Fundamentals of Algebraic Microlocal Analysis

FUNDAMENTALS OF ALGEBRAIC MICROLOCAL ANALYSIS Goro Kato California Polytechnic State University SanLuis Obispo, California

Daniele C. Struppa GeorgeMasonUniversity Fairfax, Virginia

MARCEL

MARCEL DEKKER, INC.

NEw YORK-BASEL

ISBN: 0-8247-9327-7 This bookis printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, NewYork, NY10016 tel: 212-696-9000;fax: 212-685-4540 Eastern HemisphereDistribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001Basel, Switzerland tel: 44-61-261-8482;fax: 44-61-261-8896 World Wide Web http://www.dekker.com Thepublisher offers discounts on this bookwhenordered in bulk quantities. For moreinformation, write to Special Sales/Professional Marketingat the headquartersaddress above. Copyright© 1999 by Marcel Dekker, lnc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying,microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Currentprinting (last digit) 1098765432 1 PRINTED IN THE UNITED STATES OF AMERICA

This book is dedicated to our sons Alexander Benkei and Alessandro.

Humanlife is a long arduous journey to be undertaken patiently. Convince yourself insufficiency is The Way,and then you will miss nothing. If unattainable desire presents itself to you, recall your time of most difficult struggle. If you knowonly victories without experiencing defeats, self-absorption becomes your nature. Rememberhaving quite not enoughis superior to having in excess. Persistence is the foundation for your secure and enduring life. ShSgun Iyeyasu Tokugawa

Preface This bookis an introduction to the algebraic theory of systems of differential equations, as developed by the Japanese school of M. Sato and his co-workers, mainly T. Kawai and M. Kashiwara. The book may be used for an advanced graduate course, or as a reference for someof the fundamentalaspects of this theory. Onthe other hand, we also hope that our work will be of someuse to scholars in partial differential equations whowant to acquaint themselves with the algebraic methodsof microlocal analysis. Finally, algebraic geometerswill recognize that the flavor of our workis very muchin tune with their owntaste, and maybe attracted to this beautiful subject, which has not yet received the attention it deserves. The expression "microlocal analysis" refers, from a general point of view, to that approachto the study of partial differential equations whichmovesthe problem of singularities to the cotangent bundle of the variety on which the differential equations are defined. Fromthis point of view, it is a very well developed subject, which probably needs no introduction (since the interested student should then study the fundamental work of HSrmander[89], or, at a more elementary level, the excellent workof Treves [224]); on the other hand, what we are trying to describe in this bookis the algebro-geometric approach, which can be best summarizedunder the heading of algebraic microlocal analysis. In their classical work [123], Kashiwara, Kawai, and Kimurapointedly observe that the Japanese algebraic analysis is really the algebraic analysis in the tradition of Euler; we mayadd that what we meanby algebraic microlocal analysis is the successful attempt to adapt the methodsof abstract algebraic geometryto the nomcommutative setting in which the base ring is nowthe ring of variable coefficients partial differential operators. The origin of the theory goes back to the early papers of Sato on hyperfunctions in the late 1950s, but it wasonly in the early 1970sthat the full potential of the theory of hyperfunctions and microfunctions becameevident. WhenSato introduced hyperfunctions, he was mainly guided by the belief that the natural setting for a theory of differential equations should be the analytic setting, rather than the differentiable one as in Schwartz’s theory of distributions. At the sametime, his backgroundin theoretical physics naturally led him to the development of the study of boundary values of holomorphic V

vi

PREFACE

functions, and from there to hyperfunctions. As we will show in Chapter 1, hyperfunctions are indeed (sums of) boundaryvalues of holomorphicfunctions, but, moreprecisely, they can be defined by the use of sheaf cohomology,and in this way, they present an obvious advantage over other theories of generalized functions, as they allow the use of algebraic tools for their study, and especially of sheaf theory. In a fundamental, but relatively little known,paper of Ehrenpreis [47], it was shownhowsheaf theory could play a crucial role in the algebraic treatment of differential equations; the paper (maybenot surprisingly) had little repercussion, since the sheaf of distributions was not well suited for an algebraic treatment (in particular, as we know,it is not a flabby sheaf, and therefore distribution resolutions of solution sheavesof differential equations do not carry enough information). The advent of hyperfunctions was finally necessary for Ehrenpreis’ ideas to be implemented. Weshould point out, in this regard, that the local, sheaf-theoretic definition of hyperfunctions and of microfunctions allows us to study these objects on arbitrary real analytic manifolds(in fact, we don’t even need to require orientability); however,in this bookwe have decided to stick to the euclidean case (i.e. we study only differential equations on ~n), since this allows us to avoid requiring even more prerequisites from the theory of real analytic manifolds. Readers whoare familiar with this theory will have no trouble in extending the results of this bookto the moregeneral setting. The purpose of this bookis therefore to introduce the reader to the developmentsthat in the last forty years have led fromthe creation of hyperfunctionsto the modernalgebraic treatment of microlocal analysis. Our choice of topics has been oriented to give the reader a sound foundation in the necessary tools and to ensure that all the fundamentalresults are included. This has implied that manyinteresting and even important features have not been included, because they might have led us away from the main objective, or because they would have required more prerequisites than we were willing to accept. Aninteresting feature of our work has been.the attempt to provide a historical perspective for the developmentof these ideas. This does not meanat all that we are giving precise and appropriate credit to all those whohave worked in the field. Wehave striven to be as complete as possible, but we are aware of our limitations in this respect. On the other hand, we have tried to provide the reader with a sense of the developmentof ideas, and of the motivation underlying these ideas. Wenowoutline the architecture of the book, and highlight the links among different chapters. After an introductory chapter in which we try to give a self-contained description of hyperfunctions, we define and study the sheaf of singularities of hyperfunctions, which is usually knownas the sheaf of microfunctions. The notion of microfunction is rather delicate, and to motivate it, we preface its introduction with a treatment of the differentiable case. In simple terms, a

PREFACE

vii

microfunction is a singularity of a hyperfunction (moreprecisely, the sheaf of microfunctionsis the sheaf of singularities of hyperfunctions, pulled backto the cotangentbundle of the manifoldon whichthe sheaf of hyperfunctionsis defined; in our case this last sheaf is over ~n, as we mentionedearlier). Wethen proceed to give a rather extensive treatment of microfunctions, and we prove most of the fundamental results of Sato, Kawai, and Kashiwarawhich provide the foundations for this topic. The algebraic treatment of systems of differential equations is approached in Chapters 3, 4, and 5, which can be read almost independently of the rest of the book, as long as the fundamentaldefinitions from Chapters 1 and 2 are understood. The algebraic treatment of systems of differential equations is a well honoredsubject, which goes back to the early 1960s (see the references in Palamodov’sfundamentalbook[178]), but its first application to variable coefficients differential equation is probably due to Kashiwara[108]. The point here is that, as in algebraic geometry, whereideals replace specific representations for algebraic varieties, in the theory of partial differential equationsit is possible to replace specific representations of systems by what we call :D-modules,i.e., sheaf of modulesover the sheaf ~Dof variable coefficients differential operators. Weregret to say that the treatment in these chapters is rather complicated, and some expertise in homological algebra is required. Wedo try to provide the necessary tools from the theory of derived categories, but these chapters maypresent somedifficulties for the inexperienced reader. Wehave tried to compensatefor the inherent difficulties by giving as manyconcrete examplesas we could. Wefinally come back to a more analytic point of view in the last chapter of the book, where the fundamental Sato’s structure theorem for systems of differential equations is established. This is the crowningresult of the pioneering period of the creation of algebraic microlocalanalysis, and is definitely the starting point for any serious study of this subject. We. have said something in passing, but let us nowbe more specific about the prerequisites necessary for an understanding of this book. Wehave tried to provide the reader with the basic notions that will allow him or her to progressto the moreadvancedbooksin the field [103], [123], [206]. In viewof the complexity of the topic, we have given as muchbackgroundas possible, while limiting the size of the volume. Wehave not, however,been able to provide all the necessary prerequisites, and in fact we will assumethat the reader is familiar with the content of a first course in complexanalysis (at the undergraduate level), well as with the rudimentsof the theory of several complexvariables as given, e.g., in [71], and [192]. Thesetopics are necessary evenfor the first chapter in whichhyperfunctionsare discussed. This chapter, as well as the secondone, does not require muchalgebraic know-how,but on the other hand they require some familiarity with the fundamentalnotions of sheaves and of sheaf cohomology. As we proceed to the following chapters, the reader will need someback-

viii

PREFACE

groundin algebra, as can usually be obtained from the first two algebra courses in graduate school. Moreprecisely, we will require somefundamental notions from commutativealgebra, as well as somenotions from homologicalalgebra. Wewant to express our gratitude to Professors M. Sato, M. Kashiwara, and T. Kawaifrom whomwe learned most of this subject. In fact, we want to offer this work as a modest and unworthytribute to the great contribution of the Japanese school to the creation and developmentof modernalgebraic analysis. Wefurther want to express our indebtedness to the great masterpieces of algebraic analysis, namely,[123] and [206]. In the course of our studies, we havebeen lucky enough to be in contact with manyof the people who, in one way or another, have contributed to the developmentof algebraic analysis. In particular, we wouldlike to express our gratitude to Professors C. A. Berenstein, L. Ehrenpreis, V. P. Palamodov,and P. Schapira, for manyenlightening discussions. Weare further grateful to the membersand staffs at the Research Institute for MathematicalSciences, Kyoto, and the Institute for AdvancedStudy, Princeton, for their assistance during our visits. Weare indebted to Dr. Irene Sabadini and Mr. DomenicoNapoletani for reading several preliminary versions. The first author also wishes to thank his Chairman,Professor S. Weinstein, for providing him with some release time. Wealso thank Ms. Lynn Hanson for her impeccable typesetting. On the nonmathematical side, the authors wish to express our gratitude to our spouses, Christine and Carmen, for their support while the book was written, and to the staff of MarcelDekker,Inc. for their incredible patience. Goro Kato Daniele C. Struppa

Contents Preface

3

Hyperfunctions 1.1 Introduction ............................. 1.2 Hyperfunctions of One Variable: Basic Definitions ........ 1.3 Hyperfunctions of One Variable: Main Results .......... 1.4 Hyperfunctions of Several Variables: Basics ............ 1.5 Hyperfunctions of Several Variables: Main Results ........ 1.6 Historical Notes ........................... 1.6.1 Sato’s Discovery ....................... 1.6.2 Analytic Functionals .................... 1.6.3 Generalized Fourier Integrals ................ 1.6.4 Hyperfunctions in Several Variables ............ 1.6.5 Infinite Order Differential Equations ........... 1.6.6 The Edge of the Wedge Theorem .............

1 1 2 20 42 60 72 73 75 79 80 83 85

Microfunctions 2.1 Introduction ............................. 2.2 Singular Support, Essential Support and Spectrum ....... 2.3 Microfunctions of One Variable .................. 2.4 Microfunctions of Several Variables ................ 2.5 Microlocal Operators ........................ 2.6 Historical Notes ........................... 2.6.1 Physical Origins for the Theory of Microfunctions .... 2.6.2 HSrmander’s Analytic Wave Front Set ..........

91 91 92 113 115 122 134 134 141

:D-Modules 3.1 Introduction ............................. 3.2 Algebraic Geometry and Algebraic Analysis ........... 3.3 Filtrations and Characteristic Varieties .............. 3.4 ~-Modules .............................. 3.5 Historical Notes ............................

151 151 156 169 176 179

ix

x

CONTENTS

4

Functors Associated with :D-modules 4.1 Introduction and Preliminary Material .............. 4.2 The de Rham Functor ....................... 4.3 Algebraic Local Cohomology .................... 4.4 Cohomological Properties of ~x ..................

183 183 187 200 203

5

Holonomic T)-modules 5.1 Introduction ............................. 5.2 Inverse Image and Cauchy Problem ................ 5.3 Direct Image ,. ........................... 5.4 Holonomic T~-Modules ....................... 5.5 Historical Notes ...........................

209 209 209 218 222 236

6

Systems of Microdifferential Equations 6.1 Introduction ............................. of Microlocal Operators ............. 6.2 The Invertibility 6.3 A First Approach to Bicharacteristic Strips ............ 6.4 Contact Transformations ...................... 6.5 Structure of Systems of Differential Equations .......... 6.6 Historical Notes ...........................

239 239 241 249 255 261 272

Bibliography Index

275 291

FUNDAMENTALS OF ALGEBRAIC MICROLOCAL ANALYSIS

Chapter 1 Hyperfunctions 1.1

Introduction

In this chapter we will introduce the first object necessary for the study of Microlocal Analysis, as developed by the Japanese school, namelythe sheaf of hyperfunctions. As it is knownsince Schwartz’s introduction of the notion of distribution [210], there is no hope of dealing with the subtle issues posed by the theory of partial differential equations unless one resorts to somekind of generalized functions. In the case of Schwartz, the notion which was developed is that of distribution, i.e. of a continuouslinear functional on the topological space of infinitely differentiable functions with compactsupport. Otherdifferent spaces havebeenstudied and developedfor a variety of different reasons, see e.g. [14], [17], [37], but the choice of the Japaneseschool led by M. Sato has been to employa space of functions which can be defined on any analytic manifold and whichsomehow generalizes the space of distributions itself. As we will see, the definition of this space is quite natural in the case of a single real variable (wherethis notion also has a well established history, as we shall see in the historical notes to this chapter), but, on the other hand, is quite complicatedand not too intuitive in the case of several variables. For this reason wewill begin this chapter with the case of one variable, whereall the details can be easily explained, and only after this introductory treatment we will devote ourselves to the more complex issues which stem from the generalization to several variables. Providing an introduction to the theory of hyperfunctionsis not a straightforward task, since several different approachescould be followed; in the first volume of his monumentalwork [89], HSrmanderchooses to define hyperfunctions following an approach similar to the one used by Schwartz to introduce distributions; this approachis quite interesting, and we will comeback to it as we develop the microlocalization technique in Chapter II, but it does not seem to convey the spirit of the Japanese school, and we have therefore decided to follow the approach originally used by Sato in his first groundbreakingpapers

2

CHAPTER 1.

HYPERFUNCTIONS

[195], [196], [197]. This choice, akin to the one followed by Kanekoin his beautiful book [103], seems moreapt to the study of the generalizations which will follow in the subsequent chapters. Withrespect to this last remark,it should be noted that this chapter is only introductory to the rest of our work, and that its scope is somehow limited, in view also of Kaneko’s [103], which-provides a complete, thorough and highly readable introduction to the study of the theory of hyperfunctions. As for the structure of this chapter, we provide, in section 1.2, the first definitions of single variable hyperfunctions, while the first important theorems are given in section 1.3. The extension of the definitions and of the results to the case of several variables will be dealt with in sections 1.4 and 1.5. In keeping with the rest of the book, we confine our treatment to the case of hyperfunctions defined on the Euclidean spaces ~, but we provide, at the end of section 1.5, a brief description on howthe notion of hyperfunction can be extended to the case of real analytic manifolds; the chapter ends with a rather extensive set of historical notes on the birth of the theory of hyperfunctions. For the reader interested in a more thoroughtreatment of the concepts which will be introduced and developedin this chapter, we wish to point out three basic references whichshould be kept in consideration throughout this book: the first treatment by Sato and his coworkers Kashiwaraand Kawai, [206], the treatise of Kashiwara, Kawaiand Kimura, [123], and finally the work of Kaneko, [103]. Onecommentmaybe necessary on these three references; the first one (usually referred to as SKK)is a fundamentalwork, whose readability, however, is not optimal, also in viewof the fact that it is the first introduction to a difficult subject, written as the subject itself was being developed;the secondtreatise, to whichwe will refer to as KKK,is a very comprehensivediscussion of Algebraic Analysis, in which the Japanese algebraic methodsto study partial differential equations are described; finally, Kaneko’sbook is the most readable, but his analysis stops at the notion of hyperfunction (with a treatment of the notion of microfunction) and no attempt is madeto deal with the most interesting topics concerning the algebraic treatment of systems of differential equations (namely C-modulesand T)-modules).

1.2

Hyperfunctions initions

of One Variable:

Basic Def-

Let ~ be an open set in ~, and let V be a complex neighborhood of ~, i.e. V is an open set in the complexplane(T such that ~ is relatively closed in V; by this term we meanthat ~ is compactin V and, for simplicity, the reader may fix his/her attention on ~ being an open interval (a,b) in the real line and V being an open set in ff whoseintersection with ~ is exactly (a, b), as indicated

1.2.

HYPERFUNCTIONSOF ONE VARIABLE: BASIC DEFINITIONS

3

Figure 1.2.1 in figure 1.2.1. Wecan then define the open sets

v+:={z¯v:Imz>0},v-:={z¯v:Imz= lira F-(x - is ), ~( >, ~--~0 a--~0 where denotes the duality bracket relative to the pair (~, T)), then + and F- actually glue together to yield a ]unction F holomorphicon V. Proof. It is sufficient to reduce the problemto Theorem1.2.2 by repeated integration. [] Wenowreadily see that, in viewof this result, a hyperfunctionshould be seen as a holomorphicfunction on V\~, and therefore as a pair of functions (F+, F-), with F+ in O(V+) and F- in O(V-), and with the proviso that +, F-) wi ll denote the zero hyperfunction if there exists a function F, holomorphicon all of V, such that its restriction to V± coincides with F~-; by Painleve’s Theorem this actually identifies a hyperfunction with the "difference" of the boundary values of F+ and F-, where the notion of boundaryvalues has of course to be suitably defined. As we will mentionin the historical notes, such a concept has been in the minds of mathematiciansand physicists for quite a long time, and it was only with the introduction of hyperfunctions that a full and satisfactory treatment of such notion was given. Wenowwould like to show why and how it is possible to think of hyperfunctions as generalized functions, and which operations can be defined on the vector spaces B(~). To begin with, we note that all real analytic functions can be naturally thought of as hyperfunctions, i.e. there is, for every open set in ~,

1.2.

HYPERFUNCTIONS OF ONE VARIABLE: BASIC DEFINITIONS 7

a natural embeddingof the vector space A(f~) into the vector space B(ft). a matter of fact, if f is any real analytic function on ~, then, by the definition itself of the space of real analytic functions as the inductive limit, taken on the inductive family of all complexneighborhoodsof ft, of spaces of holomorphic functions, A(~) = ind lira O(V), we have that f extends to a holomorphic function ] defined in some complex neighborhoodV of ~. If we nowchoose F+ to be ] restricted to V+, and F- to be the function identically zero, then the pair (F+, F-) defines a hyperfunction whichexactly coincides (if the boundaryvalue interpretation is used) with the analytic function f. Wecan actually say something more, since we can define a natural product betweenreal analytic functions and hyperfunctions, so that B(~) turns out be an ~(fl)-module. To do so, we simply define the product of the hyperfunction IF] by the real analytic function f to be the hyperfunction defined, in the quotient (1.2.1), by the holomorphicfunction IF, i.e. (1.2.3)

f[F] = []F].

It can be easily shownthat the definition (1.2.3) is well posed, i.e. it does not depend on the choice of the representative F of the hyperfunction F, nor on the choice of the domainof the extension ] of the real analytic function f. Such an independence, together with the independence of (1.2.2) from the choice of the complexneighborhoods chosen, can be established with a simple argument based on the Cauchy theorem. Unfortunately, it must be emphasized that hyperfunctions maynot be, on the other hand, multiplied, so that they do not enjoy an algebra structure. This is a most unfortunate fact, since one of the problemswhich physicists have been posing to mathematicianshas been the creation of a theory of generalized functions whichwouldallow multiplications; this problemhas been at the heart of several different attempts to modifySchwartz’s theory of distributions (where the kernel theorem[210] showsthe impossibility of imposingany multiplicative structure); amongthe most interesting attempts, we wouldlike to point the reader’s attention to the non-standard analysis approach by Colombeauin which multiplication between generalized functions is indeed possible. In our case, it can be easily seen whya product cannot be defined in any reasonable way; suppose, indeed, that two hyperfunctions f = IF] and g = [G] are defined on someopen set ft of ~. Then the first spontaneous attempt to define multiplication (and the only one which would be consistent with the structure of J~(~)-modulewhich we have given B(f~) in the previous few pages) wouldbe to define f .g :-- IF.G].

8

CHAPTER 1.

HYPERFUNCTIONS

It is howeverimmediatelyseen that such a definition wouldnot be invariant under the choice of the holomorphicrepresentative of the hyperfunctions. Indeed, since z - 1 is holomorphiceverywhere, we have that the hyperfunction [z] and the hyperfunction [1] are one and the same hyperfunction (both represent the identically zero hyperfunction); still, the hyperfunctions

and

are two different hyperfunctions since 1/z - 1 is not holomorphicin any neighborhoodV of the origin. Wewill see in the next chapter, whendealing with microlocalization, that, in somecases, hyperfunctions maybe multiplied; it turns out that these cases are exactly the ones of physical interest, so that we have at least a partial solution to the needsof physicists. Generalized functions are usually introduced as a device towards constructing solutions to equations for whichno solutions wouldotherwise exist. This is certainly the case with hyperfunctionsas well; as a matter of fact, the treatment of real analytic coefficients differential equations becomesparticularly transparent when hyperfunctions are employed.The deep reason for this is one of the greatest early contributions of MikioSato, and will not be completelyclear until the last chapter of this book;we can at least begin to define howlinear differential operators with real analytic coefficients act on the space of hyperfunctions. To this purpose, let P(x, ) = a~(x) i=0

be a linear differential operator with real analytic coefficients on an open set ~ of the real line, and let f -- [F] be a hyperfunction defined on the same open set; as we noticed before, each coefficient ai of P(x, d/dx) can be extended to a holomorphic function a~ on some complex neighborhood U of ~; by possibly taking an intersection, we can assume that F is actually holomorphic on the open set U\~. Wecan therefore define the action of P(x, d/dx) on f as follows: d P(x, ~xx)[fI :-- [P(z, ~z)Fl, where the operator P(z, d/dz) is defined by i ~ dzd p(z,)d = i(Zl i=O

1.2.

HYPERFUNCTIONS OF ONE VARIABLE: BASIC DEFINITIONS 9

It is not difficult to check that this definition does not dependon the several choices whichhave been made(the choice of the representative of f, the choice of the open set U and the choice of the extensions ai ). Before we proceed to the study of someelementary examples of hyperfunctions, we want to illustrate one more important concept. Wesay that a hyperfunction f = IF] defined on an open set D vanishes on an open subset t2’ _C D if, on ~’, f coincides with the zero hyperfunction (note that, in general, given a hyperfunction f, it is not possible to speak about its value at a point, and therefore we cannot just say f(x) = 0). This is of course equivalent to saying that the function F whichdefines f is holomorphicthrough the real axis at every point of t2’. It is not difficult to see that, for everyhyperfunctionf on f~, there always exists a largest open subset D’ of ~ on which f vanishes; it is indeed sufficient to take for l-l’ the union of all opens subsets on whichf vanishes. We can nowprove that f vanishes also on such a union (this mayseem trivial if we forget that f cannot be pointwise evaluated). This is howevera consequenceof the fact that the function F which defines f must be holomorphic across the real line in all points of the open sets whoseunion is D’, and therefore in all ~. points of D’. This showsthat f vanishes on all of fl For a hyperfunction f in B(t2) we define its support to be the complement in D of the largest open subset on which f vanishes. This notion will prove to be very valuable, and is strictly linked with somedeeper result which we will obtain in the next section. For K a compact subset of ~, we will denote by BK the space of hyperfunctions whosesupport is contained in K. It is possible to define a notion of integration for such functions in a rather natural way. Let f be a hyperfunction with compact support K and let F be a defining function for f which, of course, can be chosen to be holomorphicin U \ K where U is somecomplexopen set properly containing K. Let now7 be a closed simple piecewise smooth curve contained in U and surrounding K once; we will assume ~- to circle aroundK clockwise. Wethen define (see figure 1.2.2) /~f(x)dx=

f~F(z)dz

and it is easily seen that the definition is consistent with the choices of F, U and of W. This independenceis, of course, once more a consequenceof Cauchy’s Theorem. Wehave, in particular, all the elements to prove a simple yet fundamental result which,as it will be seen in the historical notes to this chapter, wasactually already knownto Fantappie’ (it was actually the beginning of his theory of analytic functionals). Theorem1.2.4 Let ]~g be the space of hyperfunctions supported by the compact set K. For any complex neighborhoodU of K, the following isomorphismholds:

10

CHAPTER 1.

