Proceedings of Conference on Hyperfunctions, Katata, 1971

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich

287

Hyperfunctions and Pseudo-Differential Equations Proceedings of a Conference at Katata, 1971

Edited by Hikosaburo Komatsu, University of Tokyo, Tokyo/Japan

Springer-Verlag Berlin. Heidelberg New York 1973

A M S S u b j e c t Classifications 1970): 35 A 05, 35 A 20, 35 D 05, 35 D 10, 35 G 05, 35 N t0, 35 S 05, 46F 15

I S B N 3-540-06218-1 Springer-Verlag B e r l i n - H e i d e l b e r g " N e w Y o r k I S B N 0-387-06218-1 Springer-Verlag N e w Y o r k • H e i d e l b e r g • B e r l i n

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer Verlag Berlin . Heidelberg 1973. Library of Congress Catalog Card Number 72-88782. Printed in Germany. Offsetdruck: Jutius Beltz, Hemsbach/Bergstr.

Dedicated to the memory of the late professor Andre MARTINEAU, who had originally planned to attend this conference. appreciated

the importance of hyperfunctions

He

for the first

time and has made the most profound contributions to the theory of hyperfunctions.

LIST OF PARTICIPANTS

Y. AKIZUKI

(Gunma University)

H. HIRONAKA

(Harvard University)

A. KANEKO

(University of Tokyo)

M. KASHIWARA T. KAWAI

(RIMS, Kyoto University)

(RIMS, Kyoto University)

H. K O ~ T S U

(University of Tokyo)

T. KOTAKE

(Tohoku University)

J. LERAY

(Coll@ge de France)

M. MATSUMIfRA

(Fac. Eng., Kyoto University)

S. MATSUURA

(RIMS, Kyoto University)

T. MATUMOTO

(Kyoto University)

T. MIWA

(University of Tokyo)

S. MIZOHATA

(Kyoto University)

M. MORIMOTO

(University of Tokyo)

Y. NAMIKAWA

(Nagoya University)

I. NARUKI Y. OHYA

(RIMS, Kyoto University) (Fac° Eng., Kyoto University)

T. OSHIMA

(University of Tokyo)

H. SATO

(University of Tokyo)

M. SATO

(RIMS, Kyoto University)

P. SCHAPIRA T. SHIROTA H. SUZUKI M. YAMAGUTI

(Universit~ de Paris) (Hokkaido University) (Tokyo University of Education) (Kyoto University)

TABLE

PART

OF CONTENTS

I

Preface

Part

CONFERENCE

I . . . . . . . . . . . . . . . . . . . . . . . .

AT K A T A T A

H i k o s a b u r o KONLATSU: An i n t r o d u c t i o n to the theory of hyperfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . Mitsuo

MORiMOTO:

Edge

of the wedge

theorem

3

and h y p e r f u n c t i o n

.

41

J e a n - M i c h e l BONY et Pierre SCHAPIRA: Solutions h y p e r f o n c t i o n s du probl@me de Cauehy . . . . . . . . . . . . . . . .

82

T a k a h i r o KAWAI: On the global existence of real analytic solutions of linear d i f f e r e n t i a l equations . . . . . . . . .

99

A k i r a KANEK0: of linear CONFERENCE

F u n d a m e n t a l principle and e x t e n s i o n of solutions d i f f e r e n t i a l equations w i t h constant coefficients

122

AT RIMS

Sunao ~UCHI: On abstract Cauchy problems in the sense of hyperfunctions . . . . . . . . . . . . . . . . . . . . . . . . .

135

Shinichi KOTANI and Y a s u n o r i OKABE: On a M a r k o v i a n property s t a t i o n a r y G a u s s i a n processes with a m u l t i - d i m e n s i o n a l parameter . . . . . . . . . . . . . . . . . . . . . . . . .

153

Hikosaburo

KOMATSU:

Ultradistributions

of

and h y p e r f u n c t i o n s . . . .

164

APPENDICES H i k o s a b u r o KOMATSU: H y p e r f u n c t i o n s and linear partial d i f f e r e n tial equations . . . . . . . . . . . . . . . . . . . . . .

180

H i k o s a b u r o KONATSU: of d i f f e r e n t i a l

192

PART

Relative c o h o m o l o g y of sheaves of solutions equations . . . . . . . . . . . . . . . . . .

II

Preface

Part

II.

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

264

Mikio SATO~ T a k a h i r o KAWAI and Masaki KASHIWARA: M i c r o f u n c t i o n s and P s e u d o - d i f f e r e n t i a l E q u a t i o n s CHAPTER

I. T h e o r y

of M i c r o f u n c t i o n s . . . . . . . . . . . . . .

265

VI

1. C o n s t r u c t i o n

2.

of the

sheaf

265

of m i c r o f u n c t i o n s . . . . . . . . .

1.1.

Hyperfunctions . . . . . . . . . . . . . . . . . . . . .

265

1.2.

R e a l m o n o i d a l t r a n s f o r m a t i o n and r e a l c o m o n o i d a l transformation . . . . . . . . . . . . . . . . . . . . .

266 273

1.3.

Definition

1.4.

Sheaves

1.5.

Fundamental

Several

of m i c r o f u n c t i o n s . . . . . . . . . . . . . .

on s p h e r e

bundle

diagram

operations

and

on C

on c o s p h e r e

277 282

operators

and m i c r o f u n c t i o n s .

2.1.

Linear

2.2.

Substitution . . . . . . . . . . . . . . . . .

2.3.

Integration

along

....

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

on h y p e r f u n c t i o n s

differential

bundle

. .

286 286

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

287

fibers . . . . . . . . . . . . . . . .

294

2.4.

Products . . . . . . . . . . . . . . . . . . . . . . . .

296

2.5.

Micro-local

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

299

2.6.

Complex

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

302

operators

conjugation

3. T e c h n i q u e s for c o n s t r u c t i o n of h y p e r f u n c t i o n s and m i c r o functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.

Real

3.2.

B o u n d a r y v a l u e s of h y p e r f u c t i o n s w i t h h o l o m o r p h i c p a r a m e t e r s and e x a m p l e s . . . . . . . . . . . . . . . . . .

CHAPTER

II.

analytic

functions

of p o s i t i v e

302

of p s e u d o - d i f f e r e n t i a l

Definition

of p s e u d o - d i f f e r e n t i a l

1.2.

Operations

on h o l o m o r p h i c

1.3.

Sheaf

1.4.

Concrete

1.5.

Adjoints,

2. F u n d a m e n t a l

properties

315 324

operators . . . . . . . . .

of h o l o m o r p h i c

composites

315

......

microfunctions . . . . . . . .

of p s e u d o - d i f f e r e n t i a l expression

operators

and coordinate

microfunctions

329

. . .

332

.

344

. .

356

transformations

of p s e u d o - d i f f e r e n t i a l

operators.

T h e o r e m s on e l l i p t i c i t y and the e q u i v a l e n c e of pseudo-differential operators . . . . . . . . . . . . . Theorems

on d i v i s i o n

of p s e u d o - d i f f e r e n t i a l

307

315

operators . . . . . . . . .

1.1.

2.2.

303

F o u n d a t i o n of the T h e o r y of P s e u d o - d i f f e r e n t i a l Equations . . . . . . . . . . . . . . . . . . . . .

I. D e f i n i t i o n

2.1.

type . . . . . . . .

356

operators

.

365

3. A l g e b r a i c p r o p e r t i e s of the s h e a f of p s e u d o - d i f f e r e n t i a l operators. . . . . . . . . . . . . . . . . . . . . . . . . .

384

3.1.

Pseudo-differential operators with holomorphic parameters . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.

P r o p e r t i e s of the R i n g of f o r m a l p s e u d o - d i f f e r e n t i a l operators . . . . . . . . . . . . . . . . . . . . . . .

3.3.

Contact

3.4.

Faithful

3.5.

Operations

structure flatness

and

quantized

contact

transforms.

384 385 . .

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

on s y s t e m s

of p s e u d o - d i f f e r e n t i a l

equations.

391 400 406

VII

4. M a x i m a l l y

overdetermined

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

419

4.1.

Definition

of m a x i m a l l y

overdetermined

systems .....

419

4.2.

Invariants

of m a x i m a l l y

overdetermined

systems .....

419

4.3.

Quantized

case -- . . . . .

427

5. S t r u c t u r e equations 5.1.

contact

transforms

- general

t h e o r e m f o r s y s t e m s of p s e u d o - d i f f e r e n t i a l in the c o m p l e x d o m a i n . . . . . . . . . . . . . . .

Structure equations

429

5.2.

E q u i v a l e n c e of p s e u d o - d i f f e r e n t i a l operators with constant multiple characteristics . . . . . . . . . . .

434

5.3.

S t r u c t u r e t h e o r e m f o r r e g u l a r s y s t e m s of p s e u d o differential equations . . . . . . . . . . . . . . . .

448

III.

theorem for with simple

429

s y s t e m s of p s e u d o - d i f f e r e n t i a l characteristics . . . . . . . . .

CHAPTER

S t r u c t u r e of S y s t e m s of P s e u d o - d i f f e r e n t i a l Equations . . . . . . . . . . . . . . . . . . . . .

I. R e a l i f i c a t i o n

2.

systems

of h o l o m o r p h i c

microfunctions . . . . . . . . .

457

457

1.1.

Realification

of h o l o m o r p h i c

hyperfunctions

......

457

1.2.

Realifieation

of h o l o m o r p h i c

microfunctions

......

462

1.3.

Real

Structure equations

"quantized"

contact

transforms

. . . . . . . . .

t h e o r e m s f o r s y s t e m s of p s e u d o - d i f f e r e n t i a l in the r e a l d o m a i n . . . . . . . . . . . . . . . . de R h a m t y p e --. . . . . .

467

469

2.1.

Structure

theorem

I -- p a r t i a l

2.2.

Structure

theorem

II -- p a r t i a l

Cauchy

Riemann

t y p e --. .

479

2.3.

Structure

theorem

III -- L e w y - M i z o h a t a

t y p e -.

.....

496

2.4.

Structure

theorem

IV -- g e n e r a l

ease -- . . . . . . . . .

Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . .

470

520

524

AN INTRODUCTION TO THE THEORY OF HYPERFUNCTIONS* Bx, Hikosaburo KOMATSU

I.

Introduction It seems to me that the theory of hyperfunctions has a rather

long and established history.

It is true that the concept of hyper-

function was introduced by M. Sato [ 3 2 ] first time.

and

[33]

in 1958 for the

However, the idea of hyperfunction has been employed most

successfully since a long time ago. For example, in the complex method of harmonic analysis one studies functions on the unit circle by representing them as boundary values of holomorphic functions on the unit disk. looking upon functions as hyperfunctions.

This amounts to

It should be remarked that

many of the deepest results in the theory of functions of a real variable have been obtained by this method.

M. Riesz' theorem [31]

on the continuity of the Hilbert transform in

L p,

I < p < ~,

and

the Littlewood-Paley theory [24] are among them. More explicitly

T. Carleman [5] showed that if

f(x)

is a

measurable function on the real line such that O(Ixl ~)

for some

~

If(t)~dt = 0 then its Fourier transform is represented as

the difference of "boundary values" of holomorphic functions.

This

is one of the earliest introductions of generalized functions. Another example is found in the proof of the spectral decomposition theorem for self-adjoint operators. operators

For a special type of

Carleman [4] proved the theorem showing that the spectral

measure is obtained as the difference of the boundary values of the resolvent.

Although many proofs have been obtained since then, this

* Sections 4 and 5~are added on June 16, 1972.

is practically

the only one method by which one can compute the

spectral measure explicitly. Titchmarsh

We recall that the deep theory of E. C.

[36] and K. Kodaira

[15] for ordinary differential opera-

tors and that of T. Kato and S. T. Kuroda

[14] for partial differen-

tial operators both rely on this method. The third example is seen in the treatment of divergent integrals. Euler's integrals Re ~ < O.

B(~

However,

, ~ )

and

~(6 )

~ , ~

integrals

the form

a(x)x~

< 0,

in the complex domain which

# 0, -i, -2, "'"

(see Whittaker-Watson

A similar method works when one justifies Hadamard's divergent

Re~

L. Pochhammer and H. Hankel showed that these

integrals have integral representations converge for all

diverge for

[8].

[37]).

finite parts of

These facts show that singular functions of

should be regarded as boundary values of holomorphic

functions. Boundary values of holomorphic P. A. M. Dirac

functions appear also in physics.

[6] introduced not only Dirac's function

~ (x)

but

also the boundary value i x + i0

v.p. ! _ ~ i x

f(x).

More recently boundary values of holomorphic

functions play an

essential rSle in the so-called dispersion relation in the quantum field theory

2.

(see Bogoliubov-Shirkov

Hyperfunctions Let

~I

[3]).

of one variable

be an open interval

Intuitively a hyperfunction

f

(a, b)

on

or an open set in

is the difference of the

boundary values of an arbitrary holomorphic outside (2.1)

~

: f(x) = F ( x + i O )

~.

- F ( x - iO).

function

F(z)

defined

In order to define hyperfunctions by this formula, we have to characterize those holomorphic functions zero.

F

Painlev@'s theorem says that if

are continuous up to

~

is holomorphic also on hyperfunctions on

~

Suppose that

V

for which the difference is F(x+iy),

y > 0

and

y < 0,

and the boundary values coincide, then ~.

F(z)

Taking this into account, 'we define

as follows. is an open set in

relatively closed set.

¢

containing

Then the hyperfunctigns on

~

~

as a

are by defini-

tion the elements in the quotient space (2.2)

~(~)

where

~(V)

=

and

~(V \ ~ ) /

(>(V),

~(V \ ~ )

are the

~

~

....a(

b

spaces of all holomorphic functions on V

and If

F

V\~

~

respectively.

F ~ ~(V \~.),

we denote by

IF]

the class of

a defining function of the hyperfunction

(2.3)

[F] = F ( x + i 0 )

choice of the complex neighborhood Theorem 2.A (Mittag-Leffler, of open sets in (i, j)

C. with

(2.4)

Gj ~

(2.5)

assume that (2.6)

U C V. ~(V\~)

U

~(~)

[ii], p.13).

does not depend on the

Let

Gij ~ ~(V i ~ Vj)

V i ~ Vj # ~

and satisfy

and

V

on

~Vj~

are given for all

V i ~ Vj ~ V k .

on

V. ~ V.. I J

be complex neighborhoods of

The restriction mappings ~ ~(U\~)

induce the linear mapping

be a family

such that

G.. = G. - G. l] j l In fact, let

We also write

Suppose that

~(Vj)

and call

V.

Gij + Gjk + Gki = 0

Then, there are

[F].

F

F(x- i0).

The following theorem shows that

pairs

V ~

and

~(V)

> ~(U)

~'~

We may

(2.7)

(~ (V \ ,,q,) / O-(V)

,~

Clearly this is injective.

0 (U \ ,,~L)/ O" (U).

Considering the family of open sets

{U, V\~I~ , we see that every

F ~ ~(U\fL)

F=G where

G e 6~(V\~_) When

~' C ~

and

-N,

N ~ O(U).

Therefore

are two open sets in

(2.8)

~,

is written

: ~(~)

IR,

(2.7) is surjective.

the restriction mapping

> ~(fg')

is defined in a natural way so that defining functions remain the same. ~

is the identity and if

' = ?n"

= P~'

holds.

~"

We write also

Theorem 2.A shows also that mappings

o~£L ~,

form a sheaf.

Theorem 2.1.

the chain rule

(f) = f I SL

~j (£L),

~ C ~,

with restriction

Namely we have :

~jl

Let

C ~'L' C ~.,

be an open covering of an open set

f~

in ~.

(i) f = 0

If

f E ~(~)

satisfies

= 0

for all

j,

then

f E

~(~)

~(~).

in (ii)

If

f.j ~

~(~j)

satisfy

f'j I ~j ~ ~ k for all

f I ~j

j,

such that

k

with

~j

~ ~k

= fk I ~ j

n ~k

then there is an

# ~ '

f I ~j = fj-

Proof. (ii).

(i) is trivial. Let

fj = [Fj]

complex neighborhood

of

with

~

Fj E ~(Vj \ ~ j ) .

V = ~Vj

is a

We have

Gij = Fj. - F.l E ~ ( V i ~ Vj). Clearly

Gij

G.. = G. - G.. lj j l

satisfy (2.4).

Hence there are

such that

Thus we have F. - G. = F. - G. l i j j

Define

G°j E ~(Vj)

F E ~(V\fL)

by

on

F(z) = F j ( z ) -

(V. ~ V.)\~.. l j Gj(z)

for

z E Vj \~Lj.

Then

f = IF] E

~ (~)

gives the desired hyperfunction.

The sheaf of hyperfunctions is flabby : Theorem 2.2. hyperfunction Proof.

Let

~'

f ~ ~(~') Choose

be an open subset of can be extended to an

.

Then any

f E ~(~).

as the complex neighborhood of

¢~'

a defining function of

~

f

~'.

Then

can be regarded as a defining function of

This is a peculiar property of the hyperfunctions which the distributions do not share. Since the hyperfunctions form a sheaf, we can talk about the support

supp f

of a hyperfunction

space of hyperfunctions on If

f ~ ~(~)

function

where

F ~

F

is

encircling It

is

We denote by

with support in

has a compact support

~(V~K).

K,

curve

in

~K(~)

the

K. f

We define the integral of

a closed

K

~

f.

has a defining f

by

V

once.

clear

defining function

that

F

the

integral

does

.

or the curve

with the interpretation

not

depend

on t h e

choice

of

Note that (2.9) is compatible

(2.1).

Suppose that (2.10)

d m P(x, ~ x ) = ~, a~(x) ~=0

is a differential operator with coefficients of real analytic functions on

~L

Then

analytically to a complex neighborhood d

(2.11)

P(z, d~z ) =

the

V

a~ in the space

a~

can be continued

of

fL

~(~)

Let

m

~ a~(z) ~=0 dz~

be the corresponding differential operator on

V.

We define the

operation of

P(x, d/dx)

(2.12)

on

~ (~)

by

P(x, d~ )IF] = [P(z, d~)F]

Theorem 2.3.

Let

K

be a compact set in

~

and let

~(K)

be

the space of germs of real analytic functions defined on a neighborhood of

K

endowed with the inductive limit topology

(2.13)

where

~(K) = lim ~(U),

U

~.(K) '

ranges over the complex neighborhoods of of

~.(K)

is identified with

~K(~)

K.

Then the dual

under the inner

product / -

(2.14)

<~

, f>=

~/L~(x)f(x) dx

~)~ is a neighborhood of

where

K

on which

~ e ~(K)

is defined.

This follows from Theorem 2.B (K~the [22]). an open set containing as above.

space

K

be a compact set in

Define the locally convex space

and

C

V

~(K)

Then

(2.15) Let

K.

Let

~(K)' ~ s ~ i.

(~(V \K)/ (~(V) .

According to Beurling and BjUrck [2], we define the

~(s)'(/L)

of ultradistributions

of Beurling type (or of class (s) ~(s)(~)

= ~ ~ E ~(~);

of Gevrey class of order

s

for short) to be the dual of ~h > 0,

3C

(2.16) sup ~ D ~ ( x ) ~ ~ C h l ~ | I ~ | ~ s

, I ~ = 0, I, 2, "'" ~

X

endowed with a natural locally convex topology.

~(s),(~),

form a sheaf under natural restriction mappings.

Similarly to the

case of distributions,

the subspace of all

f e ~(s),(~)

compact support is identified with the dual of

~

with

C ~,

£(s)(~)

= { ~ 6

E (~);

~K

compact

C ~,

V h > 0, ~ C

(2.17)

sup ID~?(~)I ~ Ch1"~l~l:s

Ixl= 0, I, 2, "'" 1

xeK Since the real analytic functions are dense in (s)(~),

~(~)

or in

it follows from Theorem 2.3 that the hyperfunctions with

compact support contain the distributions with compact support and the ultradistributions

of class

(s)

with compact support.

Extending these imbeddings, we can show that the sheaf distributions and the sheaf

~(s),

are subsheaves of the sheaf

~

of ultradistributions

of hyperfunctions.

differential operators are preserved

~'

of

of class

(s)

The operations of

(Harvey [i0], Komatsu [16], [19]).

The following theorem characterizes distributions and ultradistributions. Theorem 2.4. i < p <~, constant

(i)

A hyperfunction

~(~)

if and only if for every compact set C

~F(x + iy)~ep (K) = ( ]

for small

~y~.

(ii)

[F] E

K G ~L

L~oc(~),

there is a

~(~)

IF ( X + i y ) ~ p d x ) I / p

is a distribution if and only if for every

sup ~F(x+iy)~ ~ x~K

(2.19)

More precisely

[F]

N

C[y]

and

C

such that

-N

belongs to the local Besov space

-~0 < 6- < ~0,

I ~ p, q ~ ~

every compact

K G D,

there is a constant

~_C

K

there are constants

K G

(2.20)

is in

such that

(2.18)

compact

IF] E

(see Besov [i]),

if and only if for

and for some/any non-negative integer ~

B¢ fL) p,q,loc (

k ~

such that

E ( ~ y ~ k - ~ I D ~ ( x + iy)~Lp (K) )q

dy / ~Yl < ~ -

-~

(iii)

[F] e

~ (f~)

is an ultradistribution of class

(s) ,

s >i,

I0 if and only if for every compact set

K C~,

such that

there are

L

and

C

i

•

f e ~S-I )

(2 21)

xE~SU I F ( x + i y ) I ~ C e x p { k T ~ j

In all cases (2.22)

F(x+i0)

= s-lim F ( x + i y )

y$0 exist in the topology of respective space and (2.3) holds. (i) is due to F. Riesz [30]. (ii). The characterization of distributions by (2.19) is due to G. Ko'the [23]. If

The precise form follows from (i) by interpolation.

0 < ~" < I,

B ,~,loc(~.) is the space of Lipschitz continuous

functions of exponent exponent

6"

0- .

Thus

[F]

is Lipschitz continuous of

if and only if

(2.23)

sup ~DF(x+iy); ~ C IYl 6"-1 xeK

This has been used by G. H. Hardy [9] to show that Weierstrass' nondifferentiable function ---~ >. a k cos b k 7Cx, k=l is non-differentiable whenever differentiability when

ab > I.

0 < a ~ I,

(Weierstrass proved the non-

3 ab > I + ~ vE .)

(iii) is proved in Komatsu [18], [19].

3.

Ordinary differential equations In order to show how things become transparent in the framework

of hyperfunction, we discuss linear ordinary differential equations with real analytic coefficients. Let dm (3.1)

P(x, d )

= am(X ) dx _ _ m + ... + al(x ) d +

be an ordinary differential operator with coefficients

a0(x ) ai(x ) ~ ~ ( ~ )

11 and

a (x) ~ 0. m

We consider the most fundamental problems of existence,

prolongation, uniqueness and regularity of solutions

u

of the equa-

tion (3.2)

P(x, d~) u(x) = f(x). Existence. Theorem 3.1 (Sato [33]).

For any

f E

~(~)

-

~

there is a solution

6 (/I).

uE

Proof.

Let

neighborhood of coefficients

V

be a complex

/

£L

to which the

~

a.J

J

are continued

k

~

'

" V

+

~ '

~

We may assume that

V

and

are simply connected and that all zeros of

V

are contained in

of

f.

£L

Let

There is a solution

P(z,

(3.3) Then

u = [U]

F e ~ ( V \fL)

U ~ ~ (V \0.)

d~ ) U(z)

xV

-----

analytically. V\£L

~

am(Z)

in

be a defining function

of

= F(z)

.

gives a solution.

Prolongation. Theorem 3.2 (Sato [33]). If

f E ~ (a, b),

any solution

be prolonged to a solution Proof.

Let

(c, d)

be a subinterval of

u e ~ (c, d) of (3.2) on

~ E ~ (a, b)

on

(c, d)

(a, b). can

(a, b).

Because of Theorem 3.1 we have only to consider the

homogeneous equation (3.4) Let

P(x, d~) u(x) = 0 . V

be the complex neighborhood of

of Theorem 3.1.

(a, b)

Then W = V \ ((a, b) \ (c, d))

employed in the proof

12 is a complex neighborhood O ( V \ (a, b)) (c, d).

be a defining

P(z, d/dz) U(z)

is a function

N(z) E

(3.5) c a n be c o n t i n u e d ( a , b)

gives

Clearly

Let

U ~

~ ( W \(c, d)) =

function of a solution

is holomorphic

on

W.

u

of (3.4) on

If we show that there

such that

d/dz) U(z) - P(z,

to a holomorphic

d/dz) N(z)

function

on

V,

then

~ = [U-N]

and

V2

such that

a prolongation.

we c a n c h o o s e o p e n s e t s

V2 ~ ( ( a ,

b) \ ( c ,

simply connected

d))

and f r e e

P(z,

= #

V1

and t h a t

from zeros

~ ( V I \ ((a, b) \ (c, d)))

By Theorem

(c, d).

~(W)

P(z,

= V,

of

of

V1 \

am(Z).

V1 v V2

((a,

b) \ (c, d))

Let

NO E

is

be a solution of

d/dz)N0(z)

2.A there are

NI ~

= P(z,

(~(VI)

d/dz)U(z).

and

N0(z ) = N2(z ) - N l ( Z ) ,

N2 e

~ ( V 2)

such that

z E V1 ~ V2.

Now d e f i n e z e V 1 \ ((a, b) \ (c, d))

N(z) = $ N 0 ( z ) + N I ( Z ) '

L N 2 (z) N @ ~(W)

Then

and

,

z~V

2•

(3.5) is continued to a holomorphic

function on

V.

Uniqueness. Theorem 3.3 (Komatsu

[20]).

dim ~ u e ~ ( ~ ) ,

Let

~,

be an interval.

Then

P(x, d/dx)u = 0 }

(3.6) = m + where

ord x am(X)

~ ord x am(X ) , xE~ denotes

the order of zero at

x

of the function

am(X). To prove this, we need the following

index theorem.

13

T h e o r e m 3.A (Komatsu [20]). let the coefficients differential

a.(z) J

operator

Let

V

be an open set in

of (2.11) be holomorphic

P(z, d/dz)

: ~(V)

~ ~(V)

on

¢

V.

and Then the

has the index

~(P) = dim ker P - codim im P (3.7) = m~(V)

-

Z

ord

where

~(V)

of

of

z aO(z )

space of If

V

,

of

V

or the number of

minus the number of compact connected

¢\V.

Sketch of Proof. ~ord

(z) m

is the Euler characteristic

connected components components

a g

z~V

If

m = O,

ker P = 0

and one can easily find

polynomials which form a basis of a complementary

sub-

im P. P = d/dz,

it follows from T h e o r e m 2.A (see HUrmander

[ii],

Theorem 2.7.10) that ker P ~ H0(V, ¢)

and

coker P ~ HI(v, ¢).

Since the index of the product of operators indices,

we

have

(3.7) for

is the sum of the

P = am(z)dm/dz m.

Lastly the perturbation by

am_l(Z)dm-I/dz m-I +

.-" + a0(z )

does

not change the index. Actually we have a convenient Banach spaces.

stability theorem of index only in

Therefore we have to approximate

~(V)

by a sequence

of Banach spaces and compute the index as the limit of the indices in approximate

spaces.

We note that this theorem implies the classical Perron theorem: If

V

is connected and simply connected,

m-

Z ord z am(Z)

the homogeneous

there are at least

linearly independent holomorphic

solutions on

V

of

equation.

Proof of T h e o r e m 3.3.

Choose a complex neighborhood

as in the proof of Theorem 3.1.

V

of

~g

Then we have the commutative diagram

14 with exact rows and columns : 0

0

0

0 -'~ oP(v) ---> ~P(v\f~) --+ v~ ~P(_o.) ~

0--+

e(v)

0-~

PV $ Pv\Pa $ P fZ C>(v) --~ @(v\~l) --~ 63(£) --~ 0

(3.8)

--~ C~(v\&)

O(V)/P (Y(V)

--~ 8 ( & )

~P(a~)/7~P(v\~)

0

--~0

--* 0

0

0 (>P

where

(~P)

denotes the sheaf of holomorphic

solutions of the homogeneous Since dim ker PV\~

PV

and

equation.

PV\II have indices and

are finite,

dim ker PV

and

P~. has the index

X(P&) = ~ ( P v \ & ) =

(hyperfunction)

2m

-

%(Pv)

(m- ~

ord x am(X))

= m + ~ ord x am(X )

8 P(O.),

Note that there are two types of solutions in coming from

6~P(v\~)/ (yP(v)

and the other from

one

(~ (V)/P @(v).

Regularity. Theorem 3,4.

(a)

~P(f).) C ~.(,f~) ;

(b) (c)

Proof.

The following are equivalent :

am(X ) ¢ 0

for all

Pu ~ & ( ~ )

(a)

>

> (b),

that there are more than

If m

x • ~L ;

u e &(~)

am(X 0) = 0,

it follows from Theorem 3.3

linearly independent

solutions near

Therefore we can find at least one non-zero solution

u ~ ~P(~L)

x 0. such

15

that

u(x) = 0 (b)

for

> (c).

there are

m

x < x0,

which cannot be real analytic.

On each connected component

~i

linearly independent solutions in

of

~P(F~I).

other hand, Theorem 3.3 shows that there are exactly independent solutions in For each

f E

~(~)

solution is in (c) ~

~P(~I).

Hence

there is a solution

x0

m

clearly On the

linearly

~P(fzI) = ~P(FLI)u 6 ~(~).

Thus every

~(~).

(a)

trivially.

Next we consider the case in which Let

~

am(X )

has zeros in

~

.

be such a zero or a singular point of the equation.

We define the irregularity

~

of

x0

to be the maximal

x

2 o

0

i

2

m

J

gradient of the highest convex polygon below the points (j, ordx0 a.j(x)), If

~ ~ i,

an irregular

(b) (c)

x0

=

0 7

" " " ~

m.

is called a regular singular point and if

singular point ,,

Theorem 3.5. (a)

j

The following are equivalent: ~e(f~)

C~'(f~)

All singular points in Pu e ~ ' ( ~ )

~

;

are regular ; ---> u e ~ ' ( f ~ )

¢ > i,

16 Theorem 3.6.

Let

s > i.

P(fl) C ~ ( s ) ' ( [ l )

(a) (b)

The irregularity

Pu E ~ ( s ) ' ( J l )

Proof.

Theorems

(a)

(b).

~

> i,

Let

0

0,

a holomorphic

is a non-zero

log z

some

solution

U(z)

<

of

of

P(z, d/dz)U(z)

''-} z~p(z, log z) , p(z,

log z)

is a polynomial

are formal power series of

(see Hukuhara-Iwano

A

,

[12]).

z I/q

the solution

U(x±iO)

satisfies

~

(c).

loss of generality

ment

}

= 0

~(s)'(~)

has a solution for any

s > ~/(~

-I)

Since the problem is local we may assume without that

~

is an interval

containing

0

as the

singular point of the equation.

Let = 1

to the

the estimate

P(x, d/dx)u(x)

which does not belong to

(b)

unique

Therefore

for

Hence if we choose the

sup I U ( x + iy) l ~ C e x p I ( ~ > ~ - i x~K on the half plane.

.

rE/(0-- i)

sector either in the upper or the lower half plane according argument

~

expansion

constant and

whose c o e f f i c i e ~ s

q > 0

in the same way.

be a singular point of irregularity

U(z)--~ expl z ~ +

in

does

u ~ ~(s)'(f£).

3.5 and 3.6 are proved

which has the asymptotic

~

==~

we can find, on each sector with angle

and summit at

where

D_.

of any singular point in

s-i

(c)

= 0

;

s

not exceed

If

Then the following are equivalent :

U(z)

~ ~(V\O_)

for Theorem 3.5 and

be a defining ~

function of

= s/(s - i)

(b) means that the irregularity

~ ~

u.

We set

for Theorem 3.6.

State-

17 Now let j = O, i, "'', m-l.

Vj(z) = (z ~ d~) j U(z) , Then, the column vector

V(z) = t(V0(z),

''-, Vm_l(Z))

satisfies the

equation ( Z T d ~ + B(z)) V(z) = F(z) , where

B(z)

is a matrix whose elements are bounded near

0

and

satisfies the estimate

IF(x+iy) I ~_

I C-IY~ L

~=

I

i

'

C exp~ C ~ ) r - l l ,

for some constants

L

and

C

~>I

by the assumption.

If we choose log( _ ~ i ~ , t

ql = i

= ,

as the independent variable, the equation is transformed into (c d~ + Bl(t))Vl(t) = Fl(t) ' where denote

c

is a constant depending only on B(z(t)), etc.

Since

Bl(t)

~

and

Bl(t), etc.

is bounded, we have by the

standard technique of differential inequality the estimate C'

i

lyle'

'

T

=

i

IV(x+iy) I <= I C ' exp Hence u(x) = U(x+iO)

U(x - iO)

T>I

F(z)

18

belongs

to

~ '(fl)

(c) ~

(a)

Coroll@ry singular points

or

~(s)'(/l).

trivially.

3.7

(Meth@e

in

[28]).

Let

are regular,

dim~u

~ ~

'(~_);

~

be an interval.

All

if and only if

Pu = 0

(3.9) = m + Example

3.8.

~ ord x am(X ) xE~

Let

~

be an interval

containing

the origin and

let d d P(x, ~ x ) = x ~ x - ~<

(3.10) Since

0

exactly

is the only regular two linearly

singular point of order

independent

solutions

both of which are distributions. ing linearly independent (a)

In case

--1

[ (-z) ~]

,

2i sin ~

x~ = (-x)+ (b)

In case

= -I, -2,

{ ~(-~-i)

"'" ,

(-1)~ (-~-I) ' =

v.p. x (c)

In case

~

i where

z~

and

log~

27ti

= ~((x+i0)

= 0, i, 2,

" [z~] ' + ( x - i 0 ) ~) ;

-'- ,

x~ , x+

-

--i -2 ~ i

denote

[

equation

More explicitly we have the follow-

solutions :

----

there are

of the homogeneous

~ # integer,

i x+

i,

z ~

log (-z)]

,

their principal branches.

19 Example 3.9.

Consider

(3.11)

(x 2 d . I) u(x) dx

This has regular (0, ~ ) ,

solutions

The solution

c e -I/x

e -I/x

infinitely differentiable 0].

The prolonged

independent origin,

= 0 on the intervals

and there are no other hyperfunction

intervals.

(-~,

the equation

distribution

for the solution

any distribution

on

(0, ~ )

solution on solution

~

solutions

-I/x

and

on these to an

by extending it by zero to

is actually

on

0)

can be prolonged

the only one linearly

solution on any interval e

(- ~ ,

(-~,

0)

containing

the

cannot be prolonged

to

on such an interval and there are no distribution

solutions with support at the origin. The formula linearly

(3.9) shows, however,

independent hyperfunction

that there are two other

solutions.

We can choose,

for

example, -i/(x+i0)

f °[e'l/z

]

The first one is a prolongation

of the solution

and the second one is the only one linearly

e

-i/x

independent

on

( - ~ , 0)

hyperfunction

solution with support at the origin.

4.

Hyperfunctions

of several variables

To obtain hyperfunctions consider

the tensor product

of two independent

variables,

of (2.2) :

~(~i ) ~ ~(~)-2) (4.1)

= (~(VI\~II)/O(VI))

~ ( V I) ~

~

(~(V2\~I2)/~(V2))

~-(Vl\fLl) @ ~(V 2 \ ~ 2 ) ( > ( V 2 \ ~ 2) + ~(Vl\~.l) ~

~ ( v 2)

let us

20 Unfortunately the spaces

~(~j)

of hyperfunctions do not possess

natural locally convex topologies, so that we cannot define the space (~I X ~2 )

of hyperfunctions

of two variables as the completion of

(4.1) as in the case of distributions

etc.

It is expected, however,

that ~ ( ( V I \ ~ I) X (V2\~L2))

(4.2) ~(~i

X ~2 ) =

~(V I X ( V 2 \ ~ 2 ) ) +

~ ( ( V I \ ~ I) X V 2)

'

where the numerator and the denominator are the completions of those

of (4.1).

Similarly we can expect that n

(4.3)

~(~LIX

"'" X ~ n ) =

~ ( V # ~L)/ ~. ~(Vj)

j=l

,

where V ~ SL = ( V I \ ~ I) X

V.J

= (VI\fL I) X "-- X (Vj _ 1 \ ~

"'' X ( V n \ ~ n) ,

j _ 1) X Vj X (Vj+l\fLj+l) X "'" X (V n \,~)..n ) .

Although this is a right definition of hyperfunctions variables,

of several

the expression depends on the coordinate system strongly

and it seems very difficult even to show that they form a sheaf. To overcome this difficulty, Sato [32], [33] invented the notion of relative cohomology for open pairs. introduced by Grothendieck Suppose that

~

topological space

X

[7] independently.)

is a sheaf (of abelian groups etc.) over a and that

a relatively closed set. H~(V, ~ ),

(The same notion has been

V

is an open set containing

Then, the relative cohomology groups

p = 0, i, ... ,

are defined so that

F

as

21 F (V)

~ (V)

~ (V \ F)

LI

II

li

H O(V, ~ )

> H 0(v, ~ )

~ H 0(VXF,

~)

>~(v, ~,)

H I(V, $ )

~ H I(V\F,

~)

(4.4)

~H~(V, ~)

~

"'"

is a long exact sequence in a natural way, where HP(v \ F, ~ ) H p(V, ~ )

pv

V\F

and

are ordinary cohomology groups and the mappings ~H p(V\F,

~(V)

:

HP(v, ~ )

are those induced from the restriction

~)

(see [16], Chap. I for a precise defini-

> ~(V\F)

tion) . In the one dimensional case, we have continuation theorem.

(>~(V) = 0

by the unique

On the other hand, Mittag-Leffler's Theorem 2.A

is equivalent to (4.5)

HI(v, (>) = 0

for any open set

V

in

¢.

Hence we have the isomorphism HI(v,

(T) ~ H0(V\0_,

(4.6)

(~)/H0(V, (>)

~(~).

Extrapolating

this, Sato defines the space

on an open set

~A

(4.7)

in

~Rn

~(~_)

of hyperfunctions

by

~ (f~) = H~(V, ~ ) ,

where

V

is an open set in

¢n

containing

~

as a closed set.

By the excision theorem ([16], Theorem I.I) (4.7) does not depend on the choice of the complex neighborhood If

~ I'

~.'

is an open subset of

: H (V, ~ )

) U ,(V', ~ )

~.,

V

of

f~.

we have natural mappings

induced from the restriction.

shown that if (4.8)

H~(V,

~)

= 0 ,

p = 0, i, ''', m-1

It is

22

for every Therefore

O_ ,

then

)

H~(V,

to show that

~(fl),

form

a sheaf

~n

O_ C

,

([16], Theorem

1.8).

form a sheaf, we have to

prove Theorem 4.1

(Sato).

(4.9)

H~(V,

~)

= 0 ,

p = 0, I, ''', n-i

for every open set in (a basis of) Once this is proved, from [16], Theorem Theorem 4.A

~n.

the flabbiness

of the sheaf

follows

1.8 and

(Malgrange).

(4.10)

HP(v,

for every open set

V

~)

in

cn.

~(0_),

~C

= 0 ,

p ~ n

Thus we have Theorem 4.2.

~n,

form a flabby sheaf.

K~the's duality Theorem 2.B extends also: Theorem 4.B

(Martineau

[25]).

If

K

is a compact

set in

cn

such that (4.11)

HP(K,

~)

= 0 ,

p > 0,

then HP(¢n'K

(4.12)

~)

= f 0

,

p # n

~(K)',

p = n.

satisfies

(4.11).

I

Every compact

K

set

in

IRn

Therefore we

have the following. Theorem 4.3.

If

K

(4.13)

~ K ( I R n) ~

Similarly tions ~(~)

is a compact

~'(f6)

set in

and the ultradistributions

In order to regain

then

~(K)'

to the case of one variable,

of hyperfunctions

~n,

we can imbed the distribu~(s)'(f~)

into the space

through this duality. (4.3), let us consider

relative

cohomology

23 groups of coverings. set

V

and that

Suppose that

(~, ~')

By this we mean that

~

open coverings of

and

V

F

is a closed set in an open

is an open covering of the pair = ~ Vi; i ~ I} V\ F

and

~'

the covering

(~,

~')

=~ Vi; i ~ I' I

respectively and

We define the p-th relative cochain group

(V, V \ F). are

I' C I. cP(~J", ~g~', ~ )

with coefficients in a sheaf

of

to be the

set of all direct products of Fi0,''',i p

E ~(Vi0 , ... ,ip )

= V. ~ V. l0 z0,''',i p

defined for all non empty

..- ~ V. , i P

which are

alternating in indices and satisfy F. 10,''" 'ip

= 0

whenever all

i0, ..-, i 6 I' P

The coboundary operations : cP(9~", ~j'', j~,) are defined in a usual way. HP(~,

~',

~)

,'~cP(~u~, Q~', j~)

Then, the p-th relative cohomology group (~,

of the covering

~')

C'(~,

cohomology group of the complex

is defined to be the p-th

~j", ~ ) .

We have natural

homomorphisms (4.14)

HP(lY, ~ ' ,

~ )

> H~(V, ~ )

and these are isomorphisms if (4.15)

HP(vi0,...

and

([16], Theorem i.i0).

V. 10,''',i q

iq

~)

0

for all

p ~ 0

The Oka-Cartan Theorem B ([II], Theorem 7.4.3) asserts that (4.16)

HP(v,

~)

= 0

for all

for every Stein (= pseudo-convex)

open set

p > 0 V

in

cn

Since the intersection of Stein open sets are Stein, it follows that (4.17)

~ (~)

=~ Hn(]~, ~ ' ,

~ )

24 if

(9]', ~ " ) Let

is a Stein covering of

q~ = {U0, U I, ---, Un} U0 = VI ×

(V, V \ ~ ) .

and

9j" = {UI,

(4.17) becomes

with

... × V n ,

Uj = V I× -'. x Vj_ I ~ (Vj \ ~ j ) Then

..., Un}

× Vj+ I ~

"'' X Vn

(4.3).

More generally we have Theorem 4.C (Grauert). system of Stein neighborhoods Therefore of

~

(~,

V

in

~')

be the covering of and

~'

={VI'

V0=V Vj V0

and

IRn

has a fundamental

small Stein neighborhood

V

V ~ ~n = /~..

{V0, V I, "'', Vn}

Since

in

C n.

we can choose an arbitrary

such that Let

~A

Any open set

"''' Vnl

are Stein,

defined by

q)- =

with

,

z ~ V ' Im z.j # 0 ,

V. J

(4.18)

(V, V\~I)

j = i, -.., n

we have

(0.) --~ (~(V ~ ~)_)/ ~_~ O(Vj)

where V~

~L= Iz E V"' Im z.j # O,

VO = { z When F

V~ where

E V ; Im z k # 0, we denote by

F E (~(V ~ ~.),

and call ~_ r~

F has

j = I, -.-, n},

k # Of" [F]

the cohomology class of

a defining function of the hyperfunction 2n

connected components

= I y E ~n,

through all n-tuples of

~j yj > O,

near

IRn : V ~ ( ~ n + i r ~ ) ,

j = I, "'', n }

the hyperfunction

of boundary values:

where

IF] (x) = sign

~=

~i

and

~

runs

± i.

Intuitively we can interprete

(4.19)

[F].

"'" ~n"

~. sign ~" F ( x + i [ ' 6 0 )

,

[F]

as the sum

25

Im z

Im z 2

2

b-

Another Stein covering follows. in

~n

We choose

n+l

(~,

~')

points

of

~i'

(V, V \F)

"''' ~n+l

so that the simplex with vertices

the origin in its interior.

VO = ~z d V ;
~i'

ImZl

is obtained as

on the unit sphere "''' ~n+l

contains

Let V0=V

Then

,'~'~<~

\

Im z I

0, V I, "'', Vn+ll

,

~j>

< 0~,

and

~'

j = I, "'', n+l.

=IVI,

..., Vn+l}

form a

Stein covering of

(V, V\~.). Thus we have n+l ~ ( ~ ) -~ ~ ~(V')/~ ~(Vi j =I J i<j 'j ) '

(4.20) where

V-j = V I ~ -.. ~ Vj_ I ~ Vj+ I ~ l,j

5.

=

V I~

-'-~Vi_ l ~ V i + l ~

-.- ~ Vn+ I ,

"'" ~Vj_ I ~ V j + I~ -'-~Vn+ I

•

Hyperfunctions with holomorphic parameters Martineau

[25], Harvey

[I0] and Schapira

[35] prove Sato's

Theorem 4.1 employing Martineau's Theorem 4.B.

In this section we

give a direct proof of Theorem 4.1 and clarify the meaning of the "boundary value" wedge

F(x+ir0)

V ~(~n+i~).

of a holomorphic

(Cf. Morimoto

[28'],

function

F(z)

defined on the

[29] for a similar proof.)

26 Our proof is based on the following be a closed set in a topological we denote by H~nv(V,

~).

~(~)

space

result of Sato X.

For a sheaf

[33]. ~

Let

over

F X

the sheaf associated with the pre-sheaf

We say that

F

is purely m codimensional

relative

to

if (5.1)

~P(~) F If

relative

A to

= 0,

is a closed set in ~

,

for any open set

(5.3)

F

and

V.

is purely

codimensional

isomorphisms

0 = t

m

p<

p-m

m

HAnv(V , J { F ( ~ ) ) ,

m

p ~- m

Taking the limits of both sides, we have also

J~P(~)

(see [16], Theorems

p # m .

F

then we have natural

HPnv(v '

(5.2)

for all

tO{[

~

p< m

P'm(~F(~)),

p ~_ m

1.8 and 1.9).

We set Rj

=

~zE

C n",

z.

~

~

},

j

=

I,

2,

"'"

, n,

3

Ri0,...,i p = Ri£''" and consider

the following Cn

N Rip

sequence of closed sets :

D ~ × C n-I D ~ 2 × on-2 ~ ... ~ n Ii

Ii

i!

RI

R1, 2

R1,''',n

We prove the f o l l o w i n g theorem by i n d u c t i o n on Theorem 5.1.

RI,2

..., r

is purely

to the sheaf

of holomorphic

(5.4)

r~n-r

r

functions

r

~ = ~Rl,...,r

codimensional on

(~)

Then (5.5)

HP(v ' r ~ n - r ~ )

for every Stein open set

V

in

= O, ¢n

r.

p > 0

Cn .

relative

We write

27 Theorem 4.1 is the special case of

r = n.

If

r = 0,

this is

Theorem B of Oka-Cartan. Lemma 5.2.

If Theorem 5.1

holds for an

r

and

0 r~n-r HRr+InV(V, ~) = 0

(5.6) for all Stein open set

V

in

cn,

then Theorem 5.1 holds for

r+l

and we have r + l ~ n - r - i 6~(V) ~ r ~ n - r ~ ( V \ R r + l ) / r ~ n - r

(5.7)

for every Stein open sets V Proof.

Let

V

(5.8)

in

(~(V)

cn. Cn .

be a Stein open set in

H~r+InV(V, r ~ n - r ~ )

In fact this is an assumption for

p #l.

= 0 ,

p = 0.

If

We have

p ~ 2,

this follows

from the exact sequence : V

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

o --->,H R0

]

a v ( V , r 8 n-r C~),~~

-___r*_! . . . . . . . . . . . . .

H0(V, rdsn-r6~ ) --~ H0(V\Rr+ 1 , r~sn-r(~)

,

H1 . r n-r __~IHl(v,r~n-r6%) ---> Rr+lC~V(V, ~ (~-)

(5.9)

, 1

r n-r

--'+:H (VkRr+l, 8

I

,

0"):

i

I

__>

r+l V(V, r~ n - r ~ )

' 2 (v,r~ n---~,H 'I

because

V

Since RI," .. ,r'

(5.10)

and

V\Rr+ 1

r~ n-r(>

, .... -->~

II

,I

0

It

0

are Stein.

is a sheaf with support in the closed set

we have H~ r+inv(V, r ~ n - r O )

Since each point in

cn

neighborhoods, it follows that relative to

re )

r ~ n-r(~

= H~I,.. ",r+I~V(V' r ~ n - r ~ ) .

has the fundamental system of Stein open Rl,...,r+ 1

is purely

and hence by (5.3) purely

r+l

1

codimensional

codimensional

relative to Consequently we have by (5.2) the canonical isomorphisms

28

--~='HP(v' ~ R li, . . . , r + l (r~ n-r6~ ))

HP(v' r + l ~ n - r - i o )

=

HP+l RI, "'" ,r+I~V(V ' r ~ n - r ~ )

= HRP+ ~ inv(V, r~n-r(~).

Thus (5.7) follows from (5.9) and (5.5) with

r

replaced by

r+l

from

(5.8). In case

r = 0,

(5.6) is a consequence of the unique continuation

theorem for holomorphic functions. l~n-l~(v)

As (5.7) shows the elements in

are considered hyperfunctions of the real variable

with holomorphic parameters

z2, ".', z n.

Taking this into account, we call the elements in hyperfunctions of real variables parameters

(Zr+ I, "'', Zn).

in the closed set restriction to

(Xl, -.., Xr)

Since

RI,..., r,

RI ... r

xI

r~n-r~

×

with holomorphic is a sheaf with support

we may identify

= ~r

r ~ n - r ~(V)

r~n-r~

with its

~n-r

Clearly the induction proceeds if we prove the following unique continuation theorem for Theorem 5.3. open set in

Let

C n-r.

r~n-r~. ~

If

be an open set in ~ r

f ~ r ~ n-r O (~]-X V)

a non-empty open subset

U

of

To prove this theorem for

V,

then

and

V

vanishes on

f

a connected ~_× U

for

is identically zero.

r+l~ n-r-l~ ,

we prepare a local

version of Bochner's tube theorem. We assume that the following Hartogs' continuation theorem holds for

r~n-r@,

n-r ~ 2.

Lemma 5.4. set in for each

Cn-r-2 v

Let and

~ U C V

be an open set in

W

a (Stein) open

two connected open sets in

in the projection

~2 v

of

plane

Vv =lu

~r

(u, v)

vI

V

¢2

such that

to the second coordinate

29 is connected and simply connected, V v \ U v = ~u e ¢ ; (u, v) ~ V \ U I . is compact in

and there is a non-empty open subset of

V

~ 2 V on

V

which

V v \ Uv = Then the restriction r~n-ro(a~_×

V × W)

> r~n-r~(~_×

U ×W)

is bijective. With this assumption we can prove the following local Bochner theorem. Let

Lemma 5.5. of

0 < & ~ 1/2

and define the subsets

Fg

and

Gg

~2 by

(5 .ll)

F~ = {(u,v); Im u = O , O_-
v);

Im v = O,

0~

Im u < i,

~((Re u ) 2 + (Re v ) 2 ) + I m G~ = {(u, v);

(5.12)

Im u ~_ 0,

E(Im v) 2 < I - E }

v-

u-

£(Im u) 2 < l - E } ,

Im v _~ 0,

Im(u+v)

g((Re u)2+ (Re v) 2)+Ira u + I m Let

~_

and

W

U'

v - [ ((Imu)2+ (Imv))2
be as above and

Imu<0 where

< i,

is an open neighborhood

of

or Imv

F&

in

~2.

01, Then the restric-

tion (5.13)

r~n-r(~ (~_ X (U U GE) K W)

> r~n-r6~(~_×

U × W)

(5.7) shows the sheaf

r~n-r~

is bijective. Proof.

As the representation

admits holomorphic

coordinate transforms

introduce the coordinates Z~l

= Zr+ I

in

(Zr+ I, "'', Zn )"

We

30

z' + + ig (Zr2+ z 2 r+2 = Zr+l Zr+2 i + r+2 )

Zr+j

___

Z

! r+j

'

j 2 3 .

Then we can apply Lemma 5.4 and obtain the assertion. We denote by

I

and

H

the unit interval and the unit half disk:

i :IxE,R


Iz1
Since F½ C I × H L/ H X I , G½ D ~¢H ~( g H where

/C = I - I / ~

,

Corollary 5.6.

we have the following

Let

~

and

U = I(u, v ) £ where

U'

W

x U × W)

(U U ( ~ H

X ~.H))

be as above and

U' ; Im u < 0

is an open neighborhood

f 6 r~n-r~(~_ A×

,

of

or

Im v < 0},

I × H U H X I.

can be continued analytically

r~n-r(.~. Proof.

It is sufficient vanishes on

W

~

D

be the unit disk

to prove that if × I × D × W,

f

theorem holds for ~ z £ ¢ ; Iz| < I}

f E r+l~n-r-l~(~,

U

is a Stein open neighborhood of

~x

is a section of

F

We have to prove that

of

over F

~

×

~

I X XD

be a defining function.

r~n-r~

r+l~n-r-lo~. and

x I X AD

~ > I. X W)

is identically zero.

Choose a Stein open neighborhood

as (5.7) and let

theorem.

Assume that Theorem 5.3 and Lemma 5.4 hold for

Then the unique continuation Let

to

X W .

Now we are able to prove the unique continuation Lemma 5.7.

Then every

Then X W.

U X D x AD Represent

By the assumption

(D \I) K #~D X W

is continued analytically

to

f F

~ ~-~ D × D X W. ~.x D ~

A D × W.

31

Choose constants I × (~H+~)

~

and

~

U H~(~I+~)

so that

~ (D\l+i0)

~ ~D U D × D ,

~ H + ~ D . Then, Corollary 5.6 together with the unique continuation theorem shows that

F

can be continued analytically to

~ x mH x ( ~ m H +

Repeating the same procedure infinitely, we can prove that analytic continuation to

~x

For each

(5 14).

where and

v 6 ~2V

5.4.

g

choose the same

g

Vv

~

locally,

coincides with

g

f

on

Employing representation also in the case where

r = 0,

this is well

which encircles both

does not depend on the choice of

r > 0.

~

U

f = g

V.

on a non-empty open set.

by the unique continuation theorem.

(5.7), we can define integral (5.14) Since we have the unique continuation

theorem, we can prove the lemma for

r > 0

in the same way.

This completes the proofs of Theorems 5.1 and 5.3. ~r

u

Since we can

is a holomorphic function on

By Cauchy's integral formula we have Hence

In case

f ( ~ d, ~v,~ w) _ u

is a simple closed curve in

V v \ U v.

has an

we define

g(u, v, w) = 2~iI" ~

~

F

D × X D x W.

Lastly we have to prove L e n a known.

~) x W.

proof gives us a method to

compute

the boundary values

32

F(x+ir0)

E ~(~)

a wedge-like

= HI(V , ~ )

domain

of holomorphic

functions defined on

V ~ (~Rn + i r ) .

First we note that

r

may be assumed to be an open convex cone

because of the following Lemmm 5.8 (Kashiwara

[13]).

If

function defined on a neighborhood 0
z I < r,

' , r'

and

Im z 2 . . . . . @

to the domain

such that

F(z I, .-., Zn)

of the set

Im Zn = 0~ ,

+ ~Im z nl < ~ I m

~ .

~y E ~n ;

(see [21]).

system, we may assume that the first

Yl > O, "''' Yn > 0 I is included in the interior

We define the first boundary value ~ I F ( E I, z 2, "'', z n) e l ~ n - l ( ~ ( V

to be the cohomology ~(V

~Im z2~ + "'"

Zl~.

Taking a suitable coordinate

of

~Re z[ <

is continued analytically

0 < Im z I < r',

This is easily proved by Lemma 5.5

quadrant

~z E C n"

there are positive numbers

F(z I, • • • , Zn)

~z E ~n., IRe z~ < ~',

is a holomorphic

~ (~+ir)\

class in (5.7) of the function

R I)

~z E V ~ 0 R n + i ~ ) ;

a (~Rn+i[~) ~ R I)

which is equal to Im Zl > 0}

and

0

F

~i F E

on the positive

side

on the negative side•

Assuming that the r-th boundary value ~r~r_l'''

~iF(Xl,

..., Xr, Zr+ I, -•', Zn ) E

r ~ n-r

~(V

~ ( m n + i [ ~) ~ E l , . . . r)

is defined, we define the (r+l)-st boundary value be the cohomology % r + l ~r

''' ~ i F

to

class in (5•7) of the section

"'" ~i F e r~sn-r(}(V f~ (~Rn+i[ ~) f~ R l , . . . , r \ R r + I)

which is equal to ~Rl,...,r

~r+l

~r

"'" ~ i F

•, Im Zr+ 1 > 0~

and

on the positive 0

side

Iz & V ~ (~n+i~,)

on the negative side.

We write the n-th boundary value

33

n "'" ~ i F (x I,

Theorem 5.9 depends only on

"'', Xn) = F(x + i F 0) ~

(Martineau [27]). F

commute.

m n)

n

The boundary value

F(x+iF0)

and the orientation of the coordinate system.

particular, it does not depend on the cone Proof.

~(V

P

First we note that the operations

For example, if

n = 2,

~ 2~i

In

~i

and

~j. anti-

and

~I ~ 2

are composi-

tions in the anticommutative diagram H 0 ( V \ R I \ R 2, ~ ) (5.15)

dl~

~d 2 i (V\RI, ~) ~2\RI,2

I ~RI,2(V

\R2 , ~ ) ~ d2

dl~ H 2 (V, (~) Rl, 2

which is obtained from the commutative diagram of complexes: 0

0

0

0

> CRI,2 " (V, ~ ~)

" (V,$ ~ ) " CR2

) CR2\R I ,2 (V \ RI, 6~)

> 0

0

> CRI(V, (>)

~

>

> 0

C'(V, ~ )

C ' ( V \ R I, (>)

$ 0 -- CRI\RI,' (V\R2,~)--~C" (V~R2,~) 0

>C'(V\RI\R2,

0

~)

> 0

0

with exact columns and rows. As the definition shows, F

near the hyperplane

R I.

~i F

depends only on the behavior of

In particular, it does not depend on the

location of the first coordinate axis Yn = OI

I y E ~n.,

Yl ) 0,

as far as it remains in the positive side of

Similarly we find that the n-th boundary value depends only on the behavior of

F

Y2 . . . . .

ORn + i P ) \ R ~n "'" ~I F

near the n-th coordinate axis as

far as the orientation of the coordinate system is preserved. write

I.

Thus we

34

n

where

y

"'" ~ i F ( x )

is any vector on the n-th coordinate axis.

anticommutativity

of

~. l

and

F(x+iy0) where

y'

~ . we have j =

~(x+iy'0)

~

~ ,

Since

y'

can

be the first quadrant:

= ly ~ ~n; V

,

this completes the proof.

~ = ~y E ~n;

and

In view of the

is a vector on the first coordinate axis.

be any vector in Let

= F(x+iy0)

a Stein neighborhood

F E O(V ~ ( ~ n + i ~ ) ) j-l~l@n-j~

Yl ~ 0, ''-, Yn ~ 0 ~ 0 1

of an open set

define

(V ~ ( m n + i ~ )

yl > 0, "'', yn > 0 },

~).

in

~n

For

fj(xl,''',Xj_l,Zj,Xj+l,''',x n) E

~ RI,.. .,j_l,j+l,...,n )

by

fj = (-l)n-J ~n''" ~j+l ~j-l''" ~ I F '

j=l,''-,n.

Then we have ~j fj(x)

= F(x+iP0),

j = i, "'', n

The converse is the following edge of the wedge theorem. Theorem 5.10. another

U C V

RI,...,j_I,j+I

For every complex neighborhood

such that if ...,n),

f. ~ J - l ~ l ~ n - J ~ ( v J

j = i, "--, n, ~ifl =

then there is a holomorphic fj = (-l)n-J~n''"

where

~

there is

~ (RRn+i~)

~nfn

,

F E (~(U ~ (~Rn+ip))

such that

~j+l ~j-i °'" ~I F on

Proof.

of

satisfy

~2f2 . . . . .

function

V

U ~(rRn+i~)

n Rl,...,j_l,j+l,...,n

For the sake of simplicity, we consider only the case

n = 2. We may assume that

Stein neighborhood

of

V

is Stein.

V N (~R2+i~)

Since f~ RI,

V ~ ~z; Im z 2 > 0} we can find

is a

E = (E+, E-)

35

~(V ~{z;

Im z2 > 01 \RI)

(5.16)

such that

f2(xl , z 2) = E+(Xl+i0,

Extending

E

by zero to

z 2) -E-(Xl-i0,

V • ~z; Im zoL < 0~\RI,j

z2).

we have

E E

~ ( V \ R I \R2). Im z 2

~ ~ Im

g

E

z 1

(5.16) means that

dl(E) = (f2' 0),

homomorphism in (5.15).

where

dI

is the connecting

We have

~ifl = ~2f2 = d2(f2, 0) = d2dl(E) = -dld2(E) = -dl(~2E+, ~2 E-~., Hence there is

g E I~i~

(V ~ R2)

fl

~2 E+ ,

g =

~2 E Since

V

,

such that Im z I > 0 , Im z I < 0 .

is a Stein neighborhood of

(G+, G') ~ (~(V \R2)

V ~ R 2,

there is

G --

such that

g(zl, x2) = G+(Zl , x 2 + i 0 ) -G-(Zl, x 2 -i0). If

Im z I < 0,

we have g(zl, x2) = E (Zl, x 2 + i 0 ).

Hence on

G

can be continued analytically to a holomorphic function

V a ~ z; Im z I < 0

or

Im z 2 < 0}.

