Seminar on Micro-Local Analysis. (AM-93), Volume 93 9781400881574

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Annals of Mathematics Studies Number 93

SEMINAR ON MICRO-LOCAL ANALYSIS BY

VICTOR W. GUILLEMIN, MASAKI KASHIWARA, AND T AKAHIRO KAWAI

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1979

Copyright © 1979 by Princeton University Press ALL RIGHTS RESERVED

Published in Japan exclusively by University of Tokyo Press In other parts of the world by Princeton University Press Printed in the United States of America by Princeton University Press, Princeton, New Jersey The appearance of the code at the bottom of the first page of an article in this collective work indicates the copyright owner's consent that copies of the article may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc., P.O. Box 765, Schenectady, New York 12301, for copying beyond that permitted by Sections 107 and 108 of the United States Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Library of Congress Cataloging in Publication data will be found on the last printed page of this book

TABLE OF CONTENTS PREFACE INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS by Masaki Kashiwara SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS FOR ELLIPTIC SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS by Masaki Kashiwara and Takahiro Kawai A SZEGO-TYPE THEOREM FOR SYMMETRIC SPACES by Victor Guillemin SOME MICRO-LOCAL ASPECTS OF ANALYSIS ON COMPACT SYMMETRIC SPACES by Victor Guillemin

vii 3

39 63

79

ON HOLONOMIC SYSTEMS WITH REGULAR SINGULARITIES by Masaki Kashiwara and Takahiro Kawai

113

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES by Masaki Kashiwara and Takahiro Kawai

123

PREFACE This book is the outgrowth of a seminar on micro-local analysis sponsored by the Institute for Advanced Study during the academic year 19771978. For the benefit of the general reader we will attempt, in a few words, to put the subject matter of this volume into a historical perspective. By micro-local analysis we mean the study of generalized functions as local objects on the cotangent bundle. In a sense micro-local analysis has its roots in the work of Cauchy, Riemann and Hadamard on the relationship between singularities of solutions of partial differential equations and the geometry of their characteristics. However the theory we will be concerned with here really starts about 1970 with Sato's definition of microfunctions as localizations of hyperfunctions and with the work of Maslov, Egorov and Hormander on quantized contact transformations (or Fourier integral operators). These two closely related developments enabled one to study in much more meticulous detail than was ever before possible the singularities of solutions of partial differential equations and of generalized functions arising naturally in geometric and group-theoretic contexts. The first series of lectures in this volume are an introductory account of the theory of microfunctions. This parallels somewhat the account in [SKK]; however, here the cohomological aspects of the subject are somewhat suppressed in order to make these lectures more accessible to an audience of analysts. The subsequent lectures in this volume are devoted to special aspects of the theory of microfunctions and to applications such as boundary values of elliptic partial differential equations, propagation of singularities in the vicinity of degenerate characteristics, holonomic systems, Feynman integrals from the hyperfunction point of view and harmonic analysis on Lie groups. vii

Seminar on Micro-Local Analysis

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS Masaki Kashiwara §0. Introduction The purpose of this note is to give an introduction to the theory of hyperfunctions, microfunctions and micro-differential operators. Hyperfunctions were introduced by M. Sato (J. Fac. Sci. Univ., Tokyo, Sect. I, 8 (1959), 139-193; 8 (1960), 387-437). For Sato a hyperfunction is a sum of boundary values of holomorphic functions. In order to formulate the theory in a rigorous way, he introduced local cohomology groups and expressed hyperfunctions as cohomology classes. Here, we employ a more intuitive way of defining hyperfunctions. For a rigorous justification of our approach we refer to the article: Sato-Kawai-Kashiwara, Microfunctions and pseudo-differential equations, Lecture Notes in Math. No. 287, Springer, 1973 pp. 265-529 (abbreviated by S. K. K.).

§ 1. Hyper/unctions

1.1. Tangent Cones. We will need some geometric preliminaries. We will begin with the definition of tangent cones. Let M be a C 1 -manifold. We shall denote by TM the tangent vector bundle, T*M the cotangent bundle, r: TM

->

M, rr: T*M

cal projections. T xM is the tangent vector space at x the cotangent vector space at x

E

E

->

M the canoni-

M and T~M is

M.

© 1979 Princeton University Press Seminar on Micro-Local Analysis 0-691-08228-6/79/00 0003-36 $01.80/1 (cloth) 0-691-08232-4/79/00 0003-36 $01.80/1 (paperback) For copying information, see copyright page 3

MASAKI KASHIW ARA

4 Take a point

X

in M and a local coordinate system (xl' ... 'xe) in a

neighborhood of x . DEFINITION 1.1.1. For two subsets A and B of M, the tangent cone Cx(A; B) is the set of limits of sequences an(xn-Yn) where an> 0, xn

E

A, y n

E

B, such that xn, y n converge to x. We regard Cx(A; B)

as a subset of T xM. Set C(A; B) = U

xEM

Cx(A; B).

REMARK 1.1.2. This definition does not depend on the choice of coordinate systems. Tangent cones enjoy the following properties: a) C(A; B) is a closed cone in TM . b) C(A; B) = -C(B; A). c) C(A; B) = C(A;

B).

d) CxCA; B) = (2l ~ x

1 An i3.

e) Cx(A; B)= !O! ~ x is an isolated point of A and f) Let f:M-->N

bea C 1 -map, y=f(x).

B.