HYPERFUNCTIONS

U

Figure 1.2.2

(1.2.4)

BK ~= O(U \ K)

o(u)

Proof. Let f be in BK; for any open set D containing K, we can think of f as an element of B(f~), by simply continuing f to zero in f~ \ K. In particular, can choose f~ to be UN_~,which showsthat any element of B~can be expressed as in the quotient in (1.2.4). On the other hand, if we nowchoose a complex neighborhood U of f~ and we represent B(D) as the quotient O(U\ f~)/O(U), we immediately see that the quotient in (1.2.4) corresponds to the space hyperfunctions having compact support in K. Note that, in particular, the embeddingof BKinto B(f~) is well defined for any open set f~ containing K. The definition of integral for a compactly,supportedhyperfunctionis a special case of a moregeneral definition whichapplies to all hyperfunctions whichare, at least in two points, real analytic. Let indeed f be a hyperfunction defined on a neighborhoodof a closed interval [a, b], and supposethat f be real analytic at the points a and b (this means that the defining functions + and F- a re b oth holomorphic, or can be holomorphically continued, in a neighborhoodof a and of b); let nowT+ and T_ be piecewise smooth arcs connecting a to b in such waythat T+lies in the open set in whichF+ is defined and ~-_ lies in the open set in whichF- is defined. Then, see figure 1.2.3, the integral of f from a to b is defined by ~bf(x)dx = f~+F+(z)dz - f,_ F-(z)dz.

1.2. HYPERFUNCTIONSOF ONE VARIABLE: BASIC DEFINITIONS

11

b

a

Figure 1.2.3 It is easily seen (by using again Cauchy’stheorem), that this definition independent of the manyarbitrary choices which have been made. Wecan nowlook at some of the simplest examplesof hyperfunctions, which will lead us directly to the consideration of somedeep results in the theory of hyperfunctions. Example1.2.1 The first exampleof a (non trivial) the Dirac delta function which can be defined by

hyperfunction is given

It is clear fromits definition that ~(x) is zero in ~ \ {0}, since the function is holomorphicin ~ \ {0}; in other words, the support of ~ is the origin; it is also clear that, at the origin, the hyperfunction~ is "singular" (we havenot yet defined this notion in any precise way) since it should represent the boundary value of a function whichhas a singularity at the origin. Wehave, however,given ~ a rather famousname, since we claim that it is actually the well knowndelta function introduced by Dirac in 1936, [40]. If we look back at Dirac’s original definition, we notice that his delta was defined as a "function" which was zero everywhereexcept at the origin, and such that, for any infinitely differentiable function a(x) with compactsupport, it wouldgive (1.2.5)

/~n a(x)~(x)dx = a(O).

Now,we have already easily checkedthat the delta hyperfunction vanishes outside the origin; in view of the definition of product betweena real analytic function and a hyperfunction,as well as in viewof the definition of the integral of a compactlysupported hyperfunction, we immediatelyget the validity of (1.2.5),

12

CHAPTER 1.

HYPERFUNCTIONS

at least whena(x) is real analytic. The delta hyperfunction has therefore all the characteristics to justify its name.It has still to be remarkedthat it has to be allowed for the test functions to be real analytic instead of simply infinitely differentiable, so that the analogyis not, at least at this moment,complete. Example1.2.2 Let us take a few derivatives of the delta hyperfunction, according to the rules of differentiation for hyperfunctions;weeasily get

and, moregenerally, [(-1)~+ln! 1 ~¢~)(x)-L ~ "z(~÷l) It is easily seen that all of these hyperfunctionsare still compactlysupported, and that they behaveas the derivatives of the classical Dirac delta does; namely, for any real analytic function a(x), it is /~ a(x)5(n) (x)dx =(’~) (0). Example1.2.3 Classically, the Dirac delta function is seen as the derivative of the Heavisidefunction H(x), this being defined as identically zero for negative values of x, and identically one for positive values of x. This function, which was originally used to describe electrical switches, is not differentiable in the classical sense, but, if we allow for weakderivation (i.e. integration by parts) to take place, it has the Dirac delta as its derivative. In our case, we define the Heaviside hyperfunction to be

wherethe logarithmic function is taken in its principal value; it is then clear that H(x) = 1 for any positive value of x, while H(x) = for an y negative value of x; on the other hand H(x) is not defined at x = 0, if it is considered as a function. By applying the rules for differentiation which we have set up for hyperfunctions one immediately obtains that H’(x) = ~(x). One sees here an application of an important phenomenon,namelythat differential operators do not enlarge the set where hyperfunctions are "singular" (we will makethis concept muchmore precise in the next section). This could also be seen, of course, in Example1.2.2.

1.2.

HYPERFUNCTIONSOF ONE VARIABLE: BASIC DEFINITIONS 13

Example1.2.4 Wenow define two very simple and innocuous looking hyperfunctions, whichwill be neededin the immediatesequel; consider the following functions which are holomorphicon IT \ ~: e(z)=

1, ifImz>O O, iflmzO iflmz_ n + 1. In this case we want to use Theorem1.4.3 (part b). As a matter of fact, if U an arbitrary Stein neighborhoodof the origin, we immediatelydeduce that, for j _> n + 1, it is H~u(U,O) = 0. Therefore, by taking the inductive limit, we deduce, that the stalk at the origin of SJ vanishes. The case of j _< n - 1 is

1.4.

HYPERFUNCTIONSOF SEVERAL VARIABLES: BASICS

51

slightly more complicated, since we need to use, in this case, Theorem1.4.4, which requires us to showthat for a fundamental system {U}of neighborhoods of the origin, one has that the set ~n U can be thought of as a difference L \ of two closed analytic polyhedra. As shownin [103], this can be easily done by noticing that the real axis can be expressed as the set

-- {ze¢:leiZl 0},

v_~= v n {z ~ ¢~: Im(z~) < 0}. One immediately sees that,

with the notations used before, Vk = V+k U V_k,k = 1,...,n.

CHAPTER 1.

54

HYPERFUNCTIONS

Wecan nowconsider the relative covering of the pair (V, V \ fl) defined

v = {Y0,v±l,..., v~,}, v’ = {v±l,..., v±,}. With the same arguments discussed before, and noticing that

V+kn v_k= O, it is not too difficult to provethat the cochain spaces are given as follows:

C’~+I(Y, V’;O)= cno 2 ,

"~’

;

O)

= 0 O(V 0 ~1Ve~.. ~l----~l,...,~nmq-I

11~1,

.

,

I~1 ,

Yen.n)

and

C’~-’(V,V’;O) = (~ O(Vo n V~,.,n 9~,~.~n... n Y~,,.,d. k=l~2,...n

Fromthis characterization of the cochain spaces, it is immediateto derive the isomorphism gerO") ~--

O(Vo n V~.,n . . . n V~,.,,).

(~

Similarly, one can see that the coboundaryoperator ~ = ~,~-1 is defined (with, hopefully, obvious choice of symbols) by ~-1 (,~)~,.,,...,-.,, =~2() 1~’~,.,,...,~.,~ ........,,. k=l

where~ is a relative (n - 1)-cochain. Since, now,

v0n v~,.~n ... n v~,.n= ½,.tn ... n v~,.n= v n {~+ ir~,,...,~}, where F~, ....... is the open convexcone (with vertex at the origin) defined in by F~,....... = {y = (y~,...,y~)

~ ~ ej. y~> 0for any j = 1, .. .,n},

we deduce the following important isomorphism: B(a)= H"(]), 12’; -~ ee,=zkl... ..... 4-iO(Vr~ (j~n.~_ Suchan isomorphismexpresses the fact that every hyperfunction on the open set f~ can be represented by 2~ holomorphicfunctions in the "numerator"of the

1:4.

HYPERFUNCTIONSOF SEVERAL VARIABLES: BASICS

55

previous isomorphism,wheretwo representations give the samehyperfunction if and only if their difference is a coboundaryaccording to the definition we have ~-1. given of 6 Let nowF be one of the open cones defined before; every holomorphicfunction fonRn + iF gives a hyperfunction in a canonical way, via the following sequence of homomorphisms: (1.4.10)

o(v ~ (~" ~r)) -~ o(v n (~n"+ ir~, ....... ) --~ ~(~).

Definition 1.4.3 Let f be a holomorphic function on an open set of the form V A (/t~ ~ + iF). Then we denote by br(f) the hyperfunction obtained as (1.~.10), and we say that br(f) is the boundaryvalue off along the cone F. We will sometimeswrite br(f) f( x + iF0). Analmost immediateconsequenceof these arguments and definitions is the following reformulation of the definition of hyperfunction(we will see later on howto makethis into an even moregeneral definition): Proposition 1.4.1 With the notations given before we have the following isomorphism: (1.4.11)

13(~) = ~,... ....

(O(U ~(~"+.........

))).

~’1 =flzl,...,e=-l- 1

Proof. Immediateapplication of the explicit descriptions we have given above. It maybe interesting to note that in the case in which n = 2, and V = (T2, the description of hyperfunctions which is given in Proposition 1.4.1 amountsto say that if a hyperfunction f is represented by a holomorphic function F in O((IT \ ~)2), f (x, y)= F(x +iO, y + iO) - F(x + iO, y F(x - iO, y + iO) ÷ F(x i0 , y - i0), which is in full acc ordance with what we already intuitively proposedat the beginningof this section. One different, yet important, way to represent hyperfunctions as sum of boundaryvalues of holomorphicfunctions relies on a different representation of the cohomologygroups described above, which only employs n + 1 angular domains.The algebraic basis for this decomposition,whichwe want to describe, is the following Proposition 1.4.2 Let a be a vector in ft ~ and define the open half-space E~ = {y 6 ~n : a. y > 0}, where a ¯ y denotes their real Euclidean inner product.

56

CHAPTER 1.

HYPERFUNCTIONS

Consider now n + 1 vectors s°,al,...,an in ~n, and set Ej = E,~ for j = 0, 1, ..., n; supposenowthat these vectors satisfy (1.4.12)

Eo U E, U... U En -- ~n \ {0}.

Thenthe following statements are true: (a)

EonEln...nEn=O;

(b) any n of the vectors aboveare linearly independentover ~:t; as a consequence the intersection of the correspondinghalf spaces is a proper open convex cone~.

(c) define, accordinglyto (b), the cone rj = Eon...n.~j

n...nEn;

then its dual cone

is a closed convex set generated by the o~~ with k =¢j, and oJ belongs to -Int(F~). As a consequence, the dual cones r~,..., r~ give a decomposition of the dual space of Nn, a real Euclideanspace of dimensionn itself, consisting of closed convexpyramids; (d) Forany j andk in 0, 1,... , n, the following equality holds: rj+

r~ = Eon...nk~...~k~...nEn.

Proof. (a) Assumethe intersection is not emptyand let a be a nonzero vector in it; then -a would not belong to the union E0 ~ ... U En = ~n \ {0}, which wouldgive a contradiction. (b) Supposethat a~,..., an are linearly dependent. ~hen ~here wouldexist a nonzerovector y such that, for any j = 1,..., n, it is ai. y = 0. Then, one immediatelysees that both s° ¯ y > 0 and s° ¯ y 0,

~J’Y--~l’lC~jl>O,J--2,...,n. Becauseof (b), such a system as a solution, but then one gets O/0 "y ~ ~10~ 1 "y --

I~jl~ j

"y : --

> O,

and therefore y ~ E0 ~ ... ~ E,, which wouldcontradict (a). (d) The inclusion of F~ + F~ in the intersection of the half-spaces is obvious. Moreover,since both sides are openconvexcones, it really suffices to prove that a halfspace of the form a. y ~ 0 containing the left-hand side, always contains the right-hand side. But now, by definition, such an a must be an element of the dual cone (Fj + Fk)°; however,

(r~+ r~)° = r; n r~, and, in view of (c), this is equal to the common sides of the two cones

~ k 0, Z~k 0}. { EZ,~ iCj,k

This proves immediately our statement.

~

Wecan nowuse this result to construct a different relative coveringof the pair (V, V \ fl), whichwill allow us to see hyperfunctionsas sumsof n + 1 boundary values. Given a nonzero vector a in ~", consider as before the half-space E,~= {yE ~’~ :a.y >O}. Takenown + 1 vectors al, a2,..., in Proposition1.4.2, it is

a,~ in ~n such that, with the notations used

E~uE2u... uE~= Z~n \ {0}, and set V~ = V n(~’~ + iEj),j = 0,1,...,n. Then the covering (Y, Y~) given

V=(V, Yo,Vl,...,V,), V’=(V0,V~,...,Y,) is clearly a Stein covering to which Leray’s theorem applies, and which can therefore be used to computethe relative cohomologyof the pair (V, V

58

CHAPTER 1.

HYPERFUNCTIONS

In view of Proposition 1.4.2 (part (a)) we knowthat no nonzero relative (n + 1)-cochains exist and therefore any relative n-cochain is necessarily an cocycle; such a cochain will always be an alternating collection F = {Fj}, for j = 0, 1,...,n, with

Fj ¯ O(Vo n... r1 ~ n... n v.). In order to express the alternating dependence of Fj from its index, we can actually write F as follows:

~(z)=j=O ~~(z)VAVoA... A... where the wedgeproduct only indicates the alternating behavior of the dependence of Fj from its index. A relative (n - 1)-cochain, on the other hand, can be described, using the same notations, by a collection G = {Gj~} with Gjk

¯

O(VoN...N

~jN...N

~kN...N

V~),j,k

= 0,...,n.

Onceagain, we will write, for the sake of keeping track of the alternating relations,

c(~)= E~j~(z)V AVoh... hv~A...Av~A.., h j H~((T"\ c9~, O) --+ H~-I’~, O) -+... ... --~ H~n(e Since 0~ is compact, the classical Malgrange Theorem(see e.g. [153]) shows that H~I((Tn, O) = and therefore the flabbiness of/3 follows nowfrom Lemma 1.5.2. Wewouldnow like to show someelementary applications of the flabbiness of the sheaf/3 to the study of differential equations (both of finite and infinite order). The applications, whichare taken mainlyfrom [144] and [217], are also of someinterest as they provide a natural introduction to the algebraic treatment of systems of differential equations and therefore to someof the themes which will be discussed in Chapter III.

1.5.

HYPERFUNCTIONS OF SEVERAL VARIABLES

69

Let, in the rest of this section, S denote one of the following sheaves on j~n : ¢Z[, B, ~)t, ~ (i.e., respectively,the sheavesof real analytic functions,hyperfunctions, Schwartzdistributions and infinitely differentiable functions). Most of what will follow holds true for anyoneof these sheaves, and we will point out explicitly whenevera difference arises. Let nowP(D) be any rl × r0 matrix of linear differential operators with constant coefficients: by this we simply mean that P(D) = [P~j(D)], and that D indicates the usual differential symbol

where z = (zl,... ,zn) is the variable on the Euclidean space ~n. As we have seen in our previous pages, such a matrix defines a sheaf homomorphism P(D) : Sro whosekernel we denote by SP, i.e. se is the sheaf of solutions in ,5 r° of the homogeneoussystem of equations g(D)u = 0; u is of course a vector of solutions. Denote nowby A --~T[X1,..., Xn] the ring of polynomials in n indeterminates, with complexcoefficients; if Q(X) is any matrix with entries in A, we will denote by Q’(X) its adjoint matrix tQ(-X). If we go back to our system P(D), and we replace by Xj the element -iO/Oxj, we obtain a matrix P(X) whose adjoint P’(X) defines a~T-homomorphism P’(X) : ~’ - ~ ATM, whosecokernel is the finitely generated A-modulewhich will be denoted by Aro ~, P’(X)A As it has been shownby Palamodov[178] (for the more general case, but essentially already through the Hilbert’s basis theorem), M’ admits a free resolution (1.5.3) 0 ~ M’~- ~° P~_~2) A *, P~_x) A ~ ~-... ~- A,~_~ P’~(x) A~~ ~- 0 whichterminates for somern _< n. See [3] for a discussion of the complexityof such a procedure.

70

CHAPTER 1.

HYPERFUNCTIONS

The first important consequenceof (1.5.3) is due to several mathematicians whoobtained it independently, and looking at different special cases: we will only mentionhere someof the mathematiciansassociated to this result, namely Ehrenpreis, Malgrange, HSrmander, Palamodov, Komatsu, Harvey (see [52], [140], [144] and [217] for the precise references): Theorem1.5.6 I.f Wis a convex open set or a compactconvex set in KIn, then the sequenceo.f groupsof sections (1.5.4) 0 --~ ~qP(w)-+ ~q(W)TM P(~ $(W)r’ -+... Pm:t~D) ~q(W)rm --~ 0 is a resolution of SP. An immediate consequenceof Theorem1.5.6 (just take a cofinal family of convexneighborhoodsof a point and then take the inductive limit) is the fact that we can actually rewrite the sequenceat the sheaf level and get Theorem 1.5.7 The sequence (1.5.5) O--~SP --~ Sr° P(--~) rl ~(--~D) St2 - --4 . ..S rm-~P’~-~") Sr’~ -~0 is a resolution of the solution sheaf ~qP. In the case of hyperfunctions, i.e. when ,.q =13, resolution (1.5.5) is a flabby resolution. Remark 1.5.1 Both Theorems 1.5.6 and 1.5.7 remain true if we consider W in (T’~ and we take the sheaf (9 of germs of holomorphicfunctions. Sequences(1.5.4) and (1.5.5) are of course of independentinterest, but becomeeven morefull of significance, whenone considers the long exact relative cohomologysequence of the sheaf 8 P with respect to the pair (KIn, K), K being a real compactset. In principle, and from general definitions, one has the following sequence: (1.5.6) 0--~ H°g(KI’~,,.q P) ~ H°(KI’~;8P) -~ H°(KI’~ \ g;~qP) ~ ---~ HIK(KI’~,~qP) ~ HI(KI~; ~qP) ---~ Hi(KI’~\ K;$P) 4... It is interesting that, essentially as a consequenceof Theorem 1.5.7, sequence (1.5.6) can be actually decomposedinto short exact sequences, as follows: Theorem1.5.8 With the same notations as above, the following sequences are exact:

~, SP) -+ H°(KIn; P) -~ H°(KI ~ \ K;e) ~ HA 0 -+ H~(KI (Kn, ~) -+0 and, for p >_1, 0 ~ HP(KI~; P) - --+ HP(KI ~ \ g;s ~) ~H~K+I(KI~,$ P) ~ O.

1.5.

HYPERFUNCTIONS OF SEVERAL VARIABLES

71

Theorem 1.5.8 has a very interesting series of applications in terms of differential equations: indeed one sees that the vanishing of the relative cohomology can be given important interpretations. For example, Theorem1.5.8 showsthat if the relative cohomology group of order zero vanishes, then we have an injection 0 ~ g°(~n; SP) ~ H°(~n P) \ K; ,S whichis equivalent to the unique continuation property for solutions of P(D)u 0 from ~n \ K to all of ~n. Similarly, if the first relative cohomologygroup vanishes we have the surjection H°(~; ,S P) ~ H°(~ ~ ~ K; S P) ~ 0 whichis equivalent to the existence of such continuation. Of course, if both the 0-th and the first relative cohomologyvanish, we have existence and unicity of continuation for the solutions of P(D)u= O, as it is attested by the isomorphism H°(~n; SP) ~ H°(~~ ~ K; sP). Moregenerally (and the reader is referred to [140], [144] or to [217]), the vanishing of higher order relative cohomologiesalso has importantconsequencesfor the structure of the sheaf of solutions of the systemP(D)u0;such consequences, being less evident, are left to the interested reader. It becomestherefore interesting and important to determine general conditions which wouldimply (or be equivalent to) the vanishing of the relative cohomology.To do so, one must consider the dual sequenceof (1.5.3), i.e. 0 ~ Ar° ~ Ar~ ~ Ar~ ~ ...

~ ATM ~ 0,

which is obviously a semi-exact sequence. Now,by definition, the ~th cohomology group of this complexis what we call the ~th Ext module, i.e. the A-module Ext,(M, A), and the behavior of the relative cohomologycan be described in terms of this module. To begin with, we note that, should it be r0 = 1 and P ~ 0, then Ext,(M, A) always vanish. Moregenerally, the following algebraic results (whichwe only formulate for the sheaf of hyperfunctions) can be obtained (their proofs, by nowstandard, can be found in [144]): Theorem 1.5.9 Ext~(M,m) = 0 if and only if H~(~n;BP) = O; this last P) vanishing, in particular, is equivalent to H~0 } (~n, B = O. Theorem 1.5.10 Take p ~ 1. Then Ext,(M, A) = 0 if and only if, bounded convex subset K of ~~, it is H~(~n, BP) = O.

for any

In [217] (but see also the subsequentand related papers [11], [12], [137]) it shownhowthis same algebraic treatment can be applied to the case of systems

72

CHAPTER 1.

HYPERFUNCTIONS

of infinite order differential equations.In that case, someextra difficulties arise, whichare related to a purely algebraic matter, i.e. the possibility of constructing a finite resolution for the systemitself. In other words, the ring A of polynomials is nowreplaced by the ring of symbolsof infinite order differential equations; such a ring, i.e. the ring of functions which are obtained by replacing D = -i ~x~x~’’"’ 0 - ~ with

z= (zl,..

"’z~)

turns out to be (see also our section 2 for the one variable case) the ring of holomorphicfunctions with infraexponential growth, i.e. the ring of entire functions f such that for any e > 0 there exists a positive constant A~= A(e, f) such that If(z)l A~exp(~lzl). As it maybe easily imagined(and examplescan be easily given, see [217]), there is no general Hilbert sygyzy theoremfor such a ring. In [137] Kawaiand Struppa (see also the contribution of Meril and Struppa as described in [14]) have given a rather delicate technical condition to ensure that a sygyzy theorem can be restored. This leads to the consideration of the so called "slowly decreasing" systemsof infinite order differential equations, for whichall the previous theory can be fully restored; even though the problemis algebraic in nature, the only waywhichhas been found so far to deal with it, is strictly analytic. The slowly decreasing condition is rather complicated (even though it was shownto apply to most relevant cases), and its discussion falls outside the scope of this book. Werefer the reader to the indicated literature for moredetails on this question.

1.6

Historical

Notes

Thebirth of the theory of hyperfunctionsis particularly fascinating, since (like most theories of deep content and great relevance) it has strong relations with various and diverse preexisting theories, as well as with somequite concrete problemsfrom physics. Wehave therefore thought it useful to divide this long Appendixinto six different sections, each of whichdeals with a different historical aspect; section 1 will briefly outline the steps whichled Sato to the creation of hyperfunctionsat the end of the fifties; section 2 will deal with the notion of analytic functional which,fromthe first steps of the Italian school of Fantappie’, to the conclusive results of Grothendieck, KStheand Martineau, has paved the wayfor the creation of hyperfunctions; in section 3, we will describe a different forerunner of hyperfunctions; the generalized Fourier transform of Carleman;in section 4 we will go back to analytic functionals, to commenton the giant step which was necessary for Sato in order to extend his ideas to the case of several

1.6.

HISTORICAL NOTES

73

variables; section 5, on the other hand, will deal with the developmentof the theory of infinite order differential operators whichwe have already introduced and, as we shall see, play a crucial role in the theory of hyperfunctions and microfunctions; finally, section 6 will discuss the famousedge of the wedgetheorem, its first formulationsas well as its relations with the developmentof the theory of hyperfunctions and microfunctions. 1.6.1

Sato’s

Discovery

To begin with, Sato’s first paper on the theory of hyperfunction was [195], which appeared in the Japanese journal Sugaku;for this reason, the paper was not knownoutside Japan, and it was only with the appearance of [196] in the Proceedings of the Japan Academyof Sciences that his work becameavailable to western mathematicians. Immediately afterwards, Sato published a much richer account of his newtheory in the Journal of the Faculty of Sciences of the University of Tokyo,[197]. It was then that the connectionsof Sato’s workwith the previous achievements of mathematics becameapparent; but before we get to this topic, let us briefly attempt to recreate the frameworkof ideas within which Sato was movingat the time of his creation. It maybe said that the origin of Sato’s hyperfunctiontheory lies in the much older uneasiness of mathematicians with the pseudofunctions which engineers and physicists had been using at various stages in history. Weonly need to recall Fourier uninhibited use of a "series" expression for the delta function in his treatise on the theory of heat [61], or Poisson’s notion of dipole, [188], where the derivative of the delta appears, or finally Heaviside[76], [77], [78] and Dirac [40] with their introduction of various species of operational calculus and with a formal definition of the delta function. The problem which engineers and physicists alike were concerned with, was the difficulty of dealing, on one hand, with somesimple but hard to describe physical objects (impulsiveforces or, again, electric dipoles), and on the other hand, with very singular functions. Wehave mentionedbefore the British engineer O. Heaviside, whofound it useful to introduce the function which, today, carries his name. He neededsuch a function in order to study somedifferential equations whicharise in the study of signal transmission for telegraphic signals. The problem, of course, was that such a function could not be differentiated according to the usual rules, and yet it becamenecessary to do so, possibly by changing the meaningof differentiation. To be more precise, the Heaviside function H(x) could be differentiated everywhereexcept at the origin, but that was exactly the point in which it was interesting to understand the meaningof its derivative Hr, especially from a physical point of view. It maybe worthwhile recalling that the unorthodoxuse that Heavisidemadeof differentiations, while giving him correct results, also caused his dismissal from the LondonAcademy of Sciences,[151].

