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<0}, and the set B(f~) of hyperfunctions as the quotient (1.2.1)
o(v)
where O denotes the sheaf of germsof holomorphicfunctions and, therefore, for any given open set U in ¢, O(U)denotes the set of holomorphicfunctions on U. Before we can make any remarks on the set B(f~), we must point out apparent oddity in its definition, which seems to dependon the choice of the complexneighborhood V. This is not the case in view of a deep result in the theory of holomorphicfunctions of one complexvariable, namely the so-called Mittag-Leffler Theorem.In classical textbooks, the Mittag-Leffler Theoremis also knownas the theorem on the existence of a meromorphicfunction with assigned polar singularities and assigned principal parts. In cohomologicalterms, this theoremsoundsas follows: Theorem1.2.1 (Mittag-Leffter: cohomological formulation). For any open set U in the complex plane, the first cohomologygroup of U with coe~cients in the sheaf of germsof holomorphicfunctions vanishes, i.e. H~(U, O) =
4
CHAPTER 1.
HYPERFUNCTIONS
Proof. This is an immediate consequence, almost a rephrasing, of Cartan’s TheoremA. More directly, it can be shown, when in the form described in Theorem1.2.Y below, by a simple Cauchyintegral formula argument, which is described in detail in [103]. [] Of course, such a statement has a more direct and less abstract formulation, whichis whatwe will need to provethat hyperfunctionsare a well defined object: Theorem1.2.1 ~. (Mittag-Leffier; explicit .formulation). Let U, V be two open sets in(T. For any function F in O(UA V) there are functions F1 in O(U) F2 in O(V) such that, on the intersection U f3 V, it
Fl(Z)F2 (z) =F( Let us now show how this result can be used to prove that the quotient (1.2.1) does not dependon the choice of Proposition 1.2.1 Let U and V be two complex neighborhood of the same open set f~. Thenthere is a vector space isomorphismbetweenthe quotient spaces B(f~; U) - O(U\ ~) and B(l’~; V) - O(Y \
o(v)
o(v)
Proof. To begin with, we may assume U to contain V, since we can always prove the isomorphismbetween B(f~; V) and B(~2; U yl V) and the isomorphism between B(Ft; U) and B(~; U n V), which would prove the isomorphismbetween B(~; U) and B(~; V). Consider nowthe natural mapp, induced by the restriction mapping, and the inclusion of V into U, (1.2.2)
O(U \ ~) O(V P: o(u) ~ o(v~
It is immediatelyseen that the mapin (1.2.2) is injeetive, since the restriction of a holomorphie function F from U \ ~ to V \ ~ is holomorphic on all of only if F itself was already holomorphieon all of U. Wenowonly have to prove that p is surjective. To do so, take F in O(V \ fl), and apply Theorem1.2.1’ to the pair of open sets U \ ~ and V. Weget a function F1 in O(U\ ~) and a function F2 in O(V) such that, on the intersection (U \ ~) N V = V \ f~, it F(z) = F~(z) F2(z). Th en th e in verse image under p oftheequivalence clas [F] of F is exactly the equivalence class [F~] in B(f~; U). This result showsthat the definition of the space of hyperfunctionsdoes not dependon the choice of the open set in the defining equation (1.2.1), i.e. B(~)=
lim
O(Y\~).
o(v)
1.2.
HYPERFUNCTIONS OF ONE VARIABLE: BASIC DEFINITIONS 5
throughout this book, we will denote by IF] the hyperfunction defined, in the quotient (1.2.1), by the equivalence class of the holomorphicfunction F or, equivalently, by the pair (F+, F-), where + =FW + and F- = F~v-. Note that we can interpret what we have just described by looking at the short exact sequenceof global section functors
0 ~ r(y, v \ ~; -) --~ r(y, -) -%r(v \ whereF(V, V \ ~; -), also denoted by Fn(V, -), is the kernel of the restriction
e: r(v, -) --~ r(v \ a,-). The sequence above then becomesthe following exact sequenc~ 0 ---~ r(Y, Y \ ~; ~:) ---~ r(Y, t:) -% r(V \ ~; for £ any flabby sheaf (we will comeback later to these notions). But by the definition of flabbiness, the restriction becomesa surjective mapand then a standard homological algebra methodprovides the long exact sequence 0 ~ g°(v,Y
\ ~;0)
~ H°(V,O) H°(V \
--+ Hi(V, V \ £t; O) -~ Hi(V,
-~ HI( v
where Hi(V, (9) vanishes for all j = 1, 2,.... property for an analytic function implies that
~;
\ ~ , O) -~~ ...
Nowthe analytic continuation
H°(V, V \ ~; O) i.e. if fly\~ = 0 for f E H°(V, O) F(V, O), th en f is identically zer o in Then we obtain an isomorphism B(~) H°(V \ ~;(9) H°(Y, O)
~ H’(V, Y \ ~; O) = H~(V;
to whichwe will refer frequently later on. Nowthat we have a properly defined notion of hyperfunction on an open subset of the real line it maybe natural to look for an interpretation of such a concept, especially if we want to showthat we are really defining a sort of generalized function. To begin with, we note that because of its definition, the set B(~)actually inherits a structure of vector space over¢, eventhough, at least at this moment,we are unable to provide it with any topological structure. To understand what B(~t) stands for, we mayrecall a famous result by Painleve’ [175], actually a formulation of a special case of Schwarzreflection principle, [1921:
6
CHAPTER 1.
HYPERFUNCTIONS
Theorem1.2.2 (Painleve’s Theorem) Let ~ be an open set in the real line, and let V be a complex neighborhoodof ~. If a function f is holomorphic in V \ ~, and continuous on V, then f is holomorphicon V. Proof. Once again this is a consequence of Cauchy’s integral formula and of MoreraTheorem.For further details the reader is referred to any classical text in one complex variable. [] For later purposes, it maybe useful to point out a stronger version of this same result, in which the notion of continuity is replaced by a weaker form (actually a distributional form). To do so, note that the continuity hypothesis on F only meant that the limit ofF(x+iy), x E ~, as converges to0 f rom the right, and the limit of F(x+iy), x ~ ~, as converges to0 f rom theleft , coincide (in symbols, lin~_~0+F(x + iy) = limv_~0- F(x + iy)). Theselimits, instead of being taken in the strong sense, maybe taken in the sense of distributions, to obtain the following result: Theorem1.2.3 Let F+ and F- be two functions holomorphic, respectively, in V+ and in V-. If, for any compactlysupported infinitely differentiable function ~ in ~, i.e. for any ~ in 7)(~), it lira < F+(x + is), ~(x) >= 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, iflmz
and
g(z)={O’-1,
ifImz>O iflmz
Thenit is clear that [e] = If], since the difference e - ~ is entire. Thehyperfunction [e], clearly defined on all of the real line, can be thought as somehow the unit hyperfunction, so that we will often denote it by 1. As a matter of fact, if we think of hyperfunctions(in the naive sense) as differences of boundaryvalues of holomorphicfunctions, then it is quite natural to set [e] = 1. This hyperfunction, together with the structure of A(f~)-modulewhich we have given B(f~), also allows us to showthat every real analytic function can be thought as a hyperfunction, or, equivalently, that, for any openset f~, one has a natural injection of A(f~) into B(ft). Suchan injection is obtained by multiplying a analytic function a(x) by the unit hyperfunction 1. The resulting hyperfunction will be identified with a(x) itself. Example 1.2.5 Let F be a function holomorphic in U \ ft, where we maintain the usual notations. Wewant to give a precise meaningto the expression "boundary value of F from above and from below"; we will shownow that such a boundaryvalue can be used to define precisely the value of a hyperfunction defined by IF]. In this new framework,the difference of the boundaryvalues of F from above and from below is the value of the hyperfunction at that point. Define, to this purpose, F(x + iO)~=-IF. and
F(x- i0) = IF. With this notation, which the physicists have been employingfor quite some time to denote boundaryvalues in various weaksenses (mainly in the sense of distributions), we nowhave the intuitive equality [F(z)](x) = F(x + iO) - i0), whichjustifies our use of the term "boundaryvalue". It maybe worthwhile observing that Example1.2.5 is the first exampleto provide us with objects which were not necessarily knownbefore; indeed, Examples 1.2.1-1.2.3 gave us the Dirac distribution and the Heaviside function, while Example1.2.4 only gave us the well knownreal analytic functions; now we finally get in touch with objects which are not known.As a matter of fact (the reader is referred to section 3 in this chapter), it is well known,[210] that
14
CHAPTER 1.
HYPERFUNCTIONS
all distributions on the real line .zan be obtained as boundaryvalues (in the sense of distributions) of functions which are holomorphicin the upper halfplane; howeverit was shownby G. KSthein [138] that not all boundaryvalues of holomorphicfunctions yield distributions (this characterization of distributions is, interestingly enough, to be found in one of the papers which can be characterized as part of the prehistory of hyperfunctions, as we shall discuss in the historical notes). The following important result holds true: Theorem1.2.5 Let f~ be an open set in ~, and let U be a complex neighborhood of ~. Let F be a function holomorphicin U+. Then F(x + iO) is a distribution in t2 if and only if, for every compactK in f~, there are positive constants C, N such that
-~. s~plF(x+iy)l
[]
Pro@See [1381, [142].
Example1.2.6 As a consequenceof this result, we see that all singularities of polar nature only produce distributions, while in order to obtain boundary values which cannot be treated by distribution theory, one needs to look for more singular holomorphicfunctions, such as F(z) exp
.
Then IF(z)] is indeed a hyperfunction, supported at the origin, and which certainly not a distribution. Other examples of hyperfunctions which are not distributions will be more evident as we progress with our theory. It maybe interesting to point out the relationships whichlink distributions with polar singularities, and hyperfunctions with essential singularities; Theorem 1.2.5 somehowpoints this out. As we shall see, the connection is even stronger than it mayappear; it is, indeed, one of the important points about hyperfunctions. Example1.2.7 Let f be a hyperfunction supported at the origin, so that one can write f = IF], with F holomorphicin U \ {0}, U being a suitably small open set in 6, and containing the origin. Wehave seen before that all finite order derivatives of f are well defined hyperfunctions; one can, however,do something more, and consider what we shall call, in later pages, an infinite differential operator. Namely,we consider an operator which, at least formally, is defined
by
id +o~ P(Z, ~z) ~-~.ai(Z) dzi;-i=0
1.2.
HYPERFUNCTIONSOF ONE VARIABLE: BASIC DEFINITIONS 15
one can then try to apply P(z, d/dz) to the holomorphicfunction F(z); as long as
P(z,
~__~
+~ )F=~-~ai(z)
d~F(z)
i=0
remains a holomorphicfunction, we can claim that a suitably defined operator P(x, d/dx) has acted on f to yield a newhyperfunction. The reader is invited to comparethis situation with what happens in the distributional case, where it is well known,[210], that every distribution supported at the origin is a finite linear combination of derivatives of Dirac’s delta. Onceagain, as we have discussed before, we have this striking parallel betweenhyperfunctions as essential singularities and distributions as polar singularities. Our example provokes, however, a natural question: when is the operator P(z, d/dz) such that it actually mapsholomorphicfunctions into holomorphic functions; more simply, which growth conditions should we imposeon the series of coefficients a~(z) which defines P(z,d/dz) in order that the formal series P(z, d/dz)F converges in somepunctured neighborhoodof the origin? In order to answerthis natural question, let us first note the followingcharacterization of hyperfunctionssupported in the origin: Lemma1.2.1 Let f be a hyperfunction supported at the origin; then one can express f as
= i=0E wherethe sequence{bi} satisfies the following condition:
Proof. To begin with, let us recall that, in view of Theorem1.2.4, the space of hyperfunctions with support at the origin can be seen as O((~ \ 0)/O((~); forth, the hyperfunction f(x) has a representative F(z) which is holomorphic on g’ \ 0. Wenowconsider the Laurent expansion of F at the origin, F(z) = E aizi; of course, wheni _> O, the correspondingpart of F is entire, and therefore (by the way hyperfunctions are defined) we can say that a different representative of f(x) is given by the part of F composedonly of negative powers, i.e. there is a one-to-one correspondencebetweenhyperfunctions supported at the origin and Laurent expansions such as +~
ai E zi+I ’
16
CHAPTER 1.
HYPERFUNCTIONS
with infinite radius of convergence (rememberthat F was to be holomorphic everywhereexcept at the origin). This condition on the radius of convergenceof the Laurent expansion translates into the fact that, by the Cauchy-Hadamard theorem, one must have limi_~ ~ = 0. Wenownote that 1/z ~+1is nothing but the i-th derivative (up to a coefficient) of 1/z, and therefore, by putting b~ = one sees that the defining function of f(x) can be written as +oo
¯ Z i=O
Z
If we nowrecall the definition of the delta hyperfunction, we realize that we have just proved that +~ d~5 f (x) y~bi~xTx~ (x) i=0
with the required growthconditions on the coefficients bi.
[]
Wecan nowprove a necessary and sufficient condition for an infinite order differential operator to act on hyperfunctions as we have described before. We have: Proposition 1.2.2 Let P(z,d/dz) be defined as above with coefficients ai(z) holomorphicin an open set U; then P(z, d/ dz)F(z) is a germ of a holomorphic function at zo for any germF of holomorphicfunction at zo, if and only if, for any compact K contained in U, (1.2.6)
lii~ ~/SUpglai(z)li!=
Proof. Webegin by proving the sufficiency of the limit condition (1.2.6). Let F be a germ of a holomorphicfunction at somepoint z0. This means that F is holomorphicin someclosed disk D of radius 5 around z0, and we can call Mthe maximum of ]F(z)l in that disk. By Cauchy’sinequality we then obtain that, on D, d~F(z) i!M
~ <-~V-z
Wecan nowcouple this information with the hypothesis (1.2.6); since D is itself a compactset, the hypothesis can be rephrased by saying that for any e > 0 we can find a positive constant C~such that, for any i,
1.2. HYPERFUNCTIONSOF ONE VARIABLE: BASIC DEFINITIONS
17
By nowchoosing e = ~/2, we immediatelydeduce from the previous inequalities that the series diy(z) i=0
converges uniformly in D to a holomorphicfunction. Wenowproceed to prove the converse, i.e. the necessity of condition (1.2.6) for P to act on the space germs of holomorphicfunctions. By contradiction suppose that condition (1.2.6) is not satisfied; then, for some~ > 0 there exists a sequence zj, which we can assumeto be convergent to somez0, such that
/laj( )Y!l>2 ,j =1,2,.... Wenowapply the operator P(z, d/dz) to the function F(z)-
1 (Z-Zo-
whichis clearly holomorphicin the open disk D centered in z0 and of radius e. It is immediateto see that the series P(z,
)F(z)
+~ ai(z)(_l)ii! = ~ (;=~e~+ 1 - ~Fi(z) i=0 i=0
does not converge locally uniformly in D, because of the estimates which we have given on the F~, and therefore the theoremis completely proved. [] Weshall comeback to this result later on, to showhowit actually implies that condition (1.2.6) is a necessary and sufficient condition for P(z, d/dz) to be a sheaf homomorphism on the sheaf of hyperfunctions. In the sequel, wewill refer to operators satisfying condition (1.2.6) as local operators, the reason for this terminology being that they do not movethe support of the functions they act on. Local operators are, therefore, a special case of infinite order differential operators. It is interesting to note howother operators of infinite order do appear and have someimportant applications in theory; the most important, probably, is the translation operator defined by
exp( ~
"~DiF(z) ~
g
where D denotes the differentiation operator d/dz. Such operators are clearly unsuited to act on sheaves, since if F is only a germof holomorphicfunction at the origin, then F(z + 1) maynot even be defined. Froma historical point of view, such operators were studied since the end of last century by the Italian mathematicianPincherle. Somemore details on this will be given in section five of the historical appendixto this chapter.
18
CHAPTER 1.
HYPERFUNCTIONS
Example1.2.8 Infinite order local operators can be used to construct new examplesof hyperfunctions; in connection with Example1.2.6 (and to provide an immediateapplication of Proposition 1.2.2), we can easily see that +~ 2~ri [exp(-1/z)] = k! (k + 1) DkS(x)" k=O
This equality is immediately established by expanding in its Laurent series around the origin the function exp-1/z. Example1.2.9 Wecan also use the notion of boundary value to define other important examples of hyperfunctions. To begin with, let F be a meromorphic function on a complex neighborhood U of an open set ~ in ~; we may assume that the poles of F all lie in ~. Followingthe classical exampleof Hadamard,we can define the so called "partie finie" or finite part of F(x) as the hyperfunction defined by f.p.(F(x)) ½{F(x + iO) + F(x - i0 )}. This hyperfunction (which, of course, is actually a distribution, as the reader will immediatelysee) can be obviously defined by the pair (1/2F(z), -1/2E(z)). Acase of historical interest, and certainly well known,is the finite part of 1/x, for which the following identity (known by the physicists as the LippmannSchwingerformula) holds: 1 (x + i0) = f.p.(1/x) - ~i~(x). The expression on the right hand side of the Lippmann-Schwingerformula is also known,in the literature on distributions, as Cauchy’sprincipal value of 1/x, denoted by p.v. (~). The reasons for these terminologies are rather well known,and can be found in detail in Schwartz’streatise [210]. Example1.2.10 Other interesting examples of hyperfunctions can be defined by taking boundaryvalues of multivalued functions, such as the powerfunction of a complexvariable. To begin with, one mayconsider (x + i0) ~, whosedefining function is the pair(z", 0) and
(x - i0) ", whosedefining function is the pair(0, -z’).
Example1.2.11 Along the same lines, we have the hyperfunctions x~_, which are defined in different ways depending on whether # is an integer or not. To be precise we have, for # not an integer ~] _ 1 [-(-z) x~_ = [2isin~r#] 2isinr# {(-x + iO)t’ - (-x - i0)"},
1.2.
HYPERFUNCTIONSOF ONE VARIABLE: BASIC DEFINITIONS 19
Figure 1.2.4 and [2i sin~r#J 2i sin ~r#{(z+ iO)’ -- (--x -- i0)~’}, while, in the case in which# is an integer, we define
To conclude this introductory section on single variable hyperfunctions, we show(by essentially quoting Cauchy’sintegral formula) howto choose a specific defining function for a given compactlysupported hyperfunction; ~he choice whichwe will makeis interesting from a particular point of view, and the reader may,after reading this result, directly jumpto our historical appendix, to compare this result with someold Fangappie’stheories. Proposition 1.~.~ ~et f be ~ h~perf~nction with compact s~pport K. Define 2~i J-~ z - z Then [G] = f, G is holomoKphicin~ ~ K, and, finally, G(~) = O. Moreover, there is onl~ one definin9 function for f whichs~tisfies these two properties; we will call such ~ definin9 .function the standarddefinin9 function for f . Pro@To begin with, let us note that the function 1/(z - z) is a real analytic Nnction of z on K, for each ~ chosen in ~ ~ K; therefore, by the previous definitions, it can be multiplied by ~hehyperfunc~ionf(z); ~heresult is a hyperfunction for whichit makessense to take ~he integral as in (1.2.7). Takenow a complexneighborhoodof K, and F a representative of f which is holomorphic in U ~ K. If we choose a closed path r contained in U and which leaves z on the outside (as in fig. 1.2.4),
20
CHAPTER 1.
HYPERFUNCTIONS
T
Figure 1.2.5 we see that the definition of definite integral for a hyperfunctionyields G(z) -i~ F(O) do Wethen deduce immediately that G is holomorphicin ~T \ K, so that [G] is hyperfunction with support in K. Wenowwant to prove that [G] = IF] near K, whichwouldshowthat [G] = f. By definition of hyperfunctions, it will suffice to showthat G - F can be holomorphically extended near K. To do so, choose T’ to be a closed curve encircling z once in the positive direction, as shownin figure 1.2.5. Cauchy’sintegral formula immediatelyyields the followingseries of equalities
- , G(z)- F(z)= ~i-1~ oF(O)-zdO~i~-zl
F(O)do = _~+.~_~_~-1 , F(O)
which shows that G - F actually does extend to a holomorphicfunction to the interior of the region boundedby ~- + ~-~. This region contains K, and therefore we have proved that G is a representing function for f. The fact that G is holomorphicon all of~T~1 \ K is a consequenceof its integral representation, and so is its vanishing at infinity. Finally, if H were another representative of f with these two properties, we would have that G - F would be holomorphic everywhere on (~1, and vanishing at infinity. By Liouville’s theorem, G wouldbe a constant and hence the zero constant. This concludes the proof. []
1.3
Hyperfunctions sults
of One Variable:
Main Re-
Section 1.2 has been essentially devoted to a rather cursory description of the notion of hyperfunction, together with some fundamental examples. Wehave
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
21
also described howto operate on hyperfunctions, so that they can be concretely considered as generalized functions. This section, on the other hand, will dwell moredeeply into the properties of hyperfunctions. Wewill showthat hyperfunctions actually constitute a sheaf of generalized functions on ff~ of whichthe sheavesof real analytic functions and of distributions are proper subsheaves. This is one of the fundamentalresults in the theory of hyperfunctions. Wewill also prove that the sheaf B of hyperfunctionsis a flabby sheaf, a very useful result whendealing with systems of differential equations. Somesimple applications of these notions to the study of differential equations are included to provide the flavor and showthe powerof the method. After providing the reader with the statement and proof of the celebrated KStheduality theorem(which, more appropriately, could be called Fantappie’KSthe-Martineau-Satoduality theorem, as we shall showin the historical appendix to the chapter), this section will be concludedby a brief discussion on the possible topologies whichthe spaces of hyperfunctions can be endowedwith; this is, as weshall see, one of the weakpoints of the theory, and it will be useful to have a preliminary look at the situation in the single variable case. In what follows we have decided not to provide sheaf theory per se, since manyadequate texts can be found to fill the gap, [64], [123]. In particular, Kaneko’s monographon the theory of hyperfunctions [103] contains a rather extensive section devoted to the theory of sheaves and to the theory of cohomology,with an eye towards the necessities of hyperfunction theory. The reader unfamiliar with sheaves is therefore invited to consult one of these texts. The first result we wish to prove is the flabbiness of the sheaf of hyperfunctions, whichmarksthe first importantdifference with the theory of distributions. Theorem1.3.1 The assignment B(f~), with f~ any open set in ~:l, is a flabby sheaf of vector spaces. Proof. To prove that a presheaf, i.e. a collection of vector spaces indexed by the family of open sets of a topological space, is a sheaf one needs to prove two completenessproperties; that local vanishing implies global vanishing, and that local belongingimplies global belonging. To be moreprecise, the following properties have to be established to prove that hyperfunctionsare a sheaf on ~; let ~ be an open set in ff~, and let {~i} be an open covering of ~: (i) if an element f in B(f~) vanishes whenrestricted to each ~, then f is the zero in B(~); (ii) if a family {fi} of hyperfunctionsin B(~)satisfies the compatibility ditions f~ -- fj on f~ N ~j, then there exists f in B(f~) such that its restriction to each f~ is exactly f~. The reader will rememberthat we have already proved (i) when establishing the existence of a support for hyperfunctions, so that we only need to establish
22
CHAPTER 1.
HYPERFUNCTIONS
(ii). Theproof of (ii) is slightly deeper, as it requires the use of Mittag-Leffier Theorem. Let therefore ~, f~i and fi be as above, and let Fi be holomorphic functions in Ui \ f~i, which define the hyperfunctions fi (the Ui are just complex neighborhoods of fh). Weset U to be the union of the Ui, and, by the paracompactnessof both (P and ~, we can assume that both {fti} and {Ui} are locally finite (i.e. every point only belongsto a finite numberof opensets). The compatibility condition immediatelytranslates into the fact that the function Fq defined by Fj - Fi is holomorphicin Ui n U~, even thoughnot necessarily zero there. It is howeverimmediateto see that the family {Fq} defines a 1-cocycle on U with values in the sheaf of holomorphicfunctions. Mittag-Leffier Theorem (Theorem1.2.1 and 1.2.1’), then showsthat {Fq} is actually a 1-coboundary, and therefore there are functions Gi, holomorphicon Ui such that Fij= G~ - G~on Uin Uj. The conclusionof the proof of part (ii) nowfollows by observing that the function Fi - Gi = Fj - Gj is holomorphic on U \ ft, and since [Fi - Gi] = [Fi], the hyperfunction IF] is the desired global hyperfunction. To concludethe proof of the theorem, we only need to establish the flabbiness of the sheaf of hyperfunctions. Werecall that a sheaf ~- is said to be flabby if, given any two open sets f~ and f~’, with ~ _Df~’, the natural restriction map # : ~’(Ft) ~-~ ~(f~’) is alwayssurjective. In particular, to prove the flabbiness of a sheaf .T" on a topological space X it is sufficient to prove the surjectivity of the restriction map~’(X) ~ 5c(ft), for any open set f~ in X. In our case, let f~ be an open set in ~. Weknow, by Proposition 1.2.1, that B(ft) can represented as O(U\ f~)/O(U), for U any complexneighborhoodof f~. To prove the flabbiness, it is sufficient to take U = IT \0f~, where0Qdenotes the boundary of f~ in /R. Then every representative F in O(U\ ft) of f gives the required extension to all of ~, since U\ ft is nownothing but IT \ ft. Moreexplicitly, one has that
o(¢ = o(¢ \ 0a) Then the restriction
o(¢
o(¢,
induces a surjective map
o(¢ \ o(¢ o(¢) o(¢) and therefore from the commutative diagram
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
we obtain the surjectiveness of B(~) ~/3(fl). hyperfunctions is thus proved.
23
The flabbiness of the sheaf [~
The reader will note howthe fact that B is a sheaf allows us to speak of a support for hyperfunctions (indeed, in our previous section we had already provedhalf of the sheafness of/3, in order to introduce the notion of support); this is not different from what happens with distributions (which, too, form a sheaf), but it contrasts with the situation for analytic functionals. Indeed, see e.g. [83], an analytic functional is defined as an element# of 7-/’(~’), where the symbolN’(~T)denotes the dual of the Frechet space of entire functions (i.e. the space of linear continuouscomplexvalued functionals on the space of entire functions); analogously,if K is a compactset in,T, then one can define the space of germs of holomorphicfunctions on K as the inductive limit 7-/(K) = ind lim It is possible to showthat 7{(K), endowedwith the inductive limit topology, not anymorea Frechet space, but rather what is called a DFSspace (see [223] for moredetails on topological vector spaces), and one can easily see that for an analytic functional, there exists a compactset K in ~T such that # belongs to the dual 7{’(K) of 7/(K). This is equivalent to the following estimate on values: for every open neighborhoodU of K, there exists a positive constant Cu such that, for every ~ in 7~(U), it I< #, 9~ > I <- CusuPKI~(z)]. Whensuch a condition is satisfied, we say that f is carried by the compactset K; it occurs naturally the question of whether a minimalsuch K exists for every analytic functional. It is by nowwell knownthat this is not alwaysthe case (see e.g. [83]), and that, therefore, a notion of support cannot be defined for analytic functionals; from this point of view, our last Theoremshowsone of the crucial properties of hyperfunctions.
24
CHAPTER 1.
HYPERFUNCTIONS
Onthe other hand flabbiness, i.e. the property whichallows to extend a given hyperfunction from any open set to all of/R, is an extremely important property in the theory of differential equations, since it allows to define hyperfunctionson all of ~, regardless of the singularities which they maypresent at the boundary of their original domain: this is in striking contrast with what happenswith differentiable functions, or even with distributions, which maybe too rigid to extend beyonda given point. Anotherimportant point, as we shall soon see, is the possibility of using the sheaf B to construct flabby resolutions for sheavesof solutions to systemsof differential equations. To illustrate the powerof dealing with hyperfunctions, we prove here a well knownresult of Sato, [142], concerningthe surjectivity of differential operators on spaces of hyperfunctions. Theorem 1.3.2
Let dx
=
i=o
be a linear finite order differential operatorwith real analytic coejficients on an open set ~ in fit, and suppose that am ~ O. Then, for every f in B(~), there exists a solution u in B(~) to the differential equation
p(z,
=/(x).
Moreover,any solution u can be extended to a hyperfunction solution defined on any open set containing ~ and on whichall the coe~cients a~ and f are defined. In particular, if all of the coe~cients and f are defined on all of ~, then any solution u defined on ~ can be extended to a hyper/unction solution u* on ~. Proof. Let V be a complexneighborhoodof 9t to which all coefficients a~ can be analytically continued: By taking V suitably, we can makesure that both V and V\~ are simply connected and that all zeroes in V of the leading coefficient a,~ are actually containedin ~t. Let nowF E (.9(V \ ~) be a defining function f. By the well knownCauchy’s existence theorem, and since am has no zeroes in V\ ~, there is a solution U E (_0(V\ ~) to the (complexanalytic) differential equation
P(z, z)U(z) = F(z). It is noweasily seen that u = [U] gives the required solution. Wenowproceed to prove the possibility of extending the solution u to a solution of the same equatio n to all of ~ (we are assuminghere that both f and the ai are defined on/R, and that u is any solution of P(x, d/dx)u = f on somereal open set ~). By what wehave just proved(the existence part of the theorem), it will be sufficient to consider the case of homogeneousequations such as P(x, d/dx)u(x) = 0. Let
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
25
V be the same complexneighborhoodof f~ we have used in the first part of the proof, anal set It is easily seen that Wis still a complexneighborhoodof gt, and we denote therefore by U ¯ (9(W\f~)a representative for the solution u of the homogeneous equation on ~. Note that P(z, d/dz)U(z) is holomorphicon all of W. Wewill now show that there exists a function N(z) holomorphic on Wsuch that the function P(z, d/dz)U(z) - P(z, -~z)g(z) can be continued analytically on V; this immediatelyimplies that the hyperfunction u* defined on all of ~ as the equivalenceclass of U - N gives the required extension, and will therefore conclude the proof of the theorem. To construct such a function N we will use once again the Mittag-Leffler Theorem(Theorem 1.2.1’). To begin with, choosetwo open sets V~and V2such that VxUV2coincides with V, V~ ~ (~ \ ~) = ~, and V1\ (~ \ f~) is simply connected and contains no zeroes of a,~(z). Again by Cauchy’sexistence theorem, we obtain that there exists a function No, holomorphicin VI \ (~ \ gt), solution of the equation P(z, ff--~)No(z)=P(z,ff--~)U(z). Wecan nowapply Mittag-Leffier Theoremto find N1¯ (.9(V1) and N2¯ (.9(V~) such that, on V1N V~’, No(z) = N2(z) - N~(z). If we nowdefine the function N(z)
f No(z) N~(z), z ¯ V~ \( ~\f g2(z), z e
’
we immediately see that N is holomorphicon Wand that the theorem is therefore completely proved. [] The reader will notice howthis theorem is more powerful than the Cauchy existence theoremwhich it employs, as it does not require the non-vanishingof the leading coefficient of the differential operator. In the next chapter we will slightly refine this result to discuss the location of the singularities of u with respect to the singularities of f and to the set of zeroes of a,~; in so doing we will be able to prove a preliminary version of the celebrated Sato’s F~ndamental Principle. Wehave seen that the operator P(x,d/dx) is surjective on the space of hyperfunctions; we have therefore looked at P as an operator between vector spaces, but we mayactually look at it as a sheaf homomorphism, i.e. we have the followingsimple fact:
26
CHAPTER 1.
HYPERFUNCTIONS
Proposition 1.3.1 The operator P(x,d/dx) is a sheaf endomorphismof the sheaf A of real analytic functions and of the sheaf B of hyperfunctions. Proof. Werecall that, given two sheaves of(T-vector spaces on a topological space X, say 9v and G, a sheaf homomorphism is a collection of if-linear maps hv : Jz(U) ~ ~(U), indexed on the family {U}of all open sets of X, such that the diagram ~=(V)
-~ ~(U)
~(v) ---, ~(v) commutes for any inclusion V ~-~ U. In our case, if we consider the sheaf ..4 of real analytic functions, then the result is an obviousconsequence of the definition of partial differential operators. As for the case of the sheaf of hyperfunctions, one immediatelyobtains the sameresult if one considers that P(z, d/dz) actually acts as a sheaf endomorphismon the sheaf of germs of holomorphicfunctions, and therefore the sameresults applies to hyperfunctions. [] From general properties of sheaves and their endomorphisms(which can howeverbe immediately derived by their definitions), we obtain the following Corollary, of someindependentinterest: Corollary 1.3.1 Differential operators with real analytic coe]ficients do not enlarge the support of hyperfunctions. Proof. Immediatefrom Proposition 1.3.1, and from the general fact that sheaf endomorphismsdo not enlarge the supports of the sections on which they act.
Wehave mentioned the importance of flabbiness in extending hyperfunction solutions of differential equations; flabbiness has, however,one other important consequence, which may be worthwhile mentioning. As the reader no doubt knows,one of the features whichhave madethe theory of infinitely differentiable functions (and, as a consequence,the theory of distributions) so successful, the existence of the so called cut-off functions, i.e. the existence of compactly supported functions. This fact, which sheaf theorists summarizeby saying that the sheaf of C~ functions is a fine sheaf, is crucial in the study of differential equations, and plays a substantial role in the developmentof the study of singularities of distributions (we shall comeback to this in our next chapter, wheremicrolocalization will be discussed). Despite the attractiveness of hyperfunctions, however,this feature is missing in this case (and this was probably
1.3.
HYPERFUNCTIONSOF ONE VARIABLE: MAIN RESULTS
27
one of the reasons for not using real analytic functions as test functions, before Sato). In particular, given a hyperfunction f, it is not possible to truncate it to a compactly supported one (in the case of distributions, one achieves this result by multiplying the given distribution by a suitable cut-off function), not even if one allows somemodifications in a small neighborhoodsof the sets in which the support has to be restricted (which it has necessarily to be done in the distribution case). Onehas, however,the following remarkableresult, which we state and prove for hyperfunctions but which, as it is obvious from the proof itself, is actually a result valid for anyflabby sheaf. Proposition 1.3.2 Let f be a hyperfunction on an open set ~, and let supp(f) = A1ta A2, with A1,2 two closed sets. Thenthere exist hyperfunctionsfl and f2 on ~, with supp(fi) =Ai, i 1,2, such that f = fl + f2. The same result holds true if the support of f is given a locally finite decompositionin closed sets. Proof. Wewill give the proof for the case of two open sets A1 and A2, for which we set E~ = ~ \ A~. It is indeed sufficient to define the hyperfunctionfl by the position onE~ f l ={ fo on E1 " Note that this position is well given, since, on the intersection A1~l A~_,f is actually zero. Wetherefore have a hyperfunctionon A1taA2. Since B is a flabby sheaf, such a hyperfunctionextendsto all of ~ (and we will still call it f~). It supp(f~) C_ A1, and the result follows if we simply define f2 = f - f~. Wewish to remark that this result will be essentially used as a substitute for the existence of partitions of unity; as it is well known,partitions of unity are a crucial tool in the theory of fine sheaves(of whichthe sheaf of infinitely differentiable functions is a goodexample);the aboveresult constitutes the analogue for flabby sheaves. In a sense, even if no canonical way is provided, some advantageis gained by the fact that we will be able to obtain decompositionsin whichno modification is necessary near boundaries (which is on the other hand necessary whendealing with the customarypartitions of unity). Our next result has been already provedin the previous section, even though we did not use any sheaf terminology there: Proposition 1.3.3 ~4 is a subsheaf of B. Proof. It is sufficient to show that, for every open subset ~ of/R, we have an inclusion of A(~) in B(~), whichis consistent with the restriction maps(i.e. that the restriction, in the sense of hyperfunctions,of a real analytic function is still a real analytic function, namelythe restriction
c_
28
CHAPTER 1.
HYPERFUNCTIONS
induces the restriction on the subgroups ~ Pn’l.4(n) ~t(~) ---~ ~(~’) of analytic functions). To prove such an inclusion we resort to the use of the function ~ which we have defined in the previous section. Recall that the space of real analytic functions is defined as the inductive limit of spaces of holomorphic functions, and let ~ be a holomorphic function on some complex neighborhood U of ~, which defines (through the inductive limit procedure) a real analytic function whichwe will still call ~. Wecan nowassociate to ~ the function e~, holomorphicon U \ [~, which defines an object in the quotient space O(U\ ~)/(.9(U); but if we nowtake the inductive limit, on all open neighborhoodsof ~t, of the equivalence class of levi, we obtain the hyperfunction which we associate to ~, in our sheaf homomorphism. The fact that this map is well defined (i.e. independent on all the arbitrary choices we made)and that it actually a sheaf homomorphism is nowimmediate, and this concludes the proof.
It becomestherefore interesting to study where a hyperfunction fails to be real analytic. Indeed, we say that a hyperfunction f (which, by flabbiness, we mayassumeto be defined on all of ~) is real analytic on ~ if it belongs to if f = IF], its being real analytic really dependson the lack of singularities of the pair of holomorphic functions F = (F+,F-) which represents f. More specifically, one has that a hyperfunction f = IF] on ~ is real analytic on if and only if both F+ and F- extend analytically across ~t, to holomorphic functions in somecomplexneighborhoodof ~t itself. The reader will note that, by the definition of B, the real analyticity of a hyperfunctionf does not depend on the representing function F whichhas been chosen to test the analyticity. Real analyticity is obviously a local property (essentially because of Proposition 1.3.3) and therefore, given any f in B(~), there exists the largest open subset of ~ on which f is real analytic. The complementof such a subset will be called the singular support of the hyperfunction f, and will be denoted by the symbol sing supp (f). The choice of the terminology may look somewhat suspicious, but we will showlater in this section that the singular support of a hyperfunctionis actually the support of a specific section of a different sheaf. Example 1.3.1 It is obvious that sing supp (5) = {0}; indeed, the delta function is identically zero (and therefore real analytic) cannot be analytically continued. Example1.3.2 An analogous argument showsthat, for any positive integer n, one has sing supp (5 (~)) = {0}.
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
29
Example1.3.3 The Heaviside hyperfunction, too, has singular support equal to the origin, since H(x) is locally constant outside the origin. In different words, the function log (-z) can be analytically continued across the real axis everywhereexcept at the origin. Example1.3.4 The singular support of the hyperfunctions [c] and [c] is the emptyset, since they are both real analytic functions over the real line. Indeed, [~] can be represented by the holomorphicpair (1, 0), and both the function and the function 0 can be analytically continued across the real line. A similar argument holds for Example1.3.5 If F is a function holomorphicon, say, the upper half plane, then its boundary value F(x ÷ iO) is a hyperfunction whose singular support coincides with the closed set of all points of ~ across whichF cannot be analytically continued. Indeed, F(x + iO) = (F, 0), and 0 can alwaysbe analytically continued everywhere. Example 1.3.6 Let F be a meromorphicfunction on a complex neighborhood U of a real open set ~, and assumethat all the poles of F lie in ~. Thenwe have defined the hyperfunction f.p.(F) as the holomorphic pair (F/2,-F/2). From what precedes, it is clear that the singular support of f.p.(F) coincides with the set of poles of F. In particular, sing supp f.p.(1/x) = {0}. Example1.3.7 The singular support of the hyperfunctions x~: which we have defined in Example1.2.11 is, of course, the origin. However,one can check that sing supp ((x + i0) ~) is the emptyset if # is a nonnegativeinteger, and is the ~. origin otherwise. The sameis true for Ixl The fact that singular supports are defined in a local way allows us to immediately obtain the following result: Proposition 1.3.4 Local operators do not enlarge the singular support of hyperfunctions. Proof. This follows immediately from the fact that local operators (both of finite and of infinite order) act as sheaf homomorphisms on both Jt and B. Therefore, if a hyperfunctionis real analytic at a point x0, then so is its result after the application of a local operator. [] Because the singular support of a hyperfunction is actually the support of a section of a different sheaf on which, too, local operators with analytic coefficients act, one immediatelyobtains the previous result as a corollary of a general fact fromsheaf theory; this fact will be clear later in this section.
30
CHAPTER 1.