Then

(df )xCx(A; B) C Cy(fA; fB) . If Cx(A; B) n (dO;/ (0) C l 0!, then (df)xCx(A; B) = CyCf(A n U); f(B n U)) for a neighborhood U of x. g) If N is a submanifold of M, then CxCA; N)+ T xN = Cx(A; N) for X

EN. Notation: We set CN(A\ =CiA; N)/TxN C (TNM)x = TxM/TxN and

CN(A) = U

xEN

CN(A)x C TNM.

1.2. Definition of (f. Let M be a real analytic manifold (say an open set of Rn ), X its complexification (say an open set of en), (.') the sheaf of holomorphic functions on X , and (f the sheaf of real analytic functions on M . For x as:

E

M, the complex tangent space to X at x can be decomposed

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

so we can identify (TMX)x

=

5

TXX/TXM with v-1 TXM. We shall denote

by r the projection of V-1 TM onto M. DEFINITION 1.2.1. For a point (x 0 ,yl-1v 0 ) E V-1 TM, we say that an open set U of X is an infinitesimal neighborhood of (x 0 , yC1 v 0 ) if C(X-U) does not contain (x 0 ,yl-lv 0 ) (¢::::::":> UJX+V-1tv for !,

x-x 0 \

«

1, 0 < t

«

1 and \v-v 0 \

«

1 ). For an open cone U of

yl-1 TM, we say that an open set U of X is an infinitesimal neighborhood of n

if

u

is an infinitesimal neighborhood of any point in n .

DEFINITION 1.2.2. (S.K.K. Def. 1.3.3, p. 276). For an open cone U in V-1 TM, we set

-

ct(U)

lim l:l(U) ->

u where U runs over a set of infi!)itesimal neighborhoods of n

and l:l(U)

is the set of holomorphic functions defined on U. REMARK 1.2.3. An infinitesimal neighborhood of (x; V-1 0) is nothing but a neighborhood of x. Therefore, ct(yi-1 TM)

=

ct(M).

REMARK 1.2.4. (S.K.K. Prop. 1.5.4, p. 285). Suppose that U has con-

~ected ~ibers (i.e.' n ct(U)

=

n vCI

T XM is connected for any X E M ). Then

(f (the convex hull of U ). Here the convex hull of U is the

union of the convex hulls of n

n V-1

T XM (x EM).

1.3. Definition of hyper/unctions. Although hyperfunctions are defined by the use of local cohomology in [S], [S. K.K.], we shall give here a more intuitive definition. Let V be an open set of M. We denote by :f(V) the totality of the following data: !ni, ui!iEI where I is a finite se!, ni an open convex cone in

yC1

TM such that r(U) => V, and ui

E

ct(U). Let -

equivalence relation on :f(V) generated by the relation

be the

MASAKI KASHIWARA

6

-·{fl.· u.}. -lfl'.· 1' 1 lfl R J' u'.}. J JfI' if there are open convex cones 0 1.1. (id, J. fl')

R .

-

and wij f d(Oij) satisfying the properties (i) and (ii): (i) OijJOiUOj for ifl and jfl'. (ii) ui

= ~ wij' vj =~ wij. jd'

if!

DEFINITION 1.3.1. We define :B(V)

=

~(V)/-

and call a member of :B(V) a hyperfunction defined on V. We have the following fundamental properties of

:13.

THEOREM 1.3.2. (a) :J3 : V .... :J3(V) is a sheaf. (b) The sheaf

:13

is a flabby sheaf (i.e., :B(M) .... :B(V) is surjective

for any open set V ).

Let 0 be an open convex cone of yCI TM. Then, for any u

f

d(O)

(0; u) is a member of ~(V), so that we can define the map

-

bn: d(O) ....

:13( rn)

.

THEOREM 1.3.3. (S.K.K. Th. 1.5.2, second row of (1.5.2), p. 283). bn : d(O) ....

:13( rO)

is injective.

If we regard :B(V) as a space of generalized functions, then bn is the map assigning to each holomorphic function its boundary values. By this notation, !Oi; uilifl f ~(V) corresponds to the hyperfunction

~ bn.Ol. Set u(z)= 1/z 1 ···zn. Then We define the a-function supported at the origin by

(1.4.1)

We will show that xj o(x) = 0. We can assume j = 1. Set UE

2'

... E = !z; EJ·Imzl.>O 0=2,-··,n)!. ' n

Then

Thus

Q.E.D.

8

MASAKI KASHIWARA

In particular, we have supp o(x)

c !OI

(i.e., o(x) = 0 on

Rn -!01). Also

we have

because

(b) We give another definition of the a-function. Let l f 1 , · · ·, f n I be a set of linearly independent real n-vectors. Set n

f

0

=-fc .. ·-fn· Weset =:L (fj)vzv, where (fj)v isthe

v-th component of fj.

v=l

Set Oj = lzcCn; Im < z, fk> > 0 for Vk

(1. 4. 2)

o(x)=

I= jl. Then, we have

Ibn ( 1 Ln Ifo A···Af·A···Af A -11 n J n H· (- 2rrv-J.) j=O J

1 ) IT t:

.

k/=j

Here If 1 A··· Af nl signifies the absolute value of the determinant of the nxn matrix (f 1 , · · ·, f n) . We shall prove that this definition coincides with the one given in Example (a). Recall the following formula (Feynman's formula):

(1.4.3)

(n-1)!

J

"'t '· ··'"'n?_O

w(ry) n

1

for x 1 , · · ·, xn > 0. Here the integral is over the sphere sn-l defined by 11 1 ,. · ·, 71n ?_ 0, and w(71) is the volume element

n"1 1 dn"'2 A···Adn.,n-n"'2 dn"1 1 Adn"1 3 A···Adn.,n +···+(-1)n-ln"'ndn"1 1 A·•·AdT/ n- l" By a change of coordinate, we have

9

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

(n-1)!