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Dirac, on the other hand, carried this approach one step further, by introducing his delta function ~(x) in such a wayas to modifythe notion itself of function; once again, mathematicaland physical necessities were pushing for a different notion of function and of differentiation (the developmentof the notion of function is, after all, one of the lines whichone couldtry to follow in studying the history of modernmathematics). As it is well known,the first satisfactory solution to the difficulties posed by the objects of Heaviside and of Dirac was due to L. Schwartz who, in 1947 (see his original work[210], but also [151] and [214] for a detailed accountof the developmentsof the theory of distributions) developeda deep and far reaching theory of generalizedfunctions whichhe called distributions; finally, the equality H’(x) = 6(x) was fully justified. Within the theory of distributions, as it is well known, the basic viewpoint is essentially based on the vision (inherited by Dirac) a generalized function as a functional on somespace of test functions; in the case of Schwartz’s distributions, the space of test functions is the space T) of infinitely differentiable functions with compactsupport, and therefore the theory of distributions is essentially a theory which can be applied to the study of differential equations on arbitrary differentiable manifolds. Theseconsiderations bring us to someimportant points in the study of hyperfunctions; as we haveseen, even hyperfunctions(at least in the case of compactly supported hyperfunctions) can be seen as functionals on the space of germs of real analytic function; this was not, however,the original approach whichSato followed in his papers and in his work. Accordingto his ownrecollections [151], Sato was initially unhappywith the fact that the theory of distributions would work in the category of differentiable functions and manifolds; this fact stroke him as unnatural, and he was firmly convinced that the natural space to use as a space of test functions wouldhave to be the space of analytic functions. This point is really worth of attention, since Sato (and his coworkers) have always claimed to be analysts in the sense of classical mathematics(see, in this regard, the introduction to [123]); as we look back to the developmentof eighteenth century mathematics, we cannot but recall howanalyticity was considered the quintessential form of a function (functions, so to speak, had to be expressed as convergentpowerseries, or else there was not even the possibility to work with them; one might even recall Hilbert’s address in Paris). It is howeverclear that the space of real analytic functions could not be easily taken to be the space of test functions, essentially because we could not consider compactly supported real analytic functions; some other way had to be found, to circumventthis difficulty. It is not easy to see through the first ingenious pages in which Sato develops the theory of hyperfunctions; the results which he proves, as we shall see, are not essentially new, but what is totally newis the spirit and the far reaching approach whichhe follows; the key point

1.6.

HISTORICAL NOTES

75

in his first work is the (successful) attempt to give an operational meaning the notion of boundaryvalue of a holomorphicfunction. The importanceof this notion had already been established in physics (see, for example, the work of Bogoljubovon the edge of the wedge[22]) and it was very well knownthat all distributions could be expressed as boundaryvalues of holomorphicfunctions, as we have already mentionedin this Chapter. It was also known,however,that someholomorphicfunctions did not have boundaryvalues, at least not in the sense of distributions, since the boundaryvalue at the origin of a function such as el/z

is just too muchof a singular object to be dealt within the theory of distributions. Still, manyproblemsfrom the theory of causality and the study of dispersion relations, see e.g. [28], [30], [44], imposethe considerationof these moregeneral boundary values, and it was therefore natural for the young Sato, whohad originally

been a student

of the future

physics

Nobel prize

Tomonaga, to look

for a precise formalization of such boundaryvalues. The connectionswith dispersion relations explain, at least partially, the interest of Sato for a notion whichwouldsatisfactorily deal with boundaryvalues of holomorphicfunctions; still, it is even moreinteresting to examinethe many links between the work of Sato, and what had been done previously in totally different areas of the world. 1.6.2

Analytic

Functionals

Whenthe first papers on hyperfunctions were published in English by Sato, it was immediatelynoted by A. Weil, [196], that one of Sato’s first results (his duality theorem) had actually been knownalready to G. KSthe, who had published a very interesting series of papers on integral representations of analytic functionals [138], [139], whichculminated with the duality theoremitself. This comment,which is mentioneddirectly by Sato in the second part of [196], certainly spurred him towards the production of his totally new approach to the several variables case (in which the duality theorem acquires a muchdeeper meaningand difficulty). One must also say that this comment only gives part of the story. Indeed, the duality theoremwhichis usually attributed to KSthe(and rightly so, since he wasthe first to provide a conclusiveproof), has a rather long and by nowwell established story (see our [218], [219], where morehistorical details are given on this topic). It can be safely said that the first to look upon such a problem was the Italian mathematicianL. Fantappie’, a rather singular student of Volterra and of Severi, who,in 1924, [55], (but see also [57] and, finally, [59], for a complete bibliography on Fantappie’s workon the theory of analytic functionals) created a theory of analytic functionals, on the invitation of Severi whohad wantedhim

76

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HYPERFUNCTIONS

to understandin a general fashion the action of operators such the differentiation operator. The original definitions and notations of Fantappie’ are rather cumbersome, so we will simplify here his approach, while trying to conveyhis main ideas. 1Fantappie’ defined the space of ultraregular functions on an open set U of~T~ as the space of functions which were holomorphicin U and which would vanish at infinity, whenthe point at infinity cx~ belongsto U. Fantappie’defined therefore the space S(1) of all ultraregular functions, which(becauseof its definition) is not even a vector space. A linear region R was then introduced as a subset of S(1) closed with respect to the ~T-linear combinationsof its elements (for the sumof two functions to be well defined it was deemednecessary and sufficient that the regions where they were defined wouldhave a non-emptyintersection). The first interesting remarkof Fantappie’ was the fact that there was a bijective correspondence between linear regions R and what he called their characteristic sets A, i.e. the intersection of all the regions oftT~°1 wherethe functions of R are defined. It turns out that A is a proper, non-emptyclosed subset of tT~~. Somethingmore can be said; indeed R can be shownto coincide with (A), where, in modernterms, (A) can be defined

u(r~(U(A),O)) with U(A) varying over all possible open neighborhoods of A and F~ denoting the space of sections of holomorphicfunctions which vanish at infinity (ultraregular functions). Finally, Fantappie’ defined his funzionali analitici (analytic functionals) as the element of the dual of (A). Well, actually this wouldbe to go too far, since, as we have seen, the space (A) was not even a vector space, and even the topology which Fantappie’ built for it was not too adequate to a modern treatment of duality. Let us not forget that all of this was taking place in 1924, so that the necessary continuity conditions which had to embeddedinto the notion of "functional" were suggested to Fantappie’ by somefamous results of Poincare’ on analytic dependenceof solutions of analytic Cauchyproblems for partial differential equations. The precise definition of Fantappie’ runs as follows. For a given a linear region R, a iT-linear mapF : R --+ d~ is an analytic functional if: (a) given Y0, a continuation of Yl, it is F(y0) = F(yt); (b) if y -- y(z, ~) is holomorphicin z and ( (actually, someextra conditions technical nature are necessary), then f(() : Fz(y(z,()) is holomorphicin (, where defined.

1.6.

HISTORICAL NOTES

77

The reader will note howthese analytic functionals are, mutatis mutandis, essentially our analytic functionals in the sense of (O(A))’. As we have seen in the duality theorem, such a space is isomorphic to the space of hyperfunctions with compactsupport in A, which, on the other hand, coincide with the space of functions holomorphic in U \ A, with U some open neighborhood of A in ~T. This result had already been established, in a rough form, by Fantappie’ in [56]. Let us recall the definition of indicatrix of a functional (whichwe have already used in our treatment of hyperfunctions in the previous pages): given F an analytic functional on A (whether in our modernsense or in Fantappie’s sense), we define its indicatrix (sometimescalled the Fantappie’-Satoindicatrix) as the holomorphicfunction

It turns out that such an indicatrix is holomorphicexcept at A and that the following duality theoremholds true, [56]: Theorem1.6.1 Let A C_ (T be a compact set and let O(A) be the space holomorphic functions defined in some open neighborhood of A. Let F be an analytic functional on (9(A) and let Ugbe its indicatrix function. Then, for y in O(A), y holomorphic in some open neighborhood M of A, one has: F(y) = (2~i)-1 f~ uF(t)y(t)dt, where 7 denotes a smooth curve which separates the compact A from the set ((~ U {~c}) \ M. In other words, the indicatrix UFrepresents F via the duality integral. Webelieve the reader will notice how Theorem1.6.1 actually foreruns KSthe’s duality theorem; in both cases, indeed, one sees howanalytic functionals carried by a compactset A are in a bijective correspondencewith the space of holomorphicfunctions defined on the complement(T \ A of In a way, therefore, and with the caution which is always necessary when makinghistorical statements, we can say that Fantappie’ had already attributed a dignity to pairs of holomorphicfunctions having discontinuities along the real line and, therefore, to prehistoric hyperfunctions. He was of course lacking the idea of considering these objects as generalized functions. The path between the work of Fantappie’ and that of KSthe, however, is worthyof a short description, since it provides a case study in the development of mathematical ideas. Indeed, as we have pointed out, Fantappie’s approach was mainly a set theoretical approach (even though, at a later stage, he tried to develop a topological description for the spaces he was workingwith). This was due, partially, to the specific applications which Fantappie’ had in mind

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(interestingly enough,as weshall see, Fantappie’maininterest was, for a number of years, concentrated towards a unified treatment of the Cauchy-Kowalewsky phenomenon)and, more fully, to the fact that the spaces which Fantappie’ was constructing wereessentially limits of Frechet spaces, and dual of such spaces. In the twenties and thirties, whenFantappie’ was first developinghis approach, no duality theory was knownfor such spaces, and so no theoretical backgroundwas available to help the Italian mathematician develop a more rigorous approach to this study. With WorldWarII, however,we have one of the major revolutions in twentieth century analysis; the developmentof distribution theory by L. Schwartz. We wish to briefly describe howthis revolution altered the course of events which have (indirectly) led to the birth of hyperfunctions. As it is well described in Lutzen’shistorical analysis, [151], a rather unknown "exercise" of Schwartz was instrumental in the developmentof the theory of distributions. According to Lutzen, Schwartz, immediately after writing his doctoral thesis on series of exponentials, [209], was workingin someisolation and decided to try his hand on the problem (which he apparently regarded as devoid of any intrinsic interest) of extending the duality theory for Banach spaces to the case of Frechet spaces. He clearly succeededin his ei%rts, and one of the legacies of this period is a groundbreakingpaper which he co-authored with J. Dieudonne,[39]. This paper deals with duality theory in Frechet spaces and their limits. This work has proved one of the most influential pieces of works in the theory of topological vector spaces, in view of the large use of l~rechet spaces in modernanalysis. Fantappie’ recognized the importance of this development,and one of his best students, J.S. e Silva, wrote an important doctoral dissertation [211], under the guidance of Fantappie’ in 1950. Silva tried to place Fantappie’s theory on firm topological grounds. His attempt has been praised, but unfortunately his approach contains someserious flaws which aroused the interest of two mathematicians: KSthe and Grothendieck. They both realized the interest of Silva’s program,as well as the shortcomingsof his work, and, in the early fifties, wrote a series of papers, [69], [138], [139], in whichhis ideas were clarified. One can safely say that in these papers lie the foundations for what we nowconsider the theory of analytic functionals, and certainly contain the first complete proof of the duality theorem. The premature death of Fantappie’ in 1954and his loss of interest for mathematicsin the last few years of his life, have unfortunately prevented what could have becomea very fruitful collaboration. For the sake of completeness,we mayalso point out that someextensions of KStheresults were obtained in the early sixties by the GermanmathematicianH. G. Tillmann, [221], [222], whoessentially considered the case of functionals carried by subsets of the real line: by that time, as we know,hyperfunctions already existed. Recent historical research has shownthat Tillmann was not aware of them and that he was getting closer and closer to a completeformulation of the theory, at least for the one variable case. It has of

1.6.

HISTORICAL NOTES

79

course to be remarkedthat before Sato, nobodyever thought of looking at these functionals as generalized functions, and it is probably in this idea which lies the maincontribution of Sato (at least for the single variable case, since, as we shall see, the case of several variables is so totally newthat no mathematician can claim having been a precursor); on this topic we refer the interested reader to the workof Lutzen [151], or to the original papers by Tillmann. It might be mentionedthat we can probably trace to this series of developments the existence, and the strength, of the French and the Portuguese school of hyperfunction theory and microlocal analysis; it is somehow surprising that so little has remained, on the other hand, in Italy (where only a handful of mathematicianshave done serious research in this field) and in Germany. 1.6.3

Generalized

Fourier

Integrals

Wehave, so far, only discussed howthe theory of analytic functionals was a precursor of the theory of hyperfunctionsin one variable. In this section we will discuss Carleman’sFourier theory, as it is exposedin his book[34] (even though somecommentsare taken from our historical survey [218]). One of the most intriguing problemsof twentieth century analysis has been the attempt to define somekind of Fourier transform for functions which do not decrease at infinity. Someauthors (Ehrenpreis amongthem) even venture to hint [52] that such a theme could be taken as a path through most of the analysis which has been developedin the last sixty years or so. Finding Fourier integral representations for functions of arbitrarily large growthis, indeed, one of the motivations of the celebrated Ehrenpreis-PalamodovFundamentalPrinciple, [52], [178], whichis also one of the themesdear to the Japanese school of MicrolocalAnalysis(see the references in [123], for example). Attempts to define Fourier representations for functions which do not belong to L1 have a long history, and we should at least mention the names of Hahn [72], Wiener[229] and Bochner[20]. Fromour point of view, however,we would like to fix our attention to the work which, in the thirties, was done by the SwedishmathematicianT. Carleman,and whichis essentially collected in [34]. Carlemannoted a very simple and well knownfact: if f is a Lebesgueintegrable function on ~, then its Fourier transform g(z) can be written in the following natural way: g(z) = (2~)-½ f+_~ exp(-izy)f(y)dy

’i

= (27~)-~ exp(-izy)f(y)dy

(2~)-½ r+¢~ ex p(-izy)f(y)dy =

=gl(z)- g2(z).

If now, in this expression, one allows the real variable z to take on complex values, it is immediateto notice that gl is holomorphicin the upper half Complex

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. plane and that g2 is holomorphic in the lowerhalf complexplane, so that one sees that the Fourier transform of an L1 function can be expressed as the difference of two functions holomorphicon opposite sides of the real line in ~T. Carlemanthen stated two problems which appeared very naturally from the consideration of the identity established above. Is it always possible to decompose a function defined on ~ as the difference of two functions holomorphic on the opposite sides of the real axis? And, in the affirmative case, is this decomposition uniquely determined? It was not too difficult, for Carleman, to prove that both questions can be answeredin the affirmative, and that, therefore, there exists somekind of equivalence between functions defined on the real line (we refer to Carleman’s[34] for the details) and pairs of holomorphicfunctions as aboveor, equivalently, elementsof (.9(~ \ Kt). Nowthat the intrinsic interest of these pairs of holomorphic functions was established, Carlemanwent on to notice that a Fourier representation such as the one given above did actually makesense even for functions which had polynomial growth at infinity, rather than L1 decay. If, indeed, a function f satisfies a polynomialgrowth condition of the form /If(x)ldx

= O(Ixlk), for somek C ~W,

then its classical Fourier transform may not exist, but the functions gl and g2 introduced above are well defined, and holomorphicon complementaryhalfplanes. Wecan therefore conclude that not only functions could be represented as pairs of holomorphicfunctions, but that such pairs could also be considered as somesort of generalized Fourier transform. Upto this point, however,the theory was unsatisfactory (and Carlemanwas quick to point this out) since it was somehowlacking symmetry; Carleman thus proceeded in trying to show how to treat pairs of holomorphicfunctions as somesort of generalized functions, thus comingcloser than ever to the point of view which wouldhave eventually producedhyperfunctions. It wouldbe inappropriate here to enter the details of the work of Carleman, for which we refer the reader to the original work[34], but we content ourselves by pointing out that Carlemanactually succeeded in developing a full-fledged theory of Fourier transform for pairs of holomorphic functions, in such a waythat one could even obtain an inversion formula. It has not been explored, as far as we know,the existence of possible connections between Carleman’stheory and the theory of Fourier transform for hyperfunctions developed by Kawaiin [130]. 1.6.4

Hyperfunctions

in Several

Variables

Up to now, we have mentionedsome of the physical motivations which led Sato to the creation of hyperfunctions (essentially the dispersion relations and the study of causality), together with the work of someprecursors, which, however,

1.6.

HISTORICAL NOTES

81

had confined their ideas to the case of a single variable. This restriction is not too surprising, in view of the extreme complexityof the technical tools which becomenecessary in trying to extend Sato’s original ideas to several variables. As we have seen in this chapter, the first difficulty lies in the fact that the notion itself of boundaryvalue, which had been the guiding notion in the developmentof Fantappie’s theory, as well as in Carleman’sone, is suddenly unclear, since there are just too manydirections to take into account (and not just "above" and "below"). But other formidable hurdles seem to forbid the generalization of the notion of hyperfunction from one to several variables; to begin with, even the theory of complexanalysis in several complexvariables was not, at the beginningof the century, too well developed; also, someof the key results in one variable, are not true anymore(just think of Hartogs’ removable singularities theorem, or of the vanishing of the first cohomologygroup with coefficients in the sheaf of germs of holomorphicfunctions). In one word, one mayprobably say that the main difficulty lies in the fact that it is not true anymorethat any open set in ~n is a domainof holomorphy. Fromwhat it is knownso far, it appears that Sato was led to the discovery of the right wayto define hyperfunctionsin several variables by the appearance, whenstudying dispersion relations, of families of holomorphicfunctions which satisfied "strange" relations, which later on turned out to be just the cocycle relations needed for the definition of relative cohomology.Sato was, however, unawareof relative cohomologytheories, and so it had to develop completely from scratches such a theory for the sheaf of germs of holomorphicfunctions. Interestingly enough, at almost the same time, Grothendieck was developing a theory whichis almost exactly of the samescope, eventhoughfor totally different purposes (Grothendieckwas actually setting up his revision of the methods of Algebraic Geometry,whichwere totally algebraic in nature, so that one may say that Algebraic Geometryand Algebraic Analysis share at least this part of their past). Withrespect to these developments,it is interesting to note, [204], that Sato first realized a hyperfunctionas a relative cohomology class back in the SpringSummerof 1958. Later on, however, he left for the Institute for Advanced Studies in Princeton in 1960where, in the companyof L. Schwartz, he explained to A. Weil (in his office) his hyperfunction theory using relative cohomology. The reaction of A. Weil did not encourage Sato too much(he was not aware that Weil was not, at the time, too fond of cohomologicalmethods)and this fact apparently prevented him from writing the third paper on hyperfunctions, which wouldhave contained the theory of derived categories, including his treatment of spectral sequences and hypercohomology.Wenow knowthat the equivalent of this material is containedin Hartshorne’s[73]. Incidentally, Sato gavea series of talks on :D-modulesin the KawadaSeminar at the University of Tokyojust before his departure for Princeton. By 1960, however, the developmentof hyperfunctions was .essentially com-

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HYPERFUNCTIONS

pleted and, in fact, just before leaving Tokyofor Princeton, Sato had already switched to the study of the algebraic treatment of systems of differential equations. Wewill comeback to this topic in a later chapter. Even though, before Sato and Grothendieck, nobody had developed any notion of relative cohomology,it might be interesting to take a step back to the work of Fantappie’. As we already mentioned(but the interested reader is referred here to [58] and [215]), Fantappie’ was essentially interested in applying his theory of analytic functionals to the study of the Cauchy-Kowalewsky theorem, which he wantedto approach from a functional point of view; he actually managed[58], [60], to express the correspondencewhichassociates to the initial conditions the unique solution by meansof an analytic functional (whose argumentswere, in fact, the initial conditions); in order to computethe solution to given initial value problem,it was necessary to knowthe value of the functional for very simple functions (i.e. to knowthe Fantappie’ indicatrix of the functional). The nature itself of this problemled Fantappie’ (and someof his later students) to deal with analytic functionals acting on holomorphicfunctions of several complexvariables. Fantappie’ had to struggle very hard trying to understand the correct generalization of his work to the case of several variables; we nowknowwhyhis task was so hard, but we doubt that his difficulties were fully appreciated at the time, even though a large part of Italian mathematics was concerned with similar problems, from the more geometric point of view of Algebraic Geometry. In the multivariable case, Fantappie’ managedto mimic his ownapproach by considering [58], [215] ultraregular functions defined as analytic functions on (T~TTM whichwouldvanish at infinity (the notion of infinity in the case of several complexvariables was, itself; a debated one at the time). Even thoughthe initial steps were simple, Fantappie’ met his first challenge whentrying to prove an analogueof the duality theorem. The first difficulty was in the impossibility of easily defining an indicatrix; the difficulty is of geometricnature, since the obvious kernel which one wouldlike to consider, y(z, ~) (Z 1 -- ~1) " (Z2 -- ~ 2) " -’1 ’ (Zn -- ~n)’ does not necessarily intersect all functional regions (A). However,even when such an indicatrix is used (Fantappie’ called it the antisymmetricindicatrix), duality theorem can be proved, except under somevery stringent hypotheses on A, in whichcase the result is just a trivial restatement of the one-variablecase. Even more interesting is the phenomenon of the multivaluedness of analytic functionals. Indeed, whenintegral representations are used to describe analytic functionals, they turn out to be multivalued (even whentheir indicatrices are not) because of the different integration contours which can be used in the representation formulas. This phenomenonis interesting as well as it must have been unpleasant to Fantappie’; its explanation, however, was too deep and

1.6.

HISTORICAL NOTES

83

required too muchnew mathematics for Fantappie’. Indeed, it turns out that it was only with the brilliant work of Martineau, [157], [158], that Fantappie’s intuition was proved correct, even though one had to renounce the hope to describe a functional via a holomorphicfunction (its indicatrix). As Martineau correctly pointed out, the way out is the use of relative cohomologyclasses of the sheaf of holomorphicfunctions (and the ultraregularity phenomenonis easily dealt with, by meansof a correct choice of representative cocycles). We therefore see how,even though in a primitive way, Fantappie’s ideas were on the right track, and howit was really only the cohomological approach which was missing. Weshould take this chance to describe the tremendousrole which Martineau had in the diffusion of the theory of hyperfunctions (he maybe credited as being the single strongest force behind the great developmentof the hyperfunction/microlocal analysis school in France). This role has been "recognized" by the Japanese school itself, whodedicated to his memorythe volumeof the Proceedings of the October 1971 Katata Conference [141], which Martineau could not attend because of his premature death. Besides his influential Seminaire Bourbaki of February 1960, in which hyperfunctions are introduced to mathematicians outside Japan, Martineau had already been working for a while on the study of analytic functionals, and their applications to the theory of infinite order differential equations. Such problems (in which he was exploiting sheaf cohomologyto provide conclusive answers to questions left open by Fantappie’) madehim naturally the readiest recipient for the theory of Sato. 1.6.5

Infinite

Order Differential

Equations

Wehave seen in this chapter howinfinite order differential operators play a rather important role in the developmentof hyperfunctions; in the next few chapters we will see how essential the study of these operators is, and why they are so strongly intertwined with the theory of hyperfunctions and microfunctions. In this short section of the appendix,we wish to attract the reader’s attention to someaspects of mathematicsof the early years of this century which are not so widely known,but in which someaspects of these moderntheories showedtheir first appearances. Werefer here to S. Pincherle, one of the major Italian characters in turn of the century mathematics,whodeveloped(see [186], [187]) a rather refined theory of infinite order differential operators, whichhe called "operazioni distributive" (distributive operations), to signify their linear character. Interestingly enough (and the reader maywant to read our [220], in which someof these aspects are treated with somedetail), Pincherle’s workis actually mentionedby Fantappie’ as one of the motivation for his creation of analytic functionals, and, in particular, Pincherle’s distributive operations can be viewed as special examplesof analytic functionals.