HYPERFUNCTIONS
Proposition 1.3.4 showsthat if P is a local operator, and f is real analytic, then P(f) is, still, real analytic. Theinverse of this propertyis usually referred to as the "ellipticity" of the operator. Wewish to conclude this section with a brief (and somehow preliminary) discussion of the ellipticity phenomenon as applies to the sheaf of hyperfunctions. Let us begin with a finite order mconstant coefficients differential operator P(d/dx). Webegin by stating the following simple fact, whose proof can be foundin [142]: Proposition 1.3.5 The dimension of the space of hyperfunction solutions to the differential equation P( ~--~)f(x) is rn. Moreprecisely, let ~ be any interval of the real line: then dim{f e B(~): P(~x)f(x ) = 0} = m. As a consequenceof this result, one deducesthat, as in the case of distributions, all constant coefficients operators can be consideredto be elliptic. To this purpose, we say that an operator T acting on the sheaf B is an elliptic operator if the followingcondition is satisfied for any f in B(~): if T(f) = 0, then f is real analytic on ~. This condition is often referred to as the analytic hypoellipticity, to distinguish it fromthe differentiable hypoellipticity, in whichthe setting is the sheaf of Schwartzdistributions and the fact that f belongsto the kernel of T only implies that f is actually differentiable on ~. Fromthis point of view, it is clear that, a priori at least, analytic ellipticity is rather strongerthan differentiable ellipticity, in which a stronger hypothesis is requested (f is a distribution rather than hyperfunction) and a weakerconclusion is obtained (differentiability rather than anMyticity). Wecan indeed prove the following result: Theorem1.3.3 Every linear constant coe~cient finite order m differential eratoris elliptic.
op-
Proof. Let ~ be any open set in ~. Then, on each connected component of ~, there are m linearly independent real analytic solutions to the equation P(d/dx)f = g, with g real analytic (this is just the well knownEuler Fundamental Principle for solutions of ordinary differential equations). But, in view of Proposition 1.3.5, there are exactly mlinearly independenthyperfunction solutions to that same equation. Therefore the space of hyperfunction solutions
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
31
in ~t coincides with the space of real analytic solutions in f~, whichconcludes the proof. [~ The reader will note that, by the linearity of the operators we are working with, the ellipticity condition can be somewhatrelaxed, and one can see that it is actually equivalent to the request that every hyperfunction solution to P(d/dx)f = is actually rea l ana lytic. Suc h a c ondition immediately imp lies (and is therefore equivalent to) the ellipticity of the operator P(d/dx). In Theorem1.3.3, we have requested the hypothesis of finite order on the differential operator. In viewof the fact that we are restricting our attention to the one variable case, we can removethat hypothesis and obtain the following result: Theorem1.3.4 All local operators acting on the sheaf of hyperfunctions on fit are elliptic. Proof. See [103].
[]
Wewill use this result in the sequel to prove that our representation of hyperfunctionssupported at the origin as derivatives of the Dirac delta function can actually be extended to the case of any general hyperfunction. The situation becomesa little moreinteresting if one considers variable coefficient operators. To begin with, weshall restrict our attention to linear finite order differential operators with real analytic coefficients. Let
d~
P(z, d/dz) = ~ aj(z)-~z j j=O
be a differential operator with holomorphiccoefficients and of finite order m: we recall that the index of a differential operator P(z, d~) acting on the space O(V) of holomorphicfunctions on an open set V in ¢ is defined as the difference Ind(P) -- dim Ker(P) - codi~n Im(P). Wehave the following result, originally due to Komatsu[142]. Theorem1.3.5 Let V be an open set in ~T. Then the differential P(z, ~z) has index Ind(P) = mc(Y) - ~ ordzam(Z),
operator
Z
where c(V) denotes the Euler-Poincare’ characteristic of V 5.e. c(V) is difference bo(V) - b~(Y) of the two first Betti numbersof V) and ordzam(Z) denotes the order of zero at z of the leading coeJficient am(Z).
32
CHAPTER 1.
HYPERFUNCTIONS
Proof. To begin with let us remark that, in this particular case, the topological characteristic c(V) can also be interpreted as the numberof connected components of V minus the number of compact connected components of its complementin (T. This will be the interpretation which we will use for c(V) in the rest of the proof. Nownote that if m = 0, then Ker(P) = 0, and then one easily finds a basis for the complementof Im(P) which is constituted by exactly ~z ordzao(z) polynomials. If we now consider the case in which P(z, d/dz) = d/dz, then from the Mittag-Leffier Theoremone has that
Ker(P) H°(V,(T), and Coker(P)
= HI(v~(~).
Since bo(V) = dimH°(V,(T) and bl(V) -- dimH~(V,(T), the result follows immediatelyalso in this case. It is a general fact that the index of the product of operators is the sum of the indices, and so from what we have proved it follows also the result for the operator P(z, d/dz) = d’~/dz"~. Finally, [142], it is possible to showthat in a Banachspace a lower order perturbation does not change the index of an operator; since O(V)is an inductive limit of Banach spaces in a standard fashion [223], we can conclude the result for the general operator P(z, d/dz) acting on O(V). From this general result we can immediately obtain an extension of our previous Proposition 1.3.5 to the case of variable coefficients operators.
Theorem1.3.6 With the notations used before, and f~ any interval in ~, we have that d dim(f ¯ B(12) P(x, ~x )f = O)= my~ordxam(X).
Proof. This is a typical diagram chase proof; let V be a connected complex neighborhoodof ~ to whichall the coefficients aj can be analytically continued; restrict V in such a wayas to take both V and V \ ~ simply connected and such that all the zeroes of am(z) are containedin f~ (all of these topologicalrestrictions are necessary to simplify the application of Theorem1.3.5). Consider nowthe following commutativeexact diagram:
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
33
where 0P and BP denote, respectively, the sheaves of holomorphic and hyperfunction solutions of the associated homogeneousequation. Since Pv and Py\a have both well defined indices, and since both dim Ker(Py)anddim Ker(Py\ are finite, we deducethat also Pa has index which is given by Ind(Pn) = Ind(Py \ ~) - Ind(Py) = m + ~ ordxa,~(x). x
This concludes the proof of the Theorem. It maybe worthwhile noticing that (as it should probably be expected) the diagram shows clearly that two types of solutions are represented in BP(~): those coming from OP(Y \ ~)/oP(v), and those coming from O(V)/PO(V). This hyperfunction version of the index theoremallows us to provide a con: dition for ellipticity which is muchweakerthan the one given in our previous Theorem1.3.3. Theorem1.3.7 With all notations as before, the following statements are equivalent:
(i) P(n) (ii) am(X)# 0 for all x in (iii)
P(x, d/dx)f e .A(Q) implies f ¯
Proof. (i) implies (ii): suppose that am vanishes at a point x0 in ~; then, Theorem1.3.6, there are more than mlinearly independent solutions near x0.
34
CHAPTER 1.
HYPERFUNCTIONS
Take, therefore, a non-trivial solution f in ~P(~) whichis identically zero for all x < x0; such a solution, clearly, is not real analytic and this violates (i). (ii) implies (iii): let us assumefl to be connected(the argumentwouldotherwise apply to each connected component). Then on t2 there are exactly mlinearly independent real analytic solutions of the homogeneous equation associated to P; on the other hand, Theorem1.3.6 (and hypothesis (ii)) showsthat there exactly mlinearly independent hyperfunction solutions to that same equation; this showsthat .AP(~) coincides with BP(~’~). WenowUse the linearity of the operator to concludethat, for every real analytic function g on ~, all solutions to P(x, ~)f = must be real analytic as well. (iii) implies (i): this is immediate. Let us nowlook at a few examplesin whichwe will see the concrete meaning of solving a differential equation in the realm of hyperfunctions. Example1.3.8 Weshall start with the simplest case in which we deal with a constant coefficient differential equation of order one. Takethe equation, on all of ~, d~ If we consider f = [F], where F = (F+, F-), then df/dx = 0 translates in
~ --0, which means, by the definition of hyperfunctions, that dF+(x + iO) dz
dF-(x - iO) _ H(x), dz
where H denotes, as usual, the Heaviside function. Since F+ and F- are holomorphicfunctions in, respectively, the upper and lower half-plane, their derivative can coincide on the real axis if and only if F+ -- F- + const. This shows that the hyperfunction solutions to df/dx = 0 are exactly the constants as one expects. Example 1.3.9 Let us proceed to something slightly considering the equation (1.3.1)
more complicated, by
xf(x) =
Here the order of the equation is zero, but there is a singular point at the origin (where the leading coefficient x vanishes). Note that whatever solution we will
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
35
obtain, it has to coincide with 1/x for x ~ 0, and have a singularity at x = 0. Again, let f = IF]. Then equation (1.3.1) becomes
[zF(z)] = = from which one deduces the existence of a function G(z), holomorphicnear the origin, such that zF(z) - a(z) = ~(z). If we now expand G(z) in Taylor series around the origin, we have G(z) = Co + zGl(z), with Gl(z) holomorphicnear zero. This implies F(z) = e(z) + co + G~(z), Z
Z
and therefore (using the notations introduced in the previous section) 1 f(x) - (x + iO) + The reader will note that the dimensionof the space of solutions is, in this case, one, which is in agreement with Theorem1.3.6. Example1.3.10 Wenowcombine these two examples, to look at a first order differential equation with a singularity at the origin. Considerthe equation (1.3.2)
xdf(x)
- f(x) =
In view of Theorem1.3.6, we knowthat we should be looking for two linearly independentsolutions to (1.3.2). Weneed to consider three different cases. (i) ~ = 0, 1, 2,... are
(ii) ~ = -1,-2,... are
(nonnegativeintegers); then the two independentsolutions
(negative integers);
then the two independent solutions
p.v.(z~). (iii) a not an integer; then the two independentsolutions are
36
CHAPTER 1.
HYPERFUNCTIONS
It maybe interesting to note that the two solutions which appear in these three cases fit exactly in the schemedescribed in our commentafter Theorem 1.3.6. Let us nowgo back to the inclusion of the sheaf ~4 in the sheaf B. Generally speaking, we knowthat given a sheaf inclusion 9v ~ ~ on a topological space X, and given the family {ft} of all open sets of X, the quotient presheaf given by {6(~)/gv(f~)} is not necessarily a sheaf. The quotient sheaf of two sheaves, indicated by ~/~-, is on the other handdefined as the sheafification, see [64], of the presheaf described above. In the case of the inclusion of A into B, and at least as far as the case of one real variable is concerned,the situation is simpler, and has a nice geometric interpretation to which we already made reference previously; we have the following result: Theorem 1.3.8 The presheaf B/Jt is actually a sheaf, i.e. B/A(ft) = B(f~)/fl,(ft). Moreover, for every hyperfunction f on an open set ft, its singular support is the support of the equivalenceclass f rood Jt(~), considered as an element of B/A(f~). Finally B/A is a flabby sheaf. Proof. See e.g. [103].
rn
It is rather obviousto note that the quotient sheaf whichwe have just represented is a description of the singularities of the hyperfunctions(since the real analytic part is eliminated through the passage to the quotient). Still, this description is not fully adequate, and a complete description of the singularities of hyperfunctionswill have to wait until we introduce the process of microlocalization in the next chapter. Wehave so far proved that the sheaf of real analytic functions is a subsheaf of the sheaf of hyperfunctions;this is an importantresult, whichlegitimizes the use of hyperfunctionsas generalized functions; a related, moredelicate, issue, is the question of howlarge the sheaf of hyperfunctionsis. As it turns out, we know that every Schwartz distribution can be represented as the boundaryvalue of a holomorphic function (and in 1.2 we have proved that some growth conditions are necessary on the holomorphicfunctions in order for their boundaryvalues to be distributions); since hyperfunctionsarise essentially as the correct concept to formalize the notion of such boundaryvalues, one mayexpect that the sheaf of distributions is actually a subsheaf of the sheaf of hyperfunctions. This is indeed the case, as our next result shows: Theorem1.3.9 The sheaf T)’ of Schwartz distributions on J~ is a proper subsheaf of the sheaf B of hyperfunctions on ~. One could actually show(see e.g. [103]) that a sequence of sheaf inclusions can be given, namely
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
37
The following theorem can be found directly in [103], to which we refer the reader for any further detail on this matter. Theorem1.3.10 If we indicate with j the inclusion in the sheaf of hyperfunctions: (i) for any function f locally integrable in ~, and any real analytic function ,~ in ~, ~j(f) = j(pf); (ii) for any differentiable function f, (d/dx)j(f) = j(df (iii) for any function f, real analytic in a neighborhood of the interval [a, b]; fab f(x)dx = f~bj(f)(x)dx. The inclusion of the sheaf of distributions in the sheaf of hyperfunctions also raises the issue of the right notion of boundaryvalue; such an issue is immediatelydisposed of by the following result in which, again, j represents the inclusion of the sheaf of distributions in the sheaf of hyperfunctions: Theorem1.3.11 Let f be a distribution and let F = (F+, F-) be a holomorphic function which represents j(f). Then the function F+(x+ ie) - F-(x convergesto f, in the sense of distributions, as ~ decreasesto zero. Wewill comeback in future sections to the various notions of boundary values, as we will tackle the issue of extending these concepts to several real variables. As for now, we continue this section with one of the most important results which,as the reader will see, provide the link betweenthe moderntheory of hyperfunctions and its prehistory. Weneed to recall a preliminary result of KSthe[138], [139] on analytic functionals in the complexplane: we remind the reader that if K is a compactset in (~, then the space of germsof holomorphic functions on K is defined as the topological inductive limit O(K) = ind limO(U), U where U ranges over a cofinal family of open neighborhoodsof K. Theorem1.3.12 Let K be a compact set in the complex plane and V an open set containing K. Then
O(K)’O(V o(v) \ Proof. See Theorem1.3.14 below.
[]
Withthe help of this result we can prove the celebrated duality theoremfor hyperfunctions:
38
CHAPTER 1.
HYPERFUNCTIONS
Theorem1.3.13 Let K be a compactset in ~:t, and let .4(K) be the space germs of real analytic functions defined on a neighborhoodof K. Then the dual ,A(K)’ can be identified with the spaceBz((fft) of hyperfunctionssupported by means of the inner product < ~, f >= fa ~(x)f(x)dx, wheref is a hyperfunction, ~ is a real analytic function and 12 is a real neighborhoodof K on which ~ is defined. Proof. Immediate consequence of Theorem1.3.12.
[]
The interest of this theoremis that it establishes that hyperfunctionscan be interpreted as non compactlysupported real analytic functionals; moreover, at least whenthey are compactly supported, a fully developed theory of Fourier transform already exists, and one might hope to develop it and extend it to the case of general hyperfunctions. This has actually been done by Kawai, originally in [130], and we will comeback to this at a later stage. Theorem1.3.13 has been stated, as of now,in the frameworkof linear spaces, with the duality being understood in that sense; one may, however, question whether the isomorphismwhich it describes mayactually be a topological isomorphism, and, in that case, which topology would we be considering on the space of hyperfunctions. As a matter of fact, unlike what happens for the theory of distributions (which is heavily based on the theory of topological vector spaces), the theory of hyperfunctions has more of a linear space flavor, and indeed, so far, we have not even touched upon the problem of endowingB(f~) BK with a natural topology. Such a problem, however, will be crucial in the next section, and we therefore wish to address it right away. Werefer the reader to [223] for all notations and definitions concerningtopological vector spaces. To begin with, we shall showthat, for any compactset K in ~, the space BK of hyperfunctions supported by K can be given a quite natural Frechet space topological structure. This is a consequenceof the following result, which is nothing but a strengthening and modification of Proposition 1.2.3. Theorem1.3.14 Let K be a compact set in ~ and let hyperfunctions supported by K. Define a map
I~ K
denote the space of
(1.3.3) T: BK--+ O(~T\ by setting, for any hyperfunction f supported by K, T(f)(z)
f(x)
JI~
=/
Z--
X "
Then the mapT is injective and its image is a closed subspace of the Frechet space O((F \
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
39
Proof. Since T just provides a defining function for f, the injectivity is clear. Let us nowturn to the topological part of the result. Suppose{fj} is a sequence in B~c such that {T(fj)} converges to someholomorphicfunction F the Frechet topology of O(¢ \ K). Wewant to prove that the hyperfunction identified by the pair F, whichis clearly supported by K, is such that T(f) = This will conclude our proof. To do so, just note that T(fj) can actually be computedusing any boundary value representation for fj. This was already proved in Proposition 1.2.3. so that we can conclude that if fj = [Fj], then T(fj) can be written as the integral T(fj)
FJ(~)d4"
Now,by hypothesis, {Fj} converges uniformly (on each compactset which does not intersect the real axis) to F, and therefore this concludesthe proof. [] As a consequenceof this result we can identify BKwith a closed subspace of the Frechet space O(¢ \ K), and this maketherefore the space BKa Frechet spacein itself. It is immediateto notice that the Frechet structure whichwe have just given to the space of compactlysupported hyperfunctions is a natural consequenceof the fact that such hyperfunctions can essentially be considered as equivalence classes of functions holomorphicoutside a compactset. Also, the reader will note that, in defining the injection of BKin the Frechet space of holomorphic functions, we have used a very specific representative for the hyperfunctions involved, essentially based on the standard representative introduced in Proposition 1.2.3. As it is shownin [103] (wherethe several variables case is treated as well), this particular choice bears no consequences.This fact is particularly obvious in the one variable case, as the followingresult shows: Proposition 1.3.6 Let {fj} be a sequence of hyperfunctions supported in the samecompactset K, ft a real open neighborhood.forK and, finally, V a complex neighborhoodof ~ \ K. Let { Fj} be a sequenceof functions holomorphicin O(V) such that [Fj] = fj. Thenif {Fj} convergesto F in the Frechet topology of O(Y), one has that {fj} converges to f = IF] in Sg. Proof. The proof is immediate from the direct computation of T(fj) which one mayobtain from using their boundary value representation. [~ An immediate corollary of this argument is the following complementto Proposition 1.3.1: Proposition 1.3.7 Every linear differential operator with real analytic coefficients acts continuously on the space B~:. Moreover,every local operator acts continuously on the same space.
40
CHAPTER 1.
HYPERFUNCTIONS
Proof. For the case of finite order differential operators, the proof is an immediateconsequenceof the definition of the action of differential operators on hyperfunctionsand of the definition of the topology on the space of compactly supportedhyperfunctions. In the case of infinite order differential operators, we need to recall that such operators act continuously on the space of sections of holomorphicfunctions, in view of the estimates on the coefficients; indeed, for F a holomorphic function, the norm of P(D)F only depends on the norm of F itself. Onethen applies the same argumentas in the case of finite order to conclude the continuity on the space of hyperfunctions. [] Let us take this opportunity to go back briefly to one of the important first remarks we made on hyperfunctions; in section 2 we have shown how every hyperfunctionsupported at the origin could be written up as, formally, a series of derivatives of the Dirac ~ function; the previous result immediatelyimplies that the formal series whichone constructs is actually convergentin the topology of the space/3{0} of hyperfunctions supported at the origin. There is an important consequence of the way we have defined the topology on the space of compactly supported hyperfunctions, and we wish to point it out in a separate proposition, since it will becomenecessary later on: Proposition 1.3.8 For any compact K in the real line, the space with the topologydescribedbefore, is a reflexive space.
BK,
endowed
Proof. Indeed since, for any open complex set U, the space d0(U) is a Montel space, so is the space/3g. Its reflexivity then follows from standard arguments in topological vector spaces. [] The discussion on reflexivity brings us back to Theorem1.3.13, of whichwe can nowgive a stronger and more interesting formulation, based on the topology we have just described. Before doingso, let us quickly recall the topology of the space of analytic functionals. As the reader mayremember,given a real compact set K, the space .4(K) of germson K of real analytic functions is defined as the inductive limit lim do(U)of the spaces of holomorphicfunctions on a cofinal set {U}of open complex neighborhoods of K. As a consequenceone sees that A(K) is a limit of Frechet space, which is naturally endowedwith an LF topology; as such, as it is well known,~t(K) is not a Frechet space, even thoughmanyof its properties closely resemble those of ~echet spaces. In particular, even though one cannot define a topology on an LF space through the notion of convergent sequences, we have the following result: Proposition 1.3.9 A sequence {~j} of germs in ~4(K) converges to a germ ~ .A(K) if and only if ~j(z) convergesuniformly to ~(z) in a complexneighborhood of K.
1.3.
HYPERFUNCTIONS OF ONE VARIABLE: MAIN RESULTS
41
Proof. This is an immediate,and general, consequenceof the definition of the inductive limit topology of ~,(K). Wecan nowstate the topological version of Theorem1.3.13 as follows: Theorem1.3.15 For every compact set K in ~, one has the following topological isomorphism: A(K)’ = BK. Proof. This result follows from the preceding propositions in a standard way. The reader is referred to [103]. [] This result is of course of great theoretical interest (besides being a natural completionof KStheduality theorem), especially as it will allow us to define notion of Fourier transform for compactly supported hyperfunctions, and thus to develop someof the tools necessary for the approachto differential equations on hyperfunctions along the ideas in Ehrenpreis’ Fourier Analysis in Several ComplexVariables [52]. Evenmoreinterestingly, from this point of view, the converseis true. Indeed, one has the following result: Theorem1.3.16 For any compactset K in J~, one has the following topological isomorphism: {BK}’ = A(K). As a corollary we get that, since BKis an FS-space, then its dual A(K) is a DFS-space,which fact entails someinteresting consequences(see e.g. [102] for someapplications and a hyperfunction version of the Ehrenpreis-Palamodov FundamentalPrinciple). Onemayalso note that, as it was done in [103], Theorem 1.3.16 actually implies Theorem1.3.15, in view of Proposition 1.3.8 on the reflexivity of the space of compactlysupported hyperfunctions. Note, however, that from Theorems1.3.15 and 1.3.16 we could also directly conclude that both BKand ft,(K) are reflexive spaces. Wenowpoint out one last result on compactlysupported hyperfunctions; this allows us to display a behavior of hyperfunctions whichis completely different from the corresponding behavior of Schwartzdistributions. Proposition 1.3.10 Let K C_ L be two connected compactsets on the real line. Then the natural inclusion BK~-~ BL has a dense image. Proof. In view of the Hahn-Banachtheorem, it is sufficient to showthat if a functional ~ in {BL}’ = A(L) vanishes on ~K, then it is identically zero. Let
42
CHAPTER 1.
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therefore aeK, so that, for any multi-index a, D~5(x-a) belongs to BK. Then, if < , > denotes the duality bracket induced by Theorem1.3.16, one has: 0 =< Da6(x - a), ~(x) >=<5(x - a), (-D)%p(x) (-1 )l~l~(n)(a), and thus, by the uniqueness of analytic continuation, ~ = 0 everywhere, which concludes the proof. [] So far, we have described the topology of compactly supported hyperfunctions, and we have noted howtheir structure is not too different from what we are already used to; this is essentially the consequenceof two facts: the representation of hyperfunctions in terms of a well defined standard holomorphic function, and the duality expressed by KSthe’s theorem. Unfortunately, the situation is not at all this simple whenwe try to consider the case of non-compactly supportedhyperfunctions. As a matter of fact, one can say that it is not possible to endowthe space B(~) with any reasonable topology! The intuitive reason for this state of things is essentially the fact that for noncompactlysupported hyperfunctions there is no way of choosing a standard defining function. Even the zero hyperfunction has no canonical way to be represented, unlike what happened in the compactcase. In addition, as it has been pointed out both in [103] and in [123], we do not expect to be able to ever come out with a reasonable topology for B(~), since its dual wouldhave to somehow represent the space "real analytic functions with compactsupport". This lack of a good topological structure will have rather negative consequences, as it will be immediately apparent in the next few pages.
1.4
Hyperfunctions sics
of Several Variables:
Ba-
In this section we will try to develop and describe the first fundamentalnotions in the theory of hyperfunctions on ~n. The reader will immediately recognize that a totally different level of complexitybecomesnecessary here, and someof the reasons for this are sketchedin the historical appendicesto the chapter itself. While section 1.2 only required the most elementary notions from one variable complexanalysis, and section 1.3 used the fundamental definitions from sheaf theory, this present section, and the next one, will rely heavily uponthe theory of relative cohomologywith coefficients in a sheaf, as well as on somedelicate properties in the theory of several complexvariables. In order to help the reader’s intuition, and in keeping with our original plans, we have restricted our attention to the case of hyperfunctions defined on the Euclidean n-dimensional space; more generally (and the reader is referred to [123] for this) one can extend all of our definitions and arguments to the
1.4.
HYPERFUNCTIONS OF SEVERAL VARIABLES: BASICS
43
situation, howeveruseful and important, wouldhave eliminated the clarity and accessibility which, we hope, will makeour approachsuccessful. Let us begin by pointing out somenatural attempts to define hyperfunctions in, say, two variables, and to showthe difficulties whichare linked to these approaches. If nothing else, this will motivate the need for the introduction of a more complexpoint of view. A first, ill fated, approach, wouldconsist in a blind mimickingof the definition which we have given in one variable; let K be a compactset in ~2, f~ a real open set which contains K, and V be a complex neighborhood of ~. Then the quotient O(V \ K)/O(V), which would be suppose to define the space of hyperfunctions supported in K, would actually vanish for any choice of V and K, in view of the celebrated Hartogs’ theorem on the removability of compact singularities of holomorphicfunctions in several complexvariables. This circumstance,crucial in the theory of several complexvariables, essentially relies on the fact that (in several complexvariables) there are open sets whichare not holomorphydomains(in contrast with what happensin the single variable case); thus, essentially, the failure of a direct analogof the single variable treatmentis reduced to the fact that the Mittag-Leffler theoremdoes not hold for every open set in 07%In cohomologicalterms, it is no longer true, whenseveral variables are considered, that Hi(V, (.9) = 0. Asit will be clear in the sequel, weactually don’t need such a vanishing, which will be replaced by a muchsubtler result (Malgrange’stheorem, as it will turn out). An alternate and more successful attempt consists in looking back at the wayin whichdistributions of two variables are defined; we are then tempted to define, for a pair of open sets f~l and f~2, the space of hyperfunctionson ~1 ® ~2 as the completionof the tensor product B(~I) ® B(~2); to do so let us consider such a tensor product: (1.4.1)
B(~I) ® B(~2) =
o(v~\ ~,) ®o(v~ o(vl) ®o(v~\ ~) +o(v~\ ~) Unfortunately, as we have thoroughly described in the previous section (but see also [103]) the spaces of hyperfunctionswhichwe need in order to construct this tensor product cannot be endowedwith any natural locally convex topology, and so the idea of just completing the previous quotient space seems to fail. Still, if one looks at the quotient whichwe havejust defined, one realizes that a reasonable guess for the space of hyperfunctions on a cartesian product of two
44
CHAPTER 1.
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open sets is given by what we could call the functional interpretation of the completion of the tensor product (1.4.1); namely, one maysuggest the following definition for the space of hyperfunctionson O(V~ \ ~) × O(Y2 t~(~ x ~) = o(y~ × (v2 \ ~)) + o((v1 al ) × v2 The reader will immediatelynote that in the previous expression the numerator and the denominatorof the quotient are the completions of, respectively, the numerator and the denominatorof the quotient in (1.4.1). As we will see in while, this description actually turns out to be a correct definition for the space of hyperfunctions in two variables, but it immediately showssomefundamental shortcomings;to begin with, its description heavily relies on the use of a specific coordinate system; this is not so muchof a problem when dealing with the case of Euclidean spaces, but might becomea burden when dealing with more general real analytic manifolds; on the other hand, the definition whichwe have proposed does not seem amenable to computations, and it might prove quite difficult to showthat the object thus constructed is a sheaf on the Euclidean space. Comingback to our representation, let us restrict our attention, for the moment,to the case in which ~1 -- ~t~ = ~ and therefore we can take V~ -V~ = (~. Then we immediately see that our proposed definition of the space B(~2) is as follows:
0((¢\ ~) × (¢ s(~) o(¢ ×(¢\ ~))o((¢ ~)× ¢) This is very muchin agreementwith the following argument(which is just a more direct version of the tensor product approach); let f and g be two hyperfunctions on ~ with defining functions F and G, holomorphicfunctions in ~T outside the real axis. Then one might try to define a hyperfunction h(x, y) = f(x)g(y) by means of the holomorphic function H(z,w) = F(z)G(w). It is then clear that H is holomorphicon ((~ \ K~)x ((~ \ ~), and that it is defined modulofunctions in O((~ x (~T \ ~))) + (9((~ \ ~) x~T), in view of the fact that both defined moduloentire functions. This brings us backto our original definition for bivariate hyperfunctions. Weshould also point out that this particular definition is by no meansrestricted to the case of two variables; as a matter of fact, one may easily envision the followinggeneralization to the case of several real variables. Let ~21,~,...,~2,~ be n open sets in ~, and let V~,V2,...,Vn be n complex neighborhoods of them. Then, if we set
v#r~: (v, and ~* ~-- (Yl \ ~-~1)x... x (~-1 \ ~-1) x V~x (Y~+]\ ~-~+l) x... x (V,
1.4.
HYPERFUNCTIONSOF SEVERAL VARIABLES: BASICS
45
Im Zl(x +i0, y + i0) (x - iO, y + iO)
(x + i0, y - i0)
Figure 1.4.1 we can reasonably define the space of hyperfunctions on al x ... × an by the following quotient:
~jn=l
O(Vj*)
There is, however,at least one other natural wayto introduce hyperfunctions in two variables, and it is through the notion of boundaryvalues of holomorphic functions, which, as we have seen, are one of the key concepts that hyperfunctions were designed to deal with. Let us go back to our exampleof the hyperfunctions f = IF] and g = [G]; by using the notations which we have introduced in the previous sections, we have that I(x) = F(x + iO) - F(x - whileg(x) = G(z +iO) - G( i0); it is therefore quite natural to consider the product (fg)(x, y) = f(x)g(y) whichcan be thought of as the sumof four different boundaryvalues as follows: (fg)(x, y) = H(x+iO, y+iO)-H(x-iO, y+iO)+H(x-iO, y-iO)-H(x+iO, where the holomorphicfunction H is defined as the product of F and G, and it is therefore holomorphicon (~ \ ~T/) x (IT \ ~). This representation is correctly depicted in the following figure, whichillustrates the four directions which, in (T2, allow the approach to ~2. The reader will immediately note how this same approach would allow us to think of hyperfunctions in n variables as the sumof 2~ boundaryvalues of holomorphicfunctions. As we will see, this point of view is not too far from the truth, even thoughit only holds a partial viewof truth itself. In particular, one is immediately struck by the deeper meaning that the notion of boundaryvalue assumeswhendealing with several variables. Indeed,
46
CHAPTER 1.
HYPERFUNCTIONS
~2
Figure 1.4.2 in one variable the variable z = x + iy only admits two "directions" to approach its real part; y mayconverge to zero from the positive direction or from the negative direction; whenwe consider two variables such as.(z, w), then one has several possibilities; while our exampleabove (and figure 1.4.1) only refer the four possibilities (2~ possibilities, in general) whichcorrespondto the four quadrantsof ~2 (to the n orthants of ~7~n, i n general), i t i s not difficult t o realize that other more complexsituations mayarise, if the orthants are replaced by other moregeneral cones in the appropriate space. These remarks lead us to define somespecial open sets whichplay a crucial role in what follows, and we take this chanceto describe the necessary notations for what follows. In so doing, we moveto the general case of n variables, since, after discussing possible motivations, we are nowready to provide the correct definitions. Let g be an open cone in ~, with vertex at the origin. Thenfor every open set ft in ~ we can define a wedgein IT~ as the open set ~xiF. as shownin the following set of pictures. Suchan object, as we said, is said to be a wedge,and the coneF is called its opening; a wedgeis of course a special case of a tubular neighborhood. Given two cones F and F’ with vertex at the origin of ~, we say that F~ is relatively compactin F, and we write F~ << F, if the closed set
where Sn-1 denotes the unit sphere in ~n, is contained in the interior of F. We also say, in this case, that F’ is a proper subconeof F (note that this expression, though widely used, is somewhatmisleading, since it seems to imply that F’ is
1.4.
HYPERFUNCTIONSOF SEVERAL VARIABLES: BASICS
47
Figure 1.4.3 separated from F, while we knowthat they both have vertex at the origin, and therefore they get infinitesimally close one to another). Our next concept is that of infinitesimal wedge(which generalizes the notation x =t= i0); given an open set ~ and an open cone F in ~n, we say that complexn-dimensional open set U is an infinitesimal wedgeof type ~ + iF0 if U is contained in the wedge~ + iF and, for every proper subcone F’ of F and every ~ > 0 there exists cf > 0 such that U ~ ~ + i(r’ n {lYl < ~}), where ~ denotes the set obtained by f~ by shrinking it by ~. In this case, see figure 1.4.4, we say that ~ is the edgeof the infinitesimal wedge. Wenote, see again figure 1.4.4, that an infinitesimal wedgeis nothing but an open set contained in the wedge~ + iF which, asymptotically in the vicinity of the edge ~, approachesa wedgeof opening F; in the sequel, the symbol~ + iF0 will denote any such a set, since it will neverbe necessarythe specification of the precise set; similarly, the space O(f~ + iF0) denotes the set of functions which are holomorphicin someinfinitesimal wedgeof type f~ + iF0. In this respect, we need to quote an important result due to Bochner, and knownas the Bochnertube Theorem[103], which will be useful in what follows: Theorem1.4.1 Let U be a connected open set in Ktn, and let U^ be its convex hull. Then every function F holomorphic in ~ + iU extends analytically to h~’~ + iU^. As a consequence,if F is a cone as aboveand F^ is its convex hull, any function F holomorphicin ~2 + iF0 can be holomorphically extended to the infinitesimal wedgef~ + iF^0. Proof. Werefer the reader to [103] for detailed proofs of this and related statements. []
48
CHAPTER1.
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Figure 1.4.4 Wecan nowproceed to provide one further intuitive definition for hyperfunctions in several variables; this will turn out to be the closest to the precise definition, and will provide us with the final intuitive vision of hyperfunctions, before plunging into the formal treatment based on relative cohomology. ConsiderN open convexcones in ~ with vertex at the origin, F1, F2, ¯ ¯ ¯, FN; we are taking the Fj to be convexin view of Theorem1.4.1. Let Fj be holomorphic functions defined, for j = 1,..., N, on the infinitesimal wedges~ + iFjO, for ~ an open set in Ktn. Then the formal sum N
(1.4.2)
f(x) =~ Fj(x ir~0) j=l
defines a hyperfunction on ~. It is clear that the functions Fj which appear in (1.4.1), and which we can call the defining functions for the hyperfunction f are far from being uniquely determined, and actually the cones Fj themselves are not uniquely determined at all. It should also be clear that the definition of the space of hyperfunctions B(~) which is somehowprovided by (1.4.1) is still ambiguousand, at best, incomplete. In what follows, however, we will regard (1.4.1) as an operational way define hyperfunctions, and we will soon prove that every hyperfunction can be represented, as in (1.4.1), as the sum of a finite numberof boundaryvalues holomorphicfunctions. Wenowproceed to the rigorous introduction of hyperfunctions on £rt"; this will prove to be a non trivial step, and the reader maywish to skip this approach, on a first reading, and try to rely on our previous intuitive explanations, at least until more familiarity is acquired. Whatfollows heavily relies on the cohomologicalproperties of holomorphicfunctions in several complexvariables,
1.4.
HYPERFUNCTIONSOF SEVERAL VARIABLES: BASICS
49
as well as on the theory of derived sheaves, from which we will only recall the strictly necessaryfacts. Definition 1.4.1 Let $ be a sheaf on a topological space X, and let S be a locally closed subset. The n-th derived sheaf of $ is the sheaf associated to the presheaf given, on X, by the collection (1.4.3)
{H~nu(U,S)}.
Oneeasily notes that, by its very definition, the n-th derived sheaf is a sheaf on the topological space X; however,as a sheaf, it is localized on S, in the sense that one can show that 7{~(S)(U) = 0 whenever U and S are disjoint. As it is usual in the theory of sheaves, it is not easy to describe a sheaf as the sheaf associated to a non-completepresheaf, since it becomesquite difficult to visualize the sections of such a sheaf. In this particular case, the situation is not easier but we get a partial help by the following general result, for whichproof werefer the reader to [103]: Theorem1.4.2 The presheaf indicated in (1..~.2) is a sheaf in the following cases:
(a) when n=O; or
(b) when (1.4.4)
HJ~(S)= 0 for j = O, 1,...,n-
In this latter case its sections can be obtainedas follows:
n~(s)(u)=H~n~(u, It is then clear that the possibility of easily describing the spacesof sections of derived sheaves rests on the possibility of proving the vanishing described in (1.4.4). In sheaf theoretical terms, wesay, in such a case, that S is of codimension greater than or equal to n relative to the sheaf $. Morespecifically, if one could provethe vanishingof H~($)for all j different fromn, then we wouldsay that the set S is purely n-codimensional with respect to ~. Thus, pure codimensionality becomesa key issue for the description of sections of derived sheaves. Wewill be interested in showing,following Sato’s original approach.in [197], that ~, as a locally closed subset of(~~, is purely codimensionalwith respect to the sheaf (9 of germsof holomorphicfunctions. This fact is in somesense the best extension which one can hope to make of the Mittag-Leffier Theoremwhich, as we saw,
50
CHAPTER 1.
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does not hold any more in several variables. To prove Sato’s result, which will becomethe basis for the formal definitioa of hyperfunctionsin several variables, we will need somecohomologicalfacts on the sheaf of holomorphicfunctions in several variables; such facts are by nowstandards, but they were far from so whenoriginally provedby Martineau[156] (it should be noticed that the original papers by Sato only bear a minimal amountof details concerning the proofs; so, even if he was undoubtedly the discoverer of these theorems, one should credit people such as Martineau, Harvey, Schapira, Komatsuas being the first to actually provide complete proofs). Theorem1.4.3 (a) Let K = K~ × ... × K,~ be a cylindrical V a Stein domain. Then
(1.4.s)
closed set, and
HJKxv(~T" x V, O) = O, for j >_n + l.
(b) Let ~ be an n-dimensional real open set, and V a Stein domain. Then ~ x 1/’, (9) = O, for j k n + 1. H~xv((~
(1.4.6) Proof.
See [103].
[]
Theorem1.4.4 Let K and L be two closed analytic polyhedra, and V a Stein domain. Then (1.4.7)
j n V,O) = O, forO<_j<_n-1. H(L\K)xV(IT.
Proof. See [103].
[]
Weare now ready to prove the pure codimensionality of ~n in ¢~, with respect to the sheaf O. This fundamental result is due to Sato, but the proof which we are supplying is due to Kaneko[103]. Theorem1.4.5 The embedding~’~ ~+ (T" is purely n-codimensional with respect to the sheaf (9. Proof. Wewill directly calculate the stalks of the sheaves $~ := H~(O), for F = ~’~, and we will prove that they vanish for j different from n. Since, for any j, the sheaf SJ is homogeneous with respect to real shifts, we will simply consider the stalk at the origin. Let us begin with the case of j >_ 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<1, I -izl<1}; now, one can set
f(z) =1 - (Zl +... and define the two closed analytic polyhedra L-=~’~n{ze~T’~;[Zl I <_1,...,
[z,~[ _< 1},
K = LN{z eCT~:[ef(z)[ _< 1}. Finally we consider the fundamentalsystem of open neighborhoodsof the origin given by the dilations {rU : r > O}of the open set U = {z e(F’~ : Izl[ < 1,..., Iz,,I < 1, Re(f(z)) > 0}. It is immediate to verify that U is a complexneighborhoodof the origin and that L\K=~NU, so that Theorem1.4.4 allows us to conclude the proof of the pure codimensionality. [] Weare finally ready to provide the formal definition for the sheaf of hyperfunctions on the Euclidean n-dimensional real space; Definition 1.4.2 The sheaf B of hyperfunctions on the topological space Kt is defined as the n-th derived sheaf of (9, namely
In view of Theorem1.4.2 on the description of the global sections of the derived sheaves, and because of the pure n-codimensionality of ~ which we have just proved, we can actually use Definition 1.4.2 to precisely describe the vector spaces of hyperfunctions on open sets of ~. Indeed, we have the following immediateconsequenceof our results and definitions: Theorem1.4.6 Let f~ be an open set in ~ln, and let V be any complex neighborhoodof f2. Then ’~ \ 9; (9) H~(ITn, (9). (1.4.8) B(f2) = H"(V, V \ ~; (9) = H’~((Tn,¢
52
CHAPTER 1.