G

36

The local Bochner be c o n t i n u e d

theorem

analytically

(Lemma 5.5) proves

then that

t o an open n e i g h b o r h o o d

U

of

G

can

V ~ N2.

Then on

F = E+ - G+ + ~"

U n (~2+i[~

)

gives the desired result. Martineau

[26] has shown that if the boundary value

exists in the sense of distribution, homological tions

boundary value.

F(x + i ~ 0)

then it coincides with the co-

The same result holds for ultradistribu-

[19]. When a proper open convex cone

criteria

p

to the case of one variable

boundary value ultradistibution

F(x+iP0) etc.

At the end of

exists

(see [26],

is fixed, we have similar

to determine whether

or not the

in the sense of distribution [19]).

§ 4 we showed that every hyperfunction

is the sum of boundary values

or

of holomorphic

functions

f E ~(~)

on wedge domains.

More generally we can prove the following. Let

~I'

°'''

~m

dual cones

F I 0 ' "''' ~ m0

open set in

~n

and

every hyperfunction

V

be open convex cones in

~n

cover the dual space of is a Stein neighborhood

f ~ ~(~)

of

such that the ~n. ~

If in

~ C n,

is an then

can be written

m

(5.17) where

f(x) = ~ Fj(x+iP.0) j=l J Fj(z) e ~ ( V

Martineau's

'

~ (~n+iq)).

edge of the wedge theorem

gives the condition

for

F. J

[27]

(see also Morimoto

under which the sum of boundary values

equal to zero. If two hyperfunctions

f

and

g ~ ~(~)

f(x) = ~ F j ( x + i ~ j

O) ,

can be written

[29]) is

37

g(x) = ~_~ G k ( X + i ~ O ) k with open convex cones that

~j ~ ~

product

fg

~ ~

~I' for all

"''' ~ m l j

and

and k,

f e ~(~)

plane such that Im H

"''' ~ m '~ such

then we can define the

by (fg)(x) = j,~k (FjGk)(x+i(~J

If

~Pi'

for all

~ ~)0).

is written as (5.17) and if Im((~n+i~j)

j,

~ H)

H

is a complex hyper-

is an open convex cone

then we can define the restriction

f l~n ~ H = ~ • J

~j

!

in

H

by

f llRn ~

Fj(x+i~.' 0) . J

These theories have been developed by Sato [34], Morimoto [29] and Kashiwara

[13].

REFERENCES [I]

O. V. Besov,

Investigation of a family of function spaces in

connection with theorems of imbedding and extension, Inst. Steklov,

60 (1961), 42-81,

Trudy Mat.

Amer. Math. Soc. Translations,

Ser. 2, 40 (1964), 85-126. [2]

G. Bjorck,

Linear partial differential operators and generalized

distributions, [3]

Ark. Mat., ~ (1966), 351-407.

N. N. Bogoliubov -D. V. Shirkov, Quantized Fields,

[4]

T. Carleman,

Interscience, New York, 1959.

Sur les Equations Int@grales Singuli@res g Noyau

R@el et Sym@trique, [5]

T. Carleman, Rattachent,

[6]

Introduction to the Theory of

Uppsala, 1923.

L'Int@grale de Fourier et Questions qui s'y Mittag-Leffler Inst.,

P. A. M. Dirac,

Uppsala, 1944.

The Principles of Quantum Mechanics,

2nd ed.,

38

Oxford, 1935. [7]

A. Grothendieck,

Local Cohomology,

Harvard Univ., 1961, reprint,

Lecture Notes in Math. No.41, Springer, 1967. [8]

J° Hadamard,

Lectures on Cauchy's Problem in Linear Partial

Differential Equations,

Yale University Press, 1923, reprint,

Dover, 1952. [9]

G. H. Hardy,

Weierstrass's non-differentiable function,

Trans.

Amer. Math. Soc., 17 (1916), 301-325. [I0] R. Harvey, Equations,

Hyperfunctions and Linear Partial Differential Thesis, Stanford University, 1966.

[ii] L. H~rmander, Variables,

An Introduction to Complex Analysis in Several

Van Nostrand, Princeton, 1966.

[12] M. Hukuhara et M. lwano,

Etude de la convergence des solutions

formelles d'un syst@me diff@rentiel ordinaire lin@aire,

Funkcial.

Ekvac., ~ (1959), 1-18. [13] M. Kashiwara,

On the structure of hyperfunctions (after M. Sato),

Sugaku no Ayumi, 15 (1970), 9-72 (in Japanese). [14] T. Kato and S. T. Kuroda, function expansions,

Theory of simple scattering and eigen-

Functional Analysis and Related Fields,

Springer, Berlin -Heidelberg-New York, 1970, pp.99-131. [15] K. Kodaira,

On ordinary differential equations of any even order

and the corresponding eigenfunction expansions,

Amer. J. Math.,

72 (1950), 502-544. [16] H. Komatsu,

Relative cohomology of sheaves of solutions of

differential equations, S@minaire Lions-Schwartz, 1966, reprinted in these Proceedings, pp. 190-259. [17] H. Komatsu,

Hyperfunctions and Linear Partial Differential

Equations with Constant Coefficients, Seminar Notes No.22, Dept.

39

Math., Univ. Tokyo, 1968 (in Japanese). [18] H. Komatsu,

Ultradistibutions and hyperfunctions,

these Proceed-

ings, pp. 162-177. [19] H. Komatsu,

Ultradistributions,

characterization, [20] H. Komatsu,

I,

Structure theorems and

to appear.

On the index of ordinary differential operators,

J. Fac. Sci., Univ. Tokyo, Sect. IA, 18 (1971), 379-398. [21] H. Komatsu,

A local version of Bochner's tube theorem,

J. Fac.

Sci., Univ. Tokyo, Sect. IA, 19 (1972), to appear. [22] G. K~the,

Dualit~t in der Funktionentheorie,

J. Reine Angew.

Math., 191 (1953), 30-49. [23] G. K~the,

Die Randverteilungen analytischer Funktionen,

Math.

Z., 57 (1952), 13-33. [24] J. E. Littlewood and R. E. A. C. Paley, series and power series, 230-233,

Theorems on Fourier

(I), J. London Math. Soc., ~ (1931),

(II), Proc. London Math. Soc., 42 (1936), 52-89,

(III), ibid., 4 3 (1937), 105-126. [25] A. Martineau,

Les hyperfonctions de M. Sato,

S~minaire Bourbaki,

13 (1960-61), No.214. [26] A. Martineau, holomorphes,

Distributions et valeurs au bord des fonctions Theory of Distributions,

Proc. Intern. Summer Inst.

Lisbon, 1964, Inst. Gulbenkian CiSncia, Lisboa, 1964, pp.193-326. [27] A. Martineau,

Le "edge of the wedge theorem" en th@orie des

hyperfonctions de Sato,

Proc. Intern. Conf. on Functional

Analysis and Related Topics, Tokyo, 1969, Univ. Tokyo Press, 1970, pp.95-I06. [28] P. -D. Meth@e,

Syst~mes diff@rentiels du type de Fuchs en

th@orie des distributions,

Comment. Math. Helv., 33 (1959), 38-46.

4O

[28'] M. Morimoto,

Sur les ultradistributions

cohomologiques,

Ann. Inst~ Fourier, 19 (1969), 129-153. [29] M. Morimoto,

Sur la d@composition du faisceau des germes de

singularit@s d'hyperfonctions,

J. Fac. Sci., Univ. Tokyo, Sect.l,

1 7 (1970), 215-239. [30] F. Riesz,

Uber die Randwerte einer analytischen Funktion,

Math. Z., 18 (1922), 87-95. [31] M. Riesz,

Sur les fonctions conjugu@es,

Math. Z., 27 (1927),

218-244. [32] M. Sato,

On a generalization of the concept of functions,

Proc.

Japan Acad., 34 (1958), 126-130 and 604-608. [33] M. Sato,

Theory of hyperfunctions,

J. Fac. Sci.,Univ° Tokyo,

Sect.l, 8_ (1959-60), 139-193 and 387-436. [34] M. Sato,

Hyperfunctions and partial differential equations,

Proc. Intern. Conf. on Functional Analysis and Related Topics, 1969, Univ. Tokyo Press, Tokyo, 1970, pp.91-94. [35] P. Schapira,

Th@orie des Hyperfonctions,

Lecture Notes in Math.

No. 126, Springer, 1970. [36] E. C. Titchmarsh,

Eigenfunction Expansions Associated with Second-

order Differential Equations,

Oxford, 1946.

[37] E. T. Whittaker and G. N. Watson, 4th ed., Cambridge,

A Course of Modern Analysis,

1927.

Department of Mathematics University of Tokyo Hongo, Tokyo

EDGE OF THE W E D G E T H E O R E M A N D H Y P E R F U N C T I O N

BY Mitsuo MORIMOTO

About fifteen years ago,

theoretical physicists

of the quantized

field theory discovered a theorem of several complex variables, which they called euphoniously

the edge of the wedge theorem.

to discuss the evolution of mathematical

We are going

ideas around this theorem

stressing its deep connection with Sato's theory of hyperfunctions.

i.

Introduction

The edge of the wedge theorem was discovered by the theoretical physicists who were studying the Wightman function, Green function or dispersion relations.

It seems that Bogolyubov gave the first

statement of this theorem in 1956 (see Vladimirov Immediately after this, many p a p e r s w e r e

[37], p.825).

written on the edge of the

wedge theorem by theoretical physicists and mathematicians (Bremermann-Oehme-Taylor

[3], Dyson [6], see also the English trans-

lation of Bogolyubov-Shirkov

[2], p.678).

We skip over the physical

background of this theorem referring the reader to the standard text books (Bogolyubov-Shirkov

[2], Streater-Wightman

[36], Jost [I0]).

The edge of the wedge theorem concerns the boundary values of holomorphic

functions.

roughly as follows:

Bogolyubov's version of this theorem reads

Consider

~n = ~ n

×~2~

n.

open convex cone (with vertex at the origin) of T( ~p j )

f2

= Nn X ~-~ ~ j

Suppose ~n

is the tube domain with base

~2

~j.

be holomorphic f u n c t i o n s in truncated tube domains

~i

is an

= -

Let ~T(~1)

and

fl

and and

42 N

~ T ( F 2) open set

respectively, where u

values on

of

~n.

~

Suppose further that they have the same boundary

u: lira f l ( x + 4 U ~ y ) y+0

=

fl

and

f2

(i)

lira f 2 ( x + ~/~y) . y~O

Y6P1 Then

is a complex neighbourhood of an

YeP2

can be extended analytically

function

f

in a complex neighbourhood

~.'

~tatement

see Theorem 0 of the next section.)

to a holomorphic

of

u.

(For the precise

The limits in (i) were

in the sense of uniform convergence but it was recognized

that the

theorem is still valid if one takes the limits in (i) in the sense of Schwartz'

distribution

Bogolyubov's

~

'

theorem has been generalized by Epstein

case where the cones Martineau

topology

~I

and

P2

[7] to the

are arbitrarily placed.

Then

[18, 19] presented a version of the edge of the wedge

theorem in which several cones will state these theorems

P l , ~2'

"" " , ~ m

take

part.

We

in Section 2.

As for the proof of the edge of the wedge theorems three apparently different methods have been known. Cauchy's

integral formula after suitable coordinate

to get explicitly f2"

The first one uses

the extension

f

transformation

of the original functions

fl

The second one uses the estimate to show that the function on

which is defined to be the boundary value of real analytic. cohomology

fl

and

f2

and u

is actually

The third one due to A. Martineau relies on the

calculation and has a deep connection with the structure of

hyperfunctions.

In Section 3 the first two methods will be reviewed.

In Section 4, Martineau's will be outlined.

theory of the edge of the wedge theorem

The idea is as follows:

If a holomorphic

function

43

fl(z) defined in

~. ~ T ( ~ i ) fl(x + ~

has a distribution boundary value r l 0) =

lim

fl(x + ~

y),

(2)

y--~O Y~I then this boundary value coincides with the hyperfunction boundary value of

fl which is defined cohomologicallyo Therefore the

distribution version of the edge of the wedge theorem is derived from the hyperfunction version of this theorem, which in turn can be proved algebraically using the vanishing theorem of certain relative cohomology spaces. The notions such as hyperfunction, hyperfunction boundary value of a holomorphic function, etCo will be recalled in this section. We remark here that the hyperfunction was introduced by M. Sato [29] in 1958 in order to generalize the notion of function by means of the boundary values of holomorphic functions° The hyperfunction can be defined on any real analytic manifold of germs of hyperfunctions will be denoted by In 1969, M. Sato [30] constructed a sheaf spherical bundle

~-~ S*M

help of the sheaf

C

S'M,

~

over the conormal Mo

By the

~

on

M

~

is named micro-

defines a microfunction

~

whose support is, by definition, the singular support

of the hyperfunction sheaf

C

one can investigate the regularity of hyper-

function° Every hyperfunction ~

The sheaf

~ .

of the real analytic manifold

functions effectively. A section of the sheaf

on

Mo

~ . Morimoto [23] proved that the theory of

is closely related to Martineau's theory of the edge of the

wedge theorem. Indeed one can construct the main parts of Sato's theory of sheaf

C

using the techniques inspired by Martineau [18],

and conversely a hyperfunction version of the edge of the wedge theorem is a direct corollary of the theory of sheaf

~ .

(Section 5)

44

In Section 6 we will speak of a few applications of Sato's fundamental principle on the regularity of hyperfunction solution of partial differential equations° This principle is one of the most brilliant results of the theory of sheaf

C o

The Harvey-Bengel

theorem on the regularity of hyperfunction solution of elliptic differential equation is one of its corollaries. We will show a local version of the Bargmann-Hall-Wightman-Jost theorem of quantized field theory is also a derect corollary to Sato's fundamental principle° Section 7 deals with the relation of the support and the singular support of a hyperfunctiono We know the sheaf ~ functions over S~ function ,

M

and the sheaf

~

of germs of hyper-

of germs of microfunctions over

are flabby° But the form of the singular support of a hyper~

restricts the form of the support of the hyperfunction

and vice versa° We present some theorems concerning these

phenomena called "quasi-analyticity"o Our results can be considered as local versions of the

B(G)-hull theorem and the

B p (G)-hull

theorem of Chapter 5 of Vladimirov E38]o In the last section we will mention the Kolm-Nagel theorem and the ultra-hyperfunction case of the edge of the wedge theorem° References will be given at the end of this paper in alphabetical order°

2.

Statement of the Edge of the Wedge Theorems

We formulate now the edge of the wedge theorems of several authors. Prepare the notations° Let dimension with

n,

V ×~i-0o

V¢ = V x~-~ V

V

denote a real vector space of

its complexificationo We identify

A complex neighbourhood

~-

of an open set

u

V of

V

45

is, by definition,

an open set of

subset

we denote by

A

of

V,

VC

such that

T(A)

the tube with base

T(A) = V × ~ Z ~ A

A subset

~

of

tx E r .

Let

V

u = ~ ~ V.

For a

A,

namely

.

is said to be a cone if

(3)

x ~ ~

, t > 0

implies

s -- (v \ ( 0 ) ) / m + be the quotient

space of

V \(0)

by the equivalence relation:

x-~ y < With the quotient

topology,

S

> ~ t > 0,

x,

of

V.

For

x ~V\(0),

x = ty.

is isomorphic

will be called the space of (infinitesimal) 0

(4)

we denote by

(5)

to (n-l)-sphere and

directions at the origin

x0

the equivalence

class of

which is the direction of the half line passing through

a subset

A

of

V

we denote by A0 = ~x0;

A subcone

of a cone

subcone of

if

We denote by V¢

and by

fL

of

~(~)

r

~0 0"

A0

the subset of

S

x.

defined by

x e A \(0)}.

is said to be

is (relatively)

(6)

(relatively)

compact

compact

subset of

the sheaf of germs of holomorphic

the space of holomorphic

For

~0.

functions on

functions on an open set

V¢.

Bogolyubov's

first version of the edge of the wedge theorem

reads as follows: T h e o r e m 0 (Continuous version of Bogolyubov's be an open convex cone of V

V,

and its complex neighbourhood

neighbourhood is true: values

~.'

Suppose

of

u

f.j E ~ (

~2

= - ~i" ~

such that

theorem).

For any open set

Let u

~ i of

there exists a complex ~L' c~L

~L ~ T ( ~ j ) ) ,

and that the following

j = i, 2,

have the boundary

46

f . ( x + ~ ' ~ ~j 0) m lira f.(x+ ~ y ) J y~0 ] y6~j where for any compact set rj,

k

of

u

r.' J

there exists a function for

x e k

and

y

of

tending to

0

if one has fl(x+~f~

T(rj)

F.' J

and any compact subcone

the limit in (7) is uniform for

from

(7) '

F I 0) = f2(x+ ~ f e £Y(~_')

~2 0),

such that

(8)

f = f. J

on

~'~

j = I, 2.

This theorem has been called the continuous version of the edge of the wedge theorem of Bogolyubov.

It was remarked at the very early

stage that the limits in (7) can be considerably weakened:

it is suf-

ficient to assume the limits in (7) exist in the distribution sense. In fact, we have the distribution version of Theorem 0 (see Theorem 1 below).

We shall use the standard notations of

tion theory [34].

For example,

with compact support on ~'(u)

u

~(u)

is the space of C ~ functions

with usual locally convex topology and

is the space of distributions on

convex cone. f(x+~f~y)

f E ~(~NT(~))

For

?(x)dx

L. Schwartz' distribu-

u.

and

can be defined if

Let

~

~ 6~(u), y E ~

be an open the integral

is sufficiently

U

small.

If the limit P lim | y ~ O J1 yet

exists for every

f(x+~y)

~ e~(u),

? ,

> lim y~ yeP

then the functional f

f ( x + ~ f ~ y ) ~(x) dx

(i0)

u

is a distribution on

u,

boundary value of

and will be denoted also by

f

(9)

~(x) dx

which is, by definition, the distribution f(x+~f~0).

47 Theorem I (Distribution version of Bogolyubov's theorem). i' ~2' u, ~ Suppose

and

f.j ~ ~ ( ~

boundary values

~'

have the same meaning as in Theorem 0.

~ T(pj)),

j = I, 2,

fj ( x + ~ f ~ ~ j 0).

then there exists a function ~' ~ T ( p j ) ,

for

Let

If

have the distribution

fl(x+~FlO)

f E ~ (fL')

= f2(x+~l~20)

such that

f = f. J

on

j = I, 2.

Theorem I was generalized by Epstein [7] to the case where and

r2

are arbitrary cones.

Theorem 2 (Distribution version of Epstein's theorem). and

P2

be two convex open cones of

and its complex neighbourhood hood

~'

of

Suppose

u

such that

£a,

j = i, 2.

there exists a function on

~' ~ T ( ~ j )

hull of a subset As

A

c h ( H IU(-

of Theorem I.

V.

For any open set

Let u

r of

of

V

and that the following is true:

have the distribution boundary values If

fl(x+~i~ FI0)

= f 2 ( x + ~ f ~ F20),

f ~ 6 ~ ( ~ ' ~ T ( c h ( ~ l ~ ~2 ))) such that

for

i

there exists a complex neighbour-

~' C ~

f.j @ ~ (SL ~ T ( P j ) )

fj.(x+~f~ Fj 0),

fj.

~I

j = i, 2, where

ch(A)

f =

denotes the convex

V.

~'~I)) = V,

Martineau

Theorem 2 is indeed a generalization

[18, 19] generalized Theorem 2 considering

the situation where several cones take place. Theorem 3 (Distribution version of Martineau's theorem). j = i, 2, ''', m, V

be open convex cones of

and its complex neighbourhood

hood

~h'

of

u

such that

Suppose that functions boundary values

~_ ,

SL' c ~L

For any open set

j' u

and that the following is true: have the distribution

j = I, 2, ''', m.

If one has

m

Z

j=l

fj(x+~

~

J

0) = 0

of

there exists a complex neighbour-

fj. e ~ (fg ~ T(['j))

f.j(x+ ~f~ ~j 0),

V.

Let

in

~ '(u)

'

(II)

48 then there exist functions = I, 2 ..... m,

j ~ k

U ~k)) )

for

j, k

such that

=

fJ for

gjk ~ 6~ ( ~ _ ' N T ( c h ( ~ j

~

k ~ j

on

gjk

~-g' T ( ~j )

(12)

j = I, 2 ..... mo We modify these theorems in the form which is invariant under the

real analytic coordinate transformations of

u,

suppressing the

complex neighbourhood in the statement of theorem° For that purpose consider the space of directions at the origin of

V:

and the direct product of

(For later con-

venience, we add

~-~.)

defined by

x~V

and

denote by

r

V

and

(x, ~ yE

S:

V ~ ~

y 0)

V \(0).

denotes the point of

For a subset

the cone associated with = I X ~ V \ (0);

We define a sheaf over the space of

V ×~f~ S £(u

S.

S = (V \(0))/~R+

V×~Z~

r

r

of

V ×~

S,

S

we

o.

xO ~ F } o So

(13)

For an open set

u ×~-~F

we put ~-~

~)

= lira proj I~'CE P

lira ind ~ ( ~ _ ~.~ u

~T(~')),

(14)

where the inductive limit is taken under the restriction mappings as complex neighbourhood

~

of

u

tends

to

u

and the projective

limit is taken following the restriction mappings as relatively compact open subset open subset of

~'

u ×~

of

~ ,

p

tends to ~.

If

Ul × ~f~ ~i

is an

we can define the restriction mappings

(Ul×f F l) as the limit of the restriction mappings of ( ~ i ~ T( F 1 )), u

and

uI

where

such that

compact open subsets of

J~

and

~" D ~" 1 ~

~'i

~(~g

~ T([~'))

are complex neighbourhoods of

and

~'

and

and of

~I

such that

Because the open sets of the form

(15)

u ×~[~

~I'

are relatively ~' D

[~I'

form a base of the open

49

sets of (15)

V×~

S,

our

define a presheaf

~(u ×~

~ )

~

V x~f~ S.

over

this presheaf

~

is a sheaf.

A subset

r

of

s

r .

P

denoted also by

the intersection of the convex sets in

The convex hull

containing

It is easy to check that

is said to be convex if its associated cone

is convex. The convex hull of definition,

and the restriction mappings

~

ch

or all

ch ~

S

is, by

which contain

is the smallest convex set of

S

By the local version of Bochner's tube

So

theorem we have ~(u ~f~ If

f E ~ ( u x~i~ ~),

u

such that

x~-f-f c h p ) .

(16)

then by the definition for any relatively

compact open subset ~_of

P ) ~ ~(u

~'

of

~

there exists a complex neighbourhood

is defined and holomorphic on

f

~.~ T ( ~ ' ) o

can define, if it exists, the distribution boundary value of we denote by

f(x + ~-fTF).

f

We which

With these terminologies we can state the

following theorem which is a modification of Theorem 3o Theorem 3'.

Let

u

open convex subsets of

be an open set of S°

Suppose

distribution boundary values

V

fj ~ ~ ( u

f.(xj + ~f~ ~j)

and

FI''°''

×~J~ rj) for

~m

be

have the If

j = i, 2,.o., m.

we have m

fj(x + ~

(17)

~j) = 0,

j = i then there exists

gjk e ~ ( u X ~ f" =

~

k~j

J

Remark that

V×~

S

the real analytic manifold

c h ( r j u ~k)), gjk

on

j ~ k,

such that (18)

ux~-~ q o

is nothing but the normal sphere bundle of V

with respect to

V¢

and that the sheaf

can be defined on the normal sphere bundle of any real analytic manifold

Mo

In fact,

M

is locally isomorphic to an open set

u

of

50

V.

We shall come back to this point later° We have presented Theorem

3' although it is weaker than Theorem 3, because it is in this form that we deduce a hyperfunction

version of the edge of the wedge

theorem from the theory of sheaf

3.

~

in Section 5 (see Theorem 15) o

Classical Proofs

In this section we review several classical proofs of the edge of the wedge theorems

(Theorems 0, I

and 2),

although they have no

logical relations

to the remainder of this paper° The first proof uses

Cauchy's integral

formula after a suitable analytic transformation

(Dyson [6], Epstein [7], Streater-Wightman

~36], Vladimirov

[38],etCo),

while the second one uses the estimate to prove a theorem on separate analyticity, (Browder[4]). Martineau,

which serves as a lemma for the edge of the wedge theorem The third method of proof, which is due to A.

is the most interesting

for us. It is based on the

cohomology calculation and will be explained in the next section.

Remark first that the distribution versions can be easily deduced from the corresponding characterization

continuous versions

thanks to the following

of analyticity of distribution:

sufficient condition for a distribution analytic is that every regularization (Th~or~me XXIV of Schwartz [34])°

of

A necessary and

Te~'(u) T,

to be real

T*~

is analytic

Therefore we will speak only of the

continuous versions. When the dimension

n

of our vector space

Theorem 0 has been called Painlev~'s known since 1888 (Painlev@ [27])°

V

is equal to one,

theorem in function theory and is

51 (i)

Method of Cauchy's integral formula. As the analyticity is a local property and invariant under the

analytic coordinate transformation, where the cone

PI

Theorem 0 is reduced to the case

is of a special shape. In the proof which is

essentially due to Dyson [6],

~i

is first assumed to be the future

light cone, while in the proof given in the text book of StreaterWightman [36] the cone

~i

is taken to be the first quadrant° In

both proofs one then chooses a suitable analytic transformation and afterwards applies Cauchy's integral formula to obtain the explicit form of analytic continuation of the original function. We refer the reader to po 254 of the English translation of Vladimirov [38] for the details of the method of Dysono The proof in Streater-Wightman has been reproduced in a recent survey of Rudin [28] in a very accessible way° (ii)

Method of estimate. This method was initiated by Browder [4]° He proved first the

following theorem on separate analyticity in more restrictive form° Theorem 4 the

(Browder-Cameron-Storvick).

Let

fg. J

be a disk in

zj-plane: ~j =

I zj E ¢;

IzjJ < Rjl ,

(19)

and put: Lj = If

I zj ~ ~.j ;

Im zj = 0 I '

for

j = i, 2 ..... no (20)

f is a bounded function on the set n

~_~ j = 1 such that for any

LI× oooX Lj_ I ~ ~Lj× Lj+ I X oo.× en j

the function

(21)

52

I

Z.

J is holomorphic in

~j,

Lj.I ~ L j + I X °°° × L n function

F

~

(22)

f(z I ..... zj ..... Zn)

(z I ..... zj_ I, zj+ 1 ..... zn) ~ L I × ooo being fixed, then there exists a holomorphic

on

~ = { x~ such that

f = F

C n-

JzjJ< 3

-n

for

R.J

almost everywhere in

j = i, 2 ..... n }

(LI× .o.× Ln) ~ ~ .

For the proof, we refer the reader to Cameron-Storvick the proof is done expanding the function and estimating its coefficients°

(23)

f

[5], where

in the Legendre series

Browder's first proof in [4] relies

on the power series expansion of

fo

Recently Siciak [37] has

generalized Theorem 4 showing the boundedness conditions on

f

are

superfluous° From Theorem 4 one can deduce Bogolyubov's and Epstein's theorems

(see for example Kajiwara [ii]). The original proof of

Epstein's theorem uses Cauchy's integral formula (Epstein [7])° Very recently, Komatsu (16] proved the theorem of separate analyticity and its consequences by the method of Cauchy's integral formula° R emarko

As shown in Kajiwara [ii], we can deduce from Theorem 4

a more general edge of the wedge theorem named the Malgrange-Zerner theorem° We are going to formulate ito Let lh~ by

~

be a convex set in the vector space

the linear hull of r ~

we call

Theorem 5

r

F ,

~ I U ~2o

(Malgrange-Zerner theorem).

of

u

For any open set

u

We donote by

i.eo the linear subspace of

is relatively open if

relatively open convex cones of

Vo

V of

and V

P Let

~12

V

is open in rI

and

spanned lh~o

P2

be two

be the convex hull of

and its complex neighbourhood

there exists a complex neighbourhood

~h'

of

u

such that

53

D~'

and the following is true:

a q (T(~I)UT(>2)Uu)

If

f

is continuous on

and if the restriction

f I ~T(~j)

is holomorphic in the complex variables of

T(~j)

then there exists a continuous function

on

such that the restriction complex variables of

F

F I ~'n T(rl2 )

T(~12 )

f = F on

for

j = I, 2,

~' a ( T ( r 1 2 ) U u)

is holomorphic in the

and that

~'n(T(~I) UT(~2)Uu).

(24)

It seems to me that this theorem has not been published by Malgrange nor by Zernero See for this account po 286 of Martineau (18]o Of course, the distribution version of Theorem 5 is valid. More generally, Martineau

[19] generalized it in the form analogous to

Theorem 3.

4~

Martineau's Theory

In

his lecture at Lisbon [18], Martineau gave a very interesting

point of view on the edge of the wedge theorem. He uses the cohomology calculation to handle the situation where several cones take place° His method is closely related to Sato's hyperfunction theory° first recall

(i)

the cohomology and

(ii)

So we

the hyperfunctionso

(About these topics we refer the reader to Harvey E8], Komatsu [15], Schapira ~33]

or the original paper of Sato E29].) Afterwards we

sketch Martineau's ideas on the edge of the wedge theorem°

(i)

Cohomo!0g Y . Let

~

be the sheaf of germs of holomorphic functions on

For an open set space of

~

~

of

Cn,

Cno

one can define the p-th cohomology

with coefficients in the sheaf

O" ,

which we denote by

54 HP(~ ; ~ ) .

(See for example H~rmander

theorem, the space

HP(~;

~)

the p-th cohomology space coefficients

in

(~,

[9] .)

Thanks to Leray's

is canonically

isomorphic to

of a Stein covering

HP(~;

IlL of

~

with

6?): (25)

HP(f~; ~)~-- HP(I~; (>)o More generally,

if

~

is an

p-th cohomology space of

space of

~

If

~O = ~ ,

that is

~'

Putting

(>)

with support in

HP(~tmod ~; If

(~)o

~(~l;

HP(~mod~o~

~ ,

we can define the

3~ mod 03 with coefficients

is denoted by HP(~mod0o; is denoted also by

open subset of

~ ,

which

H P ( ~ m o d o0; ~ )

and is called the p-th cohomology F

and with coefficients

(~) =-- ~ ( / L ; F = ~9., O)

F = ~ \~,

in

~),

in

~

F = /~ \COo

(26)

we have

- HPA(~. ; (~) = H P ( ~ ; ~ ) o

is another open set such that

F

(27)

is its closed subset, we

have by the excision theorem HP(~;

~)

= ~ ( ~'; (~)o

(28)

Hence we may abbreviate as follows: HP[F] = ~ ( a

; (~).

(29)

We will use this notation in the following sections° One of the most important properties

of the relative cohomology

spaces is the long exact sequence of cohomology. ~ o'" HP(~; g~

Np+l -

F

~

HP( ~ _ \ F ;

(~)

(30)

( g ; 6v)

A pair ~ m o d ¢O

(>)

~ HP-I( ~. \ F; (>)

if

is called a relative Stein covering of

(~,

~')

~=

~Uil iE I

is a Stein covering of

o~

is a Stein covering of and

~'

is a subset of

~.,

~'

~(ioeo

=~Ui}iEl' I'C I).

55 Let

(~,

~[')

be such a pair° A p-cochain

coefficients in

CY

?

of (~,

27[') with

is, by definition, a family of holomorphic

functions ?i0,il,.oo,ipe

~(Ui0~ Uil~ oo. N U ip),

ip eI

such that i)

?i0,.i I, .... lj, . = - ~ i0 ° ..., ik, • . o ,iP ,ll,

2)

~i0,il, o..,ip

We denote by

0

if

cP(%~, ~ ' ;

"

ij,.o. " ' ,Ip

o.,ik,.o,,

10, l I .... , ip e I'

(~)

(31) (32)

the space of p-cochains of

with coefficients in ~ . The coboundary of p-cochain

(%~, %~')

~ ,

~ ~

is

the (p+l)-cochain given by p+l ( ~ ? )i0, .... ip+ I = j = 0 The eoboundary o p e r a t o r

cP+I(~,

~'', ~ )

~

maps

cP(~,

f2 = ~ o ~ :

and

(-i) j ~. ~ .(33) l0, .... ~j .... ip+ I ~';

cP(~[,

O)

into

9J[';~)--->cP+2(%~, ~J[';~)

is a zero mapping° Hence we can define the p-th cohomology space of the relative covering

(~I, 27[') with coefficients in

~

as follows:

HP(~I, ~I'; ~ ) =

If

~'

Ker ~ S : cP(~TL, ~['~ ~ ) Im ~ ~ .o cP-I(~, ~[,; ~ ) = ¢ ,

then the space

(>).

cP+l(~, ~'~ ~ ~ ~ c p ( ~ ' ~,; ~ ) }

HP(%~, ~'; ~ )

cohomology space of the covering HP(~;

~

~

(34)

is reduced to the p-th

with coefficients

in

~,

We can now state Leray's theorem for relative cohomology

spaces, which generalizes the isomorphism (29). Theorem 6. D ~O o

If

Let

~

(~, ~')

and

~

be two open sets of

Cn

such that

is a relative Stein covering of ~ m o d e ,

we

have the following isomorphism: HP(~-mod~;

~)

=

HP(~, ~ ;

~)o

(35)

56

Remark.

Up to this point,

complex manifold analytic

(ii)

sheaf

X

and the sheaf

over

'

by definition, ~

on an open set

an element

is a complex

theorem,

the spaces

hyperfunctions

may be an arbitrary

on

H~(~ u

where

of an oriented

; ~ )

of

will be denoted ; ~)

coherent

by

exists

real analytic

which cation

X

system

= Ilm z I ..... M.

Define

We will denote

by

~

fundamental

of the following Theorem

V.

7.

If

The

space of

sheaf of germs

= ~M

~)

denotes

is defined

of

on any

is such a manifold,

dimension

n

z I,

--., Zn X

there

such that

exist a neighbourhood of

X

~ on

MEX

of ~

x for

is called a complexifi-

the space of hyperfunctions

vanishing With

M

Im Zn = 03.

on

M

as follows:

= Hn[M]

(37)

the sheaf of germs ~

of hyperfunctions

is flabby

on

is a consequence

theorem.

the above

notations

~ ) = 0

to the case where V*

The

(36)

(29).

fact that the sheaf

H~(X; Return

• ~)

~

~(u):

of complex

and a local coordinate

(M) = H~(X;

The

M.

there

of

H nu ( ~

V

to the excision

~9. .

that the hyperfunction

x E M

~nM

space

= Hn[u] ,

manifold

a complex manifold

and that for any point X

space

Thanks on

vector

will be denoted by

It should be mentioned oriented

u.

do not depend

the last term is our abbreviation

hyperfunctions

space

u

of the n-th cohomology

neighbourhood

(u) = H ~ ( ~

M.

~

n-dimensional

Hyperfunctions.

where

in

may be an arbitrary

X.

A hyperfunction is

Cn

the 8ual

for M

we have

p # n

(38)

is an open set space of

V.

For

u

of the vector ~ E V*\(0),

we put

57

E~ = T({x e V; where

(

,

)

denotes the canonical inner product of

EEl,

half spaces

(x, ~ ) > 0}), Choose

V x V*.

-, E~N such that N

V¢\V For any open set

u

of

V,

complex neighbourhoods of neighbourhoods of u

u.

= ~ j =l e ~j

(39)

Stein (i.e. holomorphically convex) u

form a fundamental system of complex

We may suppose a complex neighbourhood

is Stein without loss of generality. ~[ = :U.j = j~ ~ Ego;

is a Stein covering of

~.\u.

Hn-l(~;

If

~0.

~9. of

is Stein,

j = i, 2, "'', N }

Hence by Leray's theorem we have

~)~

Hn-l(ga\u ; ~)

(40)

On the other hand we have, from the long exact sequence of cohomology (30), the following isomorphism if

n > i :

n ; (?) ~g Hu(fL ; ~ ) -= ~(u).

Hn-l(~\u

Here we have used Theorem 7 and Cartan's Theorem B. open convex cone

~

in

V.

Let

Consider now an

f e C~(fL ~ T ( r ) )

are going to define the hyperfunction boundary value

: (~(~nT(~))

~ As

V

(41)

be given. ~-~f

; ~(u).

V*

to

:-I, 0, I}

f :

(42)

is supposed oriented, we are given a mapping

n-th exterior product of

of

We

sgn

such that for any

of the a e ]R

we have sgn (a ~ i A " ' ' ~ n ) where vectors

sgn a = a/la ~ ~0'

~i'

if

a # 0

"" " , ~ n 6 V*

= sgn a- s g n ( ~ l A and

sgn 0 = 0.

''" A ~ n ), Choose

n+l

(43) non-zero

such that

n

~J

j=0

E~j = V c \ V

,

(44)

58

n

E C T ( F )o j=l ~J Define an (n-l)-cocycle of ficients in ~il,

As

~

~ = I ~ ~E~j}O=0,1

...,n

n ~ E = 4 , j =0 ~j

?

class of the cocycle

• --

A~in

)f

if

{il,''',inl

= II,''',nl,

(46)

otherwise. is a cocycle

(i.e.

denoted by

[~ ]

?

Hence via isomorphisms

(40) and (41),

on

u,

which is, by definition,

of

f,

~f.

~

= 0).

The cohomology

is in the space

[~ ]

Hn-l(~l; ~).

defines a hyperfunction

the hyperfunction boundary value

We claim the mapping

f~:

~(~LN

r(~))

~ ~(u)

(Lemme (5.7) of Morimoto [23]).

We remark that the mapping ~r : ~ ( u

~

can be extended to the mapping

× ~f~ p )

~ ~(u)

taking inductive and projective limits, where

(47) ~ = r O.

These ex-

!

tended

~F s define a sheaf homomorphism :

where

with coef-

as follows :

. = sgn(~ilA "'''Zn t0

is injective

(45)

-I~

projection

~

~ ~-I ~

,

(48)

denotes the pull back of the sheaf ~ : V × ~-fTS

~

~ V.

When we are considering the real analytic manifold situation can be formulated as follows: analytic n-manifold, cal homomorphism bundle over

M

TM X. M M,

Let

M

Let

where

TM

and

TMX = Coker (TM

with respect to we may identify

X.

M,

this

be an oriented real

be its complexification. ~ TXIM,

and

normal bundle over plexification of

X

by the

We have the canoni-

TX

denote the tangent

> TXIM)

Because

TMX ~ ~f~TM.

X

be the is a com-

Define the

59

normal sphere bundle over

M

by

SMX = ( T M X \ M ) / ~ + If we denote the tangential

(49)

.

sphere bundle over

SM = (TM \ M ) / ~ +

M

by

,

then we may identify SMX ~ ~ Our sheaf

is defined on

SM .

SMX

(50)

and we have defined the boundary

value operator

(51) where

~

: SMX

>M

is the projection.

As we have said, this

sheaf homomorphism is injective: ~

(exact)

To finish a review of hyperfunctions, distributions (u). ~(u)

on

u

Denote by

~ (u)

the space of

C ~ functions on

the space of real analytic functions

~(u)

• ~'(u)

~ g(u)

is injective~

~'(u)

the subspace of

~ (u)

that the mapping

Z

supp So the mapping

~

as follows:

on

u,

u

and by

which are endowed

As we have the natural injec-

with dense image,

elements have compact support, while

(u)

we recall how Schwartz'

form a subspace of the space of hyperfunctions

with usual locally convex topologies. tion

(52)

the dual mapping

is the subspace of ~'(u)

~ : ~'(u)

~'(u)

whose

can be identified with

whose elements have compact support.

Remark

does not augment the support: z(f) C

supp f

for any

f e ~'(u)~

can be extended to the mapping of If

f 6 ~'(u),

then decompose

(53) ~'(u)

f

into

as a locally

finite sum of distributions with compact support : f =

~

f~,

f~ ~ ~'(u)

(54)

60 ~-~ L(f~)

By (53) the sum tion.

is locally finite and defines a hyperfunc-

This hyperfunction, being independent of the decomposition

f = ~ f~,

is defined as

%(f),

i.e. we pose

%(f) = ~_. L(f~),

: ~% '(u)

This mapping

(55)

f = ~-~f~.

~ (u)

is injective and we consider by

this mapping (56)

' (u) c ~ (u) defines an injective sheaf homomorphism

Remark also

(57) (iii)

Coincidence of two notions of boundar Y values. For an open convex cone

~. of

of an open set (~ ( ~

T(~))

u

of

~

V,

of

V

and a complex neighbourhood

~,(~T(~))

denotes the subspace

whose element can be continued as distribution to

a complex neighbourhood of

u.

Martineau's first claim is the follow-

ing theorem (Th@or~me 2 of [18], p.240). Theorem 8.

A function

f ~ (>(~. ~ T ( r ) )

has the distribution

boundary value f ( x + ~f~ Fo) = lira f(x+ ~ y 0 ) y~0 ye~ (where the limit is in the topology of f E ~9,(~ If of

f,

,

~'(u))

(58)

if and only if

~ T(r)).

f E ~,(~.

~ T(P)),

then we can define two boundary values

namely the distribution boundary value

hyperfunction boundary value

(~'~f)(x).

If

f 6 (~,(.O- ~ T ( ~ ) ) ,

and the

Next theorem asserts that

these two boundary values coincide (Formule Theorem 9.

f(x+ ~ f ] ~ O )

~" of [18], p.256). then the hyperfunction

61

~f

is distribution

and we have (~f)(x)

(iv)

A hyperfunction Thanks

= f ( x + ~ f ~ ~0).

(59)

version of the edge of the wedge

to Theorems

8 and 9, in order to prove the distribution

version of the edge of the wedge

theorems

(Theorems

sufficient

to prove the following hyperfunction

theorems.

It is Martineau's

idea to decompose

of two notions

the distribution

Using the cohomology

the complicated

function

version of the edge of the wedge

calculation

situation where

version

Namely the principle

of boundary value of holomorphic

(Theorem 9) and the hyperfunction theorem.

i, 2 and 3), it is

version of these

of the edge of the wedge theorem into two parts: of coincidence

theorem.

techniques,

several cones

~i

,

one can manage

~2'

"" " , ~ m

take

place. Theorem i0 (Hyperfunction be an open convex cone of

1 of

version of Bogolyubov's

V

and its complex neighbourhood

complex neighbourhood every

V,

f~j ~ ~ ( ~

~'

of

~ T(rj)),

f e ~(~')

~.

:

of

u

there exists a ~'g' C f~

and that for

satisfying

~2(f2)

such that

Let

For an open set

u,

such that

j = i, 2,

~pl(fl) there exists

u

~ ~2 = - FI"

theorem).

in

f = f. J

on

~(u),

(60)

J$_' ~ T ( P j )

for

j = i, 2. The hyperfunction can be formulated Martineau

version of Epstein's

analogously.

and Martineau's

theorems

For the proof we refer the reader to

[20] but will sketch the proof when the dimension

of

V

is

one or two to convey the flavour of this proof. Proof.

We may suppose

V = ~n

and

~I

is the first quadrant :

62

rl = Iy ~ an; First consider the case by a pair

(~i'

~2 )'

n = i.

while

~ ~2f2

u

fl'

is given by

for

n = 2.

A hyperfunction on

u

(712'

~23'

?34'

~41 )'

( ~12'

?23'

~34'

~41 )

Put

is given by

~(fL r~ T(Ej))

As we have

u.

(62)

By the very definif = f. J

such that

on

E2 =ly 2 >0},

Then we have ~2

(63)

= E3 ~ E4 "

can be defined by four holomorphic functions ~jk ~ ~ ( 4 ~ T(Ej N Ek)) defines zero hyperfunction

~jk = ~ k f~ifl

(0, O, f2' 0).

(fl' 0, -f2' 0)

(fl' 0),

u ,

E 1 = IY ~ ~2 ; Yl > 01

there exist four holomorphic functions

~(Sh N T(PI)),

f. E ~(fL~T(~j)), J

Our assumption is :

f E (~(~_)

E 4 = ly 2 4_ 0}.

such that

If

as hyperfunction on

FI = E1 ~ E2 '

3, 4,

and

j = i, 2.

Suppose now E 3 = lyl < 0 I,

is given

(~I' ? 2 ) ~2 can be

is given by the pair

(0, -f2).

tion, there exists a function ~ £~T(~j)

~I

defines zero hyperfunction on

(fl' f2 )

u

A pair

to each other.

~ifl

(fl' 0) = (0, -f2) i°e°

T(Fj)).

if and only if

continued analytically across the boundary value of

A hyperfunction on

?j ~ ~ ( ~

defines zero hyperfunction

(61)

"''' Yn >0}"

Yl >0' Y2 >0'

~j"

and if and only if

~j E ~ ( ~ T ( E j ) ) ,

The boundary value of

is given by

(fl' 0, O, 0)

j = I, 2, fl E

while

~2f2

By the assumption of the theorem,

defines zero hyperfunction,

i.e. there exist

~j

such that fl = * 2 -

41 '

0 = ~3-~2'

-f2 = ~ 4 -

43 '

0 = ~i-~4

(64) "

63

~2 =

~3

on

~

T(E 2 ~ E3) ,

~4 =

~i

on

£~r(E

there exist, by localized Bochner's ~ (~_')

and

4 N El), tube theorem,

functions

such that V~ =

~j

on

~_' ~ T(Ej)

for

j = 2, 3;

=

~j

on

~

for

j = 4, I.

(65)

The function

f = 7~ - %

~' ~ T(E 1 ~ E 2) completes

and

~ T(Ej)

is in f =

~(~Q_')

~3 - ~4

the proof in the case

= f2

and

f =

on

~_' ~ T(E 3 ~ E4).

n > 2,

Theorem Ii.

4 1 = fl

on This

n = 2.

Remark that we have used localized Bochner's the case

~2-

tube theorem.

For

we have to extend it as follows: If

G

contain any straight

is a convex closed set of

V

which does not

line, we have

Hp (V C ; ~ ) = 0 r (G) For an open set

u

of

V

for

p ~ n.

(66)

and its complex neighbourhood

there exists a complex neighbourhood

~.'

of

u

such that

~

,

~LD ~'

and that the image of the restriction mapping H~nT(G)(~ ; ~) is zero for

theorem is a generalization

proof can be found in Martineau Remark that the vanishing

and

(67)

p # n.

This vanishing

implies,

~ H~,~r(G)(~' ; ~)

in particular,

(localized)

The

[18, 19, 20] and also in Morimoto

of the relative

the uniqueness

Bochner's

of Theorem 7.

tube theorem

analogously

cohomology

of analytic (p = i).

to the case

[23].

in Theorem ii

continuation

(p =0)

We can prove Theorem

I0 in the case

n > 2

n = i, 2

by the aid

of Theorem II.

For the details we refer the reader to Martineau

[20].

But the meaning of this proof can be fully understood by the theory of

64

sheaf

~

,

which we shall

As Martineau series

of theorems

sidered

of "the edge of the wedge"

the cones

Malgrange-Zerner a consequence

in the next

~. ]

theorem

type.

are relatively

II a

Indeed he conversion

in the

open and generalized

See also Morimoto

the

[21]

as for

ii.

Sheaf

Let

M

be an oriented

T~"I~ denotes

the cotangent

cotangential

sphere bundle:

real analytic bundle

over

where

M

is identified

to

respect >T~'M.

X,

T*~

X

M.

the zero section bundle

as the kernel

the exact

' > T*MX

S~"~ denotes

sequence

~ T*XIM

is a complexification

n.

the

(68)

The conormal

is defined

Hence we have 0

Because

of

and

of dimension

,

as usual with

be a complexification

manifold

M

S~'.]M = ( T ~ \ M ) / A +

X

from T h e o r e m

3 and its h y p e r f u n c t i o n

(Theorem 5).

of T h e o r e m

section.

in [18], one can deduce

in [19, 20] T h e o r e m

case where

5.

remarked

explain

M,

over

T'~. M

Let

with

of h o m o m o r p h i s m

of vector bundles

~ T~<~ of

of

T*XIM

over

..>.. 0 .

M: (69)

we have

T*X ~ T~'M × ~ r q ~ and can identify

(70)

T*MX ~ ~i~ T*M. Denote

the conormal

sphere bundle

over

S*MX = ( T * M X \ M ) / A +

M

by (71)

.

Then we may identify

S*MX = In 1969,

M. Sato

[30]

(72)

{':'-f

constructed

a sheaf

C

over

S "~\~X

and

65

of the sheaf

a sheaf homomorphism over

M

S*MX

~

of germs of hyperfunctions by the projection

to the direct image of the sheaf

~ :

~ M:

(73) such that the following theorem is true: Theorem 12.

o

The following sequence of sheaves over > ~

....

- ~.C

M

is exact:

> 0 ,

(74)

is the sheaf of germs of real analytic functions over

where and

: ~

~ ~

is the canonical injection.

A section of the sheaf

~

is called a microfunction.

intrinsic formulation of the theory of sheaf Kashiwara

[32].

On the other hand Morimoto

theory is very akin to Martineau's theorems by constructing techniques

~

Sato's

can be found in Sato-

[23] showed that Sato's

theory on the edge of the wedge

the main part of Sato's theory by the

inspired by Martineau

[18] and derived a hyperfunction

version of the edge of the wedge theorems from Sato's theory. vanishing theorem of local cohomology corner-stones

of Morimoto's

following Morimoto

in the euclidean space

construction as well as of Sato-Kashiwara's.

[23].

V.

The

(Theorem ii) is one of the

We are going to define the sheaf ~,~

M

M

~

and the mapping

~ :~

>

is supposed to be an open set

u

Because the theory is local, this assump-

tion is no real restriction. The conormal sphere bundle identified with

V K ~S*

,

~T~S*V

over the vector space

I

of

S*

~

> ~

is

where

S* = (V* \ ( 0 ) ) / ~ + . Denote by

V

the projection of

may be identified with the cone

V* \(0)

(75) onto

S*.

A subset

66

For an open convex set

I

of

S*,

we put

D(I) = {x E V; <x, ~ > ~ 0 D(I)

is non-positive dual cone of

open sets of

V x~f~S *

open set of

V

and

I

for all

I.

of the form

Let

~ ~ ~} .

~*

is a base of the open sets of

be the family of

u x~-i~I , where

is an open convex set of V ×~-f~S*.

£'(u × ~'ii I ) = lim ind H n [ ~

(76)

S*.

Put for

u

is an

Clearly

u ×~f~I

~*

e ~*,

~ T(D(1))],

(77)

~u

where

~

runs over the complex neighbourhoods of

u

inductive limit is taken under restriction mappings. abbreviation (29) for relative cohomologies. another element of

~*

complex neighbourhood hood

~I

of

uI

contained in ~

of

such that

u,

Let

and the Here we use the

u I x ~ f ~ I 1 be

u ×~-f~l . Then for every

there exists a complex neighbour-

g~_D ~L I

and we have

D(1) C D(II).

Therefore one can define a homomorphism u x~f~l ~ U l X ~ i ~ i i : ~ ' ( u X~I~I)

~ @'(u I x~f~Ii)

as the inductive limit of natural mappings of H n [ ~ l ~ T(D(II))]. the homomorphisms The sheaf over the sheaf

~

S*V

Hn[~_ ~ T(D(I))]

It is clear that the spaces

ux~-~ I

[UlX~f~ii

(78)

~ ' ( u ×~f~I)

to and

define a presheaf over the family

~*.

associated with this presheaf is, by definition,

of M. Sato.

It can be shown (Th@or~me (4.1) of [23])

that the space of sections of

~

over

u K~f~I E ~*

is given as

follows: ~(UK~I) where

I'

= lim proj lira indHn[~.nT(D(l'))], (79) I'C~ I ~$u runs through the relatively compact open subsets of I and

the projective limit is taken under the homomorphisms We construct now the mapping

~

ux~

I

~Ul~V.~1 ii.

For an open set

u

of

V,

67

the space of hyperfunctions

over

u

is given as follows:

(80)

~(u) = H n[~L N T(0)], where

f£

Therefore

u

is a complex neighbourhood of there exists the natural mapping

~ ' ( u x~-f~l)

for

u K~7~I

~ ~*.

and ~i

T(0) = V X ~ 7 ~ 0 ~ V . of

~(u)

to

As the following diagram is

commutative,

C' (u ~:i I)

u×~:l I

C '(u ~ C Y ( ~ a J)) (81)

C' ( u . f ~ J) we can glue together these mappings to

!

~I

to the mapping

~

of

~(u)

~ (u × ~/~S*) ~ It. £(u). We have defined all items of Theorem Ii.

we can decompose the singularities

of hyperfunction modulo real

analytic functions with the aid of sheaf ? 6 ~ (u), over

~?

= 7[-l(u).

denoted by S.S. ~

The

(decomposed)

Remark that

S.S.?

bundle over

u

singular support of ~: (82)

is a closed subset of the conormal sphere

by the projection S = (V \ ( 0 ) ) / ~ +

~f~SV = V × ~Z~S.

We have already

(Section 2). u

(i.e. microfunction)

and the ordinary singular support is nothing but the

S.S. ~

of the form

~

= supp ~ ?

Consider now

~I~SV

For a hyperfunction

is, by definition the support of microfunction S.S.~

image of

~

defines a section of sheaf

u ×~f~S*

This theorem says that

Let

×~3~r

,

is an open convex set of

~

~ V.

and the normal sphere bundle constructed

the sheaf

be the family of open sets of

where S.

7[ : V × ~ S *

u

is an open set of

V

and

The space of sections of sheaf

~

over

~T~SV

68

over

u × ~

~ e ~

~(u × ~ For

is given as follows:

~ ) = lira proj

f e ~ ( u x ~ - f ~ r ),

value

~F(f)

lira ind ~ i ~ ~ T ( ~ ' ) )

u × V~F

6 ~

S*

the hyperfunction boundary

can be defined as an element of

Sr: £in ×f~ir) is a subset of

If

,

S,

(83)

~ (u) (Section 3):

~ ~in).

(84)

we define its dual

F*

as a subset of

as follows: ~*

= ~

; ~ ~ V* \(0) for all

where we recall

x E V \(0) I

are the projections.

and

<x, ~ >

x ~ V \ (0) , xO E S

~ 0

such that

and

(85)

x0 ~ F}

~ ~ V* \(0) ;

~ ~ S *

We have the following characterization of the ~Ff

hyperfunction of the form

by the singular support (Th@or@me

(6.1) of [23]): Theorem 13.

Let

an open convex set in

be a hyperfunction S.

E ~(u)

and

~

be

Then the following two conditions are

equivalent: (a)

there exists a function

(b)

S.S.~ C u ×~f~ P*.

f e ~ (u x ~

r)

such that

? = ~pf. (86)

As a direct corollary we can prove a weak hyperfunction version of the edge of the wedge theorem of Bogolyubov's and Epstein's type. Theorem 14 (A version of Epstein's theorem). be open properly convex sets in and in

f2 e ~ (u × ~f~ F 2)

S.

Suppose that

Let

•I

and

~2

fl e ~ ( u ×~f~ r I)

have the same hyperfunction boundary value

~ (u) :

frlifl) then there exists u ~f~

~j

for

= fr2if2 ) ,

f E ~ ( u ~ < ~ f ~ c h ( F l U r2)) j = i, 2.

ch( r I • ~ 2 )

(87) such that

f = f. J

denotes the convex hull

on

69

of

FI u P2" Proof.

Consider the hyperfunction

? = Sr I (fl) = oct 2 (f2)

.

By Theorem 13, we have

s.s. ~ c (u ×q'z'f rl. ) ~ (~ ×#~-i r2. ) r', F2*).

(88)

~I*F~ ~2" = (ch(rl L# F2))*.

(89)

= u x~-

( rl*

But we h a v e

Again by Theorem 13 (or by Theorem 12 if exists

f • ~(u ×~

ch( r I ~ ~2))

rl* m ~2" = ~),

there

for which the conclusion of the

theorem is true.

(q.e.d.)

A hyperfunction version of Martineau's edge of the wedge theorem is a corollary of Kashiwara's theorem. Theorem 15 (Kashiwara [12]).

The sheaf

~

is flabby.

Theorem 16 (A version of Martineau's theorem). ..., r m for

are open convex sets of

j = i, 2, .... m

S.

We are given

Suppose

rl,

f. • ~(u × ~ 3

r2 , r j)

such that m

j~l Then there exists

gjk £ ~ ( u ×~-fY ch( ['j ~ pk )) fJ" =

Proof.

Put

k#j~ gjk

on

?j = ~Fjfj 6 ~(u).

S.S ~ J C (u ~

(90)

fro (fj) = 0. for

j ~ k

(91)

u x4 7[ r j. As we have

rj*) ~ k V j

(u×~

[~k*)

= k jk~J @ (u ×~f~ ( rj* F~ rk*)), by the flabbiness of the sheaf

such that

~,

such that

we can find

(92)

70

~'jk

for

= ~~< k j

j, k = i, 2, ''', m,

and m

(37j where

~[A]

support in

=

jk

(93)

'

denotes the space of microfunctions on A.

Thanks to Theorem 13, there exists

~(u ×~-f~ch(~j

U ~k))

u~-f~S *

with

,

gjk E

such that

~Pjkgjk where

k=l

= ~ jk

~ j k = ch( rj U p k ). ~(?J

for

j # k ,

As we have

k#j~ ~ j k g ~ k )

there exist, thanks to Theorem 12,

= 0 ,

(94)

h. E ~(u) J

such that

!

h.j =

~j-

k#j ~

&rjk gjk

(95)

If we put 1

gjk = gjk + (I/m-l)h.] then these

gjk

Remark.

(96)

satisfy the conclusion of the theorem.

(q.e.d.)

We have to say that Theorems 14 and 16 are weaker than

the hyperfunction version of Epstein's and Martineau's theorems in the form of Theorem I0. neighbourhood

~g

Taking the inductive limit as the complex of

such as (decomposed)

u

u,

we have reached the notion

singular support which may be defined for hyper-

functions on a manifold wedge theorem.

tends to

at the expense of weaking the edge of the

In Theorems I, 2, 3 or i0, it is important that the

complex neighbourhood

,~'

depends on

SL

but not on the individual

function

f.. It has been always one of the interesting questions J about these theorems to estimate ~).' from inside. For this topic we refer the reader to the classical literatures quoted in

§3.

71 6.

Application s of Sheaf

Sato's fundamental principle [30,31] on the regularity of hyperfunction solutions of partial differential equations can be stated by means of sheaf

~ •

operators of order set

u

of

V.

Let m

Pm(X,

P(x, D)

be a linear partial differential

whose coefficients are real analytic on an open ~ )

denotes its principal s y m b o l

Theorem 17 (Sato's fundamental principle). Pm (x, ~ )

be as above.

If

~ e ~(u)

Let

satisfies

P(x, D) P(x, D)~

and = 0,

then S°S.?

C

4 (x, ~

~

) • u ×~i-~ S*;

Pm(X,

g ) = 0}. (97)

This theorem is a considerable generalization of the BengelHarvey theorem on the real analyticity of hyperfunction

solutions of

elliptic equations

(see Harvey FS], Komatsu !14]).

operator

P(x, D)

is elliptic, Theorem 17 says that a hyperfunction

solution

?

of

P(x, D)~

thanks to Theorem 12, that

= 0 ~

satisfies

S.S. ~

Indeed if the

= ~ ,

which implies,

is real analytic.

We are going to indicate that a version of the Bargman-HallWightman-Jost

theorem is also a corollary to Theorem 17.

Bargman-Hall-Wightman-Jost

theorem and its meaning in the theoretical

physics see Jost [lO] or Streater-Wightman theorem, prepare the notations. (x0

i 2 x3) ~4 , x , x , 6

[36].

In order to state our

We define for a

4-vector

the Minkowski inner product

(x, x) = (x0) 2

(xl) 2 - (x2) 2 -

(x3)2

We denote the future and past light cones by =

Let

fR4n

4

As for the

o

,o,

> o}

,

be the space of n-tuples of 4-vectors X = (Xl, x2,

.-.

x )

x. • IR4

= -

x =

72 and put

p+n = p+ ~ F+ × "'" x NF+ , A point

X = (x I, "'', Xn)

of

~4n

p_n : _ p+n

(gs)

is, by definition, a Jost point n

if

for

every

Xj

~ 0

such that

~_. X .

j=l n

>0

we h a v e

J

n

( ~ ~jxj, j =i

~ )< j =i A j xj

0

(99)

n

i.e. the 4-vector points of

j_~#jxj

is space-like.

Let

be the set of Jost

J

~4n

We consider hyperfunctions on an open set A hyperfunction

?

on

u

is, by definition,

infinitesimal Lorentz transformations if

~

u

of the space

IR4n.

invariant under the satisfies 6 linear

partial differential equations: n

~ (x.0 ~__~__ k D j=l J Dx k. +x.j ~ ) ~0x J J

~(X) = 0,

for

k = i, 2, 3;

for

0 < k < ~.

n

J ~x£ j Dx k ?(X) = 0, J J With the above notations, Theorem 18. j=l

~ ~(u) a)

E.g. ~ C

b)

~

(i00)

suppose that a hyperfunction

satisfies two conditions:

u ~

((r2)*

~ (r#)*),

~np±0 ;

where r n = ±

(101)

is invariant under the infinitesimal Lorentz transformations.

Then the hyperfunction of Jost points in

~

is real analytic on

u G J

the set

u.

The proof consists in restating the condition b) by Sato's fundamental principle. Remark. hyperfunction

The details can be found in Morimoto

[24].