84

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HYPERFUNCTIONS

Pincherle’s approach to the study of infinite differential operators was, in manyways, similar to the one followed by the workers in microlocal analysis. Indeed, after his trip to Berlin in 1877-78(wherehe attended a complexanalysis course taught by Weierstrass) Pincherle was led to the study of the inversion of definite integrals, and, in particular, to the considerationof the integral equation of the first kind f~k(x, y)~(y)dy = f(x), (k, f given functions, ~ an unknownfunction, ~, a curve in the complexplane and x, y in (T) whichhe regarded as an operator A : ~ ---+ A(~) = This functional point of view was not totally new, since it had been successfully employedin the case of the Fourier and the Mellin transform (to quote just two relevant examples), but Pincherle had the intuition to extend the geometrical theory of homographiesfrom the finite dimensionalcase to the case of infinite dimension, by letting such operators act on the spaces of formal powerseries and of convergent powerseries. As we follow Pincherle’s developmentof his theory (completely described in his [187]), we can see that his operators can be interpreted as the convolution operators associated to analytic functionals with compactcarriers. Let us spend a couple of words on this aspect, which maybecomeof interest later on. Let T be an entire function in one variable (note that all of Pincherle’s theory was strictly confined to the single variable case), and let # be an analytic functional carried by the compact set K (this means that # belongs to O’(K), where the compactK is not uniquely defined, since analytic functionals do not constitute a sheaf, and therefore a notion of support is not fully defined, but is replaced by the weakernotion of carrier, see [83]). Onecan then define a convolution between # and ~, as follows:

~ ¯ ~(z)= 0; more generally a subset of U x ~n is said to be conic if it is stable u~der the transformations

(x, ~) -~ (x, p~),p Definition 2.2.2 Let f be a distribution on U. We say that f is infinitely differentiable in (xo,~o), Xo E U, ~o ~\{0}, if the re is a c ompactly sup ported infinitely differentiable function X which is identically one in a neighborhood of Xo, and there is an open cone Fo in j~n containing ~o such that: for every M > 0 there is a non-negative number CM such that, for any ~ in F0,

I(x~)(,~)l CM( 1"t-I~l)-M. The reader will note that the notion of conic set is essential in this setting, and so the cotangent bundle is often replaced by the cosphere bundle. In our flat case this correspond to replacing T*U-- U x ~n by the following quotient: S’U = U × (~" \ (0})/~ where the equivalence relation is given by (x, ~) ~ (y, 7) if and only if x = y ~ = p~ for some positive p. There is a canonical projection r of T*U \ (0} onto its quotient S’U, and a subset F of T*U \ (0} is said to be conic if and only if it is F = ~-~(~(F)). will also say that F is conically compact if ~(F) is compact (this, of course, does not mean that F itself is compact: as a matter of fact, this is never the case). Definition 2.2.3 A distribution f on an open set U is said to be infinitely differentiable in a conic subset F of T*U\ (0} if it is infinitely differentiable a neighborhood of every point of F.

96

CHAPTER2.

MICROFUNCTIONS

The set whichdescribes the singularities of the distribution f is therefore the complementof the largest set in whichf is differentiable, as given by the following fundamentaldefinition: Definition 2.2.4 Let f be a distribution in an open set U. The complementin T’U\ {0} of the unionof all conic opensets in whichf is infinitely differentiable is the wavefront set of f and will be denoted by WF(f). Before we go back to the case of hyperfunctions, we wouldlike to give some properties of this notion. In order to do so, we need to take a brief detour through the concept of pseudo-differential operator. This will be, in any case, a useful introduction for the notion of microlocal operators, whichwe will deal with in section 5 of this chapter. Webegin by constructing what will later be the symbols of the pseudodifferential operators. The notion of pseudo-differential operator is a very natural one, which stems from the time-honoredattempt to algebrize analysis. Pseudo-differential operators, which so clearly opened the road to microlocalization, have been called, [224], "... the most importantstep forward in our understandingof linear partial differential equationssince distributions". To understand the development of what might seem a difficult concept, we will begin with the notion (introduced by F. John in the fifties [100], [101]) "parametrix"of an elliptic linear partial differential equation. Let then P = P(~I,..., ~n) = P(~) be a polynomial with complexcoefficients in n real variables, and let P(D) = P(D1,...,

D,~)

be the corresponding linear partial differential operator obtained by replacing ~jby 0 Dj = -i--; Oxj we say that P(~) is the symbol of P(D). Consider nowthe equation

P(D) = with f E Z~(~~) given, and let us look for a solution ~ E 7?. The first, attempt wouldbe to formally apply the Fourier transform, so that P(~)~(~) =f(~),

and finally

naive,

2.2.

(2.2.3)

SINGULAR SUPPORT, ESSENTIAL SUPPORT

u(x) { 1~ )" f f(J ~)ei~.~d

97

¯

The integral on the right hand side of (2.2.3), however,makesusually no sense, because of the zeroes of P. Assume,however,that P(D)is elliptic as in the following definition. Definition 2.2.5 Let m = degP(~) and write P(~) = P,~(~) + Q(~), degQ(~) < m - 1 and Pm(~) homogeneousof degree m. Wewill call Pm(~) principal symbolof P(D)and we say that P(D)(or P(~)) is elliptic if Pm(~) for all~ ~ O. Since Pmis homogeneous its zero-set Vp,~ = {~ E J~’~ : Pm(~)= 0} n is a cone in fit knownas the characteristic cone of P(D)(or P(~)). Remark2.2.1 It is immediate to verify that for n = 1 all polynomials are elliptic, while for n _> 2 the class of elliptic polynomialsbecomesan important, but proper subclass of the class of polynomials.The symbolsof both the Laplace and the Cauchy-Riemannoperators are elliptic, while the symbol of the heat operator is not. Remark2.2.2 It is an easy exercise to verify that if P(~) is elliptic then its characteristic variety Vp = {~ E ff~n : p(~) = 0} is compactin ~n. This simple remark has, in fact, relevant consequences, since we maynowconsider the integral in (2.2.3) only outside of a compactcontaining Vp. More precisely, if Vp is contained in the ball centered in the origin and of radius r and X is a regularizing C~¢ function such that X(~) = 0 for I~1 r, X(~) = 1 fo I~1 > r~ > r, then we can define a sort of approximatesolution by 1 Howgoodof an approximationis v to u? It is actually quite good; in fact one immediately sees that P(D)v(x) = f(x) where R is the operator defined by f(~)(1- X(~))e~’~d~, i.e.

Rf with h the Fourier transform of 1 - X.

=h*f

98

CHAPTER 2.

MICROFUNCTIONS

By its construction 1-X is compactlysupported, and therefore h is (extendible to be) entire of exponential type (just apply the Paley-Wienertheorem mentioned above). Anotherimportant point is the fact that

[P(¢)I-> IP.~(¢)I-IQ(¢)I-> 1 for ]~] large enough, so that x/P actually defines a tempered distribution on ~n which is (by the isomorphismtheorem) the Fourier transform of a tempered distribution K. Then(2.2.4) actually reads v=K.f, and we have therefore obtained that (2.2.5)

P(D)K = 5 -

or equivalently (looking at the corresponding convolution operators) (2.2.6)

P(D)K = I -

where R : £’ -~ £ is a continuous linear operator. Wewill use (2.2.5) equivalently (2.2.6) as the definition for the notion of a parametrix K for the operator P(D). A parametrix is therefore a distribution K satisfying (2.2.5) a convolutionoperator satisfying (2.2.6) with R as above. Remark2.2.3 The theory of parametrices is rich (see e.g. [100], [224]). It will suffice here to note that one can use K to construct a fundamentalsolution for P(D). Indeed, if w is an entire function such that P(D)w = (and this is easily seen to exist by purely functional analytic arguments)then E=K+w ~s a fundamentalsolution since P(D)E = P(D)K + P(D)w = 5- h + Remark2.2.4 To summarize, we have seen that a parametrix K is essentially a clever modification of a right inverse of P(D). The next step towards pseudodifferential operatorsis the generalizationof these ideas to the case of differential operators with variable coefficients. If f~ is an opensubset of ~’~, a variable coefficients differential operator on f~ is defined by ~, P(x,D)= ~, c~(x)D

2.2.

SINGULAR SUPPORT, ESSENTIAL SUPPORT

99

with a = (al,...,an), lal = al +... + an, = (D1,..., Dn ) and where th e functions ca(x) are suitably regular in f~ (e.g. ca couldbe infinitely differentiable or real analytic in f~). Then the symbol of P(x, D) is the polynomial in 2n variables P(x,()= ~ c, ~(x)( and its principal symbolis defined as ’~. Pm(x,~) = ~_, cc~(x)~ As before one defines

VPm := {(X,~)~ ~ × ~n: Pm(Z,~) =0}, which is nowan algebrffm variety in ~ × ~n or (more generally) in T’f2. In the case of variable coefficients operators wesay that P(x, D) is elliptic if, for everyx0 Vp~(Xo):-- {~¢ ~n : Pm(xo,~) ---- 0}---- {0}. One can then try to replicate the process which was employedin the case of constant coefficients to construct operators K and R such that P(x,D)gf

= f- R

with R : 8’(f2) --~ 8(f2) continuousand linear. It is therefore natural to try construct a kernel k(x, ~) such that K can be defined by Kf(x)

x,

If we set, as before, v(x) = Kf(x), we have for any function w(x), ~::’~) =ei*’¢ P(x, D +¢)(w(x) P(x, D)(w(x)e and, therefore,

~(~, ~)~(~) = ~ f(~)e~~P(~,~ ~/~(~, ~)e~, so that we would have (2.2.7) if we could solve (2.2.8)

P(z, D)K =

P(x, D + ~)k(x, ~)

100

CHAPTER2.

MICROFUNCTIONS

In order to solve (2.2.8) note that P(x, D + ~) = P,~(x, ~) + ~ 5(x, j=l

wherethe Pj(x, ~, D) are differential operators of order j in x, whosecoefficients are homogeneouspolynomials of degree m - j in ~. It follows that in order for k(x, ~) to be a solution of (2.2.8) it is necessaryto take it as a sumof functions homogeneous in ~. But k must also be tempered in the ~ variables (in order for K to appropriately act on the space of distributions) and so our best bet is an infinite series of terms with negative degrees of homogeneity: k(x, ~) = ~ kj(x, j=0

and kj homogeneousof degree -j. Wewill not push here this attempt which we already knowcannot possibly worksince (2.2.7) cannot be exactly solved (in view of the zeroes of the symbol of P). Onceagain we need to eliminate these singularities with somesort of cut-off process (we are dealing here with an elliptic operator). However,since we nowhave an infinite series, just one cut-off function will not suffice and we will need to define our kernel by (2.2.9)

k(x, ~) = ~ X~(~)kj(x, j=O

Wewill not give all the details of the construction of the parametrix, our goal being here to motivate our subsequent treatment. It will suffice to say that suitable cut-off functions Xj can be constructed so that if k(x,~) is given by (2.2.9), then P(x, D + ~)k(x, ~) = 1 r(x, ~) with a function r(x, ~) which can be explicitly computedand which is such that the operator R defined by

is a continuous linear mapfrom 8’(~) to 8(~). This proceduretherefore provides us with a parametrixfor the case of variable coefficients elliptic operators. Our discussion so far can be summarizedby saying that the attempt to invert a differential operator h~ lead us to the introduction of a general cl~s of integral operators (of which both parametrices and differential operators are a special c~e), namely those operators which, for suitable kernels t, can be written 1

2.2.

SINGULARSUPPORT, ESSENTIAL SUPPORT

101

Wewill nowgive a formal definition for such operators, after we deal with some preliminary notions. Definition 2.2.6 Let K E T)(~ × ~) be a distribution 14 is separately regular if both

on ~ × ~. Wesay that

K: V(x) and Kt : ¢(y) --~ (K(y, mapT)(~) into ~(~). In addition, such a distribution will be said to be very regular if, besides being separately regular, it is C~ outside the diagonal of ~ × ft. The study of these kernels is crucial in view of the celebrated Schwartz’s kernel theorem (maybehis most important early result on distributions) which showsthat for any continuous linear map

K: there is a unique distribution K(x, y) in fl ~ ~ such that for all f G ~(~) K(f)(x) = (K(x, y), f(y)). Note that this condition is equivalent to requiring that K maps~(~) into 8(~) and extends as a continuous linear mapof 8’(~) into ~’(~). Definition 2.2.7 A very regular operator K is said to be regularizing if it extends as a continuouslinear mapof S’(~) into 8(fl) (rather than just ~’(~) Remark2.2.5 In order for an operator to be regularizing, it is necessary and sufficient that K is C~ in ~ ~ ~. Remark2.2.6 A typical example of regularizing operator is the operator R whichwe constructed while developing a parametrix for elliptic operators. The content of Remark2.2.6 actually originates the following Definition 2.2.8 Let P(x, D) be a differential for P is any very regular operator

such that KP - I is regularizing.

operator in fL parametrix

102

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MICROFUNCTIONS

Wehave already discussed the importance of the existence of parametrices for the surjectivity of the operator P(x, D). Onecan also easily prove that if P admits a parametrix, then it is hypoelliptic in the sense that if P(x, D)f E ~, ~. then f E C Weare nowready to discuss pseudo-differential operators (or, rather, standard pseudo-differential operators): Definition 2.2.9 For any real number m, we denote by

s (a,a) the subspaceof C°°(~ × ~ × J~Ztn) whoseelements a(x, y, e) satisfy the following estimate: for any compactsubset T of f~ x ~, and any a, ~, 7 > O, there exists C = C(a, ~3, 7; T) >0 such that [D~D~D~a(x,y,e)l _0 and every pair of multi-indices ..., ~,~), fl =(~, ..., fin), there exists C~,Z,N>_0 suchthat suzp IO$O{[X(x)a.(x,{)]I 0 such that for any F ~ O(V), with V convexset such that Zo ~ U C_{Iz - zol m and b,~(z) ~ O, we will say that Q(z, D) order m.

2.5.

MICROLOCAL OPERATORS

125

Theorem2.5.1 Let Q(z, D) be a microdifferential operator of order less than or equal to zero, defined in a complexneighborhoodof a real open set U. Then Q defines a sheaf endomorphismon B/,4 and on C. Proof. Let x E U and let f be a germ of a hyperfunction which represents an element If] in C(~,~) (a similar argumentapplies to the case of (B/A)x, and

f(x) = F+(x+ i0) - F-(x for somepair of holomorphicfunctions (F+, F-) defined near x. To define Q(x,D)[f], we apply Q(z,D) to both F+ and F- (note that the symbol D in Q(x,D) represents D~while the one in Q(z, D) represents Dz) by choosinginitial points of integration z[ sufficiently close to x and within the domainof holomorphyof F~. Wetherefore set Q(x, D)[f] := [Q(z, D)F+(x+ iO) - Q(z, D)F-(x i0)], where in particular

kti~nes

It is obvious that Q(z, D)F± are well defined holomorphicfunctions, but may lose meaningas z movesawayfrom the point x. In this case, for Q(z, D)F~- to be still defined, one needs to replace z~ with two newpoints z~. The difference between the two representations can be computedby integration by parts and is given by ~ ~Z

.. f~ F±(z)~z k=l

o ktimes

(z ¢)~-1 which, on ~ ~ ([z - z~l ~ 5} ~ (]z - z2~ ~ 5}, gives a real analytic function. This concludes the proof. ~ Wenowproceed to deal with the case of several variables and, in so doing, we will considera larger class of operators, namelythe so called microlocaloperators (whichare moregeneral than microdifferential operators). Weneedfirst to recall a few preliminary results on the product and integration of hyperfunctions (see Theorem2.2.4) and microfunctions.

126

CHAPTER 2.

MICROFUNCTIONS

Proposition 2.5.2 Let f, g be two hyperfunctions on 1~~, and assume S.S.(f ) ¢~( S.S.(g) ° :O. Thenthe product f . g is well defined and (2.5.3) S.S.(f.g) C_{(x,.~,~ + (1 -.~)r/): (x,r/) S.S.(9),0 < A < 1}. Proof. Since B is a sheaf, it is sufficient to look at a sufficiently small neighborhood U of a point x. The first step consists in decomposingthe singular spectra of f and g in such a way that we can write (with obvious meaningof the symbols)

f(x) = ~ Fj(x+ irj0), j=l m2

g(x) = ~ Ck(~+ i~k0), k=l

and (2.5.4) r~ n-A~ = (~ for any pair (j,k). Onthe other hand, (2.5.4) immediatelyimplies that, for any pair (j, k), it

and so a product between f and g can be defined as (2.5.5)

f(x).

g(x) := ~F~(x + ir~0)C~(z

i/ x~0)=

j,k

= E(F~C~)(x + i(r~ ~ ~)0). j,k

It is possible to showthat the definition given by (2.5.5) is independentof the choice of the decompositionfor f and g; moreover, from the right-hand side of (2.5.5) we deduce that

U j,k

=U

+

whichfinally concludesthe proof. Remark2.5.2 The right-hand side of (2.5.3) is what could be called the "convex linear combinationof S.S.(f) and S.S.(g) wi~h respect to fibers", i.e. at each fiber {x} x iS~o-1 one takes the union of all shortest arcs connecting all pairs (x,~), (x,~) with (x,~) S.S.(f), (x ,~?) ¯ S.S.(g); in particular we note that such shortest arcs are uniquelydefinedfor any pair ((x, ~), (x, y)) in view

2.5.

MICROLOCAL OPERATORS

127

the non antipodality of S.S.(f) and S.S.(g). For future reference we will denote such a set by S.S.(f) v S.S.(g). Proposition 2.5.2 can be easily translated in the language of microfunctions if we rememberthat a microfunction is an equivalence class of hyperfunctions, and that the support of a microfunctionis really the singularity spectrumof any representative hyperfunction: Corollary 2.5.1 Let If], [g] be germsof microfunctions at (Xo, ~o) and (Xo, respectively. If, for su~ciently small (but otherwise arbitrary) neighborhoodsF0 of ~o and A0of ~]o, the shortest arcs through~o, 7o does not intersect {({x0} × 0r0) n supp[f]} v {({x0} × 0~0) n supp[g]}, then If]. [g] is a well defined microfunctionin a neighborhood of the set {(z0, ~0)(Xo, ~?0)} \ (supp[f] (Jsupp[g]). Wehave seen in Chapter I that (under some mild regularity condition) it possible to define the definite integral of a hyperfunctionon compactsets with smooth boundaries. However,we will nowneed to define microlocal operators in the same way in which pseudo-differential operators were defined in the setting; this meansthat we need to define them as integral operators with microfunction kernels; to this purpose we need not only a notion of product for microfunctions (Corollary 2.5.1), but also a notion of integration along fibers (rather than just plain integration). Webegin by quoting the following weak version of the Edge of the WedgeTheorem(see the historical appendix to Chapter I): Proposition 2.5.3 (Martineau). Let Fj, j 1, ...,N, behol omorphic functions which, in a neighborhoodof a compactset K, satisfy N

Fj(x+ irj0) =

for suitable cones F1,..., FN. Then, for any proper subcone A~kc_ F~ + F~, there exist H~ E O(K + iA~0) such that H~k = --Hk~ and N

F (z) =

128

CHAPTER 2.

MICROFUNCTIONS

Wenowprove the existence of integration along fibers for hyperfunctions. Theorem2.5.2 Let f(x, y) be a hyperfunction of n + m variables (x, y) on openset U x V C_ j~n x j~m, andlet 7rx : U x V --~ U be the canonicalprojection; let K C_ V be a domainwith piecewise smoothboundaryand let f be real analytic on a neighborhoodof U x OK. Then one can give a good definition of

f f(x, y)dy as an element of B(U). Moreover supp (fKf(x,y)dy)

~- r~(supp(f))

and S.S. (fgf(X’y)dy)C-{(x’~):(x’Y’~)ES’S’(f)

for some y}.

Proof. Once again, since B is a sheaf, we can proceed locally and assume U to be a sufficiently small neighborhoodof a point x; we then represent f as N

f(x, = Fj ((x, ir j0), j=1

with Fj analytically continuable to a neighborhoodof U x OK. Nowset f’~ := rx(Fj) and consider infinitesimal wedgesU + if’jO C_ rz((U x K) iPj0). In accordanceto the definition of definite integral for a hyperfunctionconsider, for z E U + if’j0, a piecewise smooth continuous map~j(z) such that qoj(t)=0 for t~OK (z, t + i~(t)) e (U x ir~ 0 for t

e/~.

Then one defines Gi(z) := /g+i~,j F~(z, ~)d~ which is holomorphic in U x i~j0, and N

(2.5.6)

Jr f(x, y)dy := ~ G~(x + irj0). j=l

The estimates on the support and the singular spectrum of .(~ f(x, y)dy are an immediate consequenceof (2.5.6), while Proposition 2.5.3 allows one to show that the definition in (2.5.6) is well posed and does not dependon the various choices madeduring the construction of (2.5.6). This result has a "microfunctiontranslation" as follows:

2.5.

MICROLOCAL OPERATORS

129

Theorem2.5.3 Let [f(x, y)] be a micro]unction in (x, y) defined in a neighborhoodof

{((x,~),(~,~)):x=x0,~ ¯ v, ~ =~0~0, ~= and with compactsupport with respect to y. Then the integral / f(x, y)dy defines a micro]unctionof the variable x near (x0, ~0). This result has an immediate, and fundamentalconsequence(see [103], Chapter III): Theorem 2.5.4 Let U C_ S*J~n be an open set, microfunction defined in U × Ua such that

and let K = K(x,y) be

supp(g) c_ {(x,y; (~,~)): x = y,~ Then the map TK: f(y) --4 /K(x, y)f(y)dy defines a sheaf endomorphismon C above U. This result (see section one of this chapter) allows us to give the following definition: Definition 2.5.2 A sheaf homomorphism T~ :C-~C defined as in Theorem2.5.3 is a microlocal operator and the microfunction K(x, y) is its kernel. Definition 2.5.3 If Tg is a microlocal operator such that K is a hyperfunction such that supp(K) C_ {(x,y;~,7) : x = y}, then we say that TK is a local operator. Remark2.5.3 In view of Theorems2.5.2 and 2.5.3, it is clear that a local operator actually acts as a sheaf homomorphism on the sheaf B of hyperfunctions. Example 2.5.1 Even for n = 1, microlocal operators are more general than local ones. Consider, for example, the kernel hyperfunction

K(x,~)

x-y+ iO’

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CHAPTER 2.

MICROFUNCTIONS

then the map u(y) --~ 1--~i

/ u(y) x-y+iO is a sheaf homomorphism of C in itself at { (x, ioc)}, whereit acts as the identity map, since

=

io

Note that if we consider K as a hyperfunction, supp(K) -- ~2, and S.S.(K)

{x = y, = Example 2.5.2 operators

Animportant class of local operators is given by differential P(x, D)

whose kernel is the hyperfunction P(x, D)5(x supported at x - y. The sameapplies to infinite order differential operators. Example 2.5.3 On the other hand, operators such as D-1 (in one variable) or D~-1 for j = 1,..., n (in the case of several variables) are definitely not local, as their kernels are of the form const, where y = (yl,...,y~),

yj(x ¯ y

÷ n-l’ i0)

x = (Zl,...,x~).

Wehave therefore shownthe following sequence of sheaf inclusions

(2.5.7) for P the sheaf of (analytic coefficients) differential operators, P~ the sheaf infinite order (analytic coefficients) differential operators, £:oc the sheaf of local operators and £: the sheaf of microlocal operators. The reader can easily verify that 7’ and P¢~ are sheaves of operators; the case of £:oc and £ requires some moresheaf theory and the reader is referred to [123] for details. Sequence(2.5.7), thoughinteresting, fails to find an appropriate location for what we have already introduced, in the case of one variable, as the sheaf of microdifferential operators. There are several reasons, both theoretical and practical, for the introduction of this special class of operators. Onone hand, manipulationof microlocal operators is extremely complicated, on the other hand, microdifferential operators appear very naturally. As we shall showin Chapter VI, Sato’s Fundamental

2.5.