···· '"'1 '"'n

Here G(( 1 ,. · ·, ( n) is the closed convex cone 17); 7) = ~/j (j; tj ~ O!. Let G be an arbitrary closed convex cone. Then (n-1)!

J

w(7]) n

G

isholomorphicon T(G)=IzO forany ry

f- 0 for 7)

E

G-IO! and z

E

T(G).

Let us consider the decomposition

with Gj n Gk of measure 0 for j v

=

f- k, and consider the hyperfunction

~bT(G·)(uG). J

J

Claim. v does not depend on the choice of the above decomposition. Proof. Let Rn

=

U Gj J

=

U G'k be two such decompositions. uG.nG' is k J k

defined on T(Gj n G'k) which contains T(Gj) U T(G'k). Therefore,

Q.E.D.

10

MASAKI KASHIW ARA

Thus

If we take the decomposition

then

(c) In this example M = R. Let Q± = l z; Im z ~ 0 I, and define

and

where, for

i·

and z'\log z)m, we take branches on C- lx; x::; 0! such

that zAiz=l = 1, zA(log z)m

=

(tA)m zA. We also define

11

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

and Y(x) = [2rri ~ log (x+iO) + log (x~i0)]/2rri . (d) M = Rn, X = en . Let f(x) be a (complex valued) real analytic function defined in a neighborhood of x 0 < M. Suppose that df(x 0 ) is a real covector, and that Ref(x)

=

l ZE X; Im f(z) > ~ el Re f(z)l! for

=

{}/= 0

0 (Xt M) implies Im f(x) ;;> 0 . Set

> 0 Then, for any E > 0' nE is an infinitesimal neighborhood of (x 0 , yG v) for any v E T x M such that

nE =

E

0

0

> 0 (S.K.K. Lemma 3.1.5, p. 306). Therefore, bn (f,\) is well defined, (i.e., boundary value from the direction of Im f

E

> 0 ). We shall de-

note this hyperfunction by (f+iO),\ . (e) Set 11± = lz yllm z 2 1+···+[Im znl and f(z)

=

zi ~ z~ ~ · · · ~ z~. Then f(z)

I=

0 on 11±. Therefore, we can define

which we shall denote by ((x 1 ± i0) 2 ~ x~ (f) M = R2 , X= C2 , M

Set n

=

J

~ · · · ~ x~f.

(x,y), X

J

(z,w), z[M = x,w[M = y.

!(z, w)< X; Im Z, Im w > O!' f(z, w)

a branch of w 213

such that (y~l) 2 / 3

= Z+w 2 1 3 '

= e"yC1; 3 .

where we take

Then f(z, w) is holo-

morphic on Q and never vanishes on Q. Therefore, f,\ is also holomorphic, and bn(f,\) is well-defined. We note that

1.5. Relations with distributions. We will show how distributions on a real analytic manifold can be regarded as hyperfunctions. Let u(x) be a compact supported distribution on Rn. Set cp(z)

=

(

u(x), n

1

II (zj~xj) j=l

)

12

MASAKI KASHIW ARA

Then ¢(z) is holomorphic on Im zj

f. 0 .

We associate u with the hyperfunction

where n

El'

... E = l Zf ' n

c n;

EJ· Im ZJ·

:B,

morphism of sheaves ~)' _,

> 0!' EJ· = ± 1 . This extends to a homo-

g)' being the sheaf of distributions. This

is an injective homomorphism. A function u(z)

f

Cf(O) is called of polynomial growth if u(z) satis-

fies IIm zl N Iu(z)l < const. Then u(x + yC1 ty), t ~ 0, converges in the sense of distribution to bn(u) (Komatsu: Relative cohomology of sheaves of solutions of differential equations, Lecture Notes in Math., 287, p. 226,

1971; A. Martneau: Distributions et valeurs au bord des fonctions holomorphes, Instituto Gulbenkian de Sciencia, Lisbonne, 1964).

§2. Microfunctions 2.1. Singular Spectrum. Let

yCi

v'-1

T*M be the dual vector bundle of

TM. We will identify

v-1 T*M

Take a point (x 0 , yC1

.f0 ) f yC1

with the kernel T*XIM-> T*M. T*M

(.f0 f

T; M). 0

DEFINITION 2.1.1. A hyperfunction u(x) is called

micro-a~alytic

(x 0 , yC1

.f0 )

that u

~bn.(uj) in a neighborhood of x0 and that 2. For the moment we will just show that SS8(x) = 1(0; V-IOl when

n = 1. In fact, if SS8(x) ,! (0, yCl), then sp(x+ yCl Or 1 = (-211yC!) sp8(x) at (0, yC!), and hence (x+ yC1 or 1 is micro-analytic at (0, yC!). Since

c {(x,V-10;,; > Ol, SS(x+y=Io)- 1 c {(x; y=I,;}; ,;=ol, and hence (x + yCI or 1 is real analytic. This is a contradiction.