The condition a) is equivalent to the condition that the ?

can be represented as a difference of two boundary

values of holomorphic functions

f+ ~

~(u

X ~---~~+n)

and

73

g(u×~rlr

f

n) : ~ (X) = f + ( X + ~ f ~ C +) n - f_(X - ~

c+n).

(102)

It is in this form that the condition a) is cited in physical literatures.

7.

Interdependence of Support and Singular Support

We know that the sheaf sheaf

~

~

of germs of hyperfunctions and the

of germs of microfunctions are flabby.

support of a hyperfunction

But the singular

restricts the possible form of the

support of the hyperfunction

and vice versa.

two examples of this phenomenon.

The first theorem is due to Kawai

We are going to give

and Kashiwara, which they used to prove Holmgren's theorem. Theorem 19 (Lemma (8.5) of [32]). neighbourhood function on

u u

of the origin such that

O

Let

of

dhI0 # 0.

V

~ and

be a hyperfunction on a h

be a real analytic

Suppose the hyperfunction

satisfies following two conditions:

a)

S.S. ~ ~ (0, ~f~ (dh 1 0 ) ~ )

b)

supp ~ G { x e u; h(x) ~< h(0)l. Then

~

or

S.S.?

$9 (0, ~ ( - d h l 0 ) ~ )

;

(103) (104)

vanishes in a neighbourhood of the origin

This theorem is relevalent to the quasi-analyticity

O. of the distri-

butions of certain type studied by theoretical physicists of quantized field theory.

For example studying causal commutators,

obliged to investigate the distribution ~Rn+l

T

on the euclidean space

which is the difference of two distribution boundary values of

holomorphic functions

f+ E ~-(T(~+))

and

f

T(x) = f + ( x + ~ i ~ p+0) - f (x+~/~ p_O), where

one has been

P+

and

~_

~ ~ ( T ( ~_)): (i05)

denote the future and the past light cones,

74 namely ~+

= ix e ~n+l; Xo > O, (Xo)2 - (Xl)2 . . . . . (Xn)2 > 0},

p

=-p

(106)

One poses the following question: as (105) and

T

If a distribution

vanishes in an open set

u,

tion

T

vanish in an open set larger than

that

T

vanishes in the

T

is represented

then does the distribuu?

The first answer is

~+-convex envelope of

u

denoted by

~)~ (u) (Theorem on page 278 of the English transl, of Vladimirov [39]). Theorem 19 of Kawai and Kashiwara can be considered as a local version of this

~+-convex envelope theorem.

In fact, as we have already

remarked in the preceding section, the condition that a hyperfunction ?

can be represented locally as the difference like (105) of two

hyperfunction boundary values of ~ ( u X~I-~ p_)

f+ e ~ ( u ×~f~ ~+)

and

f

is given very well in the terminology of singular

support: S.S. ? C ~ n + l ~ ~ZT (p~'< U ~ f ) , where

~+ =

~ + and

~+ 0,

~_

in

~_ = ~ _ 0,

~

and

(107) ~ ^

are the dual of

S*.

On the other hand, as a corollary to the Jost-Lehman-Dyson integral representation one has the second answer to the above question that the distribution be larger than

T

vanishes in the envelope

B(u)

B~ (u) (Theorem on page 326, ibid.).

of

u,

which may

We are now going

to state a local version of this theorem in hyperfunction category. Let us suppose that our euclidean space A point by

(~0'

x

of

~1'

V

"'''

Theorem 20.

is represented by

V

is

~n+l

(x 0, Xl, "'', Xn)

with and

~

n ~ i. of

V*

~ n )" Let

~

be a h y p e r f u n c t i o n on a neighbourhood

u

of

75 the origin

0

of

V.

Suppose the hyperfunction

?

satisfies the

following two conditions:

S.S. ~ C u ~ , ~ ' f ~ [ ~ ;

a) b)

~ ~ V*\(0),

There exists a positive number

x0 ~

Then

vanishes

a

~0 # 0~.

such that

(lO8) implies

supp ~ ~ x

alxlJ.

in a neighbourhood

(109)

of the origin

O.

The proof of the theorem is done following Araki's argument with the aid of a theorem on l-hyperbolicity refer the reader to Morimoto Remark I.

due to Kawsi

[13].

[i] We

[25] for the details of proof.

The author is very much thankful to the referee for

drawing his attention to a work of J. M. Bony. an interesting corollary to Theorem 19.

Indeed Bony [39] gave

See also H~rmander

[40] for

this topic. Remark 2. cation of

M

Let and

M N

normal sphere bundle

be a real analytic manifold, a real analytic submanifold of S~X

which reduces to the sheaf

X

a complexifi-

M.

On the co-

we can construct a sheaf denoted ~

of M. Sato when

N = M.

~N~X'

The inter-

dependence of support and singular support of a hyperfunction

can be

understood by the unique continuation property of sections of

~N~X

with respect to the "complex" parameters

of

S~X.

These results will

be published elsewhere.

8.

Final remarks To finish this rather lengthy report, we shall give two further

remarks on the edge of the wedge theorem.

76 (i)

The Kolm-Nagel

theorem.

In Theorem 0 we assumed that two boundary values of holomorphic functions coincide on an open set

u

remarked

to assume that they and their all

in [17], it is sufficient

derivatives

coincide on a

V.

An analytic curve in

is said to be

~-like

where we identify Let Let

f

value,

u

V,

V

~(~

~ T(~)).

~ e~(1), Fs,~(?)

V

on the curve

c'(t) space

and

~0.

~ '(I),

on the

=

fo

f(c(t+~/~

where

s)+

s > 0

~I

is in Vc(t)

P of

for V

t ~ !,

at

c(t).

its complex neighbourhood.

curve

c

in

u.

~'i

~ ) ~(t)dt.

is sufficiently

is a sufficiently

r

exists and the functional

This functional

uI

small and

small complex of

u

and

p'

It may happen that the ~

,

>F0,0(?)

is

is called the distribution value of

f

c.

T h e o r e m 21 (Kolm and Nagel theorem). Put

t e I = [0, i],

r-like

compact open subcone of

F0,0(?)

continuous.

• c(t),

of relatively compact open subset

is a relatively limit

be an open convex cone

consider the integral

~i ~ T(~'),

neighbourhood

r

We are going to define its boundary

This integral has a meaning if is in

But as Kolm and Nagel

curve (see Theorem 21 below).

c : t ;

with the tangent

as an element of

Given

Let

if every tangent

be an open set in

be in

V.

F ~ l i k e analytic

Le~ us prepare the notations. in

of

= ~ '

F2 = - ~ "

Let

Suppose the boundary values of in the open set

u

for every

f.(z)3 E Df. J D =

Let

P

, u, ~.

O~(J~ O T ( r j ) ) ,

exist on some ~

be as above. J = i, 2.

~-like

curve

c

If these boundary

zll- • . ~ z ~n

n

values are equal,

i.e.

Df I = Df 2

on the curve

c

for any differential

77 operator f.(z) J

D,

there exists a common analytic continuation

holomorphic By the

in some complex neighbourhood

~-convex

f(z)

of the curve

of c.

envelope theorem (see Section 7), from the

conclusion of Theorem 21 we can deduce that the common analytic continuation

f

is holomorphic

envelope of the curve

in some complex neighbourhood of the

c

in

u.

F-convex

For the details see Kolm-Nagel

[17].

It would be interesting to ask if Theorem 21 has a hyperfunction version.

For that purpose, we will have to consider local operators,

i.e. differential operators

(ii)

operators of infinite order in place of differential

D.

Ultra-hyperfunctions. The ultra-hyperfunction

has been introduced by Morimoto

the name of "ultradistribution

cohomologique"

hyperfunction and analytic functional.

[22] in

as a generalization

We can show that the edge of

the wedge theorem can be formulated and proved in this category. fact,

a theory analogous

for ultra-hyperfunctions

of

to that of sheaf

C

In

can be constructed

so that the edge of the wedge theorem and

Sato's fundamental principle have their analogues in this situation. See for the details Morimoto

[26].

REFERENCES [I]

Araki, H.:

A generalization

of Borchers'

theorem,

Helv. Acta

Phys., 36 (1963), 132-139. [2]

Bogoliubov,

N. N. and D. V. Shirkov:

of Quantized Fields,

GITTL, Moscow,

science, New York, 1959.

Introduction to the Theory 1957;

English transl.

Inter-

78

[3]

Bremermann, H. J., R. Oehme and J. G. Taylor: relations in quantized field theory,

Proof of dispersion

Phys. Rev. (2) 109 (1958),

2178-2190. [4]

Browder, F. E.:

On the "edge of the wedge" theorem,

Canado J.

Math., 15 (1963), 125-131. [5]

Cameron, R. H. and D. A. Storvick:

Analytic continuation for

functions of several complex variables,

Trans. Amer. Math. Soc.,

125 (1966), 7-12. [6]

Dyson, F. J.:

Connection between local commutativity and regularity

of Wightman functions, [7]

Epstein, H.:

Phys. Rev°

(2) ii0 (1958), 579-581.

Generalization of the "edge of the wedge" theorem,

J. Math. Phys., ! (1960), 524-531. [8]

Harvey, R.:

Hyperfunctions and Partial Differential Equations,

Thesis, Stanford Univ., 1966. [9]

H~rmander, L.: Variables,

[i0] Jost, R.:

An Introduction to Complex Analysis in Several

Van Nostrand, 1966. The Generalized Theory of Quantized Fields,

AMS,

Providence, Rhode Island, 1965. [Ii] Kajiwara,

J.: Theory of Complex Functions,

Math. Library 2,

Morikita Shuppan, Tokyo, 1968 (in Japanese). [12] Kashiwara, M.:

On the flabbiness of the sheaf

transformation,

S~rikaiseki-kenky~sho

Kokyuroku

~

and the Radon 114 (1971), 1-4

(in Japanese). [13] Kawai, T.:

Construction of local elementary solutions for linear

partial differential operators with real analytic coefficients,

I,

Publ. R.I.M.S., ~ (1971), 363-397. [14] Komatsu, H.:

Resolution by hyperfunctions of sheaves of solutions

79 of differential equations with constant coefficients,

Math. Ann.,

176 (1968), 77-86. [15] Komatsu, H.:

Hyperfunctions and Partial Differential Equations

with Constant Coefficients,

Univ. Tokyo Seminar Notes, No.22,

1968 (in Japanese). [16] Komatsu, H.:

A local version of Bochner's tube theorem,

J. Fac.

Sci. Univ. Tokyo, Sect. IA, 19 (1972), 201-214. [17] Kolm, A. and B. Nagel:

A generalized edge of the wedge theorem,

Comm. Math. Phys., 8 (1968), 185-203. [18] Martineau, A.: holomorphes, butions,

Distributions et valeurs au bord des fonctions Proc. Intern. Summer Course on the Theory of Distri-

Lisbon, 1964, pp.195-326.

[19] Martineau, A.:

Th@or~mes sur le prolongement analytique du type

"Edge of the Wedge Theorem",

S~minaire Bourbaki,

20-i@me ann@e,

No.340, 1967/68. [20] Martinesu, A.:

Le "edge of the wedge theorem' en th~orie des

hyperfonctions de Sato,

Proc. Intern. Conf. on Functional Analysis,

Tokyo, 1969, Univ. Tokyo Press, 1970, pp.95-I06. [21] Morimoto, M.:

Une remarque sur un th~or~me de "edge of the wedge"

de A. Martineau, [22] Morimoto, M.:

Proc. Japan Acad., 45 (1969), 446-448.

Sur les ultradistributions

cohomologiques,

Ann.

Inst. Fourier, 19 (1970), 129-153. [23] Morimoto, M.:

Sur la d@composition du faisceau des germes de

singularit@s d'hyperfonctions,

J. Fac° Sci° Univ. Tokyo, Sect. IA

1_7_7(1970), 215-239. [24] Morimoto, M.:

Un th@or~me de l'analyticit@ des hyperfonctions

invariantes par les transformations de Lorentz, 47 (1971), 534-536.

Proc. Japan Acad.

80

[25] Morimoto, Mo:

Support et support singulier de l'hyperfonction,

Proc. Japan Acad., 47 (1971), 648-652. [26] Morimoto, M.:

La d@composition de singularit6s d'ultradistributions

cohomologiques, [27] Painlev@, P.:

Proc. Japan Acad., to appear. Sur les lignes singuli@res des fonctions analytiques,

Ann. Fac. Sci. Univ. Toulouse, ~ (1938), 26. [28] Rudin, W.:

Lectures on the Edge-of-the-Wedge Theorem,

Regional

Conference Series in Math. No.6, AMS, 1971. [29] Sato, M.: Theory of hyperfunctions

I, II,

J. Fac. Sci. Univ. Tokyo,

Sect. I, 8 (1959-60), 139-193 and 398-437. [30] Sato, M.:

Hyperfunctions and differential equations,

Proc. Intern.

Conf. on Functional Analysis, Tokyo, 1969, Univ. Tokyo Press, 1970, pp.91-94. [31] Sato, M.:

Regularity of hyperfunction solutions of partial dif-

ferential equations,

Actes Congr@s intern. Math., 1970, Tome 2,

pp.785-794. [32] Sato, M. and M. Kashiwara,

Structure of hyperfunctions,

S~gaku-no-

Ayumi, 15 (1970), 9-71 (in Japanese). [33] Schapira, P.:

Th@orie des Hyperfonctions,

Springer Lecture Note

Th@orie des Distributions,

Hermann, Paris, 1950-51.

No.126, 1970. [34] Schwartz, L.: [35] Siciak, J.:

Separate analytic functions and envelopes of holomorphy

of some lower dimensional subsets of

C n,

Ann. Polon. Math., 22

(1969), 145-171. [36] Streater, R. F. and A. S. Wightman: All That,

PCT, Spin and Statistics, and

Benjamin, New York, 1964.

[37] Vladimirov, V. S.:

On the edge

of the wedge theorem of Bogolyubov

81

Izv. Akad. Nauk SSSR, Ser. Mat., 26 (1962), 825-838 (in Russian). [38] Vladimirov, V. S.: Complex Variables,

Methods of the Theory of Functions of Several Nauka, Moskva, 1964:

English transl. MIT Press,

Cambridge, Mass., 1966. [39] Bony, J. M.:

Une extension du th@or@me de Holmgren sur l'unicit@

du probl@me de Cauchy,

C. R. Acad. Sci. Paris, 268 (1969), 1103-

1106. [40] H~rmander, L.:

A remark on Holmgren's uniqueness theorem,

J.

Differential Geometry, ~ (1971), 129-134.

Department of Mathematics Sophia University Kioicho, Tokyo

SOLUTIONS HYPERFONCTIONS DU PROBLEME DE CAUCHY par Jean-Michel BONY et Pierre SCHAPIRA

0.

Introduction

Soit

P(x, -~x )

un op~rateur diff@rentiel d'ordre

ficients analytiques, d~fini sur un ouvert

U

principale est hyperbolique dans une direction aucune hypoth@se sur les caract@ristiques de l'on peut r@soudre le probl@me de Cauchy l'espace des hyperfonctions, sur l'hypersurface

si

<x, N > = 0

(w) et

voisinage de cette hypersurface,

de N P).

Pu = v,

~n

m

~ coef-

et dont la partie

(nous ne faisons Nous montrons que ~(u) = (w)

dans

est un m-uple d'hyperfonctions v

une hyperfonction d~finie au

et "analytique" dans la direction

N.

La m4thode consiste ~ representer les hyperfonctions comme somme de valeurs au bord de fonctions holomorphes, ~ r4soudre le probl@me de Cauchy dans le domaine complexe, et ~ montrer que la solution obtenue admet une valeur au bord.

Les deux outils essentiels sont alors, d'une

part un th~or@me de prolongement des solutions holomorphes d'une @quation aux d@riv4es partielles, d'autre part une in@galit4 hyperbolique, qui se d4duit d'un th4or@me de MM. Komatsu et Kashiwara, version locale du th4or~me des tubes de Bochner. Nous ~tudions en m@me temps les solutions analytiques d'une 4quation hyperbolique, et montrons en particulier que les solutions de l'@quation homog~ne se prolongent ~ travers la fronti@re d'un ouvert de classe

C1

d@s que la direction normale est hyperbolique.

L'4tude des op4rateurs hyperboliques ~ caract@ristiques simples a @t@ faite dans le cadre des hyperfonctions par T. Kawai [7]. Le prolongement des solutions d'une 4quation A coefficients constants a @t4 4tudi4e par C. -0. Kiselman [8] par une m4thode enti@rement

83 diff@rente. Les r4sultats expos@s ici sont extraits d'articles ~ paraftre (cf. [i] [2] [3] [4]).

i.

Notations et rappels Dans tout cet article on d@signera par

P = P(x, ~--~)

diff@rentiel g coefficients analytiques dans un ouvert

op@rateur

un

U

de

~n

et

dont les coefficients se prolongent en fonctions holomorphes sur un ouvert

~

d@fini sur

de

En

~.

On d@signera par

On identifiera

Cn

P(z, -~z )

le complexifi@ de

pour le produit hermitien



= ~--~ zi ~i i

g l'espace euclidien

Re
Le mot hyperplan signifiera, sauf mention du contraire,

~>.

hyperplan r~el de

2n.

On ~crira

Un hyperplan d'@quation z0

si

p(z0,

d@fini sur

~ ) = O,

est caract@ristique en

~2n) •

Si

I

z = x+iy, ~>=

0

~ = ~ +i~

.

sera caract@ristique en

d@signe le symbole principal de

P,

On dira aussi que c'est le vecteur

~

qui

z 0. S n-I

(resp. S 2n-l)

est une partie de

S n-I

Le polaire de

I

la sph@re unit6 de

nous dirons que

(r~sp. propre) si le cone engendr@ par aucune droite).

muni du produit scalaire

Re
off p

~ × (Cn -{01).

On d@signera par

~2n

P

I

I

~n

(resp.

est convexe

est convexe (resp. ne contient

sera le cone ferm@ intersection des

demi-espaces passant par l'origine dont la normale ext@rieure appartient I. si

On notera souvent

~

F

est un cone convexe de

l'int@rieur du polaire de IRn,

l'ensemble (ferm4) des normales ~

son polaire ~

I.

I C S n-I

Inversement sera

~ l'origine.

Rappelons un th4orSme de [3] dont nous aurons besoin. Th~orSme I.i. localement compact,

Soit ~

~

et

~

ouvert, avec

deux convexes de ~ C ~g.

~n

, ~

@tant

Consid@rons les hyper-

84

plans dont la normale est limite de directions point

(au moins) de

qui coupe

~

coupe

voisinage de la fonction

~ f

Remarque. Zerner

~

,

en un

et supposons que tout hyperplan de ce type

~ .

et si

caractbristiques

Alors Pf

si

f

est une fonction holomorphe au

se prolonge en fonction holomorphe sur

~

sur

~ ,

se prolonge

en fonction holomorphe

Ce th@orSme

se d@montre ~ partir d'un th@orSme de M.

[15] par un argument g@om@trique d~ ~ L. HUrmander

th@orSme 5.3.3.).

On retrouvera au paragraphe

Signalons aussi que ce th~orSme syst@me d'@quations

.

(cf.

[5],

4 une situation analogue.

s'@nonce pour les solutions

f

d'un

Pi f = gi"

Ce th@orSme avait @t@ d@montr6 pour les op@rateurs ~ coefficients constants par C. -0. Kis.elman [8].

2.

In@galit@ hyperbolique Th@or@me

2.1.

Soit

coefficients holomorphes coefficient de

~

des racines en

~

m

est

Q(z,

~)

un polynSme de degr@

dans un ouvert 1

~ de

C p.

et que l'@quation

r@elles pour

Alors pour tout compact

K

z de

r@el V N ~P

m

en

%

On suppose que le

Q(z, ~ )

= 0

n'a que

(z E V = ~ ~ P ) . il existe

~ > 0, C > 0

tels que Q(z, z) = 0,

Re z E K,

IIm z l < g

Ce th@or@me r@sulte imm@diatement

~

IIm~[

~

CIIm z I.

comme nous l'a signal@ M. Kashi-

wara de la version ci-dessous du th@or@me des tubes de Bochner [6], Komatsu

[8 bis]).

Th@or@me

2.2.

L =~x+iy[ Si a < r

f

(Kashiwara

Soit

L

la partie de

~x~ < r,

Yl . . . . .

cp+l

d@finie par:

yp = 0,

0 < Yp+l ~ b I.

est une fonction holomorphe au voisinage de

il existe une constante

C > 0

telle que

f

L,

pour tout

est holomorphe

dans

85 l'ouvert ~ ={x+iy

I ~xI < a,

lyll + ' ' ' + l y p l

< C]yp+II ,

A partir de maintenant nous d6signerons par 0, i)

de

et nous 6crirons

m n

Rappelons qu'un op@rateur hyperbolique dans la direction P(Xo,

~',

~n ) = 0

Th@or@me 2.3.

N

N

et

N

~ = ( ~',

E,

~ n).

au point

x0 ~n

si l'6quation r@elles

( ~' ~ ~Rn-l).

est non caract6ristique.

et un nombre ~ E ¢ n,

(0,''',

a sa partie principale

en tous points d'un voisinage de

C > 0

Yp+l < b}.

le vecteur

Supposons la partie principale de

dans la direction

Iz-x0I<

P(x, ~-~x )

n'a que des racines

En particulier la direction

une constante

z = (z', z n)

N

0<

g > 0

p(z, ~ + ~ N )

P

hyperbolique x 0.

II existe

tels que = 0

> IIm 12 I ~< C[ [~ '[ [Yl + I~[ ] D@monstration.

Les solutions de l'~quation

p(z,

~', • )

= 0

v@rifient d'aprSs le th~or~me 2.1

CI[IYi+ I~l'l]

tlm~ t ~ pourvu que

lYl

x e K,

< e,

I~'l

= l,

I~l't

< a

On en d6duit par

homog@n@it@

IIm~l ~ C l [ l y l l pourvu que

IN'I G ~I~

D'autre part

N

~'I + 1"9'I]

'I"

6tant non caract6ristique au voisinage de

on a une majoration

des solutions de

p(z,

~ + ~N)

= 0 ,

et donc

lI~'cl-~ c3[ I r~'l + I~nl] ce qui entralne llmq~ I ~ C 4 l~I

x0,

86

1~'t > ~ 1 ~ ' t .

pour

.

Deux lemmes d e prolongement On d@signe par

a,

par

B(0, a)

B(0, a)

son adh@rence,

avec l'hyperplan

<x, N > =

On d@signera par B'(0, a)

et du point

Lemme 3.1.

Soit

que

Pf

f

B'(0, a)

0

de rayon

son intersection

0. ~ )

~a N

(de coordonn~es

P

existe une constante

et par

K(a,

l'int@rieur de l'enveloppe convexe de (0, ..-, 0, ~a)).

un op@rateur de partie principale hyper-

bolique dans la direction

fonction

la boule ouverte de centre

N

en tout point de la boule

~ > 0

telle que pour tout

holomorphe au voisinage de

a < r,

B'(0, a)

se prolonge analytiquement dans

K(a,

B(0, r).

Ii

toute

ayant la propri@t~

~),

se prolonge elle-

m@me analytiquement ~ cet ouvert. D@monstration.

Soit

$ > 0.

(ferm@) des directions P(z, ~ z

)

~ 6 S 2n-I

en un point

l'ouvert

Ix'~ ~ a,

~

A&

l'ensemble

caract@ristiques pour l'op@rateur

z = x+iy

D'apr@s le th@oT@me i.i l'hyperplan de normale

D~signons par

avec

;x~ ~ r,

IYl ~ 6 -

il suffit de v@rifier que si passant par le point

IY'I ~ E

de l'hyperplan

~ a N

~z

~ ~ AE

rencontre

= 0 I. n

Ii faut donc v@rifier que le syst@me d'@quations Re
~)=

0,

z

=0, n

Ix'i
lY'I
a une solution, sachant d'apr@s le th@or@me 2.3 appliqu@ au vecteur i~

que :

I~nl~

c[ ~1~'1

On peut par exemple supposer

~n ~ 0,

+

I ~'1] •

et d'apr~s la convexit~ des

87 in@galit@s

il suffit de trouver

x'

et

y'

avec

Ix' I < a, IY'~ < g

et <x',

~ '>-(y',

~'>

>I ~ a

~ n"

Or on peut trouver x' et y' tels que g
~ ~a

<x'

'

,

~

est un cSne de

avec la boule

B(0,

Lemme 3.2.

N

holomorphe

~' > 0

P

un op@rateur

K(a,

Pf

~') + i[ ~

telle que pour tout

se prolonge o~

l,g'

~

se prolonge

+ i [~

avec

£' > 0,

(ou du th@or@me

contenant Soit

hyperbolique Soit

pour un

£ > 0,

f ayant la

dans l'ouvert

est un cSne ouvert convexe

(de

~n)

elle-m@me dans un ouvert du type P2

un

Ii existe une

toute fonction

en fonction holomorphe

et o~

~'

K(a,

~')

est un cSne ouvert convexe

[~ '

D@monstration.

contenant

1

N > = 0.

a < r,

B'(0, a) + i p~ ,

[~ ' ,

contenant

"

son i n t e r s e c t i o n

B(O, r).

~y,

!

de

contenant 2,g' '

I 2(r+l)C

de partie principale

en tout point de la boule

au voisinage

propri~t@ que

~£

on d@signe par

cSne ouvert convexe non vide de l'hyperplan constante

{

g ).

Soit

dans la direction

~n

,

~ n

in@galit@ qui sera v~rifi@e pourvu que l'on ait Si

a

Ii r@sulte du th~or@me de Cauchy-Kowalevski

i.I) que

B'(0, a ) + i

f e/2'

est holomorphe o~

~

pr@cis@

dans un ouvert convexe

est un cone ouvert convexe

r ' ~

~'

un vecteur de norme

du lemme 3.1 que si l'on d~signait caract@ristiques plan de normale

par

en un point du compact ~ A6

I. Ag

On a vu dans la d~monstration l'ensemble

Ix I ~ r,

passant par le point

des directions

IYl ~ 6 ,

~ aN

tout hyper-

recontre

B'(0, a)

88

+ i ( l y ' I ~ g).

On en d~duit qu'il existe un nombre

que de

r ')

~

et

par le point

tel que tout hyperplan de normale

~ ~aN+i¼

~

l'int4rieur de l'enveloppe B'(0, a) + i N

rencontre

passant

Soit

£ ~ ~ aN + i[ [

M ,£

et de

~12"

en est ainsi de

i.i que

f

est holomorphe dans

M

¥,~

s'il

Pf.

La r@union des un ouvert du type

M

pour

~,£

K(a,

£ > 0

~ ') + i F2,g,

contient pour tout

~' < ~

pour un cSne ouvert convexe

P2"

Solutions analytique s Th~or~me 4.1.

~,

~ e Ag

B'(O, a) + i F'£/2.

convexe du point

Ii r~sulte du th~or~me

4.

~ > 0 (ne d@pendant

N

Soit

la normale ~

f

un ouvert de classe

~

principale hyperbolique Alors si

~

en

x 0.

Soit

dans la direction

x0,

en fonction analytique au voisinage de

x 0.

D~monstration. L'intersection de centr6e au point g

Soit H£

(x 0 - iN) + ~ a N

et de

x 0 - £N

lorsque

L '" interleur " "

H£

K(a,

~L

~ )

un point de

au voisinage de ~

et si

a

x 0.

Pf

la fonction

contient une boule

tend vers

se prolonge

f

se prolonge

<x-x0,

(dans

N> = -& .

H£)

est infiniment

B'a' grand par

0.

de l'enveloppe convexe de

sera un voisinage de

Ii suffit alors d'appliquer prolongement

N

l'hyperplan d'~quation

et dont le rayon

g

x0

un op6rateur de partie

est une fonction analytique dans

en fonction analytique au voisinage de

rapport ~

P

C I,

x0

pour

le lemme 3.1.

obtenu coincident au voisinage de

6

B a'

et du point

assez petit.

La fonction

f

x0

admet un

car

x0

syst@me fondamental de voisinages dont l'intersection avec

~9,

et le

est

connexe. Th4or@me 4.2.

Soit

~

et

~

deux convexes de

~n

~

@tant

89

localement

compact,

plans dont

la normale

partie de

~

~.

principale ,

Pf

si

de

f

lesquelles

lequel

analytique

Nous allons A

de

~ f

ConsidErons

Soit

lesquelles

la

en un point au moins

au voisinage

dans

~

,

~0,

de

~

coupe et

la fonction

f

dans

reprendre

l'adhErence

.

pour

les hyper-

de ce type qui coupe

analytique

analytique

la partie principale

point au moins dans

que tout hyperplan

en fonction

Soit

C fL .

n'est pas hyperbolique

en fonction

[5].

~

est limite de directions

P

DEmonstration. 5.3.3 du

avec

est une fonction

se prolonge

se prolonge

ouvert, N

et supposons

Alors

si

SL

dans

de

P

S n-I

du thEor@me

des directions

dans

n'est pas hyperbolique

x 0 6 ~,

est analytique.

la demonstration

~

un voisinage

Soit

xI 6 ~

, ~

en un

(connexe)

> 0,

de

avec

B(x I , ~ ) C ~ .

dont

II existe un compact

K

convexe

la normale

~

A

est compact). g

de

avec

Soit

K,

t Kg K

ne se prolonge Pour tout

x

B(Xl,

Kg

~ ) coupe

et de

(car

~ une distance soit contenu

x t = x 0 + t ( x I -x0)

Kg

K

B(xt,

£ ).

A

infErieure dans

et dEsignons La frontiSre

en un point hors de

A.

Kt

6 ~)~

cherchE

Remarque.

de

alors du thEorSme pas 8

tel que

et ses normales

pas ~

dans un ouvert

prolongement

$

tel que tout hyperplan

des points

0 ~ t ~ I,

convexe

o~

qui coupe

l'ensemble

est de classe

II rEsulte

f

Soit

l'enveloppe

n'appartiennent

de

K6

et choisissons

0 ~ g ~ ~.

par de

appartient

de

pour

t > to

est

to

en

x 0,

tel que

f

I.

on a donc obtenu un prolongement

EtoilE de

4.1 que le plus petit

contenant

x,

analytique

ce qui dEfinit

le

f.

Le th@orbme

i.i se dEmontre

exactement

comme

le th@orSme

90

4.2 ~ partir d'un th@or@me de Zerner

[15] analogue complexe du th@or~me

4.1, le th@or~me de Zerner se d@montrant

lui aussi par le m@me argument

g~om@trique que le th@or@me 4.1, le lemme 3.1 @tant alors remplac@ par le th@or@me de Cauchy-Kowalewski

5.

Rappels

pr@cis@.

sur les h y p e r f o n c t i o n s

Nous ne redonnerons pas ici la d@finition des hyperfonctions M. Sato

(cf.

[ii] ou [I0],

[14])

mais nous rappellerons

de

par contre

les propri@t@s du support singulier dont nous aurons besoin, propri~t@s incluses dans la th~orie du faisceau

C

de Sato et Kashiwara

([12],

[6], [i0 bis]). D@finition 5.1. ~n.

Soit

On dit qu'un point

support singulier de u

u

une hyperfonction

(x , ~ )

u

de

(en abr@g@

fL × S n-I S.S(u))

sur l'ouvert

f~ de

n 'appartient pas au

si, au voisinage de

est somme finie de valeurs au bord de fonctions holomorphes

d@finies dans des ouverts complexe de

x,

~

=

(~+iP~)

~ ~ ~n,

et

~

~ ,

F~

o~

~

x,

f~,

est un voisinage

est un cSne ouvert convexe

dont le polaire ne rencontre pas le point Le support singulier de Si

(x, ~ ) ~ S.S(u)

direction

~

S.S(u)

Si

5.1. I

Soit

u

l'intersection de

S.S(u)

~ S n-l.

est analytique dans la

x.

une hyperfonction

sur

~

. S n-I

et si

est valeur au bord d'une fonction holomorphe

d~finie dans un ouvert

si

u

u

~

est une partie convexe propre ferm@e de

~ ~ Xl,

en d~signant par

est donc un ferm@ de

on dira aussi que

au voisinage de

Th@or@me I)

u

~

$h+i~' ~'

est vide,

qui pour tout voisinage et d'un voisinage

complexe

l'int~rieur du polaire de u

est analytique.

I'

I'

de ~'

I

f, contient

de

~,

En particulier

91

2) et si

Si FI,''',F p S.S(u)

• -', u , P

sur

sont des ferm4s de

est contenu dans ~

F,

/g ~ S n'l

de r4union

F,

il existe des hyperfonctions

u I,

avec u =

~ ui , S.S(ui) C F i (i = i, o'', p). i Le premier th4or@me fondamental de Sato peut alors s'4noncer. g Th~or@me 5.2. Soit P(x, ~-~x ) un op4rateur diff@rentiel A coefficients analytiques sur i)

S.S(Pu) C S.S(u) U ~(x, q ) I p(x, ~) = 0}.

2)

Soit

(x, ~ )

route hyperfonction

un point de v

sur

~,

~ x S n-I

od

p(x,

q)

# 0.

il existe une hyperfonction

Pour

u

sur

avee (x, ~ )

~ S.S(Pu-v).

(On trouvera dans [3] une d6monstration @l@mentaire de ce th@or@me). Soit maintenant N = (0, --', 0, I).

S

l'hypersurface

Si

u

analytique dans la direction ~

S Soit

de

(I~)~

{N} U ~-N~

~

× (~N 1 U ~-NI)

±N),

fi

~

,

avec

dontle

(on dit que

sup-

u

on peut ddfinir la restriction de

par des parties convexes ferrules propres, ouvert f~

(~ +i~)

(de tels

sur tout ouvert relativement compact de L'hyperplan complexe

= 0

~g

I~,

~ ~ f~

f~

des

tels que

u

soit

existent au moins

d'apr@s le th@or~me 5.1).

rencontre t o u s l e s

ouverts

et on pose UI<x,N>=0 = ~ b ( f ~ ) < z ,N>=0) "

(b

est

un recouvrement fini du compl@mentaire d'un voisinage

somme des valeurs au bord des

i~

de

de la mani@re suivante.

fonctions holomorphes d a n s u n

~+

N> = 0

est une hyperfonction sur

port singulier ne rencontre pas

u

<x,

d@signe alors la "valeur au bord" dans l'espace

= 0).

92

On v@rifie

que cette d~finition

est ind@pendante

de la repr@senta-

tion choisie. L'hyperfonction fonction

de

peut alors

n-i

u

peut alors

variables

d4montrer

Stre consid@r@e

"d@pendant

que si

u

comme une hyper-

analytiquement"

est nulle pour

x

de

< 0,

Xn, u

et on

est nulle

n

sur les composantes

connexes

de

~

recontrant

l'ouvert

{x

< 0} ~ ~. n

On en d@duit

donc (voir

aux hyperfonctions peut

~galement

[7 bis]),

du th@orSme

se d@montrer

de Holmgren.

de maniSre

repr@sentation

des hyperfonctions

fonctionnelles

anal~t%ques.

Th@orSme

5.3.

caract@ristique hyperfonction

Soit

pour sur

en deux r@gions

et que

permet

utilis@e

~galement

Th~orSme

u

de

@l@mentaire

comme

sommes

Pu = 0.

soit nulle

pour d6duire

le th@or~me

Soit

oO

et

~i

est limite de directions

nulle

6.

de

)"

~

non

Soit

que

S

u

Alors

~ ~ .

Pu = 0,

,

ouvert avec

et supposons

Alors et si

hyperfonctions

On supposera

u

4.1

si u

u

~ C~L.

de

~n,

Consid4rons

~

les hyper-

caract@ristiques

que tout hyperplan

en un

de ce type qui

est une h y p e r f o n c t i o n

est nulle au voisinage

~tant

de

sur e) ,

u

sur

Solutions

une

s@pare

4.2 du th@or@me

deux convexes

la normale

solution

de

de

S.

plans dont

coupe

finies

sur l'une d'elles.

~g

~

localement

Supposons

compact,

coupe

[14] grace ~ la

P(x, ~ x

localement

de

que ce th@or@me

d'@noncer :

5.4.

point au moins

l'extension

analytique

diff@rentiel de

5.2

Signalons

une hypersurface

solution

est nulle au voisinage La m@thode

S

l'op6rateur

~

grace au th@or@me

l'op@rateur

du problSme P(x, ~

de Cauchy )

d'ordre

m.

On d@signera

est

9~

alors par

~

l'application qui $ une hyperfonction

dana la direction

±N ,

d@finie au voisinage

de

associe le m-uple d'hyperfonctions de l'hyperplan

u,

analytique

<x, N> = 0, <x, N > = 0

d@fini

par

~m-____il

Th~or@me 6.1.

On suppose la partie principale de

dana la direction stante

~" > 0

N

en tout point de

telle que,

~ a < r,

analytique dana la direction U K(a, = 0

~"),

et si

d~fini au

(w)

iN,

voisinage de

voisinage

B'(0, a) U K(a,

= v,

¥(u)

=

~

eat une hyperfonction voisinage de

~"),

B'(0, a) <x, N >

il existe une hyperfonction ~N ,

d@finie dana un

solution du probl@me de Cauchy

(w).

(I' ~)~

~y, N> = O,

Ii existe une con-

analytique dana la direction

D@monstration. Soit

v

d~finie au

B'(0, a),

u,

Pu

si

hyperbolique

eat un m-uple d'hyperfonctions de

et une seule, de

B(0, r).

P

L'unicit~ de

u

r@sulte du th6or@me de Holmgren.

un recouvrement fini de la sph@re unite

p~'

l'int@rieur du polaire de

un vecteur unitaire de

~.

3.2 d@termin@e par le couple

Soit

(~',

~

~).

S n-2

de

I~' dana cet espace, la constante du lemme

On prendra

~" ~ ~

pour

tout II existe un recouvrement de

~N~ ~ I-N}

g~

holomorphes dana

U K(a, tels que plan

~") ~

(I~)~

du compl@mentaire d'un voisinage

par des parties convexes ferm@es propres, des fonctions

tels que

V+i~,g, v

a = 0

= 0.

holomorphes dana

o~

V

eat un voisinage de

soit somme des valeurs au bord des soit le polaire

Ii existe donc des m-uples V ' + i ~ ' ~,g, ,

o~

~

de (h~)

V' = V ~ < x ,

!

I~

B'(0, a) g~,

et

dana l'hyper-

de fonctions

N> = 0

et

g'

> 0,

94

tels que

(w)

Soit

soit somme des valeurs au bord des

f~

la solution holomorphe du probl@me de Cauchy

~(f~) = (h~).

Ii r@sulte du lemme 3.2 que la fonction

valeur au bord U K(a,

(h~).

g ).

u~,

hyperfonction d@finie sur un

On posera

Remarque.

u = ~

p.

constantes,

Leray et Ohya

f~

aura une

voisinage

B'(0, a)

u~.

Nous ne faisons aucune hypoth@ses

stiques de

Pf~ = g~,

sur les caract@ri-

Dans le cas off on suppose celles-ci de multiplicit6s [9] ont montr6 que l'on pouvait r6soudre le

probl@me de Cauchy dans des classes non quasi-analytiqueso

7.

Existence et prolongement des solutions hyperfonctions Th@or@me

principale point

Soit

)

un op@rateur dont la partie

dans une direction

Pour toute hyperfonction u,

v

hyperfonctions

On peut, d'apr~s

N

au voisinage du

d@finie au voisinage de

hyperfonction au voisinage de

D6monstration. u2,

P(x, ~

est hyperbolique

x 0.

il existe

7.1.

Xo,

solution de

le th6or~me 5.2, trouver

d@finies au voisinage de

x0,

x0, Pu = v. uI

et

avec

s.s(Pu I - v ) ~ (Xo, N) S.S(Pu 2 -v) ~ En utilisant

(x0, -N).

le th@or@me 5.1 on peut aussi supposer que

(x0, -N) ~ S.S(Ul), Ii r~sulte alors

(x0, N) ~ S.S(u2).

du th6or@me 6.1 qu'il existe

u3

solution de

Pu 3 = v - Pu I - Pu 2 . On pose

u = u l+u 2+u 3

Remarque. pr@cise, B(~)

On pourrait en fait d@montrer,

qu'il existe un voisinage

l'espace des hyperfonctions

par une @tude plus

~

de

x0

tel que d~signant

sur

CO

on ait

PB(~)

= B(~).

par

95

On pourrait arguments

aussi d4duire

d'analyse

Th4or@me

Supposons N

u

v

de

x0 £ ~

7.1 par des

, de

convexe

N

P

dont

la fronti@re

la normale ~ hyperbolique

~

en

dans

~/i

x0 .

la d i r e c t i o n

x 0.

une h y p e r f o n c t i o n

x0,

un ouvert

une h y p e r f o n c t i o n

alors prolonger de

Soit

~

la pattie principale

Soit et

Soit

C I.

au v o i s i n a g e

du th@or@me

fonctionnelle.

7.2.

est de class

ce dernier r@sultat

u

sur

d~finie

fin

V

sur un v o i s i n a g e

solution de

en une h y p e r f o n c t i o n

solution de

~

V

de

Pu = v.

d@finie

x0,

On peut

sur un voisinage

P~ = v.

D@monstration.

On peut

supposer d'apr@s

le th~or@me

7.1 que

Pu

=0. Reprenons

les notations

de la d @ m o n s t r a t i o n

Ii r4sulte alors du th4or@me voisinage

hyperfonction

~

7.3.

~ C fL

g

~

en

x 0 - gN,

de

x0

co~ncidera

hyperbolique hyperplan

Soit

~O

Consid6rons

de directions

et

u

se prolonge

et solution de

alors avec

u

pour lesquelles

sur

~L

se prolonge

solution de

~ un

en

u,

P~ = 0.

d'apr@s

le

4.2.

u

dont

convexes

la normale

la pattie p r i n c i p a l e

~

de

~

,

P

et supposons

coupe

~.

une h y p e r f o n c t i o n

de m a n i S r e unique

de

en

Alors sur 3,

~

si

de

~n

est limite n'est pas

que tout v

est une

solution de

hyperfonction

sur

P~ = v.

D4monstration. du th@or@me

,

deux ouverts

les hyperplans

de ce type qui coupe

u

/L

en un point au moins

hyperfonction Pu = v,

centr@e

4.1.

de Holmgren.

Th@or@me avec

6.1 que la r e s t r i c t i o n de

d@finie au v o i s i n a g e

La r e s t r i c t i o n de th@or@me

B a'

de la boule

du th~orSme

Nous allons adapter

Soit

A

l'adh@rence

l'argument

dans

S n-I

de la d~monstratior des directions

pour

96

lesquelles point

de

la p a r t i e

principale

de

P

n'est

pas h y p e r b o l i q u e

II r @ s u l t e

de la d @ m o n s t r a t i o n

en un

£I .

Soit

x e

~_ .

(le t h 4 o r ~ m e

7.2 r e m p l a ~ a n t

relativement

compact

l'enveloppe

convexe

Soit alors relativement

(~t)t>t0

de

~o ,

de

x

aOt

compacts

le t h @ o r @ m e u

soit un syst~me

si

~0

4.2

est un o u v e r t

~ un v o i s i n a g e

de

~O 0 •

(0 ~ t < ~O

que

se p r o l o n g e

et de

dans

4.1)

du t h @ o r b m e

I) ,

une

famille

tels que

fondamental

d'ouverts

tels que

~J co t = t
de v o i s i n a g e s

convexes

to ' de

~

et tels to '

que

k3 t
~t

Soit Soit

= ~ " Lt

to

l'int~rieur

le plus p e t i t

de l ' e n v e l o p p e

t

convexe

tel que

u

7.2 que

tO = 1

de

~t

ne se p r o l o n g e

et de

pas ~

x.

Lt

pour

t > to .

Ii r @ s u l t e hors

de

~

Pour de

du t h @ o r S m e

et de tout

Pu = v

x

dans

n'appartiennent

de

~,

u

l'int@rieur

Les d i f f 6 r e n t s Holmgren

x

(th@or@me

pas ~

se p r o l o n g e de

car

prolongements

L

t

A.

donc

l'enveloppe

les n o r m a l e s

en s o l u t i o n

convexe

se r e c o l l e n t

de

tO

d'aprSs

hyperfonction et de

le t h 6 o r S m e

x. de

5.4).

BIBLIOGRAPHIE [I]

Bony,

J.

solutions

-M.

analytiques

C. R. Acad. [2]

Bony,

J.

et Schapira,

-M.

prolongement

Sci.

des

Paris,

et Schapira, pour

P.:

Existence

systSmes 274 P.:

et p r o l o n g e m e n t

hyperboliques

(1972),

non stricts.

86-89.

ProblSme

les h y p e r f o n c t i o n s

des

de Cauchy,

solutions

existence

d'@quations

et

hyper-

97 boliques non strictes.

C. R. Acad. Sci. Paris,

274 (1972), 188-

191. [3]

Bony, J. -M. et Schapira, P.:

Existence et prolongement des

solutions holomorphes des @quations aux d@riv6es partielles. Article ~ paraltre~ aux Inventiones Mathematicae. [4]

Bony, J. -M. et Schapira, P.:

Solutions analytiques et solutions

hyperfonctions des @quations hyperboliques non strictes.

Article

paraltre. [5]

H~rmander, L.:

Linear Partial Differential Operators.

Springer,

1963. [6]

Kashiwara, M.:

On the structure of hyperfunctions

Sugaku no Ayumi, [7]

Kawai, T.:

15 (1970), 19-72 (en Japonais).

Construction of elementary solutions of I-hyperbolic

operators and solutions with small singularities. Acad., [7 bis]

Kawai, T.:

On the theory of Fourier transform in the theory

Kokyuroku, RIMS, Kyoto Univ., Kiselman, C. -O.:

Surikaiseki Kenkyusho

108 (1969), 84-288 (en Japonais).

Prolongement des solutions d'une ~quation aux

d@riv@es partielles ~ coefficients constants. France, [8 bis]

Proc. Japan

46 (1970), 912-915.

of hyperfunctions and its applications,

[8]

(after M. Sato).

Bull. Soc. Math.

97 (1969), 329-356.

Komatsu, H.:

A local version of Bochner's tube theorem.

J. Fac. Sci. Univ. Tokyo, Sect. IA (~ para~tre). [9]

Leray, J. et Ohya, Y.: stricts.

Syst@mes lin@aires hyperboliques non

Colloque sur l'Analyse Fonctionnelle,

LiSge~1964,

C.B.R.M., pp.i05-144. [i0] Martineau, A.:

Distributions

et valeurs au bord des fonctions

98

holomorphes.

Proc. of the Intern. Summer Inst. Lisbon, 1964,

pp.193-326. [I0 bis] Morimoto, M.:

Sur la d~composition du faisceau des germes

de singularit@s d'hyperfonctions,

J. Fac. Sci. Univ. Tokyo, Sect.

IA, 17 (1970), 215-239. [ii] Sato, M.:

Theory of hyperfunctions,

Tokyo, Sect. I, [12] Sato, M.:

Iet

II.

J. Fac. Sci. Univ.

8 (1959-60), 139-193 et 398-437.

Regularity of hyperfunction solutions of partial dif-

ferential equations. [13] Schapira, P.:

Intern. Congress of Math.,

Theorie des Hyperfonctions.

Nice, 1970.

Lecture Notes in Math.

Springer, No.126, 1970. [14] Schapira, P.:

Th@or@me d'unicit@ de Holmgren et op@rateurs hyper-

boliques dans l'espace des hyperfonctions.

Ana~s Acad. Brasil Sc.,

43 (1971), 38-44. [15] Zerner, M.:

Domaine d'holomorphie des fonctions v@rifiant une

@quation aux d@riv@es partielles.

C. R. Acad. Sci. Paris,

272

(1971), 1646-1648.

J.-M. Bony (Universit@ de Paris VI) 66, rue Gay-Lussac PARIS (V °) P. Schapira (Universit6 de Tours) 57, rue Boissonade PARIS (XIV °)

ON THE GLOBAL EXISTENCE OF REAL ANALYTIC

S0~LUTIONS OF LINEAR DIFFERENTIAL EQUATIONS* By Takahiro KAWAI

O.

Introduction

Professor Sato initiated and developed the theory of sheaf

~

in

1969 (Sato [2], [3]), and this theory has turned out to be a very powerful tool in analysis, differential equations°

especially in the study of linear (pseudo-)

(Cfo Kashiwara-Kawai

[5], Sato [2] ~ [5], Sato-Kawai-Kashiwara [21, [31.)

[i], [21, Kawai [11

E1]o

See also Hormander

The present speaker gave a survey lecture on these subjects

at the symposium on the theory of hyperfunctions and differential equations held at Research Institute for Mathematical

Sciences last

March (Kawai ~3]), and listed there four problems to be solved.

They

were: (i)

the treatment of the case

k = ~,

where

k

appearing in Egorov [11 and Nirenberg-Treves

is the number [i] concerning the

local solvability of linear (pseudo-)differential (ii)

equations;

to extend our theory to the case where the assumption of simple characteristics

is omitted;

(iii) to extend our theory to overdetermined systems; and (iv)

to give global existence theorems° Especially he placed emphasis on problems

(iii) and (iv) at that

occasion° *)

Revised on March 12, 1972o by the Sakkokai Foundation.

This work has been supported in part

100

Concerning problems (ii) and (iii) some results have been given in Sato-Kawai-Kashiwara

Eli, and concerning problem

been obtained by the present speaker

(iv) a result has

(Kawai E4], (5]).

In this lecture we will explain how problem the local theory of linear differential

(iv) is deduced from

equations°

More complete arguments will be given in our forthcoming papers (Kawai [6]) and this lecture should be regarded as a survey°

I.

Global Existence of Real Analytic Solutions of Single Linear

Differential

Equation with Constant Coefficients

As is well known the topological

structure of the space of real

analytic functions on an open set is rather complicated, Professor Ehrenpreis, of linear differential

who initiated and completed

the general theory

equations with constant coefficients

framework of distributions with Professors Malgrange, Palamodov,

hence even

Hormander and

seems at present to have abandoned the attempt to attack

the problem of global existence of real analytic solutions° Ehrenpreis

[2], [3].)

at least when we restrict ourselves

operators

(Cfo

But we can treat this problem without much dif-

ficulty by the aid of the theory of hyperfunctions ,

in the

and that of sheaf

to the consideration of the

satisfying suitable regularity conditions which allow us to

consider the problems geometrically°

In a sense our method can be

regarded as "method of algebraic analysis" contrary to "method of func tional analysis", which is developed, Palamodov [I], Ehrenpreis

[3], etco

for example,

in Hormander

[i],

(The word "algebraic analysis"

seems to go back to Euler but it has recently been endowed with posi-

101

tive meanings by Professor Sato, who aims at the Renaissance of classical analysis°) We first examine in the special case if the theory of hyperfunctions is useful to investigate the problem of global existence of real analytic solutions°

In fact we easily understand that it is very

powerful in the following special case, ioeo, the case when the operator

P(D)

is elliptic°

Of course in this case there is a decisive result due to Malgrange [i], ioeo, Theorem (Malgrange ~i])o has a solution Here

~(~)

u(x)

in

For any open set

~(~)

for any datum

~

~n, P(D)u

in

f(x)

in

= f

6(~)o

denotes the space of real analytic functions defined on

Now we show how we can prove this deep theorem with ease if we assume that

~

is relatively compact°

The essence of the proof is,

as described below,

the flabbiness of the sheaf of hyperfunctions,

which we denote by

~

in the sequel°

Our proof is divided into two parts°

First we recall the follow-

ing lemma due to John [I]o Lemma io

If the linear differential operator

then we can find a hyperfunction

(i)

P(D)E(x) =

E(x)

defined on

P(D) An

is elliptic, satisfying

~(x)

and (ii)

E(x)

is real analytic outside the origin°

This lemma can be proved by many methods;

for example, one can

use the fact that the non-characteristic Cauchy problem in the complex domain has a unique entire solution as far as all the data given are

102 entire functions,

the linear differential

is of constant coefficients plane.

operator under consideration

and the initial hypersurface

( C f . Lersy [2] Lemma 9.1.)

is a hyper-

Then one can use the celebrated

reasonings of John (i], Chapter 3 to construct

E(x).

(Cf. John [i]

pp. 65-72.) Another proof is given in the following way: elementary solution

E0(x )

of the principal part

First construct the Pm(D)

of

P(D),

in the form i

i (-2~i) n

(Pro(~)+iO)

I~1=1

),

~ m n (<x' ~>+iO) go ( -

where

l

~j(T) and

L0(~)

(-l)-J-I (-j-l) '.q: j

=.)%j I j,. 1 og~-

(j < 0)

__i j,. (i + ... +

I) .~j

denotes the volume element of the unit sphere:

= ~n (-l) j-1 j=l ~jd~ I A °'" A d~j_l

~(~)

(j _->0)

Next construct

the required

E(x)

starting from

E0(x) , or more precisely

A d ~j+l

A "°"

A d~n

.

by the successive approximation from

1 (-2qli) n P m ( ~ ) where tively.

z

and

~

!re_n(< z, ~)),

denote the complexifications

Note that

Pm(~)

of

x

never vanishes as far as

and ~

~

is suffi-

ciently near the real unit ball

i ~ ~ jRn I I~I = ii

of ellipticity.

of the successive approximation

The convergence

easy to check, and it is also easy to verify that required properties.

respec-

by the assumption

E(x)

is

has all the

103 Secondly we use the flabbiness of sheaf function

f(x)

,

which is defined on

~n

~

to obtain a hyper-

and satisfies the following

conditions: (i)

Its support is contained in the closure

(ii)

It coincides with

Then using

~(x)

f(x)

we define

in u(x)

of

.~.

fL. by the integral

JE(x-y) f~(y)dy.

This integral is well defined as an integral along fibers (Sato [i]), since the support of

~(y)

is compact by the definition.

other hand by property (i) of property (ii) of

E(x)

is real analytic in

E(x) we have

and the property of

~.

On the

P(D)u(x) = ~(x) ~(x)

and by

we see that

u(x)

In fact it trivially follows from Sato's

lemma on regularity of integrals along fibers (Sato [4] Corollary 6.5.3), which we use essentially in this r e p o r t the restriction of u(x)

u(x)

to

~,

Thus if we consider

which we denote by

is a real analytic solution of the equation

u(x)

again,

P(D)u = f.

This proof of the existence theorem in the elliptic case teaches us the following facts: (i)

Flabbiness of sheaf

~

allows us to pass by the technical dif-

ficulties, especially it reduces all the problems to the boundary;

and (ii)

The informations which the "good" elementary solutions have (property (ii) in the above case) are used in the course of integrations and give us a good solution of

P(D)u = f.

These observations urge us to consider more general differential operators, not necessarily elliptic:

in fact we have "good" elementary

solutions for the differential operator

P(x, D x)

satisfying the

104 following conditions operator

P(x, Dx)

(i) and (2), and they exist globally if the is of constant coefficients

(I)

The principal symbol

(2)

Pm(X, ~)

Pm(X, ~)

of

P(x, Dx)

is of simple characteristics,

does not vanish whenever

(Kawai (l]):

Pm(X, ~) = 0

is real.

i e.,

grad~ Pm(X, ~ )

for any point

(x, ~)

in

the real cotangent sphere bundle. We also remark that we can treat a more general class of operators first considered in Andersson Ill (see also Kawai (3], ~5]), though in this section we restrict ourselves to the case where the differential operators are with constant coefficients, which is an easy case from the view-point of construction of elementary solutions. Now, what is the good property of the elementary solutions constructed in Kawai [ii]? Lemma 2

Let

P(D)

constant coefficients

P(D)E±(x) =

be a linear differential operator with

satisfying conditions

exist two hyperfunctions (i)

It is described in the following lemma.

~ (x)

E+(x)

and

(i) and (2).

E (x)

Then there

such that

holds;

and (ii)

S.S.E!(x)

is contained in

± t grad~ P m ( ~ ) where

S~'~n

t ~ 0

and

Pm ( ~ )

= 0~

denotes the cotangent sphere bundle of

denotes the support of Sato [4].)

with

{(x, ~) E S~'~n I x = 0

E±(x)

~n

or

x =

respectively, and

regarded as sections of sheaf

S.S E±(x) ~

.

(Cf.

Here and also in the sequel the double signs should be

chosen simultaneously. The proof of this lemma was rather implicit in Kawai Eli, especially concerning the global existence of

E±(x),

it is easy however

to prove this lemma using the successive approximation method as is

105 sketched

in the proof of Lemma

i, since the operator

P(D)

has con-

stant coefficients. We believe

that such an elementary

solution

2 is very good and that all the informations should be deduced

as is given by Lemma

about the operator

from it, and the belief in the good elementary

solution has one of its rewards as is described We first consider the solvability in

n

~

.

Here

K, i e., of

K

~(K)

limO(V),

and

~(V)

denotes where

V

denotes

in

in this report.

a_(K)

for compact

the space of real analytic

the space of holomorphic

Theorem 3.

~

that

K

real analytic

function

defined near

the boundary

~_

SL

the following geometrical operator in in

(3)

~(K)

P(D)

on

functions

on

V.

P(D)u = f

in

is the closure of the relatively

= ~x ~ ?(x) < "0},

of

functions

~.

Assume

compact open set

K

as well as it plays a role as a

lemma to our final object of solving the equation for an open set

set

runs over the complex neighborhoods

This problem has its own interests

~(~)

P(D)

satisfies

we can find

K

where

~ (x)

satisfying

is a real valued grad x ~ # 0

Suppose that the compact condition

in

K

satisfies

(3) and that the differential

conditions

u(x)

set

on

(i) and (2).

~(~)

Then for any

such that

P(D)u = f

f(x) holds

~.. For any

x0

in

~

the bicharacteristic

.

of

P(D)

issuing

from

(x0, g r a d x ~ Ix=x0)

curve

b(x0,gradx ~ ~=x0 )

never intersects

~L.

The proof of this theorem is given just in the same way as in the second part of our proof of the existence

theorem in the elliptic

by the use of either one of the good elementary

solutions

given in

case

106 Lemma 2. f(x)

In fact the smoothness of the boundary and the regularity of

permit us to extend

f(x)

to ~ n

denotes the 1-dimensional Heaviside S.S.(f(x)@(- ?(x))) ! g r a d X ?(x)l.

by

f(x)@(- ~(x)),

function.

is contained in

where

@

Note that

I (x, ~ ) ~ S*~ n I x £ ~ ,

~ =

Then we can apply Sato's lemma on the regularity of

integrals along fibers to the integral

I E(x-y)f(y)e(- ~(y))dy

and

obtain the required result. This proof of Theorem 3 needs only one of the good elementary solutions given in Lemma 2, but this contradicts metry:

We must use both good elementary

one is better than the other.

to our sense of sym-

solutions,

because neither

This belief in both good elementary

solutions is rewarded again, i.e., we can improve Theorem 3 as follows. In Theorem 3 the condition

Theorem 4.

(3) on

~

can be weakened

to the following. (4)

For any of

x0

P(D)

in

~

the bicharacteristic

issuing from

curve

(Xo, grad x ~ I x = x 0 )

b(x0,gradx~ix=x0 )

intersects

~

in an

open interval. Proof of Theorem 4. condition Pm(~)

= 0,

~ = !gradx~(X)}

the bicharacteristic ~I

f(x)@(- ~(x))

(4) we have the decomposition

l(x' ~) 6 S*~ n I x £ ~ . ,

sect

We denote

and

N

curve

Pm (~)

= O,

~I •

curve

Since sheaf

also Sato, Kawai and Kashiwara we can find hyperfunctions

~

[i].)

f~+(x) and

N+ U N_,

where

~ = ! g r a d X ?(x)

x + t grad~Pm (~)

= l(x, ~) e s * ~ n l x

~(x).

By

N = I(x' ~ ) ~ s*~nl x ~ ~ L

into the form

and half of the bicharacteristic not intersect

of

by

(t ~ 0)

~ ~fL, P m ( ~ )

does not inter= 0, ~ =±gradx~(X)

is flabby (Kashiwara

~_(x)

N+ =

and half of

x + t grad~ Pm (~)

and since

,

(t ~ 0) [i]. See

HI oR n, ~ )

such that

does

vanishes~

107

S.S.(~(x)-~(x)-~_(x)) NCN_.

= ~ ,

S.S.f~(x) ~ N C N+

and

S.S.f_(x)~

Then applying Sato's lemma on the regularity of integrals

along fibers to v(x) = we find

E+

(y)dy +

S.S.v(x) ~ S*fL = ~ .

E (x-y)f_(y)dy,

Note that the above integrals are

well defined as those of sections of sheaf ~ P(D)v(x)

= f~'(x)+g(x),

where

g(x)

Therefore we have

is real analytic in

we have used again the fact that

HI(~ n, ~_)

vanishes.

ting

containing

K

g(x)

to a closed ball

can apply Theorem 3 to find interior of ing

w(x)

B

w(x)

and satisfies

from

analytic in

B

~

v(x),

Here

Then restric-

in its interior, we

which is real analytic in the

P(D)W(E)

= g(x)

we find the required

and satisfies

~n.

P(D)u(x)

there. u(x),

= f(x)

Thus subtract-

which is real

there.

