 0 in view of the Euler formula aca, (x, ) = a, (x, 4). From the results in §17 it is clear that the solution exists for x which are close to y and, furthermore, that the corresponding neighbourhood of y in which the solution exists, may be chosen uniformly in y, 4 so that the function W (x, y, ) is defined in

some neighbourhood of the diagonal x = y.

2. It follows from Theorem 19.1 that there exists a classical symbol I(x, y, ) e CS°, which is supported on an arbitrarily small neighbourhood of the diagonal x = y and

f f 1(x,

f (y)dydt - f (x) = kf (x),

(20.23)

where k is an operator with smooth kernel. From (20.23) and (20.14) it now follows that we must have the following initial condition for q: q (0, x,

I(x,

(mod S-°°).

(20.24)

If we introduce the decomposition of 1(x, y, ) into homogeneous components

1-1°+I_,+..., then (20.24) may be rewritten in the form q-j (0, x, y, 4) = 1-j (x, y, 4),

j = 0, 1, 2, ...

(20.25)

We now write out the equations for the functions q_j, given by (20.13) and (20.25). The first order terms equal 0 in view of (20.16). For the 0-th order terms we obtain

arg0+ E

ala)(x,(gs)az40+

ja, = t

L aal(x,(p:) of q0+ia°(x,(px)go=0, I2 (20.26)

q01,=0=10,

where alai = ac" a,. This Cauchy problem allows us to define q0 (1, x, y, ) for small t. Furthermore, for any integer j z 0 we obtain the following equations, called transport equations:

§20. The Operator exp(-itA)

155

a,q-i+ E ai')(x,(,Y) axq-; 1=1=i

+

ala)(x,9

°

a 4 -;+iao(x,(ox)q-i+R,=0,

(20.27)

q-tlr=o= I_i, where R` only depends on qo, q-1, . , q-(j-1) In view of the reasoning in section 17.6, the solution of (20.27) may be defined in the same i-interval as the solution of (20.26). Hence, we f i n a l l y obtain an operator Q (t), defined f o r I t I < s, which is an FIO and for which

(D, +A) Q(t) = K(t), Q (0) =1 + k,

(20.28) (20.29)

where K(t) has a smooth kernel with a smooth dependence on t and where k is an

operator with a smooth kernel. 3. Let us now demonstrate that [U (t) - Q (t)] is an operator with infinitely differentiable kernel in t, x, y (for I t I < e). For this consider the operator

R(t) = U(-t) Q(t) -1

(20.30)

and differentiate it with respect to t:

D,R(t) _ - (D, U) (- t) Q(t) + U(-t) (D, Q) (t)

= AU(-t) Q(t) - U(-t) AQ(t) + U(- 1) K(t).

(20.31)

The validity of this computation follows from the described structure of Q (t)

and from the remarks on U(t), made at the beginning of this section (the derivative is of course taken, after having applied R (t) to some function u(x) a C°°(M)). Since U(t)A = AU(t), it follows from (20.31) that

D,R(t) = U(-t) K(t).

(20.32)

On the other hand, it is clear from (20.29) that

R(0)=k.

(20.33)

Since U(- t) K (t) is an operator with smooth kernel in t, x, y, integrating (20.32) and taking (20.33) into account, we see that R (t) has the same property and then

from (20.30) it is clear that the operator U(t) - Q (t) a kernel which is smooth in t, x, y for It I < s.

U(t) R (t) also has

Problem 20.1. Let A be a classical first order WDO on M, satisfying (20.1) but not necessarily self-adjoint. Carry out the construction of the parametrix

Chapter III. Asymptotic Behaviour of the Spectral Function

156

Q (t) for this operator and show that the Cauchy problem (20.5)-(20.6) has a unique solution and that for the operator U(t) = exp(- itA), defined by this problem, the statement of Theorem 20.1 is also true.

Hint: Obtain an integral equation of the Volterra type for U(t), using the operator Uo (t) = exp (- itA0), where Ao = (A + A*). Z

§21. Precise Formulation and Proof of the HOrmander Theorem 21.1 The singularities of the Fourier transformed spectral function near zero and estimates of the averaged spectral function. Let e (x, y, A) be the spectral function of the same (first order) operator A as in §20, U(t, x, y) the kernel of the operator exp (- itA). If Wj (x) are the eigenfunctions of A with eigenvalues 2. j , we have

e(x,Y.A) = Y gpj(x) pj(Y),

(21.1)

A,5A

U(t,x,Y) _

e-iA,I qp,(x) QPj(Y),

(21.2)

where the latter series is summed in the sense of e. g. distributions on M x M, depending smoothly on t (this is easy to prove by arguments similar to those used after Definition 20.1). It follows from (21.1) and (21.2) that

U(t,x,y) = je-1A'de(x,Y,A),

(21.3)

where the integral is understood as a Fourier transformation (from A to t) in the distribution sense. Let Q (A) E S (IR') and let Q (t) = FA_, Q (A) be the Fourier transform. Then

from the known properties of the Fourier transformation it follows that Q(t) U(1,x,y) = FA-, JQ(A-,u)

(21.4)

Now choose p (A) such that

1) Q(A)>0 for all AalR'; (0) = 1; 3) supp (t) c (- s, s), where s > 0 is sufficiently small. 2)

Exercise 21.1. Prove the existence of such a function Q(,L). Hint: This is done by analogy with the construction of X (x) at the beginning of the proof of Theorem 6.3. Let now Q (t, x, y) be the distribution kernel of Q (t), constructed (for small I t 1) in §20. In view of Theorem 20.1, we have

U(t,x,y)-Q(t,x,y)eC'((-s,e)xMxM).

(21.5)

§21. Proof of the Hormander Theorem

157

Thus the functions U(t, x, y) and Q (t, x, y) have the same singularities in a neighbourhood of the point t = 0. Now (21.4) implies

Lemma 21.1. The function

JQ(A-tu) d, e(x,y,,u) - F,-i(Q(t)Q(t,x,y))

(21.6)

is a smooth function in all variables, tending to 0 faster than any power of A as A- + oo, uniformly in x, y e M.

Let us now compute the second term in (21.6). This can be easily done, thanks to the linearity in t of the phase function tp (t, x, y, ), in the definition of the operator Q (t). First we compute formally, not worrying about convergence of the integrals. In the notation of §20 we have

Q(t,x,y) =

(21.7)

F,-.x(Q(t)Q(t, x, y)) (A) =(2n)-' -S 6(t)

ell' -Ba,(y.0+f.c=.Y.ct dtd4.

(21.8)

et.xdt.

(21.9)

Set

(2n)-'I Q(t) Then

F,_.z(Q(t)Q(t,x,y))(A) =

(21.10)

x,

Let us now note that R (A, x, y, ) is a smooth function of all variables and, furthermore since q (t, x, y, 4) is a smooth function of all variables (including 1), then R rapidly tends to 0 as IA I -- + oo and we have for any N > 0 the estimates

&x0l&aiR()., C .V

x,y,S)I

(21.11)

C CO,6N<

Here R admits an asymptotic expansion into homogeneous functions

in . From (21.11) it is clear in view of the ellipticity of a, (y, ), that R (A - a, (y, ), x, y, ) rapidly tends to 0 as I I - + oo, hence the integral in (21.10) is absolutely convergent. As for the justification of the transition from (21.8) to (21.10), which was done formally, it remains to note that it is valid for symbols q (t, x, y, ) which have compact support in t, and perform the standard passage to the limit, as in the definition of oscillatory integrals (cf. §1). Taking Lemma 21.1 into account, we see that we have proven Lemma 21.2. If R is defined via the formula (21.9), then for any N > 0

I JQ(A-p)de(x,y,lt) - JR(2-a, S CN(1+IAI)-N, where CN is independent of x and y.

I

(21.12)

Chapter Ill. Asymptotic Behaviour of the Spectral Function

158

Let us now estimate the second term in (21.12). Lemma 21.3. We have

If

(21.13)

where C does not depend on x and y.

Proof. Let us denote, as in §15: V,(2) =

J

(21.14)

Since the function a, (x, ) is homogenous of degree I in , then V, (A) = V,,(1)All .

(21.15)

Let us now utilize the obvious identity eiv(x.r.t)dd=

JR(). -aI

e1c(x.r. )dV,,(a),

both sides of which are defined due to the estimate (21.11). Taking (21.15) into account, we see now that the left-hand side of (21.13) can be estimated via M

=CNJ(]+I).-al)-Nor e-'da.

CN J(]+Iz-al) 'dVV(a) 0

0

But in view of the obvious inequality I + I a 15 (1 + 12 - a I) (1 + I A I) we have

(1 +

I,1-a1)-N a"-l da

0 (1+IAI)"-'

J (]+1,1-or p-N+'-,da= CO + JA DR-', - Go

from which (21.13) follows.

Cl

Corollary 21.1. The following estimate holds

I Jp(A-µ)d,,e(x,y,µ)I <- C(1+I where C is independent of x and y.

(21.16)

§21. Proof of the Hormander Theorem

159

21.2 Passage to estimates of the spectral function Lemma 21.4. The following estimate holds

Ie(x,x,A+1) -e(x,x,A)1 < C(1+IAI)n-1,

(21.17)

where C does not depend on x and A.

Proof. Since Q (A) is positive and non-zero on [ - 2, 2], we have A+2

JQ(.Z-p)d,e(x,x,pu)zC J x-1

where C> 0; the statement of the lemma follows now from Corollary 21.1.

11

Lemma 21.5. The following estimate holds (21.18)

Ie(x,y,2+1)-e(x,Y,2)I
where C does not depend on x, y and A.

Proof. It follows from (21.1) that

e(x,Y,2+1) - e(x,Y,2) =

9,(x) cP;(Y) A
and from the Cauchy-Bunyakovskij-Schwarz inequality we get

I e(x,Y,2+1)-e(x,Y,A)I 1/2

1/2

5 L 1<1,s1+1

Iw;(x)12]

E

I(;(Y)1Z

A<1,SA+1

=(e(x,x,A+1)-e(x,x,2))112

(e(Y,Y,A+1)-e(Y,Y,2))1n

5 C(1+IAI)"-1

by Lemma 21.4 (with the same constant Q. 0 Lemma 21.6. The following estimate holds

Ie(x,Y,2+p)-e(x,Y,2)I <_

C(1+I2I+lpI)"-1(1+IpI),

(21.19)

where C is independent of x, y, A, u.

Proof. We can prove (21.19) by partitioning e(x, y, A+µ) - e(x, y, A) into a sum of a t most 1 + 1µI terms, estimated as in Lemma 21.5. 0

Chapter Ill. Asymptotic Behaviour of the Spectral Function

160

Lemma 21.7. The following estimate holds

IJQ(A-h)e(x,y,u)dp-e(x,y,A)I
Proof. We use the fact that QeS(1R1) and obvious inequality

JQ(A)d,1=1.

With the help of the

i +IAI+IpI s (I+ IA) (1+IpI) we obtain

I JQ(A-µ) e(x,y,,u)dp-e(x,y,A)I = I J@(A-p) [e(x,y,p) - e(x,y, A)] du I = I f Q(u)[e(x, y, A +p) - e(x, y, ).)] dp I J(1+IAI+IpI)"-' (1+IpI)-"dps C(1+IZI)

as required. Lemma 21.8. We have A

-J

dAJQ(2-u)de(x,y,p)-e(x,y,A)I
(21.21)

Proof. Integrating by parts, we have

JQ(A-u)dde(x,y,u) = JQ'(A-p) e(x,y,p)du, from which A

A

J (JQ(A-,a)dde(x,y,4u))da.= J (JQ'(A-u)e(x,y,p)du)dA - 00

OD

=J (Q'(A_4u)dA) e(x,y,u)du= JQ(A-u) e(x,y,p)du, -J OD

so that (21.21) now follows from (21.20).

Taking Lemma 21.2 into account, we derive the following Proposition 21.1. We have

e(x,y,A)- JJ e<x

s C(1+IAI)"where C is independent of x, y, A.

(21.22)

§21. Proof of the Hormander Theorem

161

21.3 The Hbrmander theorem for first order operators. We would like now to rewrite the estimate (21.22) in a simpler form. Note, that in view of (21.9) +m

(21.23)

9(O,x,y, ) = I(x,y, ).

J - OD

Therefore, putting

f

J

the following estimates for R, follow from (21.11): (21.24)

axafa1R,(A,x,y, We clearly have,

if R(a-a,

x,y, i) e`r(x.r.c)dad

a
J

I(x, y, ) e`w(x.r.t)d + J R, (A-a, (y, ), x, y, ) eiv(x.r.e)d .

One can show by analogy with Lemma 21.3 that

6 C(1+IAI)" Therefore we see that the following holds Lemma 21.9. We have

e(x,y,2)-

J

eir(x.r.f)ddl 5 C(1+IA

(21.25)

where C is independent of x, y, A.

For further simplification we need Lemma 21.10. For nearby x and y we have J I

I(x,y, )

6.0.0<2

where C does not depend on x, y, A.

J a, (Y-4)< A

ea(x.r.t)d I < C(1 + 1,1

I)"-,

,

Chapter Ill. Asymptotic Behaviour of the Spectral Function

162

Proof. From the proof of Theorem 19.1, it is easily deduced that for nearby x

and y

II(x,Y, )-1 I <

(21.26)

Therefore (I(x, J (y.()<1


y,

SC f

(1+1 1)`4

f (I+IgI)-'dVy(p-)

µ<1

al <1

<

J a.(y,C)<1

1

A

CJ(1+IpI)n-2dp

CJ(1+I.UI)-1p"-tdp < 0

S

C(1+I.II)"-'

0

as required. D Note, that if the pair x, y belongs to some compact set in M x M, disjoint from the diagonal, then we may assume that I (x, y, ) = 0 and, in this case, Lemma 21.9 implies that Ie(x,y,A)1 < C(1+IA1)"'We

summarize these results in the form of a theorem. Theorem 21.1. 1) Let y' (x, y, .) be defined for nearby x and y and let

WxIx=y=4

(21.27)

Then for nearby x and y we have

e(x,Y,A)-

l

J

S

C does not depend on x, y and A. In particular, we have I

e (x, x. 2) -

Vx (1) I S C (1 + I2 )" -' .

(21.29)

2) If the pair x, y belongs to a compact set in M x M disjoint from the diagonal, then

Ie(x,Y,2)I <

(21.30)

C(1+IA1)"-).

Corollary 21.2. The following asymptotic formula for the number N (A) of eigenvalues of the operator A smaller than A, holds

IN(A)-

J a,(x.S)<1

dl dr s

C(1+I2I)"-t

§21. Proof of the HSrmander Theorem

163

21.4 The case of higher order operators. Let A be a self-adjoint elliptic differential operator of order m on a closed manifold M with principal symbol a.(x, ) Z 0. Let us construct for nearby x and y a function W (x, y, ) such that a. (x, Wx (x, y, )) = a. (y,

(21.31)

),

WxIx=y=

(21.32)

,

from which it follows that yi is homogeneous of degree one in as

W(x,y,c)=(x-y)'

and such that

x-+y.

(21.33)

Theorem 21.2. For the operator A of order m described above we have: 1) For nearby x and y n-1

e(x,y,A)-

e'r'A S C(1+IA1) a,

J

,

(21.34)

where C does not depend on x and y. In particular, a=1

e(x,x,A)

dal
J

(21.35)

a.(X.O<1

and consequently, M=1

N(.1)-

J

C(1+IAI)

(21.36)

a.(x,0<1

2) If the pair x, y belongs to some compact set in M x M disjoint from the diagonal then n-1

Ie(x,y,A)1 s C(1+IAI)

(21.37)

Proof. Let us first note that, without loss of generality, we may assume A > 0 (if this is not true for A, then, in view of Corollary 9.3, it is true for A + cl, for sufficiently large c> 0). Introduce the operator A1= Al/-. This is an elliptic

pseudodifferential operator of order i with principal symbol a1 (x, ) = a,,(x,, )1Jin. It is clear that (21.31), (21.32) for kv (x, y, ) simply coincide with

the equations (21.27). In addition it is clear that

e(x,y,A)=

coj(x) coj(y) _ 1'$1

A's nsliM

9,(x) coz(y) =e1(x,y,

where e1 (x, y, A) is the spectral function of A I. All statements in Theorem 21.2 now follow from the corresponding statements in Theorem 21.1.

164

Chapter 111. Asymptotic Behaviour of the Spectral Function

Problem 21.1. Consider the case of a homogeneous operator a (D) with constant coefficients in IR". Write down e (x, y, A) as an integral and verify the estimates (21.34), (21.35) and (21.37) directly.

§22. The Laplace Operator on the Sphere 22.1 The Laplace operator on a Riemannian manifold. 1) Let M be a manifold with a Riemannian metric, i.e. in any tangent space TIM there is given a bilinear form < , > with a positive definite quadratic form. If x', ... , x" are the local coordinates in an open set U c M, then G_X1_1

az"") is a basis for

the tangent space at all points in U. Denoting (22.1)

we obtain a positive definite matrix gi. (x). If now v = E vi i=1

11

z

e TIM, then

"

_

(22.2)

g,j (x) v' v'. i.)=1

The cotangent space T,* M is, by definition, the dual of TI M. A basis of it (for x e U) consists of the 1-forms dx', defined by the relations

(dxi

az

)=b

A metric on a vector space E induces an isomorphism of E with E. With the help of this isomorphism we may transfer the metric from E to E*. Fixing x e U, let us compute at x. Which tangent vector corresponds to dx'? Denote its coordinates by a''t, k = 1, ..., n; then we have to satisfy the condition b

Yg,,j a'k,

from which a'k = g'k, the elements of the matrix inverse to II gik II . We now have <0Y, dxi> k=r

g 'k ak , dx') = g`' , ax

(22.3)

§22. The Laplace Operator on the Sphere

i.e. for any cotangent vector a =

165

a, dx' e T* M i=1

_

gi'a,aj.

(22.4)

2) Now introduce on the Riemannian manifold a smooth density (volume) such that in the tangent space the volume of a parallelepiped in an orthonormal

system equals 1. For an arbitrary parallelepiped, defined by the vectors et,... , e", the volume equals vol {e1,

...,

e"} = I det {e;', ... , eV} 1,

(22.5)

where e,°' is the column of coordinates of e; in the orthonormal basis. Formula (22.5) follows from the fact that the volume has to be an additive, non-negative invariant of parallelepipeds.

What about the case where the coordinate basis is not orthonormal? Let us then consider the operator A, mapping an orthonormal basis e,, ... , e;, into the basis e, , ..., e.. The columns of its matrix (in the basis e, , ... , e;,), will be the coordinates of the vectors e, in the orthonormal basis e; , so that vol{el,...,e"} = IdetAl = det(A*A). Now, the matrix elements of A*A in the basis {e;} are of the form = = <e ; , ej> .

Therefore, the volume of the parallelepiped

(22.6)

Ga7 x,

... ,

equals V9_1

4__

where g = det Il g1 (x) Il and the volume of an arbitrary set F in the local coordinates of U is defined as

vol (F) = J

g (x) dxt ... dx".

(22.7)

F

It follows directly from the change of variable formula for an integral that the integral (22.7) is independent of the choice of local coordinates. We may therefore take it immediately as a definition of vol (F), defining a smooth positive density dv on M by the formula dv =

g (x) dx' ... dx"

3) For any differential operator

A: C°°(M) - C'(M)

(22.8)

Chapter III. Asymptotic Behaviour of the Spectral Function

166

there exists a unique operator A*, such that (Af, g) = U,A*g),

f, gECo (M),

(22.9)

where (22.10)

(f, g) = $ f (x) g (x) dv.

Analogous arguments hold also for operators on sections of vector bundles over M, if in each fiber of the bundle we have a hermitean metric (positive definite hermitean form), which replaces f (x) g (x) in (22.10). Let us define a scalar product on the space of 1-forms A' (M), taking on

T,* M ® C the hermitean scalar product induced by the metric on T,* M introduced above. Consider the operator (22.11)

d:

mapping a function f e C 'O (M) to its differential df e A' (M), defined by the fact that if v E T,, M, then = (vf) (x), where (vf) (x) denotes the derivative off

in the direction v. In coordinates:

df= The operator 6: A' (M)

I4dx'.

C`° (M) is defined as the adjoint of d, i.e. 6 = d*.

Definition 22.1. The Laplace (or Laplace-Beltrami) operator on functions

d: C'(M)-+C'(M) on a Riemannian manifold M, is defined by the formula

d= -6 - d.

(22.12)

Analogously, the Laplace operator is defined on p-forms AD(M) as

A = - (db+Sd): AP (M) - AP (M), but we will not consider this case. It is immediately clear that the Laplace operator has the following properties: a)

b)

if T: M--'M preserves the metric on the tangent spaces and

1': C°`(M)-+C' (M) is defined by If = f o T, then A1' = TA,

§22. The Laplace Operator on the Sphere

167

i.e. A commutes with isometrics; c) if M is closed, then (d f,J) 5 0 and from A f = 0 it follows that f= const.

4) Let us compute S and A in local coordinates. We have

(b(,jai dx', .f

adx'

j ardX`, J=I ax i=1

Jf _ x, (Vg- g 'Ja,)] &C

=i.; jgiJa' ax 1/ dx =

1

axi from which 1

"

a

(Vg_ g'' a,) Irg I. /=1 ax, and

a frgiiP4).

df=bdf=

g-

(22.14)

V9_

Example 22.1. In the Euclidean space 1R" with its standard metric, we obtain

df=

a2f axi2

Note that all invariants of d are also invariants of the Riemannian manifold, in particular, the residues and the values of the . -function. Exercise 22.1. Compute in local coordinates the principal symbol of the Laplace operator on a Riemannian manifold. 22.2 The Laplace operator on the sphere S". The n-sphere S" is the following submanifold of Ht"+':

S"= {(x°, ...,x"): i (x')2= 1 i-0 There is a metric on S, induced by the standard metric on R"+' and correspondingly a Laplace operator, which we denote by ds. There is the following method for computing ds, using the operator d on 1R"+' Proposition 22.1. Let fl o)) be a function on S" and extend it to IR"+', putting

f(x)=f(f-)

168

Chapter III. Asymptotic Behaviour of the Spectral Function

i.e. by homogeneity of degree 0. Then

43f= 'ills.,

(22.15)

(4J)(x)=r-2dsf,

(22.16)

or

where r= I x I .

Proof. The equivalence of (22.15) and (22.16) is obvious, since A jhas degree

of homogeneity -2. Let us prove (22.15). Denote temporarily by As' the operator on S", defined by the right-hand side of (22.15). The group of isometries SO (n + 1) acts on S" by restricting to S' the rotations of 1R"+'. Using this group, any unit tangent vector may be mapped into any other such vector. Therefore, the principal symbol of an operator commuting with all isometries, is constant on all unit cotangent vectors (and, consequently, is uniquely defined up to a scalar multiplier). Clearly the operators As and As commute with the action of SO (n + 1) on functions. Therefore, there exists a constant A such that the operator ds- Ads has order 1. But this operator also commutes with SO (n+ 1) and since there are no linear functions which are invariant with respect to rotations and since the multiplication term of As - .ids

is constant because of rotational symmetry, then the operator ds-.lds is a multiple of the identity, and, consequently, is zero, because dsl = A s I = 0. Hence As = Ads . From what follows, it will be clear that A = 1, on the other hand

this can be verified by computing d s and d s on any non-harmonic function on

the sphere. O

223 Eigenvalues of the operator ds . Let us compute d in polar coordinates. Let r = I x l and co = x/ I x 1, then

d (f(r) g(w)) = (df) g+f(dg) + 2(Vf) - (Vg) = (df)g+f(dg), (22.17) since (V f) (Vg) = 0 (V f is directed along the radius vector and Vg is directed along the tangent). Let us compute df (r). We have xa

rX =-, r

1

(x1)2

r

r3

rXs,=--

.

From this we obtain

a (Xi f'(r)) _ E f"(r) (r )2 a( 1=0 1=0

Af(r) = Y_"

+ o (I - (X')2 1 f' (r) = f" (r) + n f' (r) 1=0`

J

.

§22. The Laplace Operator on the Sphere

169

Therefore a2

A

n a

r14.-

r ar

1

+

ds.

(22.18)

r2

where As is applied on the unit sphere with subsequent extension by homogeneity of degree zero. Formula (22.18) is true also on arbitrary functions (linear combinations of functions of the type f (r)g(w) are dense in the space C°°(R"+l \ 0)). In particular, for f (r) = rN, we obtain

A(r"g(w)) = r-2+" [Asg+[u(u-1)+nulg]

= r-2+"[dsg+u(u+n-1)g].

(22.19)

It follows from (22.19), that the following proposition holds

Proposition 22.2. The equality d (r"g(w)) = 0 (for r + 0) is equivalent to the fact that g(w) is an eigenfunction of the operator - As with eigenvalue

A=u(u+n-1). Since all eigenvalues of the operator - As are non-negative, we may assume

that u ? 0 or u < 1 - n. Note that the quadratic function A (u) = u (u + n -1) takes all values ).;-> 0 for p > 0 and each of them exactly once. It is therefore clear

that the eigenvalues A z 0 are in a one-to-one correspondence with those u ? 0, for which there exists a non-trivial function g (w) such that r"g(w) is a harmonic

function on R"+' \ 0. But then, by the removable singularity theorem, the function r"g(w) is harmonic everywhere on R"+' and, consequently, is a harmonic polynomial by the Liouville theorem. In particular, µ is an integer. Clearly the converse is also true, i.e. the restrictions to S" of homogeneous, harmonic polynomials are eigenfunctions of the operator - d s with eigenvalues A = k (k+n-1), where k = 0, 1, 2, ... and by the maximum principle we see that a harmonic polynomial is uniquely defined by its restriction to S". Hence, we have proven

Theorem 22.1. The eigenvalues of the operator -ds are A=k(k+n-1),

... and the multiplicity of the eigenvalue k (k + n - l) equals the dimension of the space of homogenous, harmonic polynomials of degree k. where k = 0, 1, 2,

22.4 Computing the multiplicity. Let M,, be the space of homogeneous polynomials of degree k. Let us compute Nk = dim M.. A basis in MR is given by the monomials xo ... x.k-, where ko + kl + ... + k" = k. The number of ordered

partitions of k into a sum of n + 1 non-negative numbers equals the number of ordered partitions of k + n + 1 into a sum of n + I positive numbers, which in turn equals the number of ways of choosing n from n + k, i.e. equals Nk

n

n!k! = nr

(k+n) (k+n-1) ... (k+1).

(22.20)

170

Chapter M. Asymptotic Behaviour of the Spectral Function

Note that d determines a map

d:

(22.21)

and one has the exact sequence (22.22)

where H. = Ker A I k,, is the space of homogeneous, harmonic polynomials of degree k, the dimension of which we would like to compute.

Theorem 22.2. 1) The operator d : Mk -+ Mk - 2 is surjective. 2) There is a direct sum decomposition

Mk = ® r21 Hk-2l,

(22.23)

dim H. = Nk - Nk - 2 .

(22.24)

k-21a0

where r2 = xo + ...

+ X. I.

Corollary 22.1. a)

b) If fond (pare two polynomials, then there exists a unique polynomial u such that

du =f,

u11.1=1= (Pljxi=1

(22.25)

(i.e. the Dirichlet problem for the Poisson equation in the unit ball is solvable in polynomials).

c) The harmonic polynomials cannot be divided by r2.

Derivation of Corollary 22.1 from Theorem 22.2. a) The relation (22.24) follows from the fact that the sequence (22.22) can be rewritten in the form (22.26)

b) In view of part 1) of the theorem, the solvability of (22.25) reduces to the case f = 0, where it is obvious in view of (22.23). c) Also obvious in view of (22.23).