MICROLOCAL OPERATORS

131

Principle claims that if the principal symbolP,~ (x, ~) of a differential operator P(x, D) of order mdoes not vanish at a point (x, ~), then P can be inverted and its inverse is what we will call a microdifferential operator. Moregenerally, one wouldlike to identify a specific class of microlocal operators for whichthe inverse exists and for whicha truly algebraic treatment is possible. The starting point is once again John’s plane waveexpansion formula; if ~ P(x, D) = ~a~(x)D is an (analytic coefficients) differential operator with a = (al,..., an) a negative multi-index, we have already seen that P(x, D) is a microlocal operator with kernel given by K(x, y) = P(x, D)5(x But if we nowrewrite the plane waveexpansion (2.4.2) for 5(x - y) we obtain

(2.5.8)

5(x

Y)

(n- 1)!f d( (-2~ri)" J ((x - y)( +

ApplyP(x, D) (where D is obviously the multi-derivative with respect to x) obtain 1 n K(x,y) = (-~-~i) (- 1)J(3_’_+ n -_ l) ’pj (x,()d,

-

+

-"

where

:= E °ao(z)¢ is obviously a homogeneous polynomial of degree j in ~. Microdifferential operators arise whenwe generalize the pj’s to holomorphic functions. Definition 2.5.4 Definition 2.5.4. Let U be an open set of~Tn x ~T’~, and for any ~ E ~T, j ~ ~ consider a holomorphicfunction on U

satisfying:

LrO(p~+j(z,~)) i=1

(We say, with a languageabuse, that p~+j is homogeneousof degree ~ + in ~);

132

CHAPTER 2.

MICROFUNCTIONS

(ii) for every compactset K C_ U, there exists a positive constant that, for all j < 0 andall (z, ~) E IP~+j(z, ~)l < C~J(-J)!;

CK

such

(iii) for every compactset K C_ U and every ~o > O, there exists a positive constant C~(,~ such that, for all j > 0 and all (z, ~) ~ ¯ 1

Ip +j(z,
0 and by ~(~) the union Uje~$(A+ j). Finally an element of ~(~) is microdifferential op erator of fin ite ord er, while ~( ~) is microdifferential op erator of order at most A. When ~ =O, we will simply suppress it from the notation. ThroughDefinition 2.5.4, we have introduced what we have called microdifferential operators, but from the definition it is not obvious at all that the new sheaf $(~) is a subsheafof/2, the sheaf of microlocaloperators. To do so, one introduces the following sequence {~j(A)} of holomorphic functions of one variable:

~y(A)

~

...

1 (-2~ri)"F(1 -

for j > 0 ~-J

for

log~-~ k=l

j O, then (z,c~) ¯ U. ThenP(z,D) setU C_(T’~×(T determines, by taking suitable boundaryvalues, a microlocal operator on

n(u):= {(x, ~) ¯ s*~n:(x, i~) ¯ Proof.

Indeed if one takes the boundaryvalue of

(2.5.10)

~ p~(z,i~)~+~ (i~) J

along Imp-> 0, this defines a hyperfunction,in viewof Proposition4.1.1 of [123]. Set now

~ = (x and denote by K(x,y,~) the boundary value of (2.5.10). Then K ¯ B(V) for V an open set containing

{(x,y,~): ¯ = ~,(x,i~) and one can verify that S.S.(K) C_ :={(x,y,~;((,~?,p)) : ( x, y,¢) ¯ V (x-y).~=O,~=-~?=k~,

~=k(x-y)

for

some k>0}.

Define now K(x, y) = / K(x, y, ~)d~. Then since for a point in G, p = 0 implies x = y, we see that K(x,y) is a microfunction defined in

{(z,u;(~,~)): (z,~)¯ with support contained in {x = y, ~ = -~}, and it therefore defines a microlocal operator according to Definition 2.5.2. [] Remark2.5.4 It can be shownthat £ is a flabby sheaf, while $ is not. In the next few chapters we will use the sheaf $, and someother related sheaves of differential operators, to study systems of differential equations from an algebraic point of view.

134

2.6

CHAPTER 2.

Historical

MICROFUNCTIONS

Notes

Wehave two appendices to this chapter. The first deals with somephysical motivations for the theory of hyperfunctions and microfunctions, while the second describes HSrmander’s definition of analytic wavefront set, and its relationship with the singular spectrum of a hyperfunction. 2.6.1

Physical

Origins

for

the

Theory

of Microfunctions

In this section we will explore someof the physical origins of the theory of microfunctions. Unfortunately, it is beyondthe scope of this book, and certainly beyond our capabilities, to give a more comprehensive account, for which we refer the reader to [44], [94], [184]; it wouldhave seemedto us unacceptable, however, to have an introductory treatise on microlocal analysis in which the reader is not alerted to the deep physical meaningthat these theories have. We will restrict our attention, here, to the aspects related to the microanalyticity and the macrocausality of the S-matrix. Even thoughthe need for hyperfunctions (and for the study of singularities) only becomesevident whendealing with quantumfield theory, we wish to begin by showing the reader how boundary values of holomorphic functions are naturally linked to macroscopicnotions of causality. Let us start with a simple example: consider the (one dimensional) motion of a harmonic dumpedoscillator of mass m, acted upon by an exterior force mf(t) (t being the time variable). The motion equation (for x x(t) the space variable) is (2.6.1)

d2Xdt-~ + 27-~dx + w~ = f(t), where w0is the natural frequency of the oscillator, and ~, > 0 is the dumping constant. This physical system is knownto be causal, as a consequenceof the positivity of "~ (in other words, as long as f(t) = O, the systemonly admits its free oscillations); here, however,we will ignore this extra piece of information, and we derive the causality of the system from the intrinsic properties of the equation whichdescribes it. If nowwetake f(t) =- (i .e. welook at thefreeoscil lations of th e s ystem), then the general solution of (2.6.1) (2.6.2)

xo(t) = aexp(-wlt) + bexp(-wet)

where a, b are arbitrary constants depending on the initial conditions of the system, wherewl, we are the solutions of the characteristic equation w2+2i7w~, i.e. w0 (2.6.3)

wa,e:

~-(OJo2 -- ,.y2)1/2

__ i~.

2.6.

HISTORICAL NOTES .

135

A different situation occurs if we consider a harmonicexterior force, namely f(t) = F~exp(-iwt). It is then immediateto verify that the solution x(t) is given by x(t) = X~exp(-iwt), where,by (2.6.1) and (2.6.2), -1 X~ = G(w)F~, G(w) = (w - wl)(w Finally, by compositionof these elementaryharmonicforces, one can consider the "general" case, in which f(t) is assumedto be arbitrary, but represented by a Fourierintegral, i.e. f(t) = (2~r) -1/$~ F(w)exp(-iwt)dw, and, of course, F(w) =/$~ f(s)exp(iws)ds. Since we are dealing with a linear system, the solution in the general case follows, by integration, from the harmoniccase, so that x(t) = (2~r) -1/$~¢~ X(w)exp(-iwt)dw = (2~r) -~/$~ G(w)F(w)exp(-iwt)dw = (2~r)-1/$$ f(s)ds

f$] G(w)exp(-iwt)dw :

=~ ¯ f(t), where (2.6.4)

g(t)

= (2~r)-I

G(w)exp(-iwt)dw,

and ¯ denotes the usual convolution product. Remark2.6.1 Notice that if f(t) where 5(t), the Dirac distribution; then x(t) wouldcoincide with g(t) which, therefore, is nothing but the fundamental solution of (2.6.1). Wealso see that g(t) or, equivalently, its Fourier transform G(w), contains all information necessary to the study of the system; in the terminology currently used in QuantumPhysics one might say that g(t) is the transition function betweenthe input f(t) and the output x(t). Let us nowexaminea(w) = -1/(w - w~)(w- moreclosely; its i ntegral, which will provide the fundamentalsolution g(t), can be easily computedwith

136

CHAPTER 2.

MICROFUNCTIONS

Y

X

-R ~d 2

~d 1

Figure 2.6.1 the help of residues theory. By (2.6.2) one sees that both wl and w2belong I_(w) = {w ¯ ¢ : Imw< 0}, and hence G(w) can by analytically continued in the half plane I+(w) = {w ¯~ : Imw> 0}. Fix t = T < 0 in (2.6.4): then the integral (2.6.4) can be computedby closing the integration path in the half plane I+, where G is holomorphic:Since the integral of G on half-circles in I+ tends to zero as their radius R grows to infinity (indeed, IG(w)[rapidly decreases zero, whenIwl -~ +oc), we deduce g(~-) = (2~r) -1 f~o~ G(w)exp(-iwT)d~v = (271")

-1

f~

G(w)exp(-iw~r)dw

where3’ is as in figure 2.6.1. Hence

= 0 for < 0.

(2.6.5)

A similar reasoning applies if we put v > 0 in (2.6.4); this time, however, the path 3’ must be chosen(see figure 2.6.2) in the half-plane I_(w) (whereG is no longer holomorphic)if we want to kill off the integral along the semi-circles. For ~- > 0, hence, the residues theoremgives g(~-) = -2~i Z Res G(w)exp(-iw~-) = 2~ i - (wl - w2)(exp(-iw~-) exp(--iw2T)). Finally, we have x(t)

F

= ~ g(t-

s)f(s)ds=

_ g(t-

s)f(s)ds=

2.6. HISTORICAL NOTES

137 Y

-R

R

D X

Figure 2.6.2

2 _ ~2)1j2 (Wo where the upper boundof integration is t instead of +oo, because of (2.6.4). Remark2.6.2 The result just obtained can be given a quite interesting physical interpretation: the position x(t) of the systemat the instant t, only depends on the values f(t) in instant s precedingt(s < t), i.e. the systemis causal. This result is of course a direct consequenceof (2.6.4), which in turn follows from the fact that G(w)has a holomorphiccontinuation in I+(w). The computations above thus show, even thoughin a single example, that the analyticity of (and its behaviorat infinity) plays a key role in establishing the causality of the physical system which G describes. Wenowproceed to reverse the above process: more precisely we showthat an abstract assumptionof causality for a given systemleads to obtain analyticity properties on the functions (distributions) whichdescribe the systemitself. Wewill adopt here an axiomatic approach: someaxioms (inspired by physical considerations) are stated, and from them we deduce important properties of the objects which are to describe the system. In the sequel both the input f(t) and its output x(t) will be assumedto belong to a suitable class of generalized functions (e.g. the space :D’ of distributions). Three reasonable axiomswill stated: a) linearity (superposition principle), ~) invariance with respect to time translations, macroscopicprimitive causality.

138

CHAPTER 2.

MICROFUNCTIONS

The linearity axiomstates that our physical systemacts as a linear operator on a suitable space of distributions, hencethere exists a kernel g = g(t, s) such that x(t) : ~ g(t,s)f(s)ds (the integrals whichwe write are meaningfulonly insofar as f, g, x are functions; in all other cases they must be looked at from a symbolicpoint of view, as it is customaryin the theory of distributions). Axiom/~states that if the input is translated with respect to time, the same happens for the output. It is well knownthat the convolution operators are the most general operators which commutewith translations, so that from we deduce that x(t) can be obtained from f(t) via convolution with a given convolutor. In other words,

(2.6.6/

x(t) --- g(t - s)f(s)ds= g

(the integral, as usual, mayby symbolic, while g * f is perfectly meaningful,as far as at least one of the two distributions g, f is of compactsupport). Finally, the primitive causality condition implies that if f(t) vanishes for t < T, the same happensfor x(t) (in ghe examplewe examinedbefore this did not occur, because the system was endowed,so to speak, with "built-in" oscillations, which should not be consideredin our discussion). This, in particular, implies that g(t) = 0 t < 0, i.e. the support of g, considered a distribution, is containedin is the output z(t) correspondingto the input f(t) = ~(t) and of course coincides with the kernel appearing (2.6.g)). This fact has an interesting consequence: G(w)is the Fourier transformof g(t) (whichexists if g is a compactlysupported, or tempered,distribution, then the causality condition implies that G(a~) = g(t)exp(icot)dt (indeed, 9(t) = 0 for t < 0). This apparently irrelevant simplification implies in particular that G(co) admits of a holomorphiccontinuation in I+(co). Indeed, w = u +iv, v > 0, then G(w)

f0~ g t ()exp (Jut) exp -vt t

is still convergent, becauseof the factor exp(-vt) (this wouldnot be true if integral were extendedto all of ~). The preceding discussion can be madevery precise, for examplefor L2(Kt), and summarizedby a result whosedetailed proof the reader can find

[92].. Theorem2.6.1 Let G(w) ~ L2(~). Then the following conditions are equivalent:

2.6.

HISTORICALNOTES

139

(i) g(t) vanishes for t (ii) G(w) is, almost everywherein the sense of Lesbesgue, the limit, for v 0+, of a function G(u + iv), holomorphicin the half-plane I+(w) = I+(u iv), whichis squareintegrable on eachline parallel to the real axis, i.e.: I+_~lG(u +iv)12du < C (v > 0); (iii) G(w) satisfies Plemelj’s formula (or, in the languageof physicists, dispersion relation):

where P.V.

...:=

lim(/

...

e--~0 J_~

+

...)

denotes the Cauchyprincipal value of the integral. Remark2.6.3 The requirement that G belongs to L2(~T/) can often be translated into the physical assumptionthat the total energy of the systembe finite. Thoughthis seldom occurs in practice, one can equally well handle more complicated situations (e.g. whenG is of polynomial growth) with the use theorems in the same spirit as Theorem2.6.1. Remark2.6.4 A function G(w) which satisfies any of the equivalent conditions of Theorem2.6.1 is said to be a causal transform (an exampleof such a function is, of course, the transition function whicharises whendealing with the harmonic oscillator: -1 G(w) = (w - wl)(w - ~vl)’ Imwl,~ < 0. Theorem2.6.1 extends somehowto the (more interesting) case of distributions; let :D’ denote the space of Schwartzdistributions on ~, :D~_the subspace of those distributions whosesupport is contained in [0, +oc), and ,~ the space of tempereddistributions. Thenthe following result holds: Theorem2.6.2 Let G~-- ~t E $’. The gt ~ 7)+ if and only if: (i) G~is, in the sense of distributions, the boundaryvalue of a function G(u iv), holomorphicfor v > 0; 5i) for any fixed value v > 0, G(u+ iv) belongs, as a distribution in u, to ~g’ and, in this space, convergesto G~, for v --~ 0+;

140

CHAPTER 2.

MICROFUNCTIONS

(iii) given ~ > O, there exists an integer n such that

c(o.,)--

o(M") (Imw_> e > 0).

Remark 2.6.5 Both Theorem 2.6.1 and Theorem 2.6.2 show that the mathematical translation of the physical assumptionof causality leads to a natural introduction of the notion of boundaryvalue of a holomorphicfunction satisfying suitable growthconditions. In other words, if we do not rely on the fact that every distribution can be represented as boundaryvalue of a suitable holomorphicfunction (i.e. on the injection :D~ -~ B of the sheaf of distributions into the sheaf B of hyperfunctions to be thought of as the "sheaf of the boundary values of holomorphic functions"), we do not obtain a natural description of the consequencesof the causality axiom. Hence, if a formal theory of boundary valued does not seem(at this stage) unavoidable, it certainly looks natural. All the material described so far only refers to one dimensional problems. Wewish to conclude this discussion with a few remarks on the situation which arises whendealing with several variables. LetF += {x =(xl,.. ", xn)¯~n:x~ -~-2xi ~ >0, Xl> 0}, 2 r-={x¯~:x~xi~ >0, xl< 0}; 1

correspondingly one can construct the forward tube domain T+ +, -- ~ + iF and the backward tube domain T- = ~ + iF-. Causal distributions can then be defined as those tempered distributions (on ~) which, by convolution, describe causal systems, and it can be proved that u ¯ S’ is causal if and only if supp(u) C_ +. Theorem 2.6.2 e xtends t o: Theorem2.6.3 u ¯ S~ is causal if and only if its Fourier transform ~t extends holomorphicallyin the tube T+ to a function of which, in 8~, ~t is the boundary value. Wenowbriefly mention a more difficult analysis of howmicroanlyticity is related to microcausality. As it is well known,one of the historical interests of elementary particles physics was the study of collision processes and the theory of the scattering matrix (or S-matrix in brief) has been developed (essentially by a group physicists [22], [43], [44], [79], [92], [93], [94], [95], [99] in the early sixties) exactly for this purpose. Underthe assumptionthat only strong interactions are considered, and that the particles involvedare stable particles with respect to such strong interactions, one maysafely assumethat both incomingand outgoing particles are free, and can therefore be characterized by their impulse-energy real four-dimensional vector

2.6.

HISTORICAL NOTES

141

where the componentP(0) > 0 represents the energy and where the mass of the particle is given by Minkowski’s metrical relation (with the speed of light being set as c = 1)

m2 =p0) Thus, particles can actually be represented as points in the algebraic variety (the mass-variety) given by the hyperboloid M= {p E ~4 : p2 = rn2,p(o) > 0}; with p~ being defined by P~0) -/~2 and/~2

2

2

Wewill not give here the details which can be found on any initial textbook in quantumphysics, but we will just mentionthat the superposition principle allows us to say that all the information whichcan be derived fromthe collision is contained in the so called S-matrix of the process, whoseentries are complex valued distributions on the algebraic variety which is the product of the massvarieties associated to all the particles involvedin the collision. Historically, two equivalent principles have been stated in the study of the S-matrix: a mathematicalone and a physical one; the reader is referred to [93], [95], [184], [185] for further details on this topic. Let us point out that when the study of the S-matrix begun, not much was knownabout the structure of the matrix, except that somegeneral properties (related to its analyticity) were understood. Particularly important is the fact that strong relationships must link the physical interactions whichthe matrix strives to describe, and the structure of its singularities, in the sense of microlocalanalysis. In fact, it can be shownthat the physical postulate of macrocausality, is equivalent to the mathematicalpostulate of microanalyticity. This of course, is what is knownnow (and it has been knownfor maybe 20 years), but at the origin of the theory, physicists were forced to formulatesimilar postulates, such as the i~-postulate which can only be correctly expressed in the language of hyperfunctionsin several variables. 2.6~2

HSrmander’s

Analytic

Wave Front

Set

At the beginningof this chapter we have defined the wavefront set for distributions and we have used the same procedure to create an analogous concept for hyperfunctions, which we have called the singular spectrum of a hyperfunction. In the first part of this historical appendix, we have also mentionedhowphysicists have felt the need for the introduction of a similar notion (the microlocal essential support) in relation to the study of the S-matrix. Nowwe will go back to distributions to describe a notion due to HSrmander,whoin [85] introduced the concept of analytic wavefront set. As it will turn out, all these concepts

142

CHAPTER 2.

MICROFUNCTIONS

are equivalent and demonstrate the necessity of a concept which is being born in so manydifferent areas of study. It is also interesting the fact that all these notions were created independently and essentially at the same time. Let therefore f be a distribution on an open set U in ~/n. The goal of HSrmander’s notion is to study the open subset of U wheref is not real analytic; to do so, we will once again try to use the Fourier transform: however, since no compactly supported analytic functions exist, one needs (as already pointed out in section two of this chapter) to have a more complex description. The fundamentalidea is still the same; we want to say that if f is real analytic in a point x0, then one can find compactly supported distributions which coincide with f in a neighborhoodof x0 and such that their Fourier transforms satisfy suitable bounds (a similar idea is developed also in Kaneko’s[102], where "psychological Paley-Wiener theorem" for compactly supported real analytic functions is proved). To make the previous commentsprecise we follow [18] and we start by recalling the following well knowncharacterization of real analytic functions: Lemma2.6.1 A function f is real analytic in an open subset U of J~’~ if and only if it satisfies the following condition: for every compactsubset K of U, there exist constants CKand mgsuch that, for all multi-indices ~ sup lO0-~l _< I~1+~. CK(CKlal) K

The reader will note that one might replace this Lemmawith other similar results originating from the theory of quasi-analytic classes, to obtain similar notions of quasi-analytic wave-frontsets. Lemma2.6.2 There is a constant Cn such that if K is a compact subset of ~Rn and if Kr is the r-neighborhoodof K, Kr = {x e IR~ : d(x, K) 0 such that if {~vm}is the bounded sequence associated by Lemma 2.6.2 to B(xo, t), then there exist someconic neighborhoodF of ~0, a constant C, and an integer k, such that 1(~. f)(~)[ ~< (Cm)’~(1k-m for all ~ E F. It is immediateto see that if ~ is the canonical projection of the cosphere bundle on its basis, then ~r(WFA(f))is a closed subset of U. Moreover,an application of Proposition 2.6.1 showsthat f is real analytic outside this projection and therefore the analytic wavefront set satisfies the most obvious requests.

2.6.

HISTORICAL NOTES

145

Wenowwant to show that this new definition of analytic wave front set coincides with the notion of singular spectrum whenapplied to distributions; as a matter of fact, we nowhave two different ways of describing the singularity locus of distributions: one is given by the notion of analytic wavefront set introduced just now,and one (essentially provided in Section 2.2) is based the fact that each distribution is a hyperfunction as well. Fromthis point of view, we knowthat every distribution can be seen as sumof boundaryvalues of holomorphicfunctions, and as such it is a hyperfunction to which the notion of singular spectrum can be applied. If we look back at the definition of singular spectrum for a hyperfunction we readily see that if f is a distribution on U, then its singular spectrum S.S.(f), which is also referred to as the analytic singular spectrum of f, can be defined as the closed subset of U × iS~-~ which is the complement of the set of those points (x0, i~0~c) for whichthe following condition holds: there exists some open neighborhood~ of Xo and a family {f~}of function holomorphic in ~ + iF~ such that f is, in ~, the sum of boundaryvalues of the f~, and for each o~ there exists somey~ in F~ for which(y~, ~o) is negative. The proof of the equivalence of these two notions is far from being trivial, and we will follow Bjbrk’s argumentsas described in [18]. Wefirst need to notice that the distributions whicharise as boundaryvalues of holomorphicfunctions are not arbitrary; this was already proved in Chapter one, where a tempered growth condition was described to characterize such boundary values; we now translate that growthcondition in terms of the analytic wavefront set. Proposition 2.6.2 Let f be a holomorphic function in ~ + iF~, for F~ = F N {0 < [y[ < (~}, whichsatisfies the temperedgrowthcondition If(x-k +iy)] 0(QT, x(k) is just O[~,..., ~,], the ring of polynomialswith coefficient in O, where ~i = al(o-~). This showsthat the associated graded sheaf of rings ~ is canonically isomorphic to the symmetricalgebra of the holomorphictangent sheaf. Thus, through the notion of a filtration, we have obtained the commutativeobject ~ from the non-commutative object O . Wewill now define a filtration on the direct sum O’n, as follows. Let i Ui = [0,0,...,0, 1,0,...,0],i = 1,2,...,m, be the canonical generators for Om:define m

(3.3.4)

(O’~)(k) = ~ O(~-~’)Ui~ ~ O(~) (-li)U~

whereli are integers. It is easily seen that {(O~n)(~)}k=0,1 is an increasing (m). filtration on O Let nowA/[ be a finitely generated W-ModuleAJ: we need to introduce a special kind of filtration whichwewill call a "goodfiltration": such a filtration is locally defined by

3.3.

(3.3.5)

FILTRATIONS AND CHARACTERISTIC VARIETIES

(k) = :P(k-~l)ul + 73(k-~2)u2+... + 73(k-~m)um, A4

whereli E ~, i = 1, 2,..., Notice that we have (3.~.6)

171

m, and ul, u2,..., umare generators for 2t4 over 73.

M(~)

C j~(k÷l),

M: k

and

73(k)M(h)C (~+h)forintegers k and h.