This completes

the proof of Theorem 4. In an obvious way we can modify the form of Theorem 4 to obtain a result which assures the existence of solution refer the reader to Kawai Remark.

u(x)

in

topological vector space, i.e.,

~(K)

has a natural structure as a

~(K)

duality theorem holds for the pair

is a DFS-space,

(~(K),

the space of hyperfunctions with support in

~ K ), K.

theorem shows that the existence of solutions in

Serre's

where

~K

On the other hand the unique continuation

Theorem 5.

denotes

Then Serre's duality ~(K)

can be deduced

by the unique continuation theorem concerning hyperfunction

characteristics.

solutions.

theorem follows easily from

[I] in a precise form using the notion of bi-

Thus we have the following theorem. Let

K

We

[4] Theorem I' about the modifications.

Since the space

Theorem 3.3' in Kawai

~(K).

be a compact set in

~n

and the operator

108

P(D)

satisfy conditions

below holds.

Then

(5)

(x,

For any hull

ChK

Pm ( ~ ) segment

P(D) ~(K) ~ )

of

= O,

(I) and

in

K,

=

(2).

Suppose that condition

~(K)

S*~ n

holds.

such that

but not to

K,

there is a point

y

x--~ does not intersect

bicharacteristic

curve of

(5)

belongs to the convex

and such that outside

K

P(D)

x

ChK

~

satisfies

for w h i c h the

and is contained in the

issuing from

(x, ~ ).

We omit the proof of this theorem in this lecture since it essentially uses "functional analysis".

We only remark the following

two facts which are related to T h e o r e m 5.

It is possible to deduce

T h e o r e m 5 from the following Remark A. Remark A.

An analogue of Theorem 4 can be proved even if

a closure of an open set necessarily assumed.

whose regularity at the boundary

In fact it is sufficient

the following condition (6)

fL ,

curve of

P(D)

is

is not

in this case to assume

(6) instead of condition

Any bicharacteristic

K

(4):

intersects

~

in an open

interval. The validity of this statement

is obvious from the method of the proof

of T h e o r e m 4, if we remark the fact that sheaf case, however, we need not assume f(x)

belongs

flabbiness

to

~(~_),

of sheaf

~

.

f(x)

~

belongs

since we extend

is flabby. to

f(x)

to

In this

~(K)

but only

~n

using the

Hence this analogue of T h e o r e m 4 should be

regarded as an existence theorem for

~(~)

rather than

~(K).

(Cf. T h e o r e m 9 below.) Remark B. complex valued,

If we allow the principal

symbol of

P(D)

to be

then we have the following T h e o r e m 6.

Before stating T h e o r e m 6 we prepare a notion regarding bicharac-

109

teristics of

P(D).

In order to define the notion we assume in the

sequel that the principal symbol where (7)

A

m

and

grad E Am

=o,

B

Pm(~ )

has the form

Am( ~ ) + i B m ( ~ ) ,

are real valued, and that

m

and

grad~ Bm

are linearly independent whenever

Pm(~)

"~#o.

Using these assumptions on

Pm(~ )

plane

through

y~

of

P(D)

we can define the bicharacteristic (x0,

~0)

to be the 2-dimensional

(×0' go) linear variety passing through and

g r a d ~ B m l =~0,-!

where

x0

which is spanned by

Pm(~0)=

0

gradg Aml~=~0

holds.

Preparing this notion, we have the following theorem. Theorem 6. the compact set

Let the operator K

P(D) £(K) = ~(K) (8)

in ~ n

P(D)

satisfy condition (7) and let

satisfy the following condition

(8).

Then

~ ~ ( C h K - K)

has

holds.

For any bicharacteristic plane

A

of

P(D),

no relatively compact component. We have not yet proved this theorem without using the duality theorem.

A little weaker theorem is obtained by a direct method

similar to the proof of Theorem 3 usin~ the elementary solution in Kawai [4], Theorem 2. Now we go on to the problem of global existence of solutions in ~(fL) in

~R2,

for an open set ~

A complete result is obtained if

~'L is

hence we first state the theorem.

Theorem 7. coefficients

For any linear differential operator with constant

P(D)

compact open set (9)

.

we have ~L in

~R2

P(D) ~ ( ~ )

= ~ (~)

if a relatively

satisfies the following condition:

Any characteristic line of

P(D)

intersects

~

in an open

110

interval. The proof of this theorem relies on the fact that explicit construction of elementary solutions of in the 2-dimensional

P(D)

is possible for any

P(D)

case.

We also note that

de Giorgi-Cattabriga

[I], [2]

have very

recently obtained the affirmative answer concerning the global existence of real analytic solutions of

P(D)u = f

in

~(~2).

We can also prove that the converse of the theorem is true at least if

P(D)

Theorem 8.

is homogeneous. Let

P(D)

be a homogeneous

tor with constant coefficients {x I ~(x) < 0 I

in

~n,

function defined near

In fact we have the following theorem.

defined on

where ~

~ (x)

satisfying

linear differential

~n

Let

~

be the domain

is a real valued real analytic grad x ~(x) # 0

on

Assume that there exists a characteristic boundary point i.e.,

x0 E ~

satisfying

characteristic hyperplane Ixl<x-x0,

Pm(grad x ~(X) Ix=x0 ) = 0,

through

grad X ?(X)[x=XO > = 01,

x 0,

N

The existence of a special null-solution

homogeneity of

P(D)

~L x0

. of

~).,

such that the

i.e.,

intersects

compact set for any compact neighborhood

theorem and we omit the details.

opera-

(~n_ ~ ) ~ N of

of

x 0.

P(D)

Then

in a P(D) ~ ( ~ )

proves this

We hope that the assumption on

will be redundant and that the characteristics

should be replaced by the bicharacteristics,

though we have not yet

proved them because of some technical difficulties. On the contrary, we have the following Theorem 9 as an affirmative answer to the global existence of real analytic solutions. Theorem 9.

Let the operator

P(D)

satisfy conditions

(I) and (2)

111

and let a relatively following (i0)

condition

compact (i0).

open set with smooth boundary

Then

Any bicharacteristic

P(D) ~ ( ~ )

curve of

P(D)

= ~(~)

satisfy the

holds.

intersects

~

in an open

interval. The proof of this theorem is just the same as that of Theorem 4. (Cf. Remark A after Theorem 5.) Since Theorem 9 seems to require too much information the global shape

of

Theorem I0.

~

Assume

Let a relatively

, we modify Theorem 9 as follows. the same conditions

compact open set

~

function

satisfying

~

both condition ~(D) ~ ( ~ ) (Ii)

# 0

on

.

~(x)

For any point

x

that for any bicharacteristic

in

~x I ?(x) ~ 0}

defined near ~

(Ii) below,

of

{ N jlj=l P ~

curve

b(x,~ ) ~ ( ~ - {x~) ~ Nj

neighborhood

as in Theorem 9.

satisfies then

holds.

There exists a family of open sets following:

P(D)

If the open set

(4) in Theorem 4 and condition

= ~(~)

(x, ~ )

on

have the form

for a real valued real analytic grad x ~(x)

concerning

which

satisfy the

we can find some b(x,~ )

of

P(D)

is connected,

where

j

such

through Nj

is a

x.

The proof of this theorem is similar to that of Theorem 4, so we omit the details. Remark.

As is remarked before Lemma 2, we can generalize Theorems

4, 9 and i0 for a wider class of linear differential constant

coefficients

not necessarily

satisfying

We omit the details here and refer to Kawai however

emphasize

operators with

conditions

(I) and

[5] and [6] for it.

the fact that one of the advantages

(2).

We

of hyperfunction

theory appears when one tries to state the theorems using conditions

on

112

the principal

part of

P(D)

only.

lower order terms is needed. in treating

overdetermined

Erratum in Kawai

[5].

Thus in Kawai

[5] no condition

This fact is sometimes

systems with constant

remarkably useful

coefficients.

Add the following assumption

There exists a neighborhood

V

of

~L

on

to Theorem i.

for which the following

hold: (i)

For any if

(x, ~ )

(K~+x) ~

(ii) For any

in

N = I(x, ~ ) e S * ~ n I x ~ ~ L ,

# ~ ,

(x, ~ )

in

then N,

if

V~(K~+x)

Pm ( ~ )

= 0~

C ~ U ~x~.

(-K~+x) ~ ~

# ~ , then

V~(-K~+x)

I 2.

Global Existence

Differential

Solutions

Equations with Real Analytic

The reasonings elementary

of Real Analytic

of

~i depends

want to treat operators with variable the arguments

of elementary

solutions

differential ficients tions.

of Kawai

except

V

P(x, D ) x

are real analytic

set

V

[i],

in

depends

unsatisfactory,

~n,

not

~n

of good But if we

then there appears

[2] show only local existence

in some trivial cases,

e.g. a linear

part being of constant

coef-

of lower order terms being entire func-

versions

coefficient

itself,

case we must content

of Theorems

i.e., we must consider all the problems

open set

P(D).

coefficients,

By this reason in the variable

ourselves with the semi-global present,

operator

operator with its principal

and the coefficients

Coefficients

on the global existence

solutions of the differential

a difficulty:

of Single Linear

4, 9 and I0 at

in subsets of a fixed

even if the coefficients

in a larger set than

V.

on the operator under consideration. hence we will not discuss

of

Of course the open Such results are

them any more here.

However

113

there is a case where the elementary solutions exist globally, hence all arguments in of Leray [I].

31 succeed: globally hyperbolic operators in the sense

(Cf. also Bruhat [I].)

If we combine our construction

of local elementary solutions and investigations of their properties developed in Kawai [1] with Leray's penetrating study of emissions, which are closely related to bicharacteristics, then we have the following Lemma ii.

(Concerning the definition of global hyperbolicity and

the related topics we refer to Leray [I] and Bruhat [I].

See also Kawai

[5] .) Le~na ii. .........

Assume

that the linear differential operator

P(x, D X )

is globally hyperbolic on real analytic complete Riemannian manifold Then we have an elementary solution

E(x, y)

for

(x, y) e V ~ V

V.

satis-

fying the following conditions: (12)

supp E(x, y) ¢ ~ (y),

(13)

S.S.E(x, y) C{(x,y; ~, ~) ~ S*(V~V)Ix=y, ~ = - ~ I U I ( x , y ; ~, ~ ) 6 S*(V~ V) I (x, ~ ) strip of

P(x, Dx)

and

where

(y,-~)

with

~(y)

denotes the emission of

are on the same bicharacteristic

x E g(y)}.

Thus we have a global elementary solution in this case. we can prove analogues of Theorems 4, 9 and I0. and refer to Kawai [5].

y.

Therefore

We omit the details

Of course the assumption of hyperbolicity also

allows us to treat the Cauchy problems for such operators both in the framework of real analytic functions and in that of hyperfunctions.

A

remarkable fact which appears in our treatment of Cauchy problems in the framework of hyperfunctions is firstly that bicharacteristics play no part when we decide the existence domain of solutions and secondly that they play their own essential role only when we decide the domains where the uniqueness of solutions holds.

About the details we also

114

refer to Kawai

3.

[5].

Global Existence of Real Analytic Solutions of Systems of Linear

Differential Equations with Constant Coefficients The investigations

of the problems

section are still progressing,

stated in the title of this

hence we cannot give the final theorems

but only sketch two methods which are expected to give complete results and, in fact, have given results in some special cases.

Since

we want to explain the main ideas and do not try to give complete arguments

in this section, we assume some additional conditions

cerning the algebraic order to

avoid

con-

structure of the systems under consideration

technical difficulties.

in

That is, in Theorem 12 we

assume that the system of compatibility conditions has one generator and in T h e o r e m 13 we assume that the system under consideration has only one unknown function.

We remark that some trivial cases which

can be treated by just the same method as developed omitted by these assumptions:

§ I may be

a typical example is a system whose

adjoint operator is an (over-)determined operators.

in

system of linear differential

But we hope the most typical features of the system of

linear differential

operators appear clearly even if we assume these

conditions. The first approach is the one concerning the existence of solutions in

~(K)

for compact set

ly due to Ehrenpreis of Theorem 5.

[i],

K

in

~n

[3] and is a direct extension of the proof

That is, it uses the pairing of

Serre's duality theorem.

This method is essential-

(~(K),

~K )

snd

Then it is easy to reduce the existence

theorem to the problem of support of solutions and we obtain the fol-

115

lowing: Theorem 12.

Denote by

M0

a system of linear differential

operators with constant coefficients and by gives its

compatibility

operator of

MI,

pact set in

~n

Then

conditions.

MI

Assume that

has only one unknown function.

the system which tMl, Let

satisfying the following conditions

ExtI(M0 ,

~(K)) = 0

holds,

i.e. if

P(D)

the adjoint K

be a com-

(14) and (15).

is a system r0

representing

M0,

the equation

P(D)u = f

has a solution in

~(K)

r1 for any

(14)

f E ~K)

satisfying the compatibility condition.

There exists a real valued real analytic function

?(x)

which is defined in a neighborhood of the convex hull of

K,

Ch K

and satisfies

(a) and

(b) (15)

grad x ~(x) # 0

The system for any

x0

tM I

in

Ch K - K .

is hyperbolic with respect to

satisfying

~(x0)

= t

with

grad X ? ( x ) I x = x 0

i < t ~ 2.

The proof of this theorem is obtained by the method of pie-nibbling due to Ehrenpreis

[i], [3].

By the way of the proof condition

(b)

can be weakened but we will not discuss it any more in this lecture. The second approach is concerning the existence of real analytic solutions on an open set as follows:

f~

and it can be summarized schematically

if we can solve the system of linear differential equa-

tions in the space of hyperfunctions, sheaf

~

and that of sheaf

C

then using the flabbiness of

we can solve the system in the space

of real analytic functions assuming some additional "convexity"

116

conditions

on the boundary of

~

.

We hope that the solvability

in

the space of hyperfunctions will be obtained under the least restrictive conditions

on the "convexity"

give us a complete result,

of

~.

and that this method will

though we have not arrived there.

that, for example, we need no "local convexisty" the system of linear differential if the space dimension

n

conditions

Note to solve

equations with constant coefficients

is equal to

2.

By this method we have the following theorem : T h e o r e m 13.

Consider an overdetermined

unknown function. in

~n

Let

be a relatively

Assume further that

i.e. the defining ideal with

A/~

where

Then we have j=l

~

of

the generators ~(~)

if

= 0.

of f. J

~

,

~n ]

with one

V

of

M0

is real

is simple characteristic,

is reduced.

A = ¢[ ~ i' "'''

Extl(M0 , ~ ( ~ ) )

is solvable in

V

M0

M0

compact convex open set

Assume that the characteristic variety

and non-singular.

M0

~-

system

Here we identify

and

~

is its ideal.

That is, denoting by Pj(D)u = fj

(j = I, ''', d)

satisfy the compatibility

conditions.

The proof of this theorem is given by a method analogous

to

that employed in the proof of T h e o r e m 4, if we take account of Komatsu's

result that

for any convex open set

Extl(M0 , ~ ( ~ ) ) ~

in

~n.

course these forms of presentations

= 0

holds for any

(Cf. Komatsu

M0

[i], [2].)

and Of

of the theorems are very unsatis-

117

factory from the aesthetical viewpoint.

In fact we have some recipes

for generalizing these results using the notion of the bicharacteristics concerning the overdetermined

systems, but we cannot make them

applicable at present since we have almost no results concerning the global existence of hyperfunction

solutions except for Komatsu's one

or those which can be easily deduced from it only by algebraic arguments.

Hence the present

at a later occasion.

speaker

wishes to return to these problems

Please give him enough time until then.

REFERENCES Andersson, K. G.: [i]

Propagation of analyticity of solutions of

partial differential

equations with constant coefficients,

Ark.

Mat., 8 (1971), 277-302. Bruhat, Y.: fold,

[I]

Hyperbolic partial differential

Proc. of Battelle Recontres,

equations on a mani-

Benjamin,

New York, 1968,

pp.84-I06. De Giorgi, E. and L. Cattabriga:

[I]

On the existence of analytic

solutions on the whole real plane of linear partial differential equations with constant coefficients, : [2]

Una dimonstrazione

analytiche

constanti,

[i]

L.:

[I]

Bol. Un. Mat. Ital., ~ (1971), 1015-1027.

Conditions

ential operators, Ehrenpreis,

diretta dell'esistenza di soluzioni

nel piano reale di equazioni a derivate partiali a

coefficienti Egorov, Yu. V.:

for the solvability of pseudo-differ-

Dokl. Akad. Nauk USSR, 187 (1969), 1232-1234.

Some applications

to several complex variables, Princeton,

to appear.

1957, pp.65-79.

of the theory of distributions

Seminar on Analytic Functions,

118

: [2]

Solutions of some problems of division IV,

Amer. J. Math.,

82 (1960), 522-588. ........ : [3]

Fourier Analysis in Several Complex Variables,

Wiley-

Interscience, New York, 1970. H~rmander, L.: [I]

Linear Partial Differential Operators,

Springer,

Berlin, 1963. : [2]

Uniqueness theorems and wave front sets for solutions of

linear differential equations with analytic coefficients,

Comm.

Pure Appl. Math., 24 (1971), 671-704. • [3]

On the existence and the regularity of solutions of linear

pseudo-differential equations,

Enseignement Math., 17 (1971),

99-163. John, F.: [i]

Plane Waves and Spherical Means Applied to Partial

Differential Equations, Kashiwara, M.: [I]

Interscience, New York, 1955.

On the flabbiness of sheaf

e

,

S~rikaiseki

Kenkyusho KSky~roku, No.l14, R.I.M.S. Kyoto Univ., 1970, pp.l-4 (in Japanese). Kashiwara, M. and T. Kawai:

[i]

theory of hyperfunctions,

Pseudo-differential operators in the Proc. Japan Acad., 46 (1970), 1130-

1134. ..... [2]

Ibid.

Kawsi, T.:

[i]

(II),

in preparation.

Construction of local elementary solutions for linear

partial differential operators with real analytic coefficients (I) - - T h e

case with real principal symbols - - ,

Publ. R.I.M.S.

Kyoto Univ., ! (1971), 363-397.

Its summary is given in Proc.

Japan Acad., 46 (1970), 912-916

and 47 (1971), 19-24.

: [2]

Construction of local elementary solutions for linear

119

partial differential operators with real analytic coefficients II. - - T h e M.S.

case with complex principal symbols

Kyoto Univ., ~ (1971), 399-424.

.....,

Publ. R.I.

Its summary is given in

Proc. Japan Acad., 47 (1971), 147-152. : [3]

A survey of the theory of linear (pseudo-) differential

equations from the view point of phase functions

existence,

regularity, effect of boundary conditions, transformations of operators, etc.

Reports of the Symposium on the Theory of Hyper-

functions and Differential Equations,

R.I.M.S. Kyoto Univ.,

1971, pp.84-92 (in Japanese). --

: [4]

On the global existence of real analytic solutions of

linear differential equations. I,

Proc. Japan Acad., 47 (1971),

537-540. : [5]

Ibid. II,

Ibid. (1971), 643-647.

: [6]

On the global existence of real analytic solutions of

linear differential equations (I), to appear, Komatsu, H.: [i]

(II), --.,

J. Math. Soc. Japan, 2'4 (1972)

in preparation.

Relative cohomology of sheaves of solutions of dif-

ferential equations,

S@minaire Lions-Schwartz, 1966,

Reprinted

in these Proceedings. : [2]

Resolutions by hyperfunctions of sheaves of solutions of

differential equations with constant coefficients,

Math. Ann.,

~76 (1968), 77-86. Leray, J.: [I]

Hyperbolic Partial Differential Equations,

Mimeographed

notes, Princeton, 1952. -

-

: [2]

Uniformisation de la solution du probl@me lin6aire

analytique de Cauchy pros de la vari@t6 qui porte les donn~es de

120 Cauchy (Probl~me de Cauchy I),

Bull. Soc. Math. France, 85

(1957), 389-429. Malgrange, B.: [i]

Existence et approximation des solutions des

~quations aux d@riv@es partielles et des @quations de convolution, Ann. Inst. Fourier

Grenoble, 6 (1955), 271-355.

Nirenberg, L. and F. Treves:

[i]

On local solvability of linear

partial differential equations, Part II:

Sufficient conditions,

Comm. Pure Appl. Math., 23 (1970), 459-501. Palamodov, V. P.: [i] Coefficients,

Linear Differential Operators with Constant

Springer, Berlin, 1970.

Translation from the

Russian original, which was published in 1967 by Nauka. Sato, M.: [i]

Theory of hyperfunctions II,

J. Fac. Sci. Univ. Tokyo,

Sect. I, 8 (1960), 387-437. : [2]

Hyperfunctions and partial differential equations,

Proc.

Intern. Conf. on Functional Analysis and Related Topics, Univ. of Tokyo Press, Tokyo, 1969, pp.91-94. --

: [3]

Structure of hyperfunctions,

Reports of the Katata Symp.

on algebraic geometry and hyperfunctions, 1969,

pp.4-1-30

(notes

by Kawai, in Japanese). --

: [4]

Structure of hyperfunctions,

S~gaku no Ayumi, 1-5 (1970),

9-72 (notes by Kashiwara, in Japanese). : [5]

Regularity of hyperfunction solutions of partial differen-

tial equations,

Proceedings of Nice Congress, ~, Gauthier-

Villars, Paris, 1971, pp.785-794. Sato, M., T. Kawai and M. Kashiwara:

[I]

equations in hyperfunction theory,

On pseudo-differential these proceedings.

Its sum-

mary is given in Proceedings of the Symposium on Partial Differ-

121

ential Equations held by A.M.S. at Berkeley, 1971.

Research Institute for Mathematical Sciences Kyoto University

FUNDAMENTAL PRINCIPLE AND EXTENSION OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS*)

By Akira KANEK0

0.

Introduction In [ 3 ] ~ [ 6 ]

we have studied a few problems on extension of real

analytic solutions of linear partial differential equations with constant coefficients.

On the other hand, we have announced in [7] a

theorem on representation of hyperfunction solutions of linear partial differential equations with constant coefficients by exponentialpolynomial solutions.

In this report we survey these results with

sketches of proofs and give an intuitive explanation of the common idea on which they are based. Shortly speaking, we treat "real analytic functions with compact supports".

We are going to give a precise mathematical exposition to

this queer statement in the sequel.

I.

Characterization of hyperfunctions and real analytic functions b x

local operators Hereafter

~

and respectively

~

denote the sheaf of germs

of real analytic functions and that of hyperfunctions on the real Euclidean space ~(~g)

~n

of dimension

and respectively

and respectively of

~(~) ~

n.

V C C n.

n

fg C R ,

denote the modules of sections of

on

of germs of holomorphic functions on on the open set

For an open set

.

(~ Cn

and

~(V)

denote the sheaf

and the module of its sections

In the theory of hyperfunctions we employ

*) Revised on March 12, 1972.

123

the following differential

operator of infinite order:

(i.i)

,

J(D) =

i ak Dk iki= 0

D = (DI,

ing

the

coefficients

growth

•

ak

are

"',

kl Dk = D 1

kn) ,

complex

numbers

Iklrlira Vlakl.k! Ikl~

= 0,

after Sato.

This name reflects

ficient condition

in order that

is a continuous J(D)

element of

: ~

......~.........~

~ [{0I],

is of the form local operator

J(D)~

Theorem

i.

can be expressed

.

on

f

where

~

of the operator

~ ~

defined by

and so are

f

J(D)

:

we see easily that any

with supports at the origin,

is the Dirac measure and

J(D)

is a

characterize

the hyperfunctions

and

functions. Let

~

C ~n

in the form

be an open set. u = J(D)f

Then every

with an elliptic function

f

on

u ~

~(~)

local operator ~

.

Here

J(D)

in the sense of hyperfunctions.

the analytic-hypoelliptic

local operator a local operator which has property

the above result corresponds

continuous

: (~

the hyperfunctions ,

coefficients)"

(1.2) is a necessary and suf-

In particular,

and an infinitely differentiable

u 6 ~,(~n)

follow-

n

(with constant

sheaf homomorphism,

We mean by an elliptic

Thus

J(D)

local operators

the real analytic

operates

the

defined above.

Conversely,

J(D)

satisfying

the local character

In fact, we can easily show that

and

'

kn ..D n ,

Ikl = k l + " ' ' + k

We call such an operator "a local operator

~akDkf

~-~x. J

j

condition

(1.2)

J(D).

D. = ~ '

k = (kl, where

"-', D n)

can be expressed function

f,

where

(i.e.

J(D)f ~ ~

implies

f 6 ~ ).

to the fact that every distribution

locally in the form A

u = ~Nf

is the Laplacian.

with a

Note that our

124

expression holds globally. Before going to the proof we briefly remember the definition of Fourier hyperfunctions.

For the details, see [II].

directional compactification of

open set

I C ~Rn

Dn

~R n "

infinite sphere on

~n

Let

Dn

which is obtained by adding the

is homeomorphic to the unit n-disk.

which contains the origin, we consider

a complex neighborhood of

be the

~)n. The space

q(D n)

For an

D n ~4q-[I

as

of Fourier hyper-

functions is defined as the relative cohomology group with support in Dn

of the sheaf

The stalk of

~

~

on

D n X N-[~I~Rn:

at a point in

(7~ = HDn(D n

Cn = N n X ~i'[nRn C D n X ~/~R n

consists

of usual germs of holomorphic functions and the stalk at a point x+~y

6 D n ~-~n

at infinity is defined as the inductive limit

<

I)

~,I where

(~i<~ × ~f~ I)

is the space of holomorphic functions in

~×~I

with the estimate if(z)l <_ C~eg~Re zl and

~i

~6 > 0 ,

,

B C 6 > 0

runs over the conic sets which contain the direction

interior;

I

runs over the usual neighborhoods of

y.

x

in the

Thus we have,

as in the case of usual hyperfunctions, n

~(Dn)

= ~(~n× ~/~l~Dn)/

~ ~(Dn ×E k=l Namely, every Fourier hyperfunction v E ~ n ) values of a family of

2n

called a defining function, quadrant if

k

~nX~Ik

holomorphic functions where

(= ~ x + ~ y ;

Fk

and has the estimate

is the sum of boundary F = {FI, "'', F2n},

is holomorphic in the k-th

y 6 I,

~jyj > 0, j = i, "'', n]

corresponds to the family of signatures

~. = ± i ) , 3

1 ~ { y j #0; j #k}).

~ = ( ~i'

"''' ~ n )'

125

(1.3)

~ f > 0,

aCK, g

~ K f_~ I k,

> 0

IFk(Z) i ~ CK,EeflRe zl

z ~ ~n~ilK

We can localize these arguments and obtain a flabby sheaf which agrees with

~

on

extended to an element of For an element

R n . Thus every element of

~n

e~f~<x'~>v(x)dx

on

~(IR n)

v E ~ ( D n)

can be

we can define the Fourier transform. (~v)(~)

=

by lifting the integral path into the complex

domain and using a defining function in the integrand. again a family of holomorphic functions

We thus obtain

"'',

G(~ ) = {GI(~),

G2n( ~)}

which satisfy (1.3), hence defines the Fourier hyperfunction Sketch of Proof of Theorem i. that

u G ~(~n).

because

¢

and let

G(~ )

small enough.

~)n

~ (Dn).

We can calculate it by the ciassical formula I

~

Since

Further we can extend

is also flabby.

Let

~v

~

u

~v.

is flabby we can assume

to an element

v 6 ~(Dn),

be the Fourier transform of

be the defining function of

~ v.

Now fix

v

~ > 0

Then we have

IG(~ ±~g--fg)l < C~e ~1~1

~ >0

~Cg > 0

An elementary consideration shows that this can be replaced by another estimate

IG(~ +~f~)l ~ Ce l~I/~(l~t)

(1.4) where

? (r)

is a positive function monotone increasing to infinity.

On the other hand we can find an elliptic local operator

J(D)

with

the following estimate from below

IJ(~ )l 2 Ce I¢1/@(1~1)

(1.5) (where

J(~)

,

is the Fourier transform of

for

J(D)~ ).

employ (1.6)

IIm ~I ! ~

~ + .. +~2 J(~ ) =-IT ( i + ) m=l (m 9 (m))2

,

In fact we can

126

with

~

modified a little.

(For the details of the calculation see

[i0] .) Now the family of holomorphic functions another Fourier hyperfunction (1.7)

&v

w:

defines

we have

= J(~)2w

.

Passing to the inverse Fourier transform, (I.8)

G( ~ )/J( ~)2

v = J(D)2~-lw

Thus it only remains to show that

(1.7) becomes

.

~-lw

is infinitely differentiable.

For this purpose we calculate the inverse Fourier transform along the path

~ +_ ~ f ~

(1.9)

F(x+~y)

.

We obtain a defining function

F(z)

of

10:y f 0

= s g n y I -'' s g n y n ~

5 -1w

:

~-i -°" 0-n sg

~ s g n Yn

e - ~ ( < x ,~ >-~G ( ~ + ~ g~)/J ( ~ + ~ ~ ) 2d~ i'"d~ n where the summation extends over ( O-I, .

. ) ,. ,. ~-n

~j

+-- I.

2n

combinations of signs

Because of the estimates

each term of the right hand side converges even when

~" =

(i .4) and (1.5) , y

~ 0

after

any (finite) times of differentiation with respect to

x.

assume without loss of the generality that

~) ~ c ~ I Y

some

~ > 0,

Thus

F(z)

-lw .

hence

~I /~ (~

lG(~+~¢)/J(~+~f~)21

(We can with

_~ C e x p ( - c I~;¥).)

defines an infinitely differentiable boundary value

f =

q .e .d.

Remark 2.

The regularized function

f

given above satisfies the

following growth condition (I.i0)

~f(x)~ ~_ Ce ~Lxi"

as is easily seen from (1.9).

This is rather strange, since there are

continuous functions with any big rate of growth, e.g., Theorem 3.

An element

u e ~ (~g)

only if for any local operator

J(D),

belongs to J(D)u

e ex.

~(~)

if and

is a continuous function

127

in

~.. Since we do not logically rely on it in the sequel, we omit the

proof and refer to [8], [i0].

(Chou [14] has given a proof

the theory of ultradistibutions

employing

and Bang's theorem.)

Now we can deduce the following psychological theorem which explains our main idea. Psychological theorem. function

u

K C Rn ,

the

If there would exist a real analytic

with its support contained in the compact convex set Fourier

transform

u

of

u

should

be an entire

function

with the estimate (1.11) where

I~(~)I HK(~)

Proof.

~ Cexp(-al~l+HK(~)),

= sup R e < x , ~ l ~ > x~K

K.

~C > 0,

is the supporting function of

For any local operator

function with support in

36>0,

J(D),

J(D)u

K.

is a continuous

Therefore by the Paley-Wiener theorem

we conclude that (1.12) Here

~J(~)~(~)l J(~)

G Cjexp(HK(~)),

is the Fourier transform of

Vj, ~ C j > 0.

J(D)~

and is an entire

function with the estimate (i.13)

IJ(~ )I ~ Ca eg[[~ ,

V6 > 0,

~Cg > 0.

(We call hereafter this type of estimate infra-exponential after Sato.) Now, on account of (1.5), the estimate (1.12) for a continuous function is equivalent to (I.ii) as is shown by the usual argument (see [6]). q.e.d. Note true

that

the

on a c c o u n t

trivial LindelWf

entire

converse

of Theorem functions

theorem.

of 3.

with

Therefore

the

above

We know i n estimate the

above

psychological fact

that

theorem there

(1.11),

by the

theorem

has

are

is

no n o n -

Phragm6n-

no u s e

also

though

128

logically 3

it is not wrong.

serve as a substitute

But we claim that in some sense Theorems

for the psychological

two theorems are our technical some open subset

V Ccn,

really exist holomorphic the condition

tool realizing

theorem. the idea.

or on some analytic variety

Thus these Note that on N C C n,

functions with such estimate.

i,

Thus

there

localizing

(I.II) in a certain way, we hope that we will be able to

give a more direct tool taking the role of the real analytic

functions

with compact supports.

2.

Theorems

of Hartogs'

We consider ~p(~\K)/~p(~) solutions

= 0

denotes

open neighborhoods

hold?

K in

Here

the sheaf of germs of

operator

is a convex compact

set and

p

p(D)

with con-

~

is one of its

For hyperfunction

solutions

~p(~\K)/~p(~)

is an overdetermined

system

solutions,

for single equations.

~p

linear differential

~n.

occurs only when

Theorem 4.

solutions

for which operator does

of the single

For real analytic

of real analytic

the following problem:

stant coefficients,

= 0

type on continuation

however,

The answer

~p(~\K)/~p(~)

(see [5],

[12]).

this problem is meaningful

even

is: = 0

if and only if

p

has no elliptic

factor. Sketch of Proof. one of its extension p(D)[u] in

K.

compact

belongs

Let

u 6 ~p(fh\K),

(we use here the flabbiness

to the space

We claim that support modulo

p(D)[u]

~ [K]

only modulo

p ( D ) ~ [K].

[u] E ~

of hyperfunctions

'',

with compact

We first note that the ambiguity occurs

of

is "a real analytic

p(D)~,(~n)

space of all hyperfunctions

[u]

and let

where

).

~(~)

be

Then

with supports

function with

6~,(~ n)

denotes

the

supports.

of the choice of the extension Therefore

if we permit the

129

ambiguity modulo In particular, is equal to

p(D) ~ , ( ~ n )

let

0

~

K,

be an infinitely differentiable

of

K.

Then

is a good extension

operator

J(D),

(2.1)

of

K,

and equal to

v = %u,

for our purpose.

In fact,

= J(D)p(D)~u

= p(D)J(D)~u

= p(D)lJ(D)u

+

J(D) % u J(D)p(D)[u]

Since the first term

~J(D)u

= ~

p(D)[J(D)%u-

when we apply Fourier

~J(D)u].

'

rood. p ( D ) ~ , ( R n ) .

can be given a precise mathematical

in

p(D) ~ , ( R n )

function "on

we obtain a column of holomorphic

vanishes

"on

p,

N(p)".

functions

operator

to

N(p)

meaning

Since the Fourier N(p )"

where

we obtain with

Precisely

speaking,

on the irreducible

which is locally in the image of a noetherian

(i.e. the restriction

function

set, Theorem 3 implies our claim.

is the variety of roots of the polynomial

no ambiguity a holomorphic

so that

'~

transform to this element.

transform of each element

N(p)

for any local

in the right hand side is a continuous

This queer assertion

nents of

outside a

I--FJ(~)(D)uD~% 6 8.. (R n)

= p(D)(%J(D)u)

with support in a fixed compact

N(p)

I

which is regarded as zero

I~ I>O~"

(2.2)

function which

we have

J(D)p(D)v

Here obviously

we can take the extension more freely.

in a neighborhood

larger neighborhood on

,

compooperator

composed with normal differ-

entiations up to a suitable order associated with

p;

us call such a function "a holomorphic

after Palamodov

p-function"

see [13]).

and call this transform the Gru§in transform after GruNin considered obtained

such transform for

in this way satisfies

C~-solutions.

have to do is thus to clarify on what those holomorphic

N(p)

[13]

[2] who first

The holomorphic

the growth condition

Let

(i. Ii).

p-function What we

there truly do not exist

functions with such estimates

except for the trivial

130

one.

It is fairly apparent that this is the case if and only if each

irreducible component of

N(p)

has a real infinite point.

The precise

proof is carried with the aid of a lemma which extends the classical Phragm@n-LindelUf

theorem, and we refer to [4].

(Note that the proof

in [4] is presented in the way that it is applied also for quasianalytic solutions.) The same method can be applied with only technical additional complication to the case of systems sion problem.

3.

([5]) or to other cases of exten-

See [6].

Representation

of hypeFfunction

solutions by exponential-polynomial

solutions Ehrenpreis'

theory on the Fundamental Principle and the represen-

tation of solutions by exponential-polynomial

solutions deeply depends

on the concept of duality of topological vector spaces.

Therefore,

at

first sight we may think that we cannot obtain the corresponding results in the case of hyperfunctions which do not form a topological vector space.

But in one sense we can say in spirit that

the dual space of

~,(~),

compact support contained in ~,(~) some

,

~(~)

is

the space of real analytic functions with ~L .

Since the Fourier transform of

is the space of entire functions with estimates K CC SL

~(~_)

is isomorphic to its dual.

Hahn-Banach theorem and the Riesz theorem,

(I.Ii) for

Therefore,

by the

its elements should be

represented by the inverse Fourier transforms of measures with the estimates (3.1)

~C

exp(-gl~l + H K ( ~ ) ) I d / ~ ( ~ ) n

I < ~

~g>0 '

VK ~ '

~ "

The inverse Fourier transform may be interpreted in the sense of hyper-

131

functions, namely, (2vr)-n ~

(3.2)

is a collection of, e.g., (3.3)

~(21~) -n ~I

-~f~(x, ~>

n e 2n

d/¢~ ( ~ )

holomorphic functions

e-~
on

~. Imz. ~ 0

'

Ir = { ~ j R e

~j < 0,

j

J

j = I, -'', n~,

j =i '

~=

"'',n}, ,

±i,

which define a hyperfunction as a cohomology class (the boundary value). Further, if the support of

is concentrated on the variety

N(p),

the inverse Fourier transform gives a hyperfunction solution of

p(D)u

and

u

(21~) -n -~

d/~

-~<x'~>d/~(~)

is the representation

N(p) of

u

by exponential-polynomial

solutions.

Now we show the outline of the way of justifying these heuristic arguments especially for the converse implication. convex open set. (3.4)

~.

~(~L) Now take

be a

~ J(D) ~ ( ~ ) . J:elliptic

is the space of infinitely differentiable functions on u E ~p(~)

arbitrarily, where

p(D)

differential operator with constant coefficients. u = J(D)v

~ C ~n

We employ the result of Theorem i:

~ (~L) =

where

Let

with some

Then the function

v ~ ~(~)

f = p(D)v

is a given single

By (3.4) we have

and some elliptic local operator

J.

satisfies

J(D)f = J(D)p(D)v = p(D)J(D)v = p(D)u = 0 . Therefore p(D)v = 0.

f

is real analytic.

We want to modify

v

so as to satisfy

For that purpose we quote the following lemma whose proof

is given in [3], Corollary 14 and also in [i0]. Lemma 5.

Assume that no irreducible component of

Then the following system of equations

p

divides

J.

132

p (D)u = f J(D)u = g has a solution J(D)f - p ( D ) g

u E ~(~)

for any

i shows at once that we can choose

to satisfy the assumption p(D)w = f,

is real analytic u = J(D)v',

p(D)v'

it follows

N(px)

v

(1.6).

f = p(D)v.

The function

thus we have succeeded

for

v'.

Since

= k~_I

~N(p~)

v

Thus we

v' = v -w,

is a C~-solution

w

we have

in replacing

[i] and Palamodov

d~(- ~i'l x) e

are the irreducible

is the transpose meanings

= 0,

so as

See formula

Putting

is elliptic.

J

v.

of

[13] that

p(D)u v

has

by measure:

v(x)

where

J

where

from Eherenpreis

a representation

(3.5)

of this lemma.

J(D)w = 0,

since

Hereafter we write = 0,

satisfying

= 0.

The proof of Theorem

can solve

f, g E ~ ( ~ L )

of the associated

of these words see [13]).

-f~i<x,~>d/~.~(~),

components

of

noetherian

N(p)

operator

The c o l u m n s

d~(~

and of )

d = ~d~(D~) 1 p

(for the

of measures

satisfy

I

N(pk) idk(_ ~__~ x)l. (I + l~l) k exp (HK(

(3.6)

for Now apply functions.

J(D)

~k

~ K (C ~g •

to the both sides of (3.5) in the sense of hyper-

We have

j(D)dx(-~i~x) e - ~ < x , ~ with some matrix derivatives

> 0,

~ )) [d#tx( ~ )I <

of

Jl ( ~ ) J ( ~ ).

Every

e -~f~<x,~>

whose elements are linear combinations Hence we obtain for

form (3.5) with the measures Theorem 6:

> = dx(-~X)Jl(~)

Jl(~)d/~(~).

u ~ ~p(~.)

u

an expression

of

of the

Thus we have

can be expressed

in the form (3.5)

133

with

d/u-A satisfying

(3.7)

~I

Id~(-~x)l for

~

exp(-&l ~I + H K ( ~ ) ) ~ d ~ A ( ~ ) l > O,

< ~

'

~ K CC ~ ,

where, as remarked above, the integral is considered in the sense of hyperfunctions. of

Conversely,

(f~). P The converse is clear.

every such expression presents an element

~

The detailed proof of this theorem is

given in [I0] for the case of general systems.

REFERENCES

[I]

Ehrenpreis, L.,

Fourier Analysis in Several Complex Variables,

Wiley-Interscience, [2]

GruNin, V. V.,

1970.

On solutions with isolated singularities for

partial differential equations with constant coefficients Russian), [3]

Trudy Moskov. Mat. Ob§~., 15 (1966), 295-315.

Kaneko, A.,

On isolated singularities of solutions of partial

differential equations with constant coefficients S~rikaiseki-kenky~sho [4]

,

(in

KSky~roku,

(in Japanese),

108 (1971), 72-83.

On continuation of regular solutions of partial differen-

tial equations to compact convex sets,

J. Fac. Sci. Univ. Tokyo

Sec. IA, 17 (1970), 567-580. [5]

,

Ibid. II,

ibid. 18 (1971), 415-433.

[6]

,

Theorems on the extension of solutions,

Lecture at the

Symposium on Hyperfunctions and Analytic Functionals at RIMS, September, [7]

,

1971 (to appear in Surikaiseki-kenky~sho

On Fundamental Principle

kenky~sho KSky~roku,

(in Japanese),

114 (1971), 82-104.

KSky~roku).

Surikaiseki-

134

[8]

Kaneko, A.,

A new characterization of real analytic functions,

Proc. Japan Acad., 47 (1971), 774-775. [9]

,

On the representation of hyperfunctions by measures (in

Japanese),

Proceedings of the Symposium on Hyperfunctions at

RIMS, March, 1971 (to appear in S~rikaiseki-kenky~sho Koky~roku). [I0]

,

Representation of hyperfunctions by measures and some

of its applications,

submitted to J. Fac. Sci. Univ. Tokyo, Sec.

IA. [II] Kawai, T.,

On the theory of Fourier hyperfunctions and its

applications to partial differential equations with constant coefficients,

J. Fac. Sci. Univ. Tokyo, Sec. IA, 17 (1970), 467-

517. [12] Komatsu, H.,

Relative cohomology of sheaves of solutions of dif-

ferential equations,

S@minaire Lions-Schwartz, 1966/7,

reprinted

in these proceedings. [13] Palamodov, V. P.,

Linear Differential Operators with Constant

Coefficients (in Russian), Moskva, Nauka, 1967. [14] Chou, C. C.,

La Transformation de Fourier Complexe et l'Equation

de Convolutions,

Th~se de Doctorat d'Etat, Univ. Nice, 1969-70.

Department of Mathematics University of Tokyo Hongo, Tokyo

ON ABSTRACT CAUCHY PROBLEMS IN THE SENSE OF HYPERFUNCTION

By sunao ~UCHI

This paper is concerned with hyperfunction solutions of the abstract Cauchy problem i

du(t) = Au(t) dt

(A.C.P.) u(0) = a where a E

A

,

is a closed linear operator in a complex Banach space

X

and

X.

Generalized solutions of A.C.P. have been studied by many people. Especially, since Lions [6] introduced the notion of distribution semigroup, there have been many works concerning distribution solutions of A.C.P.

(Barbu [i], Chazarain [2], Da Prato-Mosco [3], Fujiwara [4],

Ushijima [Ii],

[12], [13] and Yoshinaga [14], [15] etc.).

In this paper we will consider solutions more general than distribution solutions, that is, hyperfunction solutions.

We investigate

conditions for existence, uniqueness and smoothness of hyperfunction solutions of A.C.P.

We characterize these conditions by means of

properties of the resolvent of In

~I

A.

we give elementary properties of hyperfunctions of one

variable with values in a complex Banach space. be used in the later sections.

These properties will

Hyperfunctions introduced by Sato [8],

[9] are, roughly speaking, defined as sums of boundary values of holomorphic functions.

As mentioned above, they are more general than

Schwartz' distributions.

Hence, we expect that if A.C.P. is well-posed

in the sense of distribution, then it is also well-posed in the sense

136

of hyperfunction, In

which will be shown indeed in

§2.

~ 2 we shall give the definition of well-posedness

in the sense

of hyperfunction and the condition for a closed linear operator be well-posed In

A

to

in this sense.

~ 3 we discuss

the analyticity

of the fundamental

solution of

A.C.P. In

~ 4 we characterize

solutions

of A.C.P.

the operator

is holomorphic

A

for which the fundamental

in a sector in the complex plane.

Many of the results of this paper have been announced in Ouchi

i.

Hyperfunctions with values

in a Banach space

In the later sections we shall use hyperfunctions with values in a Banach space. the norm ~(~ I

II " {IE.

, E)

Let

SL

(i.i)

~.

set.

f ~

f = [~(z)],

and

E)

E) =

~(z) e

Consider the space ~ . on

Let I

to

I

'

containing

~ ( D - I, E)

I

as a closed

ig denoted by

is called a defining function of

For E-valued hyperfunctions results

C.

functions defined on

O ( D - I, E) ~(D, E)

defined by ~(z)

be a complex Banach space with

space

is a complex neighborhood of ~(I,

of one variable

We define an E-valued hyperfunction

~(I, D

E

be an open set in

be an element of the quotient

where

Let

of all E-valued holomorphic

be an open set in

[7].

f.

of one variable we can prove the

similar to the case of scalar hyperfunctions.

We summarize

elementary results which we shall use in the following sections. (1.2)

~(I,

(1.3)

Let

E) II

does not depend on the complex neighborhood and

12

the canonical restriction

be open sets in IR ~ 12'

such that

D

I I ~ 12 .

of

I.

Then

137

~12 (Ii, E) can be defined, and

~(I,

>

(1.4)

of

The sheaf

= ~(I,

E)

(1.5)

~(E)

~(E)

~ .II

rK(l,

I

~(E)),

to

of

the support of

f

of

= ~(~,

f ~ ~{a~(~,

Let

f(z)

f

~K(I,

E),

be the set of f = [f(z)] E

~(z) E ~ ( D - K ,

is contained in

E)

K.

~(t -a),

~

E),

More precisely,

of

f(z).

is concentrated on

t = a,

can be represented as an infinite sum t = a :

means the Dirac measure at

k-th derivative of

If

then

f = ~ s(n)(t -a) @ e , n=0 n ~ (t -a)

such that the

f.

~(E))

K.

of the Dirac measure and its derivatives at

where

~(E))

the notion of support can be defined.

f E ~(~,

~(E)),

f 6 ~(I,

E)

coincides with the singularities

If the support of

that is,

I.

~ (I, ~(E))

~f~ (I, E).

coincides with

f~(z) E ~ ( D - I ,

that is, the singularities

(1.6)

I

with support contained in where

The totality

E ~(~, ~(E))

be a closed set in

sections on

The sheaf gener-

is flabby, that is, for every

For E-valued hyperfunctions K

~(E).

coincides with

there exists

restriction

Let

I

E) ,

E)'s determine a presheaf.

ated by this presheaf is denoted by of sections on

~(12,

e

E E n

'

t = a,

tensor product, and

~(k)(t -a) e

the

satisfy

n

n! lira ~n. llen[IE = 0 n~ (1.7)

E-valued Schwartz distributions are contained in E-valued hyper-

functions. These results can be proved in a way analogous to Sato [8]. case of one variable is simpler than that of many variables. prove these results with the aid of Runge's approximation E-valued holomorphic to Komatsu

functions.

[5], Sato [8],

The

We can

theorem for

For hyperfunctions we refer the reader

[9] and Schapira

[i0].

138

2.

Existence and uniqueness Let

to

F,

L(E, F) where

denoted by

of hyperfunction

solutions

be the totality of bounded linear operators from

E

and

II • lie

F

are complex Banach spaces whose norms are

and

I[ " IIF

respectively.

L(E, F)

is considered

to be a complex Banach space with the operator norm denoted by The space Let X,

L(E, E) X

E

is written

L(E)

for short.

be a complex Banach space.

we denote its domain by

D(A).

II" IIE~F.

For any linear operator

A

For a closed linear operator

in

A,

its domain becomes a complex Banach space with the graph norm, which we denote by

[D(A)].

The resolvent set

f(A)

of

A

is defined as

usual: ~(A)

= ~XE

C ;

(X-A)'IE

Now we define the well-posedness

L(X)}.

of A.C.P.

in the sense of hyper-

function. Definition 2.1. A

Let

A

be a closed linear operator in

is said to be well-posed for the abstract Cauchy problem

t = 0

in the sense of hyperfunction

exists

T E

~(~,

L(X,

[D(A)]))

(~)

support of

(~)

(~(1)(t) ~ I -

(well-posed,

I

convolution and Throughout

~

(A.C.P.) at

for short), if there

T C [0, ~ ) , f(t)~A)*r

is the identity mapping of

are the identities on

Then

satisfying the following conditions :

= f(t)~l

T * (S(1)(t) ® I - S(t) ® A ) where

X.

X

and on

X,

= ~(t) @I[D(A)] [D(A)]

[D(A)]

to

X,

IX

respectively,

and and

I[D(A)] *

means

tensor product.

this paper, we shall call

T

in Definition I.I a hyper-

function fundamental solution. From Definition i.i we deduce the following proposition.

139

Proposition I.

If a closed linear operator

then the fundamental solution Proof. of

T

T

A

is unique in

is well-posed,

~(~,

L(X, [D(A)])).

This result easily follows from the facts that the support

is contained in

solution,

[0, ~ )

and that

T

is a two-sided fundamental

q.e.d.

Now we give a criterion for the existence of the hyperfunction fundamental solution of A.C.P. Theorem 2.

Let

A

be a closed linear operator in

X.

A

is well-

posed if and only if the following conditions hold: (i) any

~

For any

6 > 0

there exists a constant

Kg

such that for

in the set

(2.i)

~-~&= ~

(~-A)-I~

L(X)

(ii)

; ReX

Z

$~[

+K E I

exists.

For any

g > 0

there exists

Cg

I~(~ - A) -i IIX~X -- C ~ e x p ( ~

(2.2) holds for

~ ~ ~

such that the estimate

~ ~I)

.

To prove Theorem 2, we give a lemma. Lemma 3.

Let

E

be a Banach space and

hyperfunction with support in the Laplace transform of

f

[a, b] : f E

f = [~(z)]

~[a,b](~, ~(E)).

< f, e x p ( - A t ) > =

where

is a curve encircling the interval > 0

-

there is

f(z)

C~

exp(-~z)

dz

,

[a, b]

counter clockwise.

such that the following estimate

holds: (2.4) where

i~l[ E _~ C H(u)

=

sup

e x p ( ~ l ~ i + H ( R e ~k)) ,

-(ux).

xe[a,b] Proof.

By (1.5),

Define

by

(2.3)

Then, for any

is an E-valued

f(z) E ~ ( ¢ - [ a ,

b], E).

Hence (2.3) is

140 independent of the choice of keeps a distance of IIF(z)IIE~_ M~

~c

~ .

Set

~ =

from the interval

for some

M~c>0 ,

and

~

,

[a, b].

where the curve On the curve

l-)kzl ~--K.I)NI+H(Re~).

~ ~K,

Hence

~(z)exp(-Xz)dzll E ~_ M~ ~exp(~l)~l+H(Re)~)) ~dz~ ~_ C~exp(~I)kl+H(Re)O~ q.e.d. Proof of Theorem 2.

Necessity.

the fundamental solution in hyperfunctions ~[O,l](m,

By

(~)

~(~,

Let L(X,

A

be well-posed and

[D(A)])).

be

Since the sheaf of

is flabby, there exists a hyperfunction

~(L(X, [D(A)])))

T

T1

such that

T1 = T

for

t < 1

T1 = 0

for

t > I.

in Definition I.i and the property of

TI,

the support of

hyperfunction S 1 = (~(1)(t) @ I - ~ ( t ) ~ A ) * T is concentrated on

t = 0

and

i.

1 E ~ (m, L(X))

Thus, from (1.6) and ( ~ )

in Defini-

tion i.I, we have

--~

S 1 = ~ (t) ~ IX + ~ ~ n=0 Since

A

n

(n)

(t-I) ~ An,

where

An E L(X).

satisfy n

lira ~nIllAnl[X~X = 0

(by (1.6)),

n-~

we have for any Hence for any

~ > 0 £ > 0

oo AnAn

(2.5)

lie

n=O Let

Tl(Z)

tt X-~X

n,.IIAnlIX~X<__ ~ n there is ~o

~ ~

where


n.

Mg such that n

Ix[ [tAn[lX~x ~- M ¢ e x p ( ~ t ~ [ ) .

n=0

be a defining function of

(~(C - [0, I] , L(X, [D(A)])). (2.6)

for sufficiently large

T I.

By (1.5),

~l(Z)

Consider the Laplace transform of

exp(- ~t)>x = - ~ ~

(z) exp(- A z ) x d z

is a curve encircling the interval

[0, i].

for

x ~ X

TI

14I The Laplace (2.7)

transform of

(X-A)KTI,

T1

satisfies

exp(-At)>x

~kn exp(- )~) A x

= x+~

n

n=O If

ReX

~glJkJ

,

+log(2Mg)

Hence a right inverse

(2.5).

and by Lemma 3

~

then

~I

JJn~=0)knexp(-~)AnJIx~x

RA

of

(~-A)

_ ~

by

exists on that domain

we have the estimate :

ljR JlX_~[D(A)]<_ 2 II<Ti' exp ( - X t)> Jlx*[D (A) ]

~_ C ~ e x p ( m l A ~ )

,

f o r any

~c~O.

Similarly we can get a left inverse on a similar domain,

considering

$2 = T I , ( ~ ( 1 ) ( t ) @ I - f(t) @ A). Therefore, and

C~

for every

E > 0,

such that resolvents

Z6 and t h a t

of

K > 0, A

First

~£+

: ~

h o l d s on

&) ~ f ( A ) .

Set

Define the path in the = 0 ,

for

: ~ = 0 ,

~

=

~+i~

for

,

for

6- >__ N~

for

o~ ~_ N&.

0~_K ff'~_ N£

~7 = - ~1q ( d - - K E ) 2

_ 6 2 ¢-2-

,

Put

"•+•(z)

=

I

j~

?~---~ 2

e

Xz

(A-A)-IdA,

and

+

1

~

Tg-(z) = 271----i

e

Xz( ~ -

A) -i

d)~.

Now we note the estimate

~Ie~Z (X - A)-IJI X - + X - C ~ e x p ( 0 ~ x - T Y +

and

)~-plane as follows :

~K__ 0-G N~

= [1 ~ (4-- Kg)2 _~2~.2

(2.8)

Kg

; Re fk Z gl~,j + K £

fix a real

(0 < g < 1 ) .

we can find constants

exist on the domain

IJ()k-A)-lJ{x~x ~ C~exp(~)kl)

Sufficiency. K~ N$ = 1---~

= {~

and

~(I°'I+I~I))

(z = x + i y ) '

142

for any

~ > 0

and

X E ~&.

On the curve

~x - ~ y + ~ ( l ~ l + l ~ l ) for sufficiently large the function Similarly ~£ T+(z)

T+(z)

T~(z)

~

.

~

~£' T+(z)

and

conunon definition domain, a holomorphic function

T~(z)

T~
x-y+~

define, as

~z ~ C; Re z < 0 or

Im z < 0}).

Thus we have an

dT+ (z) dz ~6 dT+_(z) dz

~_(z)

Differentiate

or

6

varies,

Im z > 0 }

It can be shown

are equal on the L(X, [D(A)])-valued which is the holo-

(z),

and we get

e~z

--!1

= AT+_~zj + 2 7~i - -z i

x

on

X,

and

~+£ ~z - T (z)A+ -i e I on [D(A)] . 2TCi z [D(A)] -1 -e~z $(t) @ IX = [2-~-~lZ IX] = [ ~ IX] and ~(t) @ I [D(A)] _e0~z ] T = [~(z)] = [~e ~(z)] is a hyperI[D(A)]] = [2--~--~izI[D(A)] ,

Since = [

function fundamental solution, Remark 2.1. that

and

defined on C - [0, ~ )

e c~

As

(resp. ~£(z))_

half-plane

T±(z).

Re z .

are equal on their

~+(z)

morphic extension of

+~)~

~z 6 C; Im z > i _ ~ $ 2

again by Cauchy's theorem that

T(z)

-

z ~ ¢; Im z < l_@~g2 Re z .

~z 6 C; Re z < 0

holomorphic function

,

(resp. ~£~z))_

on

{z 6 ¢; Re z < 0}.

we have

is an arbitrary positive number, so

is holomorphic in

on

6

is holomorphic in

(resp. ~£(z))_

(resp. ~_(z)

-~ (~

~E+

C~

In the proof of Theorem 2, we do not use the fact

in (2.2) is independent of

following condition (ii)' (ii)'

q.e.d.

For any

~ e ~&

Theorem 2'

So we can substitute the

for (ii) in Theorem 2:

~ > 0,

there exists

tt(X- A)-lllx~x holds for

6 .

~ cg,

CE,~

such that the estimate

exp(.,<-~ X l )

. Let

A

be a closed linear operator in

X.

A

is

143

well-posed if and only if the following condition holds: For any (i)

(X-A)

~ > 0

for any

-1

E L(X)

there exist constants

~k

and

such that

Cg

in the set

exists, and

the estimate

(ii)

Kg

II(~ - A)-III

holds for

~_ Cg exp( g [A I)

X~X

7 .

Proof.

Let the condition in Theorem 2' hold.

arbitrary positive number.

If

~ Z g,

Let

~ --

then for

~ e ~

be an ;

~ ( ~ - A)'IIIx_~x ~ C a exp( ~ ~X ~) holds.

Let

& >~.

--,e ~ --,c ~

~_~g--,= ( ~

Since

a ~)-, U ( ~ 'g a >~ic) -

is compact, there exists a constant

C6,~

and

such that

II(~-A)'IIIx~x ~ C ~,~ exp(~IX~) . Thus Theorem 2'

follows from Remark 2.1.

Remark 2.2.

q.e.d.

Comparing the result due to Chazarain [2] with

Theorem 2, we find that an operator which is well-posed in the sense of distribution is also well-posed in the sense of hyperfunction. Proposition 4.

Let

A

be a closed linear operator in

X,

which

satisfies the following condition (Ch). For any (i)

g>

for any ~=

the resolvent (ii)

0

there exist constants ~

in the set

~X

(~-A)

; ReX -I

E

Z

B (i)

g llm X ~ +

L(X)

Kg}

exists, and

the estimate

i1(k- A)-llix~x Let

Kg and M£ such that

~

be a linear operator in D(B) ~ D(A)

,

Me

Re A X

holds for

~ e ~g.

satisfying the following conditions:

I44 (ii)

For any

A + B

is well-posed

Proof.

(2.9)

C~ > 0

for

and satisfies

such that

x 6 D(A).

the above condition

(Ch).

Let us note the relation (A-A-B)

From

there exists

I~Bxltx ~ ~ ttAxll X + C~ttxt] X

(2.9) Then

> 0,

= {I - B ( A - A ) - I I ( A - A )

for

X e A e.

we have

liB( A - A) -lxllx ~ ~ ]IA(X - A ) - l x I I x + C~ ]l( A - A) -lxt] X . Since

]IA ('K -

A)-lx]] X

~ llxllx + ME. e--'--~ R I..kl

ilXllx

'

we have

IIB(~-A)-Ixl[x -m g - 2M[ "

Put

If

ReMR)~ ( ~

]AI+C L) ~ ]iX]Ix

Re A --> £ ] , k ] + 2M[C~

I

lIB(A - A)-II~x_~x ~ ~ •

Therefore,

max (Kg, 2M~C~) ,

(]<-A -B)

then

if -i

@

then on that domain

Re )k ~

£ L(X)

g ]AI

+K[

Kg

, where

exists and we have the

estimate II(4

.

A - B)-IIIX~ x

The statement of the proposition

3.

Analyticity

follows

A

is a continuously

R + = (0, ~ ) . (3.1) Namely,

the analyticity

in

the uniquely determined hyperfunction T

from Theorem 2.

q .e .d.

of hyperfunction

solutions.

Let a closed linear operator

that

2Me ~e

of fundamenta! ' solutions

In this section we shall consider fundamental

L_

Then

d--Ttx = ATtx dt the following

for any

be well-posed

fundamental

differentiable

TtX C [D(A)]

X

L(X)-valued

for every x 6 X

are equivalent:

solution.

and

t > 0

and

t > 0.

be

Suppose

function and

T

Tt

on

145

(i)

T

is a continuously differentiable L(X)-valued

function on

R+ .

(ii)

T

is a continuously differentiable

function on

T

is real analytic on

L(X)-valued function

Tt

(t > 0).

Theorem 5.

Let

A

R÷,

if

~

of

R+

where

by

Tz

(z 6 ~).

T

X.

Then,

A

is real analytic on

if and only if the following conditions hold:

For any

0 ~

is a real analytic

be a closed linear operator in

is well-posed and its fundamental solution R + = (0, 00)

T

We denote the analytic extension of

to some complex neighborhood

(i)

[D(A)])-valued

R+.

We call

Tt

L(X,

~

6 > O,

~ Lg

(~ -A) -I 6 L(X)

there exist constants

for some

L > 0

exists for any

K&

and

independent of

~

~g>

6

,

O,

such that

in the domain

(3.2) (ii) for

Ii(X -A)-Iiix_~x ~ C L e x p ( g j R e A i + ~ g i l m A i )

~ e ~g. Proof.