Proof of Theorem 22.2. Both statements are proved at the same time by induction in k. For k = 0, 1 the statements of the theorem are true. Let them be true also for all I< k. 1 °. Show that Hk n r2Mk - 2 = 0. Since by the inductive hypothesis we have

Mk-z=

r k-2/-2ZO

§22. The Laplace Operator on the Sphere

171

then

r xMk_x= k-21k0

r 21Hk-xl

1>0

and ifhkEHknr2Mk_2 then

hk=

c,r2rhk-21

Y_

(22.27)

1>0

k-21a0

Now consider the harmonic polynomial h = hk - Y_ c1hk _ 21. We have 1>0

deg h:5 k and hk is the homogeneous component of h of degree k. Since h I Ix,

_, = 0, then h =- 0 from which hk = 0, as required.

2°. From the condition Mk Hk $ r2Mk _ 21 we obtain dim H. < Nk - Nk _ 2

(22.28)

where the equality is equivalent to the decomposition Mk = Hk m r2 Mk -2

(22.29)

3°. From the exact sequence (22.22) it follows that dim H. = Nk - dim d (Mk) z Nk - Nk _ 2,

(22.30)

where the equality is equivalent to the surjectivity of the operator

d: 4°. From (22.28) and (22.30) it follows that dim H. = Nk - Nk _ 2 from which

follows the surjectivity and the decomposition (22.29) and hence, by the inductive hypothesis the decomposition (22.23) also follows.

Corollary 22.2. The multiplicity of the eigenvalue A = k (k + n - 1) of the operator - A s is equal to N. - Nk _ 2, where Nk is given by the formula (22.20).

22.5 The function N(A) for the operator -As. It is clear that

N (k (k + n - 1) + 0) _ Y dim H1= E (N,-N,_2)=Nk+Nk_1.

(22.31)

rsk

But from (22.20) it is also clear that N. is a polynomial of degree n in k with leading coefficient 1 In!. It therefore follows from (22.31) that

n!

k'-

2

n!

[k(k+n-1)]"'2.

(22.32)

Chapter III. Asymptotic Behaviour of the Spectral Function

172

From (22.31) it also follows that

N(k(k+n-1)+0) - N(k(k+n-1)-0) = Nk- Nk-2= where P"- (k) is a polynomial in k of degree n - 1. Therefore 1

-1

N(k(k+n-1)+0)-N(k(k+n-1)-0)zck"-'zc[k(k+n- 1)] 2

(22.33)

,

where c > 0. From (22.32) and (22.33) it is clear that N(A) =

2!

.l"j2(1+0(A-1/2)),

(22.34)

where the term 0 (A -1 12) is sometimes greater than cA - 'I2, i.e. the remainder estimate is the best possible (in any case as far as the exponent is concerned).

Let us verify that (22.34) is exactly the asymptotic formula from Theorem 21.2 (which, in particular, also shows that the multiplier A from the proof of Proposition 22.1 equals 1). We have to verify that

2/n! = (2n)-"V"V

(22.35)

where V. is the volume of the unit n-ball and V is the area of S" (Riemannian volume).

Clearly V. =

V.

-' and from this everything reduces to the identity

n

2 (2it)" V., =

(22.36)

(n-1)!

and prove this relation. Put I= f e-x'dx; then

Let us compute

0

P= J e-(x,+...+x:)dx1 .. dx" X'40

and in particular

P = J e-(xi+x')dx1 dx2 = f d!y f re-''dr = 4 f e-xdz = it X12

0

X"10

from which 1= 2 ,

1" =

2"

.

Now note that op

J 0

0

"

r

(n),

§22. The Laplace Operator on the Sphere

173

2nnJ2

from which we obtain

T (n/2)

As is well-known, integration by parts yields T (a+ 1) = af' (a) for a > 0, from which for n = 2k we obtain

I' ()= r(k)= (k-i)! = (;_1)!. If now n = 2k+ 1, then

f

I

2

/ (k +2(k 2(k -21...2 2 2! 21,

but f (21= 2 VVW = r, from which o \\lI r (n) n-2 n-4 1

22

2

...

"-1

=2

z

(n - 2) !!

.

Now let n be even. Then 1'

2(n1)!!

(2J

=2z )(n-2)!! 2-nj2(n-1)!! j=2 2-"(n-1)! Vn, from which

."

V,

V,

2n"/2 2 . n1"+1V2

2 2-"(n-1)! In

2(2n)'

(n-1)!'

as required. The case of odd n is considered in the same way.

Problem 22.1. Write down the expression for the Laplace operator in the Lobacevskii plane (hyperbolic plane).

Problem 22.2. Compute the eigenvalues for the Laplace operator on the n-dimensional real projective space RP" with the natural metric (induced by the metric on the sphere S" under the two-fold covering S"--+ RP").

Problem 22.3. Compute the eigenvalues of the Laplace operator on the n-dimensional complex projective space CP" with the natural Kahler metric (cf. S. S. Chern [I]).

Chapter IV Pseudodifferential Operators in 1R" §23. An Algebra of Pseudodifferential Operators in IR" The aim of the study of pseudodifferential operators on IR" is to describe various effects connected with the behaviour of functions as I x I -' + oo. A fundamental role is played here by non-local effects, so we have to give up the requirement of properly supportedness of pseudodifferential operators, unposed, without loss of generality, in the local theory (Chapter I). 23.1 The classes of symbols and amplitudes Definition 23.1. The symbol class F, (R"), where m e IR, 0 < e < 1, consists of the functions a (z) e CI (IR"), which satisfy the estimates I0sa(z)I'"-0!

I,

ZEIR".

(23.1)

Example 23.1. Any polynomial a(z) of degree m belongs to F' "(R').

Let us note immediately that if a e f'Q (1R") and b E 1 Q , (IR"), then abarQ'*"-(IR") and d=aEfQ,-°I°`I(1R!). Further, given a linear monomorphism j: 1R'-+ RN, a E 1 Q (1RN) and j * a = a o j, then j * a E 1 Q (JRI). Note that n f Q (IR") = S(JR"). Definition 23.2. Let aj e 1 Q , (IRN ), j = 1, 2,

(23.2)

..., m-4 - oo as j- + oo and

a e C1(1R"). We will write

a - Y aj , Qc

(23.3)

j=1

if for any integer r a 2 r-1 (23.4)

j=1

where in, = max mj . ji,

The following propositions are similar to Propositions 3.5 and 3.6.

Chapter IV. Pseudodifferential Operator in IR"

176

Proposition 23.1. Let a, e r (1R^ ), j = 1, 2, ... , where m j - - x as j Then there exists a function a, such that a same property, then a - a' E S (IR").

+ oo .

aj . If another function a' has the J ='

Proposition 23.2. Let aj e r;, (IRN), j = 1, 2, ... , where mj - - oo as j -. + oo. Let a e C`° (IR") and for any multiindex a let the following estimate holds for some constants pa and C": (23.5)

1,9.* a (z) 1 s Ca

Finally, let there exist lj and Cj such that I,-+ - oo as j-' + co and the following estimates hold r-1

a(z) - Y- aj(z) 5 Cr
(23.6)

j=1

M Then a - I aj. j=I

Exercise 23.1. Prove Propositions 23.1 and 23.2.

We now would like to consider operators of the form Au (x) =

JJel(x

- r) - s

a

(x, ) a rQ (IRZ"). However, as is evident from the considerations in Chap. I, it is convenient to consider at once a more general formula for the action

of the operator

Au(x) = J

(23.8)

u(y)

where the function a (x, y, 1;) is called the amplitude. We will describe the class of amplitudes, which will be useful in what follows

Definition 23.3. Let 17a (IR3") denote the a (x, y, ) a C°°(lR3"), which for some m' satisfy I a{axaya(x,y,4)1 s

set

of

Caar"-ela+$+rl <X-y>"'+Qla+p+rl,

functions

(23.9)

where z = (x, y, g) a IR3n.

It is clear that if aeIIQ(IR3n), then and if b e 11P "(IR3"), then ab a1IQ +"'. (IR3n). We have J (R3n) c I7Q (IR3"). Since l<x-y) z 1, then between the classes 17' (1R3") there are the inclusions 11e (IR3") c IIQ:: (1R3n)

for m':5; m", Q' >_ Q".

§23. The Algebra of Operators in IRO

177

If a (x, y, ) e DQ (lR3i), then a (x, x, ) E f'Q (1R2").

The most important example of an amplitude of the class 11 (]R3") is provided by the following

Proposition 23.3. Let a linear map p: 1R2"-. 1R" be such that the linear mapping (x, y) into (p(x,y),x-y), is an isomorphism. Let b (x, ) E !'Q (R2 n). Define the amplitude a (x, y, 4) E C' (]R3") by the formula (23.10)

a (x, y, ) = b (p (x, y), ) . Then a E 17 (]R3n).

P r o o f. The functions I x I + I I and I p (x, y) I+ I x- y I give equivalent norms on 1R2n. Therefore, for the proof of the proposition it remains to use the easily verified inequality

(1 + Ip(x,y) I +

I

SEIR,

I)' -

from which the estimates (23.9) follow for a (x, y, S) with m' = I m I. O

Corollary 23.1. If b e rQ (1R2n), then a (x, y, ) = b (x, t) and a (x, y, ) = b (y, ) belong to 17 (JR3e). 23.2 Function spaces and the action of the operator. Now we introduce the space Cp (1R") consisting of functions u e C°° (1R) such that (23.11)

10"u(X) I S C.

for any multiindex I al. The best constants C" in (23.11) constitute a family of semi-norms for a given function, defining a Fre chet space structure on Cb (IR"). The operator A of (23.8) is conveniently studied in the space Cs (1R"). In order to give the correct definition of the oscillatory integral appearing in (23.8),

we shall have to proceed as in §1. For this purpose, let initially a (x, y, 4) E Co (1R3"). Then the integration in (23.8) in reality is performed over a compact

set and we may carry out an integration by parts, using the identities eit:-y",

(23.12) (23.13)

where M, N are even non-negative integers. From (23.8) one obtains

Au(x) = j

x [<)

(23.14)

Chapter IV. Pseudodifferential Operator in IR"

178

If the amplitude a E l7Q (IR3n) satisfies (23.9) and if u e Cb (lR"), then clearly

for m - N < - n, m'+ m - M < - n, the integral (23.14) becomes absolutely convergent, defining a continuous function of x E W. Increasing M and N, we will obtain integrals which are convergent also after differentiation with respect to x. Hence the operator defined by (23.14), is a continuous map A: Cb (lR") - C°°(IR")

(23.15)

For m < 0, we will also obtain a map A: Cti (IR") -+ Cb (1R"). This same map could also be defined using a cut-off function X (x, y, ) e Co (1R3"), X (0, 0, 0) = I

and the formula

Au(x) = lim JJe'(x-r''r

u(y)

:.+o

(23.16)

The identity of the two definitions (23.14) and (23.16) is verified in the same way as in §1 and we leave this verification as an exercise to the reader. In particular, the operator A is defined on the space S (IR"). Let us show that it gives a continuous map

A: S(IR")-+S(1R").

(23.17)

Indeed, using the inequality

(1+IxI)'` 5 (1+Iyt)k (1+ Ix-YD",

k>0,

we see from (23.14) that

(I+IxI)'IAu (x)IsCk for any k and a similar estimate holds if we replace Au (x) by 8j (Au (x)). From

this we also have Au e S (lR") for u e S (P') with an estimate of seminorms guaranteeing the continuity of the map (23.17) (which also could have been obtained from the closed graph theorem). Finally note, that since the transposed operator 'Au(y) =

JJeilx-Y"*4

v(x)dxdd

(23.18)

by similar reasoning, maps S (IR") into S (IR"), then A can be extended by duality

to a continuous map

A: S'(1R)-+S'(1R"). Definition 23.4. The class of operators A of the form (23.8) with amplitudes a E 11Q (1R3") will be denoted by G' (IR") or simply by G,' (if the dimension n is

clear or unimportant).

§23. The Algebra of Operators in IR"

179

It is useful to have a description of the operators belonging to the intersection

G-1= n GQ . We shall show that this intersection is independent of q and consists of operators with kernels K,, (x, y) e S (IR 2n). Clearly it suffices to consider the case Q < 1. Note that the operators with amplitudes a (x, y, ) and <x -y> -" (DD>' a (x, y, ) coincide, from which we see that if A E G - °°, then A can be determined by an amplitude a(x, y, ) satisfying (23.9) with arbitrarily small (arbitrarily close to -- oo) numbers m and m'. But then A has the kernel (23.19)

KA(x,Y) =

belonging to S(1R2"). From this, it follows in particular that A defines a continuous map A : S' (IR") -* S (IR") ,

(23.20)

given by the formula (23.21)

In the general case the kernel KA(x, y) is defined by the formula (KA,gi> =

a (x,

cpeS(IR2 ),

9, (x, y)

(23.22)

and is a distribution K4 ES'(IR2"). Exercise 23.2. Denote by C'O (IR") the space of functions u e C ' (IR"), with the property that for any multi-index a one can find constants C. and u., such that

Iasu(x)16 CQ<x>"

(23.23)

Show that an operator A e G' defines a map A: CC ° (IR") -+ Cr°D (IR") .

(23.24)

Exercise 23.3. Let A E G, "(R") and KA the kernel of A. Show that KA a C°° (IR2"\ A), where A is the diagonal in 1R" x IR".

23.3 Left, right and Weyl symbols Theorem 23.1. An operator A e GQ of the form (23.8) can be written in any of the following three forms Au(x) = if ei(x-Y1

QA.r(x, )

(23.25)

Au (x) = j j ei (x - Y) . 4 UA., (Y, ) u (Y) dy dd ,

Au (x) = j j ei (x -Y)

UA.

"-

(-!-, \ u (Y) dy d . 1

(23.26) (23.27)

Chapter IV. Pseudodifferential Operator in IR"

180

Here 0,11, a4., and a4,,,, belong to I'Q (1R2"), are uniquely defined and have the following asymptotic expansions: aA., (x, a

Y

a.+., (Y, )

a!

ac!

as Dr a (x, y,

(23.28)

as(-Dz)" a(x,y, )Is=r,

(23.29)

1

ft! Y!

r

2

,(23.30)

This theorem allows the introduction of Definition 23.5. The functions a4,,, a,,,, and a,,,.. from the formulae (23.25)(23.27) are called, respectively, the left, right and Weyl symbols of the operator A.

Although we shall not use any other symbols, let us show the following generalization of Theorem 23.1, containing a parameter t e lR and also allowing us to avoid repetitions in the proof of Theorem 23.1.

Theorem 23.2. Let A e GQ of the form (23.8) be given. Then for any t e IR A may be uniquely written as

Au(x) = JJ

er,:-Y

4

b, has the following asymptotic expansion

b,

b,(x,t)^

1

/ICY.

T1B!(1-T)!Y!aB+Y(-Dx)ODYa(x,Y,(23.32) e r

Definition 23.6. The function b, (x, ) will be called the r-symbol of A. Proof of Theorem 23.2. Putting

V= (l -T)x+Ty,

tl

(23.33)

w=x-y,

we obtain

S x=v+Tw,

y=v-(1-T)w,

23.34)

a(x,y,4)=a(v+tw, v-(1-T)w,,;).

(23.35)

from which Let us now expand the right-hand side of (23.35) at w = 0 in a Taylor series: TIM

IB+rI5N-

Y!

Y

(23.36)

§23. The Algebra of Operators in IR"

181

where

rN(x,Y,)=

j(1-t)N-I

c6r(x-y)°+Y

Y,

O

19+YI=N

(23.37)

x

and cd,, are constants.

In (23.36) the expression (axay a) (v, v, ) signifies that in the function ax a; a (x, y, ) it is necessary to take v = (1- r) x + r y instead of x and y. The expression (axaya) (v+trw, v -1(1- r) w, ) in formula (23.37) has a similar meaning.

Now note, that the operator with amplitude (x-y)$+' (azaYa) (v, v, ) coincides with the one given via the amplitude (v, v,) = (- 1)191 *lYl(a{

+YDzDsa)

(v, v, ) .

Therefore it follows from (23.36) that A can be represented in the form of a sum A = AN + RN, where AN is an operator with r-symbol 1

I9+7ISN-t Pt'yt'

x191(1-r)IYIaft+Y(-D.)PD a(x,Y, )ly=x,

and RN is an operator with amplitude rN (x, y, ). Note that RN is a linear combination of a finite number of terms having amplitudes of the form

O

III+yI=N. Let us show that this amplitude belongs to the class 17 - 2N2 (IR'"). For this it sufficies to show that this is true for the integrand, with all estimates uniform in t (note that this is obvious for each fixed t + 0 and true for t = 0 by Proposition 23.3). In view of the relations

v = (1 -r) (v+trw)+r(v-1(1-r)w), tw = (v+trw) - (v-t(1 -r) w) it is obvious that C-t

<

Iv+trwI+Iy-t(1-r)wI III+ItwI


where C> 0 and C does not depend on t e [0,11. Therefore I (a{+Yaaaya) +2Qr.

Chapter IV. Pseudodifferential Operator in IR"

182

Since for m' + 2 Q N z 0 we have

it is clear that if, in addition, m' + m ?. 0 and m - 2QN 5 0, then (at+y3 aya)

s IwI)m'+2m+2°",

C(1 + IVI + IwI +

where C does not depend on t. One obtains the estimates for derivatives in an analogous way. Now let the symbol b'(x, ) a f'Q (lR2n) be such that

b' (x, ) -. Y (bN (x, ) - bN - , (x, ))

No

Then, if A' has r-symbol b' (x, ) it is clear that A - A' e G i.e. the operator A - A' has a kernel belonging to S(1R2"). Let us now verify that if A has a kernel KA S(lR2n), then it has a r-symbol b, (x, l;) e S (1R2n) and the correspondence between kernel and symbol is a one-to-

one correspondence. From formula (23.31) it is clear that this correspondence is of the form

KA(x,y) = Fr2. -rb,((1-r)x+ry, ), b,

w)

(23.38) (23.39)

((23.39) is obtained from (23.38) by a change of coordinates and the Fourier inversion formula). In particular, for any KAeS(1R2n), we can find b, (v, l;) e S (1R2") by formula (23.39).

We next show the uniqueness of the r-symbol in the general case. For this we note that (23.38) is always true when A is given via a r-symbol b, (x, ) and if the partial Fourier transform, which appears in this formula, is understood in the

same sense as the Fourier transform of distributions (cf. §1). Thereby, the inversion formula is also true, leading to (23.39) after the linear change of coordinates (23.34). Also, from formula (23.39), the uniqueness of the r-symbol is obvious, taking into account the uniqueness of the kernel KA . Corollary 23.2. The class of operators GQ coincides with the class of operators of the form (23.25) with the left symbol QA ,(x, o a r; (lR2n). The same is also true if we replace (23.25) by (23.26) or (23.27) and a,., by or,,, or QA.W.

23.4 Relations between the different symbols. The symbols of the transposed and adjoint operators. The expression for the r-symbol in terms of the r, -symbol

§23. The Algebra of Operators in Qt"

183

for a different T, can be easily obtained from Theorem 23.2 in the form of an asymptotic series. Indeed, if an operator A has the T1-symbol this signifies that it may be determined via the amplitude

But then, by Theorem 23.2 its T-symbol has the asymptotic expansion (-I)IPI

B.r

fl!y!

T191(1-T)lYl (1-T,)IBI TIY1 00+YDF+Yb,

j

`

or

Ycaa{Dxb,,(x,

(23.40)

a

where (_ 1)1al

[T(1 [(1-T)r1jlY1

C. =

(23.41)

a

and, in particular, we have co = 1. Now, transforming (23.41) using the Newton binomial formula (Lemma 3.4), we obtain

ca=-j [(I-r)T,e-T(1-T1)e]a= 1

(T,-T)la1.

where e = (1, 1, ..., 1). Thus, we have proved Theorem 23.3. Symbols b, (x, ) and b, (x, ,) of the same operator A e GQ are related via b, (x,

)-E

a (T, - T)Ial a40.2,, b,, (x, i

(23.42)

In particular, b, (x, ) - b, (x, ) E r - 2p (1R2n).

Let us now consider the transposed operator 'A, defined by the formula
u, V E S (IR") ,

where


R

From the formula


JJJe'("-r" b,((I -T)x+Ty, ) u(y) v(x)dydxdd If

b,((1-T)x+Ty,

u(y) v(x)dvdxdt

(23.43)

Chapter IV. Pseudodifferential Operator in IR"

184

it follows that if A has a r-symbol b, (x, ), then 'A has the (1 -r)-symbol 'b, _,(x, ), given by the formula bT(x, -*).

(23.44)

From Theorem 23.3 it now follows Theorem 23.4. If A e GQ , then 'A E GQ and the r-symbol'bT (x, ) of 'A can be

expressed in terms of the r-symbol b, (x, ) of A by the formula 'b, (x,

a

a!

(1-2r)!

aaDa, b, (x, - c)

(23.45)

Now let A* be the adjoint of A, defined by (Au, v) = (u, A*v),

u, v eS(1R"),

(23.46)

where

(u, v) = I u (x) v (x) dx.

(23.47)

R-

By analogy with Theorem 23.4 one can show

Theorem 23.5. If A e G', then A* a GQ and the r-symbol b* (x, ) of A* is related to the (1-r)-symbol bi (x, ) of A via the relation

b* (x, ) = b, _,(x, ) ,

(23.48)

and can be expressed in terms of the r-symbol b, (x, ) of A via the asymptotic series

b*(x,t)

Y

I 00

(1-2r)Ia1 a DxbT(x, ).

(23.49)

Corollary 23.3. If A e G', then 0A. P, (x,

) = U4,. (x, )

(23.50)

In particular, the condition A = A* is equivalent to the real-valuedness of the Weyl symbol aA,,,. (x, 0-

23.5 The composition formula Theorem 23.6. Let A' c- GQ A" E G'2. Then A' o A" E GQ " +'"= and if b* (x, ) is the rl-symbol of A' and big (x, ) the r2-symbol of A", then the r-symbol b,(x, )

of A' o A" has the asymptotic expansion bT(x,

)~

capYa(ac DPb', (x, ))(at DXb" (x, )), a.d.r.a

(23.51)

§23. The Algebra of Operators in IR"

185

where ci,ya are constants (depending on T, T, and T2) such that coooo = I and the sum runs over sets of multi-indices a, /J, y and 6 such that a + y = fi + 6. In particular, we have b, (x, ) - b,, (x, ) b," (x, ) EF '+",,-2a(IR2"),

Proof. Taking Theorem 23.3 into account, we see that it suffices to consider only one arbitrary triple of the numbers T, T, and T2. Let us take for simplicity T, = 0, T2 = 1. A" can be written, using the symbol b; (y, ), as

A"u(x) = J

ebi (y.) u(y)dydd,

or

Je'" b' (y,4) u(y)dy.

(23.52)

A' has the form A'v (x) = J J e'cx -"

bo (x, ) u (y) dy do =

J

e'x " c bo (x, ) u (4) dd .

(23.53)

From (23.52) and (23.53) it follows that A' o A"u(x) =

JJe'(x - v)- r

bi (y,4)

(23.54)

i.e. A' a A" is determined via the amplitude a (x, y, ) = bo (x, ) b i (y, i) E

M, +",, (j3") .

From this we have A' a A" E G'- +,",. Applying Theorem 23.2 we obtain for b, (x, ) bt (x, 4)

Tlal

9r

(1-

Y!

ae+' [(D. bo (x, )) C

x

l

and by the Leibniz rule (Lemma 3.3)

(-1

b,(x, ) -

0.y'.6.e.

(P +-L)-! Teal (1-r)1'l (a46Dzbo)

(a;Drb'),

(23.55)

P.7.a.c

a+.-/+y as required. Inserting into (23.55) the expressions for bo, b; in terms of b,,, b,, we obtain

formulae for the coefficients cadra in (23.51), which may sometimes be simplified. For instance, one can show by analogy with Theorem 3.4 the following

Theorem 23.7. Under the assumptions of Theorem 23.6 one has bo(x, )

ac

(t02bo(x,')) (D°bo(x, )).

(23.56)

186

Chapter IV. Pseudodifferential Operator in Iit"

Problem 23.1. Show that the left symbol a4,, (x, ) of an operator A E GQ can be expressed in terms of A by the formula

e-,x cA(erx c),

(23.57)

where A acts on the variable x.

Problem 23.2. Show that if A' E G' A" a GQ then aA'.A'.w(x, )'"

( a!1gl, of 2-12+01 (a{DxaA'.w(x, S)) 2.0

Problem 23.3. Consider the polynomial

tx+...+tx+tD +...+tD N in the variables t, t eR" with operator coefficients (xj is viewed as the multiplication operator by x;) and write it in the form N1 Y_

a TO !fl!ttAaa

la+bl=N a.

Show that Al is an operator with the Weyl symbol

§24. The Anti-Wick Symbol. Theorems on Boundedness and Compactness 24.1 Definition and basic properties of the anti-Wick symbol. Put 4io(x) = it- n/4 exp [-x2/2],

xeR".

(24.1)

Then 0o e S (R") and 11 00 II = 1, where II ' II denotes the usual norm in L' (R"), generated by the scalar product (23.47). Denote by P0 the orthogonal projection in L2 (R") onto the vector 0o . Clearly, the Schwartz kernel of this projection has the form

K(x,Y) = x-"n exp[-(x2+y2)/2].

(24.2)

Computing the Weyl symbol ao of Po, we get, using formula (23.39) with t = };

ao(x,)=F,..4K x+2 u, x-2u). 1

C

1

(24.3)

§24. The Anti-Wick Symbol

187

Now, taking into account that 2

Fexp(-av2) _ (a)",2 expl - 4«),

(24.4)

or > 0 ,

we obtain from (24.3) that "n

(24.5)

exp(_x2- 4

Now let z = (x, ), zo = (xo, o) a IR2". Consider the operator P,o with Weyl symbol u,a(z) of the form O' (z) = co (z - zo) .

(24.6)

P. =M4.T, PO T,o'Mo',

(24.7)

It is easily verified that

where Mao is the multiplication operator by e" 4- and T,o is the shift operator by xo in L2 (IR"), i. e. T,, u (x) = u (x - xo). Setting U,. = Mr, Ti,, we see that P,a can be

written in the form (24.8)

P_,= U1. PO U___1

from which it is obvious, since U,o is unitary that PTe is the orthogonal projection

onto the vector 0.. = U,a0o . We wish to consider the following operator, being a linear combination of operators P, , z eJR2n:

A = f a (x, ) P.,, dx dd ,

(24.9)

where a (x, ) e 1Q (1R2"). As for the meaning of this formula, let us note that the following can be immediately verified: if u (x) e S (IR"), then (P .4 u) (xo)

eS(lR2"t). Due to this, (24.9) makes sence if we consider it on functions u (x) e S (IR"), and one can easily ensure that A maps S (JR) into S (1R). Definition 24.1. An operator A of the form (24.9) is called operator with antiWick symbol a (x, 4).

The convenience of anti-Wick symbols is demonstrated by Proposition 24.1. If a (z) Z 0, then A ? 0, i.e. (Au, u) z 0 for u e S (1R"). Proof. This follows from the fact that P," Z 0 for any zo e IR2". Li Corollary 24.1. If a (z) is real-valued, then

IIAII < sup la(z)I,

(24.10)

Chapter IV. Pseudodifferential Operator in IR"

188

where

IIAII=

sup

UES(R7,u#0

IlAull/Ilull

Proof. The statement 11 All < M is equivalent, for self-adjoint A, to the

non-negativity of the operators M-A and A+M, which is obvious for M= sup I a (z) I from Proposition 24.1. zER='

Corollary 24.2. For a complex-valued function a (z) one has the following estimate (24.11)

I I A I I < 2 sup l a (z) I zeRz'

Proof. It suffices to use Corollary 24.1 for Re a and Im a. Remark 24.1. In fact, (24.10) holds also for complex-valued functions a (z) (cf. Problem 24.4), although we shall not use this. 24.2 Connection between the anti-Wick symbol and the other symbols.

Theorem 24.1. Let A be an operator with anti-Wick symbol a (z) = a (x, ) e rQ (1R2n). Then A e GQ and the Weyl symbol of A can be expressed as

b(z) = n-" f e-Iz-z'I' a(z')dz'.