SS(x+vCior 1

(b) SS(x±vWO)'\ = {(x,yC'I~); x~=0,±~2'01 for At0,1,2,··· = {(x,yCIO; ~=01 for A= 0, 1,2, ···. In fact, if A= 0, 1, 2, ···, (x±yC!of is real analytic and the result is obvious. If At0,1,2,···, then SS(x±iO)AC{(x,yCI,;};±~;:o:OI, by

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

15

Theorem 2.1. 5. Since (x ± iO)'\ is real analytic on x ~ 0 and not real analytic, SS(x ± iO)A must be as indicated. (See Remark 2.1.6.) §3. Products, pull-back and integration of micro/unctions In this section we will show that microfunctions have nice analytic properties-e.g., we will show that they can be multiplied, integrated, etc., ... in other words, we will show that analysis on real analytic manifolds can be based on the theory of microfunctions. 3.1. Proper maps. A continuous map f: X -. Y from a topological space X to a topological space Y is called proper if the pre-image of every

point is a compact set and if the image of a closed set of X is a closed set of Y. In particular, when Y is locally compact, f is proper if and only if the inverse image of a compact set of Y is compact. Suppose that

X and Y are locally compact. Let Z be a locally closed subset of X, y a point in Y . If Z neighborhood U of Z

n f - l (y) is a compact set, then there are open n f- 1 (y) and an open neighborhood V of y such

that f(U) C V and that U n Z -. V is a proper map. We note the following lemma: LEMMA 3.1.1.

Let X be a topological space, E -. X, F -. X two vector

bundles and f: E -. F a bundle map. Let Z be a closed cone of E. In order that Z -. F be a proper map, it is necessary and sufficient that Z does not contain any point p in E such that f(p)

=

0.

3. 2. Products of micro/unctions. Let u(x) and v(x) be two hyperfunctions on M . Suppose that

(3.2.1)

SS(u)

where SS(v)a

THEOREM

=

n SS(v)a

{(x,-v'=IO; (x,v'=IO

C E

y=I

T~M

SS(v)l.

3.2.1. (S.K.K. Cor. 2.4.2, p. 297). Under the condition (3.2.1),

16

MASAKI KASHIW ARA

the product u(x) v(x) is well defined and SS(u(x) v(x)) C SS(u) + SS( v). M

Here A+ B= !Cx,vC1Ce 1 +e 2 )); (x,Pe 1 ) , A,(x,Pe2 ) , B!. M

Proof. If u or v is real analytic, the product is well defined. If not, we define u(x) v(x) in a neighborhood of a point x0 in M as follows: The condition (3.2.1) implies that there are proper closed convex cones !Zj! and !Z'k! (which contain yCI T~M) such that SS(u) C U Zj in a neighborhood of x0 , SS(v)

c

U Zi

and (3.2.2)

Z J.

n Z'a k

C - 1-1 T*M V M .

Set Dj=!Cx,vCfv): >0 for V(x,yCit),zj!, Q'k=l(x,'l}-1v): < v,

e>

> 0 for V(x, vf-10' Z'k!. Then, (3.2.2) is equivalent to

(3.2.3) By Theorem 2.1. 7, we can write u = ~>j with SS(uj) C Zj and v =

L,vk

with SS(vk) C Z'k. By Theorem 2.1.5, we can represent uj an_d vk as a boundary value of a holomorphic functions; i.e. there are cPj ( acnj) and

rfk

'tl(Q'k) such that bn/¢j) = uj and bn,k(r/Jk) = vk. We shall define

uv by

The condition (3. 2. 3) assures us that this is well defined. Using (b) in Theorem 2.1. 7, it is easy to check that this definition does not depend on the choice of nj' Q'k' uj' vk.

-

Since the singular spectrum of bn j n Q'k ((f(Q j n Q 'k)) is contained in

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS the polar set of Uj

n

SS(uv)

17

U'k, which equals Zj ~ Z'k, by Theorem 2.1.5,

c

u j,k

(Zj + Z'k)

=

M

(U j

zj) +

M

(U

Z'k) .

k

Since this inclusion holds for any choice of IZjl and IZ'kl, we have SS(uv)

c

Q.E.D.

SS(u) + SS(v) . M

THEOREM 3.2.2. Let uj(x) be a microfunction defined on an open set

- T *M (j = 1, 2). Let 11 be an open set of v-1 T1 T *M. Let us 11 j of y-1 - T *M x y-1 - T *M onto v-1 T1 denote by p the map from y-1 T *M defined M

by ((x,vG-,; 1 ), (x, vG-.; 2))

f->

(x,

vGjo£) to be well defined. Thus, we can define f*(u) =

en j)

(cf>j of). The second theorem is also proved in the same way.

EXAMPLES. a) We can define the a-function on Rn as the product o(x) = o(x1) ... o(xn). b) If f: M -. R satisfies df ~ 0 on c- 1(0) , then we can define o(f(x)),

f(x~' (f(x) ± wt'

...

as pull-backs of the hyperfunctions o(t), t~, (t ± iO)'\ of one variable by the map f. By definition, setting n+ = bn±(r\), o(f(x))=

IX f X;

± Im f(x) >

-2~i (hn/c-1)-::_hn_l

2-

( =.,.,

2 , (x-y) ((+TJ) = (x+y) ((-TJ) =

l(x,y; (,.,.,); x 2 :::: y 2 , ( = T/ = U {(x, y; (, T/); x

O!

= y, ( = -11!

11!

U l(x, y; (, T/); x = -y,( =

and ul x=O means the pull-back of u by the map R -> R 2 (y 3.4. Property of

a;at.

Consider Rn+l

J

(t, x) = (t, x 1 ,

ja the injection Rn-> Rn+l defined by x Rn+l -> Rn given by (t, x) of

a;at

1->

O!