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Proof. The first equality is just a consequence of our remarks preceding the statement of the Theoremitself. The second one follows from the excision theoremfor relative cohomology (see e.g. [64]). The third is just the definition.
Our next goal, in this section, is to employDefinition 1.4.2, together with Theorem1.4.6, to get a more concrete description of hyperfunctions; as it will turn out, these relative cohomologyclasses may indeed be seen as boundary values of functions holomorphicon infinitesimal wedges; this will allow us to recover the intuitive definition whichwe provided earlier in this section. Concurrently, this approach will allow us to visualize hyperfunctions with families of holomorphicfunctions, thus makingeasier our task of working with them. Let us start with a first representation of the relative c~homologygroup indicated in (1.4.8). Consider the pair (V,V \ ~), and its relative covering (]2, ~’) given "~=( Vo,VI, . . . , Vn}
v’ = where and
=v n (z e ¢":I. (zj) ¢ Since the open sets V~are Stein (and their intersections are Stein as well), and in viewof Oka-Cartan’sTheorem,one has that the covering(1), Y~)is acyclic and therefore one can apply Leray’s Theorem, jointly with Theorem1.4.6, to conclude that B(f~) = Hn(v, 12’; The reader interested in the details of the previous argumentshould review the description of spectral sequencesas provided in Chapter III. The space of hyperfunctions can therefore be expressed as a ~’ech cohomology group, whichwe can nowtry to explicitly calculate. To begin with, let us recall that, by definition, and using the standard notations, it is (1.4.9)
Ker{5": C"02, ]2’; O) ~ C"+102,1¢’; 0)}
and we therefore have to compute the kernel and the image of the coboundary mappings.It is immediateto see that, because of the fact that the cardinality of P is n + 1, then C"+1(12,12’;O) vanishes, so that the kernel of 5" coincides with C~(12,1¢’;0); this last vector space, in turn, is easily seen to be isomorphic to the space of sections O(Vo n V~n . . . n V,~).
1.4.
HYPERFUNCTIONS OF SEVERAL VARIABLES:
BASICS
53
Let us nowtry to describe the image of ~,~-1; once again, in view of the definition of alternating cochains, we can see that
’~-~): ~,o(v~ Zm(~ nv~.n. . . n~n. . . nv,~). j:l
Finally, if we recall the original definition of the open sets which make up our covering, we see that the following equalities hold:
v~nv~n...nv,~ = v n (¢ \.~y, and
Vonv~n... n~ n... n Y. = v n {z E¢’~:Im(z~) ¢ O/orl~ ¢ j}. As a consequence of these equalities, together with the computations of the kernel and the image of the coboundary mappings, we can finally conclude that, for ~ an open set in ~, one has the following explicit description of the space of hyperfunctions ~(a) =
O(U n (¢ \ ~)") E~\, O(V, n . . . n ~ n . . . v.)
It is not useless to remark that, in the case in which ~ is the real plane and V the complex plane ¢2, one obtains the same description which we have intuitively given at the beginning of this section: 0( (¢ \ ~) x (¢ \ s(~) o( ¢ x (¢ \ ~)) o((¢ ~) × ¢) Wenow want to push a little further the amount of information which this relative covering of (V, V \ ~) may provide us with. As a matter of fact, we will modify this relative covering in order to show how it is always possible to express a hyperfunction on ~ as a sum of boundary values of 2n holomorphic functions; this will allow us to work with (classes of) holomorphic functions when dealing with hyperfunctions, even though it will be necessary to precisely understand the holomorphic translation of the equivalence relation which underlies this process. Let us therefore define, for any k = 1,... ,n
V+~=V n {z E¢": Im(zk)
> 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 give a contradiction. (c) Weshall prove the statement for the case j = 0. Becauseof part (b), can find nonzerocoefficients /~1,’ ..,/~n such that
1.4.
HYPERFUNCTIONSOF SEVERAL VARIABLES: BASICS
57
Weshall prove that all these coefficients are strictly negative. Indeed, if ~1 were positive, one might consider the system of linear equations in y given by
~l.y = r,> 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
The coboundaryof such a cochain is given by the usual alternating formula: ~(G) = ~-~.(~-~(-1)~Gj~(z)) ~(-1)k+~G~j(z))V A voA.. . A v ~ A.. . A v j=0 j
If we nowset Fj to be the intersection of all the Ei with the exception of Ej, and if we recall the conclusions of Proposition 1.4.2, we deduceimmediatelythat the space of hyperfunctions on ~ can be given the following representation: ~ + ~rj) n v) 0~\o H~(V, O) = @j<~ O((~n0((+ i(Fy + F~)) n nWhatwe have proved so far, shows that hyperfunctions on open sets in E~ can actually be seen as formal sums of holomorphicfunctions, subject to some equivalence relation. Before we conclude this introductory section on hyperfunctions of several variables, we wouldlike to showhowwe can actually try to define hyperfunctions by starting with such formal sums. Weare not going to describe all the necessary details of what we will express, since they can be found, e.g., in [103], but wewill try to provide sufficient evidencefor our claims. To begin with, let ~ be an open set in ~ and F a cone in the dual space E~n; define (similarly to what we did in one variable) O(~ + iF0) -- ind lim O(U)
1.4.
HYPERFUNCTIONSOF SEVERAL VARIABLES: BASICS
59
wherethe limit is taken with respect to all infinitesimal wedgesof type f~ + iF0. Wecan nowallow F to range over the family of all open convexcones (not necessarily strictly convex), and take the direct sum
(1.4.13)
o(a
+ iF0). F
This is a well defined complexvector space, let us call it X, and we can think of it as the space of all possible boundaryvalues of holomorphicfunctions, wherethe boundaryvalues are taken along all possible infinitesimal directions. Of course, if we really want to talk about boundaryvalues, we have to take a quotient whichwill eliminate the several representations which, in (1.4.13), are given for the same boundaryvalue. In particular, we define the subspace, which we will call Y, generated by all the elements in X of the form
Fl(z)+F2(z)- F3(z), where each Fj belongs to (9(f~ + iFj) for F1 A F2 _D F3, and where Fl(z) +F2(z) =F3(z) whereverall three functions are defined. It is not difficult to verify that the well defined quotient vector space X/Y is, indeed, isomorphic with the space of hyperfunctions which we have defined previously. Nowthat we have given both a formal definition of hyperfunctions in the n-dimensional Euclidean space (via the notion of derived sheaves and relative cohomology),and an intuitive one (via the formal sum of boundaryvalues holomorphicfunctions), we can finally give someconcrete examples, which will be useful for the reader’s understandingof the operations whichwewill describe in the next section. Example1.4.1 The first exampleis, of course, the Dirac 5 function. Weknow that, in classical distribution theory, the multi-dimensional5 function is defined as the product of n one-dimensional5 functions; we therefore set 5(x) = 5(xl)~(x2)...~(xn) ~=1~i=-1 ( X 1~ +i0 xj-1 iO) = n= ~ (_~r/)
sgn(o)
(xl+ion0) ¯ ... ¯ (xn+ion0)
In the last representation we have set o = (°~,... ,°~),oj = +1, and therefore if we denote the °-th orthant of the space ff~ by F~, we see that the Dirac’s function is the sumof boundaryvalues of functions holomorphicin the wedges ~ + iF~. As a matter of fact, this specific property of the Dirac hyperfunction can be used as a motivation for an informal definition of hyperfunctions as sums of boundaryvalues along wedges.
60
CHAPTER 1.
HYPERFUNCTIONS
Example1.4.2 In the case of the Dirac hyperfunction, we have a combination of boundaryvalues from all possible directions. However,one mayalso consider one-sided boundaryvalues, as in the hyperfunction 1 1 1 xl +i0 " x2 + i0 " "" " x~ +i0’ Moregenerally, the function 1/z whichwe have used in this case, maybe replaced by other holomorphicfunctions F1,..., F~ which have singularities on the real axis. Example 1.4.3 Finally, another simple example which can be given is the following bidimensional hyperfunction (whichwe will use again in our next section): F(x~ + iO) . G(x2+ F(z) = ~ 2~i(2k)’~, G(z) - ~ ~:~(~k+~)’z. k=O
1.5
Hyperfunctions Results
k=O
of Several
Variables:
Main
In this last section of the first chapter, we will showhowto use hyperfunctions in several variables as if they were actually "generalized functions". In a way, this section is the several variables parallel of section 1.3; still, in its complexity, it will require muchmoreattention on the part of the reader and, unfortunately, the results whichwe will describe are muchless intuitive than the corresponding ones for a single variable. To begin with, we shall use somebasic facts from the theory of sheaf cohomologyto obtain the fundamentalproperties for the sheaf of hyperfnnctions. Let us recall that if we have a sheaf homomotphism
between two sheaves $" and G, then such a homomorphismcan be uniquely lifted to a homomorphism between the complexes derived from the canonical flabby resolution of the sheaves themselves (see any text on sheaf theory for this fact, e.g. [64] or [123]); since it is knownthat any flabby resolution (and in particular the canonical one) can be used to determine the cohomologyof sheaf, we deduce the existence of an induced homomorphism h~. : H~(X, .~) ~ H~(X,
1.5.
HYPERFUNCTIONS OF SEVERAL VARIABLES
61
between cohomologies. Such an induced homomorphism,in particular, is an isomorphism if h was, originally, an isomorphism. This simply means that H~(X,-) is a functor. The first application of this general fact is to define the action of partial differential operators on hyperfunctions;indeed, let P(z, Dz) be a linear partial differential operator with holomorphiccoefficients defined on a complexopen set U. Then P(z, Dz) induces naturally a sheaf homomorphism, let us call it h, on the sheaf (9 of germsof holomorphicfunctions and therefore, for any real open set ~ contained in U, one defines the action of P(x, D~)on B(~) as the induced maph. n, acting on the relative cohomology.The reader is invited to go back to section 1.4 and to checkhowthe action is explicitly defined; indeed, this abstract definition we have just given corresponds precisely to have h act directly on the holomorphic functions which define the corresponding hyperfunctions. We should also point out that by taking the inductive limit on a cofinal family of open sets, the action described before actually induces a sheaf homomorphism, so that we can say that linear partial differential operators with real analytic coefficients act as sheaf homomorphisms on the sheaf of hyperfunctions. Let us incidentally observe howthis fact has a remarkable interest in the theory of differential equations; indeed (more details will be given before the end of this section), one can even consider a complexof systems of linear partial differential operators with holomorphiccoefficients (i.e. a complexof :D-modules as described in Chapter III) defined on the sheaf of holomorphicfunctions; such a complexinduces a complexof differential operators on the sheaf of hyperfunctions, and important consequencescan be drawnfrom this parallel treatment; we recall here the workof Komatsu[140], [144] and of one of the authors [1], [2], [217]; as we havesaid, we will be backlater to this topic. Asa special case of this definition, wesee that it is therefore possibleto multiply hyperfunctionsby real analytic functions, and that we can take derivatives of such hyperfunctions; from a practical point of view, we will seldomregard hyperfunctions as cohomologyclasses, but we will rather look at them as formal sumsof holomorphicfunctions; fromthis point of view, the definitions of derivative and of multiplication by a real analytic functions are natural and need no further explanation. As we already pointed out whendealing with hyperfunctions in one variable, the issue of the product of hyperfunctions is a more delicate one, which we postpone to the next chapter; it will suffice to say, for the time being, that the product of two hyperfunctions is not alwaysdefined (as, however,physicists knowfairly well, and as it also happensin the case of distributions). The next important issue wewant to tackle is the definition of definite integrals for compactlysupported hyperfunctions or, even more, for hyperfunctions which are at least sufficiently smoothat the boundaryof the integration set. Sincethe issue is not totally evident, wewill first look at an intuitive definition, based on the naive notion of hyperfunctions as sums of boundary values. The
62
CHAPTER 1.
HYPERFUNCTIONS
precise cohomologicaldescription of this process will be given in a later stage. Let K C_ D be a real compactset with piecewise smooth boundary, and let f be a hyperfunction defined on ~; we wouldlike to give a meaningto the symbol K f ( X )dx; as we already did in the case of one variable, we will assume(even though this is not strictly necessary) that f is, actually, real analytic near the boundary OKof K. By adopting the intuitive definition of hyperfunction which we have given before, we can assumethat f can be expressed as a finite formal sum of boundary values such as N
f(x) = ~ Fj(z + irj0), j=l
at least near K. Of course, because of the analyticity hypothesis og f near the boundaryof K, we can assumethat each Fj has been chosen so to be analytically continuable to the real axis wheref is real analytic. Wecan nowdefine, for each j = 1 ..... N, some integration paths K + iQ = {x÷ iQ(x) : x E K}, where the continuous functions ~j(x) are defined on K so that: if x ~ OK,
ej (x) =
x + ie~(z) K+ir~0, if x C In t(K). Weare nowin the position to define the integral of f on K by the following setting: N
One might observe that in this definition we have madea certain numberof arbitrary choices, whichmight alter the result of the integration; as a matter of fact, the independenceof the integral from the choice of the integrating paths is a consequenceof Cauchy’stheorem, but the more delicate issue is the independenceon the choice of the boundary value representation for f." That the integral does not dependon the choice of such representation is a consequence of a classical theoremin complexanalysis, whosehistory is strictly intertwined with the history of hyperfunctions: the edge of the wedgetheorem. This theorem gives the right condition for a sum of boundaryvalues to represent the zero hyperfunction, and one maydeduce from it that if two representations are given for the same hyperfunction, then their difference (representing the zero hyperfunction) has to satisfy certain conditions whichimply, in particular, that its integral vanishes. Since the history and the developmentsof the edge of the wedgetheoremare so muchrelated to the developmentsof the theory of hyperfunctions, we refer the reader to our last historical appendixto this chapter; in
1.5.
HYPERFUNCTIONS OF SEVERAL
VARIABLES.
63
there we have given several versions of the edge of the wedge theorem; it will not be difficult, for the reader, to extract the version which would be needed here. In order to proceed further, we now show how real analytic functions form a subsheaf of the sheaf of hyperfunctions: in so doing, we will also obtain a different characterization of the integration process on hyperfunctions. Our first result is nothing but a reformulation of the n-codimensionality of the sheaf B together with Theorem 1.4.3 (some work would be needed to make this evident, and the reader is referred to section 1.4). Theorem 1.5.1 Let K be a closed set in Ktn. The space BK Of hyperfunctions with support contained in K can be represented by the relative cohomology group H~(¢~, O) = H’~(¢’~,¢ ’~ \ K; O). In particular, if K = K1×. ¯ ¯ × Knis a cylindrical set, then for any cylindrical neighborhood U = U1 × ... × Un, the space above can be represented (with the notations already used in section 1.4) by: O(U#K) Ej\, o(u; ) Let us point out that, in this theorem, K is not necessarily a compact set, but it suffices that it is closed; we will actually always use the case in which K is indeed compact. As a consequence of Mayer-Vietoris theorem, we also obtain the following result: Theorem 1.5.2 Let K be a closed set in ~ which is the union of finitely many cylindrical closed sets Kj,j = 1,...,N. Then every hyperfunction f whose support is contained in K can be decomposed as the sum f=f~+...+fg, where the hyperfunctions fj have supports contained in Kj. Moreover, if an open set ~ is the union of finitely many open rectangular solids (i.e. products of open intervals), then every hyperfunction on B(~) can be extended to a hyperfunction on ~ whose support is contained in the closure of f~. Wecan now define the notion of definite integral for a hyperfunction as follows: let us start with the case in which f is a hyperfunction supported by a cylindrical real compact set K1 × ... × K~. If g is an entire function, then one define the cohomological integral f~, f(x)g(x)dx
64
CHAPTER i.
HYPERFUNCTIONS
as follows: take U to be a cylindrical neighborhoodof K and choose F to be a function in O(U#K),which represents f in the cohomologygroup H~((~ nowset /l~, f (x)g(x)dx = (-1)n ~l " . ~, F(z)g(z)dZl where the curves are paths in Uj \ Kj which encircle the compactsets Kj once in the positive direction. If, on the other hand, the support of the hyperfunction f is not contained in a cylindrical compactset, but is covered by the union of a finite numberN of such sets, then we can decomposef as in Theorem1.5.2, i.e. f = f l +... + f N, and therefore define N
It is not too difficult (and we leave this to the reader) to verify that this definition of integral coincides exactly with the intuitive one whichwe have provided earlier; one can also easily showthat the eohomologiealdefinition does not depend on the manyarbitrary choices which have been necessary in the process of the definition. This definition of integration is useful, in particular, as it allowsus to explicitly embedthe sheaf of locally integrable functions in the sheaf of hyperfunctions (as a consequence,using the description of distributions as locally finite sums of compactly supported distributions, one obtains the embeddingof the sheaf of distributions in the sheaf of hyperfunctions): Theorem1.5.3 The following inclusions are sheaf homomorphisms: A ~-~L~,~oc~-~:D’,-~B. Proof. Weprovide here the .main ideas of this result which is standard and does not require any special knowledgeto be obtained. Webegin by describing howto associate a hyperfunction to a compactly supported L~ function f. If K is a compactset which contains the support of the integrable function f, we define a hyperfunction j(f) in ~Kby setting j(f) = [G], where (1.5.1)
G(z)
-n (-2~i) fJg (z~ --
f(~)
wl) .... dw (zn - w~)
is an element of It is immediateto verify that j is a(T-linear mapwhichrespects the supports, i.e. iff = f~+...+fg as in Theorem1.5.2 then, for each k = 1,...,N, it is supp(j(fk)) C_ so t hatthe s upport of j( f) is ac tually conta ined in th e support of f. Wenowknowthat every function in Ll,toc(~) can be decomposed into a locally finite sum f(x)= ~ f,(~)
1.5.
HYPERFUNCTIONS OF SEVERAL VARIABLES
65
of Ll-functions with compactsupports. One can then define j(f)
= ~j(ft), t
whereeach j(ft) is defined as before. Now,the sumon the right hand side of this definition is still locally finite because of the remark we have madeconcerning the good behavior ofj with respect to the supports. Wecan verify that this map is independent of the waythe function f is represented and actually induces a sheaf homomorphism between Ll,toc and B, which is an injective homomorphism. Since A naturally embedsinto the sheaf of locally integrable functions we have a chain of inclusions, as stated in the Theorem. The only missing inclusion is the one of 7)t into B (since the inclusion of locally integrable functions into distributions is well known). As for this case, the proof proceeds exactly in the samewaywith the caveat that (1.5.1) has to be interpreted in the sense distributions (i.e. it is only formally an integral, but, moreproperly, it is the action of the distribution f on the Cauchykernel); also, in this case werecall that any distribution can be expressed as a locally finite sumof compactlysupported distributions. This concludes the proof. D Before we can concludethis section (and this chapter) with someapplications of (flabby) sheaf theory to differential equations (for its intrinsic interest, addition to its motivationalrole), we needto discuss the flabbiness of the sheaf of hyperfunctions(whichwe already proveddirectly in the case of hyperfunctionsin one variable) and of someassociated sheaves. The proof of the flabbiness of B is a consequenceof somerather deep properties from functional analysis, whichwe have mentionedwhendealing with the topology of the space of hyperfunctions and we will only sketch it here, since we do not wish to duplicate what has already been discussed elsewhere (e.g. in [103]). Wedevote sometime, on the other hand, to a simpler result, namelythe flabbiness of the quotient sheaf B/~4. This result, besides its intrinsic interest, will be of great importancein what follows (namelythe theory of microfunctions, as described in Chapter II). To begin with, we will need to state a Lemma,which is essentially due to Grauert [66], even though we only saw it explicitly mentioned and proved in Kaneko’s[103]: Lemma1.5.1 Let ~ be an open real set~ U a complex neighborhood of ~ and F an open convex cone. Let F be a holomorphic function defined on the open set (~ + iF) A U. Then there exists a function G that is holomorphic on an infinitesimal wedgeof type ~n + iFO, and is analytically continuable to and a function H holomorphicon a neighborhoodof ~ such that, wherever they are both defined, F(z) = G(z) - g(z). Wenowhave the tools to provethe first flabbiness result, as follows:
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CHAPTER 1.
HYPERFUNCTIONS
Theorem1.5.4 The quotient sheaf B/A is a flabby sheaf. Proof. The proof of this statement goes through the understanding of the way sections of B/A are made.As we already mentioned,a quotient presheaf is not, in general, a sheaf, and so the quotient sheaf is only the sheaf associated to the quotient presheaf. In this case however(as we also saw in the case of a single variable), the situation is particularly simple, and wecan begin by provingthat, for any real open set ~.
(1.5.2)
(~-) (~) - ~(n) ~(~).
This equality is immediatelyderived from the short exact sequence of sheaves B O ~ A---+ B ~ -~ ---~ O. Indeed, by taking the associated long exact cohomologysequence, and recalling that, for any open real set ~, Malgrange[153] has proved that HI(~; A) = one obtains (1.5.2). Wenowuse (1.5.2) to prove the flabbiness of the quotient sheaf. Let If] be an element of B/Jt(~); by (1.5.2) we can represent If] globally by some f in B(~). By what we saw in the previous section, we can certainly choose a complexneighborhood U of ~ (which we maytake to be Stein without loss of generality) and express f as a sum of boundaryvalues as follows:
f(x) =~sgn(z)F~(z ir ~0), for F~e O((n + ir~)n U). By Lemma1.5.1 we see that for each Fo there exist both a function G~ (.9(~ ’~ + iF~0) and a function H~E fl,(~) such that, wherethis makessense, F~(z) =G~(z) -H~(z). If we now define g and h to be, respectively, the boundary values of these functions,i.e. g(z) = ~, sgn(a)G~(z ir ~0) e B(~n), and we get that the element [g] defined in B/A(~~) is the desired extension of If] = If + hi. This proves the flabbiness and concludes the proof of the Theorem. []
1.5.
HYPERFUNCTIONS OF SEVERAL VARIABLES
67
Notice also that the flabbiness of B implies the flabbiness of B/A as follows. Consider the commutative diagram 0
t(a)
0 ~
0
o As abovelet If] be an arbitrary elementof (B/~A)(~t), wheref is a hyperfunction on f~. Since B is flabby, one can find an extension f* of f defined on ~. Then the image [f*] of f* under the canonical map 13(~) -~ (B/c4)(Kt ’~) is the required extension of If]. Let us nowrecall a result from the theory of sheaves, whichwe leave unproved (the proof makes use of the Zorn’s Lemma,and can be found in [103], Lemma 4.2.2). Lemma1.5.2 Let J: be a sheaf on a topological space X, and let {Us}be an open coveringof X. If each sheaf ~vj is flabby, so is the sheaf .T. Theorem1.5.5 The sheaf 13 of hyperfunctions is a flabby sheaf. Proof. In view of Lemma1.5.2 we will showhowits restrictions to the open sets of a suitable covering are indeed flabby. The covering we will use will consist of boundedopen sets (e.g. the balls centered at the origin and of radius j = 1, 2, 3,...). Wewill actually showthat if fl is a (connected) boundedopen ~ set, then any hyperfunction on ~ can be extended to a hyperfunction on /R whosesupport is contained in the closure of ft. Let us then start by considering an exhausting sequence {£tk} of open sets for ~, where each flk is the union of rectangular solids as in the second part of Theorem1.5.2. Fromthat Theorem we knowthat every fl~ has an extension fk to the whole space, and that the support of such extension is contained in the closure of ~k. The (natural) idea is to construct the required extension ] as the limit of the sequence{fk}. This cannot be achieved in a straightforward manner, since the corrections whichthe {fk} mayneed to makethe sequence convergent maynot provide an extension of the original hyperfunction;this difficulty is essentially related to the fact (which we have already discussed earlier) that the space of hyperfunctionsis not at all localizeable; as a matter of fact, if K and L are two compactsets with K C_ L, then we knowthat the natural inclusion 13K ’--~ BL has a dense image. The reader is invited to contrast this with the situation which occurs whendealing with Schwartzdistributions, where the imageis actually a closed subspace; all of the differences in the treatment of convergenceand topology can be seen as stemmingfrom this fundamental difference. To overcomethis problem, we set
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CHAPTER 1.
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Kk = l~ \ ~k, in such a way that Kk is a decreasing sequence of compact sets and supp(A+I- .fk) C_ ~k+l \ ~k C_ As we saw whendescribing the topology of has a structure of Frechet space, and the by the inclusions of the compactsets are be done inductively) a countable family of topology of/3K and such that
the space of hyperfunctions, each/3K natural inclusions which are induced continuous as well. Choose(this can seminormsI " Ik,z which generate the
and Choose now g~ in/3(~) such that
IA+~ - A-
gkl -<
(~)k;
this can be done because the space of hyperfunctions on the boundary of ~ is dense in each BK. The proof of the theorem can now be concluded by defining
k=l
Yet another wayto see the flabbiness of B is the following. Let D be an open boundedset in ~T/n. For the triple
¢" \ f~ c¢~ \ a~~ c¢ there is an induced long exact sequence n, O) --+ H~((T ’~, O) --> 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.
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CHAPTER 1.
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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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CHAPTER 1.
HYPERFUNCTIONS
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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CHAPTER 1.
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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
80
CHAPTER 1.
HYPERFUNCTIONS
. 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-
82
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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
CHAPTER 1.
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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)=<~, t --+~(z+ ~) such a definition makes#. a continuous linear operator from (9(¢) into itself. Therefore we see that Pincherle’s operations were isomorphic to analytic functionals with compactcarriers, i.e., ultimately, to compactly supported hyperfunctions. Even moreinteresting is the fact that Pincherle proceededto develop a calculus for these operators, whichwouldallow him to deal with both local and non-local operators (e.g. with the translation operator), in order to eventually solve the inversion problem. Wehave not discussed in detail the inversion problemin this chapter, but it is interesting the fact that Pincherle managedto introduce a rather refined calculus for negative powersof the basic differentiation operator ~z, which in somesense is akin to the one developedindependently by Heaviside, [77], [78].
1.6. 1.6.6
HISTORICAL NOTES The
Edge
of
85
the
Wedge
Theorem
As we have pointed out in the last section of this Chapter, a key instrument in the understanding of whena sum of boundary values of holomorphicfunctions gives the zero hyperfunction is the so called edge of the wedge theorem, whose purpose is exactly to determine when such a formal sum gives rise to a unique holomorphic function. As such, the edge of the wedgetheorem is a typical holomorphiccontinuation theorem(see e.g. Vladimirov’s [226]) but its interest, at least from our point of view, lies in its manyconnections with the theory of hyperfunctions, and with the fact that it arose (as hyperfunctionsdid) from considerations from theoretical physics. Thus, a complete paper on the edge of the wedgetheoremwouldhave to have at least three components:its physical backgroundand significance, its mathematical role, and finally its connections with the theory of hyperfunctions. In this section of the historical appendix,wewill only briefly touch uponthe history of such a theorem(its original formulation has been greatly expandedand modified in the years immediatelyfollowingits first statement), and we will naturally discuss its connections with Sato’s theory. As far as its physical meaning, we refer the reader to our references [29], [99], [146] while its mathematicalimportance is discussed within the frameworkof more general analytic continuation results in [226]. A very interesting and accessible survey on this theoremand related results is also providedby Rudin’sseries of lectures [191]. It is fair to say that the very first version of the edge of the wedgetheorem was formulated, for the very simple case of one complexvariable, by Painlev~, back in 1888. This case was already discussed in this chapter as Theorem1.2.2 and, as we already noticed, it is nothing but a special case of Schwarz’sreflection principle. Whenonly one complexvariable appears, however,not only the result is fairly immediate, but also no wedgeappears to justify its name. In order to find a first extension of Painlev~’s theoremto the case of several complex variables, we probably have to wait until 1956, whenBogolyubovfirst gave a statement (we drew this information from the later work of Vladimirov[227]), justifying the interest of the results with the (at that time) intense researches into the Wightman function and the dispersion relations (interestingly enough, the samematerial which spurred Sato’s ideas, and the sametime period as well). That this theoremwas of great relevance for theoretical physicists was attested by the manypapers on the topic (see our previous references) and to those refer the reader interested in exploringthis’ relevance. But let us nowprepare the notations which are necessary for the statement of the edge of the wedgetheoremin its first version (as given by Bogolyubov); for A any subset of ~", we will denote by T(A) the tube with base A, i.e. T(A) = fit ’~ × iA; for x in ~n \ {0}, we will write (as in the previous part of this section) x0
86
CHAPTER 1.
HYPERFUNCTIONS
denote the direction of the half line through the origin and passing through x; finally, for A again a subset of ff~, we will denote by A0the set A0 = {x0: x~A \ {0}}. Wecan now state Bogolyubov’s theorem: Theorem1.6.2 Let F1 be an open convex cone in ~, and set F2 = -Ft. For any open set f~ in ~:~ and any complex neighborhoodU of it, there exists a complex neighborhoodU’ c_ U such that the following is true: let fj ~ O(UC~ T(Fj)), j = 1, have the foll owing boun dary values fj(z + iFj0) = lim f~(x + iy), where the limit is intended to be uniform on compact sets containing x and compactsubconescontaining y. If the limits coincide in the sense that f,(x + iF,0) = A(x + iF20), then there exists a function f in O(U’) such that f=fj onU’fqT(Fj),
j = 1,2.
Withoutgoing into the details of the proof, it might suffice to recall that the original proof (see e.g. [169]) relies, not surprisingly, on Cauchy’sintegral formula. First one reduces the theorem to the case in which the cone F is of special shape (for examplethe first quadrant). Onethen applies a suitable analytic transformation and finally uses Cauchy’sintegral formula to obtain the explicit analytic continuation. This approachis quite natural, especially in view of Painleve’s original result, and can be found in Rudin’s [191]. It maybe worthwhile recalling that this theorem has stimulated manyother mathematicians (see our reference list), and so several alternate proofs have been given; all of them, deal with the issue of separate analyticity, and the key ingredient is always Cauchy’sintegral formula. The most general result in this framework is the so called Malgrange-Zernertheorem(despite the fact that it only appears in Martineau’s [159], and that neither Malgrangenor Zerner seem to have published it): in it we need the notion of relatively open cone which is given as follows: a cone F in f~’~ is said to be relatively open if it is openin its linear hull (i.e. in the linear subspace of N’~spannedby F). Withthis definition can state Malgrange-Zerner’stheorem as follows: Theorem1.6.3 Let F1 and F2 be two relatively open convex cones in ~’~ and let F12 be the convex hull of their union. For any open set f~ of fd ~ and a complexneighborhoodU of its, there exists a complexneighborhoodU’ C_ U of [2
1.6.
HISTORICAL NOTES
87
such that the following analytic continuation property holds. If f is a continuous function on the set V N (T(F1) U T(F~)U andif i ts rest rictions to U~ (T(Fj are holomorphicin (Fj),j = 1,2, then there exists a continuous function F on U’ N (T(F12) t~ ~), such that its restriction to U’ ~ T(F~2)is holomorphicin T(F~2) and that f =F on U’ ~ (T(I~I) (_J T(F2) Weeasily see, from Theorem1.6.2 to 1.6.3, an improvement due to the generality of the relative positions of the cones, and of course the weakerassumptions of continuity. That the continuity hypotheses (i.e. the coincidence of the two boundary values, which in Bogolyubov’sstatement is to be taken in the space of continuous functions) could be considerably weakened,was immediately apparent; as a matter of fact, it was almost immediatelyshownthat Theorem1.6.2 could be formulatedin the frameworkof the theory of distributions. Since most distributions of interest in physics arise as boundaryvalues of holomorphicfunctions, this extension had an impact which was not just mathematicallysignificant. Let us recall the notion of boundaryvalue in the sense of distributions, so to be able to state the distribution version of Bogolyubov’stheorem. Let f~ be as usual an open set in ~, and U a complexneighborhoodof its; let 7:)(~) and 9’(~) be, respectively, the spaces of compactlysupported infinitely differentiable functions on f~, and the space of distributions on 12 (the first space endowedwith its usual locally convextopology); finally let F be an open convex cone; for f E (.9(U ~ T(F)) and g E 9(~), the integral /~ f(x + iy)g(x)dx can be defined for any (sufficiently small) y in F. If the limit lim f f(x + iy)g(x)dx yEF
exists, for every g in 7)(f~), then such a limit defines a distribution whichwill be denoted by f(x + irO), and this notation is consistent with the samenotation used in the case of the limit in the space of continuousfunctions. Wecan therefore state the following result (whichwe provide here directly in the moregeneral version due to Epstein [53], in whichthe position of the conesis general): Theorem1.6.4 Let F~ and [’2 be two open convex cones in J~. For any open set f~ (and a complexneighborhoodU) there exists a complexneighborhoodI C_ U such that the following is true: if
fje O(U~ T(rj)), = 1,2,
88
CHAPTER 1.
HYPERFUNCTIONS
have both distribution boundaryvalues fy(x + iFj0) such that fl(X + irlO) : f2(x + it20), then there exists a function f in O(U~ N T(FI:))) which coincides with fj on =1, 2.
u’ c~T(r~)for j
a, Note that whenF1 = -P2 then the convexhull of their union is all of/R which shows howTheorem1.6.4 actually contains Theorem1.6.3. The next great advancein the theory of the edge of the wedgetheorem came with the realization (due to Martineau) that in order to deal with the case which several cones were appearing, a more abstract approach to the problem was necessary. In particular, the introduction of the notion of boundaryvalue in the sense of hyperfunctions becameimperative. In order to state the results weare interested in, we needfirst to recall, from section 1.5, the basics on boundaryvalues in the sense of hyperfunctions; let ft and U be as above and let P be an open convex cone in ~/~a. As we know, the vector space B(ft) is defined by H~(U,(.9); however,one mayrecall the long exact sequence for relative cohomology(analogous to the one which is well knownfor singular cohomology),which gives ¯ ..
-~ Ha-I(u \ ~; O) ~ H~(U, (9) -~ Ha(U; ~ H~(U \ f~; O) ~ H~+I(U,
Because of Grauert’s Theorem [66], we can assume that the neighborhood U has been chosen to be Stein. Wecan now use the vanishing of the relative cohomologyin dimension different from n, together with Cartan’s Theorem B in the above long exact sequence, which shows the vanishing of the n-th cohomologyof Stein open sets, to obtain an important isomorphism: Hn-t(U \ ~; O) ~ B(fl). Wecan therefore consider a holomorphic function f E O(U~ T(F)), and associate to it its hyperfunctionboundaryvalue br(f), whichis the element of B(ft), defined by the (n - 1)-cocycle whichcan be naturally associated to f (see construction in section 1.5). At this point, one maydefine a space of holomorphicfunctions defined near Na, for which two different notions of boundaryvalues exist; a distribution notion and a hyperfunction one. Following Martineau’s [156], we define the subspace Ov(U fiT(r)) of O(U ~ T(r)) to be the subspace of those holomorphic functions whoseelements can be continued as distributions to a full complex neighborhoodof ft. Martineau succeeded to showthe following two fundamental facts:
1.6.
HISTORICAL
89
NOTES
Theorem 1.6.5 A function f E O(U ¢~T(F) ) has a distribution boundary value (as defined before) if and only if it actually belongs to the subspace Ov(U T(F)). In this latter case, its distribution boundary values coincide with its hyperfunction boundary value, i.e. br(f)(x)
: f(x it 0).
It was using this last result, that Martineau was able to prove its full version of the edge of the wedge theorem (he actually used the hyperfunction version to deduce, as a corollary of Theorem1.6.4, the distribution version). Weconclude this Section of the appendix by giving Martineau’s hyperfunction edge of the wedge theorem. Theorem 1.6.6 Let F1, F2,..., F,~ be open convex cones in ~’~. Then for any real open set ~ and its complex neighborhood U, there exists a complex neighborhood U’ C_ U such that the following is true: given functions fj in O(Un T(Fj)) such that, in B(~), ~-~ br(fj) = j=l then there exist holomorphic functions gjk in O(U’ NT(Fj,k)) for j, k = 1,..., and j = k, such that fj = ~.gjk on
m,
U’ nT(Fj),j = 1,...,m.
and gj~ + gkj = O. Wepoint out at this point that the story of the edge of the wedge theorem does not quite ends here; as a matter of fact, the theorem itself is a key element (and motivation) for the introduction of the sheaf of microfunctions; this aspect, however, we will leave to the next chapter, where microfunctions will be treated.
Chapter 2 Microfunctions 2.1
Introduction
The last chapter should have provided the reader with a sufficiently amplebackground on the general theory of hyperfunctions (even though we restricted our attention to the case of hyperfunctions defined on Euclidean spaces, purposely avoiding the moredelicate aspects whicharise fromthe necessity of dealing with general real analytic varieties). This chapter, on the other hand, will introduce the other great player in the field of Algebraic Microlocal Analysis, namelythe sheaf of microfunctions. Even though someof the algebraic machinerywhich is needed has already been introducedin the previous chapter, we still wishto warnthe inexperiencedreader since the theory of the sheaf of microfunctionsis definitely a delicate issue. The mainreason is that microfunctionsdo not arise naturally as generalized functions on the Euclideanspace but, rather, describe the .singularities of such generalized functions. In dealing with hyperfunctions, the reader could always rely on its experience with the space of distributions, whosebehavior (in manyinstances) was mimickedby the space of hyperfunctions. Of course the interest in studying hyperfunctions arose exactly from those situations in which the behavior was so markedlydifferent, and we hope we succeeded in conveying this aspect in our previous chapter. Whendealing with microfunctions, however, the reliance on the infinitely differentiable case seemsto falter (at least in mostof the classical introductions to the subject) and this makesits understanding moredifficult. In this chapter we will try to rely on the infinitely differentiable analogiesas muchas possible, at least as a wayof introducing the topic, and we will then showthe powerof the sheaf of microfunctions as constructed by Sato. Also in this case, of course, the creation and the advancement of the theory is essentially due to Sato and his coworkers, but we will see that similar ideas were brewing in manydifferent areas of mathematics,so that the interplay betweendifferent readings is even moreinteresting here than it was for hyperfunctions. 91
92
CHAPTER 2.
MICROFUNCTIONS
The chapter is structured as follows: in section 2.2 we describe different notions of singularities for hyperfunctions, and, in particular, the crucial notion of singular spectrumand its relations to differential operators. This introduction lays the groundfor the construction of the sheaf of midrofunctionswhichis done (along the lines of ChapterI) in twosteps; the single variable case is treated section 2.3 while the several variables case is dealt with in section 2.4. Finally, section 2.5 goes back to the study of microlocal operators, whichhad only been cursorily discussed in the first chapter. Nowthat microfunctions are available, it is the right momentfor a more complete discussion of this topic and of its interest within the theory of differential equations. This chapter also ends with a historical appendix (which is also an excuse for somealternative treatment of the topics discussed in the main text); this time the appendix deals with the introduction and the role of microfunctions in physics as well as with the treatment which HSrmanderhas given of a notion very similar to the singular spectrum, namelyhis notion of analytic wavefront set. With the equipment provided by these two chapters, the reader should now be capable to dwell into the mainpart of this book, i.e., the algebraic treatment of systemsof differential equations.
2.2
Singular Support, Spectrum
Essential
Support
and
In the previous chapter, we have examinedhyperfunctions as generalized functions which form a sheaf properly containing the subsheaf of real analytic functions. Thus, in particular, one maywant to knowthe location of the singularities of hyperfunctions,i.e., the set in whicha given hyperfunctionfails to be real analytic; such a set, whichcan be easily defined, is the so called singular support of a hyperfunction. Its definition (see also Chapter I) reads as follows: Definition 2.2.1 Let f be a hyperfunction on an open set U on ~:t n. The singular support of f, indicated by sing supp (f), is the complementin U of the largest open subset of U on whichf is real analytic. The reader will note that such a definition is well given, because .4 is a subsheaf of B, and therefore there exists the largest open subset of U in whichf is real analytic (in other words, the property of being real analytic is a local property). This definition, however, does not seem particularly useful, since it only pinpoints the singular locus of a hyperfunction, without explaining the differences betweenvarious singularities. As an example(whichreally originates from physics) one mayask whymultiplication of distributions (hyperfunctions) is possible in somecases but not in others. It is well known,e.g., that the square of
2.2.