(24.12)

Further, any T-symbol bz (z) of A has the asymptotic expansion (24.13)

bT (z) ^- > ca as a (z), a

where ca are constants (depending on r), such that co = I and ca = O for odd I a I . In

particular (24.14)

b, (z) - a (z) a rQ - 2 a (IR2E) .

Proof. The relation (24.12) is obvious from (24.9), (24.5) and (24.6). In view of Theorem 23.3 it suffices to show (24.13) for T = Z, i.e. for the Weyl symbol. Let us expand a (z') in the Taylor series at z: a W) =

1( (aa Ial
(24.15)

a (z)) (z' - Z)a + rN (z', z) ,

where

ca (z' - z)s f as a (z+ t (z'- z)) (1- t)"

rN (z', z) _ IaI =N

and where c;' are constants.

0

dt ,

(24. l 6)

§24. The Anti-Wick Symbol

189

Inserting (24.15) into (24.12) we obtain

b(z) _ Y caaoa(z) + RN,

(24.17)

lal
where ca

= a!n 1 n

Jzae - I=I'dz

(24.18)

(from which obtain in particular, co = 1 and ca = 0 for odd I RN(z) =

7r-n

I ).

and

(e-Iz-z'I'rN(Z',z)dz'.

Let us show that RN (z) E 1 Q -QN(IR2N) (from which we have, in an obvious manner, the required expansion (24.13)). It is convenient to rewrite RN (z) in the form

RN (z) = Y ca Jdt Jdw woe-Iw1'(aaa) (z+tw) (1 - t)N -, 0

lal=N

implying that aYRN(z) has the form of a sum of terms like Jdt Jdw

woe

(a'a) (z+tw) (1

-t)N- ,,

0

where I /3 I = N + I y 1. Clearly it suffices to estimate the expression

Iafl(z) = fdw wae-1"'hl(8fla) (z + tw).

(24.19)

uniformly in t E [0,1 ]. To estimate I (z) we decompose it into the sum of two integrals: 1;,, (z)

over the domain I w I < I z 1/2, I

I > IzI/2

For 1.'O (z) one has the estimate IIaP (z)I

CP (zy^-QCN+IrI

J<w>' e-IWI'dw = CCm-QlN+IYI>, (24.20)

and IQA(z) can be estimated as follows: 5 Ca@

J

e-IWI'<W>N+m-k

(24.21)

IWl>1:1/2

for an arbitrary k. From (24.20) and (24.21) we immediately have the required estimate for 1.0(z).

Chapter IV. Pseudodifferential Operator in Ill"

190

Theorem 24.2. Let A' e GQ G. Then there exists an operator A E GQ, such that A

is given by the anti-Wick symbol

A - A' e G -' ,

and

a (z) E f'Q (IR2n)

(24.22)

i.e. the operator A - A' has the Schwartz kernel K4 -,,. (x, y) E S (1R2n).

Remark 24.2. Not every operator A e GQ has an anti-Wick symbol a E r7 (this is clear for instance, from the fact that the Weyl symbol b (z) as defined by formula (24.12), must be a real-analytic function in z E IR2n). The actual finding of the anti-Wick symbol from a given Weyl symbol requires the solution of the inverse heat equation. Theorem 24.2 shows that if one disregards symbols in S(IR"), this process becomes possible. Proof of Theorem 24.2. Let A' have the Weyl symbol b' (x, ). Consider the operator A0 with anti-Wick symbol ao (x, ) = b' (x, ) and put A i = A' - Ao. Then A, c- G_0 -2Q by Theorem 24.1. Denote by A, the operator with anti-Wick

symbol a, (x, ), equal to the Weyl symbol of A, We have

A'=Ao+A,+A2,

AZEGQ-ae.

Continuing by induction, we may construct a sequence of operators Aj, j = 0, ..., with anti-Wick symbols aj (z) e re" - 2j? (1R2"), such that

1, 2,

w-1

A'- Y

AJ,EGm-2eN

Q

(24.23)

j=0

Let a (z) E Fe' (1R2") be such that a - Y_ 00 aj. j=o

Then if A is the operator with anti-Wick symbol a (z), (24.22) follows from (24.23) as required. L] 24.3 Theorems on boundedness and compactness

Theorem 24.3. The operator A E GQ can be extended to a bounded operator on L2(IR"). Proof. In view of Theorem 24.2 and Corollary 24.2 the statement reduces to the case A E G- ', where it is obtained from the obvious estimate 11 A II 2 <_ i,(I K,, (x, y) 12 dx dy,

(24.24)

resulting from the Cauchy-Bunjakovskij-Schwarz inequality (cf. also Appendix 3).

§24. The Anti-Wick Symbol

191

Theorem 24.4. The operator A E GQ for m < 0 can be extended to a compact operator on L2(IR"). Proof. First note, that the operators in G- °° are Hilbert-Schmidt operators (cf. Appendix 3), hence they are compact. Now let A e G", m < 0. Using Theorem 24.2, we may assume that A has the anti-Wick symbol a (z) E I'o (1R2n). Let X (z) E Co (IR2"), X (z) =1 for I z I S 1. Put

aL(z) = X (z/L) a (z) and let AL be the operator with the anti-Wick symbol aL(z).

Then sup I a (z) - a L (z) I - 0 as L -' + oo and in view of Corollary 24.2 we z-R'' therefore have II A - A L II -+ 0 as L -. + oo. But AL is compact for any L since ALE G

From this the compactness of A

also follows. 0 24.4 Problems Problem 24.1. Prove the boundedness theorem in the same way as Theorem 6.1 was proved.

Problem 24.2. Prove the compactness theorem 24.4 without using the theory of the anti-Wick symbol. Hint: Verify with the help of polar decomposition of A, that compactness of A is equivalent to compactness of A*A; show that compactness of A*A is equivalent to compactness of B" = (A *A)" and obtain the latter for large N from the fact that K8 (x, y) a L2 (fl mn).

Problem 24.3. Consider the system of vectors {4a=}sER,. defined in 24.1. Show that the map I: f - (f, 0_) defines an isometric embedding of L2 (IV) into L2

(IR2"), i.e.

111112 = (2n) " J I (f, -0) I2 dz

(24.25)

R'.

(one says in this case that the vectors V. constitutes an overcomplete system with

respect to the measure (2n)'"dz in the space IR2"). Problem 24.4. Show that an operator A with an anti-Wick symbol a (z) can be written in the form

A = I*M"I,

(24.26)

where M. is the multiplication operator by a (z) in L( R'") and the operator I: L2 (1R11) -' LZ

(1R2") was introduced in Problem 24.3. Derive from this the validity

of (24.10) for complex-valued functions a (z). Problem 24.5. Introduce the Wick symbol c (z) of an operator A E GQ with a Weyl symbol b (z) by the formula

c(z) = x-"Je-1''='I'b(z')dz'.

(24.27)

192

Chapter IV. Pseudodifferential Operator in IR"

Show that c (z) can be expressed in terms of b (z) via the asymptotic series

c(Z) E caa°b(z) ,

(24.28)

where co =1 and cQ = 0 for odd I al. Verify that the Wick symbol c (z) of A can be expressed by the formula

c(z) = (A45=,0..).

(24.29)

Derive from this that sup I c (z) I < II A II .

zest"

Problem 24.6. Let a (z) - 0 as I z I -i + oo. Show that the operator A, defined by the formula (24.26) is compact. Problem 24.7. Let A be a compact operator in L2 (IR") and let c (z) be defined by the formula (24.29). Show that c (z) -+ 0 as I z I - + x. Problem 24.8. Let A e Ge" for some m e IR and let b (z) be the Weyl symbol of

A. Show that the boundedness of A is equivalent to sup Ib(z) I < oo. zea"

Hint: Use Corollary 24.2, Problem 24.5 and the construction from the proof of Theorem 24.2. Problem 24.9. Let A and b (z) be as in the foregoing problem. Show that the compactness of A is equivalent to the condition b (z) -+0 as I z I -, + oo.

Hint: Use Problems 24.6, 24.7 and the construction in the proof of Theorem 24.2.

Problem 24.10. Show that a differential operator with polynomial coefficients in IR" has a polynomial in z = (x, ) as its anti-Wick symbol. Hint: On polynomials one can solve the inverse heat equation, since the

Laplacian is nilpotent on the polynomials of a given degree; if b (z) is a polynomial Weyl symbol of A, then its anti-Wick symbol has the form ' °c a (z) = e 4 4 b (z) = kE

4Y

1

(_-) b (z).

k

(24.30)

Problem 24.11. Compute the coefficients ca in formula (24.13) for i = 2i. e. in expressing the Weyl symbol through the anti-Wick symbol. Show that this series can be written in the form e

b (z)

1

e 4 a (z) _

i

k=o

d ()ka(z). 4

(24.31)

Hint: In computing the coefficients cQ one may assume that a (z) is a polynomial.

§25. Hypoellipticity. The Fredholm Property

193

Problem 24.12. Let A be a differential operator with polynomial coefficients on 1R". Show that A may be uniquely expressed in any of the two forms

A = Y c"p as (a+)p ,

(24.32)

4.0

A=

(24.33)

ccp(a+)paa,

a.#

where a+ = x

a

ax

,...,x"-ax a=lxt+axa,...,x"+axa

,andx,J3

are n-dimensional multi-indices, the sums (24.32) and (24.33) are finite and cap , cap are complex constants. Show that in this case the anti-Wick symbol a (x, ) and the Wick symbol of A are given by the formulae

Y c"p (x+ i )a (x - i )p ,

a (x,

(24.34)

".p

c(x, )= Y_ c,p(x-i0p(x+ioa

(24.35)

ff.P

§25. Hypoellipticity and Parametrix. Sobolev Spaces. The Fredholm Property 25.1 The class of hypoelliptic symbols and operators Definition 25.1. We shall write a (z) a Hf'Q '- (1R"), if a (z) e C a' (1RN) and there is an R such that the following estimates hold for I z I > R

Clzl"°5 la(z)I < C, Izl",

(25.1)

l a"a(z)I < Cala(z)I Izl-01"I,

(25.2)

where C, C,, C. are positive constants. From this it follows that a (z) E f'Q This class of symbols has properties close to those of the symbols in §5. Definition 25.2. By HGQ A, given by the T-symbols

(1R") or HGQ- "'° we denote the class of operators

b,aHr -'°(1R:"). Clearly, HGQ -

c GQ .

For the justification of Definition 25.2 we need Proposition 25.1. If it is true for some T E 1R that b, c- H FQ. m also for all T e lR.

then it is true

Chapter IV. Pseudodifferential Operator in IR"

194

Proof The proof is based upon Theorem 23.3 and the following Lemma proved in the same way as Lemmas 5.1-5.3 and Propositions 5.2-5.4.

Lemma 25.1. 1) The classes Hr, ,.,.(R,) satisfy: a) if a (z) e Hr', m - m (I z I > R) and (a° a)/a e rQ o l°I R);

Uz

b) if aeHf'

a'EHF then aa'EHr,, c) if a E HrQ m and ref' where m 1 < mo, then a + r e Hr-. 2) The classes HGm me satisfy:

d) if A e

e) if A

HGQ. I-, A' c- HGQ m", then A A' e HGQ +m'.

m;.

then `AEHGQ"m and and ReGQ- where m1 < mo, then A +

Exercise 25.1. Prove Lemma 25.1. Exercise 25.2. Prove Proposition 25.1. 25.2 The Parametrix and regularity. By analogy with Theorem 5.1 we can prove Theorem 25.1. If A E HGQ- m then there is an operator B E HGQ m -' such that

BA=I+ R1,

AB=I+R2

(25.3)

where Rt e G - , j = 1, 2. If B' is another operator in GQ' for which either B'A

-IeG-°° or AB'-IEG-°°, then B'-BeGExercise 25.3. Prove Theorem 25.1. Corollary 25.1. Let A E HGQ Then the following statements hold: a) If u c- S' (W) and Au c- S (IR"), then u e S (IR"). b) If u e S' (R') and Au c- CC ° (IR"), then u e C,' (W).

Proof. Follows in an obvious way from Theorem 25.1 and the results in 23.2.

25.3 Sobolev spaces. Consider for arbitrary s E IR the operator L differing from the operator with the left symbol b (x, ) = (1 + I f I2 + I x 12)S 2 by an operator in G,', s' < s. It is easily seen that L, E HG-',- S.

Definition 25.3. The space QS = Q3 ()R") consists of the distributions uES'(IR') for which L,uEL2(IR").

By analogy with Theorems 7.1 and 7.2 we can prove

Theorem 25.2. 1) If A e GI then A maps A:

(25.4)

§25. Hypoellipticity. The Fredholm Property

2) If A E HG,'-

195

u c- S' (]R") and Au a Q', then u e Q'+MO

Exercise 25.4. Prove Theorem 25.2. Since the class U G,' contains all differential operators with polynomial coefficients, it is clear that Theorem 25.2 implies Corollary 25.2. I Q5 = S (JR ") ,

I

U Q' = S' (J R ") s

s

Now we introduce a Hilbert space structure on Q'. Note that we may assume that L _, is a parametrix of Ls in the sense of Theorem 25.1. Then

L_,L,=I+R

R,eG-°°.

Let p > s, p an even integer. Put

(x'D'R,u, x'D'R,v).

(u, v), = (L,u, L,v) +

(25.5)

I'I+I$ISp

From the representation

u = L_,L,u - R,u

(25.6)

it is clear that (25.5) defines a pre-Hilbert structure on Q'. By analogy with Proposition 7.2' one verifies that Proposition 25.2. The scalar product (25.5) defines a Hilbert space structure on Q'.

Exercise 25.5. Prove Proposition 25.2.

Finally, by analogy with the argument in §7, one proves the following statements.

Proposition 25.3. T h e scalar product ( , ) in V (W) induces a duality between Q' and Q -s (the exact formulation as in Theorem 7.7).

Proposition 25.4. The operator A e GQ can be extended to a continuous -`for e > 0. The operator A: Q'-+ Q-' and to a compact operator A: embedding operator Q' c-. Q' _', E > 0, is compact for any s e 1R. Exercise 25.6. Prove Propositions 25.3 and 25.4. 25.4 The Fredholm property. By analogy with Theorem 8.1 is proved Proposition 25.5. If A E HG,, M, then A e Fred (Q', Q' - M) for any s e lR. The space Im (A I Q.) in Q3 -M is the orthogonal complement to Ker A * with respect to the scalar product (- , - ) in L2 (lR").

Note that Ker (A I Q.) = Ker (A I s. (a.) = Ker (A I scat) for any A E HGQ MN.

(25.7)

196

Chapter IV. Pseudodifferential Operator in IR'

To extend Proposition 25.5 to operators A eHGQ''-(with mo < m), it is but as an operator in the topological vector spaces S(IR"), S'(IR") and similar spaces or, as an unbounded operator necessary to regard A not as an operator from Q3 into Q'

A..:,: Q'- Q',,

(25.8)

where s'2: s - mo, with the domain D,," consisting of those u e Q' such that

AueQs. Definition 25.4. Let E, and E2 be two topological vector spaces, A an unbounded operator from El into E2 with the domain DA. The operator A is called Fredholm operator if the following conditions are fulfilled: a) dim KerA < + oc ; b) ImA in a closed subspace in E2; c) dim Coker A < + co. Theorem 25.3. 1) The operator A eHGQ defines a Fredholm operator from S (IV) into S (W) and from S' (W) into S' (IR"). 2) The operators As,.. of the form (25.8) defined by A are, jor s' ? s - mo, also Fredholm operators.

Remark 25.1. We consider the weak topology in S'(IR"). Since in Definition 25.4 the topology appears only in b), it is clear that the Fredholm property also holds in all stronger topologies.

Proof of Theorem 25.3. Let the duality between S(IR") and S'(IR") be given by the extension of the scalar product ( , ) from L2 (1R"). Note that the finitedimensionality of Ker A and Ker A* follows from Theorem 25.1 since due to the inclusion

KerAc Ker BA a Ker(I+R1) the question reduces to the case A = I+ R1 for which everything is obvious. We shall now consider the inclusion A (S' (W)) z AB (S' (R")) = Y+ R2) (S' (W))

For the operator I+ R. the Fredholm property on S' (1R") follows from Proposition 25.5. Therefore the subspace A (S'(P.")) is closed in S'(IIt") and codim A (S' (IR")) < + co which proves the Fredholm property of A in S' (1R"). Let us prove the Fredholm property of A in S(IR"). It suffices to verify only conditions b) and c) in Definition 25.4. We shall show that A (S (IR")) _ {u: u e S (IR"), ul KerA* } ,

(25.9)

where orthogonality is in the sense of L2(IR"). First, note that

A(S'(IR"))= {u:ueS'(R"), ulKerA*},

(25.10)

§26. Self-Adjointness. Discreteness of the Spectrum

197

since A (S'(1R")) is closed in S'(lR"), and S(lR") is the dual of S'(IR"). But now (25.9) follows from (25.10) since A (S (IR")) = A (S' (1R")) n S (IR")

in view of Corollary 25.1. (25.9) shows the Fredholm property of A on S(IR"). Finally let us verify the Fredholm property of A,,,. for s'? s - mo . Once again, it only remains to verify that

Im A,.,. = {u: u eQ", ul KerA*) .

(25.11)

Let ueQ3, ue(KerA*)l. Then u = Av, where vES'(IR") in view of the already proven relation (25.10). But from Corollary 25.1 we then obtain that v E Qi.e. v e Q8, since s< s'+ mo. This shows (25.11). [Ti By analogy with Theorem 8.2 one proves Theorem 25.4. Let A E HGQ and Ker A = Ker A (0). which is the inverse to A. Then there is an operator A ' E HGQ

Exercise 25.7. Prove Theorem 25.4. Problem 25.1. Show that the operator A c- HG'

is Fredholm in the space

C,°° (Dt").

Problem 25.2. Show that if a differential operator A with polynomial coefficients has a r-symbol a (z) elliptic in z = (x, ), then the symbol of its parametrix B has an asymptotic expansion in terms of homogeneous functions in z for IzI> 1.

§26. Essential Self-Adjointness. Discreteness of the Spectrum 26.1 Symmetric and self-adjoint operators. Let H1 and H. be Hilbert spaces and suppose we are given an, in general unbounded, operator

A: H1-H2.

(26.1)

As usual, DA denotes the domain of A (it is understood that this domain is given with A, which is then a linear map from the linear subspace D,, into H2; note that writing (26.1) does not imply that A is defined on all of H1). The adjoint operator

A*: H2 - H1

(26.2)

is defined if D,, is dense in H1 and, in this case, D,,. consists of all v E H2, for which

there exists a vector g e H1 with

(Au, v) = (u, g),

u E DA,

(26.3)

Chapter IV. Pseudodifferential Operator in IR"

198

(on the left-hand side of (26.3) is the scalar product in H2 and on the right-hand side that in H,). It is clear that g is uniquely defined and by definition A*v = g. In particular, we have the identity (Au, v) = (u, A* v),

(26.4)

Definition 26.1. Let H be a Hilbert space. An operator A: H-+H is called .symmetric if (Au, v) = (u, Av),

Definition 26.2. An operator A:

u, v EDA

(26.5)

is called self-adjoins if A = A*.

It is obvious that a self-adjoint operator is symmetric. The converse is in general not true. Definition 26.3. An operator A: H, -+ H2 is called closed, if the graph GA, consisting of all pairs {u, Au}, where u EDA, is a closed subspace in H, 9 H2.

Exercise 26.1. Show that if A* is defined, then it is closed.

Exercise 26.2. Let an operator A be bounded, i.e. there exists a constant C > 0, such that IlAu HH S C ]lull, u E D,, . Show that A is closed if and only if D,,

is a closed subspace of H, . The well-known closed-graph theorem (cf. Rudin [11) states that if DA = H, and A is closed, then A is bounded. Obviously the same holds if DA is a closed subspace in H,. Let an operator A: H, -H2 be given. We say that A has a closure A, if the closure G,, of the graph GA is again the graph of (closed) operator, which we denote by A. In particular, any symmetric operator A: H-+ H has a closure if DA is dense. Indeed, it is enough to verify, that if u" is a sequence of vectors in DA, such that lim u" = 0 and lim Au = f, then f = 0. But for v ED,, we obtain R1

ac

R. 00

(f, v) = lim (Au., v) = lim (u", Av) = 0, a- cc

R 00

from which we have f = 0. Note that if A is a symmetric operator, then so is A. Definition 26.4. An operator A: H-' H is called essentially self-adjoins if DA is dense in H and A = A*.

In particular, A* is then an extension of A and, hence, A is symmetric. A criterion for essential self-adjointness is given by

Theorem 26.1. A symmetric operator A: H-+H with dense domain is essentially self-adjoins if and only if the following inclusions hold Ker (A* - il) c D; F,

(26.6)

Ker (A* + il) c D; F.

(26.6')

§26. Self-Adjointness. Discreteness of the Spectrum

199

Proof. 1. The necessity of (26.6) and (26.6') is obvious. To verify their sufficiency, let us first note that since A* is an extension of A, it follows from

(26.6) that Ker(A*-il) = Ker(A-il). But Ker(A-il)=0 since A is symmetric. Therefore, from (26.6) it follows that

Ker(A*-il)=0.

(26.7)

Similarly, from (26.6') we find that

Ker(A*+iI) = 0

(26.7')

2. Let us now verify that (X- il) -' (defined on (.4- i1) (H)) is bounded. We have II

(X_ il)f 112 = ((X_ il) j, (A- il)f) = II Af II 2 +

11f112,

(26.8)

since (Af, f) is a real number in view of the fact that X is symmetric. It follows from (26.8) that 111112 s II(A-il)f1I2, i.e.

(((A-iI)-'gI) 5 Ilgh

,

ge(A-il)(H).

3. It is clear that A - it is closed. Therefore (X- il) - 'is also closed and since

is bounded, its domain (A - if) (H) is closed in H. However the orthogonal complement of (A - il) (H) is obviously equal to Ker (A - il)* = Ker (A* + il) = 0. Therefore (X- il) -' is everywhere defined. By similar reasoning, (A+il)-' is also everywhere defined.

4. Let us verify that (A- il)and (A+iI)

are adjoint to each other. We

obviously have

u, veD2.

((A-il)u, v) = (u, (A+ if) v),

Denoting (A-il) u = f and (A+il) v = g, we obtain the required relation

(f,(A+il)-'g) =

((A-il)-'f

g),

f geH.

5. Let us finally verify that X= A*. We will use the following easily verified

fact: if B is an operator in H, such that (B-')* and (B*) -' are defined, then (B-')* = (B*)-'. We have

A* =A* _ (A+il)*+i1= {[(A+i1)-']-`}*+il +i1=A-it+i1=A, = {[(A+il)'']*} as required.

200

Chapter IV. Pseudodifferential Operator in IR"

26.2 Essential self-adjointness of hypoelliptic symmetric operators. In this section we shall denote by A * the operator which is formally adjoint to an operator A e GQ , i.e. the operator A * E GQ , such that (Au, v) = (u, A+v),

u, v E Co (1R") .

In the preceding sections we have written A* instead of A+, but here the notation A* will be reserved for the adjoint operator in the sense of section 26.1.

Theorem 26.2. Let A E HGQ '"o, where mo > 0 and A=A. In L' (NV) consider the unbounded operator Ao, defined as the operator A on the domain Co (IR"). Then Ao is essentially self-adjoint and its closure coincides with the restriction of the operator A (defined on S'(lR")) to the set D,,,, = {u: u e L2 (IR"), Au E L2 (IR")}.

(26.9)

Proof. 1. Denote by D the right-hand side of (26.9). Since (Au, v) = (u, Av),

uES(IR"),

veS'(IR),

(26.10)

it is clear that D (_- D4. and in addition

AID=AoID. Let us verify that indeed D = DA;. Let v ED, i.e. v e L2 (IR") and for some f e L2(lR") the identity

(Au, v) = (u, f),

u E Co (IR"),

(26.11)

holds. But it follows from (26.10) that the same identity holds if we replace f by A. Therefore Av = f, i. e. v e D as required. Thus we have demonstrated that the right-hand side of (26.9) equals D.,.. 2. In order to now use Theorem 26.1, we will verify the inclusion

Ker (Ao - iI) cDuo .

(26.12)

From what we have already shown, it is clear that

Ker(A**-iI) _ {u:uEL2(]R"), (A-if) u=0}. Taking into account that A - iI e HG' m0, it follows from Corollary 25.1 that Ker (A o - if) c S (RR"), from which (26.12) follows, since A maps S (IR") into S(IR") continuously and Q '(R") is dense in S(IR"). Similarly one proves the inclusion Ker (Ao +if) c D,o , which concludes the proof of Theorem 26.2. C]

§26. Self-Adjointness. Discreteness of the Spectrum

201

26.3 Discreteness of the spectrum

Theorem 26.3. Let A E HG;- '

where mo > 0 and A + = A. Then A has

discrete spectrum in L2 (IR"). More precisely, there exists an orthonormal basis of eigenfunctions (pj (x) E S (IR"), j = 1, 2, ... , with eigenvalues Aj E IR, such that Al I -+ + oo as j - + ao. The spectrum a (A) of X = A* in L2 (IR") coincides with the set of all eigenvalues }At}.

Proof. The proof is similar to that of Theorem 8.3. In view of the separability of L2 (IR"), there exists a number Ao E 1R\ a (A). But then Theorem 25.4 implies

and, in particular we see that (A - AO !) -' is that (A - A0!) E HG; compact and self-adjoint in L2(IR"). The remainder of the proof is a verbatim repetition of the proof of Theorem 8.3. D Problem 26.1. Let A E GQ be such that there are numbers A* E C, such that Im A+ >0, Im A_ <0 and

and for some mo a IR. Show that if A+ = A, then A is essentially self-adjoint.

Problem 26.2. Let H,, H2 be Hilbert spaces and let A: H, -+H2, and A+: H2 -. H, , be such that (Au, v) = (u,A+v),

uED,,, vEDA..

Show that if A+A is essentially self-adjoint, then A+ = A* and X= (A+)*. Hint: Consider in H, ® H. the operator defined as the matrix 10

=(O A A

0

Then the conditions A+ = A* and A = (A+)* are equivalent to the essential selfadjointness of the operator W. Compute Ker (91 ± il).

Problem 26.3. Let A mo > 0. Denote by Ao the operator A restricted to Co (1R"). Show that Ao = Ao and Ao = (Ao)*. Hint: Use the result of Problem 26.2, after extending the operators AO and A+ to S (IRO).

Remark 26.1. The result in Problem 26.3 means that the "strong and weak extensions coincide" for an operator A EHGQ '- for mo > 0: if u e L2 (1R") and Au E V(1111), then there exists a sequence u; E Co (IR") such that u; --* u and Aul -+ Au as j -+ + oo in the L2 (1111)-norm.

Problem 26.4. Prove analogue of Theorem 8.4 on the structure of the spectrum, eigenfunctions and associated functions for operators A

mo>0.

Chapter IV. Pseudodifferential Operator in IR"

202

Problem 26.5. Let an operator A have the anti-Wick symbol a (z) a 1'Q (lR2n

and let Aa be self-adjoint in L2 (1R"). Let a (z) - + oo as I z I - oo. Show that Ao has discrete spectrum in the sense of Theorem 26.3 and that A; -+ +oo as

j-++o0. Problem 26.6. Let A eGQ be such that Ao is self-adjoint in L2 (IV) and has discrete spectrum such that A,--j, + oo as j -+ + oo. Let c (z) be the Wick symbol of

the operator A. Show that c (z) -+ + oo as I z I -. + oo.

§27. Trace and Trace Class Norm 27.1 The trace and the Hilbert-Schmidt norm expressed in terms of the symbol. Here we make use of notations and facts concerning Hilbert-Schmidt and trace class operators which are presented in Appendix 3. Let us begin with the formal expression for the trace in terms of the r-symbol. Let A e G', let b, (x, ) be the i-symbol of A and K4 its kernel. We have formally

K4(x,Y)= ei(:-A-4

(27.1)

from which

K4(x,x)= and Sp A = $b1 (x, ) dx dd .