T = IT(z)] function on

Necessity. corresponding R+,

such that Taylor's

some

to

that is,

valued analytic function.

{z = t + i s ;

Let the hyperfunction A

~z-g{ < 2 ~ I

be an L(X)-valued real analytic

T = T t = [~(z)], Then for any

expansion of .

fundamental solution

Tt

at

where

~ > O, t = g

We may assume that

Tt

is an L(X)-

there is

~L > 0

converges on

R E=

0 <

for

~& ~ L ~

L. Since

Now restrict

supp T C [0, ~ ) , T(z)

T(z) e

(>(¢ - [0, ~ ) ,

on the upper half plane

and the lower half plane

¢_ = {z = t + i s ;

L(X, [D(A)])).

¢+ = {z = t + i s ;

s < 0 } respectively,

set

T+(z) = ~/ T(z) I¢+

s > 0

(the restriction on

¢+),

and

146

~_(Z) = ~(Z)l¢ From the assumption that that Taylor's expansion of

(the restriction on

T = Tt

Tt

at

¢_).

is real analytic on

t = 6

converges on

~+ ~g

and , T'~+(z)

can be extended holomorphically across the positive real axis. denote the holomorphic extension of

T'~+(z) again by

~+(z)

is holomorphic on

~_(z)

as a holomorphic function defined on

~+(z).

Let us Hence,

¢+ U ~j ~ £ . Similarly, we can extend T_(z) g>0 holomorphically across the positive real axis. Therefore, we can regard

is holomorphic on

¢ -[0,

~),

extensible to the half plane

T+(z)

and

~ z = t+is;

¢_ U ~ ~g. 6>0 T_(z)

t < 01,

Since

~(z)

are analytically and coincides with

each other on that domain. Thus we get a two-valued holomorphic function on which is the holomorphic extension of also by ~(z)

~(z).

T(z).

=

U ~, ~>0

We denote this extension

In this proof, we shall essentially use the fact that

is a two-valued holomorphic function on Define the paths

~£± =

= ~+(~)

~_

~

= ~g_(~)

~£~(~)

as follows (fig. i):

: (1-2/4._) & + i ~

:

/)_ .

0 ig~_l,

- 6 + i(3-27¢) ~£

I ! /~-2,

(2/~-5) g - i ~ g

2 ~_g<_3,

E + i(2~-7)~

3 ~_4.

E +i(2K-l)~

0 ~_~_I,

(3-2g) g + i ~ e

i~-2,

-g + i(5-2~)fg

2 L_~-3,

(2~-7)e -if e

3~g~4.

147 s

I

i\\

.....

-~-ig£

I ~

~£-i&\\

(Fig. I) Now set R+(A)x = -/r~+ e -AzT(z)xdz

(3.3)

for

x e X.

Then, we have ()% - A)R+()~)x = - ( ~ - A);p£+ e-AZ~(z)x dz

(3.4)

I

r~+

+

= e-X(8+iS~){~( ~ + i ~8) -~(( g+ige)e2~i)} x

-

= e

;

e-XZ~(z)-AT(z)}x

dz

q+

-&(g +iga )T

I

;_

E+iggx+2~l

e -Xz

_j~

z

xdz

+ = e-X(g+i2~)T6+isax+x T£+i~g ={~(~+i;6> -~<<£+i;£)e2~i)}.

where

, Thus for any

)~ in the

domain

~+

=~ )%; gRe ~ >_ ~&(Im~)+K6+},

K~+ = log 2{ITE+ig6~lX_~X,

we have ~e" X(£+i~ which implies

()k-A)-i 6 L(X)

< I )T£+i/£II X-~X -- ~ ' exists on

~£+

and

[l(~k- A) -lllx_~x _< 2]IR+( A )I[X_+X _~ C&' exp(& {ReA~+ffgllm~]) for

>~ E ~£+. Next consider

148

(3.6)

R_(X)

= f-I

e-XZT(z)x~ dz .

Discussing as above, we find that

(X-A)

; eRek

~g_ ={A

-I

~ L(X)

exists on

Z - gg(Im A) +K8_ }

and If(X -A) for

-lllx~x <-- 211R_()~)J]X_~X --~C6" exp(elRe AI+ £& llmAl)

X e ~ e_. Hence we conclude that for any

Cg

such that

( X - A ) -I 6 L(X) ~g={)~

and for

~

~e

g > 0

there exist

~[ , K&

and

exists on the domain

; gRe}k

>_ S g l i m ) ~ + K g }

the estimate

I~(X-A)-II~x_+x ~_ C&exp( & I R e ~ l +

,~&tlm A 1)

holds. Sufficiency. ( ~ ( ¢ - [0,

~0 ),

Set for S±(z)x

-

Obviously

L(X,

[D(A)]))

A

is well-posed.

Define

T(z) e

just in the proof of Theorem 2 (see (2.8)).

g > 0 2mi

X -A)-ix dX

-

2~i

K_i

e

-A) - i d X

,

where

and

tan @ = - $g

+ L E : &(Re X) = - [ g ( I m A ) + K & ,

Im A ~_ O,

Lg : 6(Re A) = fg(Im • ) + K 6 ,

Im ]X ~_ 0,

(~<

@ < ~).

It is easy to check that

S+(z)

is

holomorphic on the domain

& A6+ = ~ Z and

Sg(z)_

= {z = t+is;

S+(z) (resp. S_(z))

T (z))

S > ---~-t+2~& }

is holomorphic on the domain A~

and

= t+is ;

s < %t+2£~I

,

is the holomorphic extension of

defined in the proof of Theorem 2.

T+(z) (resp.

Since Taylor's expansion

149

of

~+g(z)

T+(z)

at

z = 3S

converges in

is holoraorphic in

~ g = Iz = t + i s ;

~z- 36J < Sg ~ ,

U ~.) ~ g . Similarly £>0 ~_(z) is holomorphic in ~z; Ira z = s < 0 I U U ~ 6 , r~'+(z) (resp. 6>0 _(z)), the restriction on ~z; !m z = s > 0 I (resp. ~ z; Ira z = s 0 )

z; Im z = s > 0

of the defining function

T(z)

of the fundamental solution

is extensible across the positive real axis, so on

R+

is analytic

= (0, 0o) .

In fact,

T = [T(z)] = T t

Tt = lira ~0 {~+(t+iq)

(3.7)

T = [~(z)]

T

= ~&0 lira ~ I

is represented by the formula

-~

(t-iq)}

~ L E + e %(t+i~) (#~- A)-id~ - ~

e A(t-i~) (~-A) "idA} £-

_

for

i [ eAt(x _ A)-I d2~, 2 ~ i JL6+U(-L£_)

t > 2g . Remark 3.1.

q.e.d. The criterion which corresponds to Theorem 4 in the

case of distribution solutions was obtained by Barbu [I].

.

Holomorphic hyperfunction fundamental solutions Definition 4.1.

posed.

Let a closed linear operator

The hyperfunction fundamental solution

T

A

in

X

be well-

is called holomorphic

if it satisfies the condition: T

is an L(X)-valued function

Tt

which is holomorphically

extensible to the sector A~ =

z; larg z~
Holomorphic fundamental solution

Remark 4.1. corresponds tion

The n o t i o n

to that

serai-groups.

T = Tt

,

< ~

.

is denoted by

of holoraorphic

of holoraorphic

0
fundamental

serai-groups

Tz

for

z E

solutions

or holoraorphic

distribu-

150 Theorem 6.

A closed linear operator

fundamental solution {z; larg z~ < ~

,

< ~

is well-posed and its

is holomorphic in the sector

T = [~(z)] 0 <~

A

if and only if

A

~=

satisfies the fol-

lowing conditions:

(i) ( 4 -A)

For any

-I

& > 0,

L(X)

(ii)

Necessity.

Let

ei@A

note the fact that T0

~S

such that

exists on the domain

~(K-A)-IIIx~x ~ C g e x p ( g I ~ )

Proof.

solution

there exists a real

is

T = [~(z)]

(~e~ <0~)

[T(zel@)].

for

k C ~,g .

be holomorphic in

~

.

We

is well-posed and its fundamental

So the necessity follows from Theorem 2

or Theorem 2'. Sufficiency.

It follows just in the proof of Theorem 5.

q.e.d.

From Theorem 6, we have Proposition 7.

Let a closed linear operator

A

be well-posed.

Then the followings are equivalent. (i)

The fundamental solution ~

(ii)

=~z;

The operator

Remark 4.2.

T

larg z I < ~<~ , e ieA

is holomorphic in the sector

0<~<

is well-posed for

~2

; I01 < ~ .

For holomorphic distribution semi-groups, similar

results were obtained by Da Prato-Mosco [3], Fujiwara [4] and Ushijima [13]

.

REFERENCES [I]

Barbu, V.,

Differentiable distribution semi-groups,

Ann. Scuola

Norm. Sup. Pisa, 23 (1969), 413-429. [2]

Chazarain, J.,

Probl@me de Cauchy et applications ~ quelques

probl@mes mixtes,

J. Functional Analysis, ~ (1971), 386-446.

151

[3]

Da Prato, G. and U. Mosco,

Semigruppi distribuzioni analitici,

Ann. Scuola Norm. Sup. Pisa, 19 (1965), 367-396. [4]

Fujiwara, D.,

A characterization of exponential distribution semi-

groups, ~J. Math. Soc. Japan, 18 (1966), 267-274. [5]

Komatsu, H.,

Relative cohomology of sheaves of solutions of dif-

ferential equations, [6]

Lions, J. L.,

S6minaire Lions-Schwartz,

Les semi-groupes distributions,

1966-67. Portugal Math.,

1-9 (1960), 141-164. [7]

Ouchi, S.,

Hyperfunction solutions of the abstract Cauchy problem,

Proc. Japan Acad., 47 (1971), 541-544. [8]

Sato, M.,

Theory of hyperfunctions I,

J. Fac. Sci. Univ. Tokyo,

Sec. I, 8 (1959), 139-193. [9]

Sato, M.,

Theory of hyperfunctions II,

[I0'] Schapira, P.,

ibid., 8 (1960), 387-437.

Th@orie des Hyperfonctions,

Lecture Notes in Math.,

126, Springer, 1970. [ii] Ushijima, T.,

Some properties of regular distribution semi-groups,

Proc. Japan Acad., 45 (1969), 224-227. [12] Ushijima, T.,

On the abstract Cauchy problems and semi-groups of

operators of linear operators in locally convex spaces,

Sci.

Papers College Gen. Ed., Univ. Tokyo, 21 (1971), 93-122. [13] Ushijima, T.,

On the generation and smoothness of semi-groups of

linear operators,

J. Fac. Sci. Univ. Tokyo, Sec. IA, 19 (1972),

65-127. [14] Yoshinaga, K.,

Ultra-distributions

and semi-group distributions,

Bull. Kyushu Inst. Tech., I_~0 (1963), 1-24.

152

15] Yoshinaga, K.,

Values of vector-valued distributions and smooth-

ness of semi-groups distributions,

Bull. Kyushu Inst. Tech., 12

(1965), 1-27.

Department of Mathematics Faculty of Science Tokyo Metropolitan University Tokyo 158, Japan

ON A MARKOVIAN

PROPERTY

OF STATIONARY

GAUSSIAN

PROCESSES

WITH A MULTI-DIMENSIONAL PARAMETER

By Shinichi KOTANI and Yasunori OKABE

I.

Introduction

K. Urbanik [13] and H. P. McKean, Jr.

[5] gave the definition of

a new Markovian property more general than the usual Markovian property in stochastic processes.

Our problem is to determine all stationary

Gaussian processes with such a Markovian property. In one dimensional case, K. Urbanik [13] was the first who gave an example of stationary Gaussian processes with such a Markovian property and N. L e v i n s o n - H .

P. McKean, Jr.

[4] resolved the problem

completely by giving necessary and sufficient conditions for that Markovian property.

In multi-dimensional case,

H. P. McKean, Jr. [5]

gave the same definition of Markovian property as K. Urbanik [13] and proved that Brownian motions with odd dimensional parameter spaces had such a Markovian property.

Considering this problem in the framework

of the theory of differential equations, G.M. Molchan alternative proof of the result of H. P. McKean, Jr.

[6] gave an [5] with the aid

of Hilbert spaces with reproducing kernels associated with the stochastic processes. Recently, G.M. Molchan

[7], L. D. Pitt [i0] and Y. Okabe [8]

characterized the processes with Markovian property by a locality condition of the Hilbert space used by G. M. Molchan, and by a local property of the operator associated with the space.

G° M. Molchan and

L. D. Pitt resolved this problem in the case of what is called a finite multiple Markovian property by using the theory of L. Schwartz's

154

distributions. McKean,

Jr.

Y. Okabe simplified the results of N. Levinson -H. P.

[4] from the operator-theoretical

advanced the results of H. D y m - H .

P. McKean,

theory of M. Sato's hyperfunctions

[12].

point of view, and Jr.

[i] by using the

The key in [8] consists in

regarding an entire function of infra-exponential

type as a local

operator in the space of hyperfunctions. In this paper,

following the idea in [8], we consider stationary

Gaussian processes with a multi-dimensional

parameter and give a

sufficient condition for the Markovian property. We wish to thank Mr. T. Kawai for the discussion with us about the theory of M. Sato's hyperfunctions.

2.

Reproducing kernel space and Markovian property Let

~ = (X(x,

a probability

space

its value in

~

(2.1)

x ~ ~d,

(~ , B, P)

~ ~ ~)

be a stochastic process on

with its time parameter in

n J~=l cjX(xj)

(cj ~ ~, xj

has a Gaussian distribution with mean

for each vector

~d

and

satisfying the following conditions:

any finite linear combination n e ~)

(2.2)

~);

x, y

and

z

in

(X(x+z), X(y+z))

~d,

~d,

0;

the distribution

does not depend upon

E

of random

z.

Such a stochastic process is called a stationary Gaussian process and we discuss such a process in this paper. Let (2.3)

R

be the correlation function of

R(x-y)

= ~

X(x, ~)X(y, ~ ) d P ( ~ )

As shown easily from (2.3), In this paper, we assume that

R

~ :

for

x

and

y

in

~d

is a positive definite function on R

has a spectral density

~d.

~ , namely,

155

P

(2.4)

R(x) = J~d e-iX'y ~(y)dy

We note that

~

x

for

is symmetric since

in ~ d

is a real valued function.

R

Then, we introduce the Hilbert space R.

The space

~

~

with reproducing kernel

is a subspace of the space of bounded continuous

functions containing

R(.-x)

(x ~ ~d)

as its element and it possesses

the following properties: and

u E ~

x ~ ~d,

(2.5)

for each

(2.6)

{finite linear combinations n ~ RNI

is dense in ~

n ~ c jR(--x j)" cj E ~, j=l '

~

= if ~ e 2 ( ~

~ );

~d xo 3

Z~

by

f(x) = f(-x)~. is

Then, it is shown by (2.4), (2.5) and (2.6) that the space represented by the space (2.8)

~

(2.9)

as follows:

= l u; u = ~

(u, v)~ =

where

~m

d fgz~dx

for some for

f ~ ~},

u = fA

and

v =

,

/~, denotes the Fourier transformation: ~( ~ ) = ~ d e- i~.x g(x)dx

(2.10) Similarly,

~-

~(D)

of

(2.11)

~

for

g ~ L I (~d, dx).

means the inverse Fourier transformation.

Next, we define for each open set

D

in ~ d

D

~(D)

in

the closed subspace

by = the closed linear hull of

~R(.-x); x E D~.

Then, we define the following closed subspaces of set

'

.

We define a real Hilbert space (2.7)

(u, R(--x))~ = u(x);

~g(D)

for each open

JRd .

Definition 2.1.

go(D)

For each open set

u--O

D

in

~d,

in D],

we set

t56

(the future),

+(D) = ~ ~ ( ( D C ) n ) n=l ~-(D)

= ~

~ ( D n)

(the past),

n=l ~o

@~(D) where

= (~ ~ ( ( O D ) n=l

n)

(the present),

= Ix ~ ~d., distance(x, G) < nI } for any set

G

G

in fRd

and

n

nE~q.

From (2.5), it follows that ~0(D)

(2.12) Let

8-

= ~.(D)~"

be the projection of

clear that for each open set (2.13)

(the orthogonal complement).

~(D)

D

C ~+(D)

in

~

onto

~-(D).

Then, it is

]Rd

~ ~-(D)

C p_~+(D).

Now, following H. P. McKean, Jr. [5], we can state the definition of the Markovian property. Definition 2.2. property in

D

We say that the process

~(D)

the Borel sub-fields of

D

= ~_~+(D).

Though we defined the Markovian property of

means of the Hilbert space

Remark 2.2.

has the Markovian

if

(2.14) Remark 2.1.

~

~

,

by

we can also state it in terms of

~ (see [8], [i0]).

Roughly speaking,

~

has our Markovian property in

if and only if the prediction of the future under the condition

that we know the past depends only upon the informations of arbitrary small neighbourhood of the boundary

3.

~

K

in an

Z D.

Main Theorem

Before stating our theorem, we recall some notations and results of the theory of Fourier hyperfunctions introduced and developed by M. Sato

157

[12] and T. Kawai [2]. We denote by

Dd

the radial compactification of

~d

and by

the sheaf of Fourier hyperfunctions which coincides with the sheaf ~d

of hyperfunctions on

Let

~,

decreasing holomorphic functions on

be the space of rapidly

D d,

then the following identifi-

cation can be shown ([2]): (3.1)

~ ( D d) ~ £ ~

For any bounded open convex set of

~ ( R d)

function of

.

D,

we denote by

whose support is contained in D :

HD(Z) ~ sup I m ( z , ~6D

D

~ D

and by

~ >

HD

the subspace the support

We will use the next

Lemma 3.1 of Paley-Wiener type in the space of hyperfunctions

([2],

[3]). Lemma 3.1.

The Fourier image of

space of entire functions any

~ > 0,

~

I?(z)[ ~ cee

coincides with the sub-

satisfying the following property: for

there exists a constant

(3.2)

~D

~IzI+HD (z)

c

> 0

for any

such that z ~

cd

.

Now we impose the following Assumption A on the density Since our object in this paper is to give a sufficient condition under which the process

~

has the Markovian property, Assumption A is

reasonable if we consider the results in one dimensional case ([4],

[8]). Assumption A. entire function

P

estimate (3.2) with

The spectral density

~

is the reciprocal of an

of infra-exponential type, i.e.

P

satisfies the

D = ~0 I.

By (2.8), the following integral is well-defined: (3.3)

= ~ d

u(x) ?(x)dx

for

u ~ ~

and

~

~,.

Moreover, by our Assumption A, the following integral is well-defined

158

also: (3.4)

< f' ?> =

f(x) ? (x)dx

for

f ~ Zm

and

? ~ ~,.

By these correspondences, we can regard the spaces

and

the subspaces of

by (3. i).

~

and so the subspaces of

It follows immediately from (3.3) that if a u = fA , then

u e~

(3.5)

for

~u, ?>

=f~d

f(X)p(~

~(x)dx

~(D d) and

Z~

f e ~A

as

satisfy

~ ~ ~,.

Then, we have Lemma 3.2.

(i)

If

u d~

and

f ~ Zm

P(i~ )u = f (ii)

If

and if

u, v ~ P(i ~ )u

and

f, g e ~

in

u = f~ ,

u = fz~

and

v =

is a hyperfunction with compact support in ~d,

Let us consider any

6 ~,.

then

~0Dd).

satisfy

(P(i~)u) * v = ( ~-~)A p Proof.

satisfy

in

then

~(E)d)

Then, by (3.3), (3.4) and

(3.5), we have

= = = This implies (i). (i),

d f(x)

(x)dx =4 f, ? >

(ii) can be proved as follows: by (3.3), (3.5) and

we have <(P(i~)u) ,v, ? >

=
(P(i~)u)(--) , ? >

=

f~d g(x)~(x) P(x) (~ ( x ) d x

:

>.

159

In the above calculation, f(x)

we have used

P(x) = P(-x)

which follow from the symmetry of

~

and

and

f(-x) =

(2.7), respectively. (Q.E.D.)

Lemma 3.3. we have

For any bounded open set

u e ~-(D)

whenever

P(i~)u

D

= 0

in in

~Rd

and any

~)d _ ~

u ~

,

as a Fourier

hyperfunction. Proof. in

Let

D d -D

u

be any element in

P(i~)u

v = 0

in

D

n

by

and

v ~ ~ ( D n )~.

([12]),

(P(i ~ ) u ) , v

Thus, by Lemma 3.2 (ii),

bourhood of the origin.

But

Therefore,

the arbitrariness

D

and

(2.12), we have, by the theory of integration

hood of the origin.

= 0.

n E ~

P ( i ~ )u = 0

is a hyperfunction with compact support in

the theory of hyperfunctions

~g~

such that

as a Fourier hyperfunction and let

Since

~d

~

of

n

fg~

and

= 0

on a neighbour-

fg A = 0

on a neigh-

is an element of

by (2.9),

(u,

v e ~(Dn),

in

V)~ = 0. u E ~

LI@R d)

and so

Consequently, (D)

from

and this proves

Lemma 3.3.

(Q.E.D.)

In the sequel, we put the following additional Assumption B on f~ which is stronger than purely non-deterministicness dimensional

(i)

T(t)

i

A(x)

(ii) to

There exist a positive number

(t ~ [to, ~ )) ~ eT"~ ;xl) T(t) dt l+t2

or

T(t)

to

such that for

x ~ Rd,

Ixl --> t 0 ,

( ~o

We assume further either (iii)

X

in one

case.

Assumption B. function

of

is non-negative and increasing,

and a continuous

160

(iii)'

there exists a constant

T(t l + t 2 )

c > 0

~ log c + T ( t l ) + T ( t 2)

such that for

t I, t 2 e [t o , ~ ) .

In the proof of our main theorem, we use the following Lemma 3.4 proved by O. A. Presniakova Assumption B.

~

[9] under the above

This Lemma 3.4 holds without Assumption A.

For any open set of

[ii] and O. A. Orebkova

D

in

~d,

we define a closed subspace

Z~(D)

by

(3.6)

~(D)

e i 'x ; x ~

= the closed linear hull of

D} .

Then we have Lemma 3.4.

For any bounded open convex set

with the subspace of all f

f

in

~

Zm(D)

coincides

satisfying the next property:

can be extended to an entire function

the estimate

D,

cd

on

f

which satisfies

(3.2).

Then, we shall show Lemma 3.5.

For any bounded open ~onvex set

with the subspace of all ~d _~

~

such that

coincides

P ( i ~ )u = 0

in

By Definition 2.1, Lemma 3.3 and the sheaf property of

it suffices to prove that for any

(3.7)

P ( i ~ )u = 0

in

Let

f E ~

be such that

the

space

~(Dn)

f

in

~-(D)

as a Fourier hyperfunction. Proof.

,

u

D,

~d -Dn

Since by

isometrically

is contained in the latter space.

3.2 (i) and Lemma 3.4, we have

(2.4),

(2.11) and

to the space

Therefore,

(3.6)

~(Dn),

by Lemma 3.1, Lemma (Q. E. D.)

we are able to prove our main

Under Assumptions A and B,

Markovian property

u ~ ~(Dn),

(3.7).

Now, after above preparations, Theorem.

and any

as a Fourier hyperfunction.

u = f~ .

corresponds

n £ ;N

the process

in any bounded open convex set in

% ~d

has the

161

any

Proof.

Let

D

n ~ ~.

Then,

be any bounded open convex set in there exists a positive

and a positive number y-x

6 (~D) n

Therefore,

S

integer

~d

m

and fix

larger than

n

such that

for any

x ~ ~d,

the same consideration

Ix[ < ~

and

y 6 (~D)m.

as in the proof of Lemma 3.3 implies

that

(3.8)

if

u ~ ~

satisfies

P(i ~ ) u

Fourier hyperfunction, Next,

let us consider any

(3.9)

R(--x)

Since

P ( i ~ )R(.-x) (D),

= 0

in

D

D.

since

(3.II) from (3.10),

that

P(i~ )u I = 0

R('-x)

u 2 e ( ~ - ( D ) ) m.

by (2.12) and

uI ~ ~

in (D), in

C

(D)m

C

D d - (~D)m

Since

n

is arbitrary,

since

x

is any point of

of

that ~(D) C

P(i~ ),

"

we have, by Lemma 3.5, D d-~.

(3.11) and the sheaf property in

as follows:

by (3.9), we have

P(i~)u I = 0

Thus,

and

Moreover,

P(i ~ )u I = 0

On the other hand,

D)n).

and so, by the local property Therefore,

(3.I0)

as a

by Lemma 3.2 (i), it follows

D d - (DC)m .

in

g3d - ( ~ D ) m

and decompose

u I e ~-(D)

~('-x)

in

u2 = 0

P ( i ~ )u 2 = 0

=

in

u ~ ~((~

x E (DC)m

= Ul+U2,

P(i~ )R(--x)

then

= 0

and so

this implies (DC)m ,

of

~

u I e ~((~

, it follows D)n )

by (3.8).

a~.~-(D)R('-x) 6 ~ ( D ) .

by the continuity

of

~

Moreover, _

and (D)

(2.11) , we have

~

~ ( ( Dc) m ) C ~ ( D ) .

_

This implies

(D) ~

~+(D)

C ~(D)

by Definition

2.1 and completes

the proof of

- (D)

theorem.

(Q.E.D.)

Remark 3.1. and 3.5. sumption

Our proof of theorem depends

Therefore, ([9],

[ii]).

only upon Lemmas

we might replace Assumption

3.4

B by a weaker as-

162 4.

Examples. We give some examples of the densities

~

satisfying the

Assumptions A and B. Let us consider any positive sequence (4. I)

(tn)n= 1

such that

~_ tn n=l

Then, we define

P0(z)

by 2

(4.2)

P0(z) = n=l]~ (l+--~)z n

It is shown easily ([13]) that

P0(z)

(z ~ ¢),

is an entire function of infra-

exponential type and satisfies the following / (4.3)

(4.4)

Ixt n Next, we define

P(z) 2

log P0 (x) 2 dx < ~ l+x (x)dx < ~

(4.3) and (4.4): ,

for any

n E ~.

by 2

z I + ... + z d

(4.5)

P(z) = I T (i+ n=l

Then, noting that

2 t

)

(z = (Zl,

--, Zd) E cd).

n

P0(x) (x e IR)

proved by (4.3) and (4.4) that

is monotone increasing, 1 ~ = ~

it can be

satisfies Assumptions A and B.

REFERENCES [i]

H. Dym and H. P. McKean, Jr.:

Application of de Branges spaces

of integral functions to the prediction of stationary Gaussian processes, [2]

T. Kawai:

Illinois J. Math. 14 (1970), 299-343. On the theory of Fourier hyperfunctions and its

applications to partial differential equations with constant coefficients, 467-517.

J. Fac. Sci. Univ. Tokyo, Sect. IA, 17 (1971),

163

[3]

H. Komatsu:

Theory of Hyperfunctions and Partial Differential

Operators with Constant Coefficients,

Lecture Notes, Univ. of

Tokyo, No.22, 1968 (in Japanese). [4]

N. Levinson and H. P. McKean, Jr.: RI

approximation on

with application to the germ field of a

stationary Gaussian noise, [5]

H. P. McKean, Jr.: time,

[6]

[8]

Acta Math., 112 (1964), 99-143.

Brownian motion with a several dimensional

Theor. Probability Appl., 8 (1963), 357-378.

G. M. Molchan:

On some problems concerning Brownian motion in

L@vy's sense~ [7]

Weighted trigonometrical

Theor. Probability Appl., 12 (1967), 682-690.

G. M. Molchan:

Characterization of Gaussian fields with Markov

property,

Dokl. Akad. Nauk SSSR, 197 (1971), 784-787.

Y. Okabe:

On

stationary Gaussian processes with Markovian

property and M. Sato's hyperfunctions, [9]

(in Russian).

O. A. Orebkova:

to appear.

Some problems for extrapolation of random fields,

Dokl. Akad. Nauk SSSR, 196 (1971), 776-778 (in Russian). [i0] L. D. Pitt:

A Markov property for Gaussian processes with a

multidimensional parameter,

Arch. Rational Mech. Anal., 43 (1971),

367-395. [Ii] O. A. Presniakova:

On the analytic structure of subspaces gener-

ated by random homogeneous fields,

Dokl. Akad. Nauk SSSR, 192

(1970), 279-281 (in Russian). [12] M. Sato: Theory of hyperfunctions I, II,

J. Fac. Sci. Univ. Tokyo,

Sect.l, 8 (1959), 139-193, 387-437. [13] K. Urbanik: ter,

Generalized stationary processes of Markovian charac-

Studia Math., 21 (1962), 261-282. Department of Mathematics Faculty of Science Osaka University

ULTRADISTRIBUTIONS

AND HYPERFUNCTIONS

By Hikosaburo KOMATSU

In the last conference

of March,

1971, the speaker announced

the

following theorem and applied it to the theory of ordinary differential equations with real analytic coefficients. Theorem.

Let

f = [F]

be a hyperfunction

with a defining function

F.

Gevrey class of order

of Roumieu type

s

Then

only if for every compact interval there is a constant

C

f

on an interval

is an ultradistribution (of Beurling type)

K C (a, b)

(there are constants

and every L

and

C)

(a, b) of

if and

L > 0 such that

1 sup I F ( x + i y ) I ~ C e x p I ~ j x6K

}.

In this lecture we develop the theory of ultradistributions give a proof of the theorem in a generalized

i.

Ultradifferentiable Let

form.

functions

M , p =0, I,-'', P

finitely differentiable

and

be a sequence of positive numbers.

function

called an ultradifferentiable

f

on an open set

function of class

~

M

in ~ n

An inwill be

of Roumieu type

(of

P Beurling type) if for every compact set h

and

C

(i)

(and for every

h>0

K

in

~

there are constants

there is a constant

IID~fI~C(K) -~ C h P M p '

C)

such that

I~I = P = 0, I , 2 , "'"

We will impose the following conditions

on

M

: P

(M. i)

(Logarithmic

2 M p ~_ M p-I Mp+l ,

(2) (M.2)

convexity)

(Stability under convolution)

such that

p = i, 2, "'" There are constants

A

and

H

165

(3)

M

~ A Hp P

(M.3)

min M M , 0~q~p P-q q

p = 0, I, 2, ''"

(Strong non-quasi-analyticity)

There is a constant

A

such

that 0~ (4)

M.

pMp

,,~

j =p Mj+I In some problems


- -

-

Mp+ I

,

p = i, 2, 3, "'"

(M.2) and (M.3) may be replaced by the follow-

ing weaker conditions: (M.2) '

(Stability under differentiation)

(5)

Mp+I <_ A H P M p ,

(M,3) '

p = 0, I, 2, "'"

(Non-quasi -analyticity)

~_ >~

(6)

Mj <~

j=0 Mj+I It is easy to check that the Gevrey

(7) where

M

= (p~)S,

pSp

and

sequences

~(l+sp)

,

P s > I,

satisfy these conditions.

These sequences determine

the same class of ultradifferentiable'functions class of order

called the Gevrey

s.

It is convenient

to relate the above conditions with the behavior

of the associated function PM 0

(8)

M(~ ) = log sup P (M.I) is equivalent

P

to M

(9)

M

p

M0

~P

= sup ~>0 exp M ( ~ )

Under this condition (lO)

where exceed

m( A ) X

is the number of ratios

m. = M./M. J 3 3-i

which does not

166

(M.2)

is equivalent to

(ii)

2M(~) ~_ M(H ~) + log(AM0) . (M. 3) implies m

(12)

dX ~ A M ( f )

for large

f

On the other hand, (M.2)' is equivalent to

re(X) ~ :og(A/A')

(13)

-

log

H

(M.3)' is equivalent to M

(14)

Definition I.

d~

=

Let

K

We denote by

h >0.

f ~ C~(K)

be a regular compact set in ~n

g{%I,h(K)

and

the Banach space of all functions

in the sense of Whitney such that

(15)

llfi|{Mplk,h

= sup ]D~f(X) l < ~o , (K) ~,x h~IM i~I

Mp~ ,h and by

~K

the Banach space of all functions

f e C~(RRn)

support in K which satisfy (15). Mp} ,h K may be looked upon as a closed subspace of proposition 2.

If

h < k,

{Mp} ,h(K )

the injections

~{Mp} ,h

(16)

with

(Mp} ,k (K) C ~

(K)

{KMP} ,h C~{KMP} ,k

(17) are compact.

If

Mp

satisfies (M.2)'

in addition and if

k/h

is

sufficiently large, then the injections are nuclear. Definition 3. set in ~Rn.

K

be a regular compact set and

~

an open

We define the spaces of ultradifferentiable functions of

Roumieu type ~(Mp) (K)

Let

and

~, 1~e~l(K),

~{e~'" ~(~)

~(Mp) (0_)

by

and those of Beurling type

167

(18)

{Mp}

(19)

(Mp}(~)

~{Mp} ,h

(K) = l im h-~

(K) ,

= lira g{Mp~(K) , KCC~

(2o)

,h (K) ,

(K) = lim

h+%

d(MP)(~)= lim

(21)

~(Mp) (K)

Kf~fL IMp} It follows from Proposition 2 that and

~(Mp)(K)

(M.2)',

and

(Mp)(fg)

~

(K)

are (FS)-spaces.

is a (DFS)-space If

M

P

satisfies

these spaces are all nuclear.

Similarly the spaces of ultradifferentiable functions with compact support are defined in the following way : (22)

~ { M P ~ = lira ~{Mp}'h K --~ ~)K ' h~

(23)

~{Mp} (~) = lira ~ M p --+ K

(24)

(Mp) ~)K

(25)

~(Mp)

~Mp}

~( Mp~, h ¢--- '~K h->O lim

(Mp) K

(SI) = lira K¢c~

~{Mp

and

(Mp) ~(~)

K

space and

=

'

(~.)

are (DFS)-spaces,

~K

is an

(FS)-

is an (LF)-space as the strict inductive limit

of a sequence of (FS)-spaces.

Hence all spaces are Hausdorff, com-

plete, reflexive and bornologic.

If

M

P

satisfies (M.2)', then all

spaces are nuclear. A subset

B

of

. ~Mp~ ~Mp}(~) is bounded if and only dgK {Mp} °r,h if it is contained in a ._~K and bounded there, while a subset

168

B

of

~ K (Mp)

contained in a

(Mp) or ~ (0.) (Mp) ~K for a K

is bounded if and only if it is ( ,h and bounded in all ~ K Mp}

It is well known I) that ~ K Mp~~ = ~ K{Mp" ~, where greatest logarithmically convex sequence such that in case

Mp

(Mp

is logarithmically convex,

~K

M'P

is the

M'p ~- M p

# 0

and that

if and only if

M

satisfies (M.3)' Conversely suppose that M satisfies (M.I) P P and (M.3)' Then for any ball K of radius ~ > 0 there is a function

~g e o~ {KMp} such that

~a(x) _2 0

and

f f (x) dx = i.

~(Mp~ Hence i t

follows that

(~_)

is dense in

~ (fl)

and t h a t

{ Mp}

there exists a partition of unity by functions in ordinate to any open covering of If

~

satisfies (M.I) and (M.3)', there is P (M.I), (M.3)' and lim P~

Thus t h e same r e s u l t s

If

M

M ........0 p

sub-

for any

M' P

which satisfies

h > 0. (Mp)

as above h o l d f o r

satisfies (M.I),

P

(~_)

.

M

(26)

~

the spaces

~{Mp~ (K),

~{Mp} (~-),

g(Mp)

(Mp) (K)

(~)

and g~{

Mp~ spaces

K

are stable under multiplication and the

'

(Mp)

(Mp) Mp~ (fl), ~ K

under multiplication by functions in

and

~

~ M p ~ (K),

(~)

are stable

~ M p } (~I),

~(MP)(K)

(Mp) and

(~_)

respectively and the multiplications are hypo-conti-

nuous. If tion and

(M.2)' holds, the above spaces are stable under differentiaD~

is continuous for any

I) See Mandelbrojt [8], [9], Roumieu [I0], [Ii] and Lions-Magenes [7] for the results up to the end of this section.

169 The spaces of Roumieu type have been discussed by Roumieu [I0] and [ii].

However,

it is not clear whether or not the topologies he

employed coincide with the above natural topologies which have been introduced by Lions-Magenes

[7].

The spaces of Beurling type have been discussed in Bj~rck [i] from a little different point of view and in Lions-Magenes

2.

The Paley-Wiener theorem for ultrad,ifferentiab!e functions Theorem 4.

that

Suppose that

K

(for any

~ ]

( ~ K (Mp))

h > 0

there is (~)

(27)

~

M

satisfies

P is a compact, convex set in

belongs to

of

[7].

~n.

(M.I) and (M.2)' and

Then a function

if and only if there are

C)

~(x) h

such that the Fourier-Laplace

= ~?(~)

=

and

C

transform

ne -ix{ ?(x) dx

satisfies

(28)

I~(~)I

~ Cexp(-M(I~I/h)+HK(~))

,

where

HK(~) = sup Im <x, ~ >.

(29)

x~K A subsets B

•{Mp}

of

K

can choose constants uniformly for

h

and

C

)

(for any

is bounded if and only if we h > 0

a constant

C)

? 6 B.

A sequence of functions and only if for some

h > 0

converges uniformly on where

~p) ~K

jRn

?j

6 ~K

(for any

(Mp) p~ (~ K )

h > 0)

converges if

expM(l~/h)

or equivalently on a strip

~j(~ )

~Im ~I <

a,

0 < a < ~.

~p) Since of semi-norms

~

K

is a Fr@chet space, this shows that the families

170 (30)

k = I, 2, "'"

sup lexp(M(k I~I) - H K ( ~ )) ~( ~)I, ~e¢ n

and sup lexp M(k ~ I ) ~e~ n

(31)

k = i, 2, "''

~(~)~, (Mp) ~ K

determine the topology of

In order to find a family of semi-norms

~ ~Mp~ K ' we imbed the Fourier-

which determines the topology of {MPl

Laplace transform of Let

similar to (30) or (31)

in a (DFS*)-space.

K

be fixed and consider the sequence of Banach

i < r <

spaces Yh = ~

(32)

~ Lr (¢n). e x p ( M ( I ~ / h ) + H K ( ~ ) ) ioc '

~(~)

6 Lr(¢n)}

h = i, 2, ''' with the identity mappings

Yh

~Yh+l"

Since

Yh

are reflexive

Banach spaces, this forms a weakly compact sequence and its limit Y = lim ---> Yh

is a (DFS*)-space.

A modified form of Morera's theorem shows that (33)

X h = { ~ ~ Yh ;

is a closed subspace of (34)

Yh"

,

is entire on

Cn

We can prove that

~ { M P ~ K ~ = li~ X h h~m

including the topology. ~''~MP~K

is closed in

Since

~Mp} ~K

topology of

~lgp}

the relative

Y

and that

X h = Yh ~ ~ { K Mp~''"

i s a Montel s p a c e , i t i s proved t h a t the o r i g i n a l induced by t h a t of

t o p o l o g y induced by t h a t of

Theorem 5. topology of

Morera's theorem proves also that the set

~

~Mp} Y

Under the same assumptions

t

(cf.

coincides with [5] Theorem 7).

as in Theorem 4 the

is determined by the family of semi-norms

171

(35)

sup. lexp(M( g (J ~ I)) -HK(~)) ~ ( ~ )l

when

g(~)

runs through the increasing functions on

[0, ~ )

satisfying (36)

g(f ) = 0 .

lim

From the Paley-Wiener theorem (Theorem 4) we get easily the following Suppose that

Theorem 6.

M

satisfies (M.I), (M.2) and (M°3)'

P

Let Oo

(37)

J(~)

=

~. a~ ~ I~l=0

be an entire function with the growth order that for any is

C

(there are

L

and

IJ(~)I

(38)

~C

C)

L > 0

there

such that

exp M(LI~I),

Then, for any compact convex set

K

~E¢ n in ~n

the differential

operator of infinite order J(D) =

(39)

~

a~ D ~

I~t=O maps

~Mp} (Mp) ~ K (~ K )

continuously into itself.

Moreover, the right

hand side of 60

(40)

J(D) ?(x) =

~

a~D

~(x)

I~i=0 converges absolutely in the topology of

(Mp) holds for any

~6~K

in a bounded set of

p~ ( ~ K

)"

~Mp}

(~ (Mp) )

K

K

the partial sums of (40) are

contained in an absolutely convex bounded set

B

and the series

converges absolutely in the normed space generated by An entire function multiplier for the class

J(~)

is contained

More precisely if

imp} (Mp) ~K ~ K ),

and (40)

B.

satisfying (38) will be called a

IMp} ((Mp)).

It is easy to see that (37)

172 is a multiplier

for

{Mp}

there is a C (there are (41)

((Mp))

L and

if and only if for any

C)

such that

la~}~_-CLI~I/MI~ I , Proposition 7.

(M.3).

plier for

{Mpl

~

Suppose that

((Mp))

= 0, i, 2, "'"

M

satisfies

P J(~ )

Then an entire function

L > 0

(Mol), (M.2) and

of one variable is a multi-

if and only if it has Hadamard's factoriza-

tion ([2], p.22) (42)

J(~)

and for any

L > 0

(43)

= a ~

no

there is N(@)

= f=

~ (I-~) j=l J C

(there are

n(A)

is the number of

C ,

c.j with

transforms of ultradifferentiable

the original topology, Theorem 8. and that

[3].

Icj ~ ~ ~ . of the Fourier-Laplace

Since (40) converges absolutely

in

ours may be said a better characterization.

satisfies (M.I), (M.2) and (M.3) P is a compact convex set in ~Rn. Then a function ~(x)

K

Suppose that

{ Mp}

M

(Mp)

~K

(~K

transform

~ (~ )

satisfies

(44)

s~plexp(-HK( ~ ))J(~) ~ ( ~ )

for any entire function

)

J( ~ )

if and only if its Fourier-Laplace

~ <

of the form

(45)

J(~ ) = J0(Sl~ i) ... Jo(Sn~n),

(46)

J0(~ ) = - ~ (i+" j=l m.J si

such that

functions with compact support in a

belongs to

where

C)

0 < ~ <~,

Finally we obtain a characterization

way similar to Ehrenpreis

and

n(A) - n o A dX

~ M(L~ ) + l o g where

L

is

+I

or

-I

and

~j

)

'

is a sequence of positive numbers

173

converging to zero

is a positive constant).

Moreover, the family of semi-norms (44) determines the topology ~Mpl (Mp) ~ K (~K )"

of 3.

( ~j

Ultradistributions Definition 9.

Suppose that

is an open set in ~n

tha t

the strong dual space of

((Mp)(f~))

and call its

of Roumieu type P (Mp)) for short.

{Mp~

((Mp)(~)

~'(~)

~-~p~'(f~)~ ((Mp)' (~))

of class

M

is a dense subspace of ~ (~) ~Mp }'

and the injection is continuous, the distributions

satisfies (M.I) and (M.3)' and

We denote by

•

on

(Beurling type) or of class ¢~ M p ~ ( ~ )

P

~ M p ~ (~-

elements ultradistributions

Since

M

(~)

(~

(Me)'

(~))

contains

as a dense subspace.

On the other hand, since the real analytic functions on Mp~_ continuously and densely contained in follows that every ultradistribution

6

~

are

(Me) (~_)

(~

(~)),

it

is a hyperfunction.

g~ Mp~ (Mp) ~ Me }' If a E (~) (~ (6)) and f 6 (~_) (M)' ( 6 P (fL)), the product af is defined by ~Mp (47)


(48)

M

P

=

,

? E

satisfies (M.2)', the derivative

Similarly if Mp

for ~Mp~_

(Me ) }(~)

((Mp)),

,

D~f

? E~MP~(g)

satisfies (M.2) and J(D)f

(~

J(~ )

(~L)). is defined by (~

is a multiplier

is defined by ~Mp}

(49)

(~L)).

~J(D)f, ? > =
, ~ ~

(Mp) (~.)

(~

(~)).

As in the case of distributions, the existence of partition of

1 74

~{Mp~'

(M)'

unity implies that

(~)

natural restriction mappings

(~

~

(f~)),

~ C ~n,

form a soft sheaf on

~n

with the In particular,

the notion of support is defined. If

S

butions

f

is a closed set in ~ , the subspace of all ultradistri~ ' (~ (Mp)'( in ~ (~) ~)) with supp f C S is closed.

Multiplications homomorphisms.

a.,

differentiations

D~

and

J(D)

are sheaf

Namely they do not enlarge the support. ~Mp}'

(Mp)'

The dual space

(~)

~(Mp)

(g

(~_))

of

(~)

(Mp) (~

(~))

' .,{Mp~ ( ~ )

is identified with the subspace composed of all

( (

Theorem 10.

Np) ( ~ ) )

with compact support.

Suppose t h a t

Then, a hyperfunction ~Mp}'(~_)

f

M P

f ~ C(G)'

is

(there are

C

or L

G

in ~a

f IG =

~ D~f~, l~l=0

Lr(G),

I ~ r ~ ~,

and

~

(M.1), (M.2)' and (M.3)' in

~n

belongs to

if and only if its restriction

any relatively compact open set (50)

satisfies

on an open set

(~(~P)'(~))~

where

f E

'

C)

f IG

to

can be written

and for any

L > 0

there

such that

(51)

llf~l ~ C - MI~I ~ ' (50) converges strongly in Mp~ (G)

'

~{ A subsets B

of

only if constant(s)

(and

(G)).

(M)'

Mp~ (~) C

(Mp)' (~

L)

(~

P

(~'L)) is bounded if and

in (51) can be chosen uniformly in

f ~ B. Roumieu [i0], Chap. I, th@or@me 1 gives a stronger statement. By the Phragmen-Lindel@f

theorem we can show that the semi-norm

(44) in Theorem 8 is equivalent to

175 sup ~n JJ( ~ ) ~ ( ~

)I

•

Hence we obtain another structure theorem: Theorem ii.

Suppose that

Then, a hyperfunction

satisfies (Mol), (M.2) and (M.3). P on an open set ~a in ~n belongs to

f

M

!

~{Mp}'(~)

(~(Mp)

convex open set ((Mp))

(~))

G

there is a multiplier

and a finite measure

(52)

lu.

on

G

J(~)

for the class

~Mp}

such that

f ~G = J ( D ) ~ A subset

if there is 4.

if and only if for any relatively compact

B

z { ~}'(fl)

of

J(D)

, Mp) (~))

(

independent of

f E B

is bounded if and only

II/~11

and

are bounded.

Characterization of ultradistrubutions In this section we consider only the case where

n = 1

for the

sake of simplicity. When

M

P

is a sequence satisfying

~PP! M 0 ...... , Mp

(53)

M*(~)

(54)

~P M~ = M 0 sup ~>0 exp M*(~)

If

mp/p

= log sup p

(M.I), we write

is increasing, we have

Theorem 12.

Suppose that

Then, a hyperfunction

f = [F]

M* = M /p} . P P M satisfies (M.I), (M.2) and (M.3). P on an interval (a, b) belongs to

(M), (a, b) ( ~ K

in

(a, b)

(a, b))

and for any

L > 0

such that the defining function (55)

if and only if for any compact interval there is F

C

(there are

L

and

C)

satisfies

sup ~F(x+iy)~ ~ C exp M * ( ~ y ~ ) x~K

for sufficiently small A subset

B

of

IY~~{MP}'(a,

b) (~(MP)'(a,

b))

is bounded if and

176 only if the constant(s)

C

Sketch of Proof.

G+

and

Then

J+(~ )

can be chosen uniformly in F

and

satisfies J_(~ )

which are bounded near

F(x+ iy) =

(56)

(~

G

L)

Suppose that

We will find multipliers tions

(and

(c, d) y > 0

[ J_ (D)G_ (x + iy),

y<

- J_(D)G_(x - i0)

K = [c, d].

and holomorphic

f J+(D)G+(x + iy),

f = J+(D)G+(x+i0)

(55) for

f ~ B.

func-

such that

0.

belongs to

~{~Pl'(c,-- d)

(M)' P (c, d)).

Let

y > 0

and

(57)

J+(~ ) = (i+ ~ )2

where

~.J

constant).

( I + - -~J -~.), j=l J

is a positive sequence converging to zero (a positive Since

-i

J+(~)

is infra-exponential

except on the

negative real axis, i

(58)

G+(z) = 2 ~

defines a holomorphic

0

j+(~ )-i eiZ~d

function on the Riemann surface

-~

< arg z

<271.

z

for (59)

Choose a point

z0

z

-~+&

in the cone

in the upper domain <

G+F(z) = f

arg(z - z0) ~ -£ C ~ + G+(z -w)F(w) dw

P

V+

,

of

F

and define

177

where slit

is a simple closed curve starting [z, z 0]

counterclockwise.

(60)

z0

and encircling the

Then we have

J+(D)G+F(z)

= F(z)

By deforming the contour

we have

G+F(z) = if0t g+(-iv)F (z + iv)dv + " " " ,

(61) where

i g+(-iy) - 2 ~ i

(62)

f ~+i~ J~-i= J+(~+ i~)-I

eY(~+i~) d~ .

Taking it into account that ~_

eY~

Ig(-iY)t

2

(63)

jrll + m. I

eY ~

J

C inf ~>0 exp ~(~) where ~I''" ~p ~ PM0 (64)

~(~)

= log sup p

we can choose a sequence bounded.

4

M

P

so that the first term of (61) is

The remainder is also bounded.

Hence we have (56).

The proof shows that if the estimate then

Jjc(D) -I

L~(c, d),

constructed above map

and hence

B

(55) is uniform in

F(x±i0)

into a bounded set in

is bounded.

}, Conversely

suppose that

f ~ ~Mp

(a, b)

f E B,

(

(Mp)'(a, b)).

It

follows from Theorem I0 that f I (c, d) = ~, DP f p=O

fp ~ C([c, d])' P

and

'

Lp

llfpllC([c,d] ), ~ C M P

Let

F

P the estimate

be

the

standard

defining

function

of

Then we have

f P

178

LPp! sup IDPF (x+iy) J ~ C ly(p+iMp x~[c,d] P = 2Vg I CA <-- 2p 2TriM0

(2HL) p+I (p+l) ~M0 sup p

p+l IYl

Mp+ I

Therefore F(x+iy) is a defining function of

= ~ DPF (x+iy) p=0 P f

and it satisfies

JF(x+iy)l ~_ ~ sup x~[c,d]

CA

exp M*

( 2HL [yj ]

It is clear that if a defining function satisfies

. (55), any other

defining function satisfies it also. For the Gevrey sequence of order

s,

M*(~)

is equivalent to

I @s---~. Therefore the theorem in the introduction is a special case of Theorem 12.

REFERENCES [i]

G. Bj~rck,

Linear partial differential operators and generalized

distributions, [2]

R. P. Boas, Jr.,

Ark. f. Math., ~ (1966), 351-407. Entire Functions,

Academic Press, New York,

1954. [3]

L. Ehrenpreis,

Theory of infinite derivatives,

Amer. J. Math.,

81 (1959), 799-845. [4]

L. Ehrenpreis,

Fourier Analysis in Several Complex Variables,

Wiley-lnterscience, [5]

H. Komatsu,

New York.London-Sydney'Toront,

Projective and injective limits of weakly compact

sequences of convex spaces, 366-383.

1970.

J. Math. Soc. Japan, 19 (1967),

179

[6]

H. Komatsu,

Hyperfunctions and Partial Differential Equations

with Constant Coefficients,

Seminar Notes No.22,

Dept. Math.

Univ. Tokyo, 1968 (in Japanese). [7]

J. -L. Lions -E. Magenes, et Applications, Vol.3,

[8]

S. Mandelbrojt, Applications,

[9]

Dunod, Paris, 1970.

S@rie Adh~rentes.

R@gularisation des Suites.

Gauthier-Villars, Paris, 1952.

S. Mandelbrojt, Applications,

[I0] C. Roumieu,

ProblSmes aux Limites Non Homog~nes

Fonctions Enti~res et Transform@es de Fourier, Math. Soc. Japan,

Tokyo, 1967.

Sur quelques extensions de la notion de distribution,

Ann. Sci. Ecole Norm. Sup. 3 set., 77 (1960), 41-121. [ii] C. Roumieu,

Ultra-distributions d@finies sur

classes de vari@t~s diff6rentiables,

Rn

et sur certaines

J. Analyse Math., i0 (1962

/63), 153-192.

Department of Mathematics University of Tokyo Hongo, Tokyo

HYPERFUNCTIONS AND LINEAR PARTIAL DIFFERENTIAL EQUATIONS* B y Hikosaburo KOMATSU

Hyperfunctions

In a series of papers

[20, 21] published between 1958 and 1960

M. Sato introduced a new class of generalized functions.

While distributions

are closely connected with the C ~

structure of the space, hyperfunctions analytic

structure of the space.

tions on any paracompact restrict our attention space

~n

are connected with the real

In fact, one can define hyperfunc-

real analytic manifold.

However, we will

to those defined on open subsets of Euclidean

for the sake of simplicity.

Suppose containing

~ ~h

is an open set in

HI(V,

~n

and

as a relatively closed set.

of hyperfunctions group

functions called hyper-

~)

on

~

is by def£nition

= Hn(v, V - ~ ,

of holomorphic

functions.

~

)

is an open set in

Then the space

V-~

relative cohomology group is different

~ (~.)

in the sheaf

is open in

V,

from that discussed

the case in which

Actually the theory of relative cohomology

Cn

the n-th relative cohomology

with coefficients

Since

[8] by Godement as he considered

V

V-~

the in the book is closed.

is much simpler in the

*) This is a paper read at the Conference on Generalized Functions held at Katowice,

Poland on August 3 0 ~ S e p t e m b e r

lation into Japanese has been published Kokyuroku,

Kyoto Univ.

A trans-

in Surikaiseki Kenkyusho

22 (1967), 127-139.

also at the International

4, 1966.

A r@sum@ was reported

Congress of Mathematicians,

under the joint name of R. Harvey and the author.

Moscow,

1966,

181

open case as was shown by Sato [21] and later by Grothendieck

[9].

Sato [21] states the following results:

a) borhood

~(~)

does not depend on the choice of the complex neigh-

V.

~(g),

b)

~c~n,

are two open sets in ~

: ~(~)

then

~f,

~ ~(~')

= ?~,

~

~n,

is defined'

:

If

A hyperfunction

f'l ~ ~ ( ~ i ) that

~c~n

.

~D~'

D

f ;

f ~ ~(~) ~

and if a compatible

;

is zero if and

If

~il

is an

family of hyperfunctions

Finally each hyperfunction

can be extended to a hyperfunction Distributions

if

are open,

is given, then there is a hyperfunction

fi = ~ i

Namely,

the natural restriction mapping

only if it is locally zero at each point in open covering of

~n

form a flabby sheaf on

f

f

on

~

such

on sny open set

on the entire space

~n

form a sheaf but they do not possess the last

property that is flabbiness. We denote by two sets and if

~ $

space of sections of K(~) c) which

the sheaf of hyperfunctions.

is a sheaf on ~

over

B,

B

K

~ K @ R n)

with support in

is compact,

~K ~n)

is a Fr@chet-Schwartz K(~ n) ~

where

we denote by

is the space of hyperfunctions If

~(K)'

If

on

~

A C B

~A(B) A.

are the

In particular,

with support in

K.

has a natural topology with (even nuclear)

space and

~(K)',

denotes the strong dual space of the space

all real analytic functions on a neighborhood

of

K

~(K)

of

with the natural

locally convex topology. d)

The distributions

~'(~)

of Gevrey class of functions on

~g

and more generally the dual space are naturally contained in

182

~(&). Although a) is trivial, prove.

In his long paper

left so many details Unfortunately

the other b), c) and d) are not easy to

[21] Sato gave all necessary

that it is almost

impossible

[19] gave another approach However,

to hyperfunctions

with property

c).

baki Seminar,

he had to leave many gaps, too.

Intuitively

a hyperfunction

then by Leray's

V

theorem

The most complete

thesis

[ii].

is a sum of boundary values of a

function defined on an adjacent

if we choose for

starting

since it was only a lecture at the Bour-

exposition will be found in R. Harvey's

fact,

to understand.

his subsequent paper has never appeared.

Martineau

holomorphic

tools but he

complex open set.

a Stein open set such that ~ (~)

In

V N ~n = ~

,

is identified with the quotient

space: n

(1) j = 1

where

V # ~. = I z E V '• Im z.j ¢ 0

Im z k ¢ 0 e ~(V

for ~ ~.)

respective

k # j}.

Since

is actually

connected

boundary values

2n

j}

has

2n

V ~

holomorphic

components.

? (x l + i O ,

for all

Suppose

and

V. = Iz ~ V" 3 connected components,

functions defined on

that they have all regular

"'" , x ! i 0 ) .

Then it is shown that

n

belongs

to

6" i0) = 0, n

denote by

~(Vj) where [ ?]

if and only if 6-

~

sign 6 7 ( X l +

ranges over all sequences

the hyperfunction

then the above fact makes

represented

it possible

of

by

~i. ~

to interprete

sign 6-~(x I + ~li0

'

"'"

'

xn

+

~

iO), n

If we

~ ( V # ~), as the

formal sum [~ ] = ~

6-ii0, "'"

?

183 If the boundary values

exist in the sense of distribution,

show that the hyperfunction hand, ~n, f

Ehrenpreis

is equal to the sum.

On the other

[7] has proved that for each distribution

one can find a holomorphic

function

f

~ ~ ~(cn ~n)

on

such that

is equal to the sum of the boundary values of If

P(x, D) = ~ a ~ ( x ) ~ J ~ I

with real analytic operator V

[~ ]

one can

[71,

coefficients

P(z, D) =

~0..

of

/ ~x ~

~a~(z)

The class

on

is a linear differential ~

,

it can be extended to the on some complex neighborhood

~l~I/~z~

[P(z, D) ~ ]

operator

is determined

only by the class

and we define

P(x, Differential Let constant

P(D)

D)[ ~ ] = [ P ( z ,

equations with constant

be an

rl× r 0

coefficients.

(2)

D)~ ] . coefficients

matrix of differential

We consider the differential P(D)u(x)

assuming

that the known

and in

~ (~)

f(x)

operators with equation

= f(x)

and the unknown

u(x)

are in

~(~) r I

r0 Petrowsky, are

respectively.

Ehrenpreis,

C ~ functions,

distributions

the following results Theorem I.

Malgrange

There

The same equation was discussed by and HUrmander when or real analytic

f(x)

and

functions.

u(x) We have

similar to theirs. is another

system

PI(D),

called the compati-

bility system, such that for any convex open set ~ (2) has a solution r0 r1 u ~ ~(~_) if and only if f e ~(~L) satisfies

Pl(D) f

= 0

r Theorem 2. (3)

If any hyperfunction P(D)u

= 0

solution

u ~

~(~n)

0

of

184

is a distribution elliptic. r0 ~(~)

If

on a neighborhood

P(D)

is elliptic,

of the origin,

then

any hyperfunction

P(D)

solution

of (2) is real analytic on the open set on which

is u

f

is

real analytic. Theorems

1 and 2 for single equations

given by R. Harvey as his thesis work

[I0],

of Theorem 2 was proved also by Bengel The existence to Ehrenpreis [14].

r 0 = r I = I)

[II].

The second part

[6] and has been proved by Malgrange system

PI(D)

For single equations

in Theorem 2 is in the sense of H~rmander

He proves there that if any C m solution on in a neighborhood

of the origin,

is elliptic,

then

any distribution

analytic on the set on which

f

]Rn

P(D)

system

P(D)

hypoelliptic

is elliptic and that if

solution

u

of (2) is real

is real analytic.

if all distribution

The second part Hormander calls a

solutions of (3) are

The first part of T h e o r e m 2 shows that hypoellipticity

meaning for hyperfunctions. Chou [5] and Bj~rck

[4]

[12].

of (3) is real analytic

of T h e o r e m 2 is stronger than that of H~rmander.

C ~.

from

PI(D) = 0.

The ellipticity

P(D)

is due

[18] and H~rmander

is found algebraically

and the same system works in any case.

we have

are

[2], [3].

theorem for C ~ functions and distributions

The compatibility

P(D)

(i.e.

loses its

This fact has also been pointed out by Chou shows that if

then there is a non-distribution

solution

u

P(D)

is not elliptic,

of (3) which belongs

to

the dual space of a Gevrey class of functions. The detailed proofs of Theorems

1 and 2 are given in [15].

In

view of (I) T h e o r e m 1 follows from the existence theorem in the case where

f

and

u

are holomorphic

functions and the latter is proved

185

by the standard method since Malgrange [16].

The proof of the second

part of Theorem 2 is reduced to the fact that any holomorphic solution of (3) in the domain

~z;

Izl< I, Im zj > 0

analytic extension across the real space all [6].

for all

Jl

has an

Izl < i, Im z. = 0 for J Harvey proved this by the fundamental principle of Ehrenpreis

j].

{z;

The proof in [15] is more elementary.

Bengel [2, 3] employs his

theory of P-functionals and reduces the proof to a much simpler extension problem.

Resolutions of sheaves of solutions Let us denote by

~ , 6',

~

and

~

the sheaves of hyper-

functions, distributions, C ~ functions and real analytic functions, respectively, on

~n

,

and generally by

~

one of the above.