SINGULAR SUPPORT, ESSENTIAL SUPPORT
93
the delta function is not a well defined distribution, while it is possible to consider the square of 1/(x ÷ i0). Since real analytic functions do indeed multiply amongthemselves (and of course we can multiply a real analytic function by hyperfunction), it is somehow clear that the problemof multiplication resides with the singularity locus of hyperfunctions; on the other hand, the singular support of both the delta and 1/(x ÷ i0) consists of the origin, and so a more refined description seemsnecessary. It turns out that the key instrument for the understanding of this and other related phenomena,is the notion of microlocalization of a sheaf; by this term one usually refers to the fact that the study of singularities is nowremovedfrom the original manifold, and is transferred to its cotangent bundle (or spheric cotangent bundle); the necessity of such a behavior is common knowledgein the communityof differential equations scholars, where the symbolof an operator is exactly defined on such bundle. The procedure for such microlocalization is not at all immediate, and maybe advantageous to begin by exploring a similar situation, whichcan be used as a motivational tool, namelythe differentiable case. Let us therefore begin by taking a Schwartz distribution f in ~n, and let us consider the issue of its differentiability at a point x0. If f were compactly supported, we could use the Paley-Wiener Theoremfor the space ~ of compactlysupported distributions to study the differentiability of f in terms of the growth of its Fourier transform. On the other hand, if f is not compactly supported, no such theorem exists (in other words, we do not have a good intrinsic description of the differentiability of f, for the precise reasonthat it does not in general admit a Fourier transform). The way to approach this situation, therefore, is to exploit the fineness of the sheaf of differentiable functions and to multiply our distribution f by a compactlysupported function, say Xx0= X, which is identically one in a neighborhoodof x0, and quickly vanishes outside it (we can take for X any cutoff function). Then, the product f ¯ X is a compactly supported distribution which coincides with f in a neighborhoodof x0. Note that one can actually represent f as the limit, in a suitable topology, of a sequence {f ¯ Xn}of compactlysupported distributions obtained by considering the cutoff functions Xnassociated to the spheres B(x0, ~) centered in x0 and radius sn convergentto zero. If f is infinitely differentiable, then each element of the sequenceis infinitely differentiable as well. Alsonote that whilethe differentiability of f. Xdependsonly on the differentiability of f in a neighborhoodof Xo, the differentiability of the sequenceof distributions associated to a sequence of spheres of a decreasing radiuses, only dependson the differentiability of f at x0. One can nowexploit the compactsupport of the sequence to determine its differentiability in terms of the growthof its Fourier transform. Let us recall here, for the sake of completeness, the versions of the Paley-Wienertheorem which we need to employ: Theorem2.2.1 (Paley-Wiener for ~’). The vector space of compactly sup-
94
CHAPTER
2.
MICROFUNCTIONS
ported distributions £~ is algebraically isomorphic, via the Fourier transform, to the space of entire functions of exponential type which, on the real axis, have polynomial growth. In other words, the space of compactly supported distributions is isomorphic to the space PW((Tn) := (F e 7/((Tn): where A and B are positive
IF(z)l
<_ g(1 + Izl)S
exp(BlImzl)}
constants depending on F.
Theorem 2.2.2 (Paley-Wiener for T)). The vector space of compactly supported infinitely differentiable functions Z) is algebraically isomorphic, via the Fourier transform, to the space of entire functions F which satisfy the following estimate: there is a positive constant B such that for any integer M > O, there exists a positive constant CMsuch that F(z) <_ CM(1+ IZl) -M exp(SlImzl). It is a highly non-trivial fact, essentially due to Ehrenpreis [52], that the isomorphisms above can be made into topological isomorphisms. This fact has some very important consequences which we will not explore in this setting. Werefer the interested reader to Ehrenpreis’ fundamental work [52]. Based on the theorems stated above, we realize that we certainly have that the products f ¯ X,~ satisfies the weaker condition (2.2.1)
If.~Xn(z)l
<_ d(1 + Iz]) B exp(Sllmz]);
however, if the distribution f is infinitely differentiable at the point x0, then the products f - Xn also are infinitely differentiable, and therefore they satisfy the stronger condition
This fact gives us a useful insight; if a distribution fails to be differentiable, we can "measure" the degree of this failure by looking at where, in the frequency plane, its Fourier transform fails to satisfy condition (2.2.2). This consideration allows to distinguish, for example, between the delta function and the boundary values of l/z, i.e. 1/(x + iO) and 1/(x - iO). As a matter of fact (we are considering here only the one-dimensional case), one can easily see that the Fourier transform of the delta function (considered as a compactly supported distribution) is given by eix~
which, clearly, does not satisfy condition (2.2.2) in either direction (in one mension, of course, there are only two directions to be considered: ~ > 0 and ~ < 0). On the other hand the Fourier transform of 1/(x + iO) turns out to be (for H(~) the Heavisidefunction)
2.2.
SINGULAR SUPPORT, ESSENTIAL SUPPORT
95
ei~. H(~) which satisfies condition (2.2.2) for ~ > 0. This shows that the singularity of is somehowmore serious than the singularity of 1/(x + iO). Wewill see how to use this information to justify the possibility of taking the square of the second and not of the first. For the case of compactly supported distributions, we can use these natural remarks to introduce the process of microlocalization as follows. As we mentioned before, we are going to consider distributions defined on an open set U of the n-dimensional Euclidean real space, while the microlocalization will take place on the product U x ~, to be regarded as the cotangent bundle of U. This approach lends itself to a quick modification when dealing with a more general manifold M (to be assumed differentiable when dealing with distributions, and real analytic when dealing with hyperfunctions). For the next definition, we recall that a cone F in ~" is a subset of ~ stable under the dilations ~ --~ p~ for p > 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.
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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
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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, ~)
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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
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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 <_ C(1 + [el) "~-I"l on T × ~n. The elements of Sm(~, ~) are called amplitudes of degree less than or equal to m. These amplitudes will be the symbolsof our pseudo-differential operators (this is not quite accurate, as wewill see later, but we can assumeit at least for the moment). Definition 2.2.10 Let a ~ ,~’~(~, ~). Wewill denote Op(a): T~(a) -~ T~’(f~) the operator defined by Op(a)(f)(x) (~- ~)
1 ~
f
Moreover,a linear continuous operator A: 8’(~) -~ ~’(~) is said to be a pseudo-differentialoperatorof orderrn if there exists a ~ Sin(Q,~) such that A = Op(a). Wewill then write A ~ ~’~(f~). Remark2.2.7 Anydifferential operator P(x, D) is a pseudo-differential operator with a(x, y, ~) = P(x, ~). Also, any regularizing operator A with kernel A(x, y) is a pseudo-differential operator with a(x, y, ~) = A(x, -~(~-~)’~ y)x(~)e
2.2.
SINGULAR SUPPORT, ESSENTIAL SUPPORT
103
for X E 7)(IRn), f X(~)d~ Conversely,if we set rnE/R
we can showthat if A E ~-~(f~), then A is regularizing. Before we proceed to relating pseudo-differential operators to the notion of wave-front set, we want to clarify an earlier commentwe made regarding amplitudes and symbols. The problem which one obvi,ously needs to deal with is two-fold: on one hand, the role of the variable y seemsto be more"technical" than really necessary; on the other hand, a given pseudo-differential operator can be obtained through infinitely manydifferent amplitudes, while we wouldlike to have somesort of one-to-one correspondencebetween symbolsand operators. This can be achieved through a quotient process: Definition 2.2.11 For any m, £’~({2) denotes the subspace of £m(f~, {2) consisting of amplitudes whichare independentof y. Also, we set
Definition 2.2.12 The space of symbols of degree less than or equal to m is defined by
Remark2.2.8 All of the spaces introduced above can be endowedwith suitable Fr~chet topologies. Since we will not need them, we refer the interested reader to [224] for further details. The fact that Definition 2.2.12 is the correct one follows because the map a(z, ~) -+ Op(a) from~q’~(f~)to ~’~({2), defined Op(a)(f)(x) -nf := (27r) actually yields an isomorphismbetween~’~({2) and ~.~({2)/~-oo(f~). This space will be denoted by Wecan nowrelate these concepts to the notion of wave-front set of a distribution; to do so we will microlocalize the notion of regularization and regularizing operators (see also Definition 2.2.2).
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Definition 2.2.13 A pseudo-differential operator A in U is said to be regularizing in (Xo, ~o), Xo¯ U, ~o ¯ Kin {0}, if there is a compactly supported infinitely differentiable function X supportedin U whichis identically one near Xo, and an open cone F0 in ~n containing ~o such that the symbol a(x, ~) of A satisfies the following property: for any N >_0 and every pair of multi-indices ..., ~,~), fl =(~, ..., fin), there exists C~,Z,N>_0 suchthat suzp IO$O{[X(x)a.(x,{)]I <_Ca,~,N(1 + I~l) -N for any f ¯ F0. Accordingly, we say that a pseudo-differential operator A on U is regularizing in a conic subset F of U × (~n \ {0}) if it is regularizing in the neighborhoodof every point of F. Definition 2.2.14 Let A be a pseudo-differential operator A on U. Wecall micro-supportof A the set #-supp(A)defined as the complementin U × of the union of all conic open sets in whichA is regularizing. Notethat, just like in the case of the wave-frontset of distribution, the microsupport of a pseudo-differential operator is a closed conic subset of the set U × (~n \ {0})o The link between these two notions is contained in the following result whoseproof appears, for example, in [224]: Proposition 2.2.1 Let f be a distribution in U, and let (x0,~0) be a point of
u × \ {0}). Thenthe following properties are equivalent: (i) (Xo, ~o) ¢ WF(f); 5i) there exists a differentiable function a(z, ~) in U × (~’~ {0}), positivehomogeneous of degree zero, equal to one in someneighborhoodof (x0, ~0) such that if A is any properly supportedpseudo-differential operator in ~ with symbol a(x,~), then A(f) Coo(U); (iii) there exists a conic open neighborhoodF of (Xo, ~o) in U × (~’~ {0)) such that
B(f)¯ C°°(U) for any properly supported pseudo-differential operator B in U whosemicrosupport is contained in F. This proposition has several important consequences which we summarizein a single Proposition:
2.2.
SINGULAR SUPPORT, ESSENTIAL SUPPORT
105
Proposition 2.2.2 Let U be an open set in J~. Then: (i) if f ¯ 7Y(U), then the base projection of WF(f)coincides with the singular support of f; if A is a properly supported pseudo-differential operator on U then A is regularizing in a conic open subset F of U x (J~ \ (0}) if and only if Af is infinitely differentiable in U .for all distributions f ¯ 79~(U). let ~ : U --+ U~ be a surjective diffeomorphismbetweentwo open sets in J~,and let ~+: T*U-+ T*U’and ~. : 7:)’(U) --+ Z)’(U’) be the induced diffeomorphism; then WF(~,f) = ~p+(WF(f) for any f ¯ 7Y(U); (iv) if A and f are as in (ii), WF(Af) C_ WF(f) fq (# - supp(A)); in particular, WF(Af) C_ WF(f). Beforeproceedingto microlocalize the notions of distribution and wave-frontset, we showwhichare the ideas behind the definition of the trace of a distribution on a submanifoldand of the product of two distributions. To this purpose, let us split the coordinate x in ~ as z = (x’,x"),
z’ ¯ ~P, x" ¯ ~n-p,
with x~, x" non-empty. Assumefurthermore, the following condition on a distribution f ¯ (2.2.10) there exists s > 0 such that for every M> 0 there is CM> 0 such that I~(~)1 < CM(1+ I~1) -M if I~’1 < where~ and ~" are, respectively, the dual variables of x’ and x". Since, naturally, there exist C, m> 0 such that
I}(~)1_
ICI-
for somenewconstant C > O,
I.f(,C)l< C(1+ -"-’ 14’1)"+"+~(1 + ICI)
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and (2.2.11) also holds true when[~"[ >_ ~l~, as it follows immediately from (2.2.10). This estimate showsthat \2~]
f ]((, ~")d~"
is tempered and its inverse Fourier transform is what can be regarded as the trace of f on x" = 0. But now,it is not difficult to see that (2.2.10) actually means that WF(f) n {~ e ~": ~’ = 0} = 0. By(iii) of Proposition 2.2.2 we therefore obtain that if Y ~ ~ is a differentiable manifold, and f G D~(~), then the trace of f along Y (or, if one prefers, restriction fi~) is well defined wheneverthe wave-frontset WF(f)of f does not intersect the conormal bundle of Y in ~. In particular, if WF’(f) denotes the image of WF(f) under the symmetry (x, ~) ~ (x,-~), and since, for f, g distributions on
f(~). ~(~) is nothing but the trace of f(x)@g(y) along the diagonal of ~ x ~, one obtain the following important result: Proposition 2.2.3 U f, g ~ D’(~) satisfy
WF(I)~ WF’(~) then the product f ¯ g is a well defined distribution on ~. As our last remarkon the differentiable case, we want to microlocalize the sheaf of distributions. Wetherefore construct the following presheaf on T*~~ ~ {0}: for any open set U ~ T*~n ~ {0), we denote by c.sp.(U) its conic span, i.e. smallest conic set containing U, and we define V’(~") ~(U) := {f e V’(~"): f e C~(¢.~.(V))}" It is immediateto verify that the functor V ~ Y(U) gives a presheaf on T*~~ ~ {0}, and the associated sheaf is what will be called sheaf of microdistributions. Clearly every distribution f ~ D’(~~) defines a section of this sheaf whosesupport (in the sense of sheaf theory) is actually the wave-frontset of f (we will see later on howthe situation for hyperfunctionsis quite analogous).
2.2.
SINGULAR SUPPORT, ESSENTIAL SUPPORT
107
Remark2.2.9 The reader will have certainly noticed howthis analysis is all local in nature and therefore could be repeated for distributions defined on any differentiable manifold. Remark 2.2.10 An analogous microlocalization process could have been applied to the space of pseudo-differential operators to build, on T*~n \ (0}, the sheaf of pseudo-differential operators. Nowthat the reader is acquainted with the infinitely differential case, we will try to see what can be done to modify this approach to fit the case of hyperfunctions. Historically, as we will discuss in the appendixto this chapter, there is a very interesting intertwining betweendifferentiable analysis, analytic analysis, hyperfunction analysis and theoretical physics, and it is not always appropriate to discuss about priorities for concepts which have been arising almost simultaneouslyin different contexts and with different techniques. The first remark one can makeconcerns the first step we took in defining the notion of singularity for a distribution; the key tool is obviously the fineness of the sheaf of distributions, actually of C°~ functions, whichallows us to localize the problemvia the use of compactlysupportedinfinitely differentiable functions. This same procedure is clearly unacceptable for the analytic case, since no compactly supported analytic functions exist. There are some ways around this which we will describe in our historical appendix, when dealing with HSrmander’s analytic wave-front set, but we nowwant to show howthe Japanese school of Sato dealt with the problem. Let us begin with the one variable case, and let therefore f = IF] be a hyperfunction defined on the real line (or on an open subset of it); we know that we can really think of F as a pair of holomorphicfunctions, say (F+, F-), with F+ holomorphicin someopen sets of, respectively, the upper and the lower half plane. One maythen observe that, in order for f to be real analytic, one needs that the "difference of the boundaryvalues" of F+ and F- along the real axis is real analytic. One can easily showthat for f to be real analytic it is necessary that f admits a representative F for which both F+ and F- extend analytically beyondthe real axis. Here we see, therefore, that the obstruction to the analyticity of f is given by the obstruction to the holomorphiccontinuation of the two functions which represent f. Analyticity can therefore be described in terms of points of the real axis (where f mayor maynot be analytic) and the directions (along which the representatives can be analytically continued); in the one dimensional case there are only two such directions, a positive one (representing the upper half plane) and a negative one (representing the lower half plane). Thus(as before for the differentiability of distributions) the natural locus for the study of the analyticity of hyperfunctionsis not the real line (or an open set U of the real line), but its cotangent bundle deprived of the zero section or, even better, the cosphere bundle S*U. If we confine our attention
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to the single variable case, we see that S*Uois a trivial bundle whosefiber is S or, better, iS°~ = {/co,-/co}, and the following definitions are therefore quite natural: Definition 2.2.15 A hyperfunction f is microanalytic at Xo + ico if f has a representation f = (F+, F-) in which + extends a nalytically i n acomplex neighborhoodof xo. Definition 2.2.16 A hyperfunction f is microanalytic at Xo - ico if f has a representation f = (F+, F-) in which F- extends analytically in a complex neighborhoodof Xo. It is then immediate to reformulate our previous argument by saying that f is real analytic in x0 if and only if it is microdnalytic both at x0 + ico and x0 - ico. It is also obviousthat tliese definitions do not dependon the choice of the representative for f. Let us point out that being microanalytic also implies, for a given hyperfunction, the existence of a rather convenient representation, since we immediately have Proposition 2.2.4 If the hyperfunction f is microanalytic at the point xo + ico, then we can express it as the boundaryvalue from below of a holomorphic function, i.e. there exists an open set V in the complexplane containing Xo and a function F, holomorphic in U- such that f(x) = F(x - iO), at least in a neighborhoodof xo. The same statement (with obvious modifications) holds if +ico is replaced by -ico. Definition 2.2.17 Let f be a hyperfunction on an open set U of the real line. The set of points of f wheref is not microanalytic is called the "singular spectrum" off and is denoted by S.S.(f). By our definitions, S.S.(f) is a subset of S*~ = ~ × iS°~, while the singular support, sing supp(f) defined in Chapter I, is a subset of The following two results are an immediate consequence of the definition given above: Proposition 2.2.5 Let 7~ : S*fft -+ Kt be the natural projection, and let f E B(~). Then 7~(S.S.(f)) = sing supp(f). Proposition 2.2.6 Linear differential operators with real analytic coe~cients do not enlarge the singular spectrum of a hyperfunction.
2.2.
SINGULARSUPPORT, ESSENTIAL SUPPORT
109
Proof. Immediate.
[]
The notion of singular spectrum is the analog, for hyperfunctions, of the notion of wave-frontset (this link will be completelyexplainedin the appendix); one therefore expects somerelevance with respect to the problemof the definition of the product of two hyperfunctions. This expectation is fulfilled by Theorem2.2.3 Let f, g be hyperfunctions on ff~, and assume that S.S.(f) and S.S.(g) have no antipodal points (i.e. there is no (x,~) e S*~ such (x,~) ¯ S.S.(f) and (x,-~) E S.S.(g)). Then the product f .g is well and (2.2.12)
s.s.(f . g)¢_s.s.(f)~s.s.(g).
Proof. The problem of defining the product f - g is clearly local, and we therefore showhowto define under the condition that there is no ~ ¯ iS°~ for which
Assume
f(x) =F÷(x+iO)- F-(x and
~(~)=G÷(x + io) - G-(xwith obvious meaningfor the symbols. Since S°~ only consists of two points we can consider the followingcases: (i) x ~ ~r(S.S.(f)), in whichcase we have no restrictions on g; (ii) f is microanalyticin x0 + ic~, and g is, as well, microanalyticin x0 + icx). (There are of course two other cases which are, however,totally symmetric). the first case, f is real analytic at x and therefore we have nothing to prove. In case (ii), both f and g can be represented (Proposition 2.2.4) as boundary values from below:
f(~)=F-(~- i0), ~(x)= ~-(~- i0), for suitable F- , G-. Then one defines
f. g(~):=(F-.~-)(~-
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CHAPTER 2.
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It is immediateto verify that such product is well defined, and that (2.2.12) holds. [] Remark2.2.11 It is immediate to verify that such product satisfies usual algebraic properties of functional products.
all the
Example 2.2.1 (i) S.S.(5) {0} x iS°~; in deed, 1/z does no t ha ve ananalytic ext ension either waythrough the real axis; (ii) S.S.(5 (n)) = {0} x iS°~, for the samereason as in (ii); (iii) S.S.(H) {0} x iS°~, si nce H is defined as thediff erence of t he boundary values of log z (in the complexsense) for whichno analytic continuation across the origin can be given; 1
1 = {0 ± ice}, since ~ is actually
the boundary value
from one side. Example 2.2.2 As a consequence of Theorem2.2.3, and of Example 2.2.1 we obtain the well knownfact that the Dirac delta cannot be multiplied by itself or by 1/(x ± iO), while it makessense to define
Whendealing with distributions, we have discussed (Proposition 2.2.2) the invariance of the notion of wave-front set with respect to diffeomorphism.Wewill nowprove a similar result for the real analytic category. Westill confine our attention to the case of one variable: Definition 2.2.18 Let U, Ur be two open sets in’K~; a real analytic .function ¢ : U -~ U~ is said to be a real analytic isomorphismif it is a bijection and ¢~(x) ~ 0 at any point x E Definition 2.2.19 Let ¢ : U --~ U’ be a real analytic isomorphism, and let f = [F+,F-] E B(U’). Then the pull-back ¢*f is the hyperfunction on U defined by ¢*f = (F+(¢),F-(¢)) if ¢’(x) and by ¢*f = (-F-(¢),-F+(¢))
if ¢’(x)
2.2.
SINGULAR SUPPORT, ESSENTIAL SUPPORT
111
It is an immediate consequenceof the definition the fact that the following formulas hold: supp(¢*f) = ¢-l(supp(f)) sing supp(¢*f) = ¢-l(sing supp(f))
s.s.(¢*f)(¢-lx’(¢’))(s.s.(f)), where
¢-1x’(¢’) : u’
u× is0
maps(¢(x), () to (x, ¢’. Weare nowon the brink of microlocalizing the sheaf of hyperfunctions, to obtain the sheaf of microfunctions. This will be done in sections 2.3 and 2.4; before doing so, however,we need to study the singularities of hyperfunctions of several variables. Since, as we haveseen, B is a sheaf, werecall the definition of the support of a hyperfunctionf as the complementof the largest open set on whichf vanishes, while its singular support is the complementof the largest open set on which f is real analytic. Weneed to observe, however,that unlike what happensin the case of a single variable, it is not necessarily obvious to check whether or not a hyperfunction vanishes or is analytic in a neighborhoodof a point. A good characterization can howeverbe obtained as a (highly non-trivial) consequenceof the flabbiness of the sheaf/3. Werefer the reader to [103], but we quote here the result which is of interest to us: Proposition 2.2.7’ Let U be an open set in Ktn, F a cone in ~:~ and let f(x) B(U) be represented as a unique boundaryvalue
f(x)= F(x+ r0). Then f is zero (or real analytic) in U if and only if F is zero (or analytically continuable) in U. Proof. This Proposition is nothing but a very special case of the Edgeof the WedgeTheorem. [] Wecan nowproceed to attempt to analyze in a microlocal fashion the notion of analyticit:~ for a hyperfunction. Clearly, whenonly one variable is involved, analyticity (or lack of) can be interpreted in terms of continuationacross the real axis, either from aboveor from below, so that only two directions are involved, and this is reflectedin the fiber S~ of S*~. Whenthe numberof variables grows, however, S°~ is replaced by S~-1 and the spheric cotangent bundle becomes ¯ S*~ n=~×~S~
n-1
.
112
CHAPTER 2.
MICROFUNCTIONS
Definition 2.2.20 Let f E B(IRn). We say that f is microanalytic at (x, i~oc) 6 S*J~n or, equivalently, at (x, ~) 6 J~nx ’~-t i f f admits a representation as sumof boundaryvalues (see Chapter1) N
f(x) = ~ Fj(x+ irj0) j=l
with Fj holomorphicfunctions defined on infinitesimal wedges ~ + iP jO such that, for any j = 1,...,
N,
r~ n {ye ~: (~, y) < 0}# The set of all points in ~:~ × iS~-1 (i.e. in S*J~~) wheref is not microanalytic is called the singular spectrum of f and is denoted by S.S.(f). Note that, by its very definition, S.S.(f) is a closed subset of S*~n whose projection onto ~’~ should indeed be exactly sing supp(f). That this is the case is indeed once again a non-trivial consequenceof the Edge of the Wedge Theoremand we quote it for future reference. Proposition 2.2.8 Let 7r : S*IR~ -4 ~ be the canonical projection. Then for every hyperfunction f on IR~ one has 7r(S.S.(f)) = sing supp(f). Note that, in general, 7r-l(sing supp(f)) is muchlarger than S.S.(f). Example 2.2.3 (i) S.S.(5) -- {0} x iS~-~; (ii) S.S. (~) = {xj = O} x {~(J)}, where ~(J) = (O,O,...,i~x~,...,O), i~c being at the j-th position; (iii)
S.S.
1 + 1 = {zj = O} x {~(~)} U {zk = O} x {~(k)}.
Wewould like to show that a product amonghyperfunctions can be defined if the singular spectra are suitably placed with respect to each other. To do so, we state the following :result, whoseproof is postponedto section 5 (see also in [10a],[12a]). Weneeda preliminary definition:
2.3.
MICROFUNCTIONS
OF ONE VARIABLE
113
Definition 2.2.21 I.fF is a subset of S*J~~, then the symbol F° denotes the set of its images under the map
(x, ~)-~ (x,-~), where (x, ~) E
Theorem 2.2.4
Let f, g be hyperfunctions S.S.(f)
on ~, and suppose that
n (S.S.(g)) ° = ~.
Then the product f ¯ g is well defined and it is
supp(f,g) c_supp(f)n supp(g). 2.3
Microfunctions
of One Variable
In this short section we will finally microlocalize the sheaf of hyperfunctions, muchin the same way in which we have microlocalized the sheaf of distributions. This section is devoted to the case of one variable, in which all results and proofs are quite transparent. Werecall that in Chapter I we studied the singularities of hyperfunctions by looking at the (flabby) sheaf I~/A which is however, still a sheaf on ~ (or and therefore cannot contain any information on the spectra of hyperfunctions. The crucial idea, here, consists in pushing up B/~4 from ~ to S*~. The standard way in which this is obtained goes through the introduction of preliminary sheaf of, we could say, microanalytic functions. To do so, consider the basis of open sets for S*h~ given as follows: for U1, U2 open sets in ~, we consider their disjoint union
u1Hu2:= (vl x {ioo})v (u~× {-ioo})c_ which is clearly an open set in S*~ and is such that (for 7r again the canonical projection ~r : S*~T/--~
(~-1~)(u1 [I u~)= ~(u~)¯ Nowwe define on S*~ the sheaf of microanalytic functions by setting ~4*(U111U~) :-- {f E B(U~) S. S.(I) ~ (U~ x {i cx~}) = O}
¯ {f ~ s(u~)s.s.(f) n (u2 x{-io~}) = for any pair of open sets U~, U~ in ~. The reader will immediately note that an element of
A*(u,~ v~) is essentially a pair of boundary values of holomorphic functions, one from below, and the other from above. And so we give our fundamental definition:
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CHAPTER2.
MICROFUNCTIONS
Definition 2.3.1 The sheaf C of microfunctions is defined by C .--
(2.3.1)
~[, ¯
Note that since both 7r-l~ and Jr* are sheaves on S*~, so is C. Note also that by (2.3.1) we meanthat C is the sheaf associated to the presheaf F--~
(~-l~)(r) ~*(r)
FC S*~ open.
Indeed, at least in general, the sheaf built as quotient of two sheaves is not a sheaf per se, but needsto sheafified. In our case, as it will becomesoonapparent, the situation is simpler (at least as long as weare dealing with just one variable). Remark2.3.1 It is an immediate consequence of the previous definition that the following short sequence of sheaves on S*~is exact: (2.3.2)
O -+ A * ~ 7~- ~ B --+ C -~ O.
This sequence is knownas Sato’s fundamental sequence. Theorem2.3.1 C is a flabby sheaf and, for any open set F C S*~,
(2.3.3)
c(r)- (~-,~)(r)
Proof. There is, of course, a map T: (~r-IB)(F) whose kernel is A*(F). To prove (2.3.3) it will be enough to showthat actually surjective. Note that the flabbiness of C follows immediately from (2.3.3) and the flabbiness of B. Assume,to this purpose, that F = U (this causes no loss of generality). The proof which follows is essentially Mittag-Leffier type argument.Let then {Uj} be a locally finite covering of U and consider f E C(U). Supposethat f is defined locally asfj mod,4*(U~ × {i~}) for some fj~ B(U~). By the way we have defined J[*, there are holomorphic functions Fj such that fj(x) = Fj(x + iO) modJI*((Uj × {i~x~}) and from the compatibility condition
~ = E~(z+ io) - F~(~+ iO)e A*((U~ ~ U~)
2.4.
MICROFUNCTIONS OF SEVERAL VARIABLES
115
wededucethat fjk is in fact a real analytic function, and, therefore,
{fj~}¯ Essentially by Mittag-Leffier Theoremwe deduce the existence of functions gj in fl,(U~) such that
f~,(x)=g~(x)- g~(x). The global hyperfunction f on U can now be constructed by patching up the collection
F~(x +io) g~(~), and this concludes the proof of the theorem.
[]
Remark2.3.2 For any open set U C_ ~, one can define the spectral decomposition mapping sp: B(U) C(~-I(U)) as follows U(U)--)* (71-lu)(T-I(u)) : U(U)(9 ~(U) f --~ f (9 f --~ fmod.4*(U x {icx)}) (9 fmodflt*(U × {-ic~}). It is immediateto verify that this spectral decompositionmappinginduces (globally) the following short exact sequences: 0 -~ .a(U) -~ ~(U) -~ C(~-I(U)) and (locally) O--~ A ~ B ~ r,C ~ O, wherethe last sequence of sheaves is on ~, and ~. denotes the push down(also called direct image)of a sheaf under~r.
2.4
Microfunctions of Several Variables
In this section we showhowto extend to the case of several variables the construction of the sheaf of microfunctions. Wewill not give all the necessary proofs, for whichwe refer the reader to [103], [123], [206]. Our first approachto the study of singularities of hyperfunctionsof several variables was given in Theorem1.5.4, where we proved that the sheaf B/A is a flabby sheaf and that, for any open set U
(-~) (u) - ~(u). In order to lift this sheaf to S*~’~ = ~:~’~ × iS~-1 we introduce, as we have n. already done in section 2.3, the sheaf A*of microanalytic functions on S*~
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CHAPTER 2.
MICROFUNCTIONS
Definition 2.4.1 The sheaf ~4" is the subsheaf of ~r-lB associated to the presheaf defined by
n u x r {f ¯ s(u): s.s.(f) n (v x It is clear that the presheaf introduced in Definition 2.4.1 clearly represents hyperfunctions which are microanalytic in given directions; moreover, by the definition of inverse imageof a sheaf, it also followsthat if U and F are connected, then A*(U× r) -- {f ¯ B(U) S. S.(I) A (Ux F) Wecan define the sheaf of microfunctions as the quotient sheaf ~-1~
A* so that, by definition, the following is an exact sequence of sheaves on S*~: (2.4.1)
0
~ A* ~ ~-IB
~ C ~ 0.
Nowthe germof a microfunction at a point (z, ~) is simply the residue class a germ of a hyperfunction at the point ~, modulothe germs of hyperfunctions which are microanalytic at (z, ~). Note that (~-x~)z+ie~ = Before being able to push down sequence (2.4.1) to ~ we will need take a detour on the cl~sical notion of "plane wavedecomposition", essentially introduced by F. John in [101], and first used in connection with the theory of hyperfunctions by Kataokain [95] (see also [103]). The classical starting point is the well knowndecompositionof the delta function as (2.4.2)
5(x) = (-2~i) Ys~-~ (x5 +
which, of course, only makessense in the frameworkof the theory of hyperfunctions. The classical, formal, proof for (2.4.2) is obtained as follows: by inverse Fourier transform one h~ 1 ~(x) = ~ f~ e~dw. Nowwrite w = r~ (polar coordinates with r > 0, ~ ~ Sn-l) and integrate by parts with respect to r to obtain --2~1~f ei~dw
:
~1£~-~ d~~+~e~’~rn-~dr
(n- 1~ d~ ’~’ -- ~--~/-~ fs~-, (x~ ÷ i0)
2.4.
MICROFUNCTIONS OF SEVERAL VARIABLES
117
wherein the last step we have used the fact that, for Im(y)> 0, it +°° /0
eiYr rn-l
dr =
(n- ~)!(-~)" iny n
It is clear that (2.4.2) provides an integral decompositionof the hyperfunction 6(x) in terms of the boundaryvalues whichrepresent it. In order to extend this idea to all hyperfunctions, one needs to introduce the kernel (n- 1)!.
Wo(z,~) := U-~~ (z~). ~, for (z,{) eg’" xG z{ := z1{1 +...+ and its "twisted" version (n - 1)! (1 w(z,~):= (-2~i)-
iz{) ~-1 - (1 - iz{)"-2(z ~ - 2) (z{)
(z~+i(z~- (<)~))~
’
wherenow{ is a real unit vector. Remark2.4.1 The reason for introducing the twisted kernel (due to Kashiwara [107]) is that W0has singularities along the real hyperplane x{ = 0, while the only real singularities of Wappear at the origin. Let nowF be an open convexcone, and denote by p0 its dual (hence a closed convexcone); set
w(z,p0):= £0nso-,W(z,~)d~. It is easy to verify that W(z, po) is a holomorphicfunction on an infinitesimal wedgeof type £r/,~ + iF0 which induces a hyperfunction W(x, F°) on tR’L This hyperfunctionallows us to give a precise (and generalizable) proof of (2.4.2) follows: Proposition 2.4.1 Let Fa°., j = 1,..., F~’s have no commoninterior. Then
N, be a covering of Sn-~ such that the
N
(2.4.3)
a(x) = ~ w(=, j=l
Proof. First we note that the right hand side of (2.4.3) provides a uniquely defined hyperfunction on ~ regardless of the choice of the cones Fa°.. Wecan therefore choose the representation which makesour computationssimplest. For a= (a~,...,a,),
aj =+l,
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CHAPTER 2.
MICROFUNCTIONS
define sgn(a) := ala2.., a. and the a-th orthant F~ := {7 E ~" :a2V2 > 0,j = 1,...,n}. First consider the case of W0(z,F°). Since (n-l)!
d~
+¢~.
1 ~
= (~’l) nr.[~,~ eiZ~dw=sgn(a)~eiZ~(2~) ~ Z1 ¯.. n
¯ Z
we then obtain, by taking the boundaryvalues, that a
Zl ¯ ¯ ¯ Zn ~z=x+iFa
which,by definition, is exactly ~(x). If we nowlook at the c~e of W(z, ~), one can see that, for suitable boundary integrals Ba(z), it W(z, ~) = Wo(z, ~) + ~(z), where the B~(z) can be explicitly computed(see [95], [103], [107]). Whenthe sumover a is taken, however,and the boundaryvalues are considered, the terms B~(x) add to zero and we therefore obtain
Weare nowready for the proof of the following fundamentalresult: Theorem2.4.1 Let U, F be open sets in ~n and iS~-1 respectively. (2.4.4)
Then
A*(u×
In particular, the sequence (2.4.5)
0 --~ .A(U) --~ 13(U) C(~r-l(U)) __~
is exact, and therefore (2.4.6) 0 ~ A --+/3 -+ ~.C -~ 0 is an exact sequenceof sheaves. Moreover,if f is a hyperfunction, then (2.4.7)
S.S.(f) supp(sp(f)).
2.4.
MICROFUNCTIONS OF SEVERAL VARIABLES
119
Proof. Wefirst note that all the assertions follow from (2.4.4). Indeed (2.4.5) is nothing but a restatement of (2.4.4) for F =iS~o-1. Similarly, (2.4.6) follows by (2.4.5) by taking inductive limits with respect to U. Finally, (2.4.7) follows from the definition of singular spectra of hyperfunctions. Wewill therefore limit ourselves to the proof of (2.4.4) for the special case F = Sn-l, since the general case presents someadditional technical difficulties. To do so, it will be sufficient to provethe exactness of 0 --~ A*(U× iS~-1) -+ I~(U) --~ C(U× -1) --~0 and in particular we only need to prove the surjectivity of ~(U) --~ C(U× iS~-1). Let then f E C(Ux iS~-l). By definition of C, one can consider the map
:r : f -~ f~of(u)W(:~ - ~, ~)du which is easily seen to be a (non-surjective) sheaf homomorphism B
T: C-~-~. The image T(f) belongs to B/.A(U x iS~o-1); let g ~ B(U x iS~-~) be a global representative of T(f). Such a g exists in view of Theorem1.5.4, and we will now show that the hyperfunction g(x) := Is,-, g(x, ~)d( is a representative for f in the quotient (2.4.4). Takethen (x, ~) ~ U x n-1 and suppose fcan berepresented, in a neighborhood U0× S~-1 of (x, () by a compactlysupported hyperfunction ]. Then the function h(x, ~) := g(z, ~) - T(]) is analytic in U0 × S~-1; by taking the boundaryvalue we obtain that g(x) is analytic and g is the required representative.
[]
Weconclude this section by stating, without a proof, the fundamentalresult on microfunctions due to Kashiwara(see e.g. [103] for a sketch of the proof): Theorem2.4.2 The sheaf C of microfunctions is flabby.
120
CHAPTER 2.
MICROFUNCTIONS
Remark2.4.2 Before we begin a new section on microlocal operators, we will nowgive a more cohomological construction of the sheaf C of microfunctions: the reader will need to refer to our following chapters for someof the concepts we use here. Let us begin with the following diagram: 7S S~
+~ S~t (~n ~
S,~n
1 where :~S*SI~’~ is a half of the fiber product of S~~ and S*~~ over ~ defined
by
~s*S~n~={(x,~,~): <~,~)> 0},
~ = x+i~oe, f/= x+ir/0, and where(~~ is the disjoint union(~~ ~, = (ff~\~)uS~’~ a blow-upin(F’~ along/~’~. Then~--10 is a sheaf over S~’~, and S~t’~ is purely 1codimensionalfor ~--10, i.e. ?-/}~n~(r-~O)= 0 for j ¢ 1, (see [123], Proposition 2.1.1). Movingclockwise in the above diagram, ~r-17/~n,(7-10) is a sheaf over ~. Then the sheaf 7r-~Tt~s~(T-~O)is purely (n -- 1)-codimensionalf~r
~S*S1R
~ -+ S*IR~, that is R~-,(r-~7{~s~n~(~--~O))= 0 for j ¢ n- 1, the mapT : ~S*S~ (see [123], Proposition 2.1.2’). Theabstract definition of C is therefore
where the symbol RJT, denotes the higher direct image functor. Fromthis observation, one mayexpect the sheaf C to be somehowthe sheaf of "cotangential hyperfunctions". Thenthe definition should be 7/}.~n~ (~r-~O). As we will see, the abovedefinition of C coincides with 71}.~, (~r-~O). To verify th~s, we apply Leray spectral sequence to the projectmn mapS*~~ x~ --~T . ~, respectively, we consider the That is, for ~r and V*open sets in (/)~ and S*N induced Leray spectral sequence of relative cohomology E~’q =H~, (~], Rq~r,(71 "-1
(T-10)))
abutting to H~t~(~ x V*, rc-1 (~--~ (9)). Since V*can be taken to be contractible, HP(V*, ~’-~0) = 0 for p # O.
MICROFUNCTIONS OF SEVERAL VARIABLES
121
for q # 0 and
Therefore the above E~’q-term vanishes for q ~ 0. Namely, E~,°
=
Furthermore, the purely 1-codimensionality of S/R~ for T-10 implies the following isomorphism:
~-’n’~o(~--’o) ~
-’ -’
’ (~--1(-,--’0)).