27.2)

Note that (27.2) means in particular, that its right-hand side is independent

of r. Proposition A.3.2 yields

IIAIIi= jIK4(x,Y)l2dxdY= jIK4(x,x+z)I2dxdz.

(27.3)

But by (27.1)

KA(x,x+z) = je'"

(27.4)

Therefore we have formally

JIK4(x,x+z)I2dxdz= 1K4(x,x+z) K4(x,x+z)dxdz =

b,(x+rz,ry)ddrydxdz

= jer:"-t) bt(x,) b=(x,n)ddpdxdz = jl je -'=s bT(x,d 12dxdz=

(27.5)

§27. Trace and the Trace Class Norm

203

(we have here used the shift invariance of the integral and the Parceval identity for the Fourier transform). As a result we obtain (27.6)

11A 112 =

where again the right-hand side is independent of r

Proposition 27.1. The correspondence between operators A E G - m and T-symbols b, (x, ) E S (JR2") extends by continuity to an isometry between S2 (L2 (IR")) and L2 (IR") such that (27.6) holds. If A E G', then the condition A E S2 (L2 (]R")) is equivalent to b, E L2 (1R2 ")for some T and this then holds for all T

and the formula (27.6) also holds in this situation.

Proof. The computations (27.3)-(27.6) are justified for A e G -'0 or, what is the same, for K4 E S (II12w). Since G - °0 is dense in S2 (L2 (IR")) and S (IR2 ") is dense

in L2(IR2"), the existence and uniqueness of the required isometry is obvious. Finally, the last statement is obvious from the uniqueness of the r-symbol. Li Corollary 27.1. If A E GQ and m < - n, then A E S2 (L2 (IV)).

Proposition 27.2. 1) If A E GQ and m < - 2 n, then A E Sl (L2 (Ill")) and for

any fixed m < -2n and T EIR there exist constants C and N, such that the following estimate holds

IIAII1 S C Y sup {I0 b,(z)I IyISN

-",+Q1Y1}.

(27.7)

z

2) For A E GQ, m < - 2n, formula (27.2) for the trace Sp A holds for any T E IR.

Proof. 1) Choose an operator P E for Ker P = Ker P` = 0, exists (the existence of an operator P of this type

so that P-1 E HG;

follows, for instance, from Theorem 26.3). In view of Corollary 27.1, we have p2 E S1 (L2(IR")). But from the obvious representation A = P2(P-2A) and

the fact that P-2A E G°Q C

(L 2(I")), it follows that A E S, (L2(lU")).

Therefore the inclusion GQ c Sl (L2 (IR")) ,

m < -2n.

(27.8)

is proved. Let us now prove (27.7). It can be obtained in two ways: either by a direct sharpening of the arguments carried out so far (from similar estimates in the composition formula and the boundedness theorem) or from the closed graph theorem. The latter route is shorter and is commonplace for many argument of

this type, although it is also rougher. We will carry out carefully the corresponding arguments. Introduce in GQ a Fr6chet topology, defined by semi-norms of the form IIAII(N)= I,ISN

sup {Ia"bT(z)I(27.9) i

204

Chapter IV. Pseudodifferential Operator in IR"

We have to show that the embedding (27.8) is continuous in the natural Banach space topology on S, (L2 (IR")). In view of the closed graph theorem (cf.

e. g. Rudin [I]) it is only necessary to show that this embedding has a closed graph. This is most conveniently proved constructing a Hausdorff space M such that GQ c S, (L2 (1R")) c M

(27.10)

where both embeddings GQ c M and S, (L2 (IR")) c M are continuous. Now as M we may, for instance, take S2( L2 (IR")), since the continuity of the embedding of GQ and S, (L2 (IR")) in S2 (L2 (R")) follows immediately from Propositions 27.1 (formula (27.6)) and A.3.7 (estimate (A.3.29)). 2) Now we will prove (27.2) for A E G,', m < - 2n. Note that both its parts are continuous on G'. But for any m' > m, G - °° is dense in GQ in the topology of Go'. Therefore, it suffices to prove (27.2) for A e G

We would like to carry out carefully the argument from A.M. This is trivial, if we present A in the form A = L t o L 2, where the operators L I and L 2 have kernels with enough continuous and rapidly decreasing derivatives. But the latter representation can be constructed by an argument similar to the one used in 1) of this proof. 0 27.2 A more precise estimate of the trace class norm in terms of the r-symbol. The estimate (27.7) is not very convenient, since it contains a weight-function increasing in z. At the same time, we see that II A II1 does not change if we shift the r-symbol by some vector zo = (xo , o) E 1R2n. Indeed, if b, (x, t) is the r-symbol of

A and if we denote by A, the operator with the r-symbol b, (x - xo, - o), then we obtain

A.u(x)= Jeu=-r' =

r)x+ry-xo, -t0)u(y)dyd' t bi((1-r) (x-xo) u ((y - xo) + xo) dy d

= Je;c,

u(y'+xo)dy'dc', where x' = x - xo , Y' = Y - Yo , ' = - to. Denote by U the unitary operator, mapping u (x) into (Uu)(x) = e-'to' x u (x+ xo), then we see that A,, = U from which IIA=,II, = IIAIII

AU,

(27.11)

An estimate of the trace class norm which is invariant relatively to the shifts of the r-symbol is given by the following

Proposition 27.3. There exist constants C and N, such that for A e Ge , m < - 2n, the following estimate holds IIAII, s C E I 161b,(z) I dz. IrISN

(27.12)

§27. Trace and the Trace Class Norm

205

Proof. It suffices to show the estimate (27.12) in the case where b, (z) E Co (R2"). First let,

suppb,c {z: I z I S Ro},

(27.13)

where Ro is some fixed constant. Then it follows from Proposition 27.2 that there are constants Cl and M (depending on Ro) such that II A II 15 Cl E sup l a= b, (z) I . IYISM

(27.14)

=

But since for b (z) E Co (lR2n) we have b(z)

_

a2"b az

az2"

.

1

(t1, ..., 12")dtl

J- I..... 2n

and consequently a2 n b (z)

sup lb(z)I

.1

aZl ... az2"

s

dz,

it follows from (27.14) that we have (27.12) (with N=M+2n), provided that (27.13) is satisfied. Now, using the invariance of the trace class norm (formula (27.11)) and the invariance of the right-hand side of (27.12) with respect to shift in the argument

of b(z), we see that (27.12) always holds, with the same constants C and N, provided diam supp b, S Ro

(27.15)

Let us finally get rid of the condition (27.15). Take a partition of unity cc

1m

(Pi (z)

i=1

such that diam supp Bpi 5 Ro, there is a number I such that any ball of unit radius

does not intersect more than I sets suppvi, and, in addition, 101ip;(z)I 5 C,, j= 1, 2, ..., with constants C. not depending on j. Introducing the operators Ai with the r-symbols coj (z) b, (z), we obtain

IIAIIIS 1] IIA,IIISCY Y fla=(rpib,)(z)Idz i=1

j=1 IYISN

Cl

as required.

I IaYbjz)Idz, lit 5N

L]

206

Chapter IV. Pseudodifferential Operator in Qt"

Problem 27.1. Let A be determined by the anti-Wick symbol a (z) a rQ. Show that IIAII1 <J

Problem 27.2. Let c (z) be the Wick symbol of A e G,. Show that J

IIAil1.

Hint: Use the polar decomposition of A and the result of Problem 24.5. Problem 273. Let A have the anti-Wick symbol a (z) E F,, m < - 2n. Show that

dxdd.

SpA =

Problem 27.4. Let A e G,-,, m < - 2n, and let c (z) be the Wick symbol of A. Show that

SpA =

dxdd.

Problem 27.5. Let M be a closed n-manifold and let A e Lp a(M), 1 - Q:5 b < Q. Show that if m < -n/2, then AeS2(L2(M)) and if m < -n, then A eS, (L2(M)).

§28. The Approximate Spectral Projection 28.1 The Glazman lemma. In this paragraph we shall describe an abstract scheme of yet another method to obtain the asymptotic behaviour of eigenvalues

which is based on the construction of an approximate spectral projection. At the basis on this method lies the following well-known variational lemma of Glazman. Lemma 28.1. Let A be a self-adjoins, semi-bounded from below operator with discrete spectrum in a Hilbert space H, i.e. A has an orthonormal basis of eigenvectors e, , e2 , ... with eigenvalues A,, A2, ..., such that At - + oo as k--* +oo. Let N(A) be the number of eigenvalues of A not exceeding A (multiplicities counted). Then N (A) =

sup

dim L

LcDA

(Au.u)$.t(u.u).ufL

(L is a linear subspace of DA).

(28.1)

§28. The Approximate Spectral Projection

207

Proof. Let E2 be the spectral projection of A. Since dim (E2H) = Sp E2 = N (A),

(28.2)

then, putting L = E2H, we see that the right-hand side of (28. 1) cannot be smaller

than the left-hand side. Next we show that the right-hand side of (28.1) cannot exceed the left-hand side.

Let the linear subspace L c D,, be such that u e L.

(28.3)

u e (I- EA) H\ {0} ,

(28.4)

(Au, u) < A (u, u), Since

(Au, u) > A (u, u),

it follows from (28.3) that

Ln(1-E2)H= 0. But then E2 is injective as a mapping of L into E2H, from which it follows that

dim L 5 N (,k),

as required. U 28.2 Properties of the spectral projections. The spectral projections operators enjoy the following properties: 1) Ex = E2; 2) E,? = E2;

3) E2(A-,11)E2<0; 4) (1-E2) (A-AI) (I-E2) > 0 (meaning that the corresponding quadratic form is strictly greater than 0 on the non-zero vectors in DA); 5) Sp E2 = N (A).

Basic in what follows is

Proposition 28.1. Let E; be a family of operators, for which E,H c DA and which satisfies conditions 1)-4). Then 5) is also fulfils, i.e. Sp E; = N (A) = Sp E2 .

(28.5)

Proof. It follows from 1) and 2) that Ex is an orthogonal projection. Putting L2 = EEH, MA = (I- Ex) H, we have, in view of 3), that (Au, u) 5 A (u, u), u e L' a, from which, by Lemma 28.1, it follows that Sp EA' = dim L, ,;g N (A). Further,

from 4), we have (E2H) n M; = 0, which implies dim (E2H) = N (A) 5 dim L2 = SpEE, proving (28.5). 1

208

Chapter IV. Pseudodifferential Operator in IR"

Remark 28.1. Note that under the conditions of Proposition 28.1 we do not necessarily have E = Ez (cf. Problem 28.1). 28.3 Approximate spectral projection operator Theorem 28.1. Let A be an operator as in Lemma 28.1 and {4') z eR a family of operators such that S,,Hc D,, and that for some c > 0, 6 > 0, we have: 10.

2°. 'Z is a trace class operator and

.ice+oo,

as

A-6)

(28.6)

where V (2) is some positive, non-decreasing function, defined for 2 z AO; 3°..9z(A-A1) eSz 5 CA1

4°. (I-8z)(A-AI) (I-tA) -CA' 5°. Sp ofz = V(A) (1 +0 (A-a)) as 2- + oo. Let us also assume that the function V (A) appearing in 2° and 5°, is such that

[V(1+CA1-`)- V(A)]/V(,1) = O(A-a)

as

2-> +co

(28.7)

for some C > 0. Then we have

N(A) = V(A) (1+O(A-a))

as

(28.8)

Proof. The idea is to apply Lemma 28.1 to the linear subspace L, spanned by the eigenvectors of 9z, having eigenvalues close to I (they are all close to either 1 or 0, as we shall see later). Let aj be eigenvalues of t. They are real by 1° and by 2° and 5° satisfy the conditions

j

aj - a;

0 (Z

V (A))

(28.9)

>a; = V(2) (1+O(A-a)). Lemma 28.2.

Y

aJ = V (A) ( 1 +0( A-5 ) ) .

(28.10)

laj- 1151/2

Proof. For I a; - I I > i we have I aj - a; I = I a; I I a; - 1 I >_ i a; I. Therefore

Ia; -ajI <2 _

IaJ152 Ian-1 I> 112

lay- 1 I> 112

which together with (28.9) implies (28.10).

J

1a;

a;I= O(X-aV(X))

§28. The Approximate Spectral Projection

209

Lemma 28.3. Let N (A) be the number of eigenvalues of 8x in the interval 3/2]. Then

R(A)= V(A) (1+O(A-8)).

(28.11)

Proof. Put Ej =1 - aj. Then 2° can be rewritten as (28.12)

E IEj - Ej I = O (A-' V (A)), J.

and the statement of Lemma 28.2 gives that

Y- (1-Ej)= V(A) (1+O(A-b)), iz,is1/2

or

('1)= V(.) (1+0(A-a))+ Y Ej. Ic,i.5t/2

But, as in Lemma 28.2, it follows from (28.12) that

Y IE;I 5 2 Y Ie -E;I = O(A-6 V(1)), {s,Is1/2

i.,I

I/2

giving also (28.11).

Let us continue the proof of Theorem 28.1. a) Let LA be the linear manifold spanned by the eigenvectors of !,, with eigenvalues a, such that I aj -11 < so that

R(1) = dimLA= V(A) (1+O(A-"))

(28.13)

by Lemma 28.3. Condition 3° implies that

(4fx(A-AI)4f,,u,u) <- Cat'-`(u,u),

uEH.

(28.14)

But since (28.15)

( u,

it follows from (28.14) that QA

if,,

!-<

uELA.

Because dx is an isomorphism of Lx onto itself, it follows that

([A-(A+4C2'-`)I]v,v)50,

veLA.

But by Lemma 28.1

N(2) = V(A) (1 +O() .-d)) < N(A+4CA'

210

Chapter IV. Pseudodifferential Operator in IR"

Putting A +4C,1'-` = t, we obtain t = A(1 + O(A-`)), which implies x = t(1 + 0(t-`)), i.e. V(t(1+O(t-`))) (1+0(t-6)) < N(t). Using condition (28.7) we see that

V(t(1 +O(t-`))) = V(t) (1+O(t-6)), hence

V(t) (I+ 0(t-6)) 5 N(t).

(28.16)

b) Let M, = (LAY. Obviously we have

(u,u) <4u,(I-.9.)u),

ueM..

(28.17)

It therefore follows from 4° that

([A-(A-4CA'-`)I]v,v)?0,

veD,r MA.

(28.18)

From (28.18) we see that (EA-scA

-a.H)r) MA=0,

for any z' > 0 which implies by analogy with the reasoning in the proof of Lemma 28.1, that

13(x.) z N('1-4CA`-E') Now, arguing as in step a) of this proof, we obtain

N(t) 5 V(t) (1+O(1-6)).

(28.19)

The estimates (28.16) and (28.19) together give the required asymptotic formula

(28.8). 0 28.4 Sufficient conditions on V(1). Condition (28.7) looks difficult to verify. We therefore give here a simpler sufficient condition. Proposition 28.2. Let V (A) be a positive, non-decreasing function defined and differentiable for ,t 'to . Assume that

-

V'(A)IV(Z) =

(28.20)

Then V (A) satisfies (28.7).

Example 28.1. The function V (a) _ Va, where a > 0 satisfies (28.20) for

Eza.

§28. The Approximate Spectral Projection

211

Proof of Proposition 28.2. Set V (A) = V, Wl V (A)

Then V(A) = V (A0) exp

(,(t)ir).

This gives V (A + CAI -`) - V (A) = V (20)

= V(.I)

{ex(

{e(

A+ CA,

-exP (f(t)dr)}

f cp (t) dt

A+CA''

f cp(t)dt - 1

(28.21)

.

A

-, then for e $ b we obtain

Since Icp(.t)I 5 CA'

CAI

A+CAI-

I

cp(t)dt 5 C, A+ f

A

A

=

t`-b- dt = C2

C22`-el(1+CA-`)`-d_1]

For E = S, we obtain the same estimate A+CA''

_<

C3)c-d, A

=

C3A-6

J

A+CA'

f cp(t)dtC, f t-'dt=C,In(1+CA-`)5C2A-`=C2a-b. A

A

It now follows from (28.21) that

V(2+CA'-`)- V(2) 5 CV(A)

(we used here that ex - I - x as

A-b

but this is the required inequality

(28.7).

28.5 The idea for applying Theorem 28.1 (an heuristic outline). Let an operator A have the Weyl symbol b(z), zeIR2". Put V(A) = (2ir)-"

f

dz.

(28.22)

b(r)
The spectral projection EA of the operator A can be defined as the operator XA(A), when

is the characteristic function of the ray (-oo, A]. Let us

It is natural to expect consider the operator XS with the Weyl symbol that Xj for large I should imitate the spectral projection operator EA. It then remains to note that SP OA = (27r) " f XA (b(z)) dz = V (A).

Chapter IV. Pseudodifferential Operator in IR"

212

The technical implementation of this idea consists in applying Theorem 28.1. However, in view of the necessity to consider the composition of operators, we have to smooth the characteristic function Xx. In general, we arrive at a situation, which show the necessity of dealing with pseudodifferential operators with parameter, giving us the possibility of verifying 1°-5°.

28.6 The exact construction. Let us introduce a function X (t,1, x), t, 1, x EIR, A ? 1, x > 0, such that

X(t'1'x)

-(1 fort_A, 0

for t?1+2x.

(28.23)

and such that the following estimate for the derivatives holds I (ala1)kx(1,1, x) I < C' x - k.

(28.24)

The existence of X (t, x, x) is easily verified, for instance, in the following

manner. Let *(t, A, x) be the characteristic function of the set ((t, )., x) t < A + x}. Then we may put X(t,1,x) = X JW(r,1,x) Xo((t-t)/x)dr, where Xo (v) E Co (W), Xo (v) = 0 for I v I > 1 and J Xo (v) dv = 1.

Let now A have a real Weyl symbol (28.25)

such that for some C > 0 and Ro > 0

b(z) z ClzI''-,

Izl

Ro

(28.26)

(it follows from (28.25), that (28.26) holds either for b (z) or for - b (z); we fix the sign in such a wav that A becomes semi-bounded from below, this fact should be

obvious from what follows. Put e (z, 1, x) = X (b (z),1, x)

and now, choosing x

(28.27)

where v > 0, define d';, as the operator with the

Weyl symbol e (z, 1) = X (b (z),),1' - Y),

(28.28)

where v > 0 will be chosen later. Let us note immediately that e (`'1)

_ (1 for b (z) 5 1,

10 forb

z).+21'-"

(28.29)

§28. The Approximate Spectral Projection

213

Now we try to estimate the derivatives in z of e(;,..). Note that estimates f o r the class Hf Q with mo > 0 can be written in the form

Ialb(z)I 5

Cb(z)t-d"Y!,

Q'>0, IzI ? Ro,

(28.30)

where as Q' one can take, for instance, a'= p/m or possibly larger values. Now, differentiate (28.28):

8:e(z,A)=

...+h-Y=61z) cr,...Y,(dY'b(z))... (a''b(z))

A,

ark (t,

IY;I>0

(28.31)

where the sum runs over all possible decompositions of y into a sum y, + with an arbitrary number of terms k,5 I y I.

... +y,

Denote by Tk(z, k) the sum of all terms corresponding to a fixed k in (28.31). It follows from (28.31), (28.30) and (28.24) that

ITk(z,.l)I <

CYb(z)k-v'Irl

. x-kd-o

(28.32)

Due to (28.29) we can replace ,L by b(z) and rewrite (28.32) in the form . Akv-wlrl

(28.33)

I6. e(z, A)I 5 Cb(z)'-Y')IYI . Acs-0171.

(28.34)

Ia, Tk(z, A)I <

Cb(z)`R-v'Ilrl

where z is an arbitary real number. Since k5 lyl, this implies that

These estimates are true for A z 1 and for I z I obvious relations

le(z,A)I 5 C,

IzI < Ro,

Ole(z,A)=0,

IzI
Ro. In addition, we have the (28.35)

J,zAo,

(28.36)

if Iy I > 0 and ko is sufficiently large.

It is obvious from (28.34) that it is advantageous to take µ such that v <.u < Q', implying the necessity of selecting v such that v < Q', which we will assume in the sequel. Under some complementary conditions this will guarantee the applicability of Theorem 28.1. 28.7 Equipotential surfaces of the symbol and the properties of V(.%). In this section, we assume that the equipotential set {z: b (z) = A} for large A, is a smooth

hypersurface and moreover that

Pb(z) * 0

for

IzI > Ro,

(28.37)

Chapter IV. Pseudodifferential Operator in IR"

214

where V b (z) =

(.P-,...,

b_) is the gradient of b (z).

Let V (1) be defined by the formula (28.22). Since the 2n-form

dz = dz) A... Adz2n is the differential of the (2n - 1)-form w

I

2n

2n1:(-1)'+'z;dz)A...Adz;A...Adz2"

(the cap on dz; denotes that dz; is omitted), we can transform the integral in (28.22) into an integral over the surface b(z) = A:

f dz = f co. b(tkx

(28.38)

b(Z)=x

Let nt be the unit outward normal vector to the surface b (z) = A at z, i.e. ns

_

V b (z)

I Vb(z)I .

Denoting by dSZ the area element of the surface b (z)

we derive from (28.38)

that V (A)

=

(

2n n

! (z n=) dS= b(z)=1

_

"

I b(z)=d

(22 n

dS"

I

V b (z) I

(z V b (z)) . (28.39)

Now, we will calculate V'(A). Note that the distance at z from the surface b (z) = A to the near equipotential surface b (z) = A +42 is equal to Therefore

V'(d) _ (2n)-"

dSi

!

I Vb(z)I

d.1 (1 + 0 (1)) I V b (z) (

(28.40)

Comparing (28.39) and (28.40), we see that V '(A)

V(A)<- [

-t mi n (z Vb (z ))1

bA

J

(28 41) .

The formula (28.41) implies

Proposition 28.3. Let

Iz Vb(z)I z Cb(z)'-",

IzI z Ro, C>0-

(28.42)

§29. Operators with Parameters

215

Then

V'(A)/V(A) = O(A"-').

(28.43)

Remark 28.2. It follows from (28.42) that Gb (z) * 0 for I z I z Ro, so it is not

necessary to require this in advance. Also, the geometric condition (28.42) guarantees that the surface b(z) = A is star-shaped with respect to the origin, i.e. any ray, starting from 0, intersects this surface in exactly one point and at a non-zero angle. Exercise 28.1. Prove that (28.42) is satisfied if b (z) is an elliptic polynomial.

Hint: Use the Euler identity for homogeneous functions. Problem 28.1. For dim H = 2 construct an operator A, such that for some A there is an operator Ex for which the conditions 1) - 4) are satisfied, but E,; + Ez. Problem 28.2. On the torus 117" = IR"/2 rz V consider the operator A = - d + Q, where d is the Laplace operator and Q any bounded self-adjoint operator on L2 ('If"), with II Q II < M. Let N (A) be the number of eigenvalues of A, smaller than A.

Using the approximate spectral projection operator 92, which equals the exact spectral projection operator for -,J, show that

No(2-M) < N(A) < No(A+M), where No (A) is the number of points of the lattice 7", belonging to the ball I x i <_ i. Derive from this the asymptotic formula N(A) = c"An12 (1 +O(A-'/2)) .

§29. Operators with Parameter 29.1. The class of symbols and operators. The estimates (28.34) obtained for e (z, A) in §28 motivate the following

Definition 29.1. Denote by f'Q a° the class of functions a (z, A) defined for ze1R2n, AAo, infinitely differentiable in z and satisfying I aYa(z,A)I < Here m, 11, Q, a e 1R, Q > 0, a

It

C7m_Q171Aµ-6171.

(29.1)

0.

is clear that if a a FQ).-.",, j =l, 2, then a, a2 C',.,

where

J

Q = min (e,, e2), a= min (a,, Further, if a e F Q . ; then a= F0 , Note that if a e f Q then for any fixed A ;2t- Ao, a (z, A.) e fQ (1R2"), which a2).

allows us to define a class of operators A (A), depending on a parameter and with

Chapter IV. Pseudodifferential Operator in IR"

216

Weyl symbols a (z, A) E fQ a". This class of operator-valued functions A (A) acts,

G.

for instance, on S(IR") and we will denote this class by GQ;; 29.2 The composition formula

j =1, 2; and let A j (A) be the corresponding Theorem 29.1. Let a j c- F Q operator functions. Then A, (A) -,42 (A) where Q = min (L')1,02),

a = min (a I , a2) and where the Weyl symbol b (z, A) of the composition B(A) = AI (2) o A2 (A) is given by

b=

(_ 1)!a!

2-l=+e! (a{DOa1)

(3 Da2) + rN ,

(29.2)

where rN EF^',o mi - N(e. + e2F",+")-N(a +e,). @.a

(29.3)

Proof. The proof could be carried out according to the scheme used for proving Theorem 23.6, introducing first the corresponding class of amplitudes and repeating the arguments from §23. However, for brevity, we will give a direct proof. To begin with, we obtain a formula for the composition B = AI o A2 of the operators A, and A2 with Weyl symbols a, (z), a2(z) a Co (IR2"). Clearly

Bu(x)= Jal( x2xlC

Y,ry)

)

x eIftx-xJ'C+(x,-r)-ql u(Y)dydxl d>7dC.

If K. (x, y) is the kernel of B, we obtain that then

(x+ x, c) a2 Ke(x, Y) =J aI I\ 2

2

\x1+Y dx, drtds .

x ei1(

(29.4)

Now using formula (23.39) (with r=), that yields an expression for the symbol in terms of the kernel b (x, ) = J e -'x, ' 1 KB (x +

2

,

X_ 2) dx2

(29.5)

Putting x2/2 = x3, we can also write

b(x,c) = 2 " Je-2;x" KB(x+x3, x-x3)dx3.

(29.6)

§29. Operators with Parameters

217

From this and (29.4) we find that

xl-x37) 2

'7-Zx,f[ dX, dX3dgds.

x

Instead of x, and x3 we introduce new integration variables x+X, +X3

x4 =

2

,

X5 =

X+X, -X3 2

so that x, = x4 + x5 - x, x3 = x4 - x5. Observing that obtain

(XI, x3) = 2", we a (x4, x5)

b (x, c) = 22" [ a, (x4, S) a2 (xs, q) x ez'[(x-x,1 ,+lx.-x['7+1x,-x.1'cl dx4 dx5 ds dq or

b(x,0 = 22n 1 a, (y, q) a2 (z, S) x

e2i[(x-z)

7+u'-x) c+(;-y)

dydzdg dd

(29.7)

Note that the exponent in (29.7) may also be written as 1

=2i E

2i

j=1

1

1

xj yj z j

j qj 'j

From the form of this exponent, the possibility of integrating by parts follows, resulting in the appearance of decreasing factors of the type <x - z> - ", - ", - ",
make the change of variables y' = y - x, z' = z - x, g' = g Omitting the dashes, we obtain b(x,

) = 22n f a, (x+y, +q) a2 (x+z, + x e2i(!' -. 7) dgdddydz.