1->

···,

1->

(0, y) ).

xn). Denote by

(a, x) and F the projection

x (aE R). We shall investigate the properties

as a micro-local operator.

PROPOSITION 3.4.1.

i)

a;at: e -> e

is surjective.

a;at u = 0

ii) if a microfunction u satisfies

at p = (t, x; i(r dt +

< (, dx> )) with r .fo 0 then u = 0 in a neighborhood of p. iii) if a microfunction u satisfies

a;at u = 0

and is defined near

(a, x; i()), then u Proof.

that

In order to show

a;at:

a/at:

=

F*(j: u) .

e .. e

is surjective, it is enough to show

~ -> ~ is surjective. Since a hyperfunction is a sum of bound-

ary values of holomorphic functions

cp

and since we may suppose that

cp

20

MASAKI KASHIWARA

is defined on a convex set Q,

a!at 0cm = 0

:13 is surjective, because

To prove ii) and iii) we must investigate more precisely the properties of

a;at

on domains in cn+l . Let f2 be a convex neighborhood of

(0, 0).

For a convex open cone V of R 1 +n, set Tn(V) = !Z£f2; Im Z£V!. Then we have a)

a;at t9(Tn(V)) = t9(Tn(V)).

b) Suppose u £ t9(Tn(V)) such that

LEMMA 3.4.2.

a;at u = 0.

Then

Let p be a point in rr- 1 (0). Then, for any hyperfunction

u such that SSu

l

a;at v = u

p, there is hyperfunction v such that

and SSv 1 p.

- T MM, * then u = 0 and hence the lemma is trivial. If Proof. If p £ y-1 pf_O, thensetting p=(O,O;v'=I(r 0 ,.;0 )), wecanwrite

with

with convex open cones Vj in R 1 +n such that r 0 t + any (t, x) £ Vj. We can solve v

=

a;at tPj = ¢j

with tPj

f

> 0 for

t9(Tn(Vj)). Then

lbv. (t/J j) satisfies the desired condition. Now iet us prove (ii). Take a point p = (0,0; i(r,,;)). Let u be a

a;at u = 0 at p. Take a hyperfunction ii such that sp u= u at p. Then SS(a/at u)! p. Therefore, by Lemma 3.4.2, there is a hyperfunction, v such that a;at u= a;at v and SSv! p. microfunction such that

Thus, replacing ii with ii- v, we can assume from the first that

a;at ii = 0,

and u = sp

u at

p. We shall write

u= lbv.C¢j) J

with

21

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

¢j

E

¢jk

tl(Tn(Vj)). Since E

a;at u = 0'

there are convex open cones vjk and

tl(Tn(Vjk)) such that vjk:) vj

u vk' vjk = vkj'

a;at ¢j = l

¢jk k

on Tn(Vj) and ¢jk = - ¢kj, by replacing Vj and !J smaller ones if necessary. We can solve ¢jk = alj!jk;at on Tn(Vjk) such that tPjk=-tPkj· Hence a;at(¢j-ltPjk)=0. Thusreplacing ¢j with k

¢j-ltPjk, we may assume from the first time

a;at ¢j = 0.

Hence ¢j

k

is a function of x and hence defines an element of tl(T 0 (F-l FVj)). Thus

b(¢j) = F*ja *(b(¢j)) and SS(b¢j) J (t, x; i(r, 0) when r -/c 0. Therefore, the singular spectrum of u does not contain such points, and

u= F*ja *ii.

This proves (ii) and (iii). 3.5. Integration of microfunctions. We shall next describe how to integrate microfunctions. Set N = R 1 +n, M = Rn and F, ja as in §3.4. Then we have p: N x yC1 T*M c_. M PROPOSITION

set of p- 1

T*N and

w:

Nx vCJ T*M .... yCI T*M. M

3.5.1. Let !JM be an open set of yC1 T*M, !JN an open

vCJ T*N,

supp u n

vCJ

w-

u a micro/unction on N defined on !JN . Suppose that

1 (!JM) .... !JM

is a proper map. Then the integral F*(udt)

( u(t, x) dt is well defined on !JM . We define v(x) =

J

u(t, x) dt

by an indefinite integral. Take a point y = (x 0 , ie 0 ) of !JM. Then, there is a hyperfunction

ii

sp u

=

and a u

SSu n

< b such that

on

w- 1(y) n

w- 1 (y)

p- 1!JN

c supp u,

SSiin {(t,x 0 ; i(r,ke 0 )); kzO,tzb or t:Sa,(r,k)~O!

0.

=

22

MASAKI KASHIWARA

By Proposition 3.4.1, there is a hyperfunction w such that

a/at w = u.

We define v at y by

It is easy to see that this definition does not depend on the choice of b,

a and w. We can also define integrations with respect to several variables as a succession of integrations with respect to one variable. THEOREM 3.5.2. (S.K.K. Th. 2.3.1, p. 295). Let M and L be two real

analytic manifolds, N = M x L, F the projection from N to M and dt a real analytic volume element on L. Let OM and ON be open sets of

yCI T*M and yCT T*N, respectively. Let u be a micro/unction on N defined on ON. Suppose that p- 1 supp u n w- 1 oM ->OM is a proper map. Then, the integral F*(udt) = ( u(x, t)dt is well defined as a micro/unction on OM. 3.6. EXAMPLES. a) The plane wave expansion of the a-function: o(x) =

where wCO = t 1 dt 2

...