In order to obtain the desired isomorphism, we need to compute the higher direct image
Since, from Remark3.2.1, we have
-2’~-~ E[’~ -~ e~+2’°= o, 0 : E~ we deduce that the abutment RP+~(~-,7~°~s.s~.)(r-~(T-~O)) is isomorphic to E~’ ~ E~p4. As we noted, ~-~T{ls~,(T-~O) is purely (n- 1)-codimensional for ~-. Hencewe need to consider only p = n - 1 in the above E~:-term. Then the abutment becomes
m(~-.n~s.~,~o)(~-l(:’o)). Since ~S*SEt"is ~--~(S*K/"), we have
R"(.r,~°~s.s~.)(~-~b--’O)) R"(n~.~,:-,)(~--~(~--’O)), where (T’-I(T-10))
~ (T-I(TI’-Io)).
Consequently, we have
obtaining the isomorphism
c = ~-~,(:~n~o(:~v)) ~ n~.~o(~-lv).
122
2.5
CHAPTER 2.
Microlocal
MICROFUNCTIONS
Operators
In Chapter I we have briefly discussed infinite order differential operators, and their action on the sheaf of hyperfunctions. As we have mentionedin our historical notes, the study of infinite order differential operators (and their inversion) is a classical attempt to reduce analysis to algebra and to create somesort of operational calculus. As we shall see in our subsequent chapters the reason for the introduction of infinite order differential operators is even stronger whenone deals with the classification of systems of differential equations. In this section we will go beyondinfinite order differential operators and we will discuss a larger class of operators, namely those operators which act as endomorphismson the sheaf C of microfunction; such operators will be called microlocal operators. The reader whohas followed us up to nowwill have no difficulty in recognizing (under this new name)the pseudo-differential operators discussed earlier on in this chapter. As usual, we begin by treating the one-variable case, where things are moreexplicit. Wewill then moveon to case of several variables. In Chapter I we have seen that if we denote by D the differentiation operator d then it is possible to considerinfinite series of the form ~ ak(x)D~, a~: E .,4 k=0
as long as lim gsup ]a~(z)lk! = k--)cx)
VzEK
on any compact K, where ak(Z) are holomorphic extensions of ak(x). The consideration of these operators has already showed howmuchricher is the theory of hyperfunctions in comparison with the theory of distributions. Our next step consists in defining and studying negative powersof D. Let us remind the reader, here, that we have shown, essentially in Chapter I, that infinite order linear differential operators with real analytic coefficients preserve microanalyticity, and they therefore define not only an endomorphism of the sheaf B but also an endomorphism of the sheaf C. Moreover,these operators are defined on an object If] in C by considering their action on a hyperfunction f whichrepresents If], and, finally, their action on f is defined by having the operator act on a holomorphicrepresentative F of f. Wewill work in a similar way to define negative powersof D. Let U be an open set in the complexplane, and let F E O(U); then we can define D-1F(z) := F(z)dz
2.5.
MICROLOCAL OPERATORS
123
where a E U, and the definition of D-1F(z) does not depend on the choice of the path ~/connectinga to z, as long as U is simply connectedand "~ c_ U. This definition readily extends to hyperfunctions:let f (x) = F+(x + iO) -F_ (xbe a hyperfunction which is real analytic in a neighborhoodof a point a. Then one can define
where "~+(z) are paths connecting a and z and contained, respectively, in the upper and lower half-plane (with the exception of a). Note that D-~f is therefore the hyperfunction defined as the difference of the two equivalence classes on the right hand side of (2.5.1). Note that the necessity of requesting analyticity at a point a stems from the wayin which we havedefinedthe integrals of hyperfunctions.Moregenerally, if a is singular for f we can alwaysuse the flabbiness of the sheaf of hyperfunctions(see Proposition 1.3.2) and decompose with supp(f~) ~ (x : x S a} and supp(f~) ~ (x : x ~ a}. This methodallows to define an operator D-~ which is defined (at the level of germs)on B modulo A, so that we have B B Suchan operator, on the other hand, is perfectly well defined on the sheaf of microfunctions, since the sheaf C already incorporates the quotient by ~, and 80
D-~:C~C act ~ a sheaf homomorphism. Oneis therefore naturally lead to the consideration of formal operators such ~ +~ k=l
and to the conditions which one must impose on the sequence {b~(z)} of holomorphic functions for such operators to be well defined. In complete analogy with whatwe provedin Chapter I for the case of (positive) infinite order differential operators, we can showthe following result: Proposition 2.5.1 Consider the formal operator +~
Q(z, D) := ~ -~ b~(z)D k=l
where the coe~cients b~(z) are holomorphic in an open set U of the complex plane. Then the following two conditions are equivalent:
124
CHAPTER 2.
MICROFUNCTIONS
(i) for any Zo E U, there exists 5 > 0 such that for any F ~ O(V), with V convexset such that Zo ~ U C_{Iz - zol <_5}, the series ~ bk (z)D-kF(z) k=l
converges locally uniformly to an element in O(V); (ii) for every compactset K C_ (2.5.2)
lim supk_,~~/SUpzegIbk (z)Ik! < +oc.
Proof. Just an application of Cauchy’sformula as in Proposition 1.2.2. Remark2.5.1 What greatly distinguishes Proposition 2.5.1 from Proposition 1.2.2 (and operators such as Q(z, D) from the usual infinite orders differential operators) is the fact that +c~
kP(z, D) := ~ ak(Z)D k=O
acts as a sheaf homomorphism on the sheaf O and therefore, as a consequence, on the various sheaves ,4, B, C. This is absolutely not the case with an operator such as Q(z, D). In particular, evenif F were holomorphicon all of U, the series ~b~(z)D-aF(z) wouldstill not be necessarily convergentfor z ¢ z0. As we will see in a moment, however, Q(z, D) still defines a sheaf homomorphism on B/J( and on C and the interest of inverting differential operators is therefore one morereason for introducing the notion of microfunctions. Definition 2.5.1 A pseudo-differential operator is a formal series
operator,
or microdifferential
~ Q(z, D) := ~ b,(z)D -oo
where {b~(z)}:~ satisfies (2.5.2) and {bk(Z)}~ satisfies is the formal powerseries
(1.2.6). symbol
Q(z,¢) := bk(z)¢ Finally, if ba(z) =- 0 for k > 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’
130
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, <
Wedenote by g~)(U) the totality of sequences {p~+j(z, ~)} satisfying (i), (ii), (iii) and we call P(z, D) := ~-~p~+j(z, D) a microditferential
operator on U. WhenU ranges over a cofinal covering
ofCn×¢n,originatesa sheaf, denotedby andcalled the sheafof microdifferential operators of infinite order. Wedenote by ~( ~) the sheaf of operators for which p~+j = 0 if j > 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<0 -
where F(a) = fo+~ e-tt~-ldt is Euler’s gammafunction. Then a standard computation (see e.g. [103], [123]) showsthat the kernel K(x,y) of a differential operator can be rewritten as (2.5.~)
K(x, ~) = P(x, D)a(x - ~)
The question, now,is whether the conditions of Definition 2.8.4 suffice to guarantee somesort of convergencefor the right-hand side of (2.5.~) if the pj’s are replaced by the holomorphicfunctions of Definition 2.5.4. It can actually be proved that this is the case (see Proposition 4.1.1 in [123]) and so we can show the followingresult:
2.5.
MICROLOCAL OPERATORS
133
Theorem2.5.5 Let P(z, D) be a microdifferential operator in $oo over an open n such thatif(z,~) ¯ U andc > 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+;
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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) <_r}, then one can find, for every r, a sequence {~,~} of compactly supported infinitely differentiable functions, such that: (i) for .every m, ~m = 1 on K and supp(~m)C_ K~; (ii) for every m and all multi-indices a of length boundedby m, it
o" c.(c~_~),.,. sup a~9~"~ I < The interest of this lemma,originally proved in HSrmander’s[85], to which we refer the reader for the proof, is that it provides a family of cut-off functions with appropriate bounds on their derivatives. Weare nowready to prove the fundamentalresult for the construction of the analytic wavefront set:
2.6.
HISTORICAL NOTES
143
Proposition 2.6.1 Let f be a distribution on an open set U in J~’~, and let xo be a point of U. Thenf is real analytic in a neighborhoodof Xo if and only if there is some boundedsequence {f,~} in ~,(j~n), and an open neighborhood of Xo such that the following conditions hold: (i) fm coincides with f in Uofor all values of (ii) there is a constant C such that, for all m andall ~ in J~n, it
1+ I 1) IL,()I _<(c)’n( Proof. Let us first assumethat f is real analytic in a ball B(x0, 5). Set t 5/3, K = B(xo, t) and apply Lemma 2.6.2 (with r = t), to construct a sequence {~,~} satisfying the conditions of the Lemma.Nowset frn = ~m"f. It is immediateto verify that f,~ = f in B(xo, t) and that {f,~} is a boundedsequence in $’(~). Moreover,for a given m and a multi-index c~, In[ < m, we have
Since f is real analytic in B(zo,5), and 2t < 5 we use Lemma2.6.1 to find constants C1, .Till such that for any ~ and any x in B(xo, 2r), ID~f(x)l <_ ~+’~. g~(C~l,~lp Since supp(f~n) C_ B(x0, 2r), and since sup [Da~.~(x)[ C.(Cn + m~l"l Va wit h [a[ < m, we can use the expression for D’~fmto find a constant C2 such that sup
Iv~f.d <_c2(c2m)l~’va with lal _<m.
(Note, in particular, that if we assume [a[ _< m the expression [al ’m can be included into C2). Standard Fourier transform properties showthat ~/,~(~) = f e-~’~(D~f,~)(x)dx, but since f,~ is compactlysupported in B(xo, 2t) we directly obtain
I}.~(~)l< ca(c,m)"(1I, ,I) -’~, m= 1,2,. .., with C3 a new constant.
144
CHAPTER 2.
MICROFUNCTIONS
Vice versa, if we assumethe existence of a boundedsequence{f,~} as stated in the Proposition then, in Uo, D~f(x) = (27r)-" / ~of,~(~)ei~:’¢d~. The finiteness of the integral on the right hand side, together with the characterization given by Lemma 2.6.1, nowshowsthat f is real analytic near x0.
The reader will have noticed that the .condition on the boundednessof the sequence{f,~} is crucial for the proof. This result is clearly the counterpart of the classical Paley-Wienertheoremand allows us to provide a natural definition for the analytic wavefront set of a distribution; as in the cases examinedearlier in this chapter, one maydefine such an object on the cotangent bundle of U or, and this is what we will do in this case, on its cosphere bundle: Definition 2.6.1 Let f be a distribution in an open set U of ~’~; then its analytic wave front set WFA(f)is the closed subset of U × iS~o-1 whosecomplement consists of those points (x0, i~0o~)satisfying the followingcondition: There exist an open neighborhood~o of Xo, an open conic neighborhood F of 4o and a boundedsequence {f,~} of compactly supported distributions such that f,,~ = f in ~o, and
Iim()l
+I 1)
for all ~ in F and some positive constant C, which is independentof ?Tt.
Remark2.6.6 It is an immediateconsequenceof the previous Proposition that a point (x0, ~0) belongs to the complement of WFA(f)if and only if there exists some t > 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)] <_C]y] for all point in 12+iF~, and let b(f) denote its boundaryvalue (which is actually a distribution, not just a hyperfuuction). If ~o in ~-~ satisfies ( y, ~ o) <0 fo some y in F, then the point (x, i~0~) is outside WFA(b(f))for all x in Proof. Let x0 E ~ and let t > 0 be such that B(x0, 2t) _C ~. Set, as before, K = B(xo, t) and construct a sequence (~m}as in Lemma2.6.2. Wenowdefine a sequenceof "almost analytic" functions as follows: for any mwe set X(~m) = g,~(x
+ iy) := ~
(x) a!
)al<m \
Clearly, the functions gmare not analytic but they do satisfy somekind of "limit" Cauchy-Riemannequations as y ~ 0, since one can easily prove the existence
146
CHAPTER 2.
MICROFUNCTIONS
of a positive constant C = C(n) such that, for z = x + iy, IO(grn) (z)102
< C(Clyl)rn-1 m!
(Cm)m t’~
m-l _<,cl(c, bl)
with a suitable choice for C1 > 0. ChoosenowYo E F~ satisfying (Y0, ~0} < 0, and let us estimate the Fourier transformof qom"b(f) along the ray {s(0 : s > 0}. If m _> k + 1 we obtain (~.~b~(f))(S~o) li y~O mf (X(e-’8~’¢°~.~)(x +iy)) f( x + J If we set y = tyo, so that t --~ 0 wheny -~ 0, we have (in view of the existence of b(f) in the sense of distributions) that
with Ul(s)=fe-iS~’~°+s~°’~°(g,~(x+iyo))
.f(x+iyo)dx
and U2(s)f /x,r)-*z+iryo f(z)e-iZ’~°~g’~ Ad In order to complete the proof, we only need to estimate the growth of U~(s) and U2(s), to be able to apply Remark2.6.1. As for Ul(S), the boundednessof f(x + iyo) in f~, together with the estimates on the derivatives of p,~ and the definition of g~, yields IU~(s)l _< (c~)m(1+ -m, s) for a suitable positive constant C. As for Us(s) we knowthat for 0 < t < 1 it is Of (z +ityo)l
t
<_
on the other hand, the growth of f(x + ityo) is boundedby hypothesis and so we obtain the desired estimates. [] Proposition 2.6.3 For a distribution f on an open set U, it is W~A(f) C_ S.~,.(f).
Proof. This is the easy inclusion. To prove it we assume f to be a compactly supported distribution; this does not implies any loss of generality as every distribution is a locally finite sumof compactlysupporteddistributions to which we can apply the following arguments. Now,for a compactly supported distribution f we can always write f = ~ b(f~) as a finite sum where each f~
2.6.
HISTORICAL NOTES
147
is holomorphic in An + iF~. If we take (x,i~ooC) ¢ S.S.(f), one can choose the decomposition above in such a way that each cone F~ can be chosen so to contain a point ya such that (y~, ~0) is negative. Then, by Proposition 2.6.2, (x, i~ooC) ~ WFA(b(f~)), for any a, and therefore (x, i~ooC) !g WFA(f), which concludes the proof. [] The inclusion of S.S.(f) into WFA(f) is, on the other hand, much more complicated, and it requires somedetailed analysis on the analytically uniform structure of specific spaces of distributions in the sense of Ehrenpreis [52]. We first need to introduce an auxiliary space of compactlysupported infinitely differentiable functions: let F be an open cone in the frequency domain(i.e. the real space with variable ~), and fix somepair (c, (i) of positive constants; we define the space .T" as the space of all compactlysupported infinitely differentiable functions g whichsatisfy the followingconditions: (i) supp(g)C_{Ixl _< (ii) [.~(()[ _< (1 + [([)-2exp(d[Im¢]) if Re(¢) (iii)
[~(()[ _< (1 ÷ [~[)-~ exp(5[Im~[+ c[Re~[) if Re(C) Note that 5r is dense in :D; we will use this fact at a later stage; the crucial lemmaswhich we need are given in Liess’ [149]: Lemma2.6.3 There are absolute positive constants A, B, C (only depending on the dimensionof the space) such that if q(~) is a non-negative real valued function whichis Lipschitz continuouswith normless than or equal to one, then there exists a plurisubharmonicfunction Q(~) in n which is Lipschitz continuous with normboundedby C and which satisfies on all of (Tn, the following chain of inequalities: q(~) _< Q(~+ iT) _< 2q(~) + A[~?[ Lemraa2.6.4 Consider a pair of cones F CCF~, and let J: be defined as above, with respect to the cone F. Then there exist positive constants D, E, m which only dependon n and on the pair (F, F’), such that every g in jz can be written as a finite sumof distributions supportedin {Ixl <_5 +E¢} g = gl + g: +"" +gs + h, wherethe distributions satisfy the following Fourier conditions: (i) [gj(~)[ _< Dj-e(1 + [~[)’~exp(2je) j/4 < [~1 < 4jand ~¯ F’ ; (ii) I~J(~)] <- DJ-~(1 + ]~[)’~ otherwise;
(iii) Ih(~)l<_D(~+s)-~(~~ for all ~.
CHAPTER 2.
148
Weare nowready to prove the equality of S.S.(f)
MICROFUNCTIONS
and WFA(f).
Theorem2.6.4 Let f be a distribution defined on an open set U of 1R~. Then S.S.(f)
= WFA(f).
Proof. Wehave already proved that WFA(f) C_ S.S.(f), so we only need to prove the reverse inequality. Todo so, let us pick a point (x0, i~0cx3) whichdoes not belong to WFA(f);we will prove that it is also outside S.S.(f). Withno loss of generality we assume x0 to be the origin and take f E 8,(~n) (just multiply f by a suitable cut-off function). Nowlet us replace f by a suitable distribution g whoseFourier transform is /(~)(1
-s, 1~12)
where s is chosen so large enoughso that
I#ff)l-
0 a suitable constant. Notice that the location of both S.S.(f) and WFA(f)are not changed as we go from f to g (just recall the definition of these sets in terms of Fourier transform). Wenowtake the compact B(xo, r) and accordingly construct a sequence of cut-off functions {~j} such that for suitable constants C1, C2 > 0, and a suitable cone -F containing ~0 one has: ](~jg)(5)l -< C2(1+ I~l) I(qo/g)(~)l
CI( CIj)J(1I1) + ~jg = g
1
-r;
in B(x0,r).
Choose now an open convex cone F0 CC F such that -~o ~ F0, and apply Lemma2.6.4 to the pair (F0, F) with 5 = r/2 and e small enough. It is not hard to see that for all functions p E ~ one has I(g,P)l g M, for somepositive constant M(this is essentially a consequenceof the estimates in Lemma 2.6.4). By the Hahn-Banachtheorem, and this last estimate, one now deduces the existence of a complex-valuedBorel measure dA(ff) such that
(g,p) for all p ~ 5r. By the estimates which define the growthof the Fourier transform of the elements in ~-, we obtain (with X the characteristic function of F0) J
2.6.
HISTORICAL NOTES
149
Nowwe note that if Ixl < 5, lYl < ~ and (y,~> < 0 for all ~ ~ F0, one has I exp(-i(x iy). and therefore the function h(x + iy) := / exp(-i(x iy) . ~)d.~(~) belongs to O(~ + i(F~)~), where ~ := S(0, 5) and F~ := {y: (y, ~) < 0 ~ ¢ Fo}. Nowit is obvious that
for all p E ~- and so, by the density of $" in :D(t2), we deducethat g = b(h) in ~. Nowit is immediateto verify (since ~0 E -F0) that there exists yo E F~ such that
(yo,~o)<
which shows that (Xo, i~o~) ¢ S.S.(g) = S.S.(I), thus concluding the proof.
Chapter 3 -Modules 3.1
Introduction
In this Chapter we will introduce the notion of a :D-module and we will provide its first fundamental properties. Wewill then also microlocalize such a notion, as to introduce the notion of £-module. Since homological algebra is an important underlying notion for the theory of :D-modules, in this introduction we will provide a concise and self-contained review of the theory of derived categories and of spectral sequences for a double complex. In section 2, we discuss the foundations for :D-modules in the spirit of algebraic geometry. In section 3, we introduce the notion of a good filtration on q-module which allows us to define the characteristic variety of a :D-module. In particular, we prove that such a characteristic variety depends only upon the module structure and not on a particularly chosen good filtration. In the last section of this Chapter we prove Sato’s fundamental theorem for the category of S-modules. The reader who is familiar with the theory of derived categories may skip the remainder of this section and move directly to Section 3.2. Let ,4 be an abelian category. Let X" be a (cochain) complex in ,4, i.e. X° = {Xn, d~}ne~ is a collection such that each Xn is an object in A and each d~ : X~ -~ X~+1 is a map in ,4 satisfying o~x+1 o d~ = 0. A map f" from X" to another (cochain) complex Y’, f" : X" --~ Y’, is a sequence f" = {f~}n~z~ maps f~ : Xn -~ Y~ such that the following diagram commutes: n,. X
lfn
d~
d~
, yn 151
. xn+~
,,...
[fn+l . yn+l
. ...
152
CHAPTER 3. :D-MODULES
i.e. we have fa+~ ~ d~ = d,~ o f’~ for each n ¯ N. Let Co(A) be the abelian category of complexesin ‘4 with mapsas defined in the above. Define the subcategory Co+(A) of "bounded below complexes" of Co(A) as follows: an object X" of Co(A) belongs to Co+(‘4) if X’~ = 0 for n < < 0. One similarly defines the subcategory Co-(.4) of "boundedabove complexes"of Co(.4). Let Cob(.4) be the subcategory of Co(.4) whose objects are boundedbelow and above, i.e., Co~(.4) = Co+(.4) rq Co-(.A). Mapsf" and g" in Co(.4) are said to be homotopicif there exists a sequence s" = {sn}nez: of mapssn : Xn --~ y,~-i such that fn _ g~ = d~-~ o sn + s"+~ o ~z as shownin the following diagram:
d~-1 ,.
Xn-1
, Xn
,.
yn-1
,.
d~
yn
,.
Xn+l
,.
yn+l
,, ...
Wewill write f" ~ g° to denote that f" and g° are homotopic. Notice that "~" is an equivalence relation. This relation allows us to construct a new category K(.4) whose objects are those of Co(.4), but whose maps are equivalence classes of maps under the homotopyequivalence relation. So, for any X" and Y" in Co(.4), it is ~ornl((.~)( X’, Y’) =7~ornco(.~) ( X’, Y’) This in particular meansthat a mapf" in Co(A) is the zero mapin K(A) if f" is homotopicto the zero mapin Co(A). Since ‘4 is an abelian category, one can well-define the cohomologyfunctor from K(A) to A. Wewill nowdefine the derived category of .4 as a localization of K(‘4). In order to do so, we define the notion of quasi-isomorphism as follows. A map ") f" : X" -+ Y" in K(.4) is said to be a quasi-isomorphismif and only if H’~(f is an isomorphismfor all n. Wedenote the totality of quasi-isomorphisms by (QIS). Thenthe derived category of .4, denoted as D(‘4), is the localization K(A) at (QIS) D(A) := K(A)(QIs), The localization functor Q from K(A) to K(‘4)(QIS) has the usual universal property whichallows us to characterize the process of localization. If a functor Q’ carries quasi-isomorphismsin K(‘4) into isomorphismsin a category D’(.4),
3.1.
INTRODUCTION
153
then there exists a unique functor U from D(A) K(A)(QIs) to D’(A) suc that Q~factors through D(~4). Namely,the following diagramof categories and functors commute,Q~ = U o Q: t:(A)
¯ D’(A)
D(A) Wecan also provide a more explicit construction for the derived category D(A). The objects in D(A)are those Co(A) while a m apin D(A)from,say, X"to Y"is a suitable equivalenceclass of pairs (f’, s’), wheref" : X"--~ Y~" a mapand s" : Y° --~ Y~"is a quasi-isomorphism,and X’, Y" and Y~"are objects of D(A). The equivalence relation between maps(f’, s’) and (g’, t’) from to Y" is defined as follows: given the diagram °X
y°
we say that (f’, °) , ~ (g’, t ’) i f t here exists a n object Ya" with quasi-isomorphismsr" and u" satisfying r" o s" = u" o t" and r" o f" = u" o g" as indicated in the diagram below. ¯ X
y°
Remark3.1.1 This is nothing but the categorical version of the familiar arithmetic property in a localized ring : f/s = g/t holds if there is a common denominator rs = ut such that f /s = r firs = ug/ut = g/t.
154
CHAPTER 3. :D-MODULES
Remark3.1.2 A mapfrom X° to Y° in the derived category D(‘4), therefore, maybe characterized precisely as the direct limit: °Z
y°
HOmD(A) (X°, Y’)= l~_~ y’°
Then the cohomology functor H" induced a map from H"(X’) to Ha(Y°): ") ~-1 o Hn(f H"(X’)
" H"(Y’)
Since spectral sequences associated with double complexesare important for what follows, we will describe these concisely here (see also Remark3.2.1 below). A double complexin an abelian category ‘4 is a family of objects and mapsin the category ,4
such that
~, cp,q+ l
(o,1) ~. cp.q
;
,. Cp+l.q
.l
3.1.
INTRODUCTION
155
diagram,and such that t.(1,0~4P+l,q ) o d~(~?0) : 0 and hold. Then let Cn = (~p+q=nv,q and define dn :Cn~ C’~+1 by
is
a commutative
d~,,q+lo d~(~l) (0,1)
p’q=(- I ~p~Ip’q‘j ~(1,0) -4- (-1)qd~((~?l.)., d’~[c It is immediateto verify that (C’, d*) is a complex.Wecan define filtrations C" as follows: Fv C’~ = (~ ¢ ’ ~ p’ +q=n and ’ F~C’~ =
q’ ~’. ~ C q+p’=n pt~_p
Fromthose filtrations,
the following spectral sequencesare induced S~,q : Cp,q,
E~’q --(0,1)
Hq(cP’*),
’ E~,q q, : pC
’E~ ’q *’p) --(1,0)Hq(C
’q =(1,o)HP((o,~)Hq(C**)), EP2 ’E~’q =(o,1)H~((1,o)Hq(C**)) which abut to E~ = Hn(C*), the n-th cohomologyof the associated complex C" defined above. For example, let us consider a double complex(cP’q)p,q~ in the first quadrant. Namely,C~,q -- 0 unless p > 0 and q > 0. Assumethat the double complex is vertically exact, i.e. ’q --(o,D Hq(Cp’*) EP~ -- 0 for q > 0. Then the complex E~’° is quasi-isomorphic to C°. This is because we have ’°) and E~’° =(1,o)H’~(E~
+~’-1= 0. 0 = E~-~’~-~ E~’° -~ E~ Therefore, we have the isomorphismsamongtheir cohomologies: E~’° ~ E~’° ,~... E~° ~ E’~ = H’~(C°). Werefer the reader interested in moredetails to [35].
156
3.2
CHAPTER 3. :D-MODULES
Algebraic sis
Geometry and Algebraic
Analy-
In this section we will give the reader a first taste of algebraic analysis. Our goal here is to introduce the reader to somebasic material, as well as to show the philosophical links whichexist betweenalgebraic analysis and algebraic geometry. Wewill, therefore, begin with somevery basics notion from classical algebraic geometry. One of the most fundamentalobjects to be studied in algebraic geometryis the zero set of a system given by finitely manyrelations amongfinitely many generators for an algebra B over a field A. Let us study this in somedetail. Let xl,x2,... ,x,~ be a set of generators for the algebra B over A. Then if we consider the polynomial ring A[X1,...,X,~] in n variables, we have an exact sequence A[X~,X2,. .., X,~] --% B ---+ O, wherethe epimorphism7~ is defined by ~(Xi) = xi for i = 1, 2,..., n. The kernel of the map~, denoted as ker ~, is an ideal in A[XI,X2,...X,~]. Since A is a field (hence noetherian), the Hilbert Basis Theoremimplies that the ideal ker is finitelygenerated. Letfl (Xl, . . . , X,~),f2(X~,. . . , X,~),. . . , fe(Z~,. . . , be a set of generators for ker ~. Then we can write the A-algebra B as follows: B = A[x~, x2,..., x,~], wherethe generators x~, x2,..., x~ satisfy the polynomialrelations
(3.2.1)
(P.R.)
fl(zl,..., x~)= f~(xl,...,z~)=
{
fdz~,...,z~) =
That is, we have the isomorphism A[X,,...,
X,~]/(fl,...,
ft) ~ B = A[xb..., x,~].
Let nowK~ be an extension field of A. Let us try to find a solution in K~ for the system(3.2.1); this is equivalent to finding a K~-rational point (xl,..., on the algebraic variety consisting of the common zeros of the system (3.2.1), and this is a process whichcan be described as follows. To find a K’-rational point on the variety embeddedin the Euclidean space K~ × ... × K~ means to find a map¯ from B to K~ so that the diagram
3.2.
ALGEBRAIC GEOMETRYAND ALGEBRAIC ANALYSIS
157
B
~ .K
A c
commutes,or, from the point of view of affine schemes(see e.g. Hartshorne[74] for the notions and notations in algebraic geometry). Spec(B)
A In terms of the relations (3.2.1), the above A-algebra homomorphism ¯ operates as follows:
¯ (f,(xl,..., xn))=f~ (~(zl),...~(x~))=~’~(c~,..., wherecj is the assigned value in K’, i.e., c~ = ~(xj) for j = 1, 2,..., n. This is exactly what a solution of the system (3.2.1) means. In algebraic analysis, the abovecoefficient field A is replaced by the sheaf :D of non-commutativerings of differential operators over X = d7~, whoseprecise definition will be given in the next section. Wealso replace the A-algebra B by a :D-ModuleA/t. Note that we write ":D-Modules"instead of "sheaves of germs of :D-modules"since no confusion can arise. Then a system of linear partial differential equations will be described as a :D-ModuleM. Let Mbe finitely generatedover :D, and let ul, u2,. ¯ ¯, u,~ be generators for Mover T~. Then we have an exact sequence of :D-Modules :P’~ ~-~ M-~ 0, where the augmentationmap-u from the direct sum :D"~ to Jr4 is defined by (A~U~~ ... @A,~U,~) ¯ u = A~ul +... + A,~u,~, i whereU~= [0,..., 0, 1,0,..., 0], i = t,..., rn, are the elementsof the canonical basis for the free Module:D"~. When:D is left noetherian, the submoduleker -u of :DTM is finitely generatedover :D. Hence,for a set of generators vl, v2,..., ve for the submoduleker .u, there exists an epimorphism :De -~ ker u ~ 0, where each vj can be written as
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CHAPTER 3.
:D-MODULES
vj =PjlUt+Pj2U2 +... + Then the map¯v is given as (Bt Vt @... (9 BeVy)¯ v = B~vt +... + Bevy, k where the vectors V~ = [0,...,0, 1,0,...,0],k = 1,...,~, form a canonical basis for the free Module:De. Therefore, we obtaii~ the following commutative diagram:
ker ¯~
Notethat the abovemapP from :D~ to :D’~ is the compositionof the inclusion ker ou ~i :Din and the epimorphism¯v : :De --~ ker ¯ u. Then observe that for e element IBm,B2,..., Be] of :D any ([Bt,...,Be]
¯ P) ¯ u = ([Bt,...,Be]v)i)u
= O,
since the image of v is contained in ker u. Then the :D-Modulehomomorphism P maybe written as an ~ x m-matrix[P~,j] with entries in :D:
[P~,#]= P~ P~ ... L Pet P~2 ...
P~,~ = v~ Pem-I
~
By writing the augmentation map ou in vector form
the composition mapP o u = 0 becomesa system of :Dx -linear relations among the generators ut,..., Umfor the :D-Module
3.2.
(3.2.2)
ALGEBRAIC GEOMETRY AND ALGEBRAIC ANALYSIS
159
PllUl -~- P12u2-~... -~- PlmUm =0 P21Ul-[- P22u2-[- ¯ ¯ ¯ ~- P2mUm =0 (D.R.) P~lUl +
Pt~u2 +... + P~.~u,~ = 0
that is, a systemof partial differential equations. Furthermore, the exact sequence
can be extended as
obtaining a free (projective) resolution of A4 . One maylike to write this resolution of A4as follows: ....
g ~’~oQ ~:D e °P g~’~------0
....
0
-O~M~O
....
....
regarding the two row sequencesas quasi-isomorphicobjects in a suitable derived category. It is important to regard a systemlike (3.2.2) as a free resolution of ModuleA4. That is, the systems equivalent to (3.2.2), i.e., isomorphicsystems as/)-Modules, provide various free resolutions for Let now(9 be the sheaf of germs of holomorphicfunctions over X. Then local solution of the W-Module A4 in (9 is a :D-homomorphism f from A4to (9. Namely,f assigns each generator u~ to a holomorphicfunction f(u~) = f~ for i = 1,2,...,n. Then, given the :D-homomorphism f, the system (3.2.2)
160
CHAPTER 3.
:D-MODULES
implies ~ f(P~juj) = ~_, P~jf(uj) = ~_, Pi~fj indicating that (fl, ¯ ¯., fn) is a holomorphicsolution in the mostorthodoxsense for the system of partial differential equations. Since we denote the sheaf of 79-homomorphismsfrom A/I to 0 by 7-lomv(JVl, O) we now can see that the sheaf 7-lomv(A/l, (9) is in fact the sheaf of holomorphicsolutions to the system A//. Later on in this chapter, we will consider the higher derived functors ~xthv(A/l, (,9) of 7-lomv(A~,O) and we will provide an analytic interpretation for them. This short discussion should have been sufficient to justify the strong links which exist between algebraic geometry and algebraic analysis. Such a correspondence can be quickly summarizedas follows:
Algebraic Geometry 1) A schemeover a base ring 2) A map between schemes X-~ Y 3) To find K-rational points on a scheme X over A is to construct an Aalgebra homomorphismfrom K to X such that X
commutes. Algebraic Analysis 1’) An analytic D-module
3.2.
ALGEBRAIC GEOMETRY AND ALGEBRAIC ANALYSIS
161
2’) A :D-linear map between iD-Modules
3’) To find solutions of a systemof partial differential equations in a sheaf S of certain functions, i.e., a "known":D-Module,is to construct a :D-linear map from A4 to S, namely an element of ~om~(A~,S). Weare nowready to give somesignificant examplesof iD-Modules. Example 3.2.1 The sheaf (.9 is a :D-Module, called de RhamModule. For germs P = ~,~ f,~(z)O c’ in :D and h(z) 6 O, the structure of the ~D-moduleis given by Ph(z) -= ~,~ fc,(z)O’~h(z) where0~ = ( 0/0z, )~l... (0/0zn a~. Let us nowgive a projective resolution for the de RhamModuleO . Since one can easily see that
o~ ~/’z:,o~+... + :Do,,, where0i = (O/Ozi), i -= 1, 2,..., n, weobtain :D~o~[_~,~]~ ~ O -~* O, wherec3[~,1] =- [01, 02,..., 0~]t. As a systemof relations, we have (3.2.3)
&u= 0 for i = 1,2,...,n.
As we will see in the section on the de RhamFunctor in the next chapter, the de Rhamfunctor ~omv(O,-) evaluated at O can be computed as follows:
R’:/-tomv(O,O),~ ~" ~ 0 ~ ~’, wheref~° is the complexof sheaves of differential forms whosecoefficients are holomorphic functions. Then the Poincar~ lemmastates that 7~q(f~") = 0 for ") ..~(A The Poincar~ lemmaimplies simply that ¢ and fl° are q ¢ 0, and "H°(~2 quasi-isomorphic. Hence, it is clear that the sheaf 7-lomv(O, O) of germs of holomorphicsolutions is the sheaf(T. Hence the derived functor ~7-lomv(O, O) ~. ~° is a resolution of~T. The left exact ~unctor 7tomv(-, O), on the other hand, takes the above projective resolution for O
162
CHAPTER 3. O---* 7-lomv(O, O)--~-*-7-lomv(/),
0
"¢
D-MODULES
/)[ ~"q ?-l omv(/)n, O)
n "O
" O
The second and third vertical isomorphismsin this diagram are induced by the :D-linear mapswhich assign the canonical generators to the elements of (9. The/)-Module (9 is a holonomic/)-Module, in the sense of Chapter 5. maximallyoverdeterminedsystem is the older terminology for a holonomicsystem or a holonomic/)-Module.) Example 3.2.2 Another extreme case of a /)-Module contrasting the /)ModuleO, is the/)-Module/) itself. That is to say the/)-module for which there are no relations. Then, since there is no restriction, the sheaf of local solutions of the /)-Module /) in the sheaf (9, will be the whole (9. This what the canonical isomorphism7tomv(/), (9) _E+means, i.e ., eac h ele ment f e 7-lornv(/), O) is mappedto f(1) e In general, for a projective resolution
of a/)-ModuleM, the left exact contravariant functor 7torav(-, (9) induces the following complex: (3.2.4)
0 -~ 7-lomv(M, O) -~ 0"* ~ t Q-~ O n ~.. .,
where ~ o ~ = nomv(Q,~) o ~omv(P,~) = ~om~(P o Q,~) Then observe that Homg(M,O) ker~ is the solu tion shea f of t he part ial differential operators ~u = 0. Therefore, the 0-th cohomologyof the complex (3.2.4) is isomorphic to the solution sheaf ~om~(M,0), and the first cohomology of (3.2.4), i.e. ker~/im#, is the first derived functor $xt~(M,O) -, O)(M). In the sense of derived categories, the complex (3.2.4) R~om~( cohomology is given by precisely ~om~ (M, 0), whose h-th
o))
def
o)
o).
For example, £xt~(M, O) = 0 means kerq/imP = O, namely if ~f = 0 for f ~ 0~, then there exists u in 0~ satisfying Pu = f. Whenthe sequence is exact, that is ~omv(-, O) is an exact functor, all the "higher" solution sheaves £xt~(M, O) vanish. Then there is an induced doubly indexed spectral sequence E~’~ with the property E~’q = HP(X,gxt~)(M, O)) = 0 for q ~
3.2.
ALGEBRAIC GEOMETRY AND ALGEBRAIC ANALYSIS
163
The sheaf of higher global solutions over X, Ext,(X, A4, O) can nowbe computed as the abutment from the above spectral sequence: 0 -- E~’1 ~ E~’° ~ E~+2’-1 = 0. Namely, E~’° = HP(X,~lomv(A/[,O)) ~- E~’° ~ ... ~ E~° -~ Ep, and ExtW(Z , A/I, O) -- HP(X,AZ*), for A/’* the complexin (3.2.4). Remark 3.2.1 Let us provide a little more background on the notion of abutment and on the relationships between spectral sequences and derived categories. Consider a doubly indexed spectral sequence in an abelian category ,4 i.e. a family of objects {E~,q, En;p, q, r, n E z~:} such that there exists a map -~+~ ~,q : E~,q ~ E~+r,q +~’q-~+~ ,q satisfying dPr o d~ = O. Furthermorethere exists an isomorphism ~’~ : ger~’q/Im~-~’~+~-~ -~ E~_~. An object E" is said to be the abutment of the spectral sequence when E" is n) ~ ... and these inclusions a filtered object: En D ... D FP(En) D F~+~(E satisfy F~(E")/F~+I(E") -~ q and E~= {~ E~q. pWq:n
The following spectral sequence, called the spectral sequence of a composite functor, is also useful. Let ,4 and B be abelian categories having enoughinjectives. Let F be a left exact functor from,4 to B and let G be a left exact functor from B to an abelian category ~. Weassume that F takes injective objects in A into G-acyclic objects in B, i.e., the higher derived functors RiG(F(I)) vanish for any injective object I in ‘4. This induces a doubly indexed cohomological spectral sequence in C beginning with E~,q = R~G(RqF(A)) such that the derived functor of the composite functor GoFEn = R~(GoF(A)) is its abutment. Onemight ask howthe theory of the spectral sequence of a compositefunctor can be rephrased in the language of derived categories. The answer is quite simple, as follows. Withthe sameassumptions given above, let X° be an object °) = 0 for sufficiently small k}, i.e. 0+(,4) in 0+(‘4) = {X° ~ 0(,4); H~(X K+(A)(om). Let I+(A) be a full subcategory of K+(A)such that every object in I+(A) is a complexof injective objects of ,4. Then for each object X° in K+(A)there exists a unique complexI ° in I(A) such that X° and I ° are quasiisomorphic. Namely,we have a functor
K(,4)
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CHAPTER 3. :D-MODULES
sending quasi-isomorphisms in K+(A)to isomorphisms in I+(A). By the universal mappingproperty of the localization, there exists a unique functor U such that the diagram
D+(‘4) commutes. Wealso have an embedding functor Q" from I+(A) to K+(A). Then its compositionwith the localization Q, Q o Q" is a functor from I+(,4) D+(A).Nowthe uniqueness of an object in I(A) corresponding to an object in K+(A),i.e., the definition of Q~, implies that Q o Q" and U are inverse to each other. That is I+(‘4) .~ D+(‘4). Let nowF be a left exact functor from ,4 to B; wecan define its right derived functor ~F from D+(‘4) to D+(B)by setting ~F(X°) = F(I°), ° ° where I = Q’X is an object in I(A). In order to compute the right derived functor of F o G, we note that (~G o j~lF)X ° = J~G(F(I°)). Since FIk is a G-cyclic object in B, we nowobtain that I~G(F(I°)) = G(F(I°)) = ° =l~l(G o F)°. Therefore we have shownthat the derived category version of the spectral sequence of the compositefunctor is simply ~RGo _~F = ~(G o F). As an application of this concept, we will nowdiscuss the notion of hyperderived functor (hypercohomology) of a left exact functor. Let F be a left exact functor from an abelian category ‘4 with enough injectives to another abelian category B. Then F induces a functor Co F from the abelian category of positively indexed complexesin Co+(‘4) of A to the abelian category of positively indexed complexes Co+(B) of B such that CoF(X°) °) = F(X ° + for any X E Co (‘4). Wetherefore obtain the following commutativediagram of categories and functors:
3.2.