(29.8)

We shall view this integral as an iterated integral with the order of the differentials dg dd dy dz. Write out the expansions

a,(x+).-,("'+q)=

Y

}'1 l3xa1(x.+g)+r)1(x,(29.9) x

Chapter IV. Pseudodifferential Operator in 1R"

218

zP

IPISN

-1 r

(29.10))

t

where

rN (x,

y°(a

C.

jol-N

6"(x,z,

(29.11)

0

Co J (1-s)N-1 dt z'(8Xa2) 191=N

(29.12)

0

Inserting these expressions into (29.8), we obtain =z9

b(x, ) = 22n

Y

J «0-1 [aza, (x, +q)] [aaa2(x, +s"")]

I`+9ISN-I

x e2i(y_C-z a) dgdC dydz+rN

(29.13)

where rN ) (x, ) has the form of a linear combination of terms of four types: r;°B (x, ), being the same type of integral as the summands in (29.13), but

with Ia+fI?N; raSN (x, ) = JY` [ax a3 (x, +q)] [r(1(x, z,

x

, C)]

e2i('-C-z-e) dgdCdydz;

(29.14)

rN 0 (x, ) = J z' [rN) (x, y, , q) ] [a a2 (x, + 6 )]

x e2i(Y-c-: q) dd d{ dy dz; (x, z, , C) e2/u' t-z*') dq dt dy dz .

rWt (x, t) = J rv' (x,Y, , q)

(29.15) (29.16)

Let us calculate one of the integrals in (29.13). For this, note first that Y°e2ly-` _ \12 D; /)a(e2ir-`), D"e

1

(e-2iz n),

(29.17)

(29.18)

and carry out the integration by parts, using these identities. We obtain

§29. Operators with Parameters

e2++r,c-= dddddydz

22"J afl!i

-

219

22n-1a+01 a

a!$!

0

2-101(-2)-1`1 J

dq d d Ydz =

xe

17

21

/

&.D{ar (X1

a.

2-101(-2)-Ial 21J

C

Adz J

X

l0«' fl2)-lal [a' D{ar

x e'rdt dy = 2

IJ

(x, ))] [a0Dfa2(x,C)],

which gives terms with compact support in the formula (29.2). Now, note that formula (29.8) and all the following computations remain valid for arbitrary symbols a, in the classes !'Q . This fact can be established by substituting the oscillatory integral (29.8) for a convergent one, but it can also be verified by a standard passing to the limit from compactly supported symbols, as in § 1. Assume now that the symbols a, and a2 are as in the formulation of the theorem. It sufficies to prove the inclusion (29.3) for each of the remainders rr41 , r ."N) , rN 0 , rN ". As far as r;4a is concerned, this inclusion is trivial, since it has the

same form as the terms of the sum in (29.13). To estimate rs,,, rN a and rN' it is convenient as before to integrate (29.14)(29.16) by parts (using (29.17) and (29.18)). As a result y° is replaced by Dc and z0

by D7 and we arrive at the situation of having to estimate, uniformly in t, , z2 a [0,1 ], symbols of the type

J[d'Dfa,(x+z;y, +q. )] [aX1Y92(x±t2z, xe2r(r

q) drydrdydz,

)]

Ia+QI Z N.

(29.19)

Differentiating (29.19) with re?l+Fhl It+ + 0444 r; * lil+ti 114 01 INK 40I4N+ F pi

this expression is a linear co444IIHH Hl IFU44

Hl II4

dryd(.

t!1' ti;:

j

_ ¢:

1

Using the identities e21frc-=n) _ (1 + 11°1`

e2flrC-:nl

= (1 + 1°1'

I I

I+11. 41 '1I

I 1-)

4++41

tt;l+! i=;r I

1 q.4I1 x e2i(r c-= - n)

+4

dl$

I

1

r1 = A

$4I

411$

220

Chapter IV. Pseudodifferential Operator in IR"

and introducing the notation =

_ (1+IYI2+

(1+IzI2+ICI2)112,

1,112)112

,

we see that (29.20) reduces to a linear combination of expressions of the form f[ox.+Y a,(x+T1Y,C+n,A)] [6X"c v.a2(X+T2Z,C+C,A)I (29.21)

where x', x", v', v" are 2n-dimensional multi-indices, such that

Ix'I?N,

Ix"IzN,

lv'I+Iv"I?Ivi+161.

(29.22)

The integral (29.21) can be estimated in absolute value by Cf<X+T, Y,b+n>>m

_Q,Ix+r'I

AMi-Q,Ix+v'I

JMt-Qtlti ti I

X <X+T2z,

X -2M -2M dp dS dy dz ,

which, due to (29.22), does not exceed the expression C# ,+Y,-N(e,+aj)-IY+61e. (<x+T1Y,S+n>'"i-Qiti-Q,W I

X <x+T2

-2M-2M

dndC dydz.

(29.23)

Here the power of A corresponds exactly to the statement of the theorem, so that it suffices to estimate the integral in (29.23) by the desired powers of <x, C>. Note, that the integral in (29.23) coincides with the product of the integrals

f

-1MdhdY, ":MdCdz.

f

(29.24) (29.25)

Let us estimate the integral (29.24). Decompose the domain of integration into two parts

Q1= {Y, q: IYI+10512 (IXI+IXI)},

Q2 = {Y, n. IYI+I'I>? (IXI+IXI)}, and denote the integral over Qj by Ij, j =1, 2. Since the Lebesgue volume of Q, does not exceed C, <x, C>" and since the integrand can be estimated over Q1 by C2 <x, C> m- - el N- e, W1, we have for I, the estimate

Ill 15 C3<x,

(29.26)

Consider now I2 . We may assume that IxI + I C I z 1 (for I x I + I C I < 1 the desired estimates are trivial). Then, using the obvious estimate (1 + Ix+TI YI + IC+n I)° 65 (1 + IxI + IXI)' (1 + I Y I + I1I)'

§29. Operators with Parameters

221

and taking M sufficiently large, we obtain (29.27)

1121 5

From (29.26) and (29.27) it follows that (29.24) can be estimated by C<x,

Taking into account that similar estimate holds for (29.25) we see that (29.23) does not exceed C<x,f>m.+m: (Q +Qi)N

5

Ap,+px-N(a.+a,)-alP+dl

Q 1.

C<x>m,+m=-N(Q,+Qi)-Qlr+b)+2a . Ap,+p,-N(°,+a,)-°1Y+d)

which guarantees the inclusion rN E I'Q

Q+m=-N(Q1+Q,)+20.p,+p,-N(a,+a,),

(29.28)

Now, increasing N and considering the additional terms which appear in the sum (29.2), we see that (29.3) follows from (29.28), thus proving the

theorem. 0 29.3 Positivity of operators with parameter Theorem 29.2. Let a (z, A) E 1'Q , a > 0, a (z, A) 2- e > 0, e a constant, and assume that the estimates

18;a(z,A)1

CYa(z,A) . AA_AO,

(29.29)

hold, where ao > 0 and ao does not depend on y. If A(1) is the operator with the Weyl symbol a(z, A), then for sufficiently large A we have A(il) > 0 (i.e. (A(.X)u, u) > 0 for u E S(lR")). For the proof, we need the following lemma which allows us to use the antiWick symbol. Lemma 29.1. Consider an operator B(A) with anti- Wick symbol a (z, A) a I'Q and let b (z, A) be its Weyl symbol. Then

a-b=

Y.

cY (c3Z a) + rN ,

0<jYj
where c=0 f o r o d d 1 y 1 and rN E I'Q s

Proof. Similar to the proof of Theorem 24.1. Exercise 29.1. Prove Lemma 29.1.

(29.30)

222

Chapter IV. Pseudodifferential Operator in IR"

Proof of Theorem 29.2. Let Bo (A) be an operator with the anti-Wick symbol a (z, A) and let bo (z, A) be the Weyl symbol of Bo (A). Consider now the operator B, (A) with the anti-Wick symbol a (z, A) - bo (z, A) and denote by b, (z, A) its Weyl symbol. By induction we may construct a sequence of operators B0, B1, B2 , .. .

such that B; is the operator with the anti-Wick symbol a - bo - b, - ... - bi_, , where bo, b...... b; _ , are the Weyl symbols of the operators B0, B1,..., B...1. It follows from Lemma 29.1 that if

Ak=A-Bo-B,-... then A k

0.6

Bk-1

eF"'-2Qk.°-2°k. Put

Qk=B0+B,+...+Bk-1 Thus, A = Ak + Qk. An induction in k shows that b, (z, A) for j > 0 is of the form bi

=

(29.31)

cr.i(0"a) + r".1+

2i5I7I<"

where cY, i are constants and r". i e I'Q ; °" °- a". Therefore the operator QA, has

an anti-Wick symbol of the form qk (z,).) = a (z, A) +

cY [8' a (z, A)] +

A) ,

(29.32)

251t[<"

where rn (z, A) e TQ Q e"- ° - a". Taking (29.29) into account, we obtain

gk(z,A) = a (z, A) (1+A-2aa s(z,))),

where s(z,A) is such that sup I s (z, A) I < C,

A > AO,

t ER"

and C is independent of A. In particular, it is clear that qk (z, A) z e/2 > 0 for sufficiently large A implying

Qk?2

(29.33)

in view of Proposition 24.1. Now note that for large k

IIAk(A)II-0

as

(29.34)

Indeed, it suffices to verify that this holds for the Hilbert-Schmidt norm II A (A) 112. But

IIAk(A)IIi = (2n)" I lbk(z,A)I2dz-+0

as

A-+oo,

§30. Asymptotic Behaviour of the Eigenvalue

223

if we choose k so large that

m-2ek<-n,

u-2ck<0.

It follows directly from (29.33) and (29.34) that A

Theorem 29.2. U

4

1,

proving

§30. Asymptotic Behaviour of the Eigenvalues Consider an operator A with a real Weyl symbol b(z) eHfQ m°, mo > 0. By Theorem 26.2, A is essentially self-adjoint and by Theorem 26.3 it has discrete spectrum. Since b (z) has no zeros for large I z I, then, changing sign if necessary, we may assume that

C,IzI'°Sb(z)

mo>0,

IzI>Ro,

C2Iz11,

(30.1)

holds, where C, , C2 are positive constants. It is easy to show that in this case the

operator is semi-bounded from below. Indeed, repeating the argument of the proof of Theorem 24.2 and using Theorem 24.1, we see that there exists an operator A' with the anti-Wick symbol

b(z) + E cy8yb(z),

(30.2)

0
such that A - A' is bounded. But then it suffices to verify the semi-boundedness from below for A', which follows from the semi-boundedness from below of any function of the form (30.2), which in turn is a consequence of the fact that

b(z) + E c,arb(z) = b(z) I + 0
cr

aYb(z)

0
b(Z)

where all the terms in the parenthesis, except the first one, tend to 0 as Izi +oo. Let q' be a positive number such that the following estimates hold

a'b(z)I S

C,b(z)'-a'171,

IzI

;-> Ro

-

(30.3)

(as we have already remarked in 28.6, one may take, for instance, e'= Q/m, although this might not be the best value for e'). Finally assume that IzI

R0,

(30.4)

where 0 <_ x < 1, c > 0. Set

V(ii) = (2n)

!

dz,

b(z)
and let N(J.) be the number of eigenvalues of A, not exceeding A.

(30.5)

Chapter IV. Pseudodifferential Operator in IR"

224

Theorem 30.1. Let the operator A have the Weyl symbol b (z) E C°° (lR2n), satisfying conditions (30.1), (30.3) and (30.4) with x < Q'. Then for any s > 0, one has the asymptotic formula

N(2) = V(2) (I+ 0(A"-0'+e))-

(30.6)

Proof. We will make use of Theorem 28.1 with the approximate spectral projection operators 01 constructed in section 28.6. Recall that d2 has a realvalued Weyl symbol a (z, 2), equal to I for b (z) 5 2 and 0 for b (z) z 2 + 2A'and if v < Q' the estimates (28.34)-(28.36) guarantee that e(z,A)eFO4.0 .9,

p>0,

a>0.

(30.7)

We have to verify that all the conditions of Theorem 28.1 are fulfilled. The condition S,* = 6,, is obvious since e (z, 2) is real-valued. The fact that e2 belongs

to the trace class follows from Proposition 27.2. Denote the Weyl symbol of an arbitrary operator A by a (A). Obviously

(9x-1)=

cas[araxe(z,2)]

[04axe(z,2)]+(e2-e)+rN,

(30.8)

where rNC-F-MQ,-2Na. Note that all terms in the sum, except for rN, are supported where A5 a(z) < 2(1 +2).-"), and if we apply Proposition 27.3 to each term, we obtain the estimate 11 g'42

- gxl1I = O(V(2+22'-")- V(2)).

But it follows from Proposition 28.3 that

V'(2)/V(2) = O(2"-'), which, by Proposition 28.2, gives the estimate

V(1+221")- V(2) = 0 (2"-" V(2)) . Therefore

II',i-'x111=O(2" V(2)).

(30.9)

In addition, it follows from Proposition 27.2 that Sp 9,1 = (2n) 'S e (z, 2) dz

= V(2)+O(V(2+22'-")- V(2)) = V(2) (1+O(2'- v)).

(30.10)

§30. Asymptotic Behaviour of the Eigenvalue

225

Note that we must take v < Q'. Choosing v = Q' -' e, where e > 0, we may rewrite (30.9) and (30.10) in the form Ii'f -gx111

(30.11)

V(A))

Spd,, = V().)

(30.12)

2. Let us now verify requirement 3° of Theorem 28.1: -fx (A -AI) S3 5 CA'

We write this inequality in the form

49x(.1-A)d,,+Cd'-">0.

(30.13)

Next we compute the Weyl symbol of 'A (A.1- A) d' . We have 47 (d'A(AI-A)) =

ca0 (axa4e)

4942 (A-a(z)))+rN.

la+oI
a (t, (Al- A) S,) = L cYY:Y, (0,1' e) (a:' e) (a:' (A - a (z))) + rN ,

(30.14)

where iN a 1'!;(01e+ 1' - Na, the sum is finite and coo0 =1 It is clear that the remainder r'N cannot affect (30.13), and therefore we will estimate the compactly supported terms. We will show that for Iy1 I + Iy2.1 * 0 and for some j > O and o > 0, we have the estimates I(aY'e)(aYe)(aY(.1-a)) I -a(b,I+IAI+IY,I).

(30.15)

To begin with let y3 = 0. Then (30.15) holds in view of the fact that 12 -a (z) I

2.1'

on the support of (aY' e) (aY, e). Let Y3 * 0, then

I (a1'e)(ay-e)(a,a) 15 C-d(h,J+IY,U 21-"

Note that Q' - v/ 1 y3 I ? Q' - v > 0. Therefore

I(ai,e)(ai,e)(a?,a)I s

CA'-"(z>-i)(IY,I+IY,q

-eoY,I+IY,D),-e,IY,I,

226

Chapter IV. Pseudodifferential Operator in IR"

where a, = Q' - v. Taking into account, that the inequalities A:5 a <-- A + 2A1

and (z)'"° S CA hold on the support of (311e)(3 "e), we obtain the estimate (30.15) (although perhaps with smaller Q and & than in (30.7)). Therefore we obtain

(ar,e)(ar,e)(ar,(A-a))er

-',

IYII+IY2I+0.

(30.16)

Now, repeating the reasoning, used at the beginning of this section to prove the semi-boundedness of A, we obtain that the L2 (1R")-norm of the operator with

Weyl symbol (ar,e)(ar1e)(07.(A-a)) does not exceed Cpl'-', so this term also does not affect (30.13). By similar arguments, one verifies that

Ya *0,

(30.17)

hence this term also cannot affect (30.13). Finally let us investigate the function e2 (A- a) = q. The function q (z,).) has the following properties:

q(z,A)z-CA'

anger°'o

IYI>0.

If P is the operator with the anti-Wick symbol q (z, A) and p (z, A) is its Weyl symbol, then it follows from Lemma 29.1 that q - p e But it is clear that P't - CA' -' and II Q - P II 5 CA' -'. (30.13) follows from this. Putting v = Q' - e in this relation, we obtain

e,,(A-Al)l2 5 CA' -Q +`.

(30.18)

3. Let us now verify that

(I - ,) (A -Al) (I - dA) + CA' -' > 0

(30.19)

for sufficiently large C> 0. Applying Theorem 29.1, we see that the symbol of the left-hand side of (30.19) has the form C.l'-'+Y_ Cr1nr,a?, (1-e)

ar,(1-e) ar,(a-A)+rN,

(30.20)

where rN r'"-"(Q+ and c 000 = 1. min(Q.l),00)-The operator RN with the Weyl symbol rN can be estimated in norm by II RN 11 :5 C.1'-"" and so cannot affect (30.19).

Let us estimate the principal part of (30.20), the symbol

q(z, A) = (1-e(z,).))2 (a(z)-A)+ Cpl' -'.

(30.21)

§30. Asymptotic Behaviour of the Eigenvalue

227

First note that q (z, 2) >_ 2' -".

(30.22)

for sufficiently large C. Now we show that (30.23)

Cq(z,2)
Ia=q(z,2)I

for some e > 0 and v > 0. We have ECY.Y..a=.(1-e)2az..(a-2).

a=q =

If y' * 0 then the corresponding term may be estimated as in step 2. of this proof. If y' = 0, then for y + 0 we have the estimate

I(1-e)2al(a-2)I 5

Ca(z)'-e'IYI

<(q(z,2)+22)(a(z))-e 11,

(30.24)

on the support of I - e, since a (z) 5 q (z, 2) + 22. Furthermore (q(z,2)+22)(a(z))-° °1Y1 5 q(z,A)(a(z))-°'IYI +22'-" 2"a(z) < q(z, q(z, 2) 2)2-e'IYU2-e'IYli2mo+

on supp (1 - e)2, since 2'

(a(z))"-e'IYI

e'IYI

(30.25)

< q (z, 2) and 2" <- a (z)". Finally

q(z,2)(a(z))"-e'IYI <


(30.26)

Now, (30.23) follows from (30.24)-(30.26). Note also that in view of the obvious estimate I q(z, 2) I <_ C
may take v = o'- c, with e > 0. Applying Theorem 29.2, we see that (30.19) holds, or

(1-.9,i) (A-21) (1- j) + C2'

-Q+

0.

(30.27)

4. To complete the proof of Theorem 30.1, it remains to note that the requirements 1°-5° of Theorem28.1 for the construction of the "almostprojection operator" 9x have already been verified ((30.11), (30.12), (30.18) and (30.27)) and property (28.7) for V(2) follows from (30.4) and Proposition 28.3. An Application of Theorem 28.1 then completes the proof of the asymptotic formula (30.6).

228

Chapter IV. Pseudodifferential Operator in IR"

Problem 30.1. Compute the eigenvalues of the operator A = _A + 1x12 and verify directly the asymptotic formula (30.6).

Hint: The operator A= _A + 1x12 is the quantum mechanical energy operator for the harmonic oscillator, and its eigenvalues may be found in any text-book on quantum mechanics.

Problem 30.2. Show that if the Weyl symbol b(z) of A is an elliptic polynomial whose principal homogeneous part does not take values in the ray arg d = To, then the complex powers A2 and the C-function C (z) = Sp A= can be

defined and the C-function admits a meromorphic continuation to the whole complex plane C. Find its poles, its residues and the values of C (z) at the points 0, 1, 2, ... . Obtain here the asymptotic formula for N (A) (without estimating the remainder

term) using the Tauberian theorem of Ikehara.

Appendix I Wave Fronts and Propagation of Singularities In this appendix we present the definition and the simplest properties of the wave front of a distribution as introduced by Hormander [6]. The concept of wave front is important in that it allows a microlocal (localized at a point of the cotangent bundle) formulation of the theorems on regularity of solutions of differential equations and also clarifies questions connected with the propagation of singularities. The wave fronts also play an important role on spectral theory, and are naturally connected with pseudodifferential operators. While leaving out many important questions of the theory of wave fronts, I nevertheless thought it useful to add this short appendix.

A.1.1 Wave front of a distribution Definition A.1.1. Let X be an open set in lR", let (xo, o) e X x (lR"\ {O}) and u e 9'(X). We shall write (xo, o) 4 WF(u) if there exists v e d°'(X) such that u = v

in a neighborhood of xo and

Ii()I S

if 1141

Idol

l <e,

(A.1.1)

for sufficiently small e > 0 and arbitrary N > 0, i. e. i ( ) is rapidly decreasing in a

conic neighbourhood of o . Thus WF(u) is a closed conic set in X x (I V\ (0)) which we call the wave front of the distribution u.

Lemma A.1.1. If rp (x) a Co (X) and (xo, 4o)4 WF(u) then (xo, o)4 WF((ou). Proof. We have to show that if 6 is rapidly decreasing in an open cone F, then so is tpv. Now

mv() = ja(-n) w(n)drl

j v( InlSR

1)

j 1,,I2R

Appendix I

230 and

I 10 IS R

IVI2 R

I (1+Inl)'-Ld1

<_ Csup IvISR

IPIaR

<- Csup I6 In IS R

Putting R = I I' 12, we see that if belongs to a cone slightly smaller than f, then - n e l- f o r large I fl I and I n I R. In addition, I - n I - I fl I and R" +"- L is rapidly decreasing in as I,I("+'-bf2 Picking a large L we see that

IfI- 00, Ef. Corollary A.1.1. In Definition A.1.1 we may put v = (p u, for cp c- CO' (X).

Proof. We may choose first v as in the definition and then cp such tP = I in a neighborhood of x0 and cpu = cpv. It remains only to apply Lemma A.I.I.

Lemma A.1.2. Let n: X x (IR"\ 0) - X be the natural projection and u e -9'(X). Then n WF(u) = sing supp u.

Proof. a) If x0 4 sing supp u pick (p e Co (X) such that (p = I in a neighbourhood of x0, V = 0 in a neighbourhood of sing supp u. Then we see that cpueCC (X), from which tpuES(IR") i.e. x04nWF(u).

b) Let x04 n WF(u). Then for any ° a 1 \ {O} there exists a function cp,e (x) e Co (X) and a conical neighbourhood f,0 of ° such that fpta (x) = I near

x° and (tp u) () decreases rapidly in F,.. Let f, , .... f,M be a covering of

=

fl (pt, we see that 0u( )decreases rapidly everywhere so j=I that cp u e Co (X) i. e. u e C °° in a neighbourhood of x0 so x0 * sing supp u.

R"\(01. Putting c

Proposition A.1.1. Let u c -!?'(X) and (x0, b0) # WF(u). Then there exists a in a classical properly supported WDO A E CL° (X) such that a,, =- I (mod S conic neighbourhood of (x0 , °) and Au e Co (X). Proof. Let cP E Co (X), cp = I in a neighbourhood of x° and suppose 1pu ( ) decreases rapidly in a conic neighbourhood of 0. Let X (!;) be supported in this fort >-1, It z I and X e C°°(X) with X (4) neighbourhood with X X decreases cp u =1 in some smaller conic neighbourhood of b0. Then X rapidly so that X (D) (cp (x) u (x)) e C'. But then W (x) X (D) (cp (x) u (x)) E Co (X) if Vi e (X). We may pick rp so that Vi (x) =1 in a neighbourhood of x0 and then

the `PDO A= *(x)X(D)V(x) satisfies all the required conditions. Proposition A.1.2. Suppose we are given u e 9' (X), (x0, 0) E X x (lR"\ {0}) and an operator A E CL"(X) with principal symbol a.(x, ). So that Au makes

Wave Fronts and Propagation Singularities

231

sense let either u e 9'(X) or A be properly supported. Finally assume that a. (xo, 4o) * 0 and Au e C°° (X). Then (xo, 4o) # WF(u).

Proof. a) By the standard construction of a parametrix in a conic neighbourhood of (xo, o) (cf. §5), we obtain a properly supported 'YDO B e CL-'(X) such that aef -I (mod S - °°) in this neighbourhood. Obviously BAu a CI(X)so that replacing A by BA we could obtain A= I (mod S - °°) in the same conic neighbourhood of the point (xo, 4o). b) Now let X =1 in a neighbourhood of o , X e C°° (IRe), and X homogeneous of order 0 in for I ' 1? 1. Let (p (x) e Co (IR"), (p = I in a

neighbourhood of xo and the supports of T and X be chosen so that (P (x) X () OA (XI 0 = (p (x) X (0

(mod S

From this we obtain X (D) (p (x) A - X (D) (p (x) e L-

(A.1.2)

and by X (D) (p (x) Au e C (X), it follows from (A.1.2) that X (D) (p (x) u e C`° (W).

(A.1.3)

X(D) (p (x) ueS(IR"),

(A.1.4)

c) Now we show that

from which it follows that $14 (b) eS(IIt") and in particular, that (pu(b) decreases rapidly in a conic neighbourhood of to as required. (A.1.4) follows from the following lemma and (A.1.3) Lemma A.1.3. Let v e t' (IR"), X

a SQ 0. Then for p (x, supp v) z I we have

I Da X (D) v(x)1:5 CQ. N I X I

Proof. Since 2 X

-N

(A.1.5)

a SQ o I_I it all reduces to the case a = 0. Further since

v= Y Dava with continuous v we may reduce to the situation where v is 1401P

continuous. We have

X(D) v(x) = 5et(x-Y)'' X(4) v(y)dydd.

(A.1.6)

Integrating by parts and using the formula

Ix_yI-2N(_A )N from (A.1.6) we obtain X(D) v(X) = f

IX-yL -21 v(y) dydd,

(A.1.7)

232

Appendix I

which makes sense for Q (x, supp v) >_ 1. Picking N so large that (-A 4)" x E S.0 -', we see that the integral (A.1.7) converges absolutely and is estimated by C<x>-2N for a(x,suppv)>_ 1. Remark A.1.1. The condition am(xo, 0) * 0 is sometimes called ellipticity of A at (x0i 0). It is easy to formulate and prove the hypoelliptic analogue of Proposition A.1.2. We leave this for the reader as an excercise. Corollary A.1.2. For AECLm(X) denote

char (A) = Then if Au = f e C°° (X) we have WF(u) a char (A). In particular, if char (A) = Q we have u e C aO (X).

Corollary A.1.3. If u e 9'(X), then

WF(u) _

(l

char (A).

(A.1.8)

AeCL°(X) Aye a C`(X)

This holds for u c- -9'(X) if we take the intersection only over properly supported A.

The importance of Corollary A. 1.3 is that (A. 1.8) shows how to define WF(u) invariantly as a closed conic subset of T'X when X is a manifold. We now generalize Proposition A.1.2 even more, by weakening the requirement Au E C°° (X).

Proposition A.13. Again let A E CLm(X), u c91'(X) and either A be properly supported or u e o°'(X). Then, assuming am (xo, o) $ 0 and (x0, b°) t WF(Au), we have (x0 , o) + WF(u). In other words,

WF(u) c char (A) v WF(Au).

(A.1.9)

Proof. By proposition A. 1.1 there exists a properly supported P e CL° (X), with ap - 1 (mod S- °°) in a conic neighbourhood of (x0, c,0) and (PA) (u) e C°3 (X). But then, from Proposition A. 1.2 it obviously follows that (xo, 0) * WF(u).

Proposition A.1.4 (Pseudolocality of `PTO). Let u e -9'(X), A e LQ 6 (X) 0 < S < Q <_ 1 and assume either A is properly supported or u E "'(X). Then if (x0, o) 0 WF(u), we have (x0,1;0) ¢ WF(Au). In other words, WF(Au) c WF(u).

(A.1.10)

Proof. The condition (x0, bo) 4 WF(u) amounts to the existence of a properly supported `PDO P e CL° (X) such that Pu e C°°(X) and ap =- 1 (mod S- °°) in a

conic neighbourhood of (x0,1,0). Now let Q be a properly supported `YDO

Wave Fronts and Propagation Singularities

233

Q E C L ° (X) such that q0 (x0 , S0) $ 0, (q0 be principal symbol of Q) aQ E S - °°

outside of some sufficiently small conic neighbourhood of (x0, So) and PQ = Q (mod L- °°)

if

QP = Q (mod L- °°) .

Let us demonstrate that QAu E C °° (X). We have QA - QAP a L- °° so it suffices

to show QAPu E C°° (X). This however, is obvious since Pu E C°° (X). Now (xo, o) * WF(Au) follows from Proposition A.1.2. Corollary A.1.4. If A e CL!"(X), then

WF(Au) c WF(u) c WF(Au) v char (A).

(A.1.11)

Corollary A.1.5. If the operator A e CL'"(X) is elliptic, then

WF(Au) = WF(u)

(A.1.12)

Exercise A.I.I. Compute the wave fronts of the following distributions: a) S (x); b) a(x') ® I (x"), x' E IRk

x" E IR" - k;

c) bs, where S is a smooth submanifold in IR"(<S, cp> is defined as the integral of the function cp restricted to the surface S with respect to the induced measure);

d) (x+ i0) -' on R; e) the indicator function of an angle in IR2 (the function which is equal to 1 in the angle and 0 outside it).