Since

differential operators are local operators, the system P(D) defines r0 rI a sheaf homomorphism P(D) : ~ ~ ~ The kernel is exactly the sheaf of solutions of equation (3), which we denote by

P

Since convex open sets form a fundamental system of neighborhoods of any point, it follows from Theorem I and the corresponding existence theorems for

~ ', 0

is exact.

~

and

~ ~p

~ ~ r

that 0 P(D) ~ rl PI (D)

Considering a compatibility system

r2

P2(D)

for

PI(D)

and

so on, we can extend this exact sequence as long as we wish, so that we obtain a resolution of 0

~ ~P

~ P

in the form:

; ~ r 0 P(D) ~ rl PI(D) ~ r 2 rm-I Pm-i (D)

~ .... r

From the Hilbert sygyzy theorem [23] it follows that we can add

~ 0

186

at the end for some

m~

n.

Theorem 2 says that

P(D)

is elliptic if and only if

and H~rmander's theorem says that if

~'P

= ~ P.

If

P(D)

P(D)

~P

= ~P

is hypoelliptic if and only

is elliptic, we have, therefore, the flabby

resolution of the sheaf of regular solutions: 0

• r 0 P(D) ~ r I PI(D~ ~ r2

~p

> ...

Applications I.

For single equations we have a seemingly stronger existence

theorem. Theorem 3 (Harvey [I0]). open set

If

P(D)

is single, we have for any

~c~n

(4)

P(D) ~(TA) = ~(~-)Proof.

Since the compatibility system

Theorem I implies (4) for convex open sets has only to extend tion

u

on

~n

f ~ ~(~)

PI(D) = 0 ~ .

For general

to a hyperfunction on

and restrict it to

This shows that any open set

in this case,

~n,

~L

one

find a solu-

~L . /icon

to hyperfunctions for any single operators

is P(D)-convex with respect P(D),

contrasting with

the cases of C ~ functions, distributions and generalized distributions of Beurling type (cf. [13], [4]). 2.

If

P(D)

with respect to

is single elliptic, any open set is P(D)-convex ~

= ~ ,

~ ',

Theorem 4 (Malgrange [16]). (5)

P(D)$(~)

for any open set Proof.

~

and

If

P(D)

is single elliptic, we have

= ~(~)

~Lc~n

It follows from Theorem 3 that for each

f ~ ~(~L)

one

187

can find a solution point

x ~.

$L'

of

x.

Let

P(D)

u'6

On the other hand, for each ~(~')

The difference

so that it is analytic. 3.

of (2).

there is a solution

neighborhood (3),

u ~ ~(~)

defined on an open

u-u'

Therefore,

satisfies equation

uE ~(~').

be again a single elliptic operator.

Then we

have the flabby resolution : P(D) 0

~P

In particular, if groups

K

H~@R , 0_P)

~3

~0

.... ~

is a subset of

~n,

.

the relative cohomology

are computed as the cohomology groups of the

complex : 0 where

~ K ~ R n)

support in K

P(D) > ~ K ( ~ n)

~ ff3K(~n)

~ 0,

K.

Consequently we have

HP(QRn , ~

H Kl(~n, ~ P )

is compact, we ca~ compute

If

K

of

HI(~Rn, ~ P ) (K)'

P(-D)v = 0

= 0

for

p > i.

Let

P(D)

be single

~- ~P'(K)'

denotes the dual space of the space of solutions

on some neighborhood of

K

p' P'(D)v = 0

Let

P'(D) = P(-D).

The sheaf

~

of solutions of

has the resolution: P' (D) o~

~p,

~,:

~..<

~

~,

o.

By Malgrange's theorem [17] we have H p(K, ~ )

= 0

Therefore, the cohomology groups

v

equipped with the natural

locally convex topology. Proof.

If

is compact,

(6) where

P)

with

in the following way.

Theorem 5 (Grothendieck, Bengel [I, 3]). elliptic.

~Rn

denotes the space of hyperfunctions on

for

p > 0.

H*(K,

~P')

are computed as the

188

cohomology

groups of the complex: P'(D) ~(K) ~

0 < K(~ n)

and

~(K)

are dual operators P'(D)

is onto.

~(K) <

0 .

are dual to each other and

to each other.

Therefore,

we have by Serre's

If we use the resolutions re

~

0

d

I

where

d

lemma

[22]

= £ P ' (K) ,

d "'"

0<

P'(D)

:

d>

>

•

and

It follows from Theorem 4 that

l n , ~ P ) "= H 0 (K, ~ P ' ) , HK(~ 4.

P(D)

<

n ~

) 0

"''~

~

is the exterior differentiation,

¢<

0 ,

we obtain the following

theorem by the same method as above. Theorem 6 (Alexander-Pontrjagin).

If

K C ~n

is a good compact

set, e.g.

1) for all

dim H p(K, e) {_ ~ 0

p,

then

(7)

H~

n, g) ~ Hn-P(K,

Especially if

¢)',

1 n, ¢) HK(~

the duality between

b n-I = dim Hn'I(K,

¢)

p = 0,1,...,n.

Theorem 7. in

~n

system

I) set.

and let P(D)

~

K

be a compact

be one of

the following

Every compact See Proc.

¢)

holds

theorem follows from Theorem 6

theorem in the case where Let

Hn-I(K,

is at most countable.

The classical Alexander-Pontrjagin and the following

and

~

set contained

, ~'

and

~

in an open set Then for any

sequences are exact :

set satisfies

Japan Acad.

P(D) = d.

this and hence is a good compact

44 (1968), 489-490.

V

189

0 --~ H~(V, }P) --+ H0(V, }P) --~ H0(V-K, }P) --~ H~(V, ~P) --> 0 0 --->HP(v, ~P) --+ HP(v-K, ~P) --~H~+I(v, ~P) --~ 0,

for

p ~ I

This is an easy consequence of Theorem 1 and corresponding theorems for

6'

and

In particular, we have Theorem 8 (Jordan-Brouwer). in an open set

V

in ~ n

Let

K

be a compact set contained

and such that

b n-I = dim Hn-I(K, ¢) is at most countable, l) V -K

is equal to the sum of

components of Proof. 0 Since

Then the number of connected components of b n-I

and the number of connected

V. Clearly

H~(V, ¢) = 0.

> H0(V, ¢)

dim H0(V, C)

components of

V

and

and

Hence we have the exact sequence:

~ H0(V-K, ¢)

~Hn-I(K,

dim H0(V-K, C)

are the numbers of connected

V -K

¢)'

> 0.

respectively, we have the desired result.

References [i]

G. Bengel:

Sur une extension de la th~orie des hyperfonctions,

C. R. Acad. Sci. Paris [2]

G. Bengel:

262 (28 f~v. 1966), 499-501.

R6gularit~ des solutions hyperfonctions d'une

4quation e]liptique,

C. R. Acad. Sci. Paris

262 (7 mars 1966),

569-570. [3]

G. Bengel:

Das Weylsche Lemma in der Theorie der Hyperfunktionen,

Thesis, Univ. Frankfurt, [4]

G. Bjorck:

1966.

Linear partial differential operators and generalized

distributions,

Ark. fWr Mat.

6 (1966), 351-407.

190

[5]

C. C. Chou: Sci. Paris

[6]

Probl@me de r@gularit@ universelle,

C. R. Acad.

260 (1965), 4397-4399.

L. Ehrenpreis:

A fundamental principle for systems of linear

differential equations with constant coefficients and some of its applications,

Proc. Intern. Symp. on Linear Spaces, Jerusalem,

1961, pp.161-174. [7]

L. Ehrenpreis:

Analytically uniform spaces and some applications,

Trans. Amer. Math. Soc. [8]

R. Godement:

I01 (1961), 52-74.

Topologie Alg@brique et Th@orie des Faisceaux,

Hermann, Paris, 1958.

[9]

A. Grothendieck:

Local Cohomology,

Seminar at Harvard Univ.,

1961. [I0] R. Harvey:

Hyperfunctions and partial differential equations,

Proc. Nat. Acad. Sci. U.S.A., 55 (1966), 1042-1046. [ii] R. Harvey:

Hyperfunctions and partial differential equations,

Thesis, Stanford Univ., 1966. [12] L. Hormander:

Differentiability properties of solutions of

systems of differential equations,

Ark. f~r Mat., ~ (1958),

527-535. [13] L. Hormander:

Linear Partial Differential Operators, Springer,

Berlin, 1963. [14] L. Ho'rmander:

An Introduction to Complex Analysis in Several

Variables, Van Nostrand, Princeton, 1966. [15] H. Komatsu:

Resolution by hyperfunctions of sheaves of solutions

of differential equations with constant coefficients,

Math. Ann.

176 (1968), 77-86. [16] B. Malgrange:

Existence et approximation des solutions des

191

@quations aux d@riv@es partielles et des @quations de convolution, Ann. Inst. Fourier, [17] B. Malgrange:

6 (1955-56), 271-355.

Faisceaux sur des vari@t~s analytiques r@elles,

Bull. Soc. Math. France, 83 (1957), 231-237. [18] B. Malgrange: constants,

Sur les syst@mes diff@rentiels ~ coefficients

S@m. Leray, Expos@s 8 et 8a (1961-62), CollSge de

France. [19] A. Martineau:

Les hyperfonctions de M. Sato,

Sdm. Bourbaki,

1-3 (1960-61), No.214. [20] M. Sato:

On a generalization of the concept of functions,

Proc.

Japan Acad., 34 (1958), 126-130 & 604-608. [21] M. Sato:

Theory of hyperfunctions,

J. Fac. Sci. Univ. Tokyo,

Sect.l, 8 (1959-60), 139-193 & 387-436. [22] J. -P. Serre:

Un th@orSme de dualit@,

Comm. Math. Helv., 29

(1955), 9-26. [23] J. -P. Serre:

AlgSbre Locale,

Multiplicit6s,

Lecture Notes

in Math., ii (1965), Springer, Berlin.

Department of Mathematics University of Tokyo Hongo, Tokyo

RELATIVE COHOMOLOGY

OF SHEAVES OF SOLUTIONS

DIFFERENTIAL

OF

EQUATIONS*

By Hikosaburo K O M A T S U

INTRODUCT ION

Suppose operators If

Y

that

defined

P = P(x, D) on a m a n i f o l d

is a subset of

tions of

Pu = 0

is a system of linear d i f f e r e n t i a l

X,

on

Y.

X

acting on a function

we denote by If

Z

~P(Y)

is a subset

space

~

.

the space of solu-

of

Y

we have a natural

restriction mapping :

P(Y)

~ ~P(z).

What we are interested

in is this m a p p i n g and,

conditions

the kernel or the cokernel

problems

under w h i c h

of the differential

questions.

For example,

is a hypersurface. uniqueness

consider

way.

The most

elliptic

Y

for given

to solutions

on

Y.

operators w i t h constant

* Lectures Footnotes

The Hartogs

functions

Z

Z

that the

to

Z.

On

that the

theorem on remov-

is also

stated in this seems to be the

such that all solutions

on

This has been solved for single coefficients

and for the exterior

given at S@minaire Lions-Schwartz, are added on March

and

is zero means

interesting p r o b l e m in this direction

one to find maximal are extended

of h o l o m o r p h i c

to these

Y = X

is zero means

the

Many

are reduced

for the Cauchy p r o b l e m relative

the fact that the cokernel

singularities

is zero.

the case in w h i c h

Cauchy p r o b l e m has always a solution. able

Pu = 0

The fact that the kernel

theorem holds

the other hand,

equation

in particular,

2, 1972.

December,

1966.

Z

193

differentiation of order

0 trivially and for the Cauchy-Riemann

system by the Oka-Cartan theory.

However, we do not know the general

answer. If the function space forms also a sheaf. particular,

forms a sheaf, the space of solutions

Thus, we can employ the sheaf theory and, in

the cohomology theory.

So, generally,

let ~

the space of sections mapping:

~

~(Y)

be a sheaf over

~(Y,

> ~(Z)

~).

(Y, Z).

Therefore,

H~_z(Y,

~)

9)

~ Hl(y, ...

q)

,HI(z,

~)

,

the restriction is one-one if and only if

and it is onto if

H~_z(Y,_ ~ )

= 0.

condition becomes also necessary if the mapping: ~)

be

are the relative cohomology groups of the pair

H$_z(Y , ~)

HI(z,

~(Y)

Then we can embed the restriction

2 (y, q ) ~ Hy_ Z

= 0,

and let

into the long exact sequence :

i (y, > Hy_ Z

where

X

HI(y,

The last ~)

>

is one-one.

Of course, this reduction is non-sense unless we have a method to compute the relative cohomology groups.

As the first step toward

solution we consider the simplest case in which mined system with constant coefficients and in

Y.

For example, if

duality between

H~(Y,

P

S

P

is an overdeter-

is relatively compact

is a nice system, we can establish the

~P)

and

H~-P(s,

~'Q)

for some system

Qo

This includes the Alexander-Pontrjagin duality and thus we can prove the Jordan-Brouwer theorem by a purely analytic method. First we describe the theory of relative cohomology developed by Sato

[9]

and by Grothendieck

[3] in a more general setting to include

194

Cartan's theory of relative cohomology

[2].

This is fundamental both

to our discussion and to the foundation of hyperfunctions. part deals with the theory of hyperfunctions [8] and Harvey functions

[4].

We follow mainly

[4].

by Sato [9], Martineau This class of generalized

is convenient for the purpose to compute the relative

cohomology groups because it forms a flabby sheaf. theory of differential Ehrenpreis [6].

The second

Then we need the

equations with constant coefficients by

[i], Malgrange

[7], H~rmander

[5], Harvey

[4] and Komatsu

Lastly we formulate our duality theorem and give a partial

solution to the problem.

I. RELATIVE COHOMOLOGY Suppose that X

X

is a topological

space and

~

is a sheaf over

of abelian groups or modules over a commutative ring.

sake of simplicity we assume that all open subsets of compact.

This is the case if

particular,

X

is a metrizable

For the

X

are para-

space or, in

a locally compact space with a countable fundamental

system of open sets.

We use this assumption to the effect that the

restriction of a flabby sheaf over

X

to any subset of

X

is also

flabby and that the restriction of a soft sheaf to any locally closed set is soft (cf. Godement Let H~(X,

S

~)

be a subset of

with support in

H~(X, ~ ) sets in

[2] Th~or~mes 3.3.1 and 3.4.2). X. S

Then the p-th cohomology group is by definition the cohomology group

with the family of supports X

contained in

S.

(Cf. [2].

~

composed of all closed

This notation has a different

meaning from that used in [3] and [9] unless

S

is closed.)

is also called the p-th relative cohomology group of the pair

H~(X, ~ ) (X, X-S)

I95

and denoted by If

S

HP(x, X-S,

is open,

of

so that

~ .

or by

HP(x rood X-S,

then the corresponding

is paracompactifying resolution

~)

H~(X,

Otherwise,

~

~)

~).

family of supports

is computed by a soft

is not paracompactifying

and

therefore we need a flabby resolution. In addition relative

to the usual properties

cohomology

groups

of cohomology

satisfy the following

groups,

fundamental

the

prop-

erties. Theorem i.I. containing S

S

is closed

(ii)

(i) (Excision theorem)

and such that any closed set in

X,

and

F N (X-Y) = ~ ,

•

l(x , ~)

More generally

the following

in

Y

contained

X in

then

~ Hl(x,

Proof.

(i.i)

0

> H x_Y (X,

X D Y ~ Z

9)

~

be a flabby resolution

....

~

...

be a triple of subsets.

0 (x '~) "Hx-z

>

>

~-...

~ £o of

I (X HX_ z , ~ )

has a fundamental

Then

HO _ z (Y'~)

olution of

over

sections of

over ~P)

X

~1 X,

~...

which we will denote by

system of paracompact

of the resolution Y.

~

over

the restriction

ps(Y,

~)

(i) Let o ~

Y

let

>

is exact:

0 "

s ~

F

is a subset of

0(x,

HS

Since

Y

We have the exact sequence :

0

(iii)

If

to

Y

can be uniquely

in

X,

turns out to be a flabby res-

Let us denote by with supports

neighborhoods

~*

in

extended

~s(X, S.

~)

the space of

Clearly any section

to a section

t e ~s(X, ~ P )

196 and any

t ~ Vs(X , ~ P )

has the restriction

s ~ Ns(Y, ~ P ) .

There-

fore:

HP(X, ~ ) =

HP(Ps(X,

£*))g

HP(['s(Y, ~ * l y ) )

=

Z

(ii) is a special case of (iii) in which (iii)

is empty.

For any sheaf

o

rx_y(X,

•

i s c l e a r l y exact.

Z)

If

For, any section

s

...

~

~ Px_z(X,

Y

Y

If the support of

is disjoint with

t.

the restriction

in

0

X

and hence to a section Z,

~*

is a flabby resolution of

from the above exact sequence for

~

t

over

X.

then evidently so is

Take any flabby resolution

~*Iy

on the right.

can be extended to a section over an

open neighborhood of

the support of

Fy_z(Y, ~ )

is flabby, we can add

over

s

Z)

= ~P

of ~Iy,

$

Since

(iii) follows

by the standard method

of homological algebra. Remark. space

X.

In case

S

is closed,

On the other hand,

condition that only

X

if

(ii) holds for any topological

S

is open,

is paracompact.

(ii) holds under the

Similar remarks apply to

what follows as well.

Dimension of sheaves We say that a sheaf m

~

over

X

is of flabby

(soft) dimension

and write flabby dim

if there is a flabby

0

~ ~ m (soft dim

(soft) resolution of

~__~£0

flabby

(soft) dim

dim ~

~ 0

~ ~ -i

if and only if

~I

~ ...

if and only if ~

is flabby

$ ~ m) of length m

~Z ~ = O,

(soft).

m :

70, flabby

(soft)

197

Theorem 1.2. (a)

The following are equivalent.

is of flabby (soft) dimension H~(X, ~ ) = 0

(b)

for

p > m

~ m.

for any (paracompactifying)

family

of supports (c)

H~I(x,

~)

= 0

(d)

The restriction mapping

(1.2)

for any closed (open) set

Hm(X, ~ )

(d) ~

in

X.

~Hm(y, ~ )

is onto for any open (closed) set Proof.

S

Y

Immediately we have

in

(a)

X. > (b) ~

(c) ---~ (d).

(a). Let 0

> ~

• ~0

~ ~i

> ...

be a flabby resolution and let

~m = d ~ m - i

~m

Hm(X,

is flabby (soft).

Hm(y, ~ )

We have

= p (y, ~m)/d p(y, ~m-1)

induced by the restriction

~ £m

~ ~m+l

>

•

l

l

We want to show that

~) = •(X, ~m)/d [~(X, ~m-l), and the restriction

f XY : P (X, ~m)

(1.2) is

~- P (Y, ?m ).

Thus

the fact that (1.2) is onto implies ~(y, ~ m ) = Since

~

m-i

~ Xy F ( X ,

m ) + dP(Y,

£m-l).

is flabby, we have d P(Y,

£m-l)

= d~ Xy V ( X ,

Z m-l)

= ~ yX d P ( X ' gm-l) Therefore p(y, ~ m ) =

~ X ~( X y

~m) '

•

Corollary. soft dim ~ Proof. m-l.

~_ flabby dim ~ ~_ soft dim ~+I.

The first inequality is immediate.

Then we have

lar, the restriction

Hm(y,

~) = 0

(1.2) is onto.

Let

for any open set

soft dim ~ Y.

~_

In particu-

198 The flabby (soft) dimension of a sheaf is determined locally. Namely if

~

is a sheaf over

X

of flabby (soft) dimension

then its restriction to any (locally closed) set (soft) dimension

~ m.

flabby (soft) dimension over

X.

~m = d~m-i

~ m

If

then

Let

Hm+l(x, ~)

~

x

in

over which

X

~

there

is of

is of flabby (soft) dimension

Let

~',

o

>

~'

and

and

Y

Y . Therefore x

~m

is flabby

~ 3.1 (~3.4). ~

and

9"

be sheaves over

~

are of flabby (soft) dimension ~"

such that

~)

~ m+l

is of flabby (soft) dimension N m .

be an arbitrary open (closed) set. Hm+I(Y,

X

,o

respectively, then

Proof.

of

is flabby (soft) on each

Theorem 1.3.

and

~ m,

Yx

x

In fact, from the proof of Theorem 2 it follows that

(soft) by [2] Chap.II

is exact.

is of flabby

Conversely if for each point

is an open (closed) neighborhood

m

Y

~ m,

Since

vanish, the rows of the commutative

diagram

Ha(x,

~)

"~ Ha(X,

~")

Ha(y,

~)

'Ha(y,

~ ,,) '" >-Hm + l (Y,

0 are exact.

> HtTrI-I(x, ~ ' )

0

'" ~- 0

~ ' ) .... ~-0

0

By Theorem 1.2 the first and third columns are exact.

Therefore the second column is exact as is shown by a simple diagram chasing. Corollary.

If the sequence of sheaves

0

,

is exact and if

~j

are of flabby (soft) dimension

is of flabby (soft) dimension Proof.

~_ m+j,

then

~_ m.

Decompose the exact sequence into short exact sequences

199

and apply Theorem 1.3 successively.

Derived sheaves associated with relative cohomology Let in

X,

X, S

and

~

be as above.

If

U D V

are two open sets

we have the natural restriction mapping: U

(v,

:

U ~V : rsau(U' ~ *)

induced by the restriction

~ ~s~v (V' £ *)"

Since the restriction obeys the chain condition, ~H~nu(U,

~ ),

~vU I forms a sheaf data (i.e. a pre-sheaf) over

We denote by

~i~(~)

~

with support in

the same sheaf the sheaf of p-distributions DistP(s, ~ ).)

The stalk

because any section in

~(~)x

~Snu(U,

is an interior point of p > O.

with the sheaf complement.

S,

~P)

~S

0(8 ), ~S

(i)

0 }~S(~)

Irsmu(U,

x

is not in

S

S

=~x

is open, over

S

and 0 ~ S(~ )

x

JiP(~) x coincides

and zero over the

S.

~ )~

is the sheaf data of sections of U

in

X

we have

0 r(U, J{ S(~ )) = r Snu(U, ~ ) . (ii)

For any family of supports 0 r~(x, ~ s ( ~ ) )

(1.4)

~

in

= F~is(X,

X ~ ),

I I S = {A 6 ~ ; A C S~. (iii)

Then

and denotes it by

is the maximal subsheaf of

i.e. for any open set

(I.3)

where

~

On the other hand, if

0 71S(~)x

if

whose sections have supports in eemma.

x.

which induces

In general

(M. Sato [9] calls

becomes zero if it is restricted

we have

In particular,

of

S.

vanishes when

to a sufficiently small neighborhood of

for

X

the sheaf associated with the data and call it

the p-th derived sheaf of

= 0

the system

Let

~ P(~)

0

~ ~

~ ~*

be a flabby resolution of

is the p-th cohomology sheaf of the complex of

~

.

200 sheaves

o

(1.5)

sO(£o)

,~

j{P(~) If

I

1)

~

....

= ,}q,P(j.{70

s (iv)

o £ ~ }~s( s (£*))"

is flabby and

S

is closed, then

Jq~(~ )

is

flabby. (v)

If

I

Proof.

is soft and

(i)

S

is open, then

~{0(~)

We can easily check conditions

is soft.

(F I) and (F 2) of [2]

Chap.ll, ~ i.i, so that (1.3) holds. (ii)

s ~ r i (X, ~{S0 ( 9 ) )

Because of (i) any section

P s(X, ~ ).

Clearly the support of

the same as that as a section of (iii)

Let

U

HP 0(u,

=

s

~

be an open set. k e r ( P (U,~

U

tend to a point

In view of (i) we have 0 £p+l) )) > P(U,~s(

(•P))

x.

tion

P U' I ) Hsnu( (iv)

s to

Let

s

as a section of X

im(J~

tends to

ker(~fSO ( £ P ) x

Any section

by zero.

S.

Since

Therefore we have (I.5) .

0 J4 S ( ~ )

over an open set U u CS

U.

by zero and then

The extension has support in

S

r (x, ~ s

s

of

regarded as a section of contained in

~

By defini-

°(1))

and therefore belongs to (v)

j~P(~)x .

can be extended to

by the flabbiness of

• ~f 0(£p+l)x)

; j~ S0(• p)x) "

(~P-I)E

be a section of ~

~ P (u,J4~(£ P)))

Since exactness is preserved under

inductive limit, the kernel tends to and the image tends to

is

Hence (1.4) follows.

im( p (U, J<~(l P-I)) Now let

0 j~ s ( ~ )

as a section of

.

is in

~ X

J~ S0(i ) over

A

over a closed set

A

may be

with support in a closed set

is regular, we can extend

s

to

A u CS

The rest of proof is the same as above.

The correspondence

I~-~P(~s )

is a functor of the category of

201 sheaves of abelian groups (modules) to itself. phism

h : ~

0%{P(~')

~ ~'

induces a homomorphism

with natural properties.

sarily exact.

If

0

, ~'

Namely, a homomorh* : ~{ p ( ~ )

The functors ~ ~

> ~"

;

~{P

are not neces-

~ 0

is an exact

sequence of sheaves, then the following sequence is exact :

,~s(~ ,)

o

l(

,)

"as ~

~)

,,)

~j{l

s(9)

' " "

This is derived easily from the long exact sequence of cohomology groups. The case where Theorem 1.4 and

~{[ = 0

S

is open is particularly simple.

If

•

for

S

is open, then the functor

is clear.

Thus if

0

sequence

~'{ S ( ~ * )

= 0

p > 0.

for

~*

If

S

family of supports

Hp(x,

Therefore

If

in

the

0

= )~P(J{S ( £ * ) )

X

0

0

J~[ ( ~ )

~,

is open, we have for any paracompactifying

<x, = Hp

Proof.

is exact

in this case, the exactness of

is a flabby resolution of

is exact.

Theorem 1.5.

(1.6)

S

p > 0.

Proof. Since ~ S 0 (~) = ~S .~{0 S

j{0

~

s(S

~*

)

p = 0, I, 2, "'"

is a soft resolution of

~ ,

then 0

~ ~S0(@)

is a soft resolution of S

is open,

~S

~]40(10,s)

~ ~(£1)

~{S0( 9 ) by hemma (v) and Theorem 1.4

is also paracompactifying in

follows from Lemma (ii) that H p~s

, ...

(x, ~ ) =

H p (r~is(x,

£ *))

X.

Since

Therefore it

202

The isomorphism

HP~|s(X' ~ ) = HP~;s(S' ~ )

is proved in the same

way as the excision theorem. In particular, the relative cohomology group

H~(X, ~ )

is

0 HP(x, ~ S (~)).

reduced to the ordinary cohomology group

Further, we have an explicit representation of the quotient sheaf ~/j{~(~). ~IX_ S

If we denote by

over

X-S

(1.7)

and zero over 0

0

is exact.

~X-S

~s(~)

the sheaf over S,

then

) ~

"

~x-s

X

which induces

,0

Actually Cartan's theory of relative cohomology starts

with this formula (cf. [2]). Since

X-S

is closed, we have

~(~, ~×-s) f o r any f a m i l y o f s u p p o r t s

~

= in

the long exact sequence associated

~[x-s (~-~, ~) X ( [ 2 ] Th6orSme 4 . 9 . 1 ) . with

(1.7)

gives

Thus

the following

resfilt. Theorem 1.6.

Let

family of supports in

X

X.

be open and let

~

~

' H~Is(S,

" H (X,

~ )

~(~ , ~) ~ )

"~l~_s(~ s, ~) > "'"

is locally compact, we denote by

compact sets in

be a paracompactifying

Then the following sequence is exact.

0 > H~Is(S, ~)

0

If

S

*

the set of all

X,

which clearly forms a paracompactifying family

If

X

of supports. Corollary.

is locally compact and

S

is open, then the

following is exact :

0

0 ~ H.(s, ~ ) 1 > H.(S, 9)

, H °(x, ~ )

~ H °(x-s, ~ )

203

The case where Grothendieck [3].

S

is closed has been discussed by Sato [9] and

Note that we do not need any assumption on

X

in

this case. Theorem 1.7. X

and let

~

Let

S

be a closed set in a topological space

be a family of supports in

X.

Then for any sheaf

there is a spectral sequence with the second term

such that the limit

E~

is the bigraded group associated with a

filtration of the graded group Proof.

H~Is(X , ~ ) .

This is a consequence of Lemma (ii), (iii) and (iv) as

[2] Th~or~me 4.6.1 shows.

In fact, let

flabby resolution with homomorphism

d"

0

• ~

where

~'~(~')

homomorphism

0 £q

~ F ~ (X, ~P(J{S ( P,q

~ ~(~*) :

))),

denotes the canonical flabby resolution of d'.

be a

and consider the double

complex associated with the complex of sheaves K =

~ ~*

~'

with

The second term relative to the first filtration

is given by 'Epq = H~(X, jgq(j{0(~,)))

because the functors

[~

and

Cp

are exact for flabby sheaves.

On the other hand, we have

q>O.

Thus "~Pq-2 =J Hp~]S(X' ~ ) , 0 ,

q = 0 q>0.

204 This shows that HPIs(X,

~ ) = "E p0 ~ "E p0 ~ H p(K).

The f o l l o w i n g t h e o r e m i s f u n d a m e n t a l in a p p l i c a t i o n . Theorem for

1.8.

Suppose that

q = 0, I, ''', m-l.

is a closed set and

m ~ H s n u ( U , ~ )}

Then

m ${ S ( ~ )

of the sections of

S

~{q(~ ) = 0

forms the sheaf data

:

m

HSnu(U,

~ )=

and for any family of supports

~

P(U, ]{ms(9) ) in

X

0,

(1.8)

HPis(X,

we have

p = O, I, --., m-I

> = F { (X, J { S ( ~ ) )

,

p = m.

If moreover, ~{q(~)

= 0

for all

q # m,

then 0 ,

(19)

HPjs(x' Proof.

definition

Therefore for

Let that

m

= H -m(x, XS( ))m , p

~{ sq(~)

= 0

~{ qr~u(91U)

the first

p<

statement

for

q < m.

= ~'[q(~ )1U

It is clear from the

f o r any open s e t

U.

f o l l o w s from ( 1 . 8 ) by s u b s t i t u t i n g

U

X. In view of Theorem 1.7, we have

sequence for q < m

and

p+q ~_ m

HI[s(X , ~). E r0m = E 20m

we have

degenerated

The case where case. ~P(~)x

Let = 0

U D S if

for

Hence it follows

that

for

(1.8).

spectral

E Pq = 0

r ~_ 2.

Comparing

(1.9) is immediate

q <

m

EPqr = 0

spectral

a

as

for Epq

the terms

r -~ 2, with

from the theorem on

sequences. S

is locally closed is reduced to the closed

be an open set in which x ~ U

and

J{P(G)xb ~

S

is closed.

= ~ { P~( @a, [U)x

if

Clearly X

E

U.

In

205

other words, we have

~{~(~)

=~f~(j~(~)).

Thus it follows from

Theorem 1.5 that for any paracompactifying

This is by Theorem 1.7 the term limit term

e~

theorem.

to

i

in X

of a spectral sequence whose

H~IU~s(U,

is the bigraded group of

which is in turn isomorphic the excision

E~ q

family of supports

H~Is(X , ~ )

by t h e same r e a s o n as

Thus Theorem 1.7 and i t s

1.8 hold also for locally closed sets

S

~t U) =H~ts(U, ~ l U ) ,

c o n s e q u e n c e Theorem

if we restrict

~

to para-

compactifying

families of supports and if all open sets of

X

are

paracompact.

Then, Theorem 1.5 is a special case of the generalized

Theorem 1.8.

Pure codimensionality A set

S

in

X

is called purely m-codimensional with respect

to a sheaf

~

if ~ ( ~ )

= 0

for all

open set is purely 0-codimensional. m-codimensional

If

q # m. S

By Theorem 1.4 any

is a locally closed purely

set, then (1.9) holds for any paracompactifying

family of supports (I.i0)

~ .

In particular,

H~nu(U , ~ )

for any open set

U

since

we have

= HP-m(u, ~ ( ~ ) ) S ~ U

is purely m-codimensional

Conversely if (i.i0) is true for any open set m-codimensional

Let

S

be a locally closed set in

m-codimensional with respect to we

then

S

U.

is purely

owing to Theorem 1.4.

Theorem 1.9.

Then

U,

in

~

and let

T

X

purely

be a subset of

S.

have 0

Proof.

If a p o i n t

x

,

is

not

in

T,

p<

m

the

stalks

over

x

of

206

both sides vanish.

If

of open neighborhood Since

J~qnu ( ~ I U >

x U

is in of

S,

there is a fundamental system

such that

x

= ~q(~)IU

S ~ U

is closed in

, we have p<

0 I by Theorem 1.8. Corollary.

-

m

Thus

(I.Ii) follows.

Let

S

sional with respect to

U.

m

p ~_m

be a locally closed set purely m-codimen~ .

Then a subset

n-codimensional with respect to

~

T

of

S

is purely

if and only if it is purely

(n-m)-codimensional with respect to

~-~(~).

Relative cohomology groups of coverings Let

X,

case where

S S

Suppose that i~ I I and

and

(~, ~'

V. 1

be as above.

We restrict ourselves to the

is either closed or open, though this is not essential. ~')

is a covering of

= {Vi; i ~ I ' I

spectively with closed,

~

I'C I.

are open ;

(X, X-S),

are coverings of

X

i.e. and

'IT = {Vi;

X-S

We assume the following : in case in case

S

is open,

V. 1

reS

is

are closed and

is locally finite. Then a relative p-cochain a direct product of sections for all non empty

Vi O'''i

~ e cP(%~,

4)

is by definition

~i 0 .. .i 6 ~ (Vi0.. .i ). = . ~(Vi . 0 P P = Vi 0 ~

P = O, ?...i...j.. +~...j...i...

~',

= 0

"°" n V.l P

and

such that

~io... i

= 0

i ' ~) P

~...i...i...

if all

i k ~ I'.

P The coboundary mapping

~p

: cP(~,

]J'', 5 )

~ cP+I(IY, V ', ~ )

is defined by (~?)io

"" "ip+l

=

p+l i 0 • ' "{j .... ~ (-i) j . IP+I j=O ~i O. ip+ I ~i0..-{.'''i J p+l

207 It is easy to prove that

~ p + l ~ p = 0. (~-, ~ ' )

The relative cohomology groups of the covering coefficients in

~

with

are by definition the cohomology groups of the

complex :

0

~cO(~J ", q)",

As usual we denote by kernel of

~p

= Z0(~,

~',

>CI(].,~,

zP(q~, I~', ~ )

and the image of

Lemma 1.2. Proof.

~)

B0(~,

~ ).

If

~',

and

~p-i

H0(%F, ~F', ~ ) =

Since

°0-' , q ) BP(~,

~ ).

~ ) = 0,

we have

j.

Slv .. i where

Thus

~

there is a

s(x) = ~i(x)

= 0.

Conversely,

V. i

with

Thus

s

i E I'

~

s

such that

is open.

which contains

has a support in

every section

s ~ ~s(X, ~ )

Proof. cP-I(~, X

~U

~i =

In the case

~',

Let £ )

~ E zP(~, with

(p-l)-cochains = ~I U"

~',

~

£),

= ~ ,

~U

in

The set of all

for all p _~ I.

so that

by

E s.

p > 0. To construct

~ E

we consider for each open set

cP-I(~aU, *I U

x,

If

determines a 0-cocycle

?i = Slv." We denote this cocycle ~ i Lemma 1.3. If ~ is a flabby sheaf, then ~', ~ ) = 0

i

S.

with

HP(~,

in

is clear if

for any

~J~ is closed and locally finite see [2] Th@or@me 1.3.1.

x e X-S,

?

s

= 0

~', ~ )

z]

determines a global section

The continuity of

H0(9~,

~il V . . - ~jlV..

l] and

I~', ~ ) the

respectively.

Ps(X,

~ ~ zO'

> "'"

~' ~ U, ~I U)

U

such that

forms an inductively ordered set

by extension. It is not empty. where if

S

In fact, let

is closed, choose a

x ~ X-S.

Let

U = V. l

V. i

and let

x

be a point.

containing

x

In the case

and such that

~ U i0'''"ip_l = ~ i

i0"°'i p -I

i ~ I'

208

Then

~U ~ Cp-I(2)" ~ U,

~ U = ~

on

neighborhood number of

U. U

i

of

x

choosing a smaller extend to = 0

on

U U.

n

U,

so small that

S

is open, choose an open

U

intersects only a finite

which intersect

V. i

U

and it is easy to show

~U )

In the case where

and all

V.

~'

U

contain

x.

we may assume that all sections in

and satisfy the cocycle condition Now define

~U i0'''i

By

~IU

~(-l)J~i0...[j...ip+l

as above and restrict them p-i

to

U.

V.

~U ~ cP-I(qPN U,

Then they form a cochain

10 • • • ip -1

q~' n U,

~IU)

~U

such that

Take a maximal element is a point

x ( X-U.

u

v.

= ?

~U"

Find

V

U. I)

Suppose that

and

Thus

on

U # X.

as above.

~V

on

?v ) = 0

Then there

We have

U n V.

If

~'~U

p = i,

this implies by the previous lemma that 1

for some

~'e

~SnUnv(U ~ V, ~ ) .

induction hypothesis that V' ~ U ~ V, ~ )

If

there exists a

such that

~U-

~'

cP-2(~

~' ~ (U U V), ~ )

again.

to an element in

on

U N V.

~s(U ~ V, £ )

U ~ V, By the or

and write the extension

Now let ~U

~UuV =

Then

~' E c P - 2 ( ~ T ~

~ V = ~ ~'

flabbiness extend (U U V),

p > I, we may assume as an

~UOV

I

x EU

'

~U +

~'

,

x 6 V,

p _~ 2

~U +

g~'

,

x 6 V,

p = I.

gives a strict extension of

I) The proof shows only that a cochain in boundary in

cP-I(%~I U,

cP-l(~ben U, ~ IU).

U)

This is a

~U ~ cP-I(ITn U' £ ~ U ) .

q}'~ U, ~ I U )

cP-I(~u~'~ U, ~

~U"

To have

we need to subtract a co-

regarded as a submodule of

Another proof will be obtained by a repeated

application of the nine lemma.

209

contradiction. such that

~

Therefore there is a cochain

~

in

cP(~,

qf', ~ )

= ? .

Theorem i. I0

(Leray).

Let

(lk, q~')

be a covering of

satisfying the conditions at the beginning of this section. (i.12)

HP(vi0'''i

(X,X-S)

If

p > 1

' 9 ) = 0, q

for all non-empty

Vi0...i

,

then

q (I.13)

HP(q~,

Proof.

V',

~) ~ H~(X, ~ ),

p g O.

"4

Take a flabby resolution of

£*

and

consider the double complex K = with homomorphisms

~ cP(I~, q)~', £ q ) P,q d' = ~ and d" = (-l)Pd.

It follows from the assumption that (cP(IF ~' 9) ,RPq = ~' H q i , ' , , -i (Vi0'''i ' ~ ) = p 0 where

~'

q = 0 q >0,

denotes the alternating direct product over multi-indices

i0'''i

such that all i. are not in I' P 3 of cohomology groups of coverings we have 'E~ q =$HP(IY, tO

~',

~)

,

Thus by the definition

q = 0, q >0.

On the other hand, the above two lemmas imply ,,Epq = J [~s(X, £ P ) -i

[ 0

,

,

q = O, q >0

and thus 0 ,

q >0.

Now the isomorphisms follow from the theorem on degenerated spectral sequences. The isomorphisms

(1.13) are given in the following way up to

210

the multiple of

±I.

Consider the diagram : 0

0

0

> rs(X ' £0)

0

--+ ~s(X ' zl)

__+ rs(X ' £2)

__+ ...

0--+ C0(V, 17',~) i---~C0(2~,IY',£0) d---~C0(l~,1~',~I) d---+C0(~,V',£2) --+ "'"

0 -+ cl(v,v '

cl(%

0 --~ C2(~,~ ' ,~) i__+

,,zl)

el( .

$ All columns and rows except the first ones are exact and the cohomology groups of the first column are the first rows are i?

= i~?

Hs(X , ~ ).

= 0, i~

= $ ~ i"

elements

~j 6 C p - j ( ~ ,

one. of

%t', ~ )

? ~ zP(~,

~',

since

~j-l)

such that

~ ).

Since

q~'' £ 0 )

~ 6 Ps(X, 6d~

= dg~

~P)

d ~j-I = ~ j "

such that

= d 2~p

= 0

d?p

and

= ~

g

is one-

but the cohomology class

is uniquely determined by the cohomology

this correspondence

V',

~i ~ cP-I(~'

is not determined uniquely by ~

and those of

In the same way we can find a sequence of

Finally there is an element d ~ = O,

Suppose

there is an element

such that

We have

HP(%~,

gives the isomorphism:

and

class of

HP(%~,

q~',

~)

The following theorems are easily proved from the isomorphism given above. Theorem i.ii. above.

If

h : ~

Let ~

(~, ~'

]~')

as well as (1.12), H p(X,

9')

lied with

,

q

~') = 0 ,

H p(%r, %y ', ~ )

as

p ~ 1

then the induced homomorphism

coincides, when

(X, X-S)

is a sheaf homomorphism and we have :

HP( Vio -.- i ' ,

be a covering of

H p(X, ~ ) and

Hp ( ~ ,

and

H p(X,

I)-', 9' )

h, : H~(X, ~) ~')

are identi-

by (1.13), with

>

211

the homomorphism

: H P ( ~ , 9f', ~ )

homomorphisms h : C q ( ~ , 9]", ~ )

> HP(gk, 9~', ~') ~ C q ( ~ , 9~', ~')

induced by the given by

(h~)i0, .... iq = h ~ !O, .... iq" Theorem 1 12 (91[, ~')

Let Y be a subset of X and let (~o~, %t') and

be coverings of (X, X-S)

and (Y, Y-T) satisfying the

conditions as above, and with the same index sets I and I' for all i E I ,

then the restriction mappings

are identical with the homomorphisms induced

by the restrictions

: HP(x, ~ )

: HP(%~, ~', ~ )

: ( ~ ~ )i0o.oip

=

If V.D W. I i > ~(Y, ~)

* H P ( ~ , ~', ~ )

~i0.°.iplWi0o..ip

II. HYPERFUNCTIONS

Theorem 1o8 Suppose that

~

gives a method to construct many flabby sheaves° is a sheaf of flabby dimension m and that S is a

closed set purely m-codimensional with respect to ~o Then the derived sheaf

~(~)

is flabby. In fact, we have by Theorem 1.8

HP~ (x, ~ ms (9)) = _p+m H~I s (X, 9 ) for any family of supports

= 0,

~ o Since S is closed,

p >0,

there is a one-one

correspondence between the sections of

~ ~( ~ )

sections of the restriction

over So Thus we can consider

S(

~(~)I

S

over X and the

) as a flabby sheaf over S in a natural way° Example io

Let X be the Euclidean space

~n

~ be the sheaf of

C~

functions on X and S be a nowhere dense closed set, Cog. ~n-lo i Then J-IS( ~ ) is flabby° First of all, flabby dim ~ = 1 since is soft and not flabby° We have Thus

J{~( ~ ) = 0

HS0 ~ U (U, ~ ) = 0

for any open set Uo

and hence S is purely l-codimensionalo

Let U be an open set° Then by Theorem 1o8

212

1

(u, ~)

1

F(U, ~ S(~ )) = HSa U Let Then

%)" = {Vo, Vll (If,

~')

and

~'

covers

= {V0~ ,

(U, U-S).

where Since

V0 = U

and

V I = U-S.

HP(vi, ~ ) = 0

for

p>0,

we have by Leray's theorem, i HI HSnu(U, ~ ) = (~,

IF', g ).

The covering has only two open sets.

Therefore a relative l-cochain

is always a l-cocycle and it has only one component (VI, ~ ).

Thus we can identify

ZI(%~, ~Y', ~ )

?01 E

with

E (U-S).

On the other hand, a relative l-coboundary has the component (~0)01

~01 =

= - ~01U-S

= O,

with some ~0 e ~(V 0, g ) Since H 0S~U (U, ~ ) V0 the restriction ~V 1 is one-one. Thus by identifying ~01

and

~0

"

we get the isomorphism : H l(~,

~',

g)

=

~(U-S)/g(U).

Consequently we have i F(u, X s ( £ ) ) I ){S ( ~ )

Thus the sheaf of

= 8(u-s)/~(u).

may be regarded as the sheaf of singularities

C m functions defined on the complement of Example 2.

Let

X

and

S

S.

be as above.

The discussion above

is based on the following two properties of the sheaf (i)

HP(u,

~)

= 0,

p > 0,

(ii)

~Snu(U, ~ ) = 0

for all open set

for all open set

In fact, (i) implies flabby dim ~ implies

0 ~S (3)

= 0.

~ i

~ = ~ :

U.

U.

by Theorem 1.2, and (ii)

The representation I

p(u, ~s(~))

= ~(u-s)/~(u)

follows also from (i) and (ii). There are many sheaves

~

over

~n

satisfying

(i) and (ii).

213

Let us denote by

~,

~,

and

~P

the sheaves of continuous

functions, real analytic functions, and real analytic solutions of a single elliptic differential equation coefficients, respectively. = ~ . ~P.

Since

(i) for

~P C ~ ~ =~

and [22] respectively.

,

(ii) holds also for

~P

~(~) mapping

~J~(~) :

and [23]

We will also give proofs later.

i ~S(~

p)

i S(~),

~ ~

~P C ~ C ~ C ~

i ~{S(~)

i ~S(~)

, the and

are not necessarily one-oneo However, the

1(

1

~4"s(~P)

on an o p e n s e t on

:

~ = ~

has been proved by Malgrange

Although we have the inclusion relation induced mappings

with constant

It is easy to check (i) and (ii) for

C ~

and

P(D)u = 0

~~ S ~ )

U satisfying

is

one-one,

P(D) ~

because

a C~

= 0 on U-S s a t i s f i e s

function

~

P(D) ~ = 0

U.

i Go Bengel [i0] discusses the sheaf ~ S ( ~ S is the hyperplane where P'(D) = P(-D).

~

n-1

C~

n

P)

in the case where

and calls its sections P'-functionals,

The reason is that the P'-functionals with

supports in a compact set K form the dual space of the space

~P'(K)

of real analytic solutions of P'(D)u = 0 on a neighborhood of Ko

He

proved this fact from the duality : ~ ( m n, ~ P )

(2.1)

essentially due to Grothendieck

= ~P' (K)'

[16j. A proof will be given later°

The hyperfunctions on the real line sections of the derived sheaf functions

on

the complex plane ¢

of Example 2 where operator

~(~)

X = C, S =

R are by definition the

of the sheaf

~ of holomorphic

This is a special case of ~ ( ~ P )

~ and P(D) is the Cauchy-Riemann

i ~ = ~[~x + i ~ ] o

The hyperfunctions of n-variables are defined in the same way to

214

be the sections of the derived sheaf holomorphic

functions of

Cn. If

true° In order to show that ~

n

(~)

of the sheaf

~

of

n > i, (i) and (ii) are no longer

n(~)

is of flabby dimension n and

~n

is flabby we have to prove that

~n is purely n-codimensional.

These

facts are derived from Malgrange's vanishing cohomology theorem [231 and Martineau's hyperfunctions

duality [8]° The latter is also used to show that the contain the distributions

and more generally the dual

spaces of Gevrey classes of functions on

~no

Functional Analysis To prove vanishing cohomology theorems such as (i) of Example 2 and duality theorem such as (2.1), we need, of course, analysis. To my regret, however,

functional

the functional analysis of today

offers us only one or two general methods for such a purpose. One is due to Hormander

(~19] and [5] Chap° 4)°

The other is formulated by

Serre E30] though it has been used by Malgrange

[22] and others

tacitly° Hormander's method is elementary and seems to be promising, though we use mainly Serre's method. A locally convex space E is said to be Fgchet-Schwartz

or (FS)

for short (Fr~chet-Schwarts* or (FS*) for short) if it is Fr~chet and if for each absolutely convex neighborhood V of zero there is an absolutely convex neighborhood ^

linear mapping [17]).

UcV

of zero such that the natural

^

: EU---~E V is compact

(weakly compact)

A locally convex space E is (FS) ((FS*))

the projective limit such that the mappings

(Grothendieck

if and only if it is

lim E. of a sequence of locally convex spaces E. j J : E. J

>E

j-I

are compact

The strong dual spaces of (FS) spaces

(weakly compact)

((FS*)spaces)

°

are said to

215

be (DFS)

((DFS*)).

A locally convex space

and only if it is the inductive locally convex spaces one-one and compact

Ej

(FS*) and

[17].

E

(DFS*).

quotient

spaces) are

spaces and inductive

(DFS).

Quotient

spaces are (DFS*)

((FS*)

(DFS*).

[17].

(FS)

However,

F

If

the Mackey

is an open set in

of (DFS*)

~n

the space

of all C ~ solutions

~(V). operator

If

K

functions

is a compact

then the space

~P(K)

of

topology

= lim --_> E P ( v

hoods of

K.

are

set in

some neighborhood : ~P(K)

K

of (DFS*)

F

topo-

on each closed

into a (DFS*) space. then the space

E(V)

is (FS) with the standard topology.

with constant coefficients

of holomorphic

of (DFS) spaces

spaces are not always

coincide

cn ,

or in

on

~(V)

Closed subspaces,

topology and the bornologic

of all C a functions ~P(v)

V

limits of sequences

sums of sequences

of (DFS*) spaces and they make

V

P(D)u = 0

separable and Montel.

((FS*)).

logy associated with the induced topology subspace

in the sense of Ptak

limits of sequences

subspaces

of

of the differential

and,

in particular,

and

P(D)

C ~ solutions

as

V

equation

of

is an elliptic of

P(D)u = 0

is a (DFS) space with the inductive )

Thus

the space

(FS) as closed subspaces ~n

are

[27].)

and projective

spaces and direct Closed

~ Ej+ 1

in the sense of Grothendieck

(FS) and (DFS) spaces are moreover

of (FS) spaces

: Ej

(For such projective

[31] and Raikov

spaces,

if

is a reflexive Banach space

spaces are B-complete

quotient

((DFS*))

of a sequence of

J

and totally reflexive

Closed subspaces,

are

(FS*) and

see Silva

(DFS*)

[26], bornologic

lim E.

(weakly compact).

limits

is (DFS)

such that the mappings

if and only if it is both and inductive

limit

E

on

limit

runs through open neighbor-

Many of the local Sobolev spaces are

(FS*) as projec-

216

tive

limits

of s e q u e n c e s

Those

spaces

Lemma

2.1

of r e f l e x i v e

are i m p o r t a n t (Serre

Banach

because

[29]).

spaces.

of the f o l l o w i n g

Let

uI E1

El,

E2

and

E3

be F r 6 c h e t

u2 . ~ E3

~ E2

spaces v

and let E2 "

uI : E1

~ E3

mappings = 0.

or d e n s e l y by

dual m a p p i n g s

Suppose

ker u 2, space

If is equal

is

to

H.

if

im uj'

Then,

(weakly*) and

;

Z. = ker u I

(FS*),

and

such that

spaces

of

(i) im u. J closed

im u 2

in

space

u2o u I

E. 3

and

is c l o s e d

the in

E~. J

are closed.

B. = im u~.

and its dual

with

(FS),

to prove.

theorems (Schwartz

space

is o n e - o n e F.

F

topology

im u I

Let

Then

Z =

the q u o t i e n t

is i d e n t i f i e d

with

H and

in

the

strong dual

space

or the b o r n o l o g i c

H'

topology

H..

and

H'

im u 2

is equal are

to

H..

c l o s e d are u s u a l l y

t h e o r e m and the f o l l o w i n g

are

available. [29]).

ioe.

(i) S u p p o s e

there

into

and onto. The

and

the M a c k e y

then so is

that

cross-section,

to

H

The Banach-Dieudonn6

the only g e n e r a l

on a F r @ c h e t

so is

the q u o t i e n t

is

2.2

then

equipped with

conditions

dimension.

is

im u I

is F r @ c h e t

The

isomorphic

the s t r o n g dual

respectively.

E2

Z/B

linear m a p p i n g s

u. J

If

Lemma

r

as a set.

E2

associated

closed

ul J

that b o t h

H = Z/B

Fr~chet

and

B = im Ul,

H. = Z . / B .

hard

of

U

u2 :

linear

defined

El J

if and only

(ii).

and

be c o n t i n u o u s

Denote

Ej+ I

> E2

lemmas.

Z

that

is a c o n t i n u o u s

H = Z/B

linear m a p p i n g

such that the c o m p o s i t i o n Then

cross-section

has a

F

B = im u I

is c l o s e d and

exists

H

if

is of finite

f

> Z H

is

217

(ii) Suppose that (DFS*) H.

E~

and

cross-section

E~

f : F.

are (DFS*).

If

~ Z.,

im u~

then

H, = Z,/B,

is closed and

equipped with the Mackey topology is isomorphic to

cross-section exists if the algebraic dimension of

has a

H.

F..

The

is at most

countable. For the details of this section see [20].

Hyperfunctions of one variable The theory of hyperfunctions of one variable relies on the following two theorems. Theorem 2.1 (Malgrange [22]). (2.2)

HP(v,

for any open set

V

in

6~) = O,

p _~ i,

¢.

Theorem 2.2 (Silva [30], K6"the [21]). in

If

K

is a compact set

¢, H Ki(¢, • ) ~=

(2.3)

The inner product between the following way. by Leray's theorem. sented by hood

U

encircles

K. K

i HK(C , ~ )

We identify Let

~ ~ ~(¢-K). of

#(K)'

[~ ] If

and

HI(c, ~ )

~(K)

with

be the element in

f E ~(K),

f

(2.4)

~ ( C - K)/ ~(C), HI(c, C~)

repre-

is defined on a neighbor-

Choose a rectifiable closed curve

counter-clockwise.

is given in

~

in

U

which

Then

( [ ~ ]' f> = - I

~(z) f(z) dz.

F Malgrange's proof of Theorem 2.1 employs functional analysis stated in the previous section. classical method.

0

However, we can also prove it by a

In fact, since

~

~

>~

~0

218

is a

soft

(2.2) for

resolution of p = 1

has a solution

,

obviously we have

means that the differential u e ~ (V)

to the Mittag-Leffler theorem

~

(cf. [5],

for any

(2.2) for

equation

f ~ ~(V).

~u

= f

This is equivalent

theorem and is a consequence

of Runge's

~ 1.4).

We have also an elementary proof of T h e o r e m 2.2. from the Cauchy integral formula that for each (2.4) defines a continuous of the choice of

~

linear functional

on

or

Conversely,

if

It follows

[ ~] e ~ ( C ,

linear functional on

~ .

~(K)

~

~)

independent

is a continuous

~(K), _

1

is a holomorphic function

on

function on a neighborhood

C -K.

of

2~iI [ Jr

-

are holomorphic

p ~ 2.

K,

If

f(z)

is a holomorphic

the Riemann

sums of the integral

tl__~dz

f(z)

functions and converge to

compact set in the open set bounded by

f(t)

~ .

uniformly

on each

Thus we can interchange

the order of the integral and the inner product and get <[ ?], Definition: sheaf

I ~-{N(~)

The sections of

f> =

~(f).

We denote by to

~

~

the restriction of the derived

and call it the sheaf of hyperfunctions

on

~.

are said to be hyperfunctions.

From the arguments at the beginning of the chapter it follows that

1 Jf~(~)

is a flabby sheaf over

is a flabby sheaf over sections of natural way

~

~.

Since

¢

concentrated on

~

is closed in

are identified with sections of

([2] Th@or@me 4.9.1).

Thus if

~

~ { ~i( ~ )

~,

~.

Thus

the in a

is an open set in

~,

219

then the space the quotient ¢

~(~)

space

containing If

of hyperfunctions ~ (V-a~.)/ ~(V),

~g

as a relatively

~ e ~(V-~),

of

~

where

is identified with

V

is any open set in

closed set.

we denote by

is identified with the class of

on

~

[~ ]

the hyperfunction

and call

~

a defining

which

function

[9]. Hyperfunctions

of several variables

If the dimension Stein open sets

n

is greater than

(i.e. pseudo-convex

theory of hyperfunctions We need the following the generalizations Theorem 2.3

theorems.

V

open sets).

This makes the

The first two may be considered

of Theorems

HP(v,

for any open set

(2.2) holds only for

of several variables much more complicated.

(Malgrange

(2.5)

I,

in

2.1 and 2.2.

[23]).

~)

= 0,

p ~ n,

cn.

Theorem 2.4 (Martineau

[8]).

If

K

is a compact

set in

cn

such that (2.6)

HP(K,

~)

= 0,

p > 0,

then HP(¢ n, ~ )

(2.7)

= 0

for

p # n

and n n HK(G , 0 " ) ~

(2.8)

Theorem 2.5 (Martineau ~n

is polynomially

[8]).

convex in

Theorem 2.6 (Grauert of

S

neighborhood

in

cn

of

S

Any compact

set

K

contained

in

cn.

[15]).

be an open neighborhood V

~(K)'

Let

in

¢ n.

S

be a subset of

~n

and

Then there is a Stein open

contained

in

U.

U

220 Contrary to the case of one variable, we do not know any complete elementary proofs of Theorems 2.3 and 2.4.

Sato [9] states that

Theorem 2.~ can be proved by the Weil-Oka integral formula but his proof is not quite clear.

A. Friedman [14] gave a proof of (2.7)

for polynomially convex compact sets

K

by the Weil-Oka integral.

Actually we need Theorem 2.3 only in the form of (2.7) for and Theorem 2.4 only for compact sets

K

in

results are almost enough for our purpose. give the duality

~n

p = n+l

Thus Friedman's

Probably his method will

(2.8) too.

Theorem 2.5 is the Weierstrass approximation theorem for and is easy to prove. set

K

in

~n

(~(K)

Because of this (2.6) holds for any compact

The proof of Theorem 2.6 is also easy for we have

now an easy solution of Levi's problem by HUrmander [5]. Theorem 2.7. sheaf

is purely n-codimensional with respect to the

Cn

over Proof.

~n

It is enough to show that

(2.9)

Hp (V, ~ ) ~nnv

for bounded open sets

V

in

Cn

By the excision theorem we have

= 0 , Let

p # n, ~

H~(V, ~ )

= ~n ~ V

and

= H~(¢ n - ~ ,

~ ~).

=~-2. Now

consider the exact sequence of relative cohomology groups associated with the triple

cn D C n- ~

0----+H~(C , e) 0 •

Since

~

" "

and

~

D C n- ~

~ H (¢n, ~)

H (C

:

> H I (¢n-~A, ~)

e)

p+l (¢n ~)_~... H~m

°

are compact sets

in

~n,

it follows from

Theorem 2.4 that H~(¢ n - ~ ,

(>) = 0

for

p # n-l, n.

221

For

p = n-i

we have the exact sequence :

o

Hn-l(¢n ~ - ~,

~

By Martineau's spaces of n

H~(¢

n

~(~)

and

n

, ~)

~ H~(C

> 6%(~7L).

dense in

~ ( ~PL )

H~(£ n, (3")

j{n

(}(~)

, 0-)

n n H~(C , 6~)

respectively,

0-(~),

to

Therefore,

~Rn.

are the dual

and the restriction

which contains

(>(¢n),

n

by Theorem 2.5, the mapping

We denote by

((})

n n > H~(C ,e).

, ~)

is the dual mapping of the restriction

Since

is one-one.

Definition. sheaf

n

n

n n H~fa(£ , (.9-) and

duality

~(~]~)

n •H~a(¢

~)

n

H~fg(¢ , (3-)

H 7 1 ( ¢ n- ~Jl , C~)

~

the restriction

The sections

of

is

= 0.

of the derived

are said to be hyper-

functions. If

~

functions

is an open set in on

open set in

~

JRn

the space

is identified with

cn

which contains

~

~

H~(V, 6~ ) ,

(~)

of hyper-

where

as a relatively

V

is any

closed set.

More precisely Theorem

2.8.

(2.10)

For any family of supports

~(~,

~

in

~

we have

:

~ ) = H-~(V, ~).

In particular, (2.11) where over

= H Sn(v, (3"-) for any subset

~S(~h) ~S(f[) ~

denotes

with support in

Proof is immediate Theorem 2.9. Proof. Theorem In fact,

the space

if

Therefore,

~ ~

~

)

of ~

of sections

, of

~'3

S.

from Theorem 1.8.

The sheaf

0"~

Theorem 2.3 implies

1.8,

PS(fl,

S

is flabby. is a bounded

the exact sequence

of hyperfunctions that flabby dim ~ _

is flabby. n.

Thus by

We can also prove it from Theorem 2.4. open set in

~n,

Hn+l~(¢n,

6~) = 0.

222 n n , ~) H~(C

n.n> H ~(¢

shows that the restriction

~(¢n)

~

the flabbiness is determined locally, We denote by

~

~ H %+i (C n, ~ )

~)

~,I7..,

~(~)

~

is onto.

Since

is flabby over

~n.

the sheaf of real analytic functions on

In other words,

~

is the restriction of

a compact set in

~n,

is identified with

the space

~(K).

Thus

~(K)

~

to

~n.

If

of sections of

~_(K)

~

Rn.

K

is over

forms a (DFS) space with

the natural inductive limit locally convex topology.