ALGEBRAIC GEOMETRY AND ALGEBRAIC ANALYSIS ~-
Co+(~)
CoF
165
. Co+(~)
o 7~
o 7~
where 7~°(X") = Ker (do : X° -+ X1) is the zero-th cohomologyof X’. Then, for the left exact functor F, we can define a functor F° from Co+(J[) to B setting F°:=
~°oCoF
= = =
Ker (Fd° : FX° --+ FX’) F(Ker(d° : ° -+ X’)) °. Fo’H
Consequently, we obtain two spectral sequences ’E~,q _=RpF(~t-lq(x*)) and
E~’q p) = RqF(X ° of F. which abut to the hyperderived functor F=X Let us now consider a complex non-singular analytic variety V. Let Oy be the sheaf of germs of holomorphicfunctions on V, and let f~ be the sheaf of exterior algebra of O%differefitials on V. Since Poincar6 lemmaimplies that 7/q(f~) = 0 for q # 0, we see that the complexf~ is acyclic. Therefore, the spectral sequence corresponding to the above ’E~’q becomes ’E~’q = H’(V, 74q(f~.)) for any q # 0. The only non-vanishing term is given by ’E~,° = HP(V,~T), sincelT = Ker(d° : Oy -~ f~ ).
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CHAPTER 3. :D-MODULES
Consider nowthe following sequence: d~-2,1
0 = ’E~’-2’i ~2
0d~,
,E~,O ~2 *’E2~+2’°-2+1 0.
From it, we deduce
whichis isomorphicto the abutmentEp_- ~p+o=pEo ~v,o = E~V’°,i.e., ’E~,° = H~(V,(T) ~ = H~(V, f~, ). Furthermore, if V is Stein, the first spectral sequencebecomes E~’q = Ha(v, ~) = 0 for q ~ 0. Consider the sequence °, E~-l,
d~1-1,0
. °E~.
d~1,0
, E~+l,o
r(v,4)
. r(v,a
Then E~’° = Ker ~’°/Im ~-~,o is nothing but the p-th cohomologyof the de Rhamcomplexof V. Since wehave, as before, E~’° ~ Ea~’°~ ... ..~ E~’°, we can compute the hypercohomologyby HP(V, ~v) ~ ° ~ E~’° which is ° ~ E~ E~ H’(F(V, ~)). Summarizingthe above discussion, we have: H~(V~)
~=
~=
Hp(r(V,~N)
°,E~, Remark 3.2.2 Let A4’ be a finitely generated 7)-Module with generators v~,..., Vm,, i.e. M~= ~v~+...+:Dv~n,.Let *v : 7)"~’ --~ M~be the augmentation mapinduced by the generators. Choose.Q : De -~ :D’~’ in such a way that Ker(.v) = Im(.Q).
3.2.
ALGEBRAIC GEOMETRY AND ALGEBRAIC ANALYSIS
167
Then the map -v induces an isomorphism from the 0-th cohomology Dm’/Im(~Q)of the top row of the diagramto the 0-th cohomologyof Ad’ in the bottom row. If the D-ModuleA~’ is isomorphic to a D-ModuleA// determined by a projective resolution
then the complexes... ~ De’ _L~ Din’ _~ 0 --~ ... and... -~ :De -~ Dm -~ 0 --~ ... are exact, except at the 0-th spot. Their 0-th cohomologiesare moreover isomorphic to each other. This shows that the notion of a D-Moduleis an intrinsic characterization of a systemof partial differential equations. In the next chapter, a single D-Module Mwill be generalized to a complexof :D-ModulesA//° replacing a projective resolution by a quasi-isomorphic complex of objects. Example 3.2.3 Let M and M~ be cyclic 7P-Modules generated by u and v satisfying the ordinary differential equations ( x D- A )u = and ( x D-A-1)v = 0 respectively, where A # 1. This can be rewritten as M = Du (xD- ~)u = O, ~1, and M~ = Dv (zD-A-I)v=O, A-~I. In terms of free resolutions, we have
{
¯ (xD - A)
¯ D
oU
- M------’O
and ....
D
*(xD - ~ - 1)
"D
*v
~ M’----" O.
168
CHAPTER 3. :D-MODULES
Weclaim that the cyclic :D-Modules.M = :Du = :D/:D(xD- A) and M’ = :Dv = :D/:D(xD - A - 1) are isomorphic. First note that Dx - xD 1 holds in :D. Now,if u is a "solution" of the ordinary differential equation (xD - A)u 0, put v = xu so to have (xD - A - 1)v = (xD - ~ - 1)xu = (xDx - Ax -
= (x(xD 1) - A x- x)u = (x 2D+ x- Ax- x )u = x(xDA)u = 0 On the other hand
Dv = Dxu = (xD+ l)u = (xD-A+ A+ I)u = (xD-A)u+(A+ (A+l )u Let now 1 D v. Then notice that we obtain 1 1 xu = x-~-~ Dv = ~-~--~xDv
1 A-i-I
and also 1 1 1 A + ~Dxu = -~-~(xD + 1)u = -~--~(xD - A + A + 1)u 1 A + 1 ((xD A)u + (A+ 1)u) = u Therefore, we obtain an isomorphism between ~4 and Mr given by VI:D(xD - A),
. :DI:D(xD- A - 1)
where - indicates the equivalence class moduloan ideal.
3.3.
FILTRATIONS AND CHARACTERISTIC VARIETIES
3.3
Filtrations
169
and Characteristic Varieties
Let X be a complex manifold of dimension n. Wewill denote by 7) the sheaf of germsof linear partial differential operators with holomorphiccoefficients of finite order over X. Thestalk 7)~ at z consists of all finite sums (3.3.1)
c’,aa(z)a P(z, O) y]~ ot
where a = (el,..., an) takes on a finite numberof n-tuples with ai _> 0, i 1,..., n, 0° = (O/Ozl)’~... (O/Oz,~)’~., and the coefficients a,~(z) are germsof holomorphicfunctions at z. Then7) has locally the followingincreasing filtration (9 = 7)(0) C 7)(1) C ... C 7)(~) ~ ... ~ 7), 7)(-1) where the O-Submodule 7)(~) is defined at each stalk V~k) = {P(z,O) e V~ I P(z,O) = ~_, a,~(z)0% I~1 = ~1 ÷...a ÷ ~n}. Notice that we have7)~ = U~07)(~). The increasing filtration {7)(a)}k>0 stalk has the following fundamentalproperties: (~+k’), 7)(~)7)(k’)CD and for P E D(~) (~’) and Q E D we have that the Lie bracket [P,Q] = PQ- QPbelongs to 7)(k+~’-1), that [D(k), 7)(~’)] C 7)(~+~’-1)holds. This secondproperty follows fromthe well-known fact that the principal parts of P and Q commute. Remark3.3.1 Locally speaking, the non-commutative Ring 7) can be written as
using a local coordinate system (zl,...,z,~,O~,...,On), and where the series ~ a~,v..~,,z~.., z,~, belongs to O. Namely,for a small neighborhoodU of a point, we have
7)(v) =o(u)
. . . , on].
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CHAPTER 3.
W-MODULES
In other words, O(U)= {P E Home(O(U), O(U)) : [... lIP, fl]f2]..-]f~] E O for somek}. In particular, for Q = fl in O(°) = O, we have [P, fl] E o(k-1); conversely, [P, fl] E O(~-1) for all f ~ O(°), then P must be in O(k). Therefore, we can also define O(k) as follows: O(~) = {P e Oil... [[P, :l]f2]-..]A] e O(0)
fk in O}. A differential operator P in O is said to be of order k if P ~ O(k) and P ~ D(k-~). Then the canonical image of P in D(k)/D(~-~) is the principal symbol of P. Namely, if P = ~lalS~a~(z)Oa is of order k, then the principal symbol of P is ak(P) = ~lal=k aa(z)~ ~, which is a homogeneouspolynomial of degree k in the cotangent coordinates ~ = (~,...,~n) and a holomorphic function in z = (z~,..., z~). Wehave the following ring isomorphism~ a~ from the associated graded sheaf of rings to the sheaf of graded rings of holomorphic functions that are homogeneous with respect to the cotangent coordinates: (3.3.2)
=
O for a11 fl, f2,...,
~a~:
~o this homomorphism satisfies (3.3.3) ak+k, (PQ) = ak (P) " a~, (~) for any P ~ O and Q ~ O(~’). Note that locally ~k_>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
<-- Or.xlOr.x ~ Since OT.x ~¢>,] 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
3.4.
g-MODULES
177
Definition 3.4.1 A sequence of holomorphicfunctions defined on an open conic subset ~ ofT*X, denotedas {pj(z, ~)}jez~, is said to be a symbolsequenceif the following conditions are satisfied. (a) Eachpj (z, ~) is homogeneous of degreej with respect to ~,
~ ~ opj(z,~)=jp~(z,
l~_t~_n~
~
~).
(b) For any ~ > 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 <_ R~(-j)!. K
Thesheaf £~ois defined as the sheaf associated to the presheaf whosesections on an open set ~ are given by P(z,O~) = ~ez~pj(z,O~) = ~je~py(z,~). For P(z,O~) = ~j~;~pj(z,~) and Q(z~O~) = ~e~q~(z,~) in ~o(~), the product (P ~ Q)(z, 0~) = ~ r~(z, ~) is defined by ra(z,~)
= ~ ~.~pj(z,~).
O~q~(z,~),
k =~+ i- I~l where ~! = c~!.., a~!, ]c~] = c~1 +... + c~ and 0~ = 01"l/0~... 0~~. The sum (P ÷ Q)(z, Oz) = ~ s~(z, is given by
s~(z,~)p~ (z, ~) + ~( q~). Wecan also define the subsheaf g~(,n) of microdifferential operators of order defining its sections as follows:
~)(~)= {~(z,o~)= ~p~(;,~) e ~(~)lp~(z, ~) =0 for Then one has
’’~. e.~=Uel
178
CHAPTER 3.
:D-MODULES
The sheaf £x is called the sheaf of microdifferential operators of finite order. Notice that we have
and
Wealso have an isomorphismfrom~x¢(m)~¢(m-1)~"x to OT.X(m) obtained by assign’~) ing the principal symbolam(P) to every element P ¯ E(x Remark 3.4.1 Let ~ : T*X -+ X be the projection defined by ~r(z,~) = The non-commutativesheaves £~ and £x have the following algebraic properties (see [19] for proofs): (1) ~x is a coherent left Ex-Module. (2) £~ is £x-faithfully fiat. (3) :D~is :Dx-faithfully flat, where:D~’is the ring of differential operators of infinite order. (4) £xis £x(0)-flat. (5) £x is ~r-lT)x-flat. Remark 3.4.2
Wealso have the following natural properties: £(k)¢(~’) ~(k+~’) X ~-’X
C ~’X
a~+~,(Po Q) = a~(P)a~,(Q), and [P, Q] = PQ - QP ¯ ’~x ak+~,-l([P,Q]) = {a~(P),a~,(Q)} for any P ¯ $(xk) k’). and Q ¯ $(x
,
EI<_~<,~ ~ o ~ o~j
- o~ " o~ )
The most fundamental result for £x is the analogue of Sato’s fundamental theorem, a cornerstone of algebraic analysis, whichwill be discussed in Chapter VI.
3.5.
HISTORICAL NOTES
179
Theorem3.4.1 Let gt be an open subset of T*X and let P(z, 0~) be an element of $(x"0. If a.~(P) ~ 0 in ~, then there exists Q(z, 0~) in $(x-’~)(~) such PoQ=QoP=I. Proof. Let P = ~j<m Pj" Then we have ao(-~oP)= ~o(~)o a(P) Hence, ~-Pm o P can be written as I - R, where R E $(x-1)(~). Since the inverse of I - R is the formal sum ~k>0 Rk, one needs to show that ~>0 R~ belongs to $x(U). Consider the element Q = ~R~ o --~Pmin $(x-m)(~). In order to the convergenceof ~>0 R~ in ~x(U), one introduces the following quasi-formal norm
Nm(P, K,t)
= ~> 2. (2n) -k.
k!
2~+l~l+lal, O~O~Pm-k t
_ o (11 + 1!(11 which (for any compactset K in ~) belongs to the formal powerring ~T[[t]]. Then,P E $(x"~) (~) if, for any arbitrary compactset K in ~ there exists eg > such that Nm(P,K, eK) is finite. Wealso have the following property: N,~(P + Q,K,t) -<-< N,n(P,K,t) + Nm(Q,K,t) for P and Q in $(x’°(~t), where A(t) -<-
3.5
Historical
Notes
The rather short history of :D-moduletheory goes back to the late 1950’s when Mikio Sato began to develop his philosophy of algebraic analysis. Wewill make
180
CHAPTER 3.
:D-MODULES
a short commenton the role of homologicalalgebra and sheaf theory, essential for the theory of :D-modulesas shownin Chapter III. The role of homological algebra, or cohomological methods, (sometimes called "General Nonsense")has been very important for algebraic analysis since the appearanceof hyperfunctions in several variables, as was mentionedin Chapter I. For example,the first chapter of SeminarNotes, No.22(in Japanese, 1968), University of Tokyo, by Hikosaburo Komatsu, begins with spectral sequences. This series of seminars began in April 1967, and continued through February 1968, for undergraduate seniors and first year graduate students at the University of Tokyo. At that time, homological algebra was not commonknowledge amonganalysis students who came to listen to Komatsu’s seminars. Among students in the audience were Kawai and Kashiwara. Sato was encouraged to becomeserious about "his old theory" of hyperfunctions by this renewedmovement by Komatsu. As we mentioned in Chapter I, even though Sato had a plan to write Part III of hyperfunction theory in which his theory of skewcategories would have been exposed, Part III was never written. His skew category theory, which is now called derived category, was developed by Grothendieck and Verdier in France. Hartshorne’s Lecture Notes, Springer-Verlag, 1966, on this subject has been a constant reference for algebraic analysis since SKK.It is worth noticing that Sato did not receive any encouragementin Japan and at Princeton to write Part III of his hyperfunction theory, possibly because such an abstract approach by Sato to hyperfunction theory and analysis might have been too radical at that time. In fact, as he mentionedto us, his ideas were not too warmlyreceived in Princeton, whenhe presented them to A. Weil. In the 1950’s, sheaf cohomologytheory was employedby H. Cartan to clarify someof Kiyoshi Oka’sfundamentalresults in several complexvariables, e.g, the nowfamous TheoremsA and B of Oka-Cartan-Serre. In algebraic geometry in the late 1950’s, it was homologicalalgebra whichnot only rephrased old notions, but also became a powerful device which allowed Grothendieck to develop a newalgebraic geometry. Sato’s algebraic analysis, an algebraic geometry-type analysis, naturally demandedhomological algebra even to define the sheaf of hyperfunctionsin several variables. Around1957, Sato beganthe cohomologicalstudy of systems of partial differential equations. The first occasion in which he publicly announcedhis workon this topic was the series of talks whichhe delivered in the Spring of 1960at the KawadaFriday-Seminarjust before his departure for the Institute for Advanced Study. During his talks, Sato put emphasis on the importance of cohomological treatments of systems of linear and non-linear partial differential equations. We can say that Sato’s algebraic analysis began in that moment.The most spectacular successes of his programare microlocal analysis and l)-module theory, the topics of this monograph. Wewill make no extended commentson Sato’s number theoretic work done around 1960, other than to mention Sato Conjecture on L-functions, the connection between Ramanujanand Weil Conjectures, and the
3.5.
HISTORICAL NOTES
181
theory of prehomogenousvector spaces. The theory of holonomicquantumfields and soliton equations on Grassmannmanifolds are also someof Sato’s work ia mathematics-physics. Note that these physics fields are rooted in his original programoutlined in 1960on the general theory of non-linear partial differential equations. The arrival of Masaki Kashiwaraestablished another main character on the /)-module theory stage. It was Kashiwarawhoestablished almost all that we have in the theory of/)-modules. In the case of partial differential operators with constant coefficients, the straightforward applications of various finiteness properties of noetherian rings can be used. In Kashiwara’sthesis, the notion of a filtration was developedto overcomethe variable coefficients case. Namely,a filtration is used as a bridge betweencommutativeobjects and non-commutative ones. Essentially all the mathematicaldevices in Chapter III camefrom Kashiwara. ~omhis very first publication on/)-modules, namelyhis astonishing Master’s Thesis of 1970, Kashiwaraestablished the methods and the fundamental results for further study of/)-modules. Within a few years, Kashiwaraobtained the most crucial theorem in :D-moduletheory, knownas the Constructibility Theoremfor a holonomic/)-module.One of the climaxes, if not the climax, is Kashiwara’s Riemann-Hilbert Correspondence Theorem.
Chapter 4 Functors Associated -modules 4.1
Introduction
with
and Preliminary
Material
In this chapter we study some fundamental functors associated with the 7)modules which we have introduced in Chapter III. Before we get to the de Rhamfunctor in Section 4.2, we remind the reader the conditions which an Ox-ModuleMmust satisfy in order to be a T)x-module.In particular, we will show that 7-lomox(~x,-) and -~ are one the inverse of the other when acting on the categories of left and right/)x-modules. The de Rhamcomplex is a well knownresolution for Ox. In Section 4.2 we construct and study the de Rhamfunctor which allows a generalization to an arbitrary left :Dx-module.In particular we discuss the important result of Mebkhoutwho has proved that the de Rhamfunctor 7tomz)x(Ox,-) and the solution functor 7-lOmox(-, Ox) are in global duality, and we showthat both Poincar~ and Serre duality are just special cases of Mebkhout’stheorem. After a quick review of algebraic local cohomology(which we introduce in Section 4.3 as a natural extension of the space of hyperfunctionsin one variable supported at the origin) we prove, in Section 4.4, two cohomologyvanishing theorems for 7)x-modules. Let X be a complexmanifold of dimension n, Ox the sheaf of germs of holomorphicfunctions, and/)x the sheaf of germs of partial differential operators with coefficients in holomorphicfunctions. Let us denote by Ox the sheaf of germs of holomorphictangent vector fields. If a and b are holomorphic functions, the chain rule O(ab)= cOa.b +a. cObgives the formula cO ¯ a = Oa+ a ¯ cO whichinduces a non-commutative ring structure on/)x; as a first example,0i ¯ zi = 1 + zi ¯COi. The repeated use of this formula provides Leibniz’s rule, i.e., for a E Ox and for 0~ the differential operator 183
184
CHAPTER 4.
FUNCTORS ASSOCIATED WITH ~)-MODULES
Oz~,] we have that ¯ a = ~ O~a¯ 0~-~. The adjoint operator of P(z,O) = Ef~(z)O~ is defined as E(-1)lal0~. f~(z) so that, for example, the adjoint of 0~ is -0~. See also Lemma 6.2.3. Since :Dx is the sheaf of rings (Ring in short) generated by Ox over Ox, the action of Ox detern~ines a :Dx-Modulestructure. \az~’’""
Definition 4.1.1 Let J~ be an Ox-Module,and let ¢ be thefT-linear map from Ox ®¢x M to M defined by ¢(8 @m) = 8. ra for all ~ @m ¯ Ox ~¢x M. If ]or all a ¯ Ox, 8, Or ¯ Ox, rn ¯ M, ¢ satisfies (1) ¢(aS~rn) = a¢(0~m), i.e.
(aO).m-= a(O.m)
(2) ¢(~ ~ am) ---- a¢(O~m)+ O(a)m, i.e.,
(3)
=
-
8 . (am) = a(0m) +
i.e.,
8 . (8’. m) - ~’(~ . m), then A4 is a left ~)x-Module. The correspondingdefinition for a right :Dx-Moduleis the following. Definition 4.1.2 Let Af be an Ox-Module,and let ~o be the~Tx-linear mapfrom OX@¢xAfto Af defined by ~(~@n) = n. 0 for all ~@n¯ Ox@¢xAf. If g~, for all a ¯ 0x,0,0~ ¯ Ox,n ¯ AI’, satisfies (1") ~(ae@n) = T(O~an), i.e.n.
(aO) =
(2*) ~(~ ~ an) = -0(a) . n + a~(O® n), i.e., (an). 0 = -O(a) (3*) ~o([0,0’]~n) = ~o(0’~o(0(~n)) - ~o(0(~o(0’@n)), i.e., = (n. 8). 8’ - (n. 0’). 8, then Af is a right ~)x-Module.
n. [0’,0]
Note that the induced Ox-Modulestructure of the left ~x-ModuleJPl coincides with the original Ox-Modulestructure. See [19]. Since n is the dimension of the complexmanifold X, the sheaf ~ of germs of holomorphicn-forms is locally free of rank one as an Ox-Module.For 81 A ... A tg,~ ¯ A’~ Ox, an n-formw ¯ ~t~ determinesan element w(~l A... A t?,~) Ox as an COx-linear mapfrom AN Ox to Ox. Wewill define a right :Dx-Module structure on ~ via Lie derivatives. Namelywe define ((Lie (0)w)(0~A...A0n):=
:= ^.../x
- [8,8,]
^ e,,).
i=1
Moreexplicitly, this meansthat the tangent vector fields 0 ¯ Oxact on Q~ by wOa~ -(Lie O)w. Thus f~} has the structure of a right :Dx-Module.In terms
4.1.
INTRODUCTION AND PRELIMINARY MATERIAL
185
of local coordinates, if w = gdzl A ... A dz,~ C Yt~, then wc3i = -(Lie vOi)w -(Oig)dzl A... A dz,~. Moregenerally, the extendedright action of ~Dxon gt~c is given by w. P(z, O) =(P(z, c~)* g)dZl A... A whereP(z, 0)* is the adjoint differential operator of P(z, 0). Notethat the right moduleaxiomsare satisfied because of the property of adjointness which gives (P(z, O)Q(z, 0))" = Q(z, O)’P(z, The right :Dx-Module~.~, a free C0x-Moduleof rank one, induces an exact covariant functor from the category of left :Dx-Modules to the category of right :Dx-Modules as follows. Given a left :Dx-Module A/f, one can construct the right :Dx-ModuleAd ~ox ~, where the right module structure is induced by
(u
0=
+
for t~ C (9 and u ® w ~ .~d ~)vx ~. Weleave it to the reader to verify that this definition make J~4 ~Vx ~ a right :Dx-module. Notice also that, more generally, for any right :Dx-ModuleAf and any left T~x-ModuleA~ one can construct a right :Dx-ModuleA~ ~ox A/" where the right module structure is defined by
=
+
Proposition 4.1.1 Let .A4 and.IV" be, respectively, a left ~Dx-Module and a right :Dx-Module. Then
omo ( M®
)~mx
and
Proof. Just note that 7-lomox(~x,Af) becomesa left :Dx-Moduleif, for ~ ~ ~-lomox(ft~x, A/’), the action of Oxis defined by (O~)w= -(qow)~+ ~(wO). The isomorphism follows since ~t~ is isomorphic to Ox as an Ox-module.[] The above proposition shows us that the compositions of functors Homox(~x,-) and -~ and the identity functors are isomorphic on objects in both the categories of right :Dx-Modules and of left :Dz-Modules.Hence, - ~Ox~x is an equivalence from the category of left :Dx-Modulesto the category of right :Dz-Modules,and vice versa 7~Omox (~, -) is an equivalence from the category of right :Dz-modulesto the category of left :Dx modules. This result can be stated in a moregeneral setting as follows:
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CHAPTER 4.
FUNCTORS ASSOCIATED WITH D-MODULES
Proposition 4.1.2 Let J~ and j~’ be left Dx-Modulesand let .hf and ]V" be right Dx-Modules. Then:
(4.1.1) J~ ~ ’ oxA~ is a left Dx-Modulewhere the action of Ox is defined by
0(~ ®.~’) = (0.~)®~’+ (4.1.2)
j~ ~ OxjY"
is a right :Dx-Modulewhere the action of Ox is defined by
(.~ ®~)o= -(o,~)®~ + (4.1.3) 7"l omox (.hA, .M’) is a left :Dx-Modulewherethe action of Ox is defined by
(o~)(m) =o(~(,~))- ~(o(,~)); (4.1.4) 7-l Omox(.hi’, Aft) is a left ~Dx-Module where the action of Ox is defined by
(o~)(~)= -~(~)o+ and finally
(4.1.5) 7t Omox( M, is a right :Dx-Modulewhere the action of Ox is defined by
(~o)(,~)=~(,~)o+~(o,~). Proof. Wewill prove (4.1.1), i.e. that, @o x jk 4’ is a l ef t 7)x -Module. Weneed to verify (1), (2) and (3) of Definition 4.1,1. To prove (1) a E Ox,O E Ox,m,m’ E (aO)m(~ m’ = ( (aO) . m)(~m’+m (~( = a(Om)(~m’+ m(~a(Om) = a((Om)(~m’) = a(Om(~m’+ m(~Om’) = a(O(m(~m’)).
+
4.2.
THE DE RHAM FUNCTOR
187
Next, (2) is established by the following sequenceof equalities: 8 . (a(m (~ m’) ) = 8((am) (~ m’) = 8(am)
= (a(8~)+ 8(a).,~) ®,~’+ = a(8=)®,~’+8(~).,~ ®,~’+~,~ = ~((8~ ®~’+,~ ®8’~’)+8(a).,~ = ~(8(,~®,~’)) +8(a).,~ ®,~’. Finally let us prove(3), here 8’ is another elementof Ox:
[8,8’](.~ ®=’) =([8,8%) ®,~’+.~®([8, ¢],~’) =(~. (~’,~)- ~’(~=)) ~’+~ ~(~(~’=’) = e. (e’~)~ ~’ - e’(e~)~ ~’+ ~ ~ e(e’~’)- ~ ~ e’(e~’) By adding
to the above equation we obtain
whichconcludesthe proof of (4.1.1). Theother eases are left to the reader.
4.2
The de Rham Functor
The structure of a :Dx-ModuleA~I is determined by a sheaf homomorphism of the cox-Algebra :Dx to End¢x (J~l). Then the restriction of this sheaf homomorphismto the sheaf of tangent vector fields Ox(considered as a subsheaf of ~z) satisfies properties (1), (2) and (3) mentionedin the Section 4.1. fore this induces an integrable connection V on the C0x-ModuleA~. Namely we can define an integrable connection V on 2vl as a Cox-linear mapfrom the tangent sheaf Oz to End~x (J~) such that for 8 E Ox we have that the element Ve = V(8) ~nd~x(.h4) satisfies
v0(~rn) O( a) .,,~ +,~v0( ~n ),
188
CHAPTER 4.
FUNCTORS ASSOCIATED WITH :D-MODULES
and
vi0.,,l(-~)= v0(v~, (.~))-v~,(v0(.~)). Note that these two properties are, respectively, properties (2) and (3) of Definition 4.1.1. The integrable connection V on A~ can now be used to obtain the de Rham complexassociated to Jr4 as the complex
(4.2.1) where
j=l
j=l
and
~s(wi~rn) = dS(w) {~rn + (-1)Sw Ad°(m) = dS(w) (~m + ~.,(dzj j=l
in this formula, the operators ds are the coboundaryoperators of the complex Notice that w A ~°(m) is the corresponding element in ~+1 ~Ox M when we take w~°(rn) in ~X@Ox(~x~Ox .M). In particular, since ~l(~°(m)) Ej~=l(E’~=l(dz~ A dzj)~Ok(Ojm)) and Ok and 0j commute,we have ~1 o~° = 0. Then since (~+1 o ~)(w~rn) = w A (d I o dO)(m) holds, we obtain ~+1~ = 0. WhenA/[ is replaced by Ox, one gets the usual de Rhamcomplex. The right :Dx-Modulestructure of ~ described in Section 4.1, allows us to obtain the augmentation map ~’ from ~ ~¢)x :Dx to ~. By replacing Min (4.2.1) by :Dx, regarded as left Ox-Modules,we obtain that the following (de Rham)complexis a free resolution of the right T)x-Module~c: (4.2.2)
~x where
4.2.
THE DE RHAMFUNCTOR
189
c/°(lv) y~dzj(~ Voj (19)Y~dzj (~ j=l d~(w A 19) = d’(w)@19- co A d°(lv), and in general ~s(co A Iv)= dS(co) 19 + ( -l)Sco A°(iv). ~ On theotherhand,sinceOx is a freeOx-Module, thesecomplexes induce the usualKoszul complex associated to theelements 01,..., Onin Ox: (4.2.3)
7)x ’fi-~x ®OxA~Ox+h-Z~x ® ox A~ Ox~-... ~Z~x@Ox~"Ox Ox where the augmentation mape" in (4.2.a) is defined by e"(l~) = 1~ Oxand ~ is induced by right multiplication by 0~,..., This procedure allows us, in particular, to obtain a free resolution of Ox. Note that ~x @oxA~Oxis at the -i-th place of the resolution for Ox. The i-th boundarymapof (4.2.g)
~x@ Ox A~-:Ox~ Vx@ ox A~ Ox is defined by i
~i(P(~(O, A... A ~)) = ~~(-1)J-’P¢ (~)(O, A...
¯
j=l
+ ~ (--I)J+kP~([~,eklAO,A...A~jA...A~kA..AOi), l<_j
where P ~ :Dx and Oj ~ (~x for j = 1, 2,... ,n. In particular, $I(P®O)= PO. Hence, we have Im ~ = ~jn=17)xOj. Then the augmentation map e" in (4.2.3) induces the isomorphisms
no(Vx ®OxA-" ex) j=l v ,oj Ox.
190
CHAPTER 4.
FUNCTORS ASSOCIATED
WITH Z)-MODULES
In terms of local coordinates (zl,. ¯ ¯, z~, 01,. ¯ ¯, 0n) for :Dx, we obtain the vector space spanned by theiT-regular sequence (01,..., 0n). Then (4.2.3) becomes
T)X ~-- ~X @¢ A1 (~=IITOq#) ~--...
+L ~)x @¢Ai ((~=IIT0#) ~--...
¯ .. +- Z)XI~¢ An ($~’=IITOj), which is the familiar
Koszul complex in which i
5;(P@O#,A... A 0#,) ----~-~.(--1)k-lp0#k ~)0#k A...A O5#~A...A 0#i ). k----1
Therefore we have obtained a locally free resolution of
Ox~ ~x/(O~,..., o~). Retook 4.~.1 Por a vector bundle Mwith an integrable connection, (namely, if Mis free of finite rank ~ an Ox-Module), sequence (4.2.3) allows us to obtain a free resolution of M ~ a ~x-Module: @Ox A" M. Ox @Ox
M ~ ~x
~or a left ~x-Module with a good filtration, we need on the other hand ghe Spencer sequence in order to construct a free resolution, get {M(~)} be a good filtration of M. Wewill show that for any ~ ~ 0 the sequence O~ M ~ ~X@Ox M(~)
(4.2.4)
~ ~x@oxOxNoxM(k-~)
... ~-:Dx@oxA~ Ox@o,~A,I(~-~) ~-0 is locally a :Dx-free resolution defined by ~(P Ig} u) Pu, and ~i(P@(Ox A... = E(-1)#-Ip0#
of flA. The augmentation map ~ in (4.2.4)
A Oi)~)u) @(01 A...
A
j=l
--
E(-1)J-~P@(01A... j=l
+ l~j
(--1)
@([Oj,
OklAO1A.
..A0A
4.2.
THE DE RHAMFUNCTOR
191
Before we explain the interplay amongsequences (4.2.1), (4.2.2), (4.2.3) (4.2.4), we will prove the exactness of Spencersequence(4.2.4). Proposition 4.2.1 Let ./~ be a left 7)x-Modulewith a good filtration {Ad(k)}. Thenthe Spencersequence(~.2.,t) associated to AAis exact. Proof. The following proof is based on Kashiwara’sideas from [108]. First we consider the case whenAz/= :Dx with the canonical filtration on :Dx as defined in Chapter III. For k = 0, we have O ~- 7)x ~ Dx +since :D(m) = 0 for m < 0. So now let k > 0 and proceed by induction. Considerthe.~ollowing short exact sequences of complexeswhere:D(~-n) denotes 79(~-n) I~’X " X 0
0
0
-+
7~x®/~"(3x®7~ (k-’O
0
~
~x@A"Ox@9 (~-")
0
~
Dx@A"Ox@D(~-"-~)
--+
...
~ 9x@Ox@D(~-’)
T ~ .,.
~ ~x@~x@~(~-x)
(~ ~ -~) .,.
~ ~x@Ox@~
~
(i)
~
(2)
~
(3)
T
0
0 0
(1)
--~
~x®~¢~)
-~
o (3) ~ :Dx@:D(x k-~) --+ ’Dx ~ 0 0 0 In view of the inductive assumption and of the induced long exact sequence of cohomologies,it is sufficient to provethe exactnessof the top rowof the diagram. On the other hand,
192
CHAPTER 4.
FUNCTORS ASSOCIATED WITH/)-MODULES
where
(~i((01
A...
A0i)@p)
i = Z(--1)J(01 j=l
A...
A~j
A...AOi)
is the Koszulcomplexassociated to the regular sequence(01, 02,..., 0n). Therefore, the above sequence is exact. Consequently, the exact functor/)x ~ Oxinduces the exactness of the top sequence in the above diagram. Wewill nowprove the case of a general/)x-module A4. By the definition of a good filtration there exists locally an epimorphism~ of filtered objects ~ :/)~ ~ A4 which induces the short exact sequence 0 ~.,’~ ~/)~ ~ ~ ~ whereAf -- ker~. By what we provedin the first part of this proof, the Spencer sequence for the free Module/)~ is exact. The long exact sequence of cohomologies associated with Spencer sequences of J%4,/)~ and Af is
n0(~)--+ n0(/)~) --~ ~ nl(JV) ~ nl(/)~) ---~ nl(M) --+ ~2(~f)~ n2(/)~) --+ n2(M)---~... Proceedingas we did in the first part of the proof, the middlecolumnvanishes. The inductive assumption that 7/j_l(Af) = 0 for k >> 0 and for any left Modulewith a good filtration implies the exactness of the sequence 0 : 7-~j(/)~)
"---9"
~{j(d~)
~ ~{j_l(J~
f)
= 0.
Therefore we have 7gj(A4) = 0 for k >> 0 and all j = 2,3,.... the (-1)-th cohomologyat f of
Since 7~_l(Af)
0 ~- ~ ~-/)x ~ Ox.h/’(k) ~- ..., the fact that A/" -- /)x ~ OxA/’(k) for k >) 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
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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)) <_2n - 1). Inparticular, for h > 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] ~_<~s~,as we mentionedin 3.2 of Chapter III. Whenf is the projection from Y = X × Z -+ X, defined by (zl,...,zn, wl,...,Wk) -+ (z~,...,Zn), the ~Dy-Modulef’34 may be regarded as not only ID~/Z)~,P but also O,~,u~ = O, i -- 1 .... ,k, whereu~ --- f*uj, j -= 1,...,m. In the case where Y ~-> 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<,~Ox0a, a free Ox-Module. Wehave, for f : Y ~-~ X,
That is, the :Dy-Module f*A/[ is isomorphic to T)p. Then we obtain 7-lom~.(f*AA, Ov) ..~ 0~. Let ~o ¯ 7-lom~x(.h~,Ox) and let h ¯ {(u) ¯ Ox, where Pu = 0. Then f-~h = hlzl=o = h(0, z’) ¯ 0~. The assignment (5.2.4)
hl~=o~ (hl~=o,O~hlz~=o,¯ ¯ ¯, "~-~h~
~,=o~~ Ogre
is nothing but the map (5.2.5)
f-xT-lOmvx(.A4, Ox) --+ 7COmfy (f*.A,4, f*Ox)
The classical Cauehy-Kowalewsky theoremstates that assignment (5.2.4) is onto and one-to-one, if Y = (zl -- 0) is non-characteristic to P. Namely,we obtain isomorphism in (5.2.5). Furthermore, the map Ox :-~ Ox is an epimorphism, i.e., for any holomorphic function f there exists u ~ Ox such that Pu = f. Therefore EXt~x(All, Ox) - 0. Next, let Y be a hyperface of codimension1 on X, and let A/[ be a coherent 7:)x-Modulesuch that Y is non-characteristic to i.e., the singular support of .£4 does not intersect with the conormalprojective bundleto T, V(AA)~ P~.(X)= (~. For a set of generators u~, u2 .... , u,~ for one can find Pj in :Dx satisfying Pjuj = 0, j = 1 ..... m. Namely,we have the followingfree resolution of
5.2.
INVERSE IMAGE AND CAUCHY PROBLEM
213
In terms of relations, the aboveexact sequence means Plul
=0 =0
P2u2
(5.2.6) 0 <~ A4 +--- $~=iDx/7)xP~<--- Af ~ O, where V(@~=t~x/Dx~) ~ P~X = ~. Notice also that $~=~Dx/~xPj is a free Ox-Moduleas we observed in the c~e of classical Cauchy-Kowalewsky theorem. In particular, the higher Tor~x (Oy, @~=~x/~P~) vanishes for all h k 1. Hence, from short exact sequence (5.2.6) we obtain, through the functor f* = Oy ~Ox -, (5.2.7)
0 ~ f*~ ~ f"
(e?=~Ox/OxPs)
~ f*~ ~
Fromexact sequences (5.2.6) and (5.2.7), the following commutativediagram induced
--"
f-~7~Omvx
(J~,
Ox) ....
[
. ,~0
" ~lOmvy(f*Af,
--.. f -l g xt~x (A/f, Ox ) ~ f -l gxti~x
Oy) ....
(~jm=I
~)X
/Vx
j,
OX ) .. ..
214
CHAPTER 5.
HOLONOMIC :D-MODULES
Then all the ’7 i are isomorphisms from the classical Cauchy-Kowalewsky theorem. From a well known lemma in homological algebra, ker 7o --~ ker ’7o -~ ker "7° is exact. Furthermore, since 3,1 is monomorphic, there exists a connecting homomorphism 6° so that ker 70 -~ ker ’7° -~ ker "70 -~ coker 70 -+ coker ~7° 0-~ coker "7 may be exact. Consequently, we obtain coker "7o = 0 since ker "70 -- 0 as well. That is, 7o is an isomorphism.Repeatingthis process, we have 7i, i=1,2,... are isomorphisms. Next, we will consider the general case. Let Y be of codimensionr in X, and let Y’ be a hypersurface containing Y defined by zl = 0. Since Y~ is locally non-characteristic for M, we have (5.2.8) as in the above, where f~ is an inclusion from Y~ to X. The equation codim (Y, Y’) = codim Y - 1 provides the proof by induction on the codimension. Since the characteristic variety of f’-lA/I is smaller than the projection of the characteristic variety of A4 on T*Y’, Y is non-characteristic for f’-IAA. Hence for f : Y "-~ Y~we get (5.2.9)
f-17iomvw(f’-lA4,0y,)
-~ 7tOmvy(f*(f’*Nl),Oy).
Note that f*(f’*J~/[) = (f’o f)*JPl holds, since, formally speaking, we have ~)yr.,.+ x ~f_iVy ’ f-l(~)y,¢_~ x (~ft_l,~ X f-id~) : O~ @f-lOy t
f-l~y, ~f_~y, f-lOy’
~(ftOf)_iO
X (f’
= (f’ o f)*~. Consequently, we have from (5.2.8) and (5.2.9) f-xf’-l~omvx
(~, Ox) = (f’
o f)-~omv~ (~,
f-Xnom~,(f"~.Or,) ~ nomv.((f’o f)
0 f)--i~
5.2.