A.1.2 Applications: Product of two distributions, trace of a distribution on a submanifold 1) Let uj a 9' (X), j =1, 2. What does u, u2 mean? It should be the ordinary product u, u2 if, for example, u, and u2 are continuous or if one of them is smooth, and it should be a natural extension (e. g. by continuity in some sense). It

will turn out that we can define u, u2 under the condition WF(u,) + WF(u2) c X x (IR"\ 0) ,

(A.1.13)

i.e. if there are no (x, ) a WF(u,) such that (x, - ) e WF(u2). Since the product is a bilinear operation, then using a partition of unity we may assume that u,, u2 are in f' (IR") and have sufficiently small supports so that

u, (c) is rapidly decreasing outside a cone F, and u2 (0 is rapidly decreasing outside a cone F2, whereby F, + F2 c IR"\{0} (i.e. F, and F2 do not contain opposite points). In the usual situation one has u2 (u, * u2) (c), where

u,*u2( )=

u2(1)dp.

(A.1.14)

Appendix I

234

In our case this integral converges absolutely also, since either I u, ( - n) I or u2 (n) I rapidly tends to zero as I n I -+ + oo. Put u, u2 =

F-'

J u1(-n) u2 (n) do .

(A. 1.15)

Why is this an extension by continuity? We can for instance show that if XeCo (IR"), fX(x)dx=1, X(x)=E-"X(E-Lx) and u1a)=u*Xt(ECo (IR")), then lira u;`I ui = u1 U2 in the topology of 9'(IR") where u, u2 is understood in c+0+

the sense of (A.1.15).

Example A.1.1. Let uk a 9'(R2),
Xk(t)ECO (IR'),

so that supp uk c {((Kx1, x2): x2 = kx1 }. Consider the product Uk . uo . We have

uk(S) = JXk(xl) e'If-=i+C,kx-)dx, = Xk(SI+kS2)

From this we see that WF(uk) is the set of all normals to the line x2 = kx1, lying over nk ' (supp Xk), where nk: (x1, kx1) - x, . Consider now the convolution uk * u0 :

(u0 *uk)( ) = f o('I -'I) Xk('l+kn2) Al dn2 = f zo(m)dm f zk(k i )dn2 = k

X0 (0) xk(0);

from which uo

uk =

k

X0(0) - Xk(O) S(x) .

The limit as k -+ 0 does not exist, which is natural, since for k = 0 condition (A.1.13) fails. 2) Let u e 9' (X), Y a submanifold of X, NY the family of all normals to Yin T*X (the normal bundle to Y.) If WF(u) n NY = 0, then the trace u I r is defined naturally in the same sense as for products. Indeed, localizing we may assume that Y= {x, _ ... = xk = 0}. For u e Co (X) we have

u I r = f u( t ,

,

WA I ... d k.

(A.1.16)

But by hypothesis vectors ofthe f o r m (i1, ... , k, 0, ... , 0) are not contained in WF(u), so u decrease rapidly in their direction and the integral (A. 1.16) is defined. In particular, if u E -9'(W +') satisfies a differential equation of order m of the form

alt,x, a a )u=Jc-C°o(W'1), et

d,

x

tEIR', xc- IV,

Wave Fronts and Propagation Singularities

235

where the hyperplane t = 0 is non-characteristic, i.e. a. (0, x, 1, 0) 4 0 then all ak the restrictions are defined and lie in This means in particular all, r=0

that the Cauchy data make sense. Thereby u I,_,o is a smooth function of to with

values in 9'(&"). A.1.3 The theorem on propagation of singularities

First, we state the simplest version of the theorem on propagation of singularities.

Theorem A.1.1. Let P E CL!"(X) have a real principal symbol p," (x, ), u E 9'(X) and either P is properly supported or u e 9'(X) so that Pu makes sense. Then if I is any connected interval of a bicharacteristic of the function p," (x, t)

not intersecting W F(Pu) then either ! C W F(u) or 11) W F(u) = 0. In other words, in the complement of WF(Pu), the set WF(u) is invariant under the shifts along the trajectories of the Hamiltonian system

(A. 1.17)

Example A.1.2. Let us prove that the wave equation

02u

r?xo

- d u = 0 in

IIt"+' cannot have solutions with isolated or compactly supported singularities. We have m = 2, p2 (x,) _ - + p 1 2. The system (A.1.17) has a solution = const, x° = - 2 dot, x = 2 t.o Let 0 e sing supp u. Then there exists a point (0, 0, bo, ) e WF(u), so by Proposition A. 1.2 it is obvious that I

Theorem A.1.1. (-2t, 2t,

b0,

j 2 = o By ) E WF(u) for any t and, in particular,

(- 2S0 t, 2 t) e sing supp u for any t which also yields the required result. We now give a proof of Theorem A.1.1. due to V.N. Tulovsky. It is based on the following proposition describing wave fronts in terms of the action of distributions on rapidly oscillating exponentials.

Proposition A.M. Let u E 9' (X). Then the condition (x0 , 0) WF(u) is equivalent to the following condition;

A. There exists e > 0 such that if 0 (x, 0) is a smooth real valued function defined for I x - x0 I < e and 0 E (IR" \ 0), is a homogeneous function in 0 of degree 1

such that ,, (x0, 00) = o for some 00 $ 0, then for an arbitrary symbol tP E CS° (( I x - xo I < E) x IR") vanishing for I x - xo I z e/2 there exists a conical

neighbourhood T of the point 00 in IR"\0 such that I
tp(x,0)e-rm(X.e)> < C"I0I-",

0c-F, IOI z 1

.

(A.1.18)

Appendix I

236

Proof 1) Let condition A. be satisfied. Take (P E Co (IR"), such that cp - 1 in a neighbourhood of x0 , qp = 0 for I x - xo 1 Z e/2 and 0 (x, 0) = <x, 0> so that N = n, 00 = 0. We then see that
'9>

= rp' u (0) decreases rapidly

in a conic neighbourhood of 0. But from this it follows that (x0 i 0) WF(u). 2) Let (x0, 40) * WF(u). We will verify that condition A is satisfied. Let P and (p be as described in the condition. Without loss of generality, we can assume

u e 8' (JR ). Express the left-hand side of (A.1.18) in terms of a (& 0)e-'4'(x.a))

(u(x), rp(x,

= f ii

(x.e)lV(x, 0)dxdz4 ,

)eilx.t_

(A. 1.19)

where the integral is understood as an oscillatory integral. Picking a small conic decreases rapidly in r, we can neighbourhood r, around 0 in IR" such that decompose this integral into the sum I, + I2 , where in I, the integral with respect to is taken over F,, and in I. over IR"\ r, . Let us estimate 1, and I2 separately. a) In I, we integrate by parts with respect to x with the help of the exponent e-'*(x.e). We put

'L=

i i¢'x

j=,

axe

to obtain the identity 'Le-'m = e-i°. The coefficients of 'L and L are homogeneous in 6 of degree -1. Since 0,,(x0, 00) = o + 0 we have I Ox (x, 0) 1 + 0 if I x - x0 I < e and 0 E r, where r is a sufficiently small conic neighbourhood of 00. We have 1, = ! LN (e'x < q (x, 0)) u

-'O(x.e) dx d

e

.

4er,

Since I LN [eix C

<0> - N,

(p (x, 8)] I G CN

then in view of the decrease of

O E r,

IBI?1

,

in r, we obtain

Ii, I S CN<0)-N,

06r,

IOI= 1 .

(A.1.20)

b) To estimate I2 it is necessary to integrate by parts with respect to x again e;[x.4-#(x.e)J. Choosing a sufficiently small conic with the help of the exponent neighbourhood r of 00 we have - 0x (x, 0) * 0, 0 e r, e lR"\ r, , I x - x0 I < e, from which grads (x

- 0 (x, 0)) + 0,

for the same x, , 0 which allows us to carry out a standard integration by parts.

In fact it is obvious that for some C> 0 0 Er,

eIR"\r,,

Ix-x0I<e.

Wave Fronts and Propagation Singularities

237

Putting M

a

'L= -iI-0x(x,e)I-2 Y1=1

ax , 1

and hence

we obtain, 12 =

!

e;lx-t-e(x.e)] [L"(p(x,0)) u() dxdd.

(A.1.21)

satisfies I u Since u C (1 + I I)" for sufficiently large C, N1 , then for sufficiently large N the integral (A.1.21) becomes absolutely convergent and for 0 e f', Ix - xo I < e can be estimated by CP <0> - P for an arbitrary p. This together with (A.1.20) yields the required statement.

Remark A.1.2. If the point (xo, o) and the functions tp and 45 depend on a parameter and all the conditions are satisfied uniformly, then the constant C" in (A.1.18) can be selected so as to not depend on the parameter.

Proof of theorem A.1.1. 1) For simplicity consider first the operator

P

/_. Let Pu = f, (xo, o) t WF(f). The bicharacteristic of the symbol x. (xo) of which passes through (xo, o), is of the form (x;, i

is the n-th coordinate of xo, xo is the collection of its (n -1) first coordinates and

t is the parameter along the bicharacteristic. Let I be an interval on this bicharacteristic containing (xo, o) and not intersecting WF(f). We will show that either I c WF(u) or In WF(u) = 0. Let qp (x, 0) and 45 (x, 0) be determined as in Proposition A.1.5. Put

45,(x,0) = 0(x',x.-t,0).

cp,(x,0) = p(x',xp-t,0),

Then supp tp, is close to (x;, (xo) + t) and near this point the function 45, is defined. It is clear that by the choice of gyp, 45 one can make rp, and 45, to be any functions for which the conditions of Proposition A.1.5 are fulfilled at (x,, (xo)

+ t, o). We have (u, at

_ x^ U,

(,p,e-+m))

_ (u,
(_/_)(4',e.)) (p,e-",>

= R(t,0).

(A.1.22)

from which, by the condition 1 n WF(f) = 0 and by Proposition A.1.5, it follows that for the values of t of interest to us, the estimate I R (t, 0) 1 5 C" <0) -",

O e r, I B I ;1

,

(A.1.23)

238

Appendix I

holds uniformly in t. Here r is a sufficiently small conic neighbourhood of 0a. But from this J
j dt
(p,,e-gym,,) I

dt 5 I t, - t2 I C"<©) -".

(A.1.24)

decreases rapidly for 0 in F, Therefore, if for some t the function
and 0, so that i OT

[(Pe

'P (p e

+m

(A.1.25)

rapidly decreases for 0 in the cone r and uniformly in r, where for r = 0 the conditions of Proposition A.1.5 are satisfied for the point (x0, o) The condition of decreasing for (A.1.25) leads to the equation for 0, 00 ar

-P1(x,OX)=0,

(A.1.26)

which we can solve (for small r) for arbitrary 0 J, 0. In doing so, it is important to note that along the bicharacteristic (x(r), fi(r)) of the function p, (x, a) we will have (X (T), 00, T) = (T)

(A.1.27)

(cf. § 17), provided this is so for T = 0, which we will assume. For the homogeneous components of cp, we obtain transport equations which are again solved for arbitrary tp I:=o by analogy with the transport equations for q-; from

§20. The solution procedure (cf. §17) shows, that the support of any homogeneous component of (p propagates along the bicharacteristics of the function p, (x, ). Thanks to this the proof in this case for small r may be ended in a

a similar way to the case of the operator C'

ax

Let us now remark, that the necessity of restricting ourselves to small r is of no importance, since it is enough to prove theorem A. 1.1 for arbitrarily small parts of the bicharacteristics, since as an obvious consequence it will then be true also for large connected pieces. 3) Finally consider the case of an operator P of arbitrary degree m. Let Q be a classical elliptic properly supported `PDO of degree (I -m) with real principal

Wave Fronts and Propagation Singularities

239

symbol q (x, ). Put P, = PQ. Then P, e CL' (X) and the principal symbol of P, is of the form p Note that in Theorem A.1.1, in view of Corollary A. 1.4, it suffices to consider

only null bicharacteristics. But from the relations (P )4 = (Pm){ q + p,,, - q{ = (Pm)X q

for Pm = 0

(PI).=(pm)xq+p.,-qx=Wm)xq

forpm=0

the null bicharacteristics of p, and pm differ only by a change of parameter. Corollary A.1.5, being taken into account the result of theorem A.1.1 for P follows from the same result about P, already proved in 2). Problem A.M. Let A be a distribution determined by an oscillatory integral = f eim(x.e) a(x,O) (p(x) dxdO,

where 0 is a non-degenerate phase function, a (x, ) a Sm (X x JR') (cf. § 1). Show

that

WF(A)c

30EIR"\0, 450'(x,8)=0,

Problem A.1.2. Let two distributions u, , u2 c--9'(X) satisfy (A.1.13) allowing them to be multiplied. Show that WF(u, u2) c [WF(u,) + WF(u2)] U WF(u,) U WF(u2) .

Problem A.1.3. Prove that for the operator D., for any positive integer k, the same theorem on propagation of singularities is true as for D. What form does Theorem A.1.1 take?

Appendix 2 Quasiclassical Asymptotic Behaviour of Eigenvalues Observables in quantum mechanics can be represented by operators of the form (A(h) u)(x) =

(A.2.1)

Y,

where the parameter h > 0 is the Planck constant; the operator A(h) is well defined on S(lR") for example, if the function b(z) belongs to F (IR2n). Classical mechanics is the limiting case of quantum mechanics, when the Planck constant can be considered to be negligible. This motivates an interest in

the asymptotic properties of operators of the form (A.2.1) as h--0; the corresponding asymptotic analysis is called quasi-classical or semi-classical. A.2.1 Basic results

The change of variables -- h -' transforms (A.2.1) into (x) = hn f

e+cx-y

cIh b

rx 2 y ,

I u (y) dy d

(A.2.2)

where the symbol no longer contains the parameter h, which now is included in the exponent instead. We will say that b(z) is the Weyl h-symbol of A (h) or, briefly, the h-symbol (in this appendix, we will not use the v-symbols of chap. IV, which avoids any confusion). Clearly, the 1-symbol is then the ordinary Weyl symbol.

Between the h- and 1-symbols the following relation exists. Making the change of variables x-- Vh_X, y-. j y, in (A.2.2), this expression

lj

becomes (A(h)u)(x h+-) = f

eux_

b(v x2 y,

v

)

(A.2.3)

In the space of functions on IR" introduce the dilatation operator

Th: f(x) - h"" f(

h-x)

V

.

(A.2.4)

Quasiclassical Asymptotic Behaviour of Eigenvalues

241

It is easily seen that TI, is unitary on LZ(IR"). Using this operator, (A.2.3) can be written as ThA(h)u = A(*)Thu or

A(,,) = T, ' AiiiT ,

(A.2.5)

where A,i)) is the operator with the I-symbol b(')(z) = b( jz). Therefore the operator with h-symbol b(z) is unitarily equivalent to the operator with the 1-symbol b('t(z).

We will be interested in the quasicalssical asymptotic behaviour of the eigenvalues.

Definition A.2.1. Let AM be a self-adjoint operator semi-bounded from below. N. (.1.) denotes the number of eigenvalues of the operator not exceeding ,l (counting multiplicities). If there are points from the continuous spectrum of A(,,)

in the interval (- oo, ).], then by definition N,,(2) = + oo. Remark A.2.1. The Glazman variational principle (28.1) remains valid also for N,,(1); the proof (cf. §28) can be taken over with minor changes to the case N (A).

To formulate the basic result, we need the following Proposition A.2.1. Let A(,,) be an operator with the real h-symbol 0. Then for any fixed h > 0 the operator A(,,) is essentially self-adjoint.

b(z) E Hr' "'O, m°

Proof. For m° > 0 the proposition follows from Theorem 26.2. An analysis of the proof of Theorem 26.2 shows that the strict inequality m° > 0 is only needed in order to ensure A ± iI E HG'-r°. Under the assumptions of the proposition, for m° = 0 and h = 1 the inclusions A ± i I E HGQ 0 follow from the estimates

Ib(z)±iI>b(z), I01(b(z)±i) 16 C, Ib(z)I IzI-1111 = C7 I b(z)±iI IzI-1I11 Ib(z)I/Ib(z)±iI 5 C, Ib(z)±ii Iz1-1111.

For h * I one has to use (A.2.5) and the fact that b(') a Hf'o m. (as for b (z) but with other constants in the estimates of the derivatives). Let A(,,) have a real h-symbol b (z) a HFQ °. As in §30 changing the sign if necessary we may assume that b (z) z C > 0 for I z I z R°. Put

V(2) = (2n)" I dz. b(g)
The main goal of this appendix is the proof of the following theorem.

(A.2.6)

Appendix 2

242

Theorem A.M. Let A(,) have a real h-symbol b (z) e Hf °, b (z) z C > O for Q I z I z Rc . Let 1a be such that V (AO) < + co. Then, for almost all A < AO and arbitrary e > 0 we have the asymptotic formula

Nh(A) = h""(V(.)+O(h`"2-`))

(A.2.7)

Remark A.2.2. Between the asymptotic formulae in h ash -' 0 and the ones in d as A - + oo there is an intimate relation which can be explicitly exhibited when b (z) is homogeneous: b (tz) = tsb(z), t > 0, s > 0. In accordance with (A.2.5) the operator with the h-symbol b(z) is conjugate to the operator with symbol Y" (z) = hs'2b(z), so that Nh(A) = N(h-snA).

Remark A.2.3. Theorem A.2.1 is analogous to Theorem 30.1. In the latter we assumed the essential inequality b (z) ? C I z I M0, c > 0, me > 0. Theorem A.2.1

states the weaker dependence of the asymptotic behaviour in h on the behaviour of the symbol at infinity.

A.2.2 The idea of proof of Theorem A.2.1. The proof of the theorem is based on the same considerations as the proof of Theorem 30.1. We will construct an approximate spectral projection .A,, in the following way. Let X,2 be the indicator function of the interval (- co, A]; we construct a family of functions Xh. a , converging to XA as h -' 0. The operator has the h-symbol X,,., (b(z)), where b(z) is the h-symbol of A,h,. We will show that the family .0h has the following properties:

1°. A*=.1h; 2°. Ah is a trace class operator and

112-Ah111 =O(h 3°.

as

A,,(A(h)-Al) . h 5 Ch";

40. (1-Ah) (AM -AI) (I-.lh)> - Ch'; 5°. Sp Ah = h-" V(2) (1+0(h"));

here 0 < x < 1/2 and the function V(2) in 5°, is a positive, non-vanishing function, defined on the interval [A, A+s] and differentiable from the right at A. Note also that Im Ah c DAM , where D., is the domain of A,h,. In the presence of a family of operators, with properties P-5', theorem 28.1, reformulated in the new terminology is fundamental in obtaining the asymptotic formula (A.2.7). For convenience we formulate the following result.

Proposition A.2.2. Let A,h, be a family of essentially self-adjoint bounded from below operators; Ah a family of operators such that Im F. c D,, and having the properties 1 °-5°. Then

N,(A)=h`(V(A)+0(h")) as

h--0

Quasiclassical Asymptotic Behaviour of Eigenvalues

243

Proof. Similar to the proof of Theorem 28.1. Exercise A.2.1. Prove Proposition A.2.2. A.2.3 Symbols and operators with parameters

To study the approximate spectral projection, it is convenient to introduce class of symbols, depending on a parameter (cf. 29.1). Definition A.2.2. Denote by L the class of functions a(z, h), defined for z E IR2n, 0 < h <_ ho, infinitely differentiable in z and satisfying the estimate I aL a(z, h) J < C,

(z)'"-o1r1hu-"IYI.

(A.2.8)

Here m, y, e, aeiR, a>0, a<1/2. Obviously, if

j =l,

2,

then a, a2 E EQ a+'", "' + "_,

where

e-min (e1,e2), a=max(a,,a2) Note that for any fixed h, 0 < h < ho, from a E EQ o it follows that a(z, h) E I'Q (IR2') and (A.2.2) properly defines a class of operators A(h) depending on a parameter and acting on S(UV) (the parameter dependence appears either in the symbol or in the exponent). The corresponding class of operators will be denoted by S. From definition A.2.2 it follows, that for a > 0 the derivatives of the symbol

can be estimated by powers of h which are increasing as h-.0. However the influence of the increasing powers of h disappears under the action of the corresponding h-symbol of the operator. In particular, one has Proposition A.2.3. Let A (h) e S,0,:11, u > 0, a < 1/2. Then A (h) is bounded in L 2 (1R) u n i f o r m l y in h f o r 0 < h S ho.

Proof. Let a (z, h) be the h-symbol of the operator A (h) and A((,)) (h) the operator with I -symbol a(") (z, h) =a (Vh- z, h). By (A.2.5) the operators A (h) and A(i)(h) are unitarily equivalent, i.e. (A.2.9)

II A (h) II = II A((')) (h) II

Let us now show the boundedness of the operator Al (h). Selecting 0 < e' < min (e, I - 2a) we have

S C,hI*2< S

(A.2. 10)

Vh_Zyalrlh#-°IYI

CYh1h1(112-o-a'I2)+KW-o'IYI

In (A.2.10) we used the obvious inequality

(1+ l/ Izl) -

"< C(j+ j IZI) "= Ch-"12-"

for

x>0.

244

Appendix 2

Thus, ash) (z, h) e fQ (IR2") uniformly in h and by Theorem 24.3 the operator

A(h) is bounded in L' (§V) uniformly in h. Let us now introduce expressions for the trace of an operator in terms of its h-symbol. It follows from (A.2.5) that SPA(h) = SP(7 ' AiiiTh) = SpA(ii.

(A.2.11)

Using (27.2) we obtain Sp A(h) = ! blh) (x, ) dx dd = h -"f b (x,) dx dd .

(A.2.12)

With the help of similar arguments the estimates of the trace class norm (27.12) can be transferred to the case of h-symbols:

Proposition A.2.4. There exist constants C and N such that for the operator A (h) with the h-symbol b (z, h) the following estimate of the trace class norm holds 11 A (h) fl

Ch -" Y h1 12

10.1 b (z, h) I dz.

(A.2.13)

IYI$N

A.2.4 The h-anti-Wick symbols

For the operators (A.2.2), where the action depends on the parameter h, we have the following analogue of the anti-Wick symbols introduced in §24. The whole construction in section 24.1 can be carried out if, instead of 00 (x) we take the function To(x) = (7rh)-"'4e-x21(2h) = (Tip) 00) (x)

as the starting point. An operator A(") with an h-anti-Wick symbol a(x, ) is defined by analogy with (24.9): A,h) = j a (x, ) Qx. S dx A,

(A.2.14)

where Qx, 4 = T, ) Px.4 T,, are the projection operators, playing the role of Px,, in defmition (24.9).

All the results from §24 can with no effort be extended to the case of h-anti-Wick symbols. In particular, it follows from (A.2.14) that an operator is non-negative whenever its h-anti-Wick symbol is non-negative. It is easy to compute that the kernel of the projection operator Q0 on the vector 4'p is equal to (n h) - "12 e- cx2 +,2)"2h), whereas the h-Weyl symbol ao (x, )

of Q0 is equal to 2"e-(x'+f')ih

(A.2.15)

Quasiclassical Asymptotic Behaviour of Eigenvalues

245

Using (A.2.14) and (A.2.15) we obtain a formula connecting the h-Weyl symbol b(z) and the h-anti-Wick symbol a(z): iz_z,i,

b (z) = (nh) -" f a (z') e

"

dz'.

(A.2.16)

From (A.2.16), by the same reasoning as in the proof of Theorem 24.1 we can obtain the following analogue of Lemma 29.1.

Lemma A.M. Let B(h) be an operator with the h-anti-Wick symbol a(z, h) E EQ; and b(z, h) its symbol. Then

a-b=

h'Yl12 c7(a=a)+rN, o
where cY = 0 for odd I y I and rN E Eo o

12-a)N

A.2.5 The composition formula

For operators of the classes S. '-j" the composition formula holds in the following form. Theorem A.2.2. Let a; E EQ;ffl j = 1. 2; let A; (h) be the corresponding operators. Then A((h) o A2(h) E So"+1Z."i+P2,

where p = min(Qi, Q2), a = max(a(, a2), where for the h-symbol of the composition b(z, h) we have (_ 1)IBI /h 1a+01

6= IQ*al
a.(fl)

f-

2

(a{Dial) (af DXa2)+hNrN,

(A.2.17)

where (A.2.18)

Proof. We may obtain the proof just by copying the proof of Theorem 29.1. So that we do not repeat the calculations of Section 29.1 here, we will, wherever possible, refer to the proof of Theorem 29.1. Using (A.2.5), we obtain: AI(h)°A2(h)= Tip '(AI(h))(liTi,°Tip-'(A2(h))((i)) Th = Tti ' [(Ai (h))((i)) ° (A2(h))ii)] Tti

For the symbol b' (x, , h) of the composition of the operators (A; with symbols a,(") (z, h) = a,({ z, h) we have according to (29.4)-(29.18) (in which

Appendix 2

246

the nature of the dependence on the parameter is not used), the following representation b' (x, C, h) =

(

1'

la+$l
I

2_

la+PI (aaDpa(hl) (0 D" az )) + r'N .

(A.2.19)

Note that (afDXaihl) (afDXa(2)) = 01 +P1 [(a{Dxa() (aflDxa2)](h) 4

Therefore, passing from the Weyl 1-symbol in the equality (A.2.19) to the h-Weyl symbol according to (A.2.5), we obtain the terms in the sum (A.2.17). The estimate of the remainder (we are talking about the Weyl 1-symbol)

is reduced, as in section 29.1, to the uniform estimate in rl, r2 E [0, 1] of integrals of the form

fe'"Y""")[aaMaM(x+T,Y,C+7,h)] x [aDsa2(x+z2z,C+C,h)] dfdCdydz,

(A.2.20)

Ia+#I > N. By analogy with the above, we obtain that the Weyl I-symbol I (x, !;) corresponds to the Weyl h-symbol J(x,C) = hla+01 $e2i1(,'c-= a( [a;Dxa1(x+t1

x [a{Dxa2(x+i2

hY, + Vh q, h))

hC,h)]d1dCdydz,

(A.2.21)

Ia+flI ? N. It is necessary to prove that RN = h-NJ E

P. Q

uniformly in t, , r2 a [0,1 ]. Differentiating (A.2.21) with respect to x and C, we see that the derivative axa'RN is a linear combination of expressions of the form hia+pI

x

[aa+y afl+a a,(x+rl [as+y"a{+6"a2(x+z2

(A.2.22)

b'+S"=b. In the same way as in Section 29.1, if we integrate by parts we obtain

decreasing factors of the type -2M, -2M It is only necessary to take into account that in differentiating a, (a2) in y and 7 (z and C) there appears a

Quasiclassical Asymptotic Behaviour of Eigenvalues

247

factor h"2. Therefore (A.2.22) lead to a linear combination of integrals of the type hI&+pl -x+1x,+,yJi2 x [Ov,

,,, a,

(x+T, y hy, + Vin, h)]

x [a='c '-a2(x+T2 Vh-z,C+ Vh-C,h)]dndldydz,

(A.2.23)

where

v, =a+fl+y'+b',

Iv, I z N+ lye+b'I;

(A.2.24)

Iv21?N+ly"+b"I

V2=a+fl+y"+b",

The absolute value of (A.2.23) can be estimated by ha J<X+T, Y .'y,S+

V h-

X <X+T2 jZ, S+ I C, h>m,-Q,N-Q,IY +6'I x-2M-22`dodCdydz,

(A.2.25)

where

s= Ia+i6I-N+IXI+X2I/2+u,-or ,IvI+X,I +u2-a2Iv2+x2I If one takes into account the relations

a,,a2!5c<1/2, la+fll_N, y'+y"=y, b'+b"=b and (A.2.24), then the exponent s can be estimated as follows:

s=u,+u2-a1 Iy'+b'1-a2Iy"+b"I-N(a,+o2) +(N-Ia+fI) (a,+a2-1)+Ix,I (1/2-a,)+IX2I (1/2-a2) ul +u2-Or ly+b1-N(a,+a2). We see that the power of h in (A.2.25) corresponds to the statement of the theorem. Next, estimate the integral in (A.2.25) in terms of the necessary power of (x. 4). Note first that it splits into the product of integrals

I = J<x+T, vh-

"I <x+T2]rhz,

+

n> -21'drl dy,

(A.2.26)

+VhC>m,-Q,(N+IY'+6'1)
(A.2.26')

We will estimate the integral (A.2.26). Assume that m, - g, N < 0; if this is not the case, then in the expansion (A.2.17) we can take N' terms so that m) - p I N' < 0 and examine the remainder rN., representing the remainder rN,

Appendix 2

248

as rN plus the finite sum within the limits N < la+#I < N'. Using the obvious inequality <x+T,

l/Y, Vh-n)%,

x>0.