Nowthat we have a goodfiltration on a finitely generated 73-moduleA/I, we can define the O-Module~Q by setting

,Q=(~ M(k)(~-~), where the/)-Module structure on J~ is given by (3.3.7)

P. ~ = P----~,

where/5 = ~,_>0 p(k) E ~),p(,) e 73(~)/73(~-1),p(}) M, v(h) ~ M(h)/M(h-U,V(h) ~ M(h), and the product ~ is defined by ~-~ = ~ P(k)v(h) ~ ./(4, P(~:)v(~) ~ M(k+~)/M(~:+~-~). Let, for another set of generators ’u~, ... , of ~[, A~~) = 73(~-~I)u’~ +... be another goodfiltration for the 73-ModuleA/[ (in particular JP[ = [J~ AJ~)). Then, for any k, one can find k~ large enoughto have A/I(0k) C A4(~’) and for any k’ one can find k small enoughto haveA/[(k) c Az[~k’), i.e., the topologies k)} are equivalent. inducedby the filtrations {A/[(k)} and {AJ~ Let Af be a 73-submoduleof ~[: one can obtain an induced filtration by simply putting Af(k) = WI(k) ~ A/’. Furthermore, for an epimorphism9~ of Modules

one can define a filtration on A/[" as A/t"(~) Obviously,for a finitely generated :/:)-Module A/[ represented by

CHAPTER 3. /)-MODULES

172

the goodfiltration (3.3.5) is given by the epimorphism .u applied to the filtration (3.3.4) on/)’~. Consider nowa cyclic/)-Module 2~4 with one relation, i.e. Pu = O, P ¯ Assumethat P is of order k, i.e., P ¯ /)(k) but P ~f/)(k-l), and let P~ principal part of P, i.e. P = P~ + p(k-1), p(k-~) ¯/)(~-1). A goodfiltration .M is of the form A/I(e) (h---~, v(h) ¯ J~4(h)/A4(h-~). Thenthe principal Let ~ ¯ JQ be arbitrary, ~ = @v part P} ¯/)(~)//)(~-~) C 7~, annihilates ~ as follows. Sincev(h) is in A/[(h), v(h) can be written as v (h) = Q(h-i)u, where Q(h-i) Hencev(h) is written as Qh-iu, whereQh-i is the principal part of Q(h-i). Then p~v(h) = pkQh-iu = Qh-ip~u = Qh-i(--P(k-~)u) holds, since Pu = (P~ P(k-1))u = 0 implies Pku = --P(~-l)u. As Qh-ip(k-1)u belongs to M(h+k-1),we have pkv(h---~ = 0 in M(h+~)/.M(h+k-~) for all h. In general, for a finitely generated/)-ModuleM, let ~7(A4)be the annihilator ideal of the q-ModuleA)[ that is ~7(M)= {/5 ¯ ~; p f[ = ~}. For another good filtration {]vi~k)}, we also obtain the associated annihilator ideal ~70(M).Then let 15 be an arbitrary element of the radical ideal ~f~(M)of ,7(A/I). For k, 15 is the principal symbolP~ of someelement p(k) in :D(k). Thenthere exists t so that 15t = (p~)~ belongs to J(A/[). Therefore, (p~)t. ~(h) is in j~(ktTh-1) for any v(h) in (h) and al l h. Hence, for any o(kt+h-~)in A4(kt+h-~) we have (pk)t. ~(kt+h-~)¯

j~(kt+h-l+kt-1)

__ j~(2kt-2+h).

Consequently, we have that (pk)2t. ~(h) belongs to Repeating this process s times, we get (3.3.8)

(pk)st~)(h)

As we noted earlier, for another goodfiltration (J~I~h)} the relations (3.3.9) (3.3.10)

J~I(h’) C 2~I~h) and A/[~~) (a C ’’) A~

hold for someh~ and h". Let ~h) be an arbitrary element in A/[~h). Then from (3.3.10) we have

3.3.

FILTRATIONS AND CHARACTERISTIC VARIETIES

173

(pk)s’~~h)e (pk)~’~.A~(h")" By (3.3.8), for any ~(h") in .A//(h") we (p~)~’~.~(h")E A//(~’~t-s’+h’’). Then we can find s ~ large enough to have ~ ~(s’kt+h-1)

~(~’~-~’+~") C ,~,0

This shows that ~s’t is in ~, and therefore we obtain the inclusion

Reversingthe role of the two filtrations, we conclude that the radical of J(M) is determined only by the D-ModuleMindependently of the choice a good filtration. That is, we have obtained an ideal ~ globally defined on T* X.

Proposition 3.3.1 For an exact sequence of finitely

generated D-Modules

we have

(~)} be a goodfiltration on A~. This induces two filtrations Proof. Let {AA .A4’ and A4" which makethe following sequence 0 ~ A//’(~) ---+ (~) -- -+ AJ"(k) -- ~ 0 exact. Consider now the commutative diagram

on

174

CHAPTER 3. :D-MODULES 0

0

0

0 --~ ~4’(~)/A/I,Ck-1)--~ Ad(~)/A4C~-1) _~g,l"Ck)/A4"(~-1)

0 ~ .AA’(k)

~-.h4(k)

0---.--,.-..,VI,(~1)

-[..)~(k-1)

’~ A/I"(~)

~ d~,,(k-1)

0

~

0

0

The nine lemmaimplies that the top row is exact. Therefore, we obtain the exact sequence of ~-Modules

This implies that ~T(AA)= 3"(A4’) f~ LT(A4")holds, and therefore we Definition 3.3.1 n. Let U be an open subset of X = (T Define V(JU[) = V(~) = {(z,~) T* U;D(z,~) = Of orallP E ~} This complexanalytic variety in T*Uis called the characteristic variety of the 7)-Module Accordingto this definition, the characteristic variety of ,~l is the closed reduced complexanalytic variety in T*X determined by the homogeneous radical Ideal ~J~4) of 7): in abstract terms V(AA) = Specan (7~/~-~-~) In particular, the characteristic variety V(A4)is a conic subset since for (z, V(A/[) and A ~(T\{0} we always have (z, A~) ~ Definition 3.3.1 is somewhatabstract, so we nowproceed to give a more direct construction for the characteristic variety of a :D-Module.Let 3Abe a Moduleover X, and let A~Ibe the O-Module defined by fl74 ----- @k wecan then define the characteristic variety of.h/l as the support of OT.X(~ .A4. Then, our earlier definition of the characteristic variety V(AJ)coincides with the above definition. Indeed, those holomorphic functions on T*X that are in the ideal OT, X ~)O, ~(/-~ are precisely those that vanish at each point

3.3.

FILTRATIONS AND CHARACTERISTIC VARIETIES

175

Supp(OT.x GO.I(4). Hence, the NullstellenSatz of Hilbert for conic complex analytic sets implies that V(A/~) = V(~-~ V(J(Supp(Ov.x GO./O )) = Supp(OT.X GO As an example, let us consider a cyclic ~-Module~, of the form Plu=O

P~u= 0 P~u= O, P~ e ~. such a ~-moduleis represented by the following exact sequence.

PC Let ,.7 =(P1,..., Pt) be the ideal in :D generatedby those differential operators. Since the following short sequenceis exact

0 --~ J ,-+ ~ ~+ M--+ 0, the canonical filtration on :D induces goodfiltrations fore, obtain the exact sequence

on Mand ~7. We, there-

0-~ d ~ :~--~ ~Q -~ 0. Apply now the exact functor Or.x~$>~, (see Chapter IV, Section 3): obtain the following exact sequence

0 --~ OT.x@~---+ OT.X -~ OT.x@Xa~ O. This implies

,] is the ideal in Ow.xgenerated by the symbols

.a(P1),o(P2),..., a(P~), the support of OT.X~ .t[4 is the zero set of the ideal

(o(P,),...,~(P~)).

176

CHAPTER 3. 7)-MODULES

In particular, for A/I = 7)xU,i.e. M = 7)x/{P e 7)x : Pu = 0}, we have V(M) = {(z,~) T*X,a(P)(z,~) = O,Pu 0}. Weconclude this section with the following result: Proposition 3.3.2 For an exact sequence 0 ~M’ ---~M --~ M" ---~

0,

of finitely generated7)-Modules, we have

v(M) = v(M’) u v(M"). Proof. implies

For 0 -~ M’ -~ M-~ M" --~ 0, the NullstellenSatz of Hilbert that ~ : ~ V~ ~ if and only if V(~/--~(~)

V(~), and in view of Proposition 3.3.1, we have concluded the proof.

3.4

~-Modules

In this section, we microlocalize the sheaf 7)x and introduce the sheaf $x. We then prove, in this setting, a version of Sato’s fundamentaltheoremfor infinite order differential operators. This is intended to prepare the reader for the more complete treatment which will be given in Chapter VI. The microlocalization, namely the process which will bring us from 7)x to ~x, or from 7)~ to 8~, is the non-commutativeversion of the localization for commutative ring. Let X be an open subset of~Tn, and let T*X be the cotangent bundle. Let us recall that for a partial differential operator P = ~lJl 0 and any compactset K in ~, there exists a positive constant Ce,Ksuch that for any j >_0

(c) For any compactset K in ~, there exists a positive constant RKsuch that for any j < 0 sup Ipj(z,~)l such that Nm(P,K, eK) is finite. Wealso have the following property: N,~(P + Q,K,t) -) 0 implies "H_I(A/’) = 0. Hence, have "Ho(JV/) = 0 for any Ad. Oonsequently, *H0(AF)= 0. F¥omthe above exact sequence, we finally obtain H~(J~) = Nowwe are ready to describe the interplay among(4.2.1), (4.2.2), (4.2.3) (4.2.4). ReplaceA4by Ox with its trivial goodfiltration in (4.2.4). Namelyset

4.2.

THE DE RHAMFUNCTOR

193

0(~) = Ox for all k k 0 and O(xk) = 0 for k < 0. Then we regain (4.2.3) as special case of the Spencersequence for a left/)x-Module. Wewill compute the derived functor ~7-lOmvx(OX,M)using (4.2.3) follows. ~7"lOmvx( Ox , M)

~- ~lomv.:(:~x @ox A" Ox,M) ~ ~omo. (~’Ox, M)

~ : NOmox(ox,~x¯ ~ ~) That is, we have proved the isomorphism J~Om~x

(ox, ~) :~ x~" ® Ox

¯

Definition 4.2.1 The functor ~lomvx (Ox, -) that takes a left T)x-Module to the de Rhamcomplex ~’x ~ Ox~[ is said to be the de Rhamfunctor. Remark4.2.2 As a consequence of our arguments above, the h-th cohomology sheaf of the de Rhamcomplex~c ~ ox ~ is the h-th extension sheaf. Namely,

n~(~®ox~) ~ CXt~x (.ox, ~), where the h-th differential

of the de Rhamcomplex

is given by

d~(~®~) = d~® ~ + E(dz~^ ~) ®0~. j=l

Notice also that ~:l?tOmvx(Ox, Dx) ,~ fl’x ~ OxT)Xis a free resolution of the right :Dx-Module~, i.e., (4.2.2). Let us now compute

~tomz,x ( Ox via (4.2.3). By what we have just seen, this complexis isomorphicto the complex

194

CHAPTER 4.

FUNCTORS ASSOCIATED WITH :D-MODULES

~lomv,,(Z~x~ox^" Ox,Vx), which is the complexof right :Dx-Modules~2~c ~ oxDx, i.e., (4.2.2). The holonomici~y of Ox, see Chapter V, implies ~he pure dimensionali~y of ~HOmvx(Ox, ~x), i.e. ~Omvx(Ox,Dx) unless h = n,

~ $Xt~x(OX,DX

) ~ Nn(~k@OxDX)

=

and ~"(~ @ oxgx) ~ ~. This shows that .~Omvx( Ox, :Dx) In] On the other hand, from (4.2.2) we have

ext~,x(S~x,Vx)~- nh(no,~,x(a’~®oxVx,Vx))~ n~(v~~Ox~" the h-th cohomologyof (4.2.3). Therefore, we obtain

~O~x(~, Vx)[~] = Ox. Consequently, we have ~nOmvx( ~Omvx ( Ox, Vx)[n], = ~n omvx (~.Vx)[n] : Ox, i.e., ~ Xt~x n (£ t~x (Ox, ~x), ~x) ~ We w ill retur n to th is isomorphism in the section on holonomic~x-Modulesin the next chapter. Retook4.2.3 In Chapter III, we constructed a flee resolution of Ox as follows:

~x ~ Ox ~ O. ~om this resolution, we obtained ~om~x(Ox, Ox) = ~x. The de Rham functor ~om~x(Ox,- ) evaluated at Ox gives the usual de Rham complex ~ ~ o~ O x , i.e., 0 ~Ox ~ ~... ~he Poincar~ lemmathen implies that

~

~0.

gXt~x(OX,OX) = 0 unless h = O, and g~t~x (Ox, Ox) ~ ~x. That is, ~om~x (Ox, Ox) = ~x.

4.2.

THE DE RHAM FUNCTOR

195

Proposition 4.2.2 For a left T~x-ModuleAA, we have an isomorphism ~xt~x ( Ox, M) ~- Tor~h (fl~x, M). Proof. By the pure dimensionality of Ox or equivalently the holonomicity of Ox (see Chapter V), we have 8Xt~x(OX,:Dx) = 0 unless h-- n. Hence(see Remark4.2.5 below), there is a spectral sequence such that E~’q = TorV~x (Szt~ x (Ox, :Px), abuts to ~xt~-xP(Ox, M). Since E~’q = 0 unless q = n, we have

0 i.e., we see that the abutmentis given by

0

El’"~ Ef’" ~ In particular, for p = n - h ~2~-h’n

="DX Tor,,_h(gXtvx(Ox, :Dx), M)~=gxt~,x (Ox, M).

Remark 4.2.4 We can also compute ~c ~xM using (4.2.2). consider the free resolution of

Namely, if

o --~ vx ~ ~x ® o,~Vx---~... --~ ~ ~ o~ Vx~ ~x ~ o we~eta complex (~ ~ o~vx[~])~ ~, i.e.,

we obtain the de Rhamcomplex

Therefore

Wehave shown before that the right-hand side is ~Omvx(Ox, M). Hence

196

CHAPTER 4.

FUNCTORS ASSOCIATED WITH :D-MODULES

a} ®#xM[--]= ~o,~=x(ox, M), namely, once again we have

vx ~

~zt~(Ox,M).

Remark4.2.5 In this lengthy remark, we will discuss the spectral sequences associated with J~7-gOmvx(.hf, :Dx)~ ~LM,where Mis a left :Dx-Moduleand Af is a right T)x-Module. Let ,4 and B be abelian categories, and let ,4 be a category with enoughinjectives. Let G be a right exact functor from A to B, and let Co(A) and Co(B) indicate the categories of complexesof objects in ,4 and B, respectively. Define a functor Co(G) from Co(,4) to Co(B) as follows. (...

-~ Aj

--+

Aj-1

"--~...)

Co(_~)

(...

---+ GAj -~ GA¢-I -~...)

and consider the diagram

Co(,4)

Co(a)

. Co(B)

In this diagram, we define Go as G o Ho= Ho o Co(G), i.e., from C0(,4) to B such that

Gois a functor

Go(A,)= G(ker(A1---~ A0)) = ker(GA~ ~ GAo). Thenthe sequence{G,,Ao} gives the derived functors of Go, that is, the functors {G,~Ao}are the hyperderived functors of G. This induces spectral sequences

’E~,q=LpG(Hq(A,)) E~,q = LqGAp which abuts to the hyperderived functor G,~A°of G. Nowlet ’A/TMbe a complexof right 7)x-Modules, and let Mbe a left Module. Then there is induced a spectral sequence

(4)..5)

’~=U’(To~_¢ E~ (’.~’*,.~))

4.2. THE DE RHAM FUNCTOR abutting to the hypertor Tor~x °, (’.hf One can begin at

197 Notice that E~’q = 0 for all q > 0.

(4.2.6) where’.Alp = ’Af-p. Fromthe aboveE~,q we get

i.e., cohomologically

EU= nPTor~_~ (’~*,.~) Next, replace the above complex ’Af ° of right Dz-Modules by J~7-{Omvx(Af, Dz), where iV" is a left :Dx-Module.Namely,we will consider

~-~o,~,x(~, ~ ) ®#x~, where.A~ is a left :Dx-Module, not necessarily finitely presentedas a :Dz-Module, i.e. not necessarily :Dx-coherent.In the case whereA~is of finite presentation the cohomologysheaf of J~7-{om9 x (Af, :Dx) (~ ~xA4is nothing but the usual tension sheaf s xthvx ( Af, J~4). Thisis becausewehave7-l om~ x (-, :D x ) @Vx.~ 7"{Omvx(-, $’) for a finitely generated projective ~x-Module~. First t ake a projective resolution of

Then,let ’T~"def = nomvx(P,, Px). By the definition,

~Xt~x (~, v~) Next, let ’P" be a complex of flat right :Dx-Modules ~/h(,p.) ~_+7./~(,7~.), i.e., ’P" and ’T~" are quasi-isomorphic.

satisfying

n~(’v" ® v~) is an invariant of two variables, contravariant in Af and covariant in A4. Denote this functor by Ch(Af, ~/~). Then Ch(Af, A~)is an exact connected sequence functors. (See [35] or [150] for further details). This induces a secondquadrant spectral sequence such that

(4.2.7)

’q = Tor_~ E~ (3xt~, x (Af,Vx),M);

this sequenceabuts to Cu(.hf, ,~//). Finally, let Q. ~ 2~4 be an acyclic flat resolution of A~. Thenwe have that J~7-lomvx(.hf , :Dx) @~xA~can be computedas follows:

198

CHAPTER 4.

FUNCTORS ASSOCIATED WITH :D-MODULES

def

= no,~x(~.,~x)®#x~-~’= ~¢"®~x~ = ’~’@Q’. As an application of equation (4.2.7), let ff Oxandlet ~ bea coherent left ~x-Module. Since, ~ we s~w earlier, we have: Sx xt~

(o~,~x)

f fl~ tot ¢ = n 0 for

q~n

we obtain that E2h-~’~ = Tor~_h($zt~x (Ox, ~x), is isomorphic to the abutment $Xt~x (Ox, M). That is,

Tor~_~h(O~, M)~ C~t~(Ox, M), whichis Proposition 4.2.2. Remark 4.2.6 We have introduced the de Rham functor 7-lOmvx(Ox,-) and the solution functor 7-lomvx(-,Ox) in Chapter III. In his recent work, Mebkhoutproved an important global duality theorem for these functors. We will nowquickly describe Mebkhout’sresult and we will showhowboth Poincar~ duality and Serre duality follow from his theorem. Let F~(X, ~q) denote the set of global sections of a sheaf ~q over X with compactsupports, and let F(X, 8) denote the usual set of global sections ~q over X. Note that F(X,-) and Fc(X,-) are left exact functors from category of sheaves over X to the category of abelian groups. Using the left exact functors 7{Omvx(Ox,-) and F~(X,-), we can define a left exact functor Homvx,e(Ox,-) by setting, for any left :Dx-moduleA/I, Homvx,~(Ox, M) = F~(X, 74Omvx (Ox, Similarly, we can define Homgx(.M, Ox) = F(X, 7-lom9x (M/l, Ox ) Then, there are induced spectral sequences of composite functors such that g~:q~ = H~(X, Cxt~x(Ox, JPl))

4.2. THE DE RHAMFUNCTOR abutting to Ext~,x,c(Ox, A4) de=f ~nHomgx,c(Ox ’ Ad), and

199

E~’q = HP(X, £Xt~x (Ad, abutting to Ext~x (M, Ox) = ~nHom~x (~, Ox). In terms of the theory of derived categories, these can be written ~Homgx,~(Ox, M) = ~F~(X, ~nOmgx (Ox, and

~Homvx (M, Ox) = ~r(X, ~nOm~x (M, Ox)), respectively. Then Mebkhout’sglobal duality states that the Yonedapairing

2n-~ M) "~ ~ Ox), Extvxx(Ox, Ezt~vx(M, Ox) Ex ~Vx,~vx, composedwith the trace map Extvx,~(Ox, Ox) --~ 2n-j induces a topological duality betweenExt~x,¢((.gx, .M) and Ext~ (A4, (.gx). Wewill showthat the above assertion is a generalization of both Poincar~ and Serre duality. In order to get Poincar~ duality, we choose the holonomic D-ModuleOx. Then the above spectral sequence E~ becomes:

E~P’q = H~(X, &t~x (Ox,0~) Poincar~ lemma implies that ,f.xtq~c(Ox,Ox) = fo r q ¢ 0, and ¯ £zt~, x (Ox, Ox) ~ Cx, as it was observed in Remark4.2.3. Hence, we have E~,~ = 0 for q ~ 0, and E~:°~ ~- H~(X,¢x). Similarly, we also obtain E~’° ~ H~(X,¢x). From these collapsing spectral sequences, we can compute their abutmentsas follows:

Wethen obtain the pairing H;(X,fx)

x H~(X,~Tx)

Ezt~,o(Ox,Ox) H2~(X,¢x)

which is nothing but the well knownPoincar6 duality for X.

200

CHAPTER 4.

FUNCTORS ASSOCIATED WITH D-MODULES

The other extreme of a holonomic:Dx-ModuleOx is the sheaf :Dx. Namely, as mentioned in Chapter III, :Dx is the :Dx-Modulecorresponding to the system of no equations. Its characteristic variety has dimension 2n, and Serre’s duality is obtained by putting AA-- :Dx in Mebkhout’s theorem. Then, notice that 7tOmvx(DX,-) is an exact functor. Hence, the higher cohomology SXt~x(DX, Ox) vanishes for q ~ O, and $xt°~x(Dx, Ox) ~- 7{Omvx(DX,Ox) ~Ox. Hence, the spectral sequence gives us E~’° = HP(X,£xt°vx(DX, Ox)) ~- HP(X, Ox). On the other hand, we have £xtqvx(OX,:Dx) = fo r q ¢ n, and £xt~) x (Ox, :Dx) ~ ~, as we showed in the last paragraph preceding Remark 4.2.3. Therefore,

=2,c~"-a’" :H’~-a(X, gxt’~,~,(Ox,Ox))

=H~¢ (x,~,:).

Their abutments are computedthrough those collapsing spectral sequences

{

E

xt~L{(Ox ~x) ~ E~,~j’" = H2-J(X, fl~) Sx~vx (Dx, Ox) ~ ’° = H~(X, Ox ). Then the pairing becomes H~-J(X, f2nx) × Ha(X, (gx) --+ Ezt~)nx,c(Ox, H~"(X,(~x)

---+

Thus, we have obtained Serre duality, i.e. mutually strong duality between the FS-space Ha(X, Ox) and the DFS-space H~c (X,~z). See [8] for further details.

4.3

Algebraic

Local Cohomology

Consider nowthe case in which the complexmanifold X is an open neighborhood of the origin (z = 0} in(T. Thenthe stalk COx,0at z = 0 is the ring of convergent powerseries, which we will denote by ~T{z}.The sheaf T)x of germs of ordinary differential operators maythen be written as

m>_O Let Ox[z-~] be the field of Laurent series. Thenwe have the canonical exact sequence

-~]--+Ox[z-~]/Ox 0 --+Ox--+Ox[z --+o.

4.3.

ALGEBRAIC LOCAL COHOMOLOGY

201

Since both Ox and Ox[z-1] are :Dx-Modules, we can define a :Dx-Module structure on Ox[Z-1]/Ox. Notice that the support of Ox[z-1]/Ox is concentrated at z = O. This observation gives a different explanation for the phenomenondescribed in section 2 of Chapter I. Define elements 5(m) of Ox [z-1]/Ox as follows: 5(m) = [(--1)mm!z-(m+l)],m = O, 1, 2 .... -1] under the where [~] indicates the equivalence class of an element ~o in Ox[Z canonical epimorphism defined above. It is easy to see that {5(m)},n=0,1,2,... is set of generators of Ox[z-1]/Oxover~T. Wealso have

That is, 5(’) is the m-th derivative of Dirac delta function 5 = (°). Consider nowthe case in which X is an open subset of (//~ and Y is an analytic variety defined by an ideal fl of Ox, i.e., Y is the support of For any Ox-ModuleAf, the set

~zt°o~(Ox/y~, ~)= nomox (Ox/3n, ~f) represents those germsu in iV" satisfying J~u = 0. That is, given the exact sequence

0 --~ :~ ~ Ox--~ Ox/3~ -~ 0, we have the long exact sequence of Ox-Modules 0 ~ 7-lomox(Ox/J",Af)

~ 7"lomox(Ox,Af)

~ 7tomox(J",Af)

_~ extS~(ox/3.~,z) 4_4 extg~(ox,~f) ~ e~t~(3"~,z) (4.3.1)

---~ gxt2ox(OX/~"~,Af)---~...