Cn- 1)! (-277-J=l)n

J

sn-1

dtn-t2 dt 1 dt 3

...

wCO (< x, t> + iO)n

dtn+· .. +C-1)n- 1 tndt 1 ... dtn_ 1

and sn- 1 = (Rn-! 0!)/R+ the (n-1)-dimensional sphere. This explains the formula

in Example b) §1.4. b)

ro(x) dx = 1

because dY(x) = o(x) and Y(x) = 1 (x > 0), Y(x) = 0 (x < 0). dx c) SSo(x) = !(x; ic;); x=O! because we know that SSo(x) C ! (x; it); x = 0!.

,

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

Since for g

f

23

GL(n; R) ,

SSo(x) is invariant by the action of GL(n; R). Suppose that SSo(x) 1 (0; i(1, 0 · · · 0)). Then

is micro-analytic at (0, yCI dx 1 ), which is a contradiction. Hence SS(o) ;~ (0; i(1, 0 · · · 0)) and hence any (0; i 0

d)

J

1

t - x2 + iO

dx

=

=

{(t,x)

0).

-rri(t+ i0)-7'2. Note that this integral has a

sense at (0, yCI dt). For example, u(t, x)

D

(.; ;6

f

=

(t- x 2 )- 1 is holomorphic on

C 2 ; lm t > 2\Im x\, \Rex\< 1!.

Fix 0 < a < 1 . Set v(t, x)

=

I

X

u(t, x) dx .

0

Then v is also defined on D and

F =

J

dx

t - x 2 + iO

For Im t > 0, v(t, a) - v(t, -a) aeiO (0

~ () ~

phic at t

=

av;ax =

u. Therefore, we have

= v(t+ iO, a)- v(t+ iO, -a) .

fa u(t, x) dx. Let -a

77). (See the figure below.) Then

v(t, x) dx is holomory+ u(t, x) dx for Im t > 0 where cp is the con-

= 0. Set w(t) = !f .,[t. Then

tour integral around

f

y + be the path

24

MASAKI KASHIWARA

-a

a

•

x-plane

J

dx 2

t- x + iO

= w(t+ iO)

w(t)

=-

at

(0,

vG dt) ,

~,

and hence we have

J

1

t- x

Changing

2

+ iO

dx = -rri(t+ iOrY:.

at

(0,

vG dt) .

to -i we obtain

J

dx

t - x 2 - iO

=rri(t-iorY:.

at

(0,-yCidt).

at

(0, ±yCI dt).

Changing t to - t , we obtain

J

dx

t + x

2

± iO

= rr(t±ior'h

e) Jet- x+ iO)A-l(x+ iO)IL- 1 dx = (-2rri)

!{1-A-g) (t+ iOf+f1-l. 1(1-A)l(1-f1)

25

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

First, this formula has sense at (0, idt). Assume first Re(A+Il) < 1. The integral is by definition equal to

over the contour indicated in the figure below. 0

t

0

0

-a

a

Here Ja(t-x)A-lxf.t- 1 dx is holomorphic on {tfC; Im t > -a

Re(A+Il) < 0, then [-\t- x)A-lxf.t- 1 dx and

Ol.

If

()Q (t- x)A-lxf.t- 1 dx

are

a

-00

holomorphic on t at the origin. Therefore, the integral is equal to

By the analytic continuation on A and 11 we get the desired result.

f)

r

(t+< Ax, x>+ iOtdx

JRn

=

e-Crri/2)q rrn/2 1(-A-n/2) (t+ iO)A+n/2 \detA\y, r{-A)

at (0, idt), where A is a non-degenerate symmetric matrix and q is the number of negative eigenvalues of A. By a coordinate transform, we may assume that A is a diagonal matrix. Then, by the succession of integrals, it is enough to show

J(t e = 1 (a> 0), e

+ ax 2 + 1·o"\d J· x

=

-i (a< 0).

,for(-A-1/2) (t + 1. 0 ,,x,+V> r \a\ Y> 1(-A) E

= --

26

MASAKI KASHIW ARA

Changing x r> Ia[-'l'2x, we may assume a

=

± 1. Then we know

already

Therefore, we have

-

". Jet-s+ iOtCs+ iOrVo ds 2TTl

E

§4. Micro-differential operators

4.1. Micro-local operators. Let M and N be manifolds. Fix a real analytic density dy on N. We denote by p 1 the projection

yC1 T*(M x N) -. yC1 T*M (resp. p~ the projection y-1 T*(M x N) -. y-1 T*N) defined by (x, y; i(~, 71)) r> (x, i 0

(resp. (y, -i71) ).

Let Q be an open set in y-1 T*(M x N), Z a closed set of Q .

PROPOSITION

set of

4.1.1. Let QN be an open set of yCl T*N, QM an open

yCI T*M. Suppose that

1s a proper map.

Then for a micro/unction K(x, y) defined on Q and a

microfunction v(y) definedon QN, u(x)=JK(x,y)v(y)dy iswell defined on QM if supp K C Z.

This follows easily from Theorems 3.2.2, 3.3.2, and 3. 5.2.

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

27

COROLLARY 4.1.2. Suppose that (4.1)

and

are homeomorphisms. Let be the map nM -> nN defined by these two

homeomorphisms. Then, for any microfunction v(y) defined on an open subset U of ON , u(x)

= {

K(x, y) v(y) dy is well defined on - 1 (U),

i.e. this defines a sheaf homomorphism - 1 (t'Nj0 ) -> t'M. N

Consider the particular case N Set

z = vG

yC1

T*M by p 1

T~ (M X M) .