(A.2.27)

a special case of the inequality

<(1+IXI+I

0m(1+lyl+1171)1-1,

and applying (A.2.27) with x = 1 m 1- Q 1 N - Q, I y' + b' I , we obtain for the integral (A.2.26) the estimate I

I S <x, x>-,-Q,N-Q,Ir'+8'1 Jx-ZMd17dy <x,S>'",Jx-2Md17dy

>-,-Q,N-Q,li+s't

=Cl<x,

(A.2.28)

Combining (A.2.28) with the corresponding estimate for (A.2.26'), we obtain the

necessary power of <x, > in the estimate of (A.2.25) so that the inclusion (A.2.18) is proved.

F-I

A.2.6 Proof of Theorem A.2.1 The plan of the proof is as follows: first we construct an approximate spectral projection and successively verify the properties needed to apply Proposition A.2.2. 1. Let z (t, A, b) be the function introduced in Section 28.6:

za)-

1 fortS.i, 0 fort 2+26,

I (a/at)kX(t,A,6)1 S Ck8-k.

(A.2.29) (A.2.30)

Set x= 1/2-z, e>Oand e(z,h) = X(b(z),A,h").

(A.2.31)

(We omit the unimportant argument d of e (z, h)). The operator Ah is defined as the operator with h-symbol e (z, h). The function e (z, h) is infinitely differentiable with respect to z and e(z'h)_

1 for b(z)5A, 0 for b(z)>A+2h".

Let us estimate the z-derivatives of e (z, h). Differentiating (A.2.31) with respect to z gives:

Quasiclassical Asymptotic Behaviour of Eigenvalues

249

k

a a (z, h) =

t" (t, , h) I, = NO

c,...... ,, (a'' b (z)) ... (ay- b (z))

r.+...+r,-r

(A.2.32)

The summation in (A.2.32) runs over all possible decompositions of y into a sum yt +... +yk, where k 5 l y I. Taking (A.2.30) and b E H I'Q 0 into account, we obtain for an individual term in (A.2.32) the estimate

(arb) ... (5 .b)

a:k

r =b(z)

/

Ch-kxlblk

I<

k

x li lamb/bI 5

Ch-k"lblk(1+IzI)-Qlrl.

(A.2.33)

I-I

Note also that on the support of a (z, h)

IbI <,l+hx.

(A.2.34)

Therefore, from (A.2.32)-(A.2.34) we obtain the estimate S Ch-k"
I a= e(z, h) I

(A.2.35)

e(z,h)EEQ-X .

2. We need now to verify, that all the conditions of Proposition A.2.2 are fulfilled for .0"k..Fh is symmetric due to the real-valuedness of the symbol, and bounded by (A.2.35) and proposition A.2.3, hence .F,,* = .F,. The fact that .4rk is of trace class follows from Proposition A.2.4.

We denote the h-symbol of an arbitrary operator A by a (A). In order to estimate the trace class norm Il Fh2 - Fk 11 1 , we need to compute the h-symbol of the operator 3r2 1, - 'Fk .

By the composition formula (A.2.17)

a(.Fk2-.Fk)= -e(z,h)+e(z,h)2+

Y

c,,hl:+sl(Dza{e)

lal+lal
(A.2.36)

All terms on the right-hand side of (A.2.36) except rN are supported on {z: A < b(z) < 2+ 2h"}, so the trace class norms of the corresponding operators by Proposition A.2.4, can be estimated by

C [V (k+2hx) - V(2)] h-".

(A.2.37)

The trace class norm of the remainder can be estimated by o (h -" + ") for sufficiently large N.

Now, note that V(A) is a non-decreasing function; by the well-known Lebesgue theorem it is almost everywhere differentiable. In what follows, we

250

Appendix 2

assume that the A under consideration belongs to the set of full measure where V is differentiable. Then (A.2.37) is transformed into the desired estimate 2

II

-

i,ll< <= C[V'(A)h"+o(h)] h-"< Ch-"+".

(A.2.38)

In addition, by (A.2.12) we have

SpAh = (21rh)-" je(z,h)dz = h-"[V(A)+O(V(A)- V(A+2h"))] = h-" V (A) (1 + O (h")) .

(A.2.39)

Consequently, we have satisfied 1°, 2° and 5° of Proposition A.2.2 (.F,* = ,S=a, (A.2.38) and (A.2.39)). 3. Now let us verify 3° of Proposition A.2.2. For this we need the h-symbol of R,, (A(,,) So we begin by computing it. We have (-1Fi,(A01)-AI)) = e(z,h)(b(z)-A)

+

Y

h1 +alca,(D,ofe) (D°a4ab)+rN,

(A.2.40)

(S IQ+PI
rN N

E",-2NF.N(1-"). p."

The operator RN, corresponding to the symbol rN, is bounded for m - 2NQ < 0 and II RN 11 = 0 (hN(' -x)) = o (h'). Let us show that q'

((-2")1a+0l'

E

Ia:+#1>0.

(A.2.41)

for (p,B=h1a+11(ax.Ore)(axafb). For this we construct in a standard way a function (p (z, h) E C-(IR2"), for which 0:5 (p(z,h) 51, (p (z, h)

1,

for zesuppa,e, for for b(z)zA+3h",b(z)SA-h"

and which, like e (z, h) in the first part of this proof, is determined with the help of the smoothed characteristic function X (t, A, A+2h") of the interval (A, A+2h")

by the formula (p(z,h)=X(b(z),A,A+2h"). By analogy with (A.2.33) and (A.2.34) we verify that (P ezeol.

(A.2.42)

(p,a = h1°+a1(axa{e) [aPa4- ((p (b-A))].

(A.2.43)

Now note that for I x+ fl I> 0

Quasiclassical Asymptotic Behaviour of Eigenvalues

251

Let us show that rp (b - A) e EQ X . Obviously I tp (b - A) 1:5 Ch" and computing

the derivative with respect to z we obtain (A.2.44)

(a°(b-A))

(aYw) (b-.?) +

a=

j°1>0

Owing to (A.2.42) for the first term we have (3Y(p(z,h))(b(z)-A)I S Ch-'111

(A.2.45)

-elYlhx;

and for the other summands of (A.2.44) we get

Ibi' I(a°b)/bI Ch-xI

Chx-x'Yl-e'Y'.

(A.2.46)

The estimates (A.2.45) and (A.2.46) show that 4p (b - A) a E0; x , from which, taking (A.2.43) into account, (A.2.41) follows. Thus, the finite sum in (A.2.40) belongs to EQ x° ' - x and the operator ,, (A,,,, -A!) can be written in the form

.F,,(A,ti,-AI) = Q, + R,

(A.2.47)

where IIR11=O(h'-x)=o(hx) and a(Q,)=e - (b-A). Using (A.2.47) we have

.tee(Aa)-A!) -'F,,=Q,.F +R,,

(A.2.48)

IIR,II =o(hx).

We will compute the symbol of Q, .F,:

a(Q, tee) = e2 (b-A) +

Y

c°ah12+61(0x8{ (e (b-A)I)

0<12+PI
x (a'a{e) + rN .

(A.2.49)

Introducing as before the function cp and using e cp (b - A) cEQ

we see that

the norm of the operator corresponding to the finite sum and the remainder in

(A.2.49), can also be estimated via 0(h'-). Consider the principal part of the symbol - a (fib

X!) viz. the function q (z, h) = - e (z, h)2 (b(z) -A). For q and its derivatives with respect to z, the following estimates hold

(A.2.50)

q (z, h) ? - Chx;

2)+2(b-2)

101'41 = E 111> o

I

<

-e'Y1 JAI

+

Chxh-IYIx-e1Yl

1.1>0

< Chx" lYI)<

QI7I

(A.2.51)

Appendix 2

252

We shall denote by P the operator having Weyl h-symbol q(z, h) and by Q the one having h-anti-Wick symbol q(z, h). From (A.2.51) it follows that hlyl"2aygEEQ Xlyl."+('/2-x)lylI

and therefore, by Lemma A.2.1, a (P - Q) a EQ. 2e.1-' and thus II P- Q Il = 0 (h' It is furthermore obvious from (A.2.50) that Q z - Ch' and since P = Q + (P- Q) it follows that P >_ - Ch". Thus, we have for the principal part and consequently for the whole operator .F,,(At,,,-1.1).Fh the estimate (A.2.52)

_f7h (A(h) - AI) .Fh 5 Ch'.

4. Now we will verify that (I - -Fh) (Ach> - AI) (I- -,Fh) Z - Ch".

(A.2.53)

The symbol of the left hand side of (A.2.53) (after getting rid of parentheses) is

(.$rh(AM -.II).Fh) - a(. h(A(h)-21)) - a ((A(h) - Al) .Fh) + (b (z) - A).

(A.2.54)

In step 3 of this proof, it was shown that in the first two summands of (A.2.54) the principal terms are distinguished e2 (b - A) and e (2 - b), and the operators corresponding to the remainders are estimated in norm by 0 (h' - x). The third summand in (A.2.54) is analogous to the second. Thus we obtain a ((I - _11,h) (AM - 21) (I - `$h)) = (1- e) 2 (b -1.) + r ,

(A.2.55)

and the operator R with the h-symbol r admits the estimate

IIRII = O(h`-") Now consider the operator P with h-symbol

q(z,h) = (I -e(z,h))2 (b(z)-A). Arguments similar to the ones used in section 29.3 in proving the positivity of an

operator with positive symbol show that

P=Qk+Ak.

(A.2.56)

Here the operator Qk has the h-anti-Wick symbol q k (z, h) = q (z, h) +

Y_

251I
c7h111"2 aZ q (z, h) + rN (z, h)

,

(A.2.57)

Quasiclassical Asymptotic Behaviour of Eigenvalues

253

where rN E rQ - 2oN.(1 /2-x)N; and therefore the operator with the h-symbol rN can be estimated in norm by 0 (h and therefore For the operator Ak in (A.2.56) we have Ak E SQ x II Ak II = O (ht - ") with an apropriate choice of k. We keep the notation qk for the right-hand side of (A.2.57) but without the remainder rN and Qk denotes the corresponding operator with the h-anti-Wick symbol qk. Let rp (z, h) be the function introduced in the preceeding part of this proof. Decompose qk into two parts (A.2.58)

qk = 4Pgk+ (1-(p)qk.

In step 3 it was shown that tp (b - A) a EQ°, x from which it is obvious that rp (1 - e)2 (b - A) a EQ , . With calculations, analogous to (A.2.44)-(A.2.46) in step 3 it can be shown that (P .

h1Ylr2ayq(z,h)EEQ QIYI."+(112-")IYI

Consequently, the operator having the h-anti-Wick symbol 4)qk can beestimated in norm by O (h "). Finally we will show that for small h (1- (p) qk

(A.2.59)

0

and that therefore Qk ? - Ch". For this note that

(1-9)q

(1-rp)(b-A+ Y c,0112 01b

,

zeD5,

,

z t D,, ,

J)

251YI
0

(A.2.60)

where D,, = {z: b(z,h) z A+ 2h'). Relation (A.2.60) is obvious, since (1-a(z,h))

x(1-(p(z,h))=0 for

and 1-a(z,h)= I for zeDj,. Therefore in D,,

(1 -(i)gk= (1-(P) (b-A) (1 + Y c7,h1YU2(a1b)/(b-A)).

\

(A.2.61)

Y

Now note, that in D,

b(z)/(b(z) - A) = (1 - A/b(z))-1 <_ (1 -,k/(.k + from which it follows that Ih1r1i2("Yb)/(b-A)I <

Ch1Y1;2-"-QIYI.

Ch-";

2h"))-1

<--

254

Appendix 2

Consequently, the sum on the right hand side of (A.2.61) is estimated by Ch' -"= o(l), i.e. (1-(P) 9h = (I -(p) (b-2) (1 +o(1)) > 0.

Thus we have verified the requirements 1 °-5° of Proposition A.2.2 (the relations (A.2.38), (A.2.39), (A.2.52) and (A.2.53)) and we may apply the proposition. Hence Theorem A.2.1 is proved. O

A.2.7 The behaviour of Nh (A) for V (2) = + co

Theorem A.2.1 discusses the quasi-classical asymptotic behaviour for the eigenvalues 2 < ).° , where V (2°) < + oo. The membership b (z) a HI'Q ° does not rule out the existence of 2 such that V (2) = + co (for symbols of HI'Q mo > 0, this situation cannot occur). In this case, the following theorem serves as a supplement to Theorem A.2.1;

Theorem A.2.3. Let b(z) satisfy the conditions of Theorem A.2.1 and V (2) = + oo. Then, for any e0 > 0

lim h"Nh(2+e°) = +oo. h.0

Proof For each N we will construct a space H,r such that the inequality

((A(h)-21) , ) -<

6 e H,9 ,

(A.2.62)

holds and, for sufficiently small h

dim H > h'"N.

(A.2.63)

By the Glazman lemma, we obtain then from (A.2.62) and (A.2.63) the inequality N,,(A + ch") > h-"N which implies the result of the theorem. Introduce the set Q'= {z: b(z) S A). It is obvious from the definition of V(A), that V (A) = (2tr) -"mes Q , so that under the conditions of the theorem we have

mes0''=+0o. Now let Qt be a family of open sets with smooth boundaries, satisfying the following conditions (1) 0= are bounded, 0,c QA; (2) W, = as a

i.e. the distance between 0, and 00'

is not less than e.

Now construct a smoothed characteristic function of Q. (this is possible along the lines of the construction in 28.6).

Quasiclassical Asymptotic Behaviour of Eigenvalues

255

Let 2hx < e (as always 0 < x < 1/2) and *E(z, h) the characteristic function of the (hx)-thickening of the set . QE. Put Xa(z,h)=h-2"xJWs(Y,h)

Xo((y-z)h x)dy,

where Xo (v) a Co (I2"), x, Z 0, Xo(v)=O for I v I > i and J Xo (v) dv =1. It is obvious that

suppX,c W.

(A.2.64)

In addition, it is easily verified that 10= X, (z, h) I < Ch - xIYl, and from this estimate

and the compactness of the support of X, we have X, (z, h) E EQ, x . 0.

(A.2.65)

Let Fb be an approximate spectral projection as constructed in A.2.6. Denote by F,, the operator with the h-anti-Wick symbol e(z,h), by d' the operator with the h-symbol X. (z, h) and by Et, ,, the operator with the h-anti-Wick symbol L(z,h). The following relations hold between the operators . h, F,,, 8,.,,

and E,.,,: F,, - F,, =

RESQ.XQ.1 -2x,

IIRII S Ch' -2x;

(A.2.67)

F,, > E,.,, ; R'eSQ.;°.1 -2x,

8,. ti-E,.,, =

(A.2.66)

Ch'-2x.

IIR'II <

(A.2.68)

Here (A.2.66) and (A.2.68) follow from Lemma A.2.1, whereas (A.2.67) is obvious since X,< e due to (A.2.64). Now consider the operator e'L,,, . In the same way as in part 2 of the proof of Theorem A.2.1 looking at the h-symbol of the operator 02 - fL ,, we obtain the

II e2 - 01.,,II1 <

C(Wt.,,-W.)h-",

(A.2.69)

where W6,,, is the volume of the (2h)-thickening of the set 11, . But W, ,, - W, = 0 (h') since the open set 0, is bounded and has smooth boundary. Therefore SC2

Similarly we obtain

Spdit,b=h-"W,(1+O(hx)).

(A.2.71)

From (A.2.70) and (A.2.71), as in Lemmas 28.2 and 28.3, we may obtain an belonging to

asymptotic expression for the number 9 of eigenvalues of [1/2, 3/2]: N = h -"W, (1 + O (h x))

.

(A.2.72)

256

Appendix 2

The space spanned by the corresponding eigenvectors is denoted by .#°A . Now we will prove that HR = satisfies (A.2.62) and (A.2.63).

Let 7 e 0p, then for A,, we have the estimate (2Fi,7, 7) Z (1/2+O(h' -")) (7, n).

(A.2.73)

Indeed, using (A.2.66)-(A.2.68), we get (FhJl, 7) + (R7, 7) > (EE. kn, 7) + (R7, 7)

_ ('.h1,1)+((R+R')1,7) z (1/2+O(h'-"))(7.7) By the Cauchy-Schwarz inequality we have

(.e7,'1)

(9Fh1,-F,,7)(1,1)

from which, by (A.2.73), it follows that

'i,7)/ (7, 7) ?

(7,7) (I/2+O(h'

or [1/4+ 0(h' Now let eHR; then = W , , 7, where 7 E *' obtained in proving Theorem A.2.1,

(A.2.74) .

We recall the inequality,

.F,,(A(h)-11) F,:5 Ch"

(A.2.75)

(which is independent of the behaviour of V(1)). From (A.2.74) and (A.2.75) it follows that ((A(k) - Al)

((A(h) - Al) -Vi,7, Jlhn) 5 Ch" (7, 7)

(4+ 0(h'-x))

Ch' = 0(h)

Thus, on HR, we have A(,,) - (A+ 0(h")) 15 0.

In addition, from (A.2.74), it follows that F. is injective on .*'R, hence

dim HR = dimirrR = h-" W`(1+O(h)). The proof of the theorem is now completed since the volume WW may be

chosen as large as we like. 0

Appendix 3 Hilbert-Schmidt and Trace Class Operators A3.1 Hilbert-Schmidt operators and the Hilbert-Schmidt norm

Definition A3.1. Let H, and H2 be two Hilbert spaces. A bounded linear

operator K:

is

called a Hilbert-Schmidt operator if for some

orthonormal basis {e2} in H, we have (A.3.1)

Y_ IIKe,II2<+co. 0

The set of all Hilbert-Schmidt operators K: H, - H2 is denoted by S2 (H, , H2), or S2 (H) in case H, = H2 = H. The following proposition describes

the basic features of these operators. Proposition A.M. 1) The left hand side of (A.3.1) is independent of the choice

of orthonormal basis {e2} (the square root of the left-hand side is called the Hilbert-Schmidt norm of the operator K and denoted by IIKII2) 2) IIK'112=IIK112 3) IIKII S IIK112, where IIKII is the usual operator norm. 4) every operator K e S2 (HI, H2) is compact.

5) if K is a compact self-adjoint operator in the Hilbert space H, then IIKII2 =

where A,, 22,

J=1

(A.3.2)

A; ,

... are all the non-zero eigenvalues of K counting multiplicities.

6) If {e1} is an orthonormal basis in H,, then the scalar product (K, L)2 = > (Kea , Le.)

(A.3.3)

with K, L S2 (H1, H2) is independent of the choice of basis (e.) and defines on S2 (HI, H2) a Hilbert space structure with the corresponding norm II .112

7) If .S, (HJ ), j = 1, 2, is the algebra of all bounded operators in HJ, then S2 (HI, H2) is a left Se (H2)-module and a right 2' (H,)-module, moreover IIAKII2 < IIAII IIKII2,

Ae2'(H2),

IIKBII2 < IIKI1211BII,

Be 2'(H1),

KES2(HI,H2), KeS2(H,,H2).

(A.3.4)

(A.3.5)

258

Appendix 3

S2 (H) is in particular, a two-sided ideal in Y (H).

Proof. Let { fo) be an orthonormal basis in H2. Then IIKeaII2= a.B

a

I(Kea,fv)I2=

8.P

I(ea,K*fe)I=

IIK*f,112,

(A.3.6)

B

from which 1) and 2) follow. If now x = E xa ea a H1, then IIKeaII)2

2<(

IIKx112 =

Ixa! a

a

from which 3) follows. To prove 4) we approximate K e S. (H, , H2) by finitedimensional operators (operators of finite rank). Namely, let e, , e2 , ... be all the vectors of an orthonormal basis {e8}, such that Kea * 0 (it follows from (A.3.1) that this set is at most countable). Then clearly

Kx =

(A.3.7)

(x, ej) Ke,, j=1

N

putting

and

KN x = Y (x, ej) Kej

we

obtain

11 K - KN II2 5 II K - KN it 2

j=1 OC

_j=N+1 Y

as N-.oo as required.

Statement 5) follows from 1) if we choose a basis of eigenvectors of K. Now we will prove 6). Note first that i(Kea,Le.)I < IIKea11 11Le.II5

i (11Kea112+IILe.112),

from which the convergence of the series (A.3.3) follows for K, L E S2 (HI, H2). It is clear that (K, K)2 = II K II2 and hence (K, L)2 is independent of the choice of basis. Note that the algebraic properties of the scalar product are clearly satisfied by (A.3.3) and to proof 6) it only remains to demonstrate the completeness of S2 (H, , H2) with respect to 11. 112. For this it is most convenient to establish an

isomorphism between S2 (H1, H2) and 12 (M1 X M2), where Mj is a set of cardinality dimH,. The space 12(M) for an arbitrary set M consists of the functions on M different from zero in at most countably many points and such that i f(a) IZ < + oo. We may view this space as L2 (M) if M has the a-algebra aeM

generated by one-point sets with measure I for each one-point set.

Hilbert-Schmidt and Trace Class Operators

259

The required isomorphism between S2 (H, , H2) and 12 (M, x M2) is established by associating with each K ES2 (H, , H2) its matrix K.,= (Keg , f ), where {eg}, {f} are orthonormal bases in the spaces H, and H2. The fact that this association is an isometric isomorphism is clear from the calculation (A.3.6). Now we will prove 7). Since 11A Keg II S II A 11 11 Keg 11, the estimate (A.3.4) and

therefore the fact that S2 (H, , H2) is a left £ (H2)-module, are clear from the definitions. To prove (A.3.5), guaranteeing the possibility of introducing on S2 (HI, H2) a right 9 (H,)-module structure pass to the adjoint operator: IIKBII2= II(KB)*112= IIB*K*112 < IIB*II IIK*112= IIBII IIKII2,

as required. U Now let X and Y be two spaces with positive measures and H, = L2(Y), H2 = L2(X). In this situation, the operators KES2 (HI, H2) are described as follows

Proposition A.3.2. The operators KES2 (HI, H2) are exactly those which can be represented as

(A.3.8r

(Kf)(x) = f K(x,Y) f(Y)dy Y

with a kernel K (x, y) E L2 (X x Y). We then also have (A.3.9)

I J K 112 = j f I K(x, y)12 dx dy XXr

(in these formulas dx and dy denote the measures on X and Y respectively).

Proof. Let {eg(y)} and (f,I (x)) be orthonormal bases in L2(Y) and L2(X), and K E S2 (H, , H2). Note that {fa (x) eg (y) } constitutes a complete orthonormal

basis in L2 (X x Y) and if we put

K(x,Y) = > (Keg,f) fe(x) eg(Y),

(A.3.10)

g. ft

then K(x,y) E L2 (X x Y) and the operator of the form (A.3.8) coincides with K, since these operators both have the same matrices in the bases {eg} in H, and {f} in H=. The Parseval identity guarantees (A.3.9). Conversely, if K (x, y) e L2(X x Y), then decomposing K (x, y) in the basis {4(x) eg(y)}, we obtain

K(x,y) _

cae f (x) eg(Y) , a. 9

E I cga 12 < + ac

.

a. 0

But from this it is obvious that in the base {eg} and {f}, the matrix of K is cae=(Keg, f) which implies that KeS2(H,,H2). O

260

Appendix 3

A.3.2 Trace lass operators and the trace Definition A.3.2. Let H be a Hilbert space and S2 (H) the ideal of HilbertSchmidt operators on H. Let S, (H) = (S2 (H))2 be the two-sided ideal in .V (H), the square of S2 (H), consisting of operators which can be written as finite sums

A = >B;CJ, Bt, CieS2(H)

(A.3.11)

The ideal S, (H) is called the trace class and the elements of S, (H) are called trace class operators on H.

Proposition A3.3. 1) Let A eS, (H) and {e,,} an orthonormal basis in H. Then (Aea, ea) I < + oo

(A.3.12)

and

(Ae., e,)

SpA

(A.3.13)

is independent of the choice of orthonormal basis {e1}. This expression is called the

trace (Spur in German) of the operator A. The trace is a linear functional on S, (H) with Sp A z 0 for A ? 0. We may rewrite the scalar product (K, L)2 using the trace, as

(K, L)2 = Sp(L*K),

K, LeS2(H)

(A.3.14)

2) If A is a compact self-adjoint operator with non-zero eigenvalues A, A2, .. . (counting multiplicity), then A eS, (H) if and only if OC

IA I < +ao.

(A.3.15)

and

SpA= Y,it.

(A.3.16)

i=1

3) If A e S, (H), then

SpA*=SpA

(A.3.17)

4) If A e S, (H) and B e.0 (H), then Sp (AB) = Sp (BA)

Proof. 1) if A is expressed in the form (A.3.11) then

Y(Bj i

Y(CCe., B* e,), i

(A.3.18)

Hilbert-Schmidt and Trace Class Operators

261

which implies (A.3.12) and

Y(Aea,ea)=Y- (C;,B!)2,

(A.3.19)

j

a

from which it is obvious, in particular, that Sp A is independent of the choice of basis. (A.3.14) is obvious.

2) Now let A = A. If A e S, (H), condition (A.3.15) and (A.3.16) holds, because we may take for {ea} a basis of eigenvectors. Conversely, if (A.3.15) holds and if {ea} is a basis of eigenvectors, Aea = Aaea, then defining B and C by the formulas Be. = VIA. I ea ,

Cea = Aal VIA. I ea ,

we see that B, C C- S2 (H) and A = BC so that A E S, (H).

3) Let us verify (A.3.17). Writing A in the form (A.3.11), we have A* = E C,t Bj and from (A.3.19)

SPA"

(Cj, B*)2 = Sp A,

(B', C1)2 = 1

I

which proves (A.3.17). 4) Finally we will verify (A.3.18). First let B be unitary. Then AB and BA are unitarily equivalent since AB= B-1(BA) B. Hence (A.3.18) for B unitary is a consequence of the independence of the trace on the choice of basis. To prove

(A.3.18) in general, note that both parts of (A.3.18) are linear in B and the following statement holds Lemma A.3.1. An arbitrary operator B eY (H) can be expressed as a linear combination of four unitary operators.

Proof. Since we may write

B=B1+iB2,

B, =B1=B2B+,

B2 =B2=B

it suffices to verify that a self-adjoint operator may be expressed as a linear combination of two unitary ones. We may assume that IIBII 51. But then the desired expression takes the form

B=2[B+i 1-B2]+2[B-1 T

2].

Therefore Lemma A.3.1 and hence Proposition A.3.3 are proved. O

Appendix 3

262

A.3.3 The polar decomposition of an operator Let H, and H2 be Hilbert spaces. Recall that a bounded operator U: H, -.H2 is called a partial isometry if it maps isometrically (Ker U)1 onto Im U. It follows

that U*U = E,

UU* = F

(A.3.20)

where E is the orthogonal projection onto (Ker U)1 in H, and Fis the orthogonal projection onto Im U in H2 (in this case Im U is a closed subspace of H2). If Ker U = 0 and Im U = H2, then U is a unitary operator.