Theorem 2.4

gives the following as a special case. Theorem 2.10.

If

(2.12)

K

is a compact set in

~n,

~ K ( ~ n) ~ ~_(K)'

Thus the strong topology in space.

~(K)'

makes

~ K ( ~ n)

into an

(FS)

This topology behaves, however, quite differently from that

of the space of distributions.

If we choose a point

connected component of

K,

~_~--

is dense in

aJ kf(k) ( x - x j )

x. J

from each

the set of the elements of the form :

~K(N n).

Therefore any hyper-

j k=0 function with support in

K

can be approximated by a sequence of

hyperfunctions with support in { xjl-

If

in

can be expressed as

~(~n)/

~(~n)

quotient topology is trivial because

~ A @ R n)

is dense in

~n,

~ (fh)

~

is a bounded open set but the ~(~n).

Differentiation and multiplication by real an@lytic functions Let

~

be an open set in

t P(x, D) = ~ a~(x)(-i) l~l I~I=0

~n

and let I~1

o~I o~n Zx I -.- ~ x n

be a linear differential operator with real analytic coefficients

223

a~(x) E ~ ( ~ ) .

Then there is an analytic P(z, D) = ~ a ~ ( z )

on an open set

V

in

Cn

a sheaf homomorphism We define

extension

~ ~

the operation of

induced homomorphism

~

----->~

T h e o r e m 2.11. real analytic compact

~)

Let

subset of

on hyperfunctions ~).

P(x, D)

P(x, D)

:

~K(~)

by the

Since the analytic ~

,

the induced

gives also a sheaf

be a differential

on an open set

,

gives

~ .

P(x, D)

~

P(z, D)

V.

> H~(V,

P(x, D).

over

coefficients

is continuous,

over

Clearly

is unique on a neighborhood of

homomorphism depends only on homomorphism

~a .

P(x, D)

P : H~(V,

P(z, D)

Dz

containing

P : ~>

extension

~

in

operator with ~n

If

K

is a

then the mapping ~ ~K(~)

and coincides with the dual mapping of the formal

adjoint acting on

~ (K).

Proof is omitted.

Hyperfunctions Let

~'~

as classes of holomorphic

be an open set

a Stein open set

V

j = 0, i, "'', n,

in

¢n

V-~

respectively.

Stein,

= I V 0, V I,

on

such that

V N ~n

Im z., #j 0 ~

"'', V n ~

= ~ .

Define

V., J

and

V'

j = 1,2,''',n. = {V I, "'', Vnl

Since intersections

the covering

hyperfunctions

By Theorem 2.6 we can choose

and

Vj = ~z E V ; V

£n.

by

V0 = V

Then

in

functions

(~, ~

~')

satisfies

cover

V

of Stein open sets are (1.12).

Therefore

are identified with the elements

in

the

and

224 Hn(q]", 2Y', ~ ) . n-cochain

~

There are only

V # ~.

open sets in

is always an n-cocycle ~(v

~O,l,''',n We denote

n+l

V0 q V1 ~

Then

with only one component

0 n

vI n

---

~',

~)

= ~(V#

n ). for all

~

has

n

n e C~(v In ...n~.j n-..nv n) ,

].

~12"''n = O.

{z £ V •, Im z k # 0

for

Denote

k # j}

k}

by

A).

On the other hand, an (n-l)-cochain

and a component

n v

"'" m V n = {z ~ V ; Im z k # 0

zn(qk,

?01

2)-. Hence an

components j = l,''',n,

V l q "'" q v .3~ V j"

simply by

"'" q V n

Then

n

cn-l(7~, qJ~', (3-) = If

~e

(~ C~(gj). j=l

cn'l(9~, i)-', (~), =

_

( ~)Ol'''n Therefore

+

? 0 2 " ' ' n + ~ 013 .

Bn(~, ~k', ~)

n

~= ]~

. . .

n.

+

.

(~(gj).

.

(-l)n? . Ol

.

"n-I

We have thus

j=l Theorem 2.12. n

(2.13)

~(~.)

Definition. function on of

~.

If

~ ~(V~_)/

~ ~(Vj). j=l

~ e ~(V#~),

represented by

we denote by ~ ,

and call

~

[~ ]

the hyper-

a defining function

[?]. Theorem 2.13.

Let

P(x, D)

real analytic coefficients on extension of (2.14) Proof.

P(x, D)

~

be a differential operator with .

If

P(z, D)

to a Stein neighborhood

P(x, D)[ ?]

= [P(z, D ) ? ] .

This is clear by Theorem I.Ii.

is an analytic V

of

~L ,

then

225

Standard defining functions Suppose that where

LI× --.x L n condition Vj

(2.6).

are Stein,

cover

L

Cn

is a compact set in L.J

Let and

and

are compact in V0 = cn,

¢.

of the form :

Then

L

Vn~ .

respectively.

and.

L =

satisfies

V.3 = cj-i X (C - Lj) × C n-j.

q)" = {V 0 . V I , - '.-

cn _ L

Cn

V'

={V 1

V0

and

"'"

V n}

Thus by the same reasoning as

above we have n

HL(C

n

, ~)

= Hn(~) -, q~', ~ )

(2.15) = (~(]~(¢ - L j ) ) /

where

~_~(Vj)

,

V. = V k = (~ -LI) × "'" × C × "'" × (C -L n). J k#j

the inner product between terms of a Cauchy-like Theorem 2.14 ~(L). D.

J

[4]).

?D.. J

(2.16)

<[?],

Let

D = D I~ .-.x Dn

i s a b o u n d e d open s e t

boundary

and

(Y(L)

is expressed

in

integral:

(Harvey

Take a set

each

H~(C n, ~)

In this case,

~ ~ O(-[[(¢ - Lj)) such that

containing

L. J

f e(~(D) and w i t h

and

f

and that smooth

Then

f> = (-l)nf~

DlX..-×~D ~ ( z )

f(z)

dz 1

-..dz

n '

n

where ?

[~ ]

under the isomorphism

connect

integral

duality

Hn(~d", ~ d " ,

coincides

with

0-)

which is represented by

(2.15).

The p r o o f i s l a b o r i o u s that

H Ln(cn,

is the element of

(see ~)

[4]).

and

the bilinear

We must c h a s e t h e i s o m o r p h i s m

n n HL(¢ , C~)

and show t h a t

the

f o r m g i v e n by M a r t i n e a u ' s

(2.8).

Theorem 2.15. an element in

Let

~(K)'

K Then

be a compact set in

9Rn

and let

~

be

226

(2.17)

~(z) = (~-i)

~t (~(tj_zj)

gives a defining function of the hyperfunction corresponds to Proof. contains

~

~

an element of

which

by isomorphism (2.12).

Take a compact set

K.

u • ~ K ( ~ n)

L = LlX "''× Ln

in

~(-[[(¢ -Lj)).

Thus

is actually in ~L(~ n) = ~(L)'

Let

f ~ ¢t(L).

~n

which

[~ ]

gives

Then by Theorem

2.14 we have <[ ~ ]' f> = (-l)n~DlX'''×~Dn~(Z) f(z) dz

1 n<~ x--:. ~t(( 2ml ) DI×'''×~Dn

=

= }t(f(t)) Since

(~(L)

is dense in

Definition.

f (z)dz IT(zj-t j) )

= < ~[ , f > . ~(K),

[9 ] = } "

We call the function

~

defined by (2.17) the

standard defining function of the hyperfunction to

u

corresponding

~ . Example; ~n xn ~

The standard defining function

of

~l~I/ ~Xl ...

is given by !

(2.18)

(-i) n+l~j ......~] ........~''" . ~n" ~n+l ~(z) = (2~i)n Zl~i+i "''Zn

Embedding of distributions Lemma 2.3. compact space Denote by and

~ ,

morphism

Let X

~(X)

~'

and

be soft sheaves over a locally

with a countable fundamental system of open sets. and by

~,(X)

respectively, over h, :

~

~(X)

X

> ~,(X)

the spaces of sections of with compact support. satisfies

~'

If a homo-

supp h,(s) C supp s

227

for any ~'

t(x]-~ 7,-

s e

~ ~

which

then there

induces

supp he(s ) = supp s Proof. exists

holds

We have

Write compactly

any

hU

show

compact

s ~

this

whose

if and only if

~,~(X).

that for each open set

: ~'(U)

~'(U)

)

~(U)

as a locally

sections

s. ]

U

in

X

there

which

extends

hel U

sum

s = ~sj

of

V

= ~_~h,(sj).

of

supports

finite

and define

is well-defined,

neighborhood

sections

s ~

h:

w i t h restriction.

supported

that

sheaf h o m o m o r p h i s m

is one-one

for any

hu(s) To

h

to prove

a homomorphism

and is compatible

h.~.

is a unique

let

x

in

intersect

s = 0

and

x e

U

and

let

Sl,

V.

The

sum

t = s I+

U.

Take

a

be

the

• '' , sp

"'" + s

is P

in

~(U)

and

vanishes

on

a neighborhood

of

x.

h,(Sl)+

''" +h,(Sp)

vanishes

on

support

of

section

s. J

does

not

intersect

for

any

x.

Thus

vanishes clear

any

at

x.

that

Sp

for t

as

neighborhood at

x

and

is

with

supp

= supp

above.

hence

s

2.16.

true

compatible

s e

x.

generalized

h,(s)

for

= 0,

hu(t)

follows

by

assumption

at

V,

the

hu(S )

= 0.

It

for

each

If

x ~ U,

vanishes

s I,

on

that

t

We

--',

a is

~'

of

the

Beuring

classes

[28])

are

strict

consider

only

the

case

distributions classes subsheaves

and

zero

[Ii]

~' (or

of

~

of in a

way.

Proof.

is

x.

sheaves of

Since

s ~ ~(X).

hu(S)

vanishes

find

any

Since it

x.

=

restriction.

s

can

h..~(t)

hu(S)

we

Thus

The

the

of

~(U),

distributions

Denjoy-Carleman natural

an

of

Theorem of

is

that

= 0 and

This

hU

Suppose hu(s)

other

a neighborhood

Thus

in which

Ooe~f~

c

[ii].

the

228

Martineau [24] shows that

~,(~n)

We know that the injection

is the dual space of

~(~n)

~ ~(~n)

has dense range by Weierstrass' theorem. i : ~

(RRn)

~ ~,(~n)

supp i(u) C supp u, inclusion let

is one-one.

for any

f 6 ~@R

n)

It is also clear that

u £ ~ ' @Rn). be such that

To prove the converse

supp f n supp i(u) = ~ .

f. e @_(~n) J

in

(supp i(u)),

and

f. J

~ 0

in


~ '

is continuous and

Hence the dual mapping

If we have a sequence of functions g~(~n)

~.(~n).

f.> = 0. j

such that

f. 3

~ f

then we obtain

Therefore,

supp f N

For example, let fj(s) = g(x, j-l) , f(x),

where

G(x, t)

is the Gauss kernel.

The infinite sum only if

~

a~D~$

;~ I =0 . =

-liraI~I 41a~l 4' I~I~

element in

0

).

functional on any

0.

n)

~01

u = ~---~ (k log k=2

Thus

However,

~(R

belongs to

u

(~n)

k)-2k

Ak

if and ~

is an

can not be a continuous linear

or any non-quasi-analytic family of

C~

functions of Denjoy-Carleman type because it does not belong to the dual space of

c{k-log kl

which is quasi-analytic (cf. [ii]).

Hyperfunctions as sums of boundary values of holomorphic functions A hyperfunction

[ 9]

on an open set

A9_ in

~n

is interpreted

intuitively as the sum of boundary values of its defining function in

O ( v ~ ~_)

(2.19) where

:

[ ~] (Xl,''',Xn) = ~.~ sign 6" ?(xl+i~-i 0,''',xn+i0-n0), 6"

ranges over all n-tuples

(6"1, ..-, 6-n)

of

!l

and

229 sign

~=

There are many evidences which support this

6-1"'" ~n"

interpretation.

For example,

compact

K I~ '''x Kn,

support

?

ing function tion

~DI×

in

let

[~ ]

where

~(]T(¢ -Kj))

"'" X ~D n

be a hyperfunction

Kj C ~.

with

If we choose a defin-

and let the domain of integra-

in (2.16) tend to the real space, then

(2.16)

becomes (2.20)

< [ ~ ], f> = ~ s i g n o-

Many classical

integral

~(x+i6-0) f(x) dx .

~" S KI× • • •x K n

formulas which are proved by various methods

are shown to be merely two representations this type

n

~

in

of its defining

function

containing

~

Consequently, vanishes,

then

variable

x

~

K

as a relatively

if

~'

~(x+iy)

converges

~ E ~(V- ~),

the limits

V

~ =

V

is an open set

~

on which

function

limit topology of sets in

[? ]

Let

~ (z)

then

yj ~ 0 ?

~(x)

as

y

~(~3~') = 4lim~(K),

~'. be a holomorphic

is an open set in

as a sequence ~(x - iO),

~(fh)

closed set.

to the real analytic

~ ( x ± i0) = lim ~ ( x + i y j )

bution on ~(x+i0)

where

[~ ] e

considered as a function of the real

runs through compact

~(V- ~),

where

is an open subset of

Theorem 2.18 (Painlev@). in

theorems.

coincides with the set of singularities

tends to zero in the projective where

(2.19) is almost

The support of a hyperfunction

on an open set

¢

is one, interpretation

First we note the following

Theorem 2.17.

in

of

(see Sato [9]).

When the dimension perfect.

of the same integrals

exists

¢

and

function

~ = V ~ ~.

If

in the sense of distri-

tends to zero and if

is holomorphic

on

230

This follows Th6or@me

24 of Schwartz

Now, let over i)

~R

easily from the classical

G

g

be a non-degenerate

in

G,

ii)

as

If

y # 0

y

+-0 ;

g(x) =

~(x+i0)

The function

If ~

'(j~)

SZ

belongs

to

G

when ~(x + iO)

to the limits

and we have - ~(x - i0).

the product g e G

spaces

to

aLp(N),

X g

is in

)Cg ~ G

G

g 6 G

and

is continuous.

(m) (~R) ~Lp , if

for any

L p (~R),

~ (m) , (~R) Lp

1 < p < oo (cf. Schwartz

[18]).

is an open set in

associated with such that

)~f

duce in

~ (~i)

mappings

M X : ~(~)

all

~(x+iy)

~ L'p (~R) satisfy these conditions

[29] and Hille

over

function

tends to

~ e C~(~R),

such that :

transform

is fixed, and converges

the mapping which maps

and

space of distributions

g(x) z-x

is defined ; the analytic

G

vector

the complex Hilbert

~(z) = 2 ~ i

the parameter

theorem and

[29] Chap. VI.

equipped with a locally convex topology

For any

in

Painlev6

G

---> G

we define the local space

to be the space of distribution

is in

the weakest

IR,

G

for any

~ E CO(2L).

~ (/L) f 6

We intro-

locally convex topology which makes the defined by

M%g = )~ g

continuous

for

~ e C~(£L). Obviously

the system

tion forms a sheaf degenerate,

we have

limit topology on that of

~ (/L)

~

{~ (7L),

of

~-modules

~ C ~ ~(/l)

for any

/[C~R}

C ~9'

over

with the natural restricR.

Since

is non-

We assume that the projective

is stronger than the topology ~I .

G

induced by

231

The sheaves ~'

~

of infinitely differentiable functions and

of distributions are obtained in this way, for they are sheaves

of local spaces associated with

~ L2(~R) and

The natural topology of

(~'(~))

~(fL)

topology discussed above.

where

Lp

and

respectively.

coincides with the

Other examples are the sheaves

functions m-times locally differentiable locally in

~L2(RR)

~ (m), Lp

in

L p,

~P

~(m) Lp

of

of functions

of distributions of order

m

in

~P,

I < p < ~. Theorem 2.19.

be an open set in tains

fL

Let JR

~

be a sheaf obtained as above, let

and let

as a closed set.

belongs to

~ (fh) ,

belongs to

q(/L)

?(x+i0)

and

V

be an open set in

If a hyperfunction

~(x-i0)

in

x

~(~L)

which con-

[~ ] £ ~5(/L)

then the defining function as a function in

¢

~_

~ (x+ iy) & f>(V - J~)

and converges to the limits as

y

tends to

+0

and

-0

respectively, and we have (2.21)

[~ ] =

Proof. function ~[~]

Let

K

G.

Let

defining function of (I- 0~) [ ~ ]

~ .

Choose a

which is one on a neighborhood of ~

to(~) , (-I/(2-~iz)).

Since

- ?(x - i0).

be an arbitrary compact set in

co ~ C~(fL) is in

~(x+i0)

K.

be its complex Hilbert transform

It follows from Theorem 2.15 that ~0 [? ].

Thus we have

is analytic there by Theorem 2.17. K,

then

=

to

~(x) ~(x+-i0)+ ~(x) ~l(X)

K,

K

in

G

)~ 6 CO(fL)

~(x+-i0)

as =

y

is a

~i = ~ - ~ vanishes outside

%(x) ~ (x+iy) + ~(x) ~l(x+iy)

easy to see that the limits depend on the choice of

If

~

[? - ~] = (i - 00)[ ~ ].

vanishes on a neighborhood of

%(x) ~(x+iy)

Then

tends to

~(x+_i0)+

converges +-0.

~l(X)

as far as a neighborhood of

x

It is do not is

232

contained in (/I).

K.

Therefore,

Since

= [ ? ] (x)

?(x+i0) - ~(x-i0)

on a neighborhood

Theorem 2.20. and let

~(x+iy)

~-

Let

of

~

=

converges to

K,

we have (2.21).

~R.

JR

continuous,

and that for any

morphic function borhood of

~

with

to the limits g(x) =

~(z) 6 ~ ( V -

ZL),

V ~ ~ = ~ ,

~(xii0)

in

with

Assume that

~(~)

g E ~(ft) where

V

such that

~(~_)

as

y

8- C ~ C =~ '

~ (fg)

ly convex topology which makes the embeddings ~'(~_)

in

~(x+i0) - ~ (x-i0) = ~o [ ? ] (x)

be a sheaf over

be an open set in

~(x+iO)

has a local-

C 9 (fL) ¢

there is a holois a complex neigh-

~(x+iy) tends to

converges +-0

and

~(x+iO) - ~(x-iO).

Then a hyperfunction

[~ ] £ 0'3 (~I)

belongs to

~ (71)

if and

is sequentially weakly compact in

q (fL)

as

only if

~ (x+iy)

y

In this case the limits

) 0.

~ (x +- i0)

exist in

~(~_)

and

we have (2.21). Proof.

Suppose that

[~]

such that

[?]

~ ~(V-~L) and a f o r t i o r i

in

~'(/h).

Theorem 2.19 for

~ = ~ '

theorem.

This implies that =

Conversely weakly in

~(x+iO)

~i = ~ - ~

suppose that

- ~/:(x-i0)

in

in

[? ] =

~(x+iy)

in

~(x+i0) - ~(x-i0)

=

in

~ (~L)

~ (/i).

Let

in

by Painlev@'s ~l(x+iy)

converges

~ (fg).

yj > 0

~(V - ~ )

fg

~(x+iy)+

~ (x+- iyj)

for a sequence

~+(x+iy)

Then there is

~(x-i0)

is analytic on

~(x+-i0) + ~l(X)

~ (~_)

are functions

= ~(x+i0)-

that

Hence

~ (x±iO)

9(/I).

On the other hand, it follows from

~'(~_).

tO

is in

converges to

~ (x+_ iO)

tending to zero. such that

There

? (x + iO) =

233

? (x+iy) - ~+(x+iy) ~+ (x+iy)

t-~+(x+iy) Then it follows that ~+(x-iyj)

> -~+(x-i0)

=

Thus

~(x+iy)

~+(x+i0) + ~+(x)

in

If

~

~ (£g) ~+

~ (~-)

~(x-i0)

Finally,

> ~(x+i0)-

as

~(x!iO)

~(x+i0)-

~(x-i0)

Painlev@'s we have

exist in =

as

theorem that

[~ ] = [~ ] =

The sheaf

~

? - ~

~ (~.).

by Painlev@'s

converges to

tends to zero.

y > 0

Of course, ~ (x-iy)

tends to zero.

~(x±i0)

exist in

?(x+i0) - ?(x-i0),

~'(~)

~(x+i0)-

0

In the same way,

suppose that the limits

is a defining function of

limits

~+(x+iy)

y > 0

~ (~)

and

weakly in

is analytic on

~ (x+i0). in

~+(x+i0)

is stronger than the topology in

= ~+(x+iy)+

the limit is the same as converges to

y < 0.

~(x+i0) - ~+(x+i0)

on each bounded set,

theorem.

y > 0

,

~+(x+iy)

Since the weak topology in '(~)

,

|

~'(~L).

then the

by Theorem 2.19 and we have

?(x-i0).

Thus it follows from

is analytic on

~ .

In other words,

~(x+i0) - ~(x-i0).

in Theorem 2.19 as well as the sheaf

~

satisfies

the assumption. If

~ (~)

is a reflexive Fr@chet space or the projective

limit

of a sequence of (DFS) spaces, then any bounded set is sequentially weakly compact and therefore the assumption ~(x+iy)

is sequentially weakly compact as

by the weaker assumption that For example,

[?]

ll~ (x+iy)~ILp

is in

~(x+iy)

~P(~L),

in the theorem that y

> 0

may be replaced

is bounded as

I < p <~,

is bounded for all compact sets

y

> 0.

if and only if K

in

(K) If

li~(x+iy)llL p (K)

is bounded,

it is easily verified that

234

l~(d/dx)m ~ (x+iy)llLp (L) = O(~yl -m) interior of

K.

Conversely if

for any compact set li~(x+iy)lILP(K)

(m+l)-st primitive is uniformly bounded in

tions are locally derivatives of functions in if and only if for each compact set

some

such that

m

(m)' ( ~ ) ~ L2

is in

I

its

Since distribu-

P,

[? ]

in

~i

is in there is

More precisely

[~ ]

if and only if

E U ~(x+iy) U 2 iy~ 2m-ldy < ~ -6 L2(K)

for each compact set

If we consider the symmetric limit instead of

K

I~~ (x+iy)l~Lp (K) = 0(I y l-m) m > O,

in the

O(ly~-m),

=

LP(K).

~'(~)

L

~(x+i0) - ~(x-i0),

K

in

lim ( ~ (x+iy) - ~(x-iy)) y+0

Theorems 2.19 and 2.20 hold for a

wider class of sheaves containing

~

i

and the sheaf of continuous

functions. Theorem 2.19 was proved for Ehrenpreis Martineau

~ = ~'

[13] in a little weaker form.

by Tillmann [32] and See also Bremermann

[12].

[25] gives a different proof.

When the dimension

n

is greater than

I,

Theorem 2.19 is no

longer true because the functions in the denominator

~_~ ~(Vj)

can

behave wildly as the imaginary parts of the coordinates approach zero°

We have, however, the following results. Theorem 2.21 (Ehrenpreis

convex open set in tion on f

~

,

~n

and let

V = ~ × i~ n.

[25]). If

then there is a defining function

such that the limits

~'(~)

[13], Martineau

for all n-tuples

~ (x+i~0) ~

of

~i

Theorem 2.22 (Martineau [25]).

f

Let

and

be a

is a distribu-

V

of

exist in

f(x) = ~ s i g n ~ ~

SL

~ 6 ~ (V # ~ )

= lim ~ (x+i£y) y .~0 J and

Let

~(x+i~0) .

be as above.

235

If

~ e ~(V # ~)

all

~ , sign~

has boundary values

then the hyperfunction

[~ ]

in

~ '(~)

for

is the distribution

?(x+i~0).

Unfortunately here.

@(x+i~0)

the proofs are too complicated to be reproduced

The following theorem is, however, an easy consequence of

Theorem 2.15. Theorem 2.23 (Harvey [4]). let

V

be a Stein open set in

is in

~(V ~ g).

component of function on

III.

with

be an open set in V n ~n = ~ . ?

~

~n

and

Suppose that

to each connected

can be extended analytically then

analytic function

Cn

~0.

If the restriction of

V @ ~ ~g ,

Let

to a real analytic

is a defining function of the real

~_~ sign ~ ( x + i ~ 0 ) .

PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

We denote by

~

, ~ ', ~ , ~

functions, distributions, analytic functions over

and

P(D)

is an

r I~ r 0

constant coefficients, r0

Rn

and holomorphic

functions,

functions over

~

matrix of differential

real

Cn

one of these sheaves. operators with

it defines a sheaf homomorphism

P(D)

:

rI ~ ~

We denote the kernel by

sheaf of solutions (3.1)

the sheaves of hyper-

infinitely differentiable

respectively as before, and generally by If

~

u 6 ~r0(w)

~ P

~ P

of the homogeneous

e(D)u = 0.

is exactly the equation

236

Existence We denote by indeterminates

A

XI,

the ring '''

X

¢[XI,

.-., Xn]

of polynomials

with coefficients

in

¢.

If

in

Q(X)

n is

n

a matrix with elements matrix

tQ(-X).

in

A,

Clearly

Q"(X)

Suppose that a system Replacing

- i; / @ x. J

nomial elements. rI

we denote by

by

the transposed

= Q(X).

P(D) Xj,

Q'(X)

of differential

we get a matrix

We regard the matrix

operators

P(X)

P'(X)

is given.

with poly-

as an A - h o m o m o r p h i s m :

r0

A

> A

and denote the cokernel by

generated A-module

r r1 A 0/p'(X)A

M'.

M'

is the finitely

By the Hilbert

syzygy theorem

[41]

we have a free resolution: r 0 P'(X)

(3.2)

0 <

A

M' <

which terminates

A P'

... <

r I P'(X)

~

Arm-i <m-I

for some

m ~ n

r


(X)

A 2<

r

A m

and with

~

0 ,

P' (X)

as the first

2)

homomorphism. The system

PI(D)

is said to be a compatibility

system of

P(D)

because of the following theorem. T h e o r e m 3.1. and let

W

Let

be a convex open

Then the differential

2)

~

be one of

~ , ~',

(convex compact)

only for homogeneous modules M'

M'.

set in

~n

~ (or ~ ) or

cn

Yoshioka,

B. O. Ra~aMo~oB,

TopsL C nOCTORHb, MH ~OB~HNHeHTaMH,

syzygy theorem asserts

this

Palamodov gave a proof for

of finite type.

keisu Senkei Bibun Sayoso, into Japanese of

and

equation

The original form of the Hilbert

arbitrary modules

~

See V. P. Palamodov, Kyoto,

1972

(a translation

~HHeAHb~e ~H~$epeHqHa~HbLe Hay~a,

HocKsa,

Teisu-

1967).

onepa-

237

(3.3)

P(D)u = f

has a solution

u

in

~

fies the compatibility (3.4)

ro

if and only if

(W)

f 6 ~rl(W)

satis-

condition :

PI(D)f = 0 .

More generally (3.5)

P

~

0

r 0 P(D)

(w) •

°

~- ~(W)

Pm-I(D)

°

~ ~(W)

~(W)

r

m

r I PI (D) ~''"

~ 0

is exact. In the case where Ehrenpreis

the case where

~

~(W)

~

for any

of [22]

= ~

this was announced by

[7], Palamodov and H~rmander Komatsu

i.e.,

f e

~(W).

r 0 = r I = i, system.

~

Thus

then we

(3.3) has

This result was proved

[22] and Ehrenpreis,

and

[6] gave a proof in

and

Malgrange and H~rmander

there) and for variants

~

as the compatibility

by Malgrange

Ehrenpreis,

~

is a single operator,

u ~

= ~

given there).

= ~ ,

PI(D) = 0

a solution for

~

P(D)

can take

and

[i] and proved by Malgrange

(see [5] and literature

If

= o~'

for

~ = ~ '

(see [35] and literature

by Harvey

[4].

by

given

All proofs are

in some sense.

Since convex sets form a fundamental any point, we have the following

system of neighborhoods at

local statement.

Theorem 3.2.

0

,

,

r° P

(3.6)

(D)

rl PI (D~

.

r m _ I Pm-i (D)

H~rmander's

of the sheaf proof

.

.

.

~ rm

.............. ~

is a resolution

r2

>0

~P

of solutions

of (3.1).

[5] of Theorem 3.1 makes use of the following

238 two theorems

(cf. also Malgrange

[7]).

Theorem 3.3 (HUrmander [5]).

For each matrix

polynomial elements, there exists a constant is a pluri-subharmonic

function on

(3.7)

?( ~')I ~-C

I?(~ )-

with constant such that (3.8)

C,

then for any

cn

N

such that if

I~" ~'I

u e ~(¢n) rl

sup Iv(~)le-~(~)(l+l~12)-N

with

satisfying

for

P ' ( ~ )v(~ ) = P'(~ )u(~ )

P' (X)

~ 1

there is a

v e ~(¢n)Ir

and <. C I sup ~ P ' ( ~ ) u ( ~ ) ~ C -?(~)

~C n

~C n

where the constant

CI

does not depend on

u.

Although the statement is simple, the proof, which is based on Cartan's theorem Let

K

B

with bounds,

is quite complicated.

be a convex compact set in

~n

or in

,

~

(3.9)

HK( ~ ) = sup Re <x, ~ > x~K

HK(~)

is said to be the supporting function of Theorem 3.4 (Paley-Wiener).

Then an entire function

U(~ ) E

Let

K

C~(¢n)

Cn

and let

C n.

K.

be a convex compact set. is the Fourier-Laplace

transform of a C = function, a distribution, a hyperfunction with support in

K

or an analytic functional with porter in

K

if and

only if the following estimate holds respectively : (3.10)

~U(~ )l <. C ~ ( I + i~i2) - w e H K ( I m ~ )

for any

u

,

(3.11)

<. C ( I + I~12) ~ eHK(Im ~)

for some

v

(3.12)

,< C~ e e l ' ~ l eHK(Im ~)

for any

g > 0 ,

(3.13)

~ C6 e gl~ eHK(-i~)

for any

6 > 0 ,

where the constants do not depend on

~

cn.

,

239 We owe to H~rmander [35] and [5] for this formulation of the Paley-Wiener theorem (cf. also Malgrange [7]).

The Paley-Wiener

theorem for analytic functionals was announced by Ehrenpreis [I] and proved by Martineau [24].

HUrmander [5] contains a simplified

proof. Since

v log(l+ 1412 ) + HE(Ira l~i)

and

~

+ HK(Im ~ )

are pluri-subharmonic functions satisfying (3.7), the following theorems follow from the above two theorems. Theorem 3.5.

, (3 14) •

If

r0~'

n

~ K (R)

K

(D) ~

is a compact convex set in

~(~n)rlP{~_(D)

.

P'm-I

(D)

""

is exact and the image of

P'(D)

(3.15)

W

is closed in

is an open convex set in

~ ,'( W )

r 0 P'(D) ~ ~(W)

is exact and the image of strong topology of Theorem 3.6. r0) (3.16)

(~(V)

rI

P'(D)

,

0

<

)

,_ n. r0 ~K(~ )

with

(g(~n)r0) '

~n, P' (D) .. .4m-i ~(W)

~

rm

n

~K(~

the topology induced by the strong topology in If

~n

is closed in

r

~ ~(W)

m < r0

0 with the

(~(w)r0) ' If

V

is an open convex set in

P' (D) rl) '~ (e(V) ' <

is exact and the image of

P'(D)

"-"

~m-1 (D)

C n, rm) (O(V)

'~

is closed in the strong dual space

( e (v)r0) ' For the details see H~rmander [5] or Komatsu [6]. Malgrange [7] proved the following by a somewhat different method. Theorem 3.7.

If

W

0

is an open convex set in

~n,

2~0

r 0 P (D) ~

~,(W)

(3.17)

~,(W)

rlPi(D)

r P~-l(D) &,.(W) ,, m

...

0

is exact. The exactness Theorems lemma. for V

Restricting

in

.

Cn

Hn(~,

~n,

= g

and

~

follows from

we obtain the exactness of (3.5) ~

V ~ ~n = W

where and

= ~

to

In the case where

~)

V0 = V ~

~

such that

~',

for

$

3.5 and 3.6 respectively by Serre's lemma and Schwartz's

~ = •

with

of (3.5) for

9~ = {V0,

V.j = ~z E V ''

on each

Vi0...i

= ~

,

take a convex open set

and represent

~ (W)

by

...

= {VI,

..., Vn 1

Vn~,

~'

Im z.j # 0 } .

,

Since

(3.5) is exact

we get easily the exactness

[6].

P T h e o r e m 3.8.

If

K

is a compact convex set in

)

P' (3.18)

~n

~K(~n)r0 ~

~K(an)

....

r

~ K ( R n) m <

0

r0 is exact and the image of is an open convex set in

(3.19)

P'(D)

is closed in

If

W

~n,

r 0 P'(D) rI ~ ~,(W) <

~,(W)

~K(ARn)

r "'" ~

~,(W)

m <

0

is exact. Proof.

(3.20)

(3.18) is dual to the sequence

~(K)

r 0 P(D) r I PI(D) > ~(K) • "'"

which is exact by T h e o r e m 3.1.

operators

~

lemma.

is flabby, we have the following theorem for single

P(D) .

Theorem 3.9

(Harvey

(3.21) for any open set

~ 0 ,

The statements of the theorem follow

from Serre's lemma and Schwartz's Since

r > ~(K) m

[4]).

If

P(D)

P ( D ) ~ (W) = ~ (W) W

in

LRn.

is single,

241

In other words, any open set to

~

W

is

P(D)-convex with respect

This exhibits a sharp contrast between hyperfunction

solutions and distribution or infinitely differentiable (3.3)

solutions of

(cf. [35]).

Regularity A system C > 0

P(D)

is said to be elliptic if there is a constant

such that [Re ~I

for any

~

cn

~< C ( I + IIm ~I )

for which

rank p(~ ) < r 0.

Theorem 3.10 (H~rmander [34]). r0 ~({0~) , then P(D) is elliptic. tic, then

If any

u E ~ P(~n)

Conversely if

~ 'P = ~ P and hence any distribution

is in

P(D)

solution

(3.3) is real analytic on the open set on which

f

is ellipu

of

is real analytic.

This is essentially given in the classical work [40] of Petrowsky and many proofs are known since then. John [36] which

employs

fundamental

We mention only

solutions, Morrey-Nirenberg

[39] which employs a priori estimates and Malgrange Ehrenpreis' Bengel

[7] which employs

fundamental principle. [i0], Harvey

[4] and Komatsu

[6] obtained the following

theorem which improves the latter part. Theorem 3.11. Proof.

If

P(D)

is elliptic,

~ P = ~ P

By Lech's theorem [38] we have only to consider single

elliptic operators

P(D).

Let

u

be a hyperfunction

(3.1) on a bounded convex open set complex neighborhood ?~ zn(~,

then

0~', O )

V

of

W.

and we have

W

Then P(D)?

in u

~n

solution of

Choose a convex

is represented by ~ Bn(~,

~',

~).

Let

242

be an (n-I) cochain such that is a solution

u

= ~

~0 e cn-l(q~, ]f', ~ )

is represented also by = 0.

P(D)~

By Theorem 3.1 there

of

?i = ~ - ~ 0

P(D) ~ 0 = ~ " Then

for which we have

If we prove that all boundary values of

is real analytic by Theorem 2.23.

~l(Z)

u

P(D) ? i

are analytic,

Thus the proof is reduced to

the following lemma. Lemma 3.1.

There is a complex open set

that all holomorphic solutions

~

U

containing

of (3.1) on the set

W

such

V+ = {z E V;

Im z. > 0 for all j ~ can be extended analytically to V + U U. J Harvey [4] proved this by Ehrenpreis' fundamental principle. The proof in [6] is based on the a priori estimate (Komatsu [37]) : (3.22) where

llVtuIIGz~ C(IIP(D)UlIG+ ~-tllulIG ), G

is any open set in

the set of points ~G

x

in

is greater than

G

S ,

W,

t

is the order of

P(D),

such that the distance between and

;luHG

is the L2-norm of

u

Gs x

is and

over

G.

Bengel's proof of Theorem 3.11 employs his theory of P-functionals and a similar but simpler extension theorem. Harvey [4] proved also the following converse. Theorem 3.12.

If

P(D)

is not elliptic, then

Thus the hypoellipticity loses its meaning

~ P # ~ 'P

as regards hyper-

functions. Theorem 3.13. (3.23)

0

~ ~P

If

P(D) ~ ~

is elliptic, then

r 0 P(D) PI(D$. Pm-i (D$ ~ ~ rl "" ~ rm

is a flabby resolution of the sheaf of

P(D)u = 0. For example,

~P

~ 0

of real analytic solutions

243

(3.24)

0

~ ¢

0

,

>~

d,

~

d r ...

d > ~ n

) 0

and

(3 25) •

•

are flabby resolutions sheaf

~

functions

,g~) (

,...

of the constant

of holomorphic

it follows

r~

sheaf

C

Cn

over

over

IR n

>0

and the

respectively•

Hence

that flabby dim C = n

and flabby dim ~ = n. Note that this implies Theorem 2.3. shows that any coherent analytic dimension

patibility

P(D)

system

(3.26)

be a single elliptic PI(D)

In particular,

~Rn.

HI(~ n,

of

~P.

HP(w,

By definition, ~.P)

~ ~

~P)

If

K

(3.28)

are of flabby

Since the com-

(3.27)

~ 0

p ~_ i,

the first relative

~ r~K(~ n, ~ )

is compact,

P(D)> ~

= 0,

is the first cohomology 0

operator.

We have therefore

P(D) (3.27)

Cn

= 0,

0 .....~ ~ P

is a flabby resolution

in

sheaves over

of Theorem 1.3

~ n.

Now let

= i.

The corollary

flabby dim

~_P

for any open set

cohomology

group

group of the complex n > PK(~R , ~ )

> 0.

is dual to the sequence

0 ~......

P' (D) ~(K) •

0~

$~

~ (K) <

0 .

Since P'(D)

~

~p, ~

0

is exact and since we have H p(K, ~_) = 0, by Theorem 2.5, the cohomology

p $ I,

groups of (3.28) are equal to the

W

244

H*(K,

cohomology groups

~P')

HI(K,

position.

= O.

~_P').

Since

In other words,

Therefore

P'(D)

and

P'(D)

is elliptic, we have

(3.28) is exact at the first P(D)

have closed ranges.

Applying

Serre's lemma, we obtain the duality : (3.29)

H~(~ n, ~ P )

= (~P'(K))'

IV. THE RELATIVELY COMPACT CASE

In this chapter we consider the relative cohomology groups H~(W,

~P)

with coefficients

(3.1) in the case where set

W

in

~n.

with the pair

set in (4.1)

W, 0

(4.2)

0

~ P

of solutions of

is a relatively compact set in an open

First we show that the long exact sequence associated (W, W-K)

Theorem 4.1. open set in

K

in the sheaf

is decomposed

Let

~n.

If

~ K

be

~ (~'

into short exact sequences. or

g )

and let

is a relatively compact

W

be an

(and locally closed)

then the following sequences are exact : '- PK(W,

~P)'"

+HI(w,

~e)

> H p(W,

~P)

7

*

P(W,

~P)

'~ p(W-K,

~P)

0 , > H p(W-K,

~P)

~ H P+I(w, K

~P)

~ 0,

p~_l. Proof.

In view of Theorem I.i (ii), it is enough to prove that 0

is exact for Let

> H p(W,

~P) ~

H p(W-K,

~P)

p ~ i. = ~ . Then by virtue of the flabby resolution

(3.6) we

have HP(w,

~P)=

[~(W, Pp_l ~rp-l)/Pp_l r(W,

~ rp'l)

245

and HP(w-K,

~P)=

~(W-K, Pp_l ~rp-l)/Pp_l ~(W-K, ~rp-l).

The restriction mapping : HP(w, ~P)

~ HP(w-K, ~P)

by the restriction : ~(W, Pp-l~rp-l) ?

be an e l e m e n t in

W-K

is cohomologous to zero in ~IW-K = Pp-l$

Since

~

Then

?I

hence I"

is flabby

~ ~(W-K, Pp_l ~rp-l).

r ~ (W, Pp-1 ~ p - l )

Let

whose r e s t r i c t i o n

F(W-K, Pp-I ~rp'l)" E~(W-K,

' where ~

has an extension

= ? - Pp-I $i & ~(W,

is induced

~ rp-l)

to

Then we have

~ re-l), ~I & ~(W,

has a support in

~rp-l). K,

and

~i

has a trivial extension to ~Rn which we denote also by r 6 ~(~Rn, p ~ p-l). Since ~n Clearly Pp ~I = 0, or II p-i

is convex, there is an element Pp-I ~2"

Thus

~2 6 [~(~n, ~rp-l)

~ = Pp_l(tl + ~2 )

In the case where

~

is in

is either

such that

Pp-I ~(W, ~'

or

~I =

~rp-l).

~ , we employ the

soft resolution (3.6) and prove that if

? ~ F(W, Pp-i ~rp-l)

the restriction in

Pp-i ~(W-K'

then

Pp-i ~(W,

Let

~rp-l).

%

be an element in

a neighborhood of the closure of p(W, Pp-I ~rp-I)

to

~ [~ (W-K, ~rp-l),

W-K then

~ rp-l),

K.

is in

C;(W)

which is one on

If the restriction of

is written ?

~

has

?I W-K = Pp-I ~

~

with

is written

? = Pp-I ((I - %) ~ ) + 1"1 ' where

(i- ~ ) t

r i ~(W, Pp-i ~ p- ) the same as above.

is extended to

W

by zero.

Clearly

and has a compact support in W.

~i

is in

The rest is

246

The long exact sequence of cohomology groups with compact support is also decomposed into short exact sequences. Theorem 4.2 (Harvey [4]). Let ~.

If

~n

K

~

be one of

~

, ~'

and

is a compact set contained in a convex open set

W

in

then the following sequences are exact : 0

~ ~,(W-K,

p.(W, ~P)

}P)

"~ F(K, ~P)

(4.3) ~.HI(w_K, (4.4)

HP(K,

0

I

~P)

~P)

HP+I(w-K,

P p+l ~"W , ~P) > H.~

~e)

) 0,

p _~I. Proof.

We prove that H~(W-K,

is exact for

p ~ I

~P)

~ H~(W, ~P) .....~ 0

or equivalently that

p.(W, Pp-I ~rp-l) C

F.(W-K, Pp-I ~rp-l) +Pp-I U*(W'

~rp-l)"

Then the theorem follows from Corollary of Theorem 1.6. Any element

~ e [~.(W, Pp-i ~rp-l)

? = Pp-l~ by Theorem 3.1. ~i

in

p.(W,

of

K.

in

Pp-I P*(W'

with

can be written

~ 6 p(W,

Since

~

~ rp-l)

which coincides with

We have

~ rp-l)

is flabby or soft, there is an element

? = Pp-i 41 + P p - l ( ~ -

~

on a neighborhood

41 ) . Obviously

Pp-i ~I

is

Pp-l)

rp-l).

~

and

Pp-I (~ - ?i )

is in

P,(W-K, Pp-i ~

The exact sequence (4.1) shows that any solution of on (a neighborhood of) and only if W,

H KI(W,

W-K

~P) = 0

P(D)u = 0

is extendable to a solution on Since

H~(W,

~P)

W

if

does not depend on

we have the following. Corollary.

Let

K

be a (locally closed) set in ~n

and let

247

V

and

W

be two open sets which contain

K

as a relatively

compact

r0

set.

If all solutions V-K

of)

in

~

can be extended

of

P(D)u = 0

to solutions

on

on (a neighborhood

V,

then all solutions

r0

in on

on (a neighborhood

of)

W-K

can be extended to solutions

W.

Ext p (M, A) To formulate A-modules

conditions

Ext,(M, A),

under which

where

H~(W,

A = C[XI, X2,

~ P) = 0,

..-, Xn].

we need

Consider

the

dual sequence of (3.2) : r 0 P(X) (4.5)

0

Clearly

~A

r I el(X) )A

this forms a complex,

homomorphisms

are zero.

is denoted by r

i.e. the compositions

The p-th cohomology

ExtP(M, A).

I,

tion

~ -X.

Here

: X

or the A-module Although

is determined uniquely by Ext0(M, A) = 0

M

of two adjacent

group of this complex

stands for the A-module

means

obtained

resolution

A = ~[X].

M'

means that

by the transformaExtP(M,A)

[2]).

that the homomorphism P(X)

This is the case if

ExtP(M, A) = 0

from

(3.2) is not unique,

M (Godement

or that the columns of the matrix

for

>0 .

r

A 0/tp(x)A

over

Pm_l(X)Ar m ~ .................

P(X)

is one-one,

are linearly independent

P(D)

P' (D) p-i

is (hypo)elliptic. is a compatibility

system

P'(D). P Condition under which The exact sequence

property of solutions

0 HK(W , ~P)

= 0.

(4.1) shows that the unique continuation

of (3.1) holds if and only if

PK(W,

~P)

= 0.

248

Theorem 4.3. (a)

Ext0(M, A) = 0 ;

(b)

r . O R n,

~P)

(c)

~,(~n,

~,p)

(d)

[,,(~n

E P) = 0 ;

(e)

[~{0~([Rn' ~3P) = 0 ;

(f)

['{01(~n'

Proof. (f)

and

(c)

= 0 •

~,P)

= 0 . (b) .

•

we have

(f) ~

(a).

ug = O.

> (e)

>

satisfy

.Nn. r0 ~,( )

belongs to Since

u&

P(D)u

tends to

and

u

in

u = 0.

For if

ExtO(M, A) # 0, A.

Let

with polynomial components such that

then the columns of

u(X)

P(X)

be a non-trivial vector

P(X)u(X) = 0.

is the Fourier transform of a distribution

point support at the origin which satisfies (f)

(b)

r ~,(rR n) 0

u~ = Je * u

Hence,

are linearly dependent over

e Cn ,

> (d),

/, ( _ n . r o u ~ ~ ....~ ) =

For let

P(D)ug = 0,

~'(Nn) r0,

~ (c)

(f).

Then its regularization

satisfies

P(D) :

= 0 ;

Trivially we have

(c) ~

(d) ~ = 0.

The following are equivalent conditions for

Then

u ( g ),

u(x)

with one

P(D)u(x) = 0.

Therefore

be a solution.

Taking

is not true. (a) ~

(b).

For let

u(x) E ~,(l~n) rO

the Fourier transform we get P(~)a(~) Since of = 0

P(X)

P(~ )

= 0 ,

~ ~

C n.

has columns linearly independent over is equal to

almost everywhere.

r0

almost everywhere.

This shows that

C[X],

the rank

Hence we have

~(~ )

u(x) = 0.

The same proof shows that if (a) is satisfied, then

P(D)u(x) = 0

249

has no non-trivial solution function on

~n

solution in

in

~Rn,

Let

K

and let

Ext0(M, A) = 0, Proof.

whose Fourier transform is a

In particular,

i ~ p ~ 2,

L p,

Corollary. W

cn.

or on

u(x)

J

or

~(¢n),

be a relatively compact set in an open set

~

be one of

then

PK(W,

Clearly

UK(W,

Conditions under which

~3, ~)',

~P) ~P) c

H~

ExtP(M, A) = 0 ;

(b)

H~(~ n, ~ P )

(c)

H~0~(~n,

(d)

H$(W,

(b)'

= 0

~P)

If

r.(~ n, ~P).

n, ~P)

= 0

(p > 0).

p ~ I. :

for bounded convex sets

K ;

for convex open sets

W.

We note that (b) is equivalent to the statement

(b).

Pp-l(D

P'(D). P

(K)rp_l

~K(~n)

rp

p

(D)

r . ~KORn) p+l

First we prove (b)' in the case where

compact convex set. system for

.

= 0 ;

U P) = 0

~K(~n) rp-1 (a) ~

and

The following are equivalent

(a)

Proof.

~

= 0.

In this section we assume that Theorem 4.4.

there is no non-trivial

(a) implies that

P' (D) p-I

is exact. K

is a

is a compatibility

Thus it follows from Theorem 3.1 that

(K)rp

is exact and that the image of

(D) P' (D) p-I

r

~(K) p+l is closed in

r ~(K) p-i

r.

Since

~KOR n) j

r.

are (FS) spaces with the strong dual spaces

~(K) j

and

P.(D) and P~(D) are continuous linear mappings dual to each J J other, (b)' follows from Serre's lemma. Next let

K

be only bounded and convex.

Any element

f(x)

in

250 the kernel of convex hull Applying

P (D) in (b)' has a support S contained in K. P Conv S of S is a compact convex set contained in

(b)' for

Conv S,

we see that

f(x)

The Ko

is in the image

Pp_l (D) ~ K~Rn) rp'l (b) ~

(c)

(c) ~

(a) o

trivially. r F(X) E A p

Vectors

with polynomial elements are

regarded as the Fourier transforms of distributions with support at the origin. ~01)

Let

F(X)

satisfy

we can find a vector

Pp(X)F(X) = 0.

u(x)

Since

IF( ~)~ ~ C ( l + I ~ 12) ~

3.3 that there is a vector

IU(~)I

~ CI(I+

satisfies E

,

for some

U(~)

C n .

it follows from Theorem

of holomorphic functions such that = F(~ )

I ~I2)V+No

The last inequality shows that mial elements.

~(~ )

= F(~),

Pp.l ( ~ ) U ( ~ ) and

(K =

of hyperfunctions with support at

the origin whose Fourier transform Pp_l ( ~ ) ~ ( ~ )

Then by (b)'

U(~)

is a vector with polyno-

Thus

A

Pp-i (X) >

rp -I

P p (X)

r A p

Arp+l

>

is exact. The proof shows that (a) holds if there is an analytic functional solution

u

P ( D ) f = O. P

of

Pp_l(D)u = f

Therefore,

(a)

for any

is valid

if

f (b)

in is

~

!

r {0}(N n) p

true

for

with

a non-void

set

K.

In particular,

(d) for a non-void

W

implies (a).

Clearly (b)

implies (d). Theorem 4.5.

The following are equivalent conditions for

P(D):

251 (a)

ExtP(M, A) = 0;

P ~ , K (~n) rp_l

(D) p-i

,

(b)'

~

~9 K (~R) p

is exact for bounded convex sets (c)

H ~ R n, ~ ' P )

(d)

H~(W,

~'P)

Proof. (c) ~

= 0 = 0

P (D) p

r

n

,

>

(a).

n

rp+l

K (~R)

K;

for bounded convex open sets for convex open sets

We shall prove

K;

W.

By the same method as above we have

(d) ~

~

(a) ~

(b)'

and

(b)' ~

(a)

Let

be compact

K

r

and convex.

For any

f E ~ ,K(ERn ) p

solution

u & ~ K ( ~ n ) rp-I

satisfies

Pp_l(~ ) ~ ( ~ )

C

and

~

of

with

P P (D)f = 0

Pp_l(D)u = f.

= ~(~ ),

there is a

The Fourier transform

~ ~ cn,

and there are constant

such that HK(Im

l~(~)l ~_ C(l+ I~12) ~

~)

e

Ho'rmander's Theorem 3.3 shows that there is a holomorphic UI(~ )

of

Pp_l ( ~ ) U I ( ~ )

= ~(~ )

solution

satisfying HK(Im ~ )

UI(~)

~- C I ( I + I ~[2) ~+N

e

r The inverse Fourier transform and gives a solution of

uI

of

UI

belongs to

n

p-I

P p_l (D)Ul = f •

The extension to general

K

(b)'4~=~(c).

is soft, the equivalence

for open set

,

~ K(nR )

Since

8'

is done in the same way as above. is immediate

K.

Consider the condition (b)

p n ,P) HK(~ , ~ = 0

for bounded convex sets K. r

(b) implies

(b)'

In fact, let

,

Then by Theorem 3.1 we can find a solution (4.6)

n

f ~ ~ K(~ )

Pp_l(D)v = f.

p

satisfy

Pp(D)f = 0.

v e ~ '(Rn) rp-I

of

252

v

is a solution of ~n

solution on Then

P

p-i

(D)v = 0

on

~ n _K.

Let

vI

whose existence is guaranteed by (b) and T h e o r e m 4.1. r , n p-i is in ~K ~ ) and satisfies (4.6).

u = v -v I

I do not know if (b) follows from (b)' or not. Ext0(M, A)

and

Extl(M, A)

vanish, we have

enough to prove this for compact = 0

on

~n

-K.

If

W

K.

Let

P(D)v = 0

~ n -W.

W

Thus

v

u

w h i c h coincides with

v -v I

(b) for

p = i.

is in

~n~

does not depend on

which are arbitrarily

close to

if both It is

be a solution of of

K,

P(D)u then

(c) that there is a solution

If there is another solution

the difference 4.3.

~n

on

However,

is a convex open neighborhood

it follows from Theorems 4.1 and 4.5 of

be an extended

u

on a neighborhood of

v I with the same property, , ~'P)

v

then

w h i c h is zero by T h e o r e m

W.

Since there are neighborhood

K,

v

is an extension of

u.

We have the following result for infinitely differentiable

solu-

tions. Theorem 4.6

(Malgrange

(a)

ExtP(M, A) = O;

(b)

H~

(c)

H~(W,

n,

~P) ~P)

= 0 = 0

In this case H{~l~n,

~P)

~2/~x~-~ P(D)u = 0

able solution on fundamental

The following are equivalent:

for bounded convex open sets for convex open sets

For example,

~2/~x~.~ ~n

[42]).

K;

W.

(a) does not necessarily follow from the fact that

= 0.

on

[7],

P(D)

be the wave operator

Then any infinitely differentiable

-{0~ ~n

let

.

solution of

can be extended to an infinitely differentiWe have, however,

solution gives a distribution

can not be extended to a solution on

~n

Extl(M, A) # 0. solution on

The

~ n _ ~0~

which

253

Duality We assume in this section that ExtP(M, A) = 0 with

m

as in (3.2).

= (~ / ~ j )

for

P(D)

is a system such that

p = O, i, "'', m-I

Single operators,

the Cauchy-Riemann

and the exterior differentiation

system

d = ( ~ / ~ xj)

satisfy this condition. Let

Q(D) = P' (D) m-I

be the sheaves

(4.7) 0

and

~

~P

and

. f0

Qp(D) = P' (D), m-p-i ~

and let

or the sheaves

P(D)> r I

~ '

and

PI(D) P ~(D) ~ ... m-i ~ ~rm

~

and

~ . Then

~0

and (4.8)

0 ~

give resolutions

- ~r0~ of

Theorem 4.7. (4.9)

~(D)

~P Let

and K

rl Qm_E(D) ... Q(D) ~ Q

<

~

Q

~

0

respectively.

be a compact set in ~n

dim H~(~ n, ~P)

~ rm

for

such that either

p = i, 2, "'', m,

or

(4.10)

dim Hm-P(K,

~Q) ~ d 0

Then the cohomology groups

for

H~(~ n, ~ P)

p = 0, I, "'', m-l. and

Hm-P(K,

~Q)

with

natural topologies are an (FS) space and a (DFS) space respectively and they are strong dual spaces to each other for Proof.

Let

~

= ~

a flabby resolution of

and ~ P

p = 0, I, "'', m.

~ = ~ . Then (4.7) turns out to be Thus

H$(~ n, ~ P )

are cohomology groups

of the complex: ~ ~K(~R n) r 0 P(D) ~ 8K(~Rn)rl

(4.11)

0

Since

K(~Rn ) J

Pm-I (D) r ~K(~Rn) m ~ "'"

~0.

r.

are (FS) spaces, it follows from Schwartz's

that the continuous

linear mappings

Pi(D)

lemma

have closed ranges if

254

(4.9) holds. Hm-P(K,

On the other hand, by Theorem 2.6

([Q)

are cohomology

groups of the complex: (4.12)

0 <

~(K)

r 0 ~ m - I (D)

~(K)

Q(D) "'' ~

rI

Schwartz's lemma shows that the ranges of holds.

Qj(D)

r ~(K)

m <

0.

are closed if (4.10)

Since (4.11) and (4.12) are dual to each other, the duality

follows from Serre's lemma. If

K

is convex and compact, we have

p . 0, . . i,. and

, m-l.

Therefore,

H~(~ n , ~

Hm-P(K, P

) = 0

~Q)

for

= 0

for

p = 0,1,''',m-i

m n P) HK(~ , ~ = (~P(K)) Similarly we have the following duality theorems. Theorem 4.8.

(4.13)

Let

K

be a bounded open set such that either

dim H~ (~n' ~'P) N- ~ 0

for

p = l,''',m

or (4.14) Then

dim Hm-P(K, H~(~ n, ~ 'P)

and

~Q) <

Hm-P(K,

for ~Q)

p = 0,1," "',m-l.

are a (DFS) space and an (FS)

space respectively and they are dual to each other for Theorem 4.9. (4.15)

Let

W

p = 0,1,''',m.

be an open set such that either

dim H.p,,(W, ~ ' P )

~- ~ 0

for

p = i, ''', m

or (4.16) Then,

dim Hm-P(w, HP(w,

~'P)

and

~Q) < ~

Hm-P(w,

for

~Q)

p = 0, I, "'', m-l.

are a (DFS) space and an (FS)

space respectively and they are dual to each other for Let = ~2n.

P(D)

be the Cauchy-Riemann

Then we have

Therefore,

Q(D) = - ~.

system

Both

P(D)

p = 0,1,''',m.

~ = ( ~ / ~ ~j) and

Q(D)

for

cn

are elliptic.

if either dim HP(¢ n, 6~) < ~0

on

p = l,''',n

255

or

dim Hn'P(K, ~ ) ~ ~ 0 then

H~(¢ n, ~ )

O,1,''',n.

and

for

Hn-P(K, ~ )

It is clear

Martineau's Theorem 2.4.

that

p = l,''',n-1,

are dual to each other for

Hn(K, ~ )

vanishes.

Theorem 4.9 for

p =

This g e n e r a l i z e s

P(D) = ~

is exactly Serre's

duality theorem [30].

The Jordan-Brouwer theorem Now let

P(D)

be the exterior differentiation

d = ( ~ / ~ xj).

Then we have : Let

Theorem 4.10 (Alexander-Pontrjagin). ~n

K

be a compact set in

such that either dim H~(~ n

C) <

for

p = I, "'', m

or

dim Hn-P(K, ¢) ~ ~ 0 Then

H~(~ n

¢)

and

Hn-P(K, ¢)

for

are dual to each other

Theorem 4.11 (Jordan-Brouwer). 2 n dim HK(~ , C)

such that either countable~ ) If

W

Proof.

W

Let

K

is finite or

is an open set containing

connected components of components of

p = l,''',m-l~ )

W-K

be a compact set in dim Hn-I(K, C) K,

is

then the number of

is the sum of the number of connected

and the dimension of

Hn-I(K, C).

It is clear from the proof of Theorem 4.10 that the

assumption implies the duality : H~(W, C) = (Hn-I(K, ~))'

3) This condition is satisfied for any compact set Theorems 4.10 and 4.11 hold unconditionally. the Alexander-Pontrjagin theorem, 489-490.

~n

K,

Thus we

so that

See H. Komatsu,

On

Proc. Japan Acad., 44 (1968),

256

have the exact sequence : 0

) F(W, C)

by Theorems

4.1 and 4.3.

of an open set C c.

~ ~(W-K,

U

in

If

~n,

c

C)

~ (Hn-I(K, C))'

~ 0

is the number of connected components

then clearly

P(U,

C)

is isomorphic

to

Therefore we have the assertion.

Non-compact

case

The detailed discussion of the non-compact elsewhere.

case will be given

We consider here only the Cauchy-Riemann

system of holo-

morphic functions. Let ordinate for

S

be a real subspace of

system,

j = s+l,

S

is written as

..., s+t,

z. = 0 J

Cn

If we choose a suitable

C s

× ~ t = ~z ~ cn ; Re z. = 0 J

for

j = s+t+l,

---, n~.

co-

The follow-

~ng result was announced by Sato [9] and is provable by the method of Martineau

[25].

T h e o r e m 4.12. (4.17)

Let

S = £s × ~ t

J~(~)

Therefore, (4.18)

if

K

H~v(V , ~)

For example, dimension

2

in

= 0

for

= O,

let

K

C n.

If

K

The exact sequence

This

K

we have for any open set

submanifold

¢n-2 x 2

> P(V,

is a removable

~)

K

may

under a suitable coordinate

I V (V, = HK~

~)

~

~)

P(V-K,

= 0

>H

for any open

av(V, ~ )

singularity.

seems to be an improvement

V.

of real co-

is not a complex submanifold,

0 HKnV(V , ~)

set

shows that

S,

be a real analytic

Thus we have

~)

Then

p = O,l,''',n-s-l,

system.

0 HKnv(V,

Cn

p # n-s.

is a subset of

be regarded as a subset of

V.

in

of the classical theorem on

257

removable singularity of holomorphic functions. Similarly let of

cn

K

be (a subset of)

a real analytic submanifold

which does not contain any complex submanifold of complex

codimension

m.

Then we have

HP(v, ~ )

~ HP(v-K,

~)

for

p = 0,1,''',m-l.

REFERENCES

[i]

L. Ehrenpreis,

A fundamental principle for systems of linear

differential equations with constant coefficients and some of its applications,

Proc. Intern. Symp. on Linear Spaces,

Jerusalem, 1961, pp.161-174. [2]

R. Godement,

Topologie Alg~brique et Th@orie des Faisceaux,

Paris, Hermann, 1958. [3]

A. Grothendieck,

Local Cohomology,

Seminar at Harvard Univ.,

1961. [4]

R. Harvey,

Hyperfunctions and partial differential equations,

Thesis, Stanford Univ., 1966, a part of it is published in Proc. Nat. Acad. Sci. U.S.A., 55 (1966), 1042-1046. [5]

L. H~rmander, Variables,

[6]

An Introduction to Complex Analysis in Several

Van Nostrand, Princeton, 1966.