=

Let 0

=

M. Let L'l be the diagonal in Mx M.

l (x, x, vG c.;, -0)!. be an open set of

microfunction defined On Q

X

Q

We identify

yC1

z

with

T*M. If K(x, y) is a

Whose SUpport is contained in 2, then,

for any microfunction v(x) defined on an open subset

u

of n '

{ K(x, y) v(y) dy is well defined on U. Therefore

K: v(x)

1->

J

K(x, y)v(y)dy gives a sheaf homomorphism t'ln-> t'in. We

call K a micro-local operator on 0

and K(x, y) the kernel function of K.

The identity operator is a micro-local operator corresponding to the n

kernel function o(x- y)

=

H o(xj- Y). A differential operator P(x, D)

J=l

is a micro-local operator corresponding to the kernel function P(x, D)o(x-y).

4.2. Micro-differential operators (real case). The class of micro-local operators is too wide a class to work with effectively. We shall introduce a class of micro-local operators, called micro-differential operators, which we can manipulate easily. This class is in some sense, a localization of differential operators.

28

MASAKI KASHIWARA

First, we shall investigate the kernel functions of differential operators.

~

a\a\

Let P(x,D) = ~ aa(x)Ijl, Da =-...:::....._ _ _ be a differential al an \al

\a\=j is bijective on

The proof of this involves constructing the inverse of P on Q explicitly. This turns out to be a micro-differential operator. Let us put

m

=

'~',\ (z)

where we take its branch on z

f

['(,\)

,..\ ,

(-zJ·

I z;

C-

When A=-m(m=0,1,2, .. ·), we set

z ~ Ol such that ,\ (-1) = ['(,\).

A(z)=-~!zmftog(-z)-i t+Y) \

j=l

where y is the Euler constant. Then (a;az),\ (z) = A+l (z). Now, consider the kernel function of P, that is, P(x, D)B(x- y). We have

8(

) _ _,(.::...n--=-1)":::! x-y- (-2rri)n

=

J

CtJ

CO

(+iO)n

~ Jn(< x-y, i~>- O)w(O.

(2rr)

Here n(< x, i~>- 0) is a boundary value of holomorphic function

from the direction Re < 0 (i.e. Im < x,

e> > 0 ).

Then we have

INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS

29

This formula suggests introducing the following class of operators (called micro-differential operators): Let (x 0 , i e-0 ) be a point of yCI T*M c T*X (McRn, xcen). Let ,\ f e. Let IPA+j(z,OijfZ be a series of holomorphic functions defined in a neighborhood

u

of (xo, ie-o) in en

X

en. Suppose that PA+/z, (;) is

homogeneous of degree (A+ j) with respect to (;. Consider the kernel function K(x, y)

=

~

(2rr)

JI

.

PA+J·(x, i e-)« n+A+J·(< x-y, i e->

- O)w(e-) .

J

To make sense of the integrand set

for Re < x-y, (> < 0. We impose conditions so that this series converges on Re < x-y, (> > -ellm < x-y,(>ll ; i.e. we assume for Ve, there is a ce such that

(4.2.1)

ce j IPA+j(z,OI < -:re J.

for

Vj > 0,

and a constant R such that IPA+j(z,()l

:s

(-j)!R-j

for Vj < 0.

Since K(x, y, (;) is essentially a Laurent series in < x-y, (> , this growth condition assures the convergence of K(x, y, (;). Thus, we can

30

MASAKI KASHIW ARA

define a hyperfunction K(x, y, i 0 = being with respect to

! (x, y, t" );

bn

(K(x, y, it")) the boundary value E

Re < x-y, it"> < 01. Therefore, we have

-

-

-

SSKC !(x,y,t";i(t",7J,p));k lbl 2 . We next choose a family of closed sets ! Zclo

l) * be the transpose of the inclusion map of l) into g .

76

VICTOR GUILLEMIN

LEMMA 6.2. The symbol of p(DH) is the restriction to g~ of the pullback under r of the polynomial function, p, on f)*. Proof. It is enough to check this for the generators, DH. = (1/f-l)LH. c- 1 1

1

of the ring J{. However, a(C- 1 ) = a(C) = 1 on g~ by (6.2) and a((1/yC'I)LH)(e,.;)==pull-backof ~i· So,for DH. theasser1

1

tion is true.

Q.E.D.

Let W be a vector space on which G acts irreducibly. By Lemma 6.1 there exists a subspace, V, of L 2 (G) on which Gx G acts irreducibly such that W sits in V as a summand of type (6.1). Let "v be the orthogonal projection of L 2 (G) onto V. Each element, p(DH), of J{ preserves V and preserves the direct sum (6.1). It also preserves each weight space, w

H2 (X). We recall that if

-. c-oo(Y) a continuous linear

operator, there exists a distributional function, e A (x, y), on X x Y, called the Schwartz kernel of A , such that

(Af)(x) =

J

eA(x,y)f(y)dy.

y

The wave-front set of A is the set of points, (x, ~. y, 7J), in T*X x T*Y such that (x,~. y,-TJ) is in the wave-front set of eA. Moreover, A is

regular if its wave-front set contains no points of the form (x, ~. y, 0) or (x, 0, y, 7J). If A is regular then it maps C~(X) into C 00 (Y), maps

87

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

0

c ""(X) into c-""(Y), and has a transpose with the same properties.

(See [9], §2.4.) Consider now the operator,

above. From the results of §2, we

TT,

can immediately conclude that the wave-front set of rr is contained in

L x L; however, we can conclude a little bit more because of Proposition 2.2. Let p: L _, a+ be the projection (x, a)-> a and let LL1 be the fiber product of this mapping with itself. PROPOSITION 3.5. The wave-front set of

TT

is contained in LL1.