Note howeverthat the groups ~xt~ox(Ox,Af) vanish for j _> 1 since 7-lomo~:(Ox,-) is an exact functor. Definition 4.3.1 The algebraic local (or relative) j-th cohomologyo.f Y is defined by indl im e zt~ ( Ox/,~ "n, Af)

202

CHAPTER 4.

FUNCTORS ASSOCIATED WITH W-MODULES

andis denoted by Hi},] (IV’). Furthermore,we define n~Nly] (J~f)

indlim £xt~ox (j m, Jr’).

Then we obtain the following exact sequence from (4.3.1) (4.3.2) 0 ~ "H~y](A/’) ---~ A/~ "H~xly](A/) --+ n~yl(JV") ~ 0, and the isomorphismfor each j = 1, 2, 3,...

hi. 1 In the case in which Ar is a ~Dx-Module,the Ox-flatness of ~Dximplies the isomorphism

"Hom~ Since we also have the :Dx-isomorphism

induced by tha inclusion

~ C ~-t, we obtain

(4.3.3) Hence, by taking ~he direct limit over varying m, one can define a ~x-Module structure on ~FNIyI(~). Namely, (4.3.3) induces a homomorphism

Since ~y] (N) is the kernel

~[xlYl are, respecn~y] 0 tively, the j-th derived functors of ~y] and ~[xly], and since the cohomology functor and the direct limit commute, n~yl(Z ) and n~xiy](~ ) become ~xModules. Rem~k 4.3.1 Since the germs of ~omox(Ox/fl~,~) are annihilated tim ~ we noted earlier, their supports are in Y. Hencethe natural map

induces a map from

by

4.4.

COHOMOLOGICAL PROPERTIES OF T)x

203

indlim 71omo~ (Ox/ ff TM, Af) ---~ 7t~(Af), where7-/~ (Af) is the trascendental local (or relative) cohomology sheaf. Consequently, we have the induced mapfrom the algebraic local cohomologysheaves to the trascendental local cohomologysheaves

Remark 4.3.2 As we indicated, the algebraic cohomologysheaves 7~y](Af) and 7/~xly](Af) are :Dx-Modules.Therefore, the long exact sequencein (4.3.1) a long exact sequenceof :Dx-Modules.IfAf° is an object in the derived category of :Dx-Modules,then we have a triangle

where ÷1 indicates the mapfrom ~7/~xw](A/’°) to ~74~y](A/")[1].

4.4

Cohomological Properties

of

In this section, we will study several properties of the derived functors of the functor 74ornvx(-, :Dx), whichwill be used in Chapter V to give a cohomological characterization of holonomic:Dx-Modules.As we observed in Chapter III, the notion of filtration was utilized as a communication device betweenthe category of non-commutativeobjects and the category of commutativeobjects. In this section we will further observethe usefulness of the notion of filtration to study the non-commutative ring First we will begin with the assertion that the functor OT.X(~ ~x is not only an exact functor, but also a faithful functor from the category of coherent ~xModulesto the category of coherent OT.x-Modules.In order to prove this, it will be sufficient to showthat at each stalk the sheaf OT.Xis flat over :Dx. But this follows immediatelysince OT.Xis the noetherian ring of convergent power series while :Dx is the polynomialring in the cotangentcoordinates~1, ~2,-.., ~. For the ideal (~1,...,~,~), these noetherian rings are analytically isomorphic, i.e., their (~,..., ~n)-adic completionsare isomorphic. Therefore, we obtain the

204

CHAPTER 4.

FUNCTORS ASSOCIATED WITH D-MODULES

flatness of OT*X over the ring ~x. See, for example, [19] for the commutative ring argument. For a 79x-ModuleA,/with a goodfiltration, we have a quasi-free resolution of jiA (4.4.1)

£.

:

...

--~

E1

--~

£0

----}

./~

~ 0.

Namely,each £i = @~k~=179x(li,k),and each Pi is a filtration preserving map. Then the contravariant functor 7-~OmVx (--, 79X) induces the following complex from the complex£. of (4.4.1): (4.4.2) Thenthe derived functor $xt~x (.~A, 79x) of 7-tomvx(.~A, 79x) can be computed as the cohomology 7-lh(Ttomgx(£.,79x)) = 7~h(’£’) ¯ With the induced filtration on ’£" defined by

we obtain a complex of ~x-Modules ,£--~ ~ ,-~ L~+ ~ __~ ....

(4.4.3)

From the canonical commutativity of the diagram ~omv(-, 79)

£. gr -~.

74omv(-, 79) .,-£"

[gr =

we observe also that complex(4.4.3) is obtained as 7-lom~x(~.,~xx), i.e.,

from

/::--~ : ... --~ ~ -~ Zl ~+~00.

(4.4.4)

Wecan also get (4.4.3) by applying the functor ~om~x(-, ~-~-x). Since ~.. a free resolution of ~, Sxt~--~(Ad, 79x) can be computedas the h-th cohomology of complex 7-lOm~x(£., ~xx) = ~ of ~--~x-Modules. The functor (-9T.X ~ ~ a faithful and exact functor from the category of coherent ~-~x-Modulesto the category of coherent OT.x-Modules. Therefore, we obtain the isomorphism (4.4.5)

OT*X

(~xxEXt~---~(~,~-~x)

= $xt~’x(OT°X

(~"-~x -~’

OT’X),

and we note that OT*X~ ~xx~ is coherent as an OT.x-Module. Then we have

4.4.

205

COHOMOLOGICAL PROPERTIESOF :Dx

codim supp($xt~r.x(OT, x (~xx ~, OT.X)) >_ from the theory of commutativealgebras. Hence, in order to prove that codim V(gxt~)x(.M,:Dx))

(4.4.6) it is sufficient to show

V($xth~x (A//, ~Dx)) C supp ($xt~T.x (OT*x(~ ~xx-~, OT.X)). As was pointed out above, those higher extension sheaves ~zt~x(M,gx ) and $xt~r. x (OT*X ~ ~, OT.X) are computed as cohomologies of complexes (4.4.2) and (4.4.3), respectively, through (4.4.5). Wehave the following Lemma 4.4.1 Let M"be a complex of filtered Px-Moduleswith good filtrations, and let OT.x ~ M" be the induced complex. Then we have v(Nh(M’))

C supp

(N~(OT*x~)).

Proof. One can define a filtration on ker(Mh ~ M~+~) by inducing it from the filtration on Mh. Then the induced map OT*X ~ker(M a ~ Mh+l)

~ OT*X

is the zero map. Outside the support of (~a(O~.x ~M*)), the sequence

is exact at the degree h. Since we have an inclusion OT*X

(~xxker(M

h ~ .M h÷l)

~~

OT*

the map

is an epimorphismand we have the split described in the diagram below: OT*

X {~

~xx Mh-1

M--hh’4"1 ------’--~OT*x ~ ~’OT* X ~XX

OT.X ~ ~xxker(2~[h --~ h+l) M

~~xM

206

CHAPTER 4.

FUNCTORS ASSOCIATED WITH/)-MODULES

Next consider ~/h(M*)as the filtered object induced by the epimorphism J~4h D ker(JV!h ~ .hAh+l) -~ 7-lh(.M*) ---+ O. Nowwe have two epimorphisms

whosecomposition is the zero mapat each stalk outside the support of 7{h(OT.X ~xx~). Therefore OT.X @ vxT~(A/~ ~ ° ) : 0 over T*X\(supp (7-lh(OT.X ~VxM __’~Z ))). Namely, we obtain V(7-lh(M°))=

supp (OT.x(~xT-lh(A/P))C

supp (Tt~((gT.X~x~)).

[] Wecan nowuse this lemmato prove a vanishing of higher cohomologytheorem for £xthvx(J~4,/)x) as follows. Theorem4.4.1 If h < codlin V(A~), then

(M,/)x)= Proof. FYomthe isomorphism in (4.4.5),

we have

=0 for all h < codim supp (OT.x ~x-~). See, e.g., [68] for its proof. Since OT.X~ ~x is faithful and since the functor assigning the/)x-Module J~d to the /)x-Module Mwith the good filtration is faithful, we have the vanishing of £Xt~x (M,/)x) for h < codim V(M). Weare nowready to prove the final theoremof this Chapter: Theorem 4.4.2

For h > n = dimX, we have

Proof. Since the characteristic variety V(.AA)is involutive (see e.g. Ohapter VI), for V(AA) ~ wehave dimV(~d) _> n , s ee [18] . Then, from (4.4. 6) we have dim Y(Sxt~x (]vl,/)x)) n weobta in dim V($xt~x (M,/)x)) _< n- 2. Consequently, we have that V($xt~x (Ad,/)x)) is a subvariety of X. However,from Theorem5.2.3 in Chapter V, there exists an integer l for whichit is

4.4.

COHOMOLOGICAL PROPERTIES OF i)x

207

(k) be a good filtration. as a :Dx-Module. Let now M-- UA4 Choose, in the complementof a nowheredense subset of X, a point x such that is a free Ox,,-Module,and let U be a neighborhoodof x over which(k-l) .M(k)/M is a locally free Ox-Module. Then,for a sufficiently large k we obtain a locally free resolution of Mover U, i.e., the first Spencersequence 0 ~ Dx(~ox A" O(~o~M(k-’~)

---~...

~ :Dx(~o~./t4 (~) ~ M ---~

0

is exact. Therefore, we have £xthvx (M/t, :Dx) = ?’l h (?-I omvx (Vx ~ Ox A’O~) Ox M(k-’), T~x)= 0 forh > which concludes the proof. []

Chapter 5 Holonomic 5.1

9-modules

Introduction

With the preparation in Chapters III and IV, we focus on basic themes in this chapter. The Cauchy-Kowalewsky theorem in the language of D-modules is given in Section 2. In Section 3, the direct image of a D-moduleis defined. The most significant theorem in the fundamental theory of holonomic D-modules, due to Kashiwara, is discussed in Section 4.

5.2

Inverse

Image and Cauchy Problem

Let f : Y --> X be a holomorphic map of complex manifolds. For the sheaf Dx of germs of holomorphic linear partial differential operators of finite order, we will define a left Dy-Module, which is also a right f-lDx-Module as follows. Definition

5.2.1

Dy~ X .:-

Oy ~.f-lOxf-lDx

Example 5.2.1 (i) For a closed embeddingf : Y ’--> X, we have

Dy~-~. X = Oy @OXly Dxly.

If Y is given by equations Dy,--~X

zl = z2 .....

z~ = O, then we can express

as

Dy,--,x Hence an arbitrary

operator

= Dx/z~Dx + ". + z~Dx. P(y, O) in Dy~x may be written as 209

210

CHAPTER 5.

where y E Y and Oj = O/Ozj, j = 1, 2,..., we get a function P(y, O)h in (.9y:

HOLONOMIC D-MODULES

n. Then for a function h ~ Ox

P(y, O)h = ~, a~(zr+l .... , z,OO~h[v. (ii) For the projection Y = X x Z -+ X, defined by (z~,..., zn, wt,..., w~) (zt,..., z~), the Dy-Module Dy-.~.X iS the sheaf of differential operators on X whosecoefficients are extended to (.9~. Namely,we get Dy-~x

=

l Dy/EDY"

O

Anelement P(z, w, Oz) in Dv-~xmaybe written as follows:

,,, &)=a (z, .....

. . . , w,)ae, . . . og.,

where (z, w) ~ Y = X x Z and 0j O/Ozj, j = 1, 2,. .., n. h in Ox, we get a function in Or as follows:

Fora fu nction

P(z,w, Oz)h = ~a,~(z,w)O’~h.

Note that for holomorphic maps Z -~ Y ~ X, we have a map ~)z~v x g-~:Dy~x ~ Dz-~x. The above map is defined by

where y = g(z) and x = f(y) for z ~ Z. That is, one can compose a differential operator from X to Y along f with a differential operator from Y to Z along g to get a differential operator from X to Z along f o g. Definition 5.2.2 Let f : Y --~ X be a holomorphic map. For a Dx-Module define the Dy-Modulef*A4 using the right f-~Dx-Module Dv-~x by (5.2.1)

f*Ad = .tAd. Dy-~x ®.r-~x ]

5.2.

INVERSE IMAGE AND CAUCHY PROBLEM

211

By the definition of :Dy~x, the T)y-Modulef*3d is also given by f’34 = Oy ®~’-~Oxf-134. Notice also that we have f*O X = Oy

and J:*~x = :Dy~x.

Let Ad be a T)x-Moduledefined by 34 = 7Pxul + .." + 7:)xum such that (5.2.2)

~ P, juj ~- O, i = 1, 2,... I.

Namely, we have an isomorphism Z)x/~x ¯ P, P = [Pij] ~_ X such that X = ~ and Y -- ((x,y) ~ a121x 0~, let 34 = iDxu satisfying Pu = O. Then f’34 is a differential equation over Y, i.e., iDy-Module. For 0~u, we have f*(O~u) = f*(O~u(x,y)) O~u(x,y)ly 0~(0, y) . Onthe othe r hand , for an e lement O~u, we ha f*(O~u) = f*(O~u(x,y)) = O~(f*u(x,y)) = The relati on a mong those elements over Y is exactly the iDy-Modulef’34. Wewill give explicit examples. Example 5.2.2 If 34 - IDx, namely there are no equations, then f’34 ~ f*T~x~ ’~)Y--~’X "~- ~)X/3:’~)X ~- ~-,OyO~yC~ax = ~)YOax ¯ See example5.2.1, (i). If Adis given by cOyu= 0, i.e., 34 = :Dx/:DxOv,then we have

Notice-that Y = {(x, y) ~ (~lx = 0} is characteristic to 34 and that Modulef’34 is not coherent. For 34 = :DxIiDxO~:, we have f’34 = ~ OyO~= iDy. That is to say the generator u for 34 satisfying the relation O~:u= 0 over X need not satisfy any further relations over Y. Before we state the Cauchy problem, in the case of embedding, we will give the iD-Moduleversion of the classical Cauchy-Kowalewsky theorem. Let

212

CHAPTER 5.

HOLONOMIC D-MODULES

Y = {zl = 0} in X, where (zl,..., zn) are the coordinates of X. SupposeY non-characteristic to a cyclic Dx-Module A/[ represented by a partial differential operator P 6 Dx of order m. Then the principal part may be written as a Weierstrass type: 0F + Pl(z, 0’)0~-~ +... + Pro(z, 0~), where z’ = (z2,..., zn), 0’ = (02,0a,..., On) Pi(z, O0 i s o f o rder i. W ith relation (5.2.3), 7)x/7:)xP = .hA is generated by 0%~ < m, over Ox. Furthermore, the generators 0%~ < m; are linearly independent over Ox. Hence we have (5.2.3)

A4 = ~l_O. Proof. Let A//= ~Dxul + ~Dxu2+ ... + Dxum. Then the Ox-coherency of Ad follows from the Ox-coherencyof :Dxuj, j = 1, 2,..., m. That is, it is enoughto prove the cyclic ~Dx-Module case. Let ~Dxu~~- ~Dx/57,where57 is ker .u s of the epimorphism~Dx "~ ~Dxuj --~ O. Since V(Duj) = V(x/~) C (~1 ..... ~,~ = 0), there exists an integer N so that we mayhave

for somePit in 57, i = 1, 2,...,n. stationary, i.e.,

Hence, the good filtration

on A// becomes

M(k)=M for k>>0~ Since 7~(k-N) is Ox-coherent, M(k) is Ox-coherent. Next we will prove .~4 ~ Otx for some I. Let Y = (0). Then we have for f:Y’-+X

where M0and Ox,o are stalks at Y = (0). Wehave

f-~NOmvx (Ad,Ox)"~= Nomv~. (f’M,Or)

=’~~om¢(o) (¢(o)®Ox.oMo ¢o)

where l is ~he number of generagors of M0over Ox,o. Hence ~here exists a non-zero element ~ in ~Omvx(M, Ox)o inducing the following epimorphism (5.2.10)

Oy ~l-’Ox f -aM ~ Oy ~l-’Ox f-lo~

~ O.

Fromthe exac~ sequence ~ ~ O~ ~ O~/Im~" ~ 0 and (5.2.10), the fibre Ov N~-~Oxf-l(O)/Im~ ") = 0 holds. Then Na~)amaAzumaya’s lemma implies O~/Imp" = 0. Namely, ~ ~ O) ~ 0 is exact. Let ~ be ker 9". Wehave the exact sequence 0~M

~O~

~0.

Since ~ and O~ are Ox-coherent, ~ is also a coherent Ox-Module.From the exact sequence

218

CHAPTER 5.

HOLONOMIC :D-MODULES

Wetherefore obtain Af = 0 and the proof is concluded.

5.3

Direct

Image

Let f : Y -~ X be a holomorphic map. Weare going to define the direct image of a left :Dr-ModuleA// via a bimodule :Dx~-y. Let also f/~ and f2~ be the sheaves of holomorphicforms of highest degrees on X and Y, respectively. First we note that for the right :Dx-Modulef2~ we have the left :Dx-Module (~’~()--1 ---~ 7-lOmox(f2}, Ox). Therefore, for the Ox-Module (f2~¢)-1, 79x ®Ox (f/~¢)-~ is a left 7)x-Module, since x is a r ig ht 7)x -Module. Con sequently, f-~(TDx ®Ox(f~¢)-~) becomes a left f-~Dx-Module. Then we define the left f-~TDx-Module 79x~: by

Vx+~,=f-’(Vx ®ox(frx) -~) ®+-,ox~. Furthermore, f-~(:Dx ®Ox(f~x) -~) ~f-lO x ~ = : Oy ®$-1~x f-~TDx ®1-179x f-~(TDx) ~l-~ox/~z((~)-l)

~ov ~

= Vv~x~-1~ f-~((n~)-l) n~. obtaining ~he relation wi~h D~x. Recall that Dv~xis not only a lef~ DrModule, bu~ also a righ~ f-lDx-Module, ~ seen in Section 5.2. On the o~her hand, we have ~hat

~v~x @~-~Vx f-~(gx ~ox (~)-~) ~o~ is f*(Dx @ox(~)-~) ~ov ~. Then, since ~he left multiplication defines a left Dx-Modules~ruc~ure on ~x @ox(~)-1, the inverse image f*(Dx @Ox(~)-~) becomes a left Dr-Module. Summarizingwh~ we have observed in ~he above is the following. The sheaf

~x~v = f-a(Dx ~Ox(~)-1) ~f-~Ox~ f*(Vx

~Ox

(~)--1)

~Oy

~

is no~ only a lef~ f-~Dx-Module, but also a right Dy-Module.Furthermore, Dx~Y is connected wi~h Dv~x ~

5.3.

DIRECT IMAGE

219

= ~lomf-lvx(f flx,~r @or Definition 5.3.1 For a left 7)y-ModuleAd, the direct image of Jgf is defined by

denoted as ff J~. Wealso write its j-th cohomologyas

~

= ~A(Z~x,-Y®~~).

Example 5.3.1 Let Y 2_> X be a closed embeddingsuch that Y is defined by zd = 0). Then ~ and ~,~-d are respectively isomorphic to Ox Y = (zl ..... and Oy. Then the right/)~,-Module and the left z-l(:Dx)-Module :Dx~y be written as

*-’(~x)®,-,OxOv= ~x®Ox Oxl(zl,z~,..., z,~), which is isomorphic to Dx/(:Dxz~+ ... + T~xZa). Recall that the left Dy-Moduleand right z-~ (Dx)-ModuleDv~xin Example 5.2.1 is isomorphic to Z)v~+x~- T~x/(ztl)x + "" + za2Px). The description of Z)x~vin the above is just as expected since the leftness and the rightness of the modulestructures of Dx~vand :Dv~x are exactly reversed. Remark 5.3.1 For a holomorphic map f from Y to X and for a right D~ModuleA/’, the direct imageof A/" for f is defined by

f~N = ~L(~ when, f] A/is a right Dx-Module. Example5.3.2 Let us consider the case where the above holomorphic mapis the projection ~r from Y = X × Z onto X defined as ~r(z, w) = z. Note that the general case is given by the composition of embeddingcase as in Example 5.3.1 and the projection case. Namely,any holomorphicmapf : Y -~ X can be factored as y

z

,XxY

X,

220

CHAPTER 5.

HOLONOMIC :D-MODULES

where z(w) (f (w), w) andTo(f(w), w) = f(wThen we have ~)X~-Y : $-I(:DX~--X×Y)

(~,-l’Dxx

Y ~)XxY~--Y"

Let ~ be a left ~r-Module. Wewi]l compute f~ ~ for r ~ follows. The right ~y-Module~x~Yhas a locally flee resolution, see [19]:

(5.3.1) 1

l

wherel = dim(Y),i.e., the relative dimensionof the projection ~r : X × Y --~ and ~/x denotes the relative p-form. This is because I)x,:-.y de~__f f*(~)X (f~)-i) @o~.~ is isomorphic f~r/x @o~. f*/)x, where f*/)x = /)Y~X which is Oy @f-lvx f-l:Dx by definition. Then the epimorphismfrom/)y induces the epimorphisme : 12~/x @or/)Y-~/)x~-Y giving a locally free resolution of the right/)y-Module/)z~-r. The only non-vanishing/-th cohomology 7t~ (~/x @or:Dr) is given by/)y/(O1/)Y "~-’’" "~ Ol/)r) as in Section 4.2. Hence, /)x~--Y is isomorphic to/)r/(O1/)y +.. ¯ ÷ O~/)y). Compareagain with/)y~x /)r/(T)yOi +.." ÷/)r0~), in Section 5.2, i.e., the reversion of leftness and rightness is observed. Note also that the epimorphism/)y-~ f*/)x = :Dy-*x induces a locally free resolution ]~Oy/x @o)./)y of/)Y-*X. Therefore, one can compute ~tTtom9r(~)Y-.X, :Dy asfollows

= /x

@o~ ~Y

whichis the locally free resolution (5.3.1) for ~x~. Next we will express the direct image fl M of a left ~y-Module M~ a hyperderived functor of f, using the resolution of ~x~y in (5.3.1). Since ~x~Y @v~M can be computed by fl~/x @or ~y[l] @~y M, we obtain

~

.A~ = .~J+t f,(~/x GO,..A~).

Notice that for A/[ = Oy we obtain

5.3,

DIRECT IMAGE

221

That is = ~ ],(~tr/x) is nothing but the usual relative version of de Rhamcohomologysheaf, on which the Gauss-Manninconnection is defined, often denoted as 7t~D+~(Y/X).Moregenerally, if a Dr-ModuleA4 is locally free of finite rank as an Or-Module,then the hyperderivedfunctor of f, evaluated at 12~./x ®oyA/l, i.e. the direct imagefI A//, is the relative de Rhamcohomology sheaf of COx-Moduleswith the Gauss-Manninconnection. See [129] for the construction. The observation of the direct image as the hyperderived functor gives the following Proposition. Proposition 5.3.1 For an exact sequence of Dy-Modules 0 --~ Ad~ --~ J~ "-~ .M"--~ O, a long exact sequence of Dx-Modulesis induced as follows

//

//

....

Remark5.3.2 1) Let f : Y -~ X be a holomorphic map of relative dimension l. As an application of spectral sequences(4.2.5) and (4.2.6) in Remark4.2.5 Chapter IV, let

{DX~-_

’AfJ _-_ 0

Y forj = 0 forj ¢ 0

for a complex’Af° of right Dy-Modules.Then (4.2.5) becomes

abutting to (h -/)-th cohomologyof the complex Dx~r ®vZL v Ad. From the locally free resolution (5.3.1):

the abutmentis isomorphicto 7-lh(9t~/x ®or All). The collapsing spectral sequence implies the following isomorphism: TorV~(Dx~_r, A4 ~ 7t~(~./x 2) Wehave the direct image f: AJ as J~ v f,(Dx~-r ®vl~

®or

CHAPTER 5.