Definition A.3.3. The polar decomposition of a bounded operator A: is the representation of A in the form A = US

(A.3.21)

where S is bounded self-adjoint and non-negative on H, and U: H, -+H2 is a partial isometry such that

Ker U = Ker S = (Im S)'

(A.3.22)

Proposition A.3.4. The polar decomposition of a bounded operator A: H, -H2 exists and is unique. Sketch of the Proof. From (A.3.21) we have A* = SU*, from which A*A = SU*US = SES. But ES = S by (A.3.22) so that A*A = S2

(A.3.23)

and hence

S=

A*A

(A.3.24)

A*A is defined by means of the spectral decomposition theorem). In fact let C= A*A and let B be any bounded selfadjoint operator in H, such that B2 = C and B S 0. We will prove that B = S, where S is given by (A.3.24). Being a function of C, S commutes with every operator commuting with C. In particular, BS = SB because BC = CB = B 3. Hence (

(S-B) S (S- B) + (S- B) B (S- B) = (S'-B') (S- B) = 0. Both terms on the left-hand side are non-negative operators so both vanish.

Hence so does their difference (S- B)3 and therefore (S- B)° = 0. This obviously implies S - B = 0 because S - B is selfadjoint. For the details concerning spectral and polar decompositions the reader may consult e. g. F. Riesz, B.Sz.-Nagy [1]. Further formula (A.3.21) defines U

Hilbert-Schmidt and Trace Class Operators

263

uniquely in view of (A.3.22) thus proving the uniqueness of the polar decomposition. To show the existence construct S by the formula (A.3.24) and define U by

Ux = 0

U(Sx) = Ax

for xl Im S

for xEH,

(A.3.25) (A.3.26)

To verify the correctness of this definition, it suffices to show that if Sx = 0 then Ax = 0. But this follows immediately from (A.3.23) together with II Sx II

2

= (Sx, Sx) = (S2 x, x) = (A *Ax, x) = (Ax, Ax) = II Ax 112 ,

which shows that U is a partial isometry. Li Definition A.3.4. If A = US is the polar decomposition of A, we will write

S= IAI. Proposition A.3.5. Let J be an arbitrary left ideal in the algebra . (H). Then A e J if and only if I A I E J.

Proof. This is clear since A = U I A 1, U*A = I A 1. `J

Corollary A.M. We have A e S2 (H)

J A I E S2 (H) ,

By using the polar decomposition, we may, as a complement to 4) of Proposition A.3.3 prove

Proposition A.M. If A, Be S2 (H), then Sp (AB) = Sp (BA).

Proof. Using the identities

4AB' _ (A+B)(A+B)*- (A-B)(A-B)* +i(A+iB) (A+iB)'- i(A - iB) (A - iB)*,

4B*A = (A+B)*(A+B) - (A-B)*(A-B) + i (A+iB)* (A+iB) - i(A-iB)* (A-iB), we see that it suffices to verify that Sp (AA*) = Sp (A *A),

A e S2 (H)

(A.3.27)

However, using the polar decomposition A = US, we see that S e S2 (H) and hence S2 6S, (H) and since

A*A = S2,

AA* = US2U*,

Appendix 3

264

then in view of part 4) of Proposition A.3.3

Sp(AA*) = Sp(US2U*) = Sp(U*USZ) = SpS2 = Sp(A*A), as required.

A.3.4 The trace class norm

Definition A.M. The trace class norm of the operator A E S, (H) is the expression (A.3.28)

IIAII, = Sp l A 1.

Proposition A.3.7. 1) The following inequalities hold IIA112

IAII,,

IIBAII, s IIBII IIAII,, IIABII, s IAII, IIBII , ISpAI < IAII1,

AeS,(H); AeS,(H), AeS,(H), AeS,(H),

(A.3.29)

Be2'(H); BE2'(H);

(A.3.30)

(A.3.30') (A.3.31)

as well as the relations

IIA*II, = IIAII, IIAII ( = sup I Sp (BA) I

,

AeS,(H);

(A.3.32)

A cS, (H).

(A.3.33)

BIY(H) IIBII 52

2) The trace class norm defines a Banach space structure on S, (H).

Proof. 1) a) Let us prove (A.3.29). Suppose A = US is the polar decomposition of A. Then IIA II, = IISII1 = SpS,

(A.3.34)

1 1A1 1i = Sp(A*A) = SpSZ,

(A.3.35)

so that (A.3.29) is equivalent to the inequality

SpSZ < (SpS)2

(A.3.36)

which becames evident if we express SpSZ and SpS in terms of an eigenbasis of S. b) To prove (A.3.32) note that A* = SU*, AA* = US' U* and l A* l = USU*, hence

SpIA*l = Sp(USU*) = Sp(U*US) = SpS= SplA1.

Hilbert-Schmidt and Trace Class Operators

265

c) Now we will prove (A.3.31). Suppose {ea} is an orthonormal basis of eigenvectors of S with eigenvalues sa , Se, = sae,

(A.3.37)

.

We have

SpA=>(USea,ea)=Esa(Uea,ea), and since I (Uea, ea) I <_ 1 , then clearly I

(A.3.38) All

a

d) To prove (A.3.30) let BA = VT be the polar decomposition of BA. Then 11 BA 111 = SpT = Sp (V*BA) = Sp(V*BUS).

Now note that 11 V* BUII 5 II B II. The remaining argument is as in c). e) The estimate (A.3.30') follows from the estimate (A.3.30), since IIABII1 = II(AB)* = IIB`A`II1

f) The relation (A.3.33) now readily follows from the estimate ISp(BA)I < II BA III 5 IIBII IIAIII,

where we have equality for B = U*. 2) a) We will prove that the trace class norm 11 ' II 1 has the usual properties of a norm:

A', A"ES,(H); AeS,(H), AeC;

IIA'+A"1115 IIA'll, + IIA"II1, IIaAIJ1= JAI IIAIII,

IIAIII=0«A=0. Here only the first relation is non-trivial. To prove it we use (A.3.33) IIA'+A"111 = sup ISpB(A'+A")I = sup I Sp (BA') + Sp (BA") I Roost

Roost

sup (I Sp(BA') I + I Sp(BA") 1) IBIS1

sup (I Sp (B'A') I + I Sp(B"A") I) = sup ISp(B'A')I 18'151 181151

IB'1<1

+ sup ISp(B"A")I= IIAIII+IIA"III 1B'1s1

b) We want to verify the completeness of S, (H) in the norm II - III . Let n = 1, 2, ... , A a S, (H) and II Ap - A", II 1 -+ 0 as n, m-+ + oo. Then by (A.3.29) and part 6) of Proposition A.3.1 there exists an operator A eS2 (H) such that

Appendix 3

266

lim II A. -A 112=0. To verify that A e S, (H), let A= US be the polar decom-

position of A and put C. = USA,,. Obviously lim

IICR-S112 = 0,

(A.3.39)

lira IICR-C,,,II, = 0.

(A.3.40)

R.+cc

Let {ea} be an orthonormal eigenbasis of S with eigenvalues s,. From (A.3.39) it follows that

s, = (Se ea) = lim

es) .

(A.3.41)

To prove that A e SI (H) it suffices to verify that Y sQ < + oc. This, in turn, follows from (A.3.41) and the estimate ' sup

I

ea) I < +D,

(A.3.42)

which holds in view of the inequality I(Ce8,ea)16IICII

CeS,(H),

(A.3.43)

which is easily derived from (A.3.33). It remains to prove that

limIIA.-AII,=0.

R. W

Let c > 0 and select N so large that IIA,R-ARII, <E,

m,n>N.

(A.3.44)

n>N.

(A.3.45)

Let us prove that

IIAR-AII,:! E

From (A.3.43) it follows that lim Sp (BAR) = Sp (BA) for any B e SP (H). R» Co

But by (A.3.44), this gives

ISp(BAR)-Sp(BA)I 5E

for n? N,

if 11B11 :5 1. It only remains to use (A.3.33) to arrive at the desired (A.3.45). U

Hilbert-Schmidt and Trace Class Operators

267

A.3.5 Expressing the trace in terms of the operator kernel To express the trace in terms of the operator kernel is very simple from the formal point of view although difficult technically (it is difficult to justify an easily discovered formal formula). In this section, we introduce a formal scheme which should be justified in detail in each concrete case. Let X be a space with measure dx and K a trace class operator on L2 (X) with

kernel K(x,y). We would like to write K in the form K = L, L2,

L. , L2 E S2 (L2 (X))

(A.3.46)

(this can for instance be done as follows: let K = US be the polar decomposition, and take L, = U VS, L2 = j/S). Denote by L, (x, y) and L2 (x, y) the kernels of

L, and L2. Formally, we have K(x, y) = f L, (x, z) L2 (z, y) dz.

(A.3.47)

Since we may also write K = (L±); L2 , where L' has kernel Li (x, y) = L, (y, x), then by Propositions A.3.3 (formula (A.3.14)) and A.3.2 justifying

(Ki,K2)2=fK,(x,y)K2(x,y)dxdv

(A.3.48)

(where K e S2 (L2 (X)), K; (x, y) is the kernel of K,) we have Sp K = (L2 , L; )2 = J L1 (y, x) L2 (x, y) dx dy = IL, (x, z) L2 (z, x) dz dx , (A.3.49)

or Sp K = f K (x, x) dx,

(A.3.50)

Actually, a justification of the formal calculations (A.3.47)-(A.3.50) is possible, for instance, when X is a compact manifold with boundary, dx is a measure on X determined by a positive smooth density and the kernel K(x, y) is continuous. An example of these kind of arguments based on the Mercer's theorem is carried out in §13 in proving Theorem 13.2. Note the basic difficulty in the justification: the kernel K (x, y) is only defined up to a set of measure zero in X x X, but in (A.3.50) there enters the restriction of

K(x, y) on the diagonal in X x X - which is a set of measure 0. Another version of this kind of argument is to try and obtain the integral

(A.3.50) as a limit of integrals on a small neighbourhood of the diagonal appropriately normalized (cf. Gohberg and Krein [1 ], chapter III, § 10). We omit

the details, since this is not used in the main text of this book. Exercise A.3.1. Show that IIABII, < IIAIII IIBII2,

A, BeS2(H).

268

Appendix 3

Exercise A.3.2. Show that

IIBII = sup ISp(AB)I. A*S1(H)

IIA631

Exercise A.3.3. Let J be a two-sided ideal in .1'(H) and assume 0:5 A S B with B E J. Show that A E J. Hint : Show that A 112 = CB"2, where C e 2 (H) and II C II < 1

A Short Guide to the Literature Here we only mention the works most closely connected with the material

covered in the book. We make no claims whatsoever on bibliographical completeness. I have tried as far as possible to avoid referring to short communications, so that mostly monographs and detailed papers or survey articles are referred to.

Chapter I The concept of a pseudodifferential operator (WDO) originated in the theory of multidimensional singular integral operators (cf. Michlin [1] and the references therein). Subsequently, singular integro-differential operators emerged (cf. Agranovich [1] and references therein). The term "pseudodifferential operator" was coined by Friedrichs and Lax [1]. In the present form `YDO appeared basically in the work by Kohn and Nirenberg [I] (where the `PDO which we call "classical" were considered). The symbol classes S,".. and the

corresponding operator classes were introduced by H6rmander [2]. The theorem on the invariance of the class of `YDO under changes of variables is also due to him (cf. H6rmander [1 ], [2]). The proof of this theorem given here is based

on ideas of Kuranishi. The Fourier integral operators (FIO) were introduced and systematically studied in H6rmander [6]. The closely related concept of a canonical operator had been studied earlier by Maslov in connection with various asymptotic problems (cf. Maslov [1 ], [2], Maslov and Fedoryuk [1 ], and Duistermaat [3]). Hormander's work was also preceeded by the works of Eskin [1 ], [2J, Egorov [11,

[2], containing the ideas developed by Hormander. In the works by Nirenberg and Treves [ 1 ] and Egorov [ 1-4] PDO and the simplest FIO were used to study local solvability and regularity of solutions for general operators of principal type. In the exposition of the theory of oscillatory integrals, FIO, and in the

construction of algebras of `PDO,

I

basically follow Hormander [6].

Hypoellipticity in § 5, is presented in the spirit of H6rmander [2]. In the same work there is given described here in detail a sketch of the theory of Sobolev

spaces. More complete information on Sobolev spaces can be found in H6rmander [7], Nikolskii [I), Sobolev [I], Besov, Il'in and Nikolskii [I], Lions and Magenes [I]. An elementary presentation of the basic facts in the theory of Y'DO can be found in the book by Wells [I].

270

A Short Guide to the Literature

Note that what we call FIO are usually called local FIO. To avoid an excessive increase of the volume of the book, I deliberately left out the considerably more subtle and complex theory of global FIO. Global FIO or their equivalent, the canonical Maslov operator, are however necessary in a series of applications (for instance in the theory of hyperbolic equations or in spectral theory). Therefore, after the reader has acquired familiarity with local FI O and their applications, he has to learn about global FIO (this might be done

for instance from the papers by Hormander [6] and Duistermaat and Hormander [I] or from the lectures of Duistermaat [fl) and Maslov canonical operator theory (for instance, from the book by Maslov and Fedoryuk [I ]). Concerning other questions of the theory of'PDO and FIO, we refer the reader to the monographs by Friedrichs [ 1 ], Eskin [3], Taylor [ I ], Duistermaat [ 1 ], Treves [ 1 ], Grushin [2] and the papers by Agranovich [ 1 ], Kumano-go [ 1 ], [2], Beals and Fefferman [ I ], Beals [ 1 ], [2], [3], and Calderon [ 1 ]. Let us mention the important results found by Calderon and Vaillancourt [1 ]

making more precise the boundedness theorem (for example they considered operators of the class L°,d with 0:!9 S < 1) (cf. also Watanabe [I] and Kumano-go [3]). As for the action of'PDO on LP-spaces, see Muramatu [1] and Illner [ 1 ], for the action on Holder classes, see Durand [ 1 J, and for the action on Gevrey classes and classes of analytic functions see Volevii; [1], Du-Chateau and Treves [ 1 ], and Baouendi and Goulaouic [ 1 ]. Operators with complex phase-function were considered by Kucerenko [I

Miscenko, Sternin and Shatalov [1], and by Melin and Sjostrand [1]. Research on the index problem, for which'PDO provided an essential tool, excerted a strong influence on the development of the theory 'PDO: cf. Atiyah and Singer [1 ], Fedosov [1 ], Atiyah, Bott and Patodi [1 ], Hormander [5], Atiyah [1 ], [2]. The technique of'PDO was used in the work by Atiyah and Bott [I ] on the Lefschetz theorem. Important concrete applications of'PDO to classical problems in the theory of partial differential equations can be found in Oleinik and Radkievi6 [11, and

Maz'ya and Paneyah [1]. Chapter II The description of the structure of complex powers of elliptic operators in terms of'PDO and the theorem on meromorphic continuation of the kernel of complex powers and of the C-function are due to Seeley [I ], [2], [3]. Variations and generalizations of this theory can be found in Nagase and Shinkai [11, Hayakawa and Kumano-go [1 ], Kumano-go and Tsutsumi [11, Smagin [11, [2]. The Tauberian technique was first exploited in the classical work of Carleman

[1]. A survey of several variants of the Tauberian technique is given in Hormander [4]. The proof of the Ikehara Tauberian theorem given here, is close

to the one given in the book by Lang [I]. Let us mention an important application of the results of Seeley, namely, the

already mentioned work of Atiyah, Bott and Patodi [11, where a new

A Short Guide to the Literature

271

presentation of index theory is given. For a self-adjoint, non-semibounded operator A, Atiyah, Patodi and Singer [1] studied the function n4 (z) = Y (sign k,) I A; I=.

We mention also the papers by Ray and Singer [1], [21 on analytic torsion close to this circle of ideas. The papers by Seeley [2], [3] also contain a study of the C-function for an operator corresponding to an elliptic boundary problem. We did not touch upon the spectral theory of boundary problems, for this the reader is referred to the monograph of Berezanskij [i ]. The meromorphic continuation of the C-function is intimately related with the asymptotic behaviour of the trace of the resolvent (as A -# oo) and with the asymptotic behaviour of the 0-function 0(t)=Ye-x,'

j

as t- +0

(cf. for instance Duistermaat and Guillemin [1 ]). These questions are considered

in the papers by Fujiwara [1] and Greiner [1]. Pseudodifferential systems, elliptic in the sense of Douglis and Nirenberg are

studied from this view point by Koievnikov [I]. The asymptotic behaviour of

N(A) as 2-. + oo (without an estimate of the remainder) for hypoelliptic operators on a compact manifold with boundary was obtained by Moscatelli and Thomson [1). A discussion of the theory of pseudodifferential boundary problems with parameters can be found in Agranovich [2]. As we have noted in the main text, the theorem on the continuation of the C-function does not allow us to obtain any substantial information concerning the eigenvalues of non-self-adjoint operators. A survey of different questions of the spectral theory of non-selfadjoint differential and pseudodifferential can be found in Agranovich [3]. Chapter III Theorem 16.1., which is due to Hormander, is proved in [4]. Chapter III is an extensive presentation of this work, supplemented with an exposition of all the indispensable auxiliary facts. The work by Hormander [4] also contains some results bearing on the case of a non-compact manifold and some results on Riesz' means (concerning this, see also Hormander [31).

The description of the structure of the operator exp(-itA) as an FIO, is essentially based on ideas from geometric optics (concerning this, see the book by Babic and Buldyrev [11). These ideas were exploited by Lax [1] to construct asymptotic solutions and a parametrix of hyperbolic systems. Hormander [4] developed the method of Lax and arrived at the indicated description of the

structure of the operator exp(-itA), which is essentially equivalent to a

272

A Short Guide to the Literature

description of the singularities ofthe fundamental solution of a pseudodifferen-

tial hyperbolic equation. Note also that for large t, the operator exp(-itA) already is a global FIO (cf. the literature guide to Chapter 1.). The possibility of

representing the exponential operator exp(-tA) and its kernel in terms of Feynman type integrals is studied in a series of works (cf. e.g. Maslov and Shishmarev [11).

Notice, that the method of obtaining the asymptotics behaviour of spectral function by considering a hyperbolic equation, was first employed by Levitan [I]. Let us also mention Levitan [2], which contains important supplements to the work of Hormander [4]. Further results on the asymptotic behaviour of eigenvalues, which connect this question with the geometry of the bicharacteristics, can be found in the papers of Colin de Verdiere [1 ], Chazarain [I ], Duistermaat [2], Weinstein [1 ], Shnirelman [1]. Note in particular the work by Duistermaat and Guillemin [1], where under certain assumptions, the second in the asymptotics of N (A) as A -+ oo is obtained for an selfadjoint elliptic operator on a closed manifold. Using FIO Rozenblyum [1] got very sharp results on asymptotic behaviour of eigenvalues for operators on a circle.

Concerning the geometry of the spectrum cf. also the book by Berger, Gauduchon and Mazet [1], an article by Molcanov [1] and interesting papers by Gilkey [1], [2], [3]. We did not touch on questions connected with the spectral asymptotics of non-smooth or degenerate operators and boundary problems. Regarding this, we refer the reader to the lectures of Birman and Solomyak [1 ] and their survey [2], where an extensive bibliography on spectral asymptotics can also be found. A survey of a number of important results concerning eigenfunction expansions can be found in the article by Alimov, 11' in and Nikiiin [1]. Chapter IV

Essentially the theory of'PDO in IR" emerged long ago in connection with mathematical questions of quantum mechanics. (cf. e.g. Berezin [1 ], Berezin and

Shubin [1], [2]). Several versions of this theory can be found in the works of Rabinovic [I ], Kumano-go [1 ], [2], Gruiin [i ], Shubin [1 ], [5), Beals [2), Feigin [2].

In Beals [3] and Shubin [5], there is discussed the structure of operators, which

are functions of PDO in IR" with uniform (in x) estimates of the symbols (such as in Kumano-go [I ]).

Various questions, related to the Fredholm properties of 'I'DO on noncompact manifolds, are considered by Cordes and McOwen [1]. Recently a number of papers have appeared devoted to'FDO on nilpotent Lie groups (in particular on the Heisenberg group). Cf. e.g. Rothschild and Stein [1]. The construction given here of the algebraof''DOin 1R" is close to the one in Shubin [I]. The concept of the anti-Wick symbol was introduced by Berezin [2] and is a variation of Friedrich's construction [i ] (see also Kumano-go [1 ], [2]).

A Short Guide to the Literature

273

Concerning applications of the Wick and anti-Wick symbol and also more general symbols, see the papers by Berezin [2], [3], Berezin and Shubin [I), and Shubin [2], [3]. With the help of inequalities for Sp exp(-i t A), the asymptotic

behaviour of the eigenvalues is obtained in the work Berezin [2] (without remainder estimate). The results of §25 and §26 are essentially contained in the

work of Shubin [I ] (see also [4]) - The method of approximate spectral projection and all the results in §28-30 are contained in Tulovskij and Shubin [1]. A significant modification of this method was offered by Feigin [1], [2]. We mention that the method ofapproximate spectral projection is essentially a variational method. Variational methods began with the classical work of H. Weyl [I ], [2]. To find the asymptotic behaviour of eigenvalues for operators in IR", one can also apply the Tauberian method, cf. the work by Kostyucenko (I]. A survey of all results on the spectral asymptotics for operators in 1R" can be found in the already cited work by Birman and Solomyak [2]. Appendix I

On the basis of the earlier concept of singular support of a hyperfunction, due to Sato [I], Hormander [6] introduced the concept of a wave-front for a distribution. The theorem on propagation of singularities in the form given here is due to Duistermaat and Hormander [11 (see also Hormander [8]). The proof given here is due to Tulovski. Another presentation can be found in the lectures by Nirenberg [i ]. In a number of later works more subtle questions connected to the propagation of wave fronts have been considered (see e.g. the work by Ivrii (1] and references therein). Appendix 2

In this appendix the results of Roitburd [1] are presented. A closely related result but without an estimate of the remainder term was obtained by Berezin [2]. Other information on quasi-classical asymptotic formulae and references to the literature, can be found in the monograph by Maslov and Fedoryuk [I].

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Index of Notation a. (x, ; A) ack)(x,

1'(X) 1'(M)

95

) 97

5

37

AZ

88

e(x,y,t)

AZ

90

e(z,A)

212

F'(M) F'(X)

12

A.(x,y)

102

b_m_ j (x, l; , A.)

98

fj

98

B_m_j(A)

A)

b--m-!(x,

99

GQ (IR"), GQ

99

B(z), B(N) bmZZ-O

i(x.)

208, 211

65

.7 242, 248

95

99

B,(Z_,

37

Fred(E), E2)

98

B(N) (x)

128

96

B(z) 1B(z) ,b(z) mz-J' mz-/(x' (N),

4)

101

G-°°

179

Gm

216

178

HGm-mo(IR"), HG;-m0 HL-' °(X) 38

bi(x,4)

180 177 C(E), E2) 68

HL---'(X, A) 79

Cn°(IR")

HSQamO(X x IR")

CLm(X) 29 Coker A 65 CL7 (X, A) 79 CSm(X x IR") 29

H'(K)

CSm(X xIR", A) C,- (UV)

Co (X) Cm

179 3

78

53

H-'(M), Hioc(M), Hcomp(M) 53 H' (IR") 55 Hfm m°(IRN), HI'Q m0 193 Im A 65 index A 65

Int(K) (set of interior points of k)

6

59

a, a° 1 (D), D; 2 DX, D4

38

HS,-.S; (X x IR", A)

79

193

2

dx, dd a1az 39

1

I,(au)

3

Ker A

65

KA

11, 179

1, (E), E2)

65

LQa(X), Loa, Lm, L-°°

15

Index of Notation

286

N(t)

A (Laplace operator) 39 A (z), c -function 112

77

LQ a.d (X , A)

114

(G, LQ(M)), (a(G, SQ6(X)) 99

Q'(IR"), Q'

112

194

A,

R,, (resolvent) 70 6 Rm S1 (H) 260 S2(H), S2(H1, H2) 257 sing supp u 6 SQa(X X IR"), So'.8, S' (X X S-00

S ,d(X x IR", A)

104

IA"(T'M)I

53

R,

x,

x, (x)

187

PZ

98

9(l;, A)

3

77

37

52 176

17Q (IR3n)

ex (y) 141 a 26 QA

27

QA,f, QA,,, 6A,w vA(x,

)

EQo 243 r-symbol 180

Sp A

202, 260

cho(x) 186 0Z 187

V(t)

116

II

V,Q)

129

243

SO -P

WF(u) z;

229

104

F- (UV) y;

175

112

yj(x)

180

19

(trace class norm of an operator) 264 II.112 (Hilbert-Schmidt norm of an operator) 257 -

III

ti (e.g., a - Ea;) 20 (e.g., u) (e.g., (x)) 1

104

2

Subject Index Adjoint operator 26, 197 Amplitude 176 Anti-Wick symbol 187 Bicharacteristic

135

Cauchy-Riemann operator

39

Change of variables in PDO 31 Classical elliptic `PDO 40

-`PDO 29

Inductive topology

Closed operator 69 Closure of an operator 198 Cohomology of a complex 73 Cokernel of an operator 65 Complete symbol 19 Composition of `PDO's 27, 184 - - - with a parameter 216, 245 Density bundle

78 65

60

Laplace operator 39 Laplace-Beltrami operator Left symbol 19 Leibniz formula 28 Multiindex

166

I

Newton binomial formula 28 Non-degenerate phase function Null-bicharacteristics 135

37 37

Differential operator 14 Discrete spectrum 201 Dolbeault complex 75 Dual symbol 27

7

One-dimensional singular integral operator 38 Operator phase function 11 Oscillatory integral 3

Eigenfunctions of an elliptic `PDO 71 Elliptic complex 74 - differential operator 39 - 4'DO 40 Essentially self-adjoint operator 198 Euler characteristics of a complex 73 Exterior differentiation operator 38

Fourier integral operator - transformation I Fredholm complex 73

- symbol 38 - symbol with a parameter

Index of a Fredholm operator

- - on a manifold 36 - - with a parameter 79 - symbol 29

- on a manifold

Hamilton-Jacobi equation 135 Hilbert identity 89 Hilbert-Schmidt norm 257 - - operator 257 h-symbol 240 h-Weyl symbol 240 Hypoelliptic operator 39

II

Pairing 62 Parametrix of a classical elliptic operator 45 - - - hypoelliptic operator 44

- - - - - with a parameter 79 Phase function 3 Polar decomposition of an operator 262

Principal symbol of a classical 4'DO 40 - - - - differential operator 39 Proper map

16

Properly supported function

18

- operator 65 c-function of an elliptic operator 112

-- %FDO

Glazman lemma

Pseudodifferential operator ('PDO) Pseudolocality 15

206

16

- - - with a parameter 78 14

288

Subject Index

Resolvent

70

- of an elliptic differential operator 86 De Rham complex 74 Schwartz kernel

- -of FIO - space

Spectrum

31

128

70 180

r-symbol Symbol of adjoint operator

26

- - composition 27 - - resolvent

Topology in C"(M)

61

60 60 Trace class norm 264 - - operator 260 - of an operator 202, 260 Transport equations 154 Transposed operator 26

II

I

- projection

198

- - H(K) 56

II

Selfadjoint operator 198 Singular integral operator Spectral function 128

Symmetric operator

94

Symbol of transposed operator

26

Wave front of a distribution Weak topology 12 Weyl symbol 180 Wick symbol 191

229

This is the second edition of Shubin's classical book. It provides an introduction to the theory of pseudodifferential operators and Fourier integral operators from the very basics. The applications discussed include complex powers of elliptic operators, Hormander asymptotics of the spectral function and eigenvalues, and method of approximate spectral projection. Exercises and problems are included to help the reader master the essential techniques. This book is written for a wide audience of mathematicians, be they interested students or researchers.

ISBN 3-540-41195-X

9"783540"411956

http://www.springer.de