INVERSE
IMAGE AND CAUCHY PROBLEM
215
For the higher cohomology groups for the general case, we repeat the inductive proof for the hypersurface case. We have obtained the following theorem which generalizes the classical Cauchy-Kowalewskytheorem. Theorem 5.2.1 For a T)x-Module .M, let Y I~ X be of an arbitrary sion that is non-characteristic for AJ. Then the induced maps f-’£xt~x(M,
Ox)~£
xt h
m.(f
* M, OY),
codimen-
h=0,1,2...
are isomorphisms.
Remark 5.2.1 Let Y = (Zl = 0) ~ X. Then, as we saw, holds. That is, we obtain the exact sequence:
~X ~ ~X ~ ~)Yc’-~X -’)" O, furthermore inducing the exactness of f-179X -’+ f-17:)X --~ f-17~y~X "~ O. Note that f-l’~gy,_~x : f-l(Oy with ®f-]z~xf-l.A/[, we get
@f-~Ox f-lDx)
= Dy~x. Tensoring the above
Namely, f*M = MIY = f-lM/zlf-lM. Let ¢ E 7tomz)x(M, Ox), i.e., for u E M, ¢ assigns ¢(u) = ¢, Ox, and le t ly nx bethe unit in :Dy~x, i.e., lynx®¢ ~ 7-~omv~(’D¥,[email protected],vx f-l M, "Dy~x®l-,vx f-~Ox) ~ 7-lomv~, (M/zlM,
If M is given by M = 7:)x/DxP, where P = 0~n + P~(z,O’)O’~ -1 + ... Pro(z, ~’) ~ 79(xm) as before, then f*M is generated by (1 ® u,..., as a/:)y-Module. That is, f*M .~ :D~ as we observed earlier. Then the corresponding ¢* ~ 7tomvv (f*.M, Oy) may be defined as follows: for generators 1 ®~u of f’A4 ¢*(l®~u)=~¢(l®u)ly~O~
,
j=
1,...,m-
1.
216
CHAPTER 5.
HOLONOMIC :D-MODULES
Theorem5.2.2 Let Y be a subvariety of X, Y ~ X, and let M° be a bounded complex of ~x-Modules, i.e., 7-lJ(M°) ~t 0 for only finitely many j. If Y is non-characteristic for all non-trivial 7tJ (.hd), then the complexOy ®~Lf-l jv[o has coherent complex cohomologies, and furthermore, we have an isomorphism ~ f_l.DX f--l ~’, Oy). f-l EtT-lomvx(JV4", Ox) -~ E~7-lomvv(Oy Remark 5.2.2 For the complex AA" concentrated at degree zero in Remark 5.2.1, the complex Oy®f-lOxmf-lAd = :Dy~x ®f-l~xzL f-lenA may be computed by the projective resolution of ~3y~x in the above remarkas follows. LL The cohomologies of Oy ®~-lOx f-~.M = :Dy~x ®l-l~x f-l Jvi are isomorphic to the cohomologiesof the complex
In particular, the 0-th cohomology v]_[o(~)y~x ®f_lVx~LL f-lj~) is isomorphicto f-lA/I/z~f-lAd. As we saw in Remark5.2.1, for A/[= T~x/~xP, P E :D(’~), the :Dy-Modulef’A//= f-lJ~4/zlf-tAd is given as :D~. Next, the 1-st cohomology -~vx (:Dy~x,f-~.h,4) is isomorphicto the sheaf 7tl (~Dy~x® l_lVxfzL -~.Ad): TOr]l Ker(f-l.M ~ f-lA//). However, in general, the ~Dy-Module Tor(-~VX(Dy~x, f-l.M) is isomorphic to Tor~-l°x((gy, f-lJv[). This is bef-~Ox -1 cause :Dx is Ox-flat. Hence Tor~ (Oy, f ~) : ~’~j(Oy ®f-lOx f-l~) -lvx 7~(:Dy~x®~’-lvx f-123~) = T°r~ (:Dy-.x, f-lA~), where O~cis a free resolution of Ad. Then, as we observedearlier, ,~4 = :Dx/:DxPis a free (9x-Module. Therefore,the higher "For~j -I:DX (~)Y~-~x, f-IAA)~=~orf-~°xj ~’~ J’-1¢~’)~A~vanishes, for j = 1, 2 ..... Proof of the Theorem 5.2.2. Regard f-l~tT-lomvx(Jvf’, of f-l’E~,q
and also regard ~:tTtomwx (Oy
= f-l~Xt~x /L
(n q (d~°), f-ld~’,
~)y)
Ox) as the abutment
Ox), as
the
abutment
’Eg’~ ~’~÷~ ~’~"~’~ Ex@v(Tor~p~°X(Oy,~lq’(f-lJv[°)), where q = p’ + q’ and ’E~’’4 ~ 7~(Oy®~Lf-i.h/[*). Therefore, by induction on the codimensionof Y and by Remark5.2.1, the proof is reduced to the case of Theorem5.2.1. [] As an application of Theorem5.2.1, we have the following theorem.
5.2.
INVERSE IMAGE AND CAUCHY PROBLEM
217
Theorem5.2.3 For a :Ox-Module.h4, i] the characteristic variety V(Jt4) contained in X, then f14 is isomorphicto Otx for some 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<0
~ : ~?:IOxUj
---~
~)(X0)~/,1
~’’’
~)(X0)’U.m rfok _>
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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,
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CHAPTER 5.
HOLONOMIC D-MODULES
See [8] for its proof. From this lemma,we will be able to prove the finite dimensionality of Exthvx( B(xo,e), All, Ox as follows. Considera free resolution of J~Ain a small neighborhoodB(x0, e’) X0
as in Chapter III. Thenleft exact contravariant functor Homvx(B(x0, e’), Ox) induces a complexof Fr4chet spaces with continuouslinear differential: O’~(B(xo,e’)) -~ Ot(B(xo,e’)) .... The restriction mapOx (B (x0, ~’)) Ox (B (x0, e)) is a com pact map for e< ~’ Then Hh(O’~(B(xo,e’)) --~...) Exth~x(B(xo, e’), ~[ , Ox)
~(o~(~(~:~, el) ~ .. .) =~t~,~(~(~o,el, is an isomorphismbetween finite dimensional complexvector spaces. Remark5.4.5 For an elliptic T~M-Module .h~ on a real analytic manifold M with its complexification X, we have the following finite dimensionality theorem of Kawai. Recall that a coherent DM-Module J~4 is elliptic if V(M)S~X = 0,where OM= ~X[M and S~X = v/-~S*M, the cosphere bundle of M with respect to X. Kawai, T. proved the following theorem. See Kawai, T. "Theorems on the finite dimensionality of cohomologygroups", I, II, III, IV and V, Proc. Japan Acad. N° 48, 70-72,287-289(1972), ° 49, 2 43-246,655-658 and 782-784 (1973), respectively. Theorem5.4.3 Let f~ be a relatively compact open subset of M so that the boundaryO~ is a real analytic hypersurface in M. Whenthe restriction of an elliptic :DM-Module ]~ to the boundaryO~is elliptic on 0~, ExtOrt,(12, .h4 , AM) is finite dimensionalover(T, where AM is the sheaf of real analytic functions on M.
5.4.
231
HOLONOMIC D-MODULES
Notice that Kashiwara’sconstructibility theoremis a vast and ideal generalization of Kawai’s. Let A/I be a holonomic :Dx-Module.Namely, we have dim V(A//) = dim X n. Fromtheorem 4.4.1, for a holonomic:Px-ModuleA~I, we have $xthvx(Ad,:Dx) = 0 for h < n = codim V(A,I). On the other hand, theorem4.4.2 implies, for h > n = dim V(fl’l),
ext x vx) = 0. For this non-vanishing n-th cohomologysheaf, we have codim V(Sxtvx(.&4, x)) from (4.4.6). Fromthe involutivity of V(fl4), we codimV(£xt~x(2~1, Dx)) Namely,£Xt~x (M/f, :Dx) is a right holonomic~Dx-Module,which is denoted Let O-~’M
-+M ~"M ~0
be a short exact sequence of left holonomic:Dx-Modules.The left exact contravariant functor 7-lomvx (-, :Dx) induces a long exact sequence of cohomologies. Then the purely codimensionality of 8xt~vx(A4,:Dx) 0, h ~n impl ies the following exact sequence
That is, 8Xt~x(-, :Dx) is an exact functor from the category of left holonomic ~Dx-Modulesto the category of right holonomic:Dx-Modules. Similarly, ,~I** = (.&4*)* is a left holonomic:/:)x-Module. One can show actually ~l** ~ gA. See the following Remark5.4.6 for its proof. In general the characteristic variety of 8xt~x(A~l, T)x) is smaller, i.e. containedin, than that of g4. Therefore, we have codim V(A/I) _< codim V(Sxt~x (g,l, Dx)), (or from (4.4.6) for h = n). Similarly, n V(~xtz~x(M,Dx))
n n 3 V(~xtvx(~xtz,x(M,
~)
x),Dx)M ~f V(M*’)
implies n _< codim V($xt~x (M/I, :Dx)) codim V(. M) <_
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Consequently, codimV(2~4) = n holds. That is, the cohomologicalstatement
~ ~ = O, h ~n" "E xt h~’x " ~JV~, ’ ¯’’ ~’x
characterize the holonomicity of Remark5.4.6 Consider a projective resolution of a :/:)x-Module M:
Then the contravariant functor ?-lomvx(-, 7:)x) induces a complexas follows.
0 --+ 7-lom~x(M,T)x ) --~ 7-lOm~x(P°,:Dx)--~ 7-lOm~x(P-~,:Dx)-+
Denote "rtom~,x(P-h, T)x) as /Sh. Then we have a Cartan-Eilenberg resolution of/5. as follows.
The above fourth quadrant double complexis taken by the contravariant functor 7tOm9x(-, ~x) to the second quadrant double complex as follows.
5.4.
HOLONOMIC~D-MODULES
233
where~-h,O=_"riOmvx( ph, :Dx). Then spectral sequences associated with the above second quadrant double complex is induced as follows. (See introduction of Chapter III).
E~"’q =
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CHAPTER 5.
HOLONOMIC :D-MODULES
Therefore, we have E~-p’° = 0 for p ~ 0 and E~°’° ~ M. That is, E~° ~ E° = H°(Q’), where Qn ~p+q=,~Qp,q. Ontheothe r hand , the 0-th term second spectral sequence
gives :
:
’-q ’E21
*’2) ~(1,0) H-q(~
’-q ’Ell
~(1,0)*’1 )H-q((~
’-q ’E~
*’°) ~-(1,o) H-q(Q
0
0
Therefore, we get
= ext x
Vx),Vx)
abutting the (p - q)-th derived functor of the identity, i.e. p-q=O. From the sequence
E° = A/[ with
I ’ d~-_~ ~+’, E~,- ’ £"-=~,E~2+2,-q-x ...--+ ’E~-2,-a+ q
0
0
¯.~p___0 we obtain p’-~ ~E = F’(.M)/F"+I(A’I), whereN[ = ~’~ ~:"-P ~oo ¯Thisis because the total degrees of ’~’-q and td~2-2,-q+l increase by +1. For a holonomicDxModule A~, we have
E~
= £Xt~x($Xt~x(.M,Dx),Dx ) ,~ E~-" ~ .M.
5.4.
HOLONOMICZ)-MODULES
235
Proposition 5.4.2 For a holonomic l)x-Module A,~ we have an isomorphism
sxt~x (M,Ox)--~sxt~,,¢(ox,.~*®Ox where - ®Ox([2") -1 is a quasi-inverse of- ® ~, i.e.
.~* ®o,~(~n)-i = nomox(~,M*) (see ChapterIV, Introduction). Remark5.4.7 Namely, the above isomorphism says that the higher solution sheaves are nothing but the de Rhamcohomologysheaves of the dual system. See Remark5.4.8. Proof. Since the de RhamModuleOx is of finite presentation as a lgx-Module, we have the spectral sequence
abutting to Exth~x(A4, Ox). See Chapter IV, (4.2.7). Since ~ is holonomic, E[’~ = 0 for q # n. Hence we have an isomorphism n ~x E~-"’" = To~,_~(~t~ x (~, v~), O~)~ ~t~(~,
Then we have E~-~" = ~o~(~ ~ +o~ ~* +o~ (~)-~, ~ ~or~(n ~, ~" ~o~ (n~)-~)
O~)
~ n_~((~"~oxVx)v.~ ~* ~ (fl~)-~), where ~" @Ox~x[n] is the projective resolution of ~ in (4.2.2). above, we have further an isomorphism
~omthe
~ ~(~* VOx~" ~ (~)-~) = ~x(O~,~"~o~(~)-~). Remark 5.4.8 Notice that by replacing ~ by ~* @Vx(~)-~ in the previous Proposition 5.4.2, we also have ~t~ (~" eo~ (~")-~,
O~) ~ e~t~ (O~, ~).
The generalization to the derived category version of the ~boveProposition becomesas follows. Let ~" be a bounded complex of left ~-Moduleswhose cohomologies are of finite presentation. Then ~Om~x (Ox, ~’) is isomorphic to
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CHAPTER 5.
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J~"~om’Dx((*]~°) * @Ox (~’~n)-I[’I~],OX), where (AA’)* is defined ~HOmvx(~’, ~x). The proof is an exercise in Derived Category:
~nO~x ((~’)* ~Ox(~")-~[~],o~) = ~o~ (~o~ (M, ~), )[ -~] ~ ~omvx (Ox, ~’). Retook 5.4.9 Let Mbe a left ~x-Modulethat is a locally free Ox-Module, i.e., there locally exists a set of generators {ui)~iS~ such that ~ = Oxu~~ ¯ "" ¯ OxUm.Then the ~x-Module structure is determined by the ~signment of the generators for each Oi = 0Oz~"Since Mis locally free as an Ox-Module, there exists holomorphicfunctions g~, gj~,...,
gj~ SOthat we mayhave
Oiuj= gjlul +gj:u2 -4- ... + g~,,u,,, j <_m. Namely,
(5.4.6)
0i ... ...
= g21 .-.
0 ai
u
g2,~
.
Lgral ’-. gram
u (See Chapter III, §2). On the other hand, the associated complexwith a connection
’~ M*®o.~ (~")-~~ (M*®o)¢ (~")-~)®o,¢W ~ ... ~ (M*®o), (~’~)-~)®o)¢~ is integrable when,for a section g~ul + ¯ " ¯ + grauraof ~A
(5.4.7)
Oi ....
=- g~ ...
g~ra
[. gral ... gram g g holds, i.e., the 0-th cohomologyof the above complex.That is, the above equations 5.4.6 and 5.4.7 correspondto the 0-th cohomologiesin Proposition 5.4.2.
5.5
8 " Ggi
Historical
Notes
The next significant step from Kashiwara’s constructibility theorem is the Riemann-Hilbertcorrespondencetheorem. First recall the concept of an equivalence of categories.
5.5.
HISTORICAL NOTES
237
Let F be a covariant functor from a category C to another category C. Then F is said to be faithful whenthe mapinduced by F: ~iomc(X, Y) --~ ~-lomc,(F(X), is injective for arbitrary objects X and Y in C. Furthermore, F is full whenthe abovemapis surjective. For a fully faithful, i.e., full and faithful, functor F is said to be an equivalence betweenC and C’ whenfor any X~ in C’ there exists X in C satisfying F(X) ~ X’. Since we have ~?~Omvx( Ox , °) - -~ . ~om¢x (7"lomvx ( °, Ox),¢ x ) for a bounded holonomic complex of ~x-Modules, (i.e., ~n(M’) being holonomic) Kashiwara’sconstructibility theoremc~n be stated through the de Rham run’tot ~ follows: ~Om~x(Ox, M*) is a bounded complex of Cx-Modules whosecohomologiesare constructible. The above de Rhamfunctor restricted to a full subcategory of the derived category of bounded holonomic complex of ~x-Modulesinduces an equivalence to the derived category of bounded constructible complexOf Cx-Modules.The desired subcategory is the derived category of regular holonomiccomplexof Modules. One of the equivalent definitions of a regular holonomiccomplexof ~x-Modulesis the following: ~Homvx (M’, Ox), ~ ~Homvx, . (M~, where 0x,~ is the completion at x, i.e.,
0x,~ = l~Ox/m~. i
The above isomorphismmaybe interpreted as formal solutions being actually holomorphic(converging) ones. Regular holonomic~x-Modulesare studied "correctly" via the microlocal view in the magnificent 166-pagepaper by Kashiwara and Kawai[120]. For example, the above definition of the regularity of a single holonomic~x-ModuleMis equivalent to the following: there exists locally a good filtration (M(~)} on Msuch that the annihilator ideal of the ~x-Module ~, i.e., J(M) = {P ~ ~x : P~ = ~}, is reduced in ~x. (See Chapter III). That is, if a~(P), P ~ ~), vanishes on the characteristic variety V(M) = V(~), then a,(P) annihilates ~, i.e., for any l PM(0 C M(~+~-~) holds. Riemann-Hilbert-K~hiwara Correspondence Theorem asserts that the de Rhamfunctor ~Omvx(Ox,M’), (or the solution functor ~omv~ (M’, Ox) = ~O~x (~Omvx(Ox, M’),¢x)) provides an equivalence from the derived category of bounded complex of Dx-Modules M"whose cohomologies are holonomic~x-Modules, to the derived category of
238
CHAPTER 5.
HOLONOMIC :D-MODULES
boundedcomplex of~Tx-Moduleswhose cohomologiesare ~Tx-constructible. See Kashiwara[116]. Kashiwaraconstructs the inverse contravariant functor to the solution functor whosecompositionsare identities for those derived categories.
Chapter 6 Systems of Microdifferential Equations 6.1
Introduction
The purposeof this final Chapteris to present to the reader a series of applications of the theory developed,so far to the study of systems of microdifferential equations. The single most remarkable result of the early theory of microfunctions, is arguably the celebrated FundamentalTheoremof Sato, which (though classical by now)was probably the first result whichclarified the need for the apparently cumbersomemachinery of microfunctions theory. As we have already seen in Chapter 2, the notion of parametrix was developed in the theory of partial differential equations with the purpose of "inverting" elliptic differential operators and, analogously, pseudo-differential and microdifferential operators have been introduced and studied with the goal of generalizing parametrices and providing inverses for a larger class of operators. Sato’s Fundamental Theorembuilds up on this theory to show the conditions under which such an inversion is always possible. Before we give the statement of Sato’s result, letus try to put it in perspective. In the theory of linear differential equations with constant coefficients, the most general and powerfulresult of these last fifty years has certainly been the so called FundamentalPrin.ciple of Ehrenpreis-Palamodov[52], [178] which has shownthat whenstudying the structure of the (e.g. distributional) solutions of a linear constant coefficients differential equation P(D)u=O P(zl,...,zn) a polynomial in n complex variables, (P = D= (-i -k-°o~l,... ,-io--~, ),° u E :D’(~t~)),then the central object to be reckoned with is the so called "characteristic variety" 239
240
CHAPTER 6.
SYSTEMS OF MICRODIFFERENTIAL EQUATIONS
V~, = {z ¯ ~T’~: P(z) = and, indeed, if multiplicities are taken into account, such a variety completely characterizes and describes the space of solutions {u ¯ V’(/Rn): P(D)u 0}. This result has of course its origin in Euler’s FundamentalPrinciple (on exponential sumsrepresentation for linear constant coefficients ordinary differential equations), and, more recently, in L. Schwartz’s work on meanperiodic functions, but before its surprising announcement in 1961by Ehrenpreis, few people (if any) really believed in the possibility of such a characterization. The problem for variable coefficients operators has always, of course, been muchmore delicate, in view of the lack of elementary solution in the form of a family of exponential solutions. The first general result in the direction of a complete characterization of the sheaf of solutions of variable coefficients equations is, indeed, Sato’s FundamentalPrinciple, which, on the other hand, does indeed require the notion of microfunction, in order to be fully stated. It might be added that from our point of view, and after the preparation in Chapter 2, its proof is rather natural, as it will essentially rely on the explicit construction of an inverse, muchin the same way in which one wouldconstruct the inverse of a polynomial. Froma historical point of view, however, this construction was neither trivial nor evident. There are two different formulations for this important result. Oneof them we have already discussed in the frameworkof ~-modules, when(in Chapter 3) we have shownthat microdifferential operators of finite order can be microlocally inverted wherever their symboldoes not vanish. Moreinteresting, the inverse is, itself, a microdifferential operator. The other form, whichis the one we will describe in this Chapter, only showsthe invertibility in the sheaf of microlocal operators. Asa consequenceof this importantresult, one sees that the structure of solutions of microdifferentialequationsis essentially trivial outside the characteristic variety, and one is therefore lead to the consideration of what this structure may be on the characteristic variety. One of the early goals of the theory developed by Sato was to show that systems of microdifferential equations could essentially be classified in a very simple way, according to someproperties of their characteristic variety. This fact maybe considered as a powerfulgeneralization of the fundamentalresult from classical mechanicswhichstates that every partial differential equation of the first order can be transformed into a standard form by meansof the Jacobi canonical transformations. In order to be able to provea result of this nature for higher order linear differential equations, we will need to generalize the class of transformations which we allow for consideration. This newclass is constituted by the so called quantized contact transformations,
6.2.
THE INVERTIBILITY OF MICROLOCALOPERATORS
241
which in someform were first introduced by Maslov[163], and later on made more accessible by Egorov[45]. The plan of this Chapter is as follows. In section 2 we state and prove Sato’s Fundamental Theorem,for the case of linear differential operators of finite order. In section 3 we discuss somespecial cases of differential equations, namelythe waveequation and hyperbolic equations, which provide a first hint of what is the microlocal aspect of the study of differential equations. Quantizedcontact transformations are dealt with in section 4, while the last section provides the final classification theoremsfor systems of microdifferential equations. A short historical appendix concludes this Chapter.
6.2
The Invertibility
of Microlocal
Operators
This section is devoted to the detailed proof of the so called weak form of Sato’s FundamentalTheorem,whichstates the invertibility for any finite order differential operator in the ring of microlocal operators. To properly state this result we first review somepreliminary definitions (see also ChapterIII): given ~ P(x,D)=
~ a~(x)D
a linear differential operator of order m, its principal symbol~(P)(x, ~) is holomorphic function defined (on T*~=) by a(P)(x,~) := ~ ’~. a,~(x)~ Notethat the principal symbolof such a differential operator is invariant under a coordinate transformation; this property is clearly not satisfied by the lower order terms of P(x,D). The set of points where the principal symbol of an operator vanishes is knownas the characteristic variety of the operator, i.e. V(P) = {(x,~) e T’~: a(P)(x,~)
0}.
The reader is invited to comparethe abovedefinition with Definition 3.3.1 and the examplesthat follow. Wecan then state Sato’s theorem: Theorem6.2.1 A linear differential operator of finite order P(x, D) is left and right invertible in the ring E of microlocal operators over T*~n\V(P) or, terms of cospherical tangent bundle over {(x,~oo) e S*~: a(P)(x,~) 0}. The proof of Theorem6.2.1 is not particularly surprising as it is essentially the microlocalization of F. John’s construction of a fundamentalsolution for elliptic
242
CHAPTER 6.
SYSTEMS OF MICRODIFFERENTIAL EQUATIONS
differential equations. This construction, on the other hand, is based on the classical superposition principle. Wewill need two simple analytical lemmas, as well as a lemmaon composition and conjugation of microlocal operators: Lemma 6.2.1 Let u = U(Xl, ~) be a real analytic .f unction of
and let H(t) be the usual Heaviside function. Then (6.2.1) 0"-~ (u(x). H(Xl))
k=l
Proof.
For j = O, the lemmasimply claims that
u(x) . H(x,) -- u(x) . and is therefore immediatelytrue. For j ---- 1, the lemmaclaims that Ox~ (u(x).
H(x~)) = u(x). H(xl) +
which, once again, is immediatelytrue by the fact that H’(xl) = 5(xl). Wenow proceed by induction, assumingthat (6.2.1) holds for a given value of j. Then
j . 0.~k
/ C.~+l
iT10,~÷l-k
+~-~~(0,o~_ ~’)~(~)=w~,(~(~)) ~ ¯ H(~)+ ~
~(0,~’)~(~-~)(~),
where the l~t equMityis a consequenceof the inductive hypothesis. The lemma is therefore proved. ~ Lemma6.2.2 Let a(P)(x,d¢(x)) satisfies
~ O. Any real analytic function u(x)
~ p(~,D)~(~) ~ ~ 0 rood(¢(~))~ also satiees P(x, D) (u(x) H(¢(x))) = H(
6.2.
THE INVERTIBILITY
OF MICROLOCAL OPERATORS
243
Proof. Since de ¢ 0, we can use a suitable coordinate transformation to obtain, without loss of generality, ¢ = xl. Wenowrewrite P(x, D) as a differential o o polynomial in ~7~’ o by setting ID = (~-~,. ¯ ’, o-~), ’~’ Aj (x, D’) = ~, a~, (x)D and, finally P(x,D) = ~ Aj(x,D’)--~. j=O
Now,since u -= 0 modx~ (hypothesis) and since j ¯ k < m, we immediately obtain
u(0,-- 0; from this last equality we deduce
Since, by hypothesis, P(x, D)u(x) = 1, the lemmais completely proved. [] Werecall here (see Chapter II) that every microlocal operator K is essentially defined by integration against a kernel of the form k(x, y)dy for k(x, xI) a suitably defined microfunction.In this case, the kernel
k(y,x)dy also defines a microlocal operator whichis said to be the adjoint (or conjugate) operator of K and is denoted by K*. Incidentally, we note that ifK is well definedin a neighborhood of (x0, i~ooo), then K*is a microlocal operator well defined in a neighborhoodof (Xo, -i~0oo). Fromthis definition we immediatelysee that (whenall these objects are well defined) (K1K2)* = K~K~ and
(K*)*=
244
CHAPTER 6.
SYSTEMS OF MICRODIFFERENTIAL EQUATIONS
For the proof of Sato’s theoremwe need to computeexplicitly the adjoint of a linear partial differential operator. Lemma 6.2.3
Let m P(x,D)=
~ a~(x)D
be a linear partial differential operatorwith real analytic coe.O~cients.Thenits adjoint operatoris the linear differential operatordefined by P*(x, D)u(x) = ~ (-1)I~ID ~ (am(x)" u(x)). Proof. Wewill proceed step by step. First we compute the adjoint of the operator induced by multiplication by am(x). Since, as a microlocal operator, this multiplication is represented by the microfunction
ao(X)~(y we see that (am(x).)* is represented, by definition, by the kernel
a.(y)~(x i.e.
(a.(~).)* =(a.(~).). Wenowproceed to computethe adjoint of the differential operator 0 whosekernel microfunction is
~ 5(y- x). Therefore, the kernel whichcorrespondsto (~-~7~)is given, in view of the definition, by
~--
so that we conclude that
(~j)~(yl--XI)’’..’(~(yn--Xn),
6.2.
THE INVERTIBILITY OF MICROLOCALOPERATORS
245
(D°)* = ~, Ox~l "’"" * 0-~n~, =
o"°)
Finally, by the linearity of conjugation, and again by the properties of the conjugate of compositeoperators, we obtain that P*(x,D)u(x)---
~ (aa(x)D~)’u(x)=
~ (D~)’(aa(x)u(x))’=
= E(-1)l-lDO(a.(x).(x)), which concludes the proof of the Lemma. Wefinally have all the tools to prove Sato’s FundamentalTheorem, which werestate here for the sake of readability: Theorem6.2.2 A linear differential operator of finite order P(x, D) is left and right invertible in the ring ~: of microlocal operators over T*~’~\V(P)or, terms of cospherical tangent bundle over ((x,~) E S*~i? a(P)(x,~) ¢ 0}. Proof. The nature of this result is obviouslylocal, and we therefore consider it in a neighborhoodof the origin x = 0. Let ~0 be a point such that a(P)(0, i~o~x~) By the well knownCauchy-Kowalewsky Theorem, see e.g. [123], there exists a real analytic function u(x, ~,p), defined in a neighborhoodof (0, ~0, 0) E n × ~, such that P(x,D)u(x,~,p) and u--0
mod (x.~-p)m;
note that we have used here in a crucial waythe hypothesis of the non vanishing of a(P). The solution u which we have constructed is a unitary solution (or also a Leray solution). Set now v(z,~,p) := u(x,~,p) . H(x. ~ -p). By Lemma6.2.2, whose hypotheses are immediately satisfied,
we have that
246
CHAPTER 6.
SYSTEMS OF MICRODIFFERENTIAL EQUATIONS
P(x, D)v(x, ~, p) = H(x. ~ If we take the n-th derivative with respect to p (which is a real variable) obtain P(x,~,p)
~p~ v(x,~,p)
= (-1)"5(’~-l)(x.~-p),
P(z,~, p)w(~:, ~,~,)=(-1)"~("-’)(x. where we have set
w(x,~,~) =~-~(v(x,~,~)), and, finally, (6.2.2)
P(x, D)w(x, ~, y. ~) = (-1)nh(n-’)((x -
once we have chosen p to be y-~. In order to obtain a fundamentalsolution for P(x, D), we nowintegrate (6.2.2) with respect to ~, in a neighborhoodU of By the plane wave decomposition of the delta microfunction (see Chapter II), we have that, in a neighborhoodof (0, 0; ~0(x - y)oo), it P(x, D)/u w(x, ~, y. ~)w(~) (- 1)" f~~(n-i) ((z
_ y) . ~)~(
Wenowtherefore define /
1
t
\n--1
so that, in a neighborhoodof (0, 0; ~0(x - y)oe), one P(x, D)E(z, y) -= 5(x Now(by the definition of the sheaf Z:) E(x, y)dy will define a microlocaloperator in £: if we can showthat, in a sufficiently small neighborhood,its singularity spectrum is contained in T*(~n x ~). As a matter of fact, it is indeed immediate to verify that E(x, y)dy wouldthen be, in £, a right inverse of P, since, quite simply,
=
(
f
-
/
6.2.
THE INVERTIBILITY OF MICROLOCALOPERATORS
247
= P(x, D)E(x, y)dy = 5(x As for the singularity spectrum of E(x, y)dy, we recall that u was taken as a real analytic function defined in a neighborhoodof (x,~,p) = (0,~0, 0); as consequence, v(x,~,p) is a hyperfunction defined in that same neighborhood, whosesingularity spectrum is contained in the set {x.~ =p,+(x~Thus, by the results in Chapter II on indefinite integrals of hyperfunctions, we obtain the desired estimate on the singularity spectrum of E(x, y)dy which therefore defines a microlocal operator. To conclude the proof, we will show that E = E(x, y)dy is also a left inverse for the operator P = P(x, D). In other words we will showthat in addition to PE = 1 we also have EP= 1. To begin with, we note that if P* is the adjoint operator of P, then (by Lemma 6.2.3) a(P*)(x, -i~) = a(P)(x, Therefore, by what we have just proved, there exists a microlocal operator E’ such that (in a suitable open set) P* E’ = 1. The adjoint (E’)* of E’ is still a microlocal operator and now (E’)*P = (E’)*(P*)* (P*E’)* = whichshowsthat (E’)* is a left inverse of P. Finally, (E’)* = (E’)*PE = (E’*P)E which concludes our proof. A simple, but quite fundamental, corollary can be immediatelyestablished, thus giving a form of Weyl’slemma(see e.g. [123]) for partial differential equations on hyperfunctions. Theorem6.2.3 Let u and f be hyperfunctions which satisfy (6.2.3)
P(x, D)u(x) = f(x).
Thenthe following inclusion holds: (6.2.4) S.S.(u) C_ {(x, ~x~) e S*~": a(P)(x, i~cx3) = 0} U S.S.(f). In particular, if P(x, D) is elliptic at a point Xo E ~, and if f is real analytic, then also u is real analytic.
248
CHAPTER 6.
SYSTEMS OF MICRODIFFERENTIAL EQUATIONS
Proof. If we consider equation (6.2.3) in the space of hyperfunctions, and lift it to the level of microfunctions (i.e. we read every hyperfunction as the microfunction it defines in the quotient space) we obtain P(x, D)sp(u(x) = sp(f(x)). From Theorem 6.2.1 we know that at those points (x,~) for which a(P)(x,i~cx~) ~ then the re exi sts a m icrolocal lef t inv erse E s uch tha t EP = 1. Therefore sp(u) (EP)sp(u) = Esp(f), whichproves the first assertion, i.e. (6.2.4). Asfor the second, it follows immediately from (6.2.4). Another more or less immediate consequence, with which we close this section, is the following: Theorem6.2.4 Let P(x, D) be an elliptic
operator at a point xo. Then
P(x, D) : o -~B,o is surjective. Proof. By ellipticity, Sato’s Fundamental Theoremimmediately implies that P(x, D) has a right inverse at each point (x0, ~). This inverse, in particular, is unique since it is also a left inverse. In turn, this implies that there exists a right inverse operator for P, in a full neighborhoodof ~r-l(x0). FromSato’s Fundamental Exact Sequence we knowthe exactness of
so that there exists h E B~osuch that Esp(f)=sp(h),
sp(Ph- f) = in a neighborhood of r-l(Xo). By Theorem6.2.2, g:=Ph-f is a real analytic function while, by Cauchy-Kowalewsky, there exists a real analytic function v, again defined in a neighborhoodof x0, such that
6.3.
A FIRST APPROACHTO BICHARACTERISTIC STRIPS
249
Pv=g. Our theorem is nowproved by taking u = h - v, so that, in a neighborhoodof X0~
Pu = f.
6.3
A First Approachto Bicharacteristic
Strips
In the previous section we have constructed the inverse of a linear differential operator as an element of ~ under the assumption that a(P)(x, i~oc) ~ The proof, as we have seen, was based on the construction of a fundamentalsolution for P. A specific case in whichthis can be done explicitly is whenP is the wave operator, according to the followingdefinition: Definition 6.3.1 The wave operator on ~ x 1~n is defined by
Q = ot~
+"" +
= ~- ~"
Weare interested in exploring the Cauchyproblemfor such an operator, namely the problemof finding a hyperfunction u(x) such that
Qu= o
~(o,~)=~o(~) ~(o,x) ~(~). The reader is of course familiar with the Cauchy-Kowalewsky Theoremwhich guaranteesthe existence of a real analytic solution if real analytic initial data are given. The situation for hyperfunctionsis quite different and, in general, we do not even knowwhether a hyperfunction solution to (6.3.1) exists. When does though, such a solution is unique in view of Holmgren’sTheorem.Let us consider the following hyperfunction on ~ x ~:
and
It suffices a simple direct computationto showthat
250
CHAPTER 6.
SYSTEMS OF MICRODIFFERENTIAL EQUATIONS
Qu+ = Qu_ = o. Since one can see that
_ ~,t;~(~-,...,~-,~=l)oo {(0,0;i(~l,...,~n,,)~):~2 then by Theorem6.2.1 (i.e. essentially have that
:~#0,t2=x~+...+zu
~+...+~,~>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~--- <
(6.3.3) then
~(~)(x) \i-----1
(6.3.4)
Xi
=-
\i-----1
~(u+I~1-<<)a~5(~)(z)-
As a consequence we obtain the following lemma. Lemma6.3.1 If u ~ B{0}(~") is homogeneousof degree ~, then:
(i) ~(~)=o il ~ ¢ -~,-~- ~,-~2,... (ii) ~(=)= E~== ~.6(-)(=)g~ = -n - .~, Pro@Since u ~ B{0}(~=), (6.3.2) holds true. By the homogeneity of have (6.3.3) and the Lemma is therefore an immediateconsequenceof (6.3.4).~ Wenowrecall that both u+(0, x) - u_(0, x) and ~(0, x) are supported the origin. Moreover,by definition, u+(0, x) -u_(0, x) is homogeneous of degree o~ (n x) is homogeneousof degree -n. By Lemma6.3.1 (-n- +1) while ot ~, obtain that
~,+(o, z)=u_(O,x) and
o~(0, x) =c~(=), where c± is a constant which can be explicitly computedand is given by (see
e.g.[103])
c~ = ±r(~)" Weare ready to prove the existence of a hyperfunction solution to the Cauchy problemfor the waveequation. Theorem6.3.1 Let ~Ooand ~o1 be arbitrary hyperfunctions on ~n. Then there exists a (unique) hyperfunction u(x) whichsolves (6.3.1).
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CHAPTER 6.
SYSTEMS OF MICRODIFFERENTIAL EQUATIONS
Proo]. Uniqueness is a consequence of Holmgren’s Theorem, so we will just prove the existence of a solution u. Followingthe computationsof the previous pages we define two new hyperfunctions on
Kt(t,x) = 2-~{~÷(t,x) - ~_(t, and
Ko(t,~)=-~K~(t, x). To begin with, we immediately.notice that QK~(t,x) =
I
KI(O,~) =
(6.3.5)
~(0, ~) ~(~)
so that K1is a hyperfunctionsolution of the Cauchyproblemwith data (0, 5(x)). Onthe other hand, (6.3.5) implies that QKo= ~--~(QK~) = Ko(O,x) = 5(x) and
oK0 (0, z) = Ot This last equality follows from the fact that
OKo- 02K~ =
2Ot Ot and since A does not depend on t, we have
OKo~ -~(0, x) = (AK,)(0, x) = A(K~(O, Wealso note (see Chapter I) that supp(Ko)~_ supp(K1)C {(t, x) e ~’~+t Ixl 2 < t 2} and we can therefore define the following hyperfunctions: u(t~ x)
:= f Ko(t,x - y)~o(y)dy + f K~(t,x
- y)~91(y)dy.
Nowwe see that u is the required solution of (6.3.1), since:
6.3.
A FIRST APPROACHTO BICHARACTERISTIC STRIPS
253
Qu(t, x) = / QKo(t, x - y)~o(y)dy + / QKI(t, x - y)~l(y)dy
~(0,x) = f ~(x- y)~0(y)dy = = ~tt (O,Y) = / ~ff-~(O,x - y)~o(y)dy + ~-~(O,x - Y)~(y)dy =/~(x -y)~(y)dy
= ~(x).
[] The next question we wouldlike to ask about the waveoperator is the structure of its microfunction solutions sheaf. Weknowthat this sheaf is trivial outside the characteristic variety V(Q) = {(t,x;i(T,~)oc)
e iS’Ktn+~ 2 = ~},
but we would like to knowwhat happens on V(Q). The answer to this question is hiddenin a concept whichis well knownin the theory of differential equations and whichshows(for the first time in this book) an interesting interplay between contact geometry and differential equations. The notion we need is that of bicharacteristic strip: Definition 6.3.2 Let P(x, D) be a linear differential operator and let p(x, ~) its principalsymbol.Anyintegral curve(x( t ), ~ ( t ) ) of the dx~
dxn
-d~
-d~n
whichsatisfies
~.(~(t),~(t)) is said to be a bicharacteristic strip for the equationP(x, D)u(x) = O. Its first projection {x(t)} is called the bicharacteristic curve. Our next theorem (again a consequence of Sato’s FundamentalPrinciple) will showthat the microfunction solutions of the waveequation only propagate along bicharacteristic strips. Weneed a preliminary computation: Lemma6.3.2 Let E(t,x) := K~(t,x) . H(t). Then: (i) QE(t,x) -- 5(t, (ii) supp(E){(t, x): Ixl <_ t};
254
CHAPTER 6.