Proof. Let (x,/;,y,TJ) be in the wave-front set of rr. By (3.3), /; and TJ

are in a+; and, by Proposition 2.2, (/;, v) , ) being the Killing form. Hence /;

COROLLARY. The operator,

TT,

=

=

(TJ, v) for every v

f

a,

Q.E.D.

TJ.

is regular and is a smoothing operator

on the set X - U Xs . REMARK. We will see later on that the wave-front set of rr is, in fact, considerably smaller than the set, LL1 . NOTATION. We will denote by Clearly (3. 5) LL1

=

U

L~,

L~

the fiber product with itself of Ls _, S.

union over the subsimplices of a+.

§4. Some facts about Szego kernels To obtain more precise micro-local information about

TT,

we will re-

quire some general facts about reproducing kernels defined by degenerate elliptic equations. The results we are about to describe are due to Boutet de Monvel and Sjostrand and can be found in §2 of the paper [2]. Let X be a smooth manifold and let D 1 ,

· · ·,

DN be a collection of

first order pseudodifferential operators on X. We will say that the system of equations (4.1)

88

VICTOR GUILLEMIN

is in involution if there exist zeroth order pseudodifferential operators, k Qij, such that

for all 1

'S

i, j, k

'S

! be the

N. Let ai be the symbol of Di, and let

set of points in T*X- 0 satisfying: a 1 (x, () each point (x,(} of !

=

0, .. ·, aN(x, ()

0. At

=

we define the Levi form of (4.1) to be the NxN

matrix (4.2) The results of Boutet-Sjostrand concern the micro-local behavior of the system (4.1) at points where the Levi form is positive definite. It is a simple exercise in symplectic geometry to show that !

is a manifold of

codimension. 2N at such points. It is also easy to see that ! plectic at such points: the restriction to !

is sym-

of the symplectic form on

T*X is non-degenerate. To describe the results of [2] we need to introduce a little terminology. Let X be a differentiable manifold and A 1 and A2 continuous linear operators on c-oo(X). Given an open conic subset, will say that A 1 and A2 are equivalent on

e

e,

(A 1 -

of T*X - 0 we A2 on

e)

every generalized function, u' with wave-front set contained in

e

if for

(A 1 - A2 )u is smooth. THEOREM 4.1.

Let (x, () be a point in !

at which the Levi form (4.2)

is positive definite. Then there exists a conic neighborhood,

e,

of

(x, () and regular operators, rr, L 1 , ... , LN such that (i)

TT -

(ii)

I -

TTt - rr 2 on TT

(iii) Dirr -

+I Li Di 0 on

e

e on

e

for all i .

Moreover, rr is uniquely determined up to equivalence by these properties.

Proof. See [2], Theorem 2.14.

89

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

A companion theorem to Theorem 4.1 describes more precisely the nature of the operator, rr. Suppose, in general, we are given a differentiable manifold, X and a conic symplectic submanifold, L, of T*X - 0. In [1] Boutet associates with the pair (X, L) a natural class of symbols }{k(X, L) which are of type S~ '" on all of T*X- 0 and are of type 72, 72

s-oo

outside every conic neighborhood of

L. These symbols are called

Hermite symbols and the space of operators associated with them:

op}{k(X,L), are called Hermite operators. Boutet proves in [1] the following facts about these operators.

1. A

E

op}{ => WF(A) is contained in the diagonal in L

2. If A

E

X

L.

op}{k and Q is an ordinary pseudodifferential operator of

order C then AQ and QA are contained in op}{k+C. 3. If A

E

op}{k and B

f

op}{e then AB

E

op}{k+C.

4. If A< op}{k, At< op}{k. In [ 4] it is shown that Hermite operators have intrinsically defined leading symbols. The theory of these symbols involves the metaplectic group and the "symplectic spinors" of Kostant. An example of a Hermite operator which "occurs in nature" is the following. Let X be a strictly pseudo-convex domain in en and let H2 be a L 2 -closure in L 2 (JX) of

ax

the space of C 00 functions on

which are restrictions of holomorphic

functions on X. Associated with the contact structure on symplectic submanifold, L, of show that the Szego projector,

T*ax-

q

E

is a conic

0. In [2] Boutet and Sjostrand

rr: L 2 (JX) -. H 2 (dX),

in fact they show that rr belongs to

ax

op}{ 0 (ax,

is a Hermite operator;

L). More generally if

C (aX) the Toeplitz operator 00

belongs to

OPH 0 (ax, L). Getting back to the system of equations (4.1)

Boutet and Sjostrand prove THEOREM

oPH 0

4.2. The projector rr described in Theorem 4.1 1s m

cx, };) .

90

VICTOR GUILLEMIN

§5. Micro-local properties of

TT

in the interior of the positive Weyl chamber

We recall that the nilpotent algebra, n, can be decomposed into a direct sum of one-dimensional pieces

such that ni is Ad(a)-invariant and (5.1) vi being any non-zero vector in ni. (See (2.4).) To each ai X be a holomorphic map, U an open set in

T*X - T~ X and W an open set in T*Y- r; Y. Let right &;y-Module defined on W. Assume that with R.S. Assume furthermore that pThen

q;*n

"'m*(p- 1

n

R.S. defined on U.

®

1

Supp

n

n

be a coherent

is a holonomic system

n n m- 1 (U) _. U

is finite.

&y__.x) is a right holonomic &;x-Module with

-1.