222

HOLONOMIC V-MODULES

Wedenote the h-th cohomology as f]~[ = ~hf,(l)N~-y ®0~J~), where the right hand side is the hyperderived functor of f,. Then(4.2.5) becomes

Ef’q = ~V,(~./x®o,.~), inducing

’~ = n’(~h(~;./x E~ ®o,. ~), and (4.2.6) becomes E~’~ =J~:l~’ f,(’d’~(a~:/x ®or

= ~,f.(To~(~x~Y, ~)), with abutment f] NI = l~+hf,(f~,/X

5.4

Holonomic

@Or~)"

D-Modules

For any :Dx-Module~1 we have dim V(JV[) > n, (see [206]). One mayask which T)x-Modulesare most strongly determined by the relations amonggenerators so that solutions maynot have any free variables. Such systems of partial differential equations are traditionally called maximallyoverdeterminedsystems. Definition 5.4.1 A T)x-Module ~ is said to be holonomic if dim V(A4) = or Ad = 0 holds. Let

be a short exact sequence of :Dx-Modules. Then we have V(J~t) = V(J~I’) V(A4"). Suppose dim V(Jt4) < n. Then we have dim V(M’) t2 V(J~I") _< Consequently, dim V(J~A’) < n and dim V(~4") < n, must hold. Conversely, if dim V(JV[’) < n and dim V(W[")< n, then the dimension of V(J~A’)tA V(N[") cannot strictly be greater than n. Namely,we have the following Proposition 5.4.1 For a short exact sequence

-Modules, M is holonomicif and only if NI’ and ]v[" are holonomic.

5.4.

HOLONOMIC:D-MODULES

223

Remark 5.4.1 Let X =~T, and let ~A = ~¢u such that

is exact. Namely,we are considering an ordinary differential equation Pu = 0 of order k, for somek _> 0. Thenthe dimensionof the zero set of the principal symbol a(P) -~ g(z)~ ~, g(z) ¯ (9¢ is one, i.e., V(A4)= 1 = dim(T. That is, the :De-ModuleA4is holonomic. classical Cauchy’stheoremstates that the sheaf of solutions 7-lom~¢(JPI, O) is locally free of finite rank as a fix-vector space outside the singularities of the operator P. As we defined in Section 4.2, an C0x-Module ~4 with an integrable connection is a T)x-Modulethat is a free (9z-Modulesatisfying the axiomsin Section 4.1. m Namely,the structure of an integrable connection on A4 = $i=lOxui is defined by an Ox-Algebra map:

Therefore, for an integrable connection ~ = ~im=lOxu i and each Oj in O, we have a representation by n equations: (5.4.1)

Ojui = P~’lUl + P~2u2+"" + Pi,~u,,, j = 1,...,

n.

System(5.4.1) gives rise to a free resolution of the left Z)x-Module 2~4, namely, the integrable connection ~.. By Frobenius theorem, there exists a base (v~, v2,..., v,~) for the free C0x-Module J~ so that system(5.4.1) is transformed into a Pfaff systemof mequations: (5.4.2) Ojv~ = O, j = 1,...,n. Since the characteristic variety V(Ox) = T~X~- X, by Example3.2.1, and any integrable connection is isomorphicto a finite direct sumof the de Rhamsystem (_9x, as observed in the above, integrable connections are holonomic. The goal of this section is to prove Kashiwara’stheoremon the constructibility of ~xth~,( (~4, OX) for a holonomic/)x-Module A4. Let us consider a T)x-Module~ that is locally free of finite rank as an OxModule. That is, A4is a vector bundle with a holomorphicintegral connection over X. That is, locally we have an isomorphism A~ ~ @~=~OxUj,where {u~,u2,..., urn} is a set of generators of A/~ over C)x. Then we can define goodfiltration on ~4 as follows: A4 k)=0 j~(k)

for

k

224

CHAPTER 5.

The induced T~x-Module A~-is annihilated by ~i

HOLONOMIC T)-MODULES

= Crl (Di),

Di =

0 That is, O--~i"

~1.... , {n belong to the annihilator ideal :Y(AA).Hence, the zero set of ~f-~, the characteristic variety V(A~),is the zero section X of ~r T*X -~X. Conversely, if a holonomic:Dx-ModuleA~satisfies V(YV~)= X, i.e. the zero section of T’X, then we have A/[ ~ O~ for some l, (compare with Theorem 5.2.3). For an integrable connection A4 we have, by the Cauchy existence and uniqueness theorem, £Xt~x(JVI~Ox ) = 7-lom~x(A4, Ox) is a local system over X, establishing an equivalence between integrable connections and local systems. Wealso have, by Poincar~ Lemma,£xthvx (Y~, COX)= 0, h = 1, 2,..., (see Section 4.2). Recall that a local system is a~Tz-Module that is locally free of finite rank. Let us state the aboveas a type of Frobenius existence theoremas follows. Theorem5.4.1 The de Rhamfunctor 7-lomvx (COx,-) and the solution functor 7-lomz~x(-, COx)induce an equivalence betweenthe category of left ~)x-Modules that are locally free of finite rank as Ox-Modules and the categoryof local systems over X as ~TX-Modules. The converse functor can be given by Ox ~x -" Remark 5,4.2 For a holonomic :Dx-Module AA, we have the duality

:~nom~(Ox, M) -~ ~no~(~nom~,x(M, Ox),¢x) in the derived category of~-constructible sheaves. Note that a ~T-Module~" over X is said to be ~-constructible if there exists an increasing sequenceof finitely manyclosed analytic subspaces O= XoC X~ c...c

X~c...c

X

such that W]x~-x~_lis isomorphicto ¢~ for someri ~ ~V’, i.e., JZ]x~_x~_~is a local systemover the locally closed Xi - Xi-~. °One can generalize the duality in Remark5.4.2 to a bounded complexY~4 of :Dx-Moduleswhose cohomologiesare holonomic. That is, we have (5.4.3) .g?,7-lomvx( cox , Jr4") --~ ~7-lomcx( ~7-.lomvx°, COx)JT x). Proof. Let ~7" be a complex of injective :Dx-Modulessuch that AJ" --~ 3"" is a quasi-isomorphism,and let ’,7 ° be an injective resolution of the de Rham :Dx-ModuleCOx. Consider the following diagram

5.4.

HOLONOMIC D-MODULES

225

As we saw in Section 4.2, the Poincar~ Lemma tells us ITx is quasi-isomorphic to ~iOmvx (Ox, Ox). Hence we can define a map ¢ of complexes:

7-loravx( Ox, ,I’) _2~7"tornex( 7"lomgx (J’, ’3"’), 7-lora~( Ox,’ J’) namely, for ~} E 7tom9x (Ox, ,]"), assign a Cx-linear map ~(#) = qSo~ for # ~om~x(J’,’J’). Next, we must show the map ~5

~no,~,,~(ox, z4") -~, ~no,.¢,, (~nom~,x (M’,ox),¢x) °) is holonomic, as in the proof of Theorem is an isomorphism. Since 7~h(M 5.2.2, it is sufficient to prove the case of a single holonomicDx-Module Ad. For an arbitrary x ~ X, we will computethe stalk at x:

where Cx denotes the sheaf concentrated at {z} whosestalk at z is IT. Rewrite the abovestalk as follows:

= mnomc(mnom¢(¢,,¢), mnom~,x (.~, Ox))x = ~T¢ornw(ITx[-2nl, ~nomv~ (~, Ox)),

= ~r~(~om~ (~, Ox))~[~] = ~no~(~om~(Ox, ~)~,¢~)~ Therefore, we have a quasi-isomorphism

~OmVx (ox, M)x--, ~nomc(~nOmvx (M, Ox),¢x)~. Weare ready to prove the most significant result for a holonomicD-Module, namely, Kashiwara’sconstructibility theorem.

226

CHAPTER 5.

HOLONOMIC 1)-MODULES

Theorem 5.4.2 For a holonoraic 1)-Module A~I, all the higher cohomology sheaves of solutions, i.e., $xthvx(A,i,Ox) are (T-constructible. That is, there exists a whitney stratification of X = t3~Xa, independentlyof each h, so that (5.4.4)

V(.A~) C U~T~X

and£ xt~)x (A/l, Ox ) lx, is a locally constantsheaf of finite rankover(T. Remark5.4.3 A partition {X.} of a complex manifold X is said to be a stratification of X = t3~Xawhen{Xa}is a locally finite partition, each Xa is a locally closed submanifold of X, and X~ c ~,, - X~, holds for Xa Whenthe last condition is satisfied, we often write X~-~ X~,. Furthermore,a stratification X -- t3Xa is said to be a Whitneystratification if

1. t~T.~ X is a closed set of T’X, and 2. for Xa -< Xa,, i.e., X~is contained in Xa, for X, Flea, :~ q), let x be any point in X~ and consider sequences {xn} C Xa and {x~} C Xa, satisfying lim,_~ xn = lim~_~ x~ = x in Xa. If the sequence of tangent spaces {T~,.X,,} converges to T C T~Xand if(T(x, - x~) converges to a line l T~X, we have I C T. Whitney proved that any stratification X = U,X~has a finer stratification X = uzX~of X (such that X~is contained in someX~, i.e., for each a there is an index set Ba such that Xa = ~ZesoX~)satisfying the above conditions (1) and (2). Proof. First wewill establish the existence of a Whitneystratification satisfying (5.4.4). Let A = V(J~[). Note that the holonomicity of ~[, i.e., V(2~4) dimension n, implies that T~(V(A4)) is Lagrangian at a non-singular point p. Since V(A/I) is an involutive analytic set in T’X, the converse is also true. As before let ~r : T*X --~ X be the projection. Then X - 7r(A) is an open dense set in X. This is because the dimension of ~r(V(A/I)) is than (n - 1). Let X~ be the set of non-singular points of ~r(A). Then we have T~XC A, and furthermore A - T~;X is also a Lagrangian analytic set in T*X. Since ~r(A - T~:~X)can not equal X~, the dimensionof ~r(A - T~,oX) is strictly less than the dimension of X~. Let X~ = ~r(A - T~X), and let A~= A - T~r~X. Then, denoting the set of non-singular points of ~r(A~) by X~, we can define inductively as follows:

5.4.

HOLONOMIC D-MODULES

227

A2 = A1 - T~cIX X~ = the non- singular locus of ~r(A2) A~+I = A~ - T~,~X X~+I = the non-singular Thenwe obtain a stratification

locus of ~r(A~+l)

by non-singular manifolds so that X : UaX~

and A C U,~T~c~X. Thenthere exists a finer stratification

{X~}than {X~},such that

X = LJ~X~ satisfying Whitneyconditions (1) and (2) in Remark5.4.3. Before we prove finite dimensionality of the locally constant sheaf 8~v (M, O)[x~ for a Whitney stratification {X~},we need to knowunder what conditions the restriction map Ext~(~’, All, Ox) -~ Ext~x (f~, M, Ox) becomesan isomorphism, where ~t and ~ are open sets satisfying ~ C ~. Lemma5.4.1 For an arbitrary point Xo in X~, there exists a neighborhoodU of xo such that for a small enough¢ the boundaryof a ball B(x’, ¢) = {x ~ X ]x - x’I < ¢}, x’ ~ Xa ~ U, is non-characteristic for M. Proof. If the statement of the above lemmais not true, there are sequences {x~} in X~ and {Yn} in X such that xn --~ x0 and yn -+ x0 as n x~ ~ Yn and such that d~.(yn) ~ V(M), where ~,(y) = Ix - y[. Then mayassume y~ ~ XZ for some X~ satisfying X~ >- X~ (take some subsequence {y~} if necessary). Consequently, we can find a sequence in¢ so that ¢(Ynconverges to I in TzoX and Tu, Xz approaches T in T, oX. Therefore, dg~,~(y~) converges to the dual vector l* of l. By the assumption, d%o~(yn)= onTu.X~. Thenl* = 0 on T, whichcontradicts to l C v. Next, we will prove the restriction mapp (5.4.5) Ext~x (B(xo, ~’), ~, Ox) --~ Ext~x (B(xo, ~), ~, is an isomorphismfor ~’ _> 6, x0 6 X~. Notice that this isomorphismimplies the following

228

CHAPTER 5.

HOLONOMIC :D-MODULES

$Xt~x (]vl, Ox) = F_~_~oExt~ x (B(x0, e), A/I, Ox)

.~ Eztg x (S(xo,e), M,

for a small enough 6. From a well knownlemmain Functional Analysis, Ext~x (B(xo, e), ~, is finite dimensional. See Lemma 5.4.5. As a consequence, at the stalk at xo, di~ Ext~x (M, Ox)~o is finite, Therefore, for x~ satisfying Ix0 - x~] < ¢ we obtain $Xt~x (j~4, Ox)~,° ~- Ext~x (B(xo, ¢), M, Ox) ~- $xt~x (M, OX)~o, provingthe finite dimensionalityof the locally constant sheaf Ext~vx(M/l, Ox)lx~. Wemust prove that the map (5.4.5) is an isomorphism. Since we have a restriction map p" from Exthvx(B(xo,#t),AA, Ox) to Exth~x(B(xo, s),./~,Ox) for #’ > e’ > e satisfying p" = p o p’, where p’ is the restriction mapfrom Ext~x(B(xo, e’~, A~, Ox)to Exth~x(B(xo, l, Jt~, Ox), { Ext~x (B(xo, e), Jr /l, Oz forms an inverse system. Considera flabby resolution of Ox by the sheaf of forms of (0, *)-type with coefficients in hyperfunctions:

Then, Ext~x(B(xo, e), A~I, Ox) is the h-th cohomologyof the complex Homv x ( B(xo, ¢), M, B(~")). By the definition of flabbiness, the restriction mapfor 6’ > ¢ Homvx ( B ( xo, ¢’), AA,’0) --~H om vx( B (xo, ¢), M/I, B’~) ) is an epimorphism.That is, the inverse system{Homvx(B(x0, 6), Ad, ’i)) } i s satisfying the followingcondition, called Mittag-Lefller condition: for any ~ > 0 there exists ¢0 so that the decreasing sequencesatisfies Im( Homz~x ( B(xo, s’), M, B(~’O)) ~-~ x ( B(xo,¢), .M, B( ’0)

= Im(Homvx(B(xo, ¢o), M, B(~’O)) 2L, Homvx(B(xo, ~), ’0)

5.4.

HOLONOMIC D-MODULES

229

for ~’ > e0. FurthermoreExt~-~l(B(xo, ~), A4, Ox) is finite dimensional. Hence, Ext~-~~ ( B(xo, c), A4, Oz satisfies Mi ttag-Leffier co ndition. Then the epimorphism Ha (Homvx(B(xo, ~), A/I, B(~"))) de__fExt~x (B(xo, ¢), A4, Ox)

~_m_E,_~Ext~x ( B(xo,~’), A4, Ox) de-4f i~__m_~,+~Hh( Homvx ( B(Xo,~’), J~4, becomes an isomorphism. Namely, we need a general lemmaon the commutativity of the inverse limit with the cohomology. Lemma5.4.2 Let (Vc}, c E 1~, be an inverse system satisfying Mittag-Le~er condition, i.e., for an arbitrary c ~ ~ there exists co ~ ff~ so that for c’ > Co, Im(V~,-~ Vc) = Im(Vco-~ Vc) holds. Then °) --~i~_cHh(V~° Hh(~_cVc ) is always an epimorphism.Moreover,when(Hh-~(V~)}, c ~ satisfies Mit tagLeffier condition, the above mapis a monomorphism. The next lemmaimplies that the restriction mapin (5.4.5) is actually an isomorphism. Lemma5.4.3 Let (Vc}ce~ be an inverse system of finite spaces. If

dimensional vector

l__i_~,>cVc,--> V~and i~__m_c, n = dim V(fl’l),

ext x vx) = 0. For this non-vanishing n-th cohomologysheaf, we have codim V(Sxtvx(.&4, x)) from (4.4.6). Fromthe involutivity of V(fl4), we codimV(£xt~x(2~1, Dx)) Namely,£Xt~x (M/f, :Dx) is a right holonomic~Dx-Module,which is denoted Let O-~’M

-+M ~"M ~0

be a short exact sequence of left holonomic:Dx-Modules.The left exact contravariant functor 7-lomvx (-, :Dx) induces a long exact sequence of cohomologies. Then the purely codimensionality of 8xt~vx(A4,:Dx) 0, h ~n impl ies the following exact sequence

That is, 8Xt~x(-, :Dx) is an exact functor from the category of left holonomic ~Dx-Modulesto the category of right holonomic:Dx-Modules. Similarly, ,~I** = (.&4*)* is a left holonomic:/:)x-Module. One can show actually ~l** ~ gA. See the following Remark5.4.6 for its proof. In general the characteristic variety of 8xt~x(A~l, T)x) is smaller, i.e. containedin, than that of g4. Therefore, we have codim V(A/I) _< codim V(Sxt~x (g,l, Dx)), (or from (4.4.6) for h = n). Similarly, n V(~xtz~x(M,Dx))

n n 3 V(~xtvx(~xtz,x(M,

~)

x),Dx)M ~f V(M*’)

implies n _< codim V($xt~x (M/I, :Dx)) codim V(. M) 0}, by Sato’s ~ndamental Theorem),

(s.s.(u@ n {t = x = 0}c {(0,0;~(a .... ,~,,,)oo): ,2 =~ +...~,~,+, The theory of traces of hyperfunctions which we have developed in Chapter II showsthat u+(t, x), as well as its derivative ~-~-¢, can be rest[icted to t = Now,if x ~ 0, one has that

~+(0,~)- ~_(0,~) and that Out

cOtIt=o so that these three hyperfunctions are defined on ~, and supported at the origin. But nowwe mayrecall that hyperfunctions supported at the origin admit a particularly simple representation as series of derivatives of the (f-function (see Chapter I), i.e., to be precise, if u ~ B{0}(~),then

(6.~.~)

~(~) = ~o~(~),

where the sum is extended to all non-negative multi-indices a, and where, for every e > 0, there exists C~>0 such that for all a,

Notice that

and therefore if u as in (6.3.2) satisfies

6.3.

A FIRST APPROACHTO BICHARACTERISTIC STRIPS

251

x~--- 0, one deduces that S.S.(E) C_ {(x,t;i(~,~)~):

2- t ~ =0,([,T) = 2(x,t)},

and so x = - ([)t. However,for t > 0, QE= 0, and so, the first corollary to Sato’s FundamentalPrinciple, ~-2 = ~2 ~ 0 which yields our result. [~ Weare nowready for the propagation of singularities

theorem:

Theorem6.3.2 Let ~ be an open subset of S*l~l n+~. Let u be a micro]unction solution of Qu = O, and let B be a bicharacteristic strip ]or the waveequation. Thensupp(u) ~ B a union of connected subsets of B

6.4.

CONTACT TRANSFORMATIONS

255

Proof. Wewill showthat ifu(t, x) is a microfunctionsolution ofQu-~ 0, defined in a neighborhood of

~ =~o~, po(o,o;i(~o,~o)~), and if u(t, x) vanishes at Po, then u(t, x) vanishesas well in P~=(t, - ( ~o)~O;i(~-o,~o) But to do so it will suffice to prove that ,2.:= u . H(t) = To showthe vanishing of fi we note that Q5 = 0 in a neighborhood of

{(t,x;i(~,~)~):~ =~0,~ = and therefore by (i) of Lemma 6.3.2 we get ~(t, x) = / 5(t - s, x - y)5(s, y)dsdy = / [QE(t - s, x - y)] ~(s, y)dsdy. The result nowfollows via integration by parts. As we shall see shortly, this relevant role of bicharacteristic strips is much more general than what the content of Theorem6.3.2 might lead to think. Before we get to its most interesting generalizations we mayjust mentionthat a result along these samelines holds for the so-called regularly hyperbolic operators (see [123], [206]for details).

6.4

Contact Transformations

As we have shownin section 6.3, a deep interplay links contact geometrywith differential equations. This interplay can be exploited to obtain extremely powerful results. In this section we will review somebasic notions from contact geometry, we will define the notion of contact transformations, and we will finally microlocalize this notion. The tools we will construct in this section are central to the classification results whichwill end our book.

256

CHAPTER 6.

SYSTEMS OF MICRODIFFERENTIAL EQUATIONS

For the sake of simplicity, we will workon a complexanalytic manifoldX but if X were taken to be real, one could repeat similar constructions with obvious modifications. Let then L be a one-dimensional subbundle of the cotangent bundle T’X, and let L* denote its dual bundle and L± its orthogonal complement,i.e. the kernel of the map TX --~ One can define a multilinear homomorphism of vector bundles L± x L± x L-+(Tx

(vl, v2,d~)~ (d~,vl ^v2), and this provides an alternating bilinear homomorphism (6.4.1)

L± x L± ®-1. -4 L

Definition 6.4.1 Wesay that (X,L) is a contact manifold if the map (6.4.1) is non-degenerate. Remark6.4.1 Definition 6.4.1 is equivalent to require that dim X = 2n- 1 (in particular, odd) and that for a nowherevanishing section w of L, the product (6.4.2)

w A ’~-~ (dw)

never vanishes. Note that this last condition is independentof the choice of w, which will be called a fundamental 1-form (or also a canonical 1-form) for X; under this definition we often write (X, wx) rather than (X, L). There is of course a strict relationship betweensymplectic and contact geometry. In particular, if we write ~ = L*\X, then for s a cross-section of )~, we define a 1-form 0 on 2 by setting

Then (dO)n never vanishes and )~ is called the symplectic manifold associated to (X, L), with canonical 1-form t?. Historically, the first examplesof contact manifoldswereborn out of classical mechanics and are given by X = ~*Y

6.4.

CONTACT TRANSFORMATIONS

257

the projective cotangent bundle of an n-dimensional manifold Y; in this case f( = T*Y\Y. In this special case, Darbouxtheorem states that for a local coordinate system (Xl,. ¯ .,xn,Pl, .. ¯

canonical 1-form can be written as w = dxn - (pldxl ÷... +p,_idxn_~). In this case, the associated symplectic manifold has a local coordinate system (x~,. . . , Xn,~, ¯ ¯ ¯, ~n) with pj =---~ n, and the symplectic structure is given by ~ = ~ldXl + "" + ~ndxn = ~w. This exampleis not only historically relevant, but theoretically important, since it is well knownthat every contact manifoldis locally isomorphicto a projective cotangent bundle. Definition 6.4.2 If (X, Wx)and (Y, wy) are two contact manifolds of the dimension, we say that a map f:X-+Y is a contact transformation if f*Wy is a fundamental 1-form for X. Manyof the fundamentalnotions in classical mechanicsare really notions from contact geometry. Let us quickly review them: Definition 6.4.3 Let f, g be functions on a symplectic manifold .~, dim f( 2n. Their Poisson bracket is defined by {f, g}(dO)~ ’~-~. := ndf A dg A (dO) Remark6.4.2 If (x~,...,

xn, ~,...,

{f,~} -

~) are canonical coordinates on )~, then

_

.

Retook 6.4.3 The Poisson bracket is sometimesreferred to as the Lagrangian bracket. Remark8.4.4 To see whybicharacteristic strips are really concepts in contact geometry, just note that they are integral curves of the so called Hamiltonian vector field

258

CHAPTER 6.

SYSTEMS OF MICRODIFFERENTIAL EQUATIONS

HI :=

. j=l

~

OXj

OXj

O~j

Bicharacteristic strips arise therefore whenconsidering the integral curves of Ha(P). Remark6.4.5 By the definitions

above it follows immediately that

= [H:, where [A, B] := AB - BA is the so called commutatoror Lie bracket of A and B. Wehave defined above the notion of contact transformation in a rather abstract way. There is, however,a canonical way to construct contact transformations, which seems to have been first utilized by Maslov[163], Egorov[45] and HSrmander[89]. Let X, Y be two complex manifolds of dimension n, and let A be the nonsingular hypersurface of X × Y defined by f~(x, y) = 0, for ~2 someholomorphic function on X × Y. Assumefurthermore that, on A, it is det[dx0 ~ dxdu~ duf~ J ]50 (where the matrix is an (n + 1) × (n + 1) matrix). Wewant to showhowto use f~ to construct a contact transformation from to h°*Y.To do so, one first consider P~(X × Y):= {(z,y;~,~)

E ~P*(X × f~(z ,y) = 0 and

(~,r/) = c. gradf~(x,y) for somec -~ Then, by the implicit function theorem, we knowthat the first projection ~rl : P~(X × Y) is a local isomorphism,and so is r2 : ~P~(X× Y) -~ ~P*Y. Finally, ~rl and ~r2 induce local isomorphisms ~rl o ~r~-~ : ~o*y~ ~*X and

6.4.

CONTACT TRANSFORMATIONS

259

1 : J~*X T¢2 0 7F~ --~ which are clearly contact transformations which will be said to have f~ as a ; generating function. Remark6.4.6 It can be shownthat every contact transformation can be expressed by composition of two contact transformations with generating functions. Remark6.4.7 The most classical contact transformation with generating function is the so called Legendretransformation. The correspondencebetween(x, ~) and (y, 7) is given x I=_~_A~

j