SYSTEMS OF MICRODIFFERENTIAL EQUATIONS
(iii)
S.S.(E)
# (0,0)}
Proof. Wefirst note that OE
OK1. H(t)
ot and therefore 02E (t,x) = O2K1¯ H(t) 5(t,x). 2Ot Ot2 Since, moreover, H does not depend on x, we have AE(t, x) = AK1. H(t), and therefore (i) follows immediately. As for (ii), this is an immediateconsequence of the fact that supp(K1) C_ {(t,x) : ~ _< t 2} and supp(H) C_ {(t,x) : t _> 0}. Finally we need to prove (iii). To begin by (i) one has that S.S(E) ~ {t <_0} C_ {t = x = 0}. Moreover,for t > 0, E(t, x) = K~(t, andther efore, by t aking the boundary value from Im t > 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.
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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
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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
xn : Y’~? yj~,~ j < n and
YJ= ~n Yn
~i I
~ j
nj -x~ j < n and the generating function is n--1
a(x,y) =z. - y. +~] x~y~. j=l
Our next step will be to showthat these contact transformations can be lifted from the manifoldson whichthey act, to the sheaves of differential (and microdifferential) operators on such manifolds. The lifting are the so called quantized contact transformations. The process of quantization can of course be done for any contact transformation but, in view of Remark6.4.6, we will state our next result for the case of contact transformations with a generating function. Weomit its proof which can be found in [185] and in [206]. Theorem6.4.1 Let X, Y be real analytic manifolds of dimension n, and let ~ satisfy the conditions stated before:
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D is non singular; det d~f~ dzd~f~ J 7~ 0 on A. Then, for any microdifferential operator P(x, D~), there exists a microdifferential operator Q(y, Dr) such that, for every microfunction (6.4.3) / P(x, Dx)5(f~(x, y))u(y)dy = / 5(f~(x, y))Q(y, Moreover, (6.4.3) induces isomorphisms of sheaves p-lgx ~_ q-l~.y (p being the mapwhich, given Q, produces P and q being the map which, given P, producesQ, according to (6.4.3)), p-~Sx(m) "~ q-~Sy(m) and finally P-~As.x = q Definition 6.4.4 The isomorphism p, q in Theorem6.4.1 are called quantized contact transformations. Remark6.4.8 If we apply this process to the Legendre transform, we obtain that the mapswhich relate P(x, D~) to Q(y, D~) are given by xj -- Oyj
0 0 Oxn and its inverse
0 0 Oyn
j
6.5.
STRUCTURE OF SYSTEMS OF DIFFERENTIAL EQUATIONS
YJ = ~xj ~x~
261
j < n
Yn = x ¯ Dz 0 0 -- -- -xj-Oyj -0 0 Oy, Oxn
j < n
The reader should note here that since n-1
~ = x~ - y,~ + ~ x~yj, the operators ~__~, (o~ )-1 and (o--~,) are well defined microdifferential operators for ~ ~ 0 and ~ ~ 0. Onemoreimportant result is the fact that quantized contact transformations are isomorphismsat the level of microfunctions. Theorem 6.4.2 Assume the same hypotheses as in Theorem 6.4.1. Then the map --~ / 5(~(x, y))u(y)dy is an isomorphism between Cx and Proof. Consider K(z, Y)
:= f
It will be sufficient to showthat K is a kernel for an invertible microdifferential operator on S*X. This follows immediately from Sato’s Fundamental Theorem as given in Chapter V (see [185], [206] for details).
6.5
Structure of Systems of Differential tions
Equa-
In this last section we want to prove the fundamentaltheoremwhichclassifies the structure of systemsof microdifferential equations at generic points. This result is similar in spirit to all classification results fromalgebraic geometryas it will showhowto reduce quite general systems to a small numberof canonical forms.
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It is in this context that we will be able to see the great powerof the theory of microfunctions versus the theory of distributions or even microdistributions. To begin with, we will deal with the simple case of systems of microdifferential equations with simple characteristics and one unknown:we will showhow to microlocally reduce it to the so called partial de Rhamsystem. Let us first recall a few very classical facts and definitions. Definition 6.5.1 An analytic subset A of a contact manifold X is said to be involutory if wheneverfib = glh = O, then {f, g}lh = O. The notion of involutory manifoldis particularly relevant as Sato has proved that the characteristic variety of an arbitrary systemof microdifferential equations is always involutory. Also note that the property of being involutory necessarily implies that if dim X -- 2n - 1, then dim A _> n - 1. Definition 6.5.2 An involutory submanifold ~ of (X, w) is regular if w never vanishes on ~. On the other extreme, we say that an involutory submanifold~ of (X, w) Lagrangianif dim A = n - 1 (and in particular this implies that wl~ ---- 0). Remark6.5.1 Lagrangian manifolds are particularly important as they are the characteristic varieties of a fundamentalclass of systems, i.e. the holonomic or maximally overdetermined systems (see section 5.3). Let us nowdescribe the general strategy which we will use to deal with the structure (or classification) theoremsfor systemsof microdifferential equations. Suppose(we use here the notations introduced in the previous Chapters) that we start with a system.hal of microdifferential equations on ~T/n of finite order and one unknownfunction. The first, classical, step consists in applying some contact transformation to ~*~’~ in such a way as to make the characteristic variety of Mlinear. This is nothing but the following famousJacobi integration result: Theorem6.5.1 Any involutory regular codimension d submanifold of a contact manifold can be expressed as
{(x,~): ~1..... ~d= 0}, for some canonical coordinate system (x, ~). One usually says that Theorem6.5.1 deals with the geometrical optics. The second step consists in applying an invertible quantized contact transformation to ~A, so to transform it into a system A~/~with linear characteristic
6.5.
STRUCTURE OF SYSTEMS OF DIFFERENTIAL EQUATIONS
263
variety. Finally, one looks for suitable invertible microdifferential operators to transform AA’into the partial de Rhamsystem, say Af. This last part is usually referred to as the treatment of the waveoptics. As we already mentioned,the treatment of the systems is quite algebraic and it relies on a formal calculus whichcan be developedfor microdifferential operators of finite order, in analogy with what is knownfor functions of several complex variables. Wewill need, in particular, the followingversion of the Weierstrass’ division theorem, whoseproof can be found in: Theorem6.5.2 Let P(x, D) be a microdifferential that, for somepositive integer p,
operator of order m such
a(P)(x,~) ~.-~{(o;~(~,o,...,o,~,,)oo) is
a holomorphic
.function
of
~n which
never
vanishes
of ~ = O. Then, in a neighborhoodU of (0;i(1, 0,..., uniquely decomposedas
in
some
neighborhood
0)~x~), P(x, D) can
P(x, D) = Q(x, D) . R(x, where Q is invertible in U and Op
p-1
R(x, D) = ~ + ~ Rj(x, j=O
, ~h
0 with D’ = (o-~,,’", o~7-1), ord (R~) <_ p - j and a(R~)(O;i(1,O,...,O)oc)
Our first result deals with the case of simple characteristics, and if we are looking at the case with only one unknown,we can use Theorem6.5.1 to note that, without loss of generalities, on mayassumea(P) Theorem6.5.3 Let P(x, D) be a microdifferential operator of the first order defined in a neighborhood of (0; i(0,..., 1)(x)), then the equation P(x, D )u = is microlocally equivalent as a left C-module,to the partial de Rhamequation ~U
i.e., .for ~ the sheaf of microdifferentialoperators, (6.5.1) ~P ~ 0--~7
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Proof. Theorem6.5.2 allows us to write
with D~ = (D2,..., Dn) and Q invertible in U. It is therefore sufficient to prove the theorem for~° _ A(x,D’). To conclude (6.5.1) we need to construct invertible microdifferential operator R(x, D’) such that ~x~ =+~R-1PR
(6.5.2)
R= Rk(x,D’) k=O
with Ro=l
(6.5.3) and
= A.
(6.5.4)
Rklxl=o=O for k_>l. But (6.5.4) can actually be considered as a set of defining equations for the Rk, and in fact one can use the well knownlemmaof Boutet de Monveland Kree (see [185] as well as Chapter V) to showthat the operator R defined by (6.5.3), (6.5.4) and the secondformulain (6.5.2) is a well defined microdifferential operator (see [185], [206] for moredetails). Onthe other hand, (6.5.3) and the second formula in (6.5.4) imply that a(R)lx,=o = 1 ¢ 0 (note, in particular, that ord (Rk) <_ 0 for any k so is ord (R)). By Sato’s FundamentalTheorem(in the form which was proved in Chapter V), R is therefore invertible. However,by (6.5.3) (6.5.4)
it also follows that
0 which concludes our proof.
oR=Ro-[]
Our next step will consist in showinghowthe general simple characteristic works(i.e. what happenswhenwe still have a non-singular characteristic variety, whosecodimensionis, however, bigger than one). Before doing so, however, we want to briefly discuss what happenswhenthe simplicity assumptionis removed. Asa matter of fact, such an assumptionis quite restrictive, and it wouldbe quite natural to try to removeit (or at least weakenit). In fact, as we will show a second, this problem was at the heart of the motivations which stimulated
6.5.
STRUCTURE OF SYSTEMS OF DIFFERENTIAL EQUATIONS
265
Sato and his coworkersto introduce infinite order differential operators. The next Example,Proposition and Theoremare taken from [206] and illustrate this argument. Example 6.5.1 tim equation
Considerthe followinglinear (constant coefficients!) differen~2 u
(6.5.5)
PI(D)u := ~ =
and (6.5.6)
02v Ov P2(D)v .- Ox~ Ox~ -
in the space of distributions. Even thoughthe principal symbols(and therefore the characteristic varieties) coincide evenup to multiplicities, the structure of their solutions are quite different. So, an immediategeneralization of Theorem 6.5.3 will have to account for this phenomenon.In stark contrast with this examplewe have the following result. Proposition 6.5.1 Equations (6.5.5) and (6.5.6) are microlocally equivalent left ~-modules, i.e.
In particular, the sheaves BP~ and Bp~ o] hyper]unction solutions to PI and P2 are isomorphic(and so are the sheaves P~ and CP~ of microfunction s olutions). Proof. Toshowsuch a result, it suffices to construct linear differential operators A~(x, D), A2(x, D), Aa(x, andA4(x, D) suchthat S P~A~= A~P~ AaP~_ P~ A4
(6.5.7) and
{
A4AI------ 1 mod A1Aa -- 1 mod In fact, if (6.5.7) and (6.5.8) hold, then the equations P~(D)u= and Pe(D)v = 0 are equivalent by the correspondence (6.5.8)
u = Aav and v = A~u. Toprovethat (6.5.7) and (6.5.8) hold, it suffices to considerthe followinginfinite order differential operators:
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A2(x, D)
A3(x, D)
"= (cosh(xl~o~2))
(l+xlo~)+sinh(xl~-~
(-~-~1+x1£)
(l+x~__~)+sinh(x~~o~
( ~x~O
A4(x, D)
= (cosh(x~f-~2))
The reader should note that all of these operators are well defined infinite order differential operators (with variable coefficients), in view of the growthconditions which we have described in Chapter I; also, the square roots are purely formal, in view of the Taylor expansion of cosh t and sinh t. The proof can now be concluded by a direct computation: [] Remark6.5.2 The proof of Proposition 6.5.1 showsthe reason why(6.5.5) and (6.5.6) are not equivalent in the frameworkof distributions; the reason is that the operators Aj(x, D), j = 1,..., 4, whichrealize the isomorphismare of finite order. Onthe other hand, in contrast with the general situation which we will outline in our next result, equations (6.5.5) and (6.5.6) are equivalent even the level of hyperfunctions (and not just microfunctions) because the operators Aj are differential operators (though of infinite order) rather than moregeneral microdifferential operators. Theorem6.5.4 Let P(x, D) be a microdifferential operator of order m defined in a neighborhood of (0;i(0,..., 1)~x3) and such that a(P) = ~. Then, in neighborhoodof (0;i(0,..., 1)cx3), the equation P(x,D)u = 0 is microlocally equivalent, as a left E-module,to the differential equation
6.5.
STRUCTURE OF SYSTEMS OF DIFFERENTIAL EQUATIONS
267
Omu
If, moreover,P(x, D) is actually a differential equation, then the equivalence holds as left :D-modules,7) being the sheaf of linear differential operators of finite order. In the first case, only microfunctionsolutions are equivalent, while in the secondcase, also hyperfunctionsolutions are equivalent. Proof. The proof follows exactly the samelines as for the Theorem6.5.3, but the computations are necessarily more complexsince it becomesnecessary to involve infinite order differential operators. [] Weare nowready to go back to the case of simple characteristic to prove the analogue of Theorem6.5.3 for higher codimension. Theorem6.5.5 Let ~ be an £-module defined in a neighborhoodof (xo, i~o~) be such that: there is a left ideal Z such that A~I = A/[/Z (i.e. we are looking at a system with one unknownfunction); let J := Urn{am(P) : P ¯ ~ $( m)} and le t Y( J) be its zero set: then V(J) is a non-singular codimension manifold in a neighborhoodof (Xo, i~oc~) and wlv(j ) ~ 0 (for w -- ~ ~jdxj the canonical form); V(J) is real; the totality of ~-homogeneousanalytic functions which vanish on V(J) is Then, via a quantized contact transformation, we can transform JP[ into the system
Oxl
+’"+
Ox.
Proof. By the commentwe made after Definition 6.5.1 we know that V(J) is involutory and therefore (6.5.10) and (6.5.11) imply that, by Theorem6.5.1, V(J) can be written as V(J) = {(x,~): ~1 .....
~d = 0},
and by (6.5.12) we can choose PI,...,Pd ¯ ~ such that a(Pj) = ~j for j = 1,..., d. In particular, by just following the proof of Theorem6.5.3, we can
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assume P1 to be Y~7~" o Wenow show,for by some induction, we can choose Pj to be __o for j -- 1,. , d. Assume that, k < d, that we have ""
0 0 0 Pl=~-~x~, P2=~-~x2,...,Pk=~-~x ~Now,since --=0
for
j=l,...,k
we may assume 0 0 Pj = Pj(x, Oxk+~,... ’cOx,~ -) for jk =1,.+ ..,n. Weierstrass’ division theorem(in its version for microdifferential operators) showsthat there exists a microdifferential operator Q~:+~(x, o ’ ¯ ¯ ¯ , o--~,)of orO~Xk+l der at most zero, and an invertible microdifferential operator R(x, i)Xk+l o ~ " " " ~__a) Oxn such that (6.5.9)
P~+I(x,
+ 1,...,
-~x~n) = R(
) + Qk+~.
From(6.5.13), and using the Sp~ith theoremin its natural version for microdifferential operators (see [123]), we can actually deduce that the generators P1,--., Pk+l can be chosen so that (6.5.10)
[ P~:+~(x,D) = R(o%~+~+Qk+l(X,
Notethat, for j = 1,...,
k, the Lie bracket
[0-~, P~+I] := j ~----~P*+I- Pk+IOx belongs to Z and so, by (6.5.14), also [o°~, Qk+~]E ~ for j = 1,.., k. again by (6.5.14), one can even show that [~,Qk+~] = 0 for j = o i.e. Qk+~only depends on (xk+~,...,x,~, 0~,+1’"" 0~, __o) Following steps as in Theorem6.5.3 we can find an invertible microdifferential ~ 0 R(Xk+~,..., x,, 0~,+~," " ~--~,) such that
Moreover, 1,...,k, the same operator
R-~p,+~= O___R-~ Since R commutes with o~ for j = 1,... ,k, if we replace a generator u of M by Ru, we can nowchoose generators P~,..., Pd for Z such that
6.5.
STRUCTURE OF SYSTEMS OF DIFFERENTIAL EQUATIONS 0
269
0
so that we have shown, inductively, that P1 -- ~7~’" Pd We now ~_o generate Z, which would conclude our proof. proceed to prove that _~o Oxl’ " " " ’ Oxd If this were not so, we could find R(x,o,~+1"", o-~) E Z, R ~ 0. Then, it would be a(R)lv(s) = 0, but since a(R) does not dependon ~1,.-., ~d, this wouldimply a(R) =- 0, which would contradict the fact that R ~ 0. [] This last result can be extendedto the case of non-simplecharacteristic by allowing the use of infinite order differential operators exactly in the sameway which Theorem6.5.4 extended the result of Theorem6.5.3. Since the treatment of higher codimensionnon-simplecharacteristics is a little morecomplicated, we will not give the precise statement here, but refer the reader to [206], Theorem 5.3.7. Wecan nowuse Theorem6.5.5 to emphasizethe role of bicharacteristics in the study of somesystems of differential equations. Definition 6.5.3 Let V be an involutory submanifold of S*h~~ satisfying (6.5.10) and (6.5.11), and suppose V --- ((x,i~) e S’M: f~(x,~) = 7)d = fd(X,~) The bicharacteristic manifold B = B(~o,~o~) associated to V and passing through (x0, i~oo~) ~ V is the dimensiond integral manifold through (x0, i~0oc) of the d Hamiltonianoperators t=~k O~tOxt ~xt ~-~t ) ’ j = l, . . . , d. The next result showsthe great powerof our previous analysis of systems of microdifferential equations. Theorem6.5.6 Let All be an g-module as in Theorem6.5.5. Then (in a neighborhoodof (xo, i~ooC)) the microfunction solution sheaf Hom~(A~,C)is ported in V and is locally constant along each bicharacteristic manifold. Moreover, Hom~(A~l, ~) is a flabby sheaf in the direction transversal to bicharacteristic manifolds, and, for j ~ O, Ex~(A~,C)= Proof. The nature of this statement is clearly invariant under quantized contact transformations and so, in view of Theorem6.5.5, we may assume that NI is the partial de Rhamsystem Ou _ Ox~
Ou
O,
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for whichthe result is clearly true. Remark6.5.3 Whenwe say that a sheaf ~q is flabby in the transversal direction to the bicharacteristic manifolds, we meanthat there is a manifold U0, a flabby sheaf ~ on U0 and a smooth morphism ~o : U A V -~ U0 such that the bicharacteristic manifolds in U A V are the fibers of ~ and Slunv Remark6.5.4 A completely analogous result holds for the case of non-simple, codimensiond, characteristics. The reader maysee Theorem2.1.8 in [206]. Remark6.5.5 Theorem 6.5.6 really shows two fundamental features. On one hand the flabbiness of the sheaf of microfunction solutions is nothing but an elegant restatement of the propagation of singularities along bicharacteristic manifolds. On the other hand, the vanishing of the higher Ext functors shows, in particular, that the system ~4 is locally solvable, given suitable algebraic compatibility conditions. Not all systems of course, have characteristic varieties which satisfy the conditions of Theorem6.5.5. There are at least (and, in. a sense, only) two more important cases whichwe need to discuss (but we refer the reader to [185], [206] for more details and proofs). Theorem6.5.7 Let J~ = $ /Z be a system of microdifferential equations in one unknownwith simple characteristics. Assume,moreover, that its characteristic variety V satisfies the following conditions: (i) V ~ V is a non-singular involutory manifold; (ii) V A V intersect transversally; (iii) wlVC~ V # O. Then the system Iv[ is microlocally equivalent to the partial Cauchy-Riemann system Af.-O~
j
+
u=O,
j=l,...,d,
where d is the codimension of V. Systems of this sort enjoy the same properties one would expect from the (partial) Cauchy-Riemann.One has, in particular, strong propagation of regularity results. To state them(their proofs being essentially obvious after Theorem 6.5.6) we need an extra definition.
6.5.
STRUCTURE OF SYSTEMS OF DIFFERENTIAL EQUATIONS
271
Definition 6.5.4 Let fld be a system of finite order microdifferential operators whosecodimensiond characteristic variety V satisfies conditions (i), (ii) (iii) of Theorem6.5.7. The 2d-dimensionalbicharacteristic manifold of V C? through (Xo, i[ooC) is called virtual bicharacteristic manifold of Remark6.5.6 Since the notion of bicharacteristic manifoldis obviously invariant under contact transformations, then the notion of virtual bicharacteristic manifoldis invariant under real contact transformations. Remark6.5.7 Let Af be the partial Cauchy-Riemannsystem of Theorem6.5.7. Its virtual bicharacteristic manifoldthrough (x0, i[0oc) is given {(x0, i[o~):xj=(x0)j
for j=2d+l,...,n;~--~0}.
Finally, we can state Theorem6.5.8 Let .M =- ~/Z. be as in Theorem6.5.’7 and let U be any open set in the virtual bicharacteristic, manifoldof Jr4. Thenevery microfunction solution o fAd whichvanishes in U also vanishes everywherein the virtual bicharacteristic manifold. Finally, the deep work of H. Lewyon linear partial differential equations with no solutions, has stimulated the study of one moreimportant case, i.e. the so called Lewy-Mizohatatype systems. Definition 6.5.5 Let V be an involutory submanifold of S*ff~n which, in the neighborhood of a point (Xo, i~oc~) is written {(x,i[ec) pl (x,i~) .. ... p~(x,i[) = 0}. Then the generalized Levi form of V is the hermitian matrix whose coeJ~cients are the Poisson brackets {pj(x,~),~k(X,~)}l~j,kSd. Remark6.5.8 Note that the signature of the generalized Levi form is independentof the choice of the defining functions pj, and is also invariant under a real contact transformation. The structure Theorem,in this case, runs as follows [185]: Theorem6.5.9 Let J~4 = ~ /Z be an ~-module defined in a neighborhood of (xo, i~ooc) and whichsatisfies (6. 5.40) and(6.5.12). If the generalizedLevi of V( J) has p positive eigenvalues and d - p negative eigenvalues at (Xo, then .~A is microlocally equivalent to the (p, 1 - p)-Lewy-Mizohatasystem
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The great interest of these results can only be appreciatedin viewof the following structure theoremfor general systems [185]: Theorem6.5.10 Let .~ be an admissible and regular system of microdifferential equations such that Y n ~ is regular, Tz(V) ~ Tx(~) = Tz(V ~ ~) for x E V, and such that its generalized Levi form is of constant signature (p, q). Then A/[ is microlocally isomorphicto a direct summand of the direct sumo.f a finite numberof copies of the system .hf which, in a suitable neighborhood,has the following form:
wherer = 2 codim(V) - codim(V ~ ~) and s = codim(V ~ ~) (V)- (p +
6.6
Historical
Notes
Mostof the results described in this last Chapter are so recent that it is quite difficult to provide a meaningful historical appendix. As we have seen in section 6.1, Sato’s FundamentalTheoremis, from a philosophical point of view, .a far reaching generalization (at least in somesense) of Ehrenpreis’ Fundamental Principle. As it stands now,and in the frameworkof hyperfunctions, it is also a very elegant result which,in a way, had been anticipated by somedeep results of, e.g., Mizohata[166], [167] and HSrmander [80]. As for the techniques, employed in its proof, they really consist in little morethan a microlocalizationof John’s workon elliptic differential equations, [100], [101]. In section 6.2, on the other hand, we dealt with the study of the Cauchy problem and of the phenomenon of propagation of singularities along bicharacteristics. The first worksof Sato
6.6.
HISTORICAL NOTES
273
in this direction, [200], had appearedbefore the birth of microfunctions, so the notations (and also somebasic concepts) were quite different. Aninteresting comment on the psychological status at those times is given in [185]. In section 6.2 we have deliberately avoided the treatment of those operators for whichit is natural to study the Cauchyproblem, namely the hyperbolic operators (whose microlocal analysis is essentially due to Kawai[135]) and, moregenerally, the micro-hyperbolic operators, introduced by Kashiwaraand Kawaiin [118] and [119]. The fact that bicharacteristic strips should be the maincarriers of singularities has, on the other hand, slowly developedfrom the worksof F. John [101], to reach its fully maturity with the creation of microfunctions. Section 4 has dealt with contact geometry which in its modernform was probably originated by Jacobi. The evolution of different geometries, according to the different needs of mechanicsis beautifully described in Arnold’s treaty on Classical Mechanics. This section, however,also contains the very important process of quantization of a contact transformation. It is usually accepted that the first to push such an idea was Maslov[163], essentially a physicist, whosework was madefamous by Egorovin [45]. Section 5, finally, has dealt with the so called structure theorems. The material we have described is taken from [185], [206]. There is an interesting difference betweenthe treatment given in [185] (where operators on ~n, or on real manifolds, are defined as restrictions of operators defined on ~n, or on complex manifolds) and the one given in [206] (which we have mostly followed, and where microdifferential operators are defined directly on S*~n). As for the partial de Rhamsystem, the simplest case (i.e. a single equation) was originally dealt with by Kawai. Of course, its corollary on the propagation of singularities along the characteristics wasof great interest even in the C~ case (wherea complete solution had to wait until the late sixties). The case of the partial Cauchy-Riemann system was (again for d = 1) treated by Kawai. The treatment of the Lewy-Mizohatais obviously influenced by the striking discovery of Lewy[148] in 1957, as well as by the later worksof Mizohata[165], [166], and [167], whofirst understoodthe role of equations such as
Finally, one should point out that according to Hitotumatu, a conjecture on the general structure theorem was formulated by Bers in 1956 during the symposium on analytic functions at Princeton. Bers’ conjecture apparently considered de Rhamand Cauchy-Riemannsystems, but failed to take into account the LewyMizohata system which would have becomewell knownonly one year later.
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Index abelian category, 151,152, 154, 163, 164, 196, 198 adjoint, 69, 184, 185, 243, 244, 247 algebraic analysis, v, viii, 2, 81,156, 157, 160, 178-180 algebraic geometry,v, vii, 81, 82, 151, 156, 157, 160, 180, 261 algebraic microlocal analysis, v-vii, 91 associated sheaf, 65, 106
Cartan’s theorem, 4, 52, 88 category, 153, 164, 183, 185, 196, 198, 203, 204, 224, 231,236, 237 Cauchy,4, 6, 16, 18-20, 65, 76, 78, 86, 124, 139, 211, 214, 249, 251,252, 272, 273 Cauchy’sexistence and uniqueness theorem, 224 Cauchy’sexistence theorem, 24, 25 Cauchy’stheorem, 7, 9, 11, 62, 223 Cauchy-Hadamardtheorem, 16 Cauchy-Kowalewsky theorem, 82,209, 211-213, 215, 245, 249 Cauchy-Riemannoperator, 97 Cauchy-Riemannsystem, 270, 271, 273 Cech cohomology, 52 characteristic variety, 97, 151, 174, 200, 206, 214, 217, 223, 224, 231,237, 239-241,253, 262265, 270, 271 coboundary, 52-55, 58, 188 cochain, 53, 54, 58, 151 cocycle, 22, 58, 81, 83, 88 coherent, 178, 197, 198, 203, 204, 211,212, 216, 217, 230 cohomology, vi, vii, 3, 21, 55, 60, 61, 64, 66, 71, 81, 83, 88, 155, 162, 165-167, 180, 183, 192, 194, 200, 201, 204, 206, 215, 216, 219-222, 226, 228-230, 233, 236 cohomologyfunctor, 152, 154, 202 cohomologysheaf, 193, 197, 203,231 commutator, 258
Berenstein, viii bicharacteristic curve, 253 bicharacteristic manifold, 269-271 bicharacteristic strip, 253-255,257, 258, 273 BjSrk, 145 Bochner, 47, 79 Bochner tube theorem, 47 Bogolyubov,85, 87 Bogolyubov’stheorem, 86, 87 boundaryvalue, v, vi, 6, 7, 11, 13, 14, 18, 29, 36, 37, 45, 46, 48, 52, 53, 55, 57, 59-62, 66, 75, 81, 85-89, 94, 107-113, 117119, 133, 134, 139, 140, 145, 254 boundary value representation, 39, 62 Boutet de Monvel, 264 C~ function, 26, 97, 107 canonicalfiltration, 175, 191 Carleman, 72, 79-81 Carleman’sfourier theory, 79 Caftan, 180, 232 291
292 complexneighborhood, 2-4, 6-10, 14, 18, 19, 22, 24, 25, 28, 29, 32, 39, 40, 43, 44, 51, 52, 65, 66, 86-89, 108, 125 complexification, 230 conormal bundle, 106 contact manifold, 256, 257, 262 contact transformation, 255, 257-259, 262, 271, 273 convex, 43, 48, 59, 70, 71, 87 convexcone, 48, 54, 56, 57, 59, 65, 86-89, 117, 148 convexhull, 47, 86, 88 convexset, 56, 124 convolution, 84, 138, 140 convolution operator, 84, 98, 138 cotangentbundle, v, vii, 93, 95, 107, 144, 176, 256, 257 D-module,vii, 2, 61, 81, 151, 157162, 166-168, 170-176, 179181, 183, 209, 211, 225, 226, 235, 267 Darboux theorem, 257 de Rhamcomplex, 166, 183, 188, 193195 de Rhamfunctor, 161,183, 193, 194, 198, 224, 237 de Rhammodule, 161, 235 de Rhamsystem, 223, 262, 263, 269, 273 delta function, 11, 12, 28, 31, 40, 59, 73, 74, 93, 94, 116 derived category, vii, 151-154, 159, 162-164, 180, 199, 203, 224, 235-238 derived functor, 160-163, 193, 196, 202-204, 234 derived sheaf, 49, 51, 59 differential operator,v, vii, 8, 12, 2426, 30, 31, 39, 40, 61, 69, 92, 96, 98, 100-102,108, 122, 130, 131, 133, 157, 162, 169,
INDEX 170, 175, 176, 181, 183, 185, 200, 209, 210, 212,-239, 241, 244, 245, 249, 253, 266, 267 Dirac, 11-13, 15, 31, 40, 59, 60, 73, 74, 110, 135, 201 direct image, 115, 120, 121,209,218221 direct limit, 154, 202 domainof holomorphy,43, 81, 125 double complex, 151, 154, 155, 232, 233 dual cone, 56, 57 dual space, 56, 58 edge of the wedgetheorem, 62, 63, 73, 75, 85, 88, 89, 111, 112, 127 Egorov, 258 Ehrenpreis, vi, viii, 41, 70, 79, 94, 147, 240 Ehrenpreis fundamentalprinciple, 272 Ehrenpreis-Palamodov fundamental principle, 41, 79, 239 Eilenberg, 232 elliptic operator, 30, 100, 101,248 ellipticity, 30, 31, 33, 248 entire function, 23, 44, 63, 72, 84, 94, 98 Epstein, 87 essential support, 141 Euler’s fundamentalprinciple, 30, 240 exact sequence,5, 66, 68, 70, 88, 115, 116, 118, 156, 157, 159, 173176, 191, 192, 200-203, 213, 215, 217, 221,222, 231 excision theorem, 52 exponential type, 94, 98 Ext functor, 270 Ext module, 71 faithful, 178, 203, 204, 206, 237 Fantappie’, 9, 19, 72, 75-78, 81-83 Fantappie’-KSthe-Martineau-Sato duality theorem, 21
INDEX Fantappie’-Satoindicatrix, 77 filtration, 155, 169-173, 181, 203205 finite dimensionality theorem, 230 finite order differential operator, 40, 241 finite resolution, 72 flabby resolution, 24, 60, 70, 228 flabby sheaf, vi, 5, 21-23,27, 36, 6567, 111, 113-115, 119, 133, 269, 270 flat, 95, 178, 197, 202-204,216 Fourier, 73 Fourier transform, 38, 41, 72, 79, 80, 84, 93, 94, 96-98, 106, 116, 135, 138, 140, 142, 143, 146, 148 free resolution, 69, 159, 167, 188190, 193-195, 204, 207, 212, 216, 220, 221,223, 230 Frobenius existence theorem, 224 functor, 5, 61, 106, 120, 152, 153, 162-165, 175, 183, 185, 192, 193, 196-198, 200, 201, 203, 204, 206, 213, 224, 231, 232, 237, 238 fundamental solution, 98, 135, 241, 246, 249 fundamental theorem, 261 germ, 3, 16, 17, 23, 26, 37, 38, 40, 50, 61, 70, 74, 81, 116, 123, 125, 127, 157, 159, 161, 165, 169, 176, 183, 184, 200-202, 209 good filtration, 151, 170-173, 175, 190-192, 204-207, 217, 223, 237 Grauert, 65 Grauert theorem, 88 Grothendieck,72, 78, 81, 82, 180 HSrmander,v, 1, 70, 92, 107, 134, 141, 142, 258, 272
293 Hahn-Banachtheorem, 41, 148 Hamiltonian operator, 269 Hamiltonianvector field, 257 Hartogs’ removablesingularities theorem, 81 Hartogs’ theorem, 43 Hartshorne, 81, 157, 180 Harvey, 50, 70 Heaviside, 73, 74, 84 Heaviside function, 12, 13, 34, 73, 94, 242 Heaviside hyperfunction, 12, 29 Hilbert, 74 Hilbert NullstellenSatz, 175, 176 Hilbert sygyzy theorem, 72 Hilbert’s basis theorem, 69, 156 Holmgren’stheorem, 249, 252 holomorphic function, vi, 3-8, 1317, 20, 22, 23, 26, 28, 31, 34, 36, 37, 39, 40, 42-46, 48-50, 52, 53, 55, 58-61, 65, 70, 72, 75-77, 80-83, 85, 87-89, 107, 108, 112-114, 117, 123, 125, 127, 131, 132, 134, 140, 145, 159, 161, 165, 169, 170, 174, 176, 177, 183, 236, 241, 258, 263 holonomicmodule, 162, 181,194, 199, 200, 203, 209, 222-226, 231, 234, 235, 237 holonomicsystem, 162, 262 homogeneouspolynomial, 100, 131, 170, 176 homotopy, 152 hyperbolic operator, 255, 273 hyperderivedfunctor, 164, 165, 196, 220-222 induced filtration, 171,204 inductivelimit, 7, 23, 28, 32, 37, 40, 41, 51, 61, 70, 119 infinite order differential operator, 14, 16, 17, 40, 73, 83, 122-124,
294 130, 176, 265, 267, 269 infinitesimal wedge, 47, 48, 52, 59, 65, 112, 117, 128 initial value problem,82 inverse image, 4, 116, 218 KSthe, 14, 37, 72, 75, 77, 78 KSthe’s duality theorem, 21, 41, 42, 75 Kaneko,2, 21, 50, 65, 142 Kashiwara,v, vii, viii, 2, 117, 119, 180, 181, 191,209, 229, 237, 238, 273 Kashiwara’sconstructibility theorem, 223, 225, 231,236, 237 Kawai,v, vii, viii, 2, 38, 72, 80, 180, 230, 231,237, 273 Kimura, v, 2 Komatsu,31, 50, 61, 70, 180 I~oszul complex, 189, 190, 192 Kowalewsky,78, 214 Kree, 264 Lagrangian, 226, 262 Lagrangian analytic set, 226 Lagrangian bracket, 257 Lagrangian manifolds, 262 Laurent expansion, 15, 16 Laurent series, 18, 200 left exact contravariant functor, 162, 230 left exact functor, 161, 163-165,198 Leray spectral sequence, 120 Leray’s theorem, 52, 57 Lewy, 271,273 Lewy-Mizohata, 271, 273 linear finite order differential operator, 24, 31 Liouville’s theorem, 20 local operator, 17, 18, 29-31, 39, 84, 129, 130 local system, 224 localization, 152, 164, 176 locally closed, 49, 50, 224, 226
INDEX locally finite, 22, 27, 64, 65, 114,146, 226 Lutzen, 78, 79 Malgrange,66, 70, 86 Malgrange’s theorem, 43, 68 Malgrange-Zerner’s theorem, 86 Martineau,50, 72, 83, 86, 88, 89, 127 Maslov, 258, 273 maximallyoverdetermined system, 262 Mayer-Vietoris theorem, 63 Mebkhout, 183, 198, 199 Mebkhout’stheorem, 183, 200 Mellin transform, 84 microanalytic, 108, 109, 112, 113, 115, 116 microdifferentiMoperator, 124, 125, 130-133, 176-178, 239, 240, 259-261, 263, 264, 266, 268, 271,273 micro]oca] analysis, v,I,79,83,84, 134,141,180,273 microlocal operator, 92,96,120,122, 125,127,129-133, 240-247 microloca]ization, 1,8,26,36,93, 95,96,107,176,241,272 microsupport, 104 Mittag-Leffier theorem, 3,22,25,32, 43,50,115 Mizohata, 272,273 Moreratheorem, 6 n-thderived sheaf, 49,51 noetherian ring,181,203 non-characteristic, 212,214-216,227 non-local operator, 84 order (ofdifferential operator), Painleve, 5,85,86 Painleve’s theorem, 6,85 Pa]amodov, viii,70 Paley-Wiener theorem, 93,98,142, 144
INDEX paramterix, 96, 98, 100-102, 239 Pincherle, 17, 83, 84 plane wave decomposition, 116, 246 Poincare, 76 Poincare duality, 199 Poisson, 73 Poisson bracket, 257, 271 presheaf, 36, 49, 66, 106, 114, 116, 177 principal symbol, 97, 99, 131, 170, 172, 178, 223 projective resolution, 159, 161, 162, 167, 197, 216, 232, 235 pseudo-differential operator, 96, 98, 102-105, 107, 122, 124, 127 purely n-codimensional, 49-51 quantized contact transformation, 240, 241,259-262, 267, 269 quotient sheaf, 36, 65, 66, 116 real analytic category, 110 real analytic function, 6, 7, 11, 13, 26-29,37, 38, 42, 61, 69, 93, 110, 115, 125, 142, 230, 242, 245, 247, 248 relative cochain, 54, 58 relative cohomology,42, 48, 52, 57, 59, 61, 63, 70, 71, 81-83, 88, 120 relative covering, 52-54, 57 residue, 116, 136 resolution, 70, 161, 183, 189, 220 Riemann-Hilbert correspondence theorem, 181,236 Riemann-Hilbert-Kashiwara correspondence theorem, 237 right derived functor, 164 ring of differential operators, 178 Sato, v, vii, viii, 1, 2, 8, 24, 27, 49, 50, 72-75, 79-83, 85, 91,107, 179-181,239, 240, 262, 265, 272
295 Sato’s fundamental exact sequence, 248 Sato’s fundamentalprinciple, 25, 131, 240, 253, 254 Sato’s fundamental sequence, 114 Sato’s fundamentaltheorem, 151,176, 178, 239, 241, 245, 248, 250, 261, 264, 272 Sato’s structure theorem, vii Sato’s theorem, 241,244 Schapira,viii, 50 Schwartz,v, 1, 7, 18, 30, 36, 41, 67, 69, 74, 78, 81, 93, 139, 240 Schwartz’s kernel theorem, 101 Serre, 180 Serre duality, 199, 200 sheaf, vi, vii, 1, 3, 17, 21-31,36, 37, 42, 44, 49-51, 60, 61, 63-71, 81, 83, 84, 89, 91-93, 106, 107, 111, 113-116,119, 120, 122-126, 128-130, 132, 133, 140, 157, 159-163,165, 169, 170, 176-178, 180, 183, 184, 187, 193, 197, 198, 200, 203, 209, 210, 216, 218, 221, 223, 225-228,230, 240, 246, 253, 263, 267, 269, 270 sheaf of microdistributions, 106 singular spectrum,92, 108, 109, 112, 119, 126-128,134, 141, 145, 246, 247 singular support, 28, 29, 36, 92, 93, 105, 108, 111, 212 skew category theory, 180 slowly decreasing, 72 Sp~ith theorem, 268 spectral decompositionmapping,115 spectral sequence, 52, 81, 151, 154, 155, 162, 163, 165, 166, 180, 195-197, 199, 200, 221,233235 spectral sequenceof a compositefunctot, 163, 164, 198
296 spheric cotangent bundle, 93, 111 stalk, 50, 51,169, 200, 203, 206, 217, 225, 228 standard defining function, 19, 42 Stein covering, 57 Stein domain, 50 Stein neighborhood, 51, 88 Struppa, 72 subsheaf, 27, 36, 63, 92, 116, 132, 177, 187 support of a hyperfunction, 111 symbol of an operator, 93 symbols, 103 symplectic geometry, 256 symplectic manifold, 256, 257 symplectic structure, 257 tube domain, 140 tubular neighborhood, 46 universal property, 152 Verdier, 180 virtual bicharacteristic manifold, 271 waveoperator, 249, 253 wedge,46, 47, 58, 59, 85 Weierstrass, 84, 212 Weierstrass’ division theorem, 263, 268 Weil, 75, 81, 180 Weyl’s lemma, 247 Zerner, 86 Zorn’s lemma, 67
INDEX