H. Komatsu,

Resolution by hyperfunctions of sheaves of solutions

of differential equations with constant coefficients,

Math. Ann.,

176 (1968), 77-86. [7]

B. Malgrange, constants, (1961-62).

Sur les syst~mes diff@rentiels ~ coefficients

S~minaire Leray, Coll~ge de France, Expos@s 8 et 8a

258

[8]

A. Martineau,

Les hyperfonctions de M. Sato,

S@minaire Bourbaki,

13 (1960-61), No. 214. [9]

M. Sato,

Theory of hyperfunctions,

J. Fac. Sci. Univ. Tokyo,

(1959-60), 139-193 and 387-436. [i0] G. Bengel, tionen,

Das Weyl'sche Lemma in der Theorie der Hyperfunk-

Math. Z., 96 (1967), 373-392,

main results are an-

nounced in C. R. Acad. Sci. Paris, 262 (1966), Set.A, 499-501 and 569-570. [ii] G. Bj~rck,

Linear partial differential operators and generalized

distributions,

Arkiv f. Mat.,

[12] H. J. Bremermann, Transforms,

Distributions, Complex Variables and Fourier

Addison-Wesley, Reading, Mass., 1965.

[13] L. Ehrenpreis, tions,

~ (1966), 351-407.

Analytically uniform spaces and some applica-

Trans. Amer. Math. Soc., i01 (1961), 52-74.

[14] A. Friedman,

Solvability of the first Cousin problem and vanish-

ing of higher cohomology groups for domains which are not domains of holomorphy. II, [15] H. Grauert, manifolds,

Bull. Amer. Math. Soc., 72 (1966), 505-507.

On Levi's problem and the imbedding of real analytic Ann. of Math., 68 (1958), 460-472.

[16] A. Grothendieck,

Sur les espaces de solutions d'une classe

g6n@rale d'@quations aux d@riv@es partielles,

J. d'Anal. Math.,

(1952-53), 243-280. [17] A. Grothendieck,

Sur les espaces (F) et (DF),

Summa Brasil.

Math., ~ (1954), 57-123. [18] E. Hille,

Analytic Function Theory, II,

[19] L. H~rmander, operator,

Ginn Co., Boston, 1962.

L 2 estimates and existence theorems for the

Acta Math., 113 (1965), 89-152.

259

[20] H. Komatsu,

Projective and injective limits of weakly compact

sequences of locally convex spaces,

J. Math. Soc. Japan, 1-9

(1967), 366-383. [21] G. K~the,

Dualit~t in der Funktionentheorie,

J. reine angew.

Math., 191 (1953), 30-49. [22] B. Malgrange,

Existence et approximation des solutions des

@quations aux d@riv@es partielles et des @quations de convolution,

Ann. Inst. Fourier, 6 (1955-56), 271-355.

[23] B. Malgrange,

Faisceaux sur des vari@t@s analytiques r@elles,

Bull. Soc. Math. France, 83 (1957), 231-237. [24] A. Martineau,

Sur les fonctionnelles analytiques et la trans-

formation de Fourier-Borel, [25] A. Martineau,

J. d'Anal. Math., 9 (1963), 1-164.

Distributions et valeurs au bord des fonctions

holomorphes,

Proc. Intern. Summer Course on the Theory of

Distributions, 1964, Lisbon, pp.193o326. [26] V. Ptak,

Completeness and the open mapping theorem,

Bull. Soc.

Math. France, 86 (1958), 41-74. [27] D. A. Raikov, spaces,

Completely continuous spectra of locally convex

Trudy Mosk. Math. Ob., ~ (1958), 413-438 (Russian).

[28] C. Roumieu,

Ultra-distributions d@finies sur

~n

certaines classes de vari~t@s diff~rentiables,

et sur

J. d'Anal. Math.,

i_O0 (1962-63), 153-192. [29] L. Schwartz,

Th~orie des Distributions I e t

II,

Hermann, Paris,

1950-51. [30] J. P. Serre,

Un th@or@me de dualit@,

Comm. Math. Helv., 29

(1955), 9-26. [31] J. S. e Silva,

Su certi classi di spazi localmente convessi

260

importanti per le applicazioni,

Rend. di Math. Roma, 14 (1955),

388-410. [32] H. G. Tillmann,

Darstellung der Schwartzschen Distributionen

durch analytische Funktionen, [33] L. H6"rmander, operators,

Math. Z., 77

(1961), 106-124.

On the theory of general partial differential

Acta Math., 94 (1955), 161-248.

[34] L. H@rmander,

Differentiability properties of solutions of

systems of differential equations,

Arkiv f. Mat., 3 (1958),

527-535. [35] L. H@rmander,

Linear Partial Differential Operators,

Springer,

Berlin, 1963. [36] F. John,

The fundamental solution of linear elliptic differen-

tial equations with analytic coefficients,

Comm. Pure Appl.

Math., ~ (1950), 273-304. [37] H. Komatsu,

A characterization of real analytic functions,

Proc. Japan Acad., 36 (1960), 90-93. Narasimhan's theorem, [38] Co Lech, ideal,

A proof of Kotak@ and

Proc. Japan Acad., 38 (1962), 615-618.

A metric result about the zeros of a complex polynomial Arkiv f. Mat., ~ (1958), 543-554.

[39] C. B. Morrey -L. Nirenberg,

On the analyticity of the solutions

of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math., ~ (1954), 505-515. [40] I. G. Petrowsky,

Sur l'analyticit@ des solutions des syst@mes

d'@quations diff@rentielles, [41] J. P. Serre,

Alg~bre Locale,

Mat. Sb., ~ (1938), 1-74. Multiplicit@s,

Lecture Notes in

Msth., II, Springer, Berlin, 1965. [42] B. Malgrange,

Syst@mes diff@rentiels ~ coefficients constants,

261

S@minaire Bourbaki, 15 (1962-63), No.246.

Department of Mathematics University of Tokyo Hongo, Tokyo

CHAPTER I.

i.

THEORY OF MICROFUNCTIONS

Construction of the sheaf of microfunctions

i.i.

Hyperfunctions.

manifold and

X

M

be an n-dimensional real analytic

be a complex neighborhood of

determined by

M

We denote by

~X

~M

Let

M.

the sheaf of holomorphic functions on

definition,

where

~ : M C_~X

M,

~M

the sheaf of orientation of

to

~M

is locally isomorphic to

morphism

~MIU

~IU

that is,

M.

~,

on an open subset

giving an orientation of

X

and by ~M

U

~M

and giving an iso-

of

M

is equivalent to

U.

Definition I.I.I. ~M

A section of

=

is isomorphic

As in Sato [I], we define the sheaf of hyperfunctions on

(i.I.I)

M.

is the canonical injection.

We denote by ~(~).

is uniquely

if we pay attention only to a neighborhood of

the sheaf of real analytic functions on

-10~x by

X

The sheaf

~M

= ~M(6~X ) ~M

M:

is by definition

~M"

is called a hyperfunction.

As stated in Sato [i],

~(~X)

constitutes a flabby sheaf on

= 0

for

i # n

and

~M

M.

We first recall the following general lemma : Lemma 1.1.2.

Let

topological manifold

Y

X

be a d-codimensional submanifold of a

of dimension on

complex of sheaves)

X,

n.

Then, for any sheaf (or

we can define the following homomor-

phism (1.1.2) where and

~Iy COy/x = ~ d ( Z x )

~v

~ ~y(~)[d]

~

~Y/X '

is the orientation sheaf of

denote respectively

Y C X

and

~R

the derived functor in the derived

266 category and the functor of taking the subsheaf with support in Y

of Hartshorne [i]. Proof.

Since

~y(~x)

= ~y/x[-d],

we obtain the desired homo-

morphism as the composite of the following:

y/x dl • ~ y ( ~ ) e ~y/x[d]

•

q.e.d. We apply this lemma to our case where X

and

M

respectively.

(1.1.3)

~, X, Y

correspond to

~X'

Then we obtain the sheaf homomorphism ~ M

~ ~M

'

which will be proved to be injective later.

This injection allows us

to consider hyperfunctions as a generalization of functions.

The

purpose of this section is to analyse the structure of the quotient sheaf

~M/~M

1.2. tion.

from a very new point of view.

Real monoidal transformatio~ and real comonoidal transforma-

Now consider the following situation, although we apply it to a

special case in this section. Let

N

and

real analytic map. bundle of

M

be real analytic manifolds We denote by

N (resp. M) and by

bundle over

N (resp. M).

and

f : M ---~N

be a

TN (resp. TM) the tangent vector

T*N (resp. T'M) the cotangent vector

We can define the following canonical homo-

morphisms: 0 --->TM --+TN ~ M --~TMN --~0

(when

f

is an embedding)

(1.2.1) where

T * M K - - - T * N x M ~ - - T ~ N ~-- 0 N TMN (resp. T~N) is the normal (resp. conormal) fiber space.

We denote by (TM-M)/~÷

SM (resp. S~"~M, SN, S'N, SMN, S~N) (resp.

(T~"~i- M ) / ~ +, .-.),

where

~+

the spherical bundle is the multiplica-

267

rive group of strictly positive real numbers.

S~N

is not necessarily

a fiber bundle. Then, S~N C-~S*N x M N

and we have a projection (1.2.2)

: S*N × M - S~N M N

Suppose moreover that

t : M

provide

the

disjoint

union

~ N

is an embedding.

MN = (N - M )

real analytic manifold with boundary in

the

M N

the

same way as monoidal real

monoidat

of

a set of coordinate patches of I "'', (xj,

n xj)

N

N

U SMN

SMN.

transforms

transform

~ S*M

with

Then we can

a structure

of

Since this is constructed

of complex manifolds,

we c a l l

with

{Uj}

center

M.

Let

with a local coordinate

be

x. = J

such that M N Uj = {xj ~ U j, xjI

xjm = 0

~

Let V

(1.2.3)

x~j = fjk(Xk)

~ = re+l, -.., n ,

(1.2.4)

~ x ~ g~"~ J~"~, (xe) xj = y~=l

m

be a coordinate

transformation.

U'. J = {(xj , ~j);

((xj,

~j), t) I

We glue together (Xk, ~k ) e U k (1.2.4) and

We put

xj = (x~ ,''" ,xj) ~ Uj , x. J

=

group

~+

such that The multiplicative

P = i, ''', m

. J

~ ,

We denote by

in the following manner:

are identified

= i,

,m,

xj ~j~ _ 01 .

of positive numbers operates on

) (xj, t ~j). ~j

for

if

xj•

and

xk

~.j

U! by J the quotient U!/~R +.3

(xj, satisfy

~ j) e Uj

and

(1.2.3) and

268

m ~j =

~gjk,~(Xk)

We denote by by gluing

Uj.

~

~ = l,''',m .

the real analytic manifold with boundary obtained

~:

N --~ N

is the projection defined by

Uj ~ (xj, ~j)

~-I (M ) is isomorphic to the normal spherical ~--~x. e U.. Then, J J bundle SMN, and seen to be the boundary of MN. Moreover, ~ gives an isomorphism we denote by DMN

~-SMN x+

~0

---~N -M.

the corresponding point of

is a subset of the fibre product

{( ~ , ~ ) ~ SMN ~ S~N; set

~+

= (N -M) U DMN

the topology of a point that

N -M

x ~ DMN C MN +,

U N DMN

topology of --~ M+ N

For a tangent vector

< ~, ~ > ~ 0~.

DMN

~+

of

of

defined by

x

is an open set and

is the usual one, and for x

is a subset

U

U

~(x).

under the projection

0~ :

We note that the topology of

is not Hausdorff. Let 9I :

M~N* be a disjoint union of N* --'~'N

be the

canonical

(N-M)

and

projections,

S~N,

~ : ~+ will

In this way we obtain a diagram of maps of topological

MN+

(I .2.5)

+_~

~_O

DMN

SMN

N

N* ~_~ SMN

=

~

M

Note that all horizontal

-->

be equip-

under

ped with the quotient topology of

i)

such

with respect to the usual

and that the image of

is a neighborhood

SMN ~ S~N

N -M C MN+

a neighborhood of

is a neighborhood

SMN C ~ .

We define the topology on the

as follows: induced from

~ ~ TMN X - ~0},

inclusions are closed embeddings;

spaces:

269

2)

i~

can be considered as a closed subspace of

i~ x

* ;

N

3)

~

--~N

and

Remark.

MN+ - - ~ *

The map

separated if

X

are proper and separated.

f : X --~Y

is closed in

of topological

X x X.

spaces is said to be

is said to be proper if every

f

Y fibre of

f

is compact and

closed set in

X

by

f

f

is closed

is closed in

(that is, the image of a

Y).

The following lemma is

used frequently in this note. Lemma 1.2.1. a sheaf on

X.

Let

f : X --+Y

be separated and proper,

Then, for every point R k f , ( ~ )y

y

of

be

the homomorphism

If-l(y))

~ Hk(f-I (y) ;

is isomorphic for every integer

Y,

~

k.

For the proof, we refer to Bredon

[i].

In the sequel, the notion of derived category will be of constant use.

We refer to Hartshorne

[I] as to derived category.

We will not

distinguish the sheaf, the complex of sheaves and the corresponding object of the derived category. Proposition 1.2.2.

Let

~

be a complex of sheaves on

N

more precisely an object of the derived category of sheaves on

(or N).

Then we have an isomorphism

rSMN ( Proof.

)

)

~,

At first note that -i

-i

is an isomorphism. in

DMN ,

hoods of

N~DM N

This follows from the fact that for every point

the family ~(x)

( ~-I-i

,

NPSMN(~

~ U - SMN } where

U

runs through the neighbor-

is equivalent to the family

through the neighborhoods

of

x.

V -DMN

where

V

runs

x

270

Now we have a triangle:

/

IRT.rR~DMN(T-I~-I~)~

~ ~RFS~N(T-I~)

(See Hartshorne [I] for the notion of triangle.) Since

T : MN+

> MN*

is proper and separated with contractible m ~g-a~-.r(~-lz)

= o

fiber,

.

This proves the isomorphism

-1~-i~) ~rS~N(~-I~).

N~*~DMN(~

q. e. d. Remark. sheaf on

Let

Y.

and

f : X

Let

f-l~

~ Y

be a continuous map, and

__~ ~.,

with

~"

f-l~ ~ f,~'.

__~.

~

be a

be flabby resolutions

Hk(y 4--X ~ ~ )

of

is defined to

be the k-th cohomology of the simple complex associated with the double complex

r(Y ; ~')

--+ P(X ; ~ ' ) •

~ k ( f ~)

is defined to be the

k-th cohomology of the simple complex associated with the double complex logy. I)

~" - > f,~'.

They are a generalization

of realtive cohomo-

They have the following properties:

Hk( Y ~--X ; ~

) (resp.

~ ( ~ ) )

transforms a short exact

sequence into a long exact sequence. 2)

If

g : Z --+X

is given then we have a long exact sequence

"--~ H k ( y ~ x ~ 5) --+Hk(y~-z; ~) --~ H k ( x * z ; f-i ~) __~ Hk+l(y ~ X ; ~) _+. We denote by functors.

~(Y

(Cf. also Komatsu

Proposition point

<---X; ~ ) ,

x E S~N,

1.2.3.

~Z~Yf(~)

the corresponding derived

[i].)

Let

~kS~N ( ~ - l ~ ) x

~

be a sheaf on ~

N,

lim_~ H~(Ng ; ~),

through a family of locally closed sets Z

i

of

N

then for every where

Z

such that

runs

i

271

i)

Z ~ M

2)

x

is

is a neighborhood of not

Proof.

contained

in the

~(x)

in

closure

of

M , Z - M C NN*

At first, we define the homomorphism

(1.2.6)

~Fz(N:

~) - - + ~ P S ~ N ( ~ - I ~ ) E

There is an open neighborhood is a closed subset in

V

S~N ~ V.

of

x

in

~*

such that

v~-l(z)~ V

We define (1.2.6) by the composition

~rz( N : ~) _+ ~rS~N~V( v : - l ~ )

-~r

S~N(~-l~)x

Now let us prove the obtained homomorphism k (-i~) lim___+H (N ~ ~ ) ---~{S~N

(1.2.7)

x

Z is an isomorphism.

We have

0° -1 0 ---+~{S,¢N(~ ~ ) x ---~~ x --> li~ H0(V-S~ N;~r-l~) ___>~ISMN(~ -I ~)x

0

lim_~ H0(N

, -~ H0~W-Z ," !~,)

--~ ~x

Z

-~ 0

---~ lira__+ (N ; ~) --~ 0,

Z

Z

k i_~-i lira H k-I (V-SMN ; 7~-i~ ) -~ ~ SM N ~ )x for lira__+Hk-I(w - Z ; ~ ) Z where

W

subset in

-~ lira__+Hkz(N ; ~ ) Z

is an open neighborhood of W.

Remark.

k > I

m(x)

such that

Z

is a closed

These diagrsms imply the desired result,

q.e.d.

We can also prove the following proposition, but do not

give the proof here because we will not use it in this paper and its proof is a little more complicated. Proposition 1.2.4.

Let

propre open convex subset 3%-i(x) ~ U

is convex and

U

~

be a sheaf on

in

S~N

# ~-l(x))

Hk(u; ~ S ~ N ( K - I ~ ) )

<~

N,

then, for every

(i.e. for every point we have li__mH~(N ; ~ ) Z

x E M,

272

where

Z

runs over a family of locally closed subsets

Z

in

N

such

that I)

Z

contains

~ (U),

2)

the closure of

Z -M

Proposition 1.2.5.

in

~*

is disjointed from

Suppose that

~

U.

is a sheaf on

N.

Then we

have a triangle

j-.< ~rM(~)[d]~ where

d

~OM/N

~ ~.m~SMN(~-I

is the codimension of

M

in

N,

~ )[d]

and

~M/N

~M/N = ~{d(ZN)

= ~M ~ ~N* We consider the triple of spaces

Proof.

Then the following triangle is deduced from this triple:

~*~( $ )

m~M(3) Since

T

is proper and separated, ~(~)x

If

> ~T.~SMN(T-I~)

x ~ N-M,

= ~(T-l(x)

then

--+ {x} ; ~ x )

T-l(x) --+~x}

is an isomorphism;

~aL~!-%(~)x = 0 If

x E M,

then

-l(x) ~--- SMN x

for every

for is a

x ~ N.

therefore

x e N-M. (d-l)-dimensional

sphere.

Therefore ~(T

-I

(x) --~{x}

;

~ x )-~ ~ x [-d]

and this isomorphism has an ambiguity of sign and uniquely determined by the orientation of by the isomorphism

SMN x.

Since the orientation of

~ M / N , x ~ ~'

we have

SMN x

is given

273 IRIP(T-I(x) - - " { x } : ~x ) -~ (g, ~ OOM/N)x[-d]. It follows that

~Z~#~(~) -~ (~ ® ~0M/N)[-d] On the other hand, from the preceding Proposition 1.2.2 NT.N~'SMN(T -I ~ ) -- ll~.llTr. ~ - I N ~ s M N ( T - I ~ ) --~~('= 7£). 7~-I~RrSMN(T-I ~) _~ ~l(TgT). 75-i II['SMN(T -i~ ) ~" PR~r.IRq:.7~-IgRFSMN(I; -I 9%) ----.IIII.N~S~N(~ -i~ ) .

q.e.d. 1.3.

Definition of microfunctions.

original situation. manifold and

X

Suppose that

M

Now we will come back to the

is an n-dimensional real analytic We denote by

is its complex neighborhood,

sheaf of holomorphic functions defined on

O"X

the

We have the following

X.

isomorphisms TXIM -~TM • Vq~TM , T*XIM -~ T~,~M~ ~/~ T~"~M by the complex structure of

by

X.

Hence

TMX -~ TM ,

SMX --~ SM ,

T~

S~

-~ T ~ ,

~--S~]~

~ ' f ~ 4. "; g Taking account of this fact, we denote

and

~/[S'~M,

respectively.

is frequently denoted by where

~ ~ TxM- {0} Remark.

bundle

TX

If

X

The point of ~ S M

x+~0

(resp.

S~

and

S~

by ~/~SM

(resp. ~ S * M )

(resp. (x, ~-[< ~ , dx > ~)),

~ E TEN- ~0}).

is a complex manifold, then the tangent vector

of the complex manifold

X

and the tangent vector bundle

274

TX~

of the underlying real analytic manifold

ly isomorphic.

X

with the cotangent vector bundle

real analytic manifold T'X,

X

X~

by the inner product

is a local coordinate system of

Xn' YI' ..., yn) z~ = xp +~i-~y~

of the on

Re < ~ , ~ >

X

e T*X~.

and if

is a local coordinate system of ,

of the

T*X T%

in other words, by T*X ~ ~ e-~Re ~ = ~ ( w + ~ )

(Zl, -.., Zn)

are canonical.

We identify the cotangent vector bundle

complex manifold

and

of

XR

X~

TX If

(Xl, ...,

given by

then the above isomorphisms are explicitly given by

the following relations: TX=~ ~-~ (

T*X ~ dz~ (

>--~-3xv6

T~

~

,

TXIR ,

'-dxv E T*X~ ,

We use the preceding general discussions to this special case. We denote DM : {(~C~ ~ , ~ I ~ ) ~ V C ~ S M x v r f ~ S * M ; M M We have the following diagram:

<~f~

~f~):

-<~,~>

' )

MX+ <

R 0} '

>0}.

'

-3

DM

~

M

/(1.3.1)

X

Theorem 1.3.1.

k J~SM

<

(•

-i

(~X) = 0

for

k ¢ I,

where

"C :

MX ---~X . Proof. = cn.

Let

The question being local, we can suppose that x~'0 be a point of

~f~SM.

Let

(x I, ''', Xn )

M =~n C X be the

275

coordinates of

M

such that

~ = (0, ..., 0) + f~_-!-~ 0 ~1

u

Then ~ k ~/[ SM(~-I ~ X ) ~ 0-~ ~9--xx01im H k -l(u - SM; O-X)

0 ....>..X where

U

SM

(~-i~ X)

x~0 --~ ~x,0

for

k ~ i,

~ ~ lira H 0(U- v ~ S M U~--~°

runs over the neighborhoods of

x 0.

Since

; ~X),

U -v£f~SM ~ +

,

is injective by the property of unique continuation of holomorphic functions • Therefore

~ 0 fZ~SM(% -I O X)~0 = 0.

there is a fundamental system of neighborhoods -SM

is a holomorphically convex subset.

to take I. U~ = ~z = x + ~ f ~ y U{x+i~0

E X-M;

E ~SM;

~zjl < g , Ixjl < g ,

On the other hand, {~}

of

x0

such that

In fact, it is sufficient

( I Y 2 [ + ' ' ' + lYnl) < &Yl} (I~21+ "''+ l~nl ) < £~i }.

It follows from Theorem B of Oka-Cartan that k ~TT

SM('g-IOx)~0

= 0

for

k > I q.e.d.

The following theorem is the most essential one in the theory of microfunctions.

This is deeply connected with the theorem of

"Edge of the Wedge". Theorem 1.3.2. ~ $ Ik where

11: : ~ * Proof.

Let

s~ (-I~x

) = 0

for

--->X.

The question being local, we may take

xg ~ Z ~ S * M .

k ¢ n,

We choose a coordinate system

that i x~ = (0, ~ d x

I oo ) .

M = ~n C X = C n. (x I, "--, x n)

such

276

Then, by Proposition 1.2.3 ~{k

-i

rS*M

~

lim H~(X ; ~ X ) Z

where

Z

runs over the family of

Z =~z=x+v~y

~X;

Izj~ < g,

yl ~ - g(ly21+ "''+~ynl) } •

Moreover, lira__>H Zk(X ~ ~X)

k = li~ g ~ G ( O X ) 0

Z

where

G

,

G

runs all over the family of G =~z=x+~f~y

EX ;

Yl ~ - g (IY2~+ "''+lYnl)}"

By the theorem of the edge of the wedge (see the following remark), we have k ~f G ( ~ X ) 0

= 0

for

k # n .

Therefore k_~

-i S,M('~

~X ) = 0

for

k # n q. e. d.

Remark.

The following theorem is proved in Kashiwara

also Martineau subset in

Cn

[i] and Morimoto [i].) and

x

be a point in

d-dimensional linear subvariety a neighborhood of

x

in

L.

(i.3.2)

~a

O'X' ~ M

~M

for

x

such that

L ~ G

is

k ~ n-d.

-i

the antipodal map a

~M

a

~x ) ~ ~M ~/---~S*M

of the sheaf

is called a microfunction. by

be a closed convex

Suppose that there is no

through

~is~',~(~

the inverse image under

section of ~M'

a

G.

We define the sheaf

~M = ~n

where we denote by

G

Then we have

~k(C~X) x = 0 Definition 1.3.3.

L

Let

[i]. (See

on

~S*M

by

' ~S*M on

~

, and by S~'qM .

We define the sheaves

The

277

(1.3.3)

~ M = ~ i _ ~ SM ( ~ - l ~ x ) (~X = J * ( O X [ X - M )

(1.3.4) where

~M j

: X-M

c.~Mx ,

By P r o p o s i t i o n Proposition

,

'

= ~XIFi~ SM ' ~

1.2.2

:

"-~X,

n: :

and Theorems

--*X

1.3.1

and

.

1.3.2,

we h a v e

1.3.4.

Rk T . ~ ' I ~

I

eM 0

M ~ a3 M =

a

for

k = n-I ,

for

k # n-i

Proposition 1.3.5. Rkm.£M

= Rk+n-IT.~

M ® 09M = 0

for

k # 0,

and we have the exact sequence (1.3.5)

0

Proof.

Rk~,~M

~ ~M = R k+n-I

~ ~M ~*~M

of the preceding proposition. tion 1.2.5

1.4.

~ ~M

~

0

"

is the trivial corollary

The triangle obtained in Proposi-

implies irmnediately

exact sequence

> T C ,, "£M

Rk~,~

= 0

for

k # 0

and yields

(1.3.5).

q.e.d.

Sheaves on sphere bundle and on cosphere bundle.

We

consider the following situation. Let dimension S*

X

be a topological

n and

V*

space,

S = (V-X)/IR +,

We set

is the equivalent class of

to

V

V. and

We denote by V*

S

of

and

respectively,

S* = (V*-X)//R +.

D = { ( ~ , ~ ) ~ S X× s ,

I = ~(~,~) eS~S*;

be a (real) vector bundle

be a dual bundle of

the sphere bundle corresponding

that is,

V

<~,~>Z

(~ , ~ ) E (V-X)

< ~ ,~> > 0 1 and

0},

where (i'

~ (V*-X) X

E = I(~,~): S × S * " < ~ , ~ > X '

= 0~.

278

D

(1.4.1)

I

S

S*

S

S*

X

X

We denote by

n

~(

z

~

the invertible

X ) = O ~ X ~(z V) =

Proposition on

S

S n

~-module

@~-

(Zx) =

~ Xn(ZV.) "

1.4.1.

The derived category of abelian

and that of

on

S*

sheaves

are equivalent under the following

correspondences: = ~.

Remark. spaces

and

Let $

f : X --+Y

~)

X.

= {s E ~(X;

is defined

~ ~ In-l]

,

be a continuous map of topological

be a sheaf on

~f_pr(X; f,(~)

~-l~

We set

~);

supp(s) --+Y

to be the sheaf

-i YD Rkf,(~)

is proper 1

UI

~ [~(f~U)_pr(f

is its k-th derived

). (U); ~ If_l(u)

functor.

The following

lemma is fre-

quently used in this paper Lemma 1.4.2.

Let

f : X --~Y

i)

f

is separated,

2)

f

is locally proper,

and

exist a (not necessarily hood Let

V

of

f(x)

g : Y' --->Y

X' --~Y'

be a continuous map satisfying

that is, for every point open) neighborhood

such that

U

U ~ f-l(v) --~V

be a continuous

map.

and

g' = g × X : X' --~ X . Y the homomorphism g-iRkf, ( ~ )

Set

of

x 6 X, x

there

and a neighbor-

is proper.

X' = X x Y ' , Y

f' = f x Y ' Y

Then for every sheaf

___>Rkf,,(g, - i ~ )

~

on

: X,

279 is an isomorphism. For the proof, we refer to Bredon [i]. Proof of Proposition 1.4.1. DXl S* (1.4.2)

D

I

3

s X

Let

~

be a complex of abelian sheaves on

S.

We set

= IR'C. TC-1J'r I~ oO[n-l] We chase diagram (1.4.2) ~,-i ~ = ~ g , - I ~ . T g

~

00[n - I]

= ~ T . .g, -i 7C-I ~ ® ~ [ n - 1 ]

Therefore ~R7E', q;,-l~

= ~TC',IR~.q~'-ITE-I$. @ ~o[n-i] = m(~'

o ~),(xr')-l~@~[n-1]

= R~2.~T£'~71'-Iq;;I~®

~[n-

i] .

Now note the following lemma. Lemma 1.4.3. ~' and

Let

: D × I --->S ~ S S* X I ~

S.

~

be a complex of abelian sheaves on

be the canonical projection defined by

D ~-~S

Then

tR-~:',-#I:'-I~, Proof.

S K S, X

3Z'

= ~tSxS S

@ tO[l-n]

is separated and locally proper.

, The fibre of

3z'

is the intersection of an open hemi-sphere and a closed hemi-sphere. Therefore,

for every

x 6 S ~ S,

dimensional open hemi-sphere if

qT'-l(x) x 6 S × S S

is homeomorphic to (n-l)and

to

(n-l)-dimensional

280 euclidean half space or

~

if

~ ~

(~,, ~,-I ~)x

!

x

x ~ S × S - S × S. X S

[I- n]

0

for

x 6 S × S

for

x ~ S ~ S . S

s

The above isomorphism has only the ambiguity determined by the orientation (~,,

~,-I~

of fibre.

of

signature

which is

Therefore for every

~ x ~ ~ [i - n]

)x ~

Therefore

x¢=S

×S. S

It follows that

/RII ' , ~ '

-I ~

~IS&

S ~

oo[l -n]

S This completes

the proof of Lemma 1.4.3.

Now return to the proof of Proposition

1.4.1.

We have from

Lemma 1.4.3:

= T ~ ,

T,-I~

= ~2,(T~I~s×

l

Is

s , s

S) = S

Conversely,

let

time, we chase diagram

~

be a complex of sheaves on

S*.

In this

(1.4.3).

I ×D S

(1.4.3) S

S*

S*

X We set

3

,~

.

Then we have ,

~ R T , TC-I$

= N~,,TE-I

ql, -I

: ~.E,IR~,,

-lq~ - I ~

= ~2,~,

= ~Ir2~' ( ~ { i ~

I S * K S *) ~ ~0 [I - n] S~

'T

281

= ~ ~

/~ [I-n]

Hence

IR17. 7E -i ~

~=

g0[n-I] . q. e. d.

We denote by deduced from

a

S

the involutive automorphism of

V 9 g ~

(or

s*)

~ - g & V.

Proposition 1.4.4.

Let

~

be a sheaf on

= ~,7~-l~[n-l] = IR~,~[n-l]

~

~ ~

cO

S.

Set

,

= ~vr, ~

We have then the canonical triangle

is the inverse image of

where Proof.

~

by

We chase the diagram (1.4.4). DXD S* ~T

a. Consider a triangle of relative cohomology of

2 ~i ~

D

(i .4.4)

S

qxs

~

S*

D

with respect to

: D x D --+S × S; S* X

S

@ ~ = ( ~ i I- 9~)

_Cl-i ~ Put

a

A S

=

{(~ , ~

~ a

) e S >~ S; X

JR-x. ?[-I~ i i >

~ 6 S}.

Since every fibre of TC

is an intersection of two closed hemi-sphere, to (n-l)-dimensional disc when

x E S ~ S

.

~g-l(x)

is homeomorphic

z~as and to (n-2)-dimen-

282

a x ~ ~S.

sional sphere when

Therefore

I (T ~l~)x[l _ n]

for

x ~a

~

~(~

s

)x =

0

for

a

x ~ ~S

"

Therefore

nR~(~ll~)--~ We operate the functor

TII~I~ S ~ 60[l-n].

~R~2,

on diagram

~R'C2,('II~I~S)~O[

aT2,TII~

,

(1.4.5).

l-n]

a T 2 , ~R-~,I~-i T i I

It is clear that -i ~T2,(T I ~a)

~a

~ ~O[1-n] =

~

~0 [1-n]

,

S ~T2.TI I ~

=

~R~g2. ~ r . ~-i ~ 1 1

= aR(~2 o ~Ii),(~ I ~ ) - I ~ = ~ R ( ~ o T 2 ) ~(~i~i)'I ~

-i

= ~71.1RT2.~I 1 -I

= ~RTr.]I-I~ ~0[l-n].

-I~

Thus we obtain the desired result. 1.5.

Fundamental diagram on

q.e.d. ~

We will apply the arguments

in the preceding section to a special case. tion 1.4.4 to the situation = C M,

~ = Tg.e M.

Proposition

~ = ~ a M'

At first we apply Proposi-

X = M,

S = ~f~SM.

Then

We obtain

1.5.1. Rk~,.~-l~ ..

M

= 0

for

k # 0

and we have the exact sequence 0

> ~M

~ T-I~*£M

Now, we apply the same proposition Thus we obtain a homomorphism

> ~,T-ICM

~ 0 .

to the case where ~

= ~M"

283

(1.5.1)

~M

> ~-iRn-I T * ~ M @ ~

~M

= aJ*(~Xl X-M)I~-T SM

where

j : X - M c.~M~,

which implies that Rn-l~,~

= R n - l ( ~ o j),(~Xl X_M ) •

Hence we can define the canonical map R n-I T , ~

~ ~ ~

It yields, together with (1.5.1),

a homomorphism

> ~r -I~M"

~M

Summing up, we have obtained Theorem 1.5.2. of sheaves on

V~SM

We have the following diagram of exact sequences : 0

0

~ ~a-l~ M

0

~ -C-i ~ M

0

~0

II

(1.5.2)

m-l~M

~ ~-I]~.~M

-1

-1

w

M

,=

0

Proof.

>0

M 0

It has already been proved that the rows are exact.

right column is exact by Proposition 1.5.1.

Hence it follows that the

middle column is exact, Let us transform

q.e.d. diagram (1.5.2)

a diagram of the sheaves on where

T',

-PC' are ~

= to®

7,)[1-n]

B y Proposition 1.4.1,

v~S~'~M

projections

For a sheaf

(~ ~

The

on

M,

of the sheaves on by the functor

IM --+~-~S*M

we have

= 911-n]

and

~@~'

~T~SM !

~,-i

IM -->~-~SM.

to

284

LO~ IR ~' ,TO ,-I TC.~ -I ~ M = oo ~ fiR77'' 76'-I[R]1~. ~ -I £ M By operating

fir~' ,v~'-i Rk C,

on exact columns in (1.5.2), we obtain

,7~

Rk_~,

= ~ M [I -n] "

,-I~

M

,-i ~ '~

= 0

for

k # n-I ,

= 0

for

k # n-I .

~M

We define the sheaves

~M

and

~M

S*M

on

by

Rn- I '

~M

M

'

= Rn-IT',qT'-I~M

~ gO

Then, in this way, we obtain the following theorem. Theorem 1.5.3. on

We have the diagram of exact sequences of sheaves

~7~S*M 0

0

~ ~-lil~M

0

~ ]r-l~ M

0

M

~'

!M ~-I~

(1.5.3)

eM

--~0

1 CM

=

0

£ M 0

and diagram (1.5.2) and diagram (1.5.3) are mutually transformed by the functors

~ ~

~',gr'-l[n-l]

The homomorphism hyperfunction

u,

and denote it by where

u

and

9 T - I ~ M --9 ~ M

~TU, T - I is denoted by

we call the support of sp(u) S -S(u).

-H7(S-S(u))

sp.

For a

the sinsular support

is evidently the subset

is not real analytic.

We will give

a direct

application

of Theorem 1.5.2, which gives

a relation between singular support and the domain of defining function of hyperfunction. A subset

Z

of

Wq~S*M

is said to be convex, if each fiber

285 Z

X

= Z ~'[

-i

(x)

is convex.

joining two points in

It means, by definition, is contained in

ZX

antipodal points is understood to be in

f~SM,

-I

(x).

An arc joining two For every

subset

we call the smallest convex subset containing

convex hull of by

~

Zx .

that any arc

Z.

~(x, ~ f ~ )

The polar 6 ~ S ~

Z°

is the subset of

; <~ ,~>

_2 0

Z

fZ~S*M

for every

x+~f~

Z

the defined

~0 ~ Z}.

By using these notions we can state the following proposition. proposition 1.5.4. convex fiber, i)

If

V

~ )

6 ~(U; ~ ) ~(V;

~)

Proof.

U

be an open subset of

be a convex hull of

~ E P(U; ~ ) ,

~(~U;

2)

Let

then

U.

S-S(f) C U °,

such that

f = a (~).

~

~(U;

~)

with

Then we have

S - S ( ~ ( ~ ) ) C U °.

satisfies

~Z~SM

Conversely,

if

f(x)

then there exists a unique

is an isomorphism.

Consider the exact sequence 0

> ~

~ T-I~

' 7r,T-l£

>0.

From this, we have the following diagram

o

~ r(v; ~ )

,,,

~ F(v;T'l~)

> F(v;

~.z

-i

£)

<154) 0--~ with exact rows.

F(u;

Since

V ~

~)

~- F(u; ~ - 1 ~ ) ~V = ~U

and

F(u; U

> ~U

7r.z -I C )

are open

mappings with convex fibers, F(v; T'iv

-l~)= ~ ~-lv

P(u; "17"10'3)=

F(~U;

= 4rY s"~ - v ° = 4ri S*M - u °

is an open mapping with connected fiber.

F(V; ~ . T - 1 C )

05 ) ,

_-

This implies that

F(~-lV; ~ - 1 ~ ) =~ F(.c~-lv; ~ )

-- F(4-C~s"~-u°; d ). On the other hand,

[~(~S~'~M-U°; ~)

> ["(TC-Iu; "c-l~ a) = P(U; 7r.'E-l£ a)

286

is injective.

Summing up, the middle arrow in diagram (1.5.4) is an

isomorphism and the right one is injective. left one is isomorphic. 0--" is

exact,

In the

k # 0

flabbiness

~ ) --+ F ( ~ r i s " ~ - U ° ;

~ )

same way a s a b o v e , we can p r o v e t h a t

and any open c o n v e x s u b s e t

of the

t h e same r o l e

2.

~ r(~U;

which completes the proof.

for

several

Moreover,

P(U; ~ )

Remark. = 0

Hence it follows that the

sheaf

~

.

Therefore,

V

of

SM,

Hk(v; ~ ) by u s i n g

t h e open c o n v e x s u b s e t

the plays

as a domain of holomorphy in the t h e o r y of f u n c t i o n s

of

complex v a r i a b l e s .

Several operations of hyperfunctions and microfunctions In this section,

we will show that hyperfunctions and micro-

functions behave like "ordinary" functions. 2.1.

Linear differential operators.

manifold and

X

Let

be its complex neighborhood.

M

be a real analytic

Recall that the sheaf ~ X

of differential operators is defined by ~X

--dimX

~x

=

(0,dim X) ~XxX

where

(0,dim

(Ox×x

is the sheaf of

x)) ,

(dim X)-forms in the second

variables with holomorphic functions as coefficients and tified with the diagonal of

X × X.

We denote frequently the sheaf a left

~x-MOdule,

is a left with

~

-i

%

and

~M-MOdule.

~M

Module,

IYM

is a right

and

~M

~

~M

~-l~M-MOdule.

is iden-

(Cf. Sato [2]). ~XI M

by

are left

We denote by

real analytic coefficients.

X

Since

are right

I~M

~M"

Since

~M-MOdules

and

~X

is

CM

the sheaf of densities

o dim X "-X

is a right

~M-MOdules

and

CM

~X~ ~-i~ M ~-l~M

287

2.2. and

X

N --->M

Substitution.

Let

f

~:

be a real analytic map and

M~X* --*X

N

and

M

respectively,

fc : Y --+X

Y

f :

be a holomorphic map

f.

~ : N × v~S*M-~S~M --~f~S*N M --->~f~S*M are canonical maps induced by

N x~S*M-~S~ M

(cf. ~1.2).

and

be real analytic manifolds,

be complex neighborhoods of

w h i c h is the extension of and

N, M

We denote by

~N

respectively

and

~M

the projections

to avoid confusions.

Ny,

and

Ny, __~y *

are

related with each other by the diagram Ny, ~

y x X

-

)

(2.2.1) Y

If

u(x)

Y

is a hyperfunction

we can define the hyperfunction under additional

>

X

(resp. microfunction)

on

(resp. microfunction)

.

M,

u(f(y))

several lemmas.

Lemma 2.2.1.

space,

f : E

> F

on

condition.

To explain this operation , we prepare

vector bundles on

then

Let X

X

be a topological

with fiber dimension

n,

m

E, F

be two

respectively and

be a surjective map of vector bundles.

We define the

spaces SE = ( E - X ) / ~ + DE = { ( ~ , ~ )

S*E = S(E*) ~ SE × S*E

;

Sf = S(Ker f)

<~,

~>

X

~ 0~ --

and the maps E : DE --->SE ,

~ E : DE --~S*E

t : SE - S f ~ S E

,

q : S~

¢-~S*E .

We set finally n

W E = ~ X(~E )Then for every sheaf

on

SF,

we have

,

p : SE - Sf --~ SF,

N

288

-i -i 7EFI T E , ~ E t ,P ~ ® ~E = q*~R'[F* ~ ® ~)F [m-n]

(2.2.2) Proof.

We chase the following diagram : (SE-Sf) × DEC-,,,,,, t ~ DE

/ /

SE-:

L

SE

(2.2.3) DF

P

~tE

I p C__

-

S F ~, S * E

SF

C

--~

S*E

q

We have BRTE * ?~EI %,p-l~ The fiber of space or

~

space over

p

over DF.

= ~ C E * I,.TEEIp-I~

-i ~ -I ~.

= ~*~P!P

is an (n-m)-dimensional closed euclidean half

SF × S*E -DF X

and an (n-m)-dimensional euclidean

Hence we have mp,p-l~-i ~ = ~-I~IDF~

~E ~ ~F[m-n]

Therefore ~R'~.~Rp~p-ITc-I~

= nR T.(lt-I~IDF) ~ W E ~ ~F[m-n] = q.~F.~FI~

@ COE ~COF[m-n]

This completes the proof of the lemma. Lemma 2.2.2. every sheaf (2.2.4)

~

Under the same assumption as in Lemma 2.2.1, for on

S'E,

we have

~Rp, %-lIRa] q:EI ~

Proof.

COE = IRTCF, T F I q - I ~

~0F[m-n ]

This is proved in the same way as above.

ERp, % - I ~ E . TE l ~ = ~p~IRTCE, t - I ~ E I ~ = E~TC,~p,p-iT-l~ : ~R"}~.("~"-1 ~ IDF) ~} tOE ~) COF[m_n] : ~ ~F* TFlq-l~®~E~)°0F[m-n] " q.e.d.

289

Lemma 2.2.3.

Let

Y, X

closed submanifolds of

Y,

be real analytic manifolds and X

respectively.

smooth real analytic map such that also smooth. ~*

~

X

We denote by

respectively.

~N'

~:

Let

f(N) ~ M ~M

f : Y --+X

and that

the projections

N x S$~ --> S ~

N, M

be

be a

N --~M

is

l~y, _ + y

and

denotes the canonical

M

projection.

We identify

N x S~,i with the submanifold of

SN*Y. Let

M

be a sheaf on

X.

Then we can define a canonical isomorphism

~ - I [ R r s ~ ( q ~ M I ~ ) ~ ~oM/x[codimxM] (2.2.5) -~ ~R~S~f(~NIf-I~ ) ® ~N/y[codimy N] . Proof.

Consider the following diagram : Ny ~

(2.2.6)

XTN i

~ Ny_ ~(f ~M) • ~ ~ MX J

I~

Y

f

I"

Y

~X .

At first we show that

j ,~-i ~FS~(~MI~)

(2.2.7)

- -l~) --~SNY(TNIf

is an isomorphism. The question being local, we can suppose that Y = {(x,y,z,u)~n+d+rn+£}, X = {(x,z) e~n+m}, Let

q 6 ~ )

N = {(x,y,z,u)~Y; z = u = 0 } ,

M = {(x,z) EX; z = 0 } ~ SNY.

of

f(TN(q)).

{f(~-SNY) 1

f(x,y,z,u) = (x,z).

Then it is easy to see that there is a

fundamental system of open neighborhoods is convex and

and

U

of

q

such that

U -SNY

is a fundamental system of neighborhoods

Therefore,

i ~ Hk(~ - SNY ; f-l~) ~_ l~__~mHk(f(~ - SNY) ; ~ ) U U { ~ f (TN(q)) 0

for

k = 0

for

k # 0

290

This implies that jlk Nlf -I SNY(~ ~)q To see that homomorphism (2.2.7)

= 0 .

is an isomorphism, it is sufficient

to show that (2.2.8)

f-IIRrsMX('gM 1 5 )

is an isomorphism. Lemma 2.2.4.

--+ j - I ~ s N Y ( ~ N I f - I

~)

This is a direct consequence of the following: Let

f : X --~Y

be a continuous map of topological

spaces satisfying i) ii)

f

is open For every point

U --+ f(U)

x ~ X,

the neighborhoods

of

x

such that

is proper and separated with contractible fiber, form a

fundamental system of neighborhoods of Then, for every closed set on

U

Z

in

x. Y

and for every sheaf

o~

Y, f-I O i.k Z(~)

is an isomorphism.

___~k 1 (f-l~) f ~Z

(We refer to Sato-Kashiwara [5],Appendix, Corollary

(4.2) .) Now,

isomorphism (2.2.5)

is obtained from

(2.2.7) by Proposition 1.2.2 and Lemma 2.2.1.

isomorphism

This completes the

proof of Lemma 2.2.3. Lemma 2.2.5.

Let

be submanifolds of the projections analytic map

Y

Y, X and

X

Ny, __+y

such that

be real analytic manifolds, respectively.

and

~*

--+X.

Let

N

and

M

We denote by

7rN, 711M

f : Y --+Y

be a real

f(N) C M.

~: N ~ s ~ - s ~ ~ s ~ , ~: N ~ s ~

s~x ~ s ~

M

are the maps induced by

f.

Then, for every sheaf

on

X,

we can

291

define a functorial homomorphism IR ~ - I R ~ s , ~ (~~AInM

) ® ~M/x[codimx M]

(2.2.9) ---> ~ p S ~ Y ( ~ N l" fI Proof.

~N/y[codimy N]

The proof is divided into two steps.

(i-st step) N = M ~ Y

~)

The case where

f : Y --+X

(it means moreover that

T

is an embedding and

X× N -->T~Y

is surjective

everwhere). We have the diagraml

F %

X

D Ny

~

D

Y

f

Therefore we can define "f-I[RrSMX(TM I ~ ) - - ~ _ I ( S M X ) ( [ - I ~ M I ~ )

= IR~SNY('rNIf'I
By Lemma 2.2.2 and Proposition 1.2.2, we obtain the desired homomorphism. (2-nd step)

In the general case, set Z = Y × X,

Let

g : Y --~ Z

projection.
be the graph map of

f

and

h : Z --~X

be the second

Note that s

z-s

z)

L

= N X M

By L e n a

L = N × M.

2.2.3 and (1-st step),

M

we get

~f, ~If-l~rs~<~ M 1 ~) ~ WM/x[codim X M] _ ~

j-I N~SLZ( ~ L l h - l ~ ) ~

-->IR~s~y N l f(7[ - N

~L/Z [codimz L]

-I ~ ) (9 ~N/y[c°dim Y N] q.

e. d.

292 Now, we will consider how we can define restrictions of hyperfunctions. of

Let

vC~ SM.

Let

u

where

N

be a submanifold of

sect with

is a section of ff~SN,

we can take

~i

Then

~M"

u

u

~'N"

is represented as

If the domains of all UIN

does not intersect with

so that the domains of all

~j

~b(~j),

~j

by

inter-

~b(~jlN).

~S~,

then

intersect with

In this way, we have reached the following theorem. Theorem 2.2.6.

Let

M.

is a closed subset ~ M ;~2T SN ~

then it is natural to define

If the singular support of

~SN.

v~SN

We have the canonical homomorphism

be a hyperfunction on ~j

M.

~'

Let

f

be a real analytic map from

N

to

M.

be the kernel of f-l~M

where the last

f

--+ ] l , ( f - l ~ M ~ I ~ S ~ )

signifies the map

,

N × ffI~S*M - - > ~ S * M .

Then we

M

have canonical homomorphisms of (2.2.10)

f* :

(2.2.11)

f*

:

substitution ; ~'

--~ ~ N

'

~! ~i~-i~ M ---~ ~ N

"

These two homomorphisms are consistent. Proof.

The second homomorphism

applying Lemma 2.2.5. three steps.

Set

(2.2.11) is obtained directly by

Now let us define the first homomorphism in

n = dim N

and

(i-st step) The case where

m = dim M. f : N --->M

is smooth (i.e.

S~

We obtain (2.2.10) as the composite f-l~M

= f - l ~ - ~ M ( e X ) ~ O M _ ~ f ~ Im( M

) (fcIO'x) IN ~ ~ M

m ((>y)iN©60M - - ~ N ( ~ Y Y ) ~ .~0-[f~l (M)

C°N = ~ N

where the last arrow is defined by Lemma 1.1.2. (2-nd step)

The case where

f : N --+ M

is an embedding.

'

= ~).

293

We can assume

0 < n = dim N < m = dim M

without loss of

Recall the exact sequence

generality.

0

-I~ M

~ ~M

of sheaves on

~ 0

From this sequence we have the homomorphism

~l~ S*M.

By virtue of Lemma 2.2.2, the right hand term

]I-.(~MI~fTS~)

is isomorphic to

R n ~r,.( ~

l~g-~SM×N-~-~ SN ) ~ U°N" M

Consider the following exact sequence R n-I T . ( ~ )

--~ R n-I Z.< ~MIV-I~-SN) --~ Rn'f, (~MIq~S~N_~j~ SN ) n -->R T . ( ~ M )

We denote by

~

the kernel of

Rn~,(~MI~J~SM~.~J~SN)

which is isomorphic to the cokernel of

--~Rn~.(~M ),

Rn-l-r.(~M)

R n-I "g.(~Ml~f~ SN ) • The composition R n T.( ~ M ) @ uJN

~'

~ Rn Y ~ ( ~ M ~ - T SM~N-~TI SN ) ® ~ N

is always the zero homomorphism.

In fact,

R n-g.(~M ) = 0

The case

n = m-l.

~ M ~ ~M"

~ '

for

We have

~ ~ M ~u~M

n # m-l. S~

= N U N

and

Rnql.(~M ) =

is the composite

2 ~' --~ -~.(]l-l~Mlq-f~ s~M) = £~M --~ ~ M ~ U~M/N ' where the last homomorphism is given by shows that

/3'

~ R n T.~(~'~M) ~ ~oN

we obtain the homomorphism

~'

The canonical homomorphism homomorphism

aM I ~ SN --->~N"

Rn-i T * ( ~ ' M I ~ SN )® ~ N ~

~N

(fl' f2 ) ~

fl- f2"

This

is the zero homomorphism.

Thus,

> ~ . (>XIY --->~ y

induces the natural

From this we obtain the homomorphism as the composite

R n-I T.(~MI~T~ S N ) ~ N

---~Rn-i ~ N *(~N) "~ @ ~ON --~ Rn-I q~N. ( T N I ~ N ) ~ ~N -'~ ~ N °

294

Now, let us prove the composite Rn-i T * ( ~ M I ~ I T S N ) ~ ~ON - - > ~ N R n-I -Z,(~M ) = 0. Rn-I q : * ( ~ M I ~ S N

Suppose

=

~ * ( ~ M ) ~ °JN given by

Rn-i ~ * ( ~ M I ~ I T S N ) ~ (flIN) + (f21N).

Rn-I T * ( ~ M )

n-i # 0,

and

The first homomorphism

~*(~Ml~-f~ SN ) ~ O~N = £ M • a M

is

and the second homomorphism

It follows that

--+ ~ N

is given by

Rn-i-C*(~M)

Thus the homomorphism

The desired homomorphism

(3-rd step)

locally.

= ~'M

If

R n-I

~

~N = ~M ~ ~M

zero homomorphism.

L --+M

n = I.

~M

f H-+ f ~ (-f),

We set

is the zero homomorphism.

) @ U°N = £ N @9 (IN'

R n-I

Rn-i z * ( ~ M ) @ ~ N

~'

--> ~ N

~

~ON --> ~ N

is the composite

fl+f2 > ~N

! is the

is obtained. £~' --> ~ --~

The general case.

L = N X M.

Let

g : N --+L

be the graph map and

h :

be a projection.

Let

~"

g-lSN

be the kernel of

LISNL) By

the first

step, we have

(2.2.12)

g

-lh-I

~ M --+ g I ~ L

•

By the consistency of (2.2.12) and (2.2.11), the image of homomorphism

(2.2.12)

is contained in

~".

~'

by

In this way, we can

define the desired homomorphism as composite

~' --~ ~"

--~ ~N q.e.d.

2.3. in ~2.2.

Integration along fibers ,. Let

microfunctions) M

u(y)

be a density on

as coefficients.

We use the same terminology as N

with hyperfunctions

Then we can define a density

by (f.u) (x) =

-i (x)

u

(resp. f,u

on

295

under additional conditions. We denote by coefficients,

q)-M the sheaf of densities with real analytic

q~M

Theorem 2.3.1.

^ dim M ~%M @WM"

is isomorphic to

We can define the canonical homomorphisms of

integration along fibers : (2.3.1)

f* : f ' ( ~ N ~

(2.3.2)

f, : ~ , ~ - I ( ~ N ~ N ~ N )

(Cf. Sato [l],Kashiwara-Kawai Proof. tively.

Let

n

and

m

~N ) --+ ~

QOj~ M~M M '

--~ ~ M ~ ~ VM M .

[I].) be the dimensions of

N

and

M

respec-

Recall that there is a canonical homomorphism (cf. Hartshorne

[i]) n

(2.3.3)

m

~f¢!fiy[n] --+ fLx[m]

This induces n

~f,~R~N(~y[n])

--~Rf¢,~Rr

"

"

i fc

n

(~ (M)

n])

Y[

m

--+ ~ ~M(~f¢, gyn[n]) --~ ~ PM(f~X Ira]) Taking the 0-th cohomology, we obtain (2.3.4)

n n f,.~fN(~y) - - ~

m m M(fZX)

Note that n

n

~fN(~Y ) =~N(~Y)®~N

~M(~X)

m

= ~M~

= ~N ~M M

"

Thus we obtain homomorphism (2.3.1) which coincides with (2.3.4). Homomorphism (2.3.2) is obtained in the same way as above. diagram (2.2.1).

Recall

We obtain homomorphism (2.3.2) by taking the 0-th

cohomology of the following homomorphism:

296

n ~R~O,.~ -I~ PS Ny(~[N-i ~_y[n]) --~o~c,R~. --+ ~ a ) C ' . ~

(~ C-i ~ Nlfhny[n]) --*~rS~(fl~0C,

1

~- ~ S ~ ( ~

n n]) ( ~ ~i 71N I fLy[

~-ifty[n])

Mlmf¢,.F~Y[n])n _ _ + ~ S ~ ( I E M-IfgX[m] )m q. e. d.

2.4.

Products. Let

points of and

M I,

u2(x2)

M2

be two real analytic manifolds;

M2

are denoted by

Xl,

are hyperfunctions on

the hyperfunction up(xp)

M I,

Ul(Xl)U2(X2)

is a microfunction on

M1

on My,

x2

respectively.

and

M2,

M 1 x M2; then

If

Ul(X I)

then we can define

in the same way, if

Ul(Xl)U2(X 2)

is a micro-

function, although we must pay a sufficient attention as seen in the following theorem. Theorem 2.4.1.

Let

MI,

M2

be two real analytic manifolds.

Then: I)

we can construct a canonical bilinear homomorphism -i -i Pl ~M 1 × P2 ~ M 2

(2.4.1) where

p~ : M 1 × M 2 2)

~ ~MIxM 2 '

~ M p are canonical projections,

If we denote by

q~

the projection of

~LTS*(MI~M2) -MIX~T ~ S *M 2 -~f~S*MI~ M 2

onto ~2~ S~'qMp (V =i, 2),

then we can define the canonical bilinear homomorphism (2.4.2)

qlICMI

q~l £ M 2 --~ £ M ~ M21~I~ S*(MIXM2) -MIX~f~ S'M2 -~f~ S"~MIXM2

Two bilinear homomorphisms defined in i) and 2) are consistent and possess properties which "product" is expected to possess. Proof. phisms.

Let

borhood of

We only explain the definition of the bilinear homomorn~ M u.

be the dimension of We also denote by

pp

M~

and

X V be a complex neigh-

the projection

X I X X 2 --+Xp.

297 By the sheaf cohomology theory, we obtain a bilinear homomorphism -i nl -l~:n2 nl+n2 -i -i Pl 0~[MI(~X I) × P2 ~M2(C~X 2) --+~fMIXM2(Pl ~ X l @ P2 ~X 2) nl+n 2 --~ g~MlX M 2 (O~Xl× X 2) -I -i OOMIXM2 = Pl ~°MI~ P2 ~ M 2 yields the bilinear homomor-

The relation phisms

plI~MI x p21~SM2 --~ ~5 MI×M 2 To define bilinear homomorphism (2.4.2), it is convenient to pay attention to the continuous map I

2(XIX X2)* - S~IX I ' ~ ~ M 2 - M I X SM2X 2.

h > iXl. X

M2 ~i×7[2 X2. ~X 2 × X2

From this continuous map, we obtain ql-l~nl~tSMl* XI(75~IO-x I) ~ q2-1~jZSM2Z2n2* (~ ~I~x2) I nl+n2 h- "J~S~IXI×S'~M2X2 (h-i (Tt ii~ X I ~ v~ 21 69"X2))

2l+n M S~2X2( M ( ~ i × ~2 )-Iig-XI~X 2 ) h -i'~nS~iXlX nl+n 2 ---~~ / [ S*(MI×M2) (Vn IO-XI~X2)I~I~ S,(MIX M2)_~S~,~MIXMm_MI×~/T S,M2 , where

~'

~i' 7n2 are the projections MI 2)* --~ X I × X 2 ,

k,. M2 -,~ XI" --->XI, X2" --->X2. q. e. d.

Corollary 2.4.2. Let a real analytic manifold product

u.(x), j = i, 2, be two hyperfunctions on J M. If S -S(u 1)__ ~ S -S(u2 )a = ~, then the

U l(X)U2(X) can be defined in a canonical way so that the

298

singular support of Here the set If

A

and

in

run over S'M,


is the union of b

Ul(X)U2(X )

is contained in

<S-S(Ul) , S-S(u2)>.

is defined as follows:

are two closed sets in a sphere

A, B

and of arcs which join

A

and

B

respectively.

then

~A,

B~

If

A

is defined to be

S,

a

and

and

B

then b


where

B>

a

and

are closed sets

~ . x6M

The corollary is easily proved by applying Theorem 2.2.6 and Theorem 2.4.1: hyperfunction

we consider

Ul(X)U2(X)

Ul(Xl)U2(X2)

on

M x M

Before ending this section, we

by

S~

(or

n

and

M U V~i S ~ )

quotient topology.

Set

X

induced from

~*

MX*

C

Let

MX, = S ~ X

*

> X.

If

x 6 M

contains

x ~ X.

S~

x*.

x*~ ~S~,

x~M

is

with the fol-

MX*,

~X)

~ ~MJS~

where S~

S~X

U

is

coincides

S~.

= 0

for

k # n = dim M.

= ~ M

and

~ j ~ M M = ~ M"

then every neighborhood of

sp : ~ M , x

MX*

and the

{~-I(u)~

It is clear that

~{k~x( ~

We denote

and denote by

~*

Therefore, we obtain the restriction map

which coincides with

M K M.

with the

is an open subset of

= ~n ~-i J~ S~X ( ~ eX) ~ ~M" and

M.

is an open subset, the topology of

It is easily proved that ~M

We equip

and the induced topology of

with the quotient topology.

We set

T~/~+

U (X-M) = X H ~f~ S'M,

~MX~

the neighborhoods of MX*

of

be a real analytic mani-

coincides with the original one of

a closed subset of

M

be a complex neighborhood of

fundamental system of neighborhoods of runs over

M

the topological space

the canonical projection lowing topology:

to the diagonal

give one remark on the relation

of hyperfunctions and microfunctions. fold of dimension

as the restriction of the

x

in

S~

~ M , x --> ~M,x*'

--+~M,x*"

The employment of these terminologies makes Theorems 2.2.6,

299 2.3.1 and 2.4.1 between N --~M

~

transparent.

and

~

.

Especially,

it reveals the connection

We will mention the modified theorems.

is a real analytic map,

If

f :

then we have the canonical continuous

maps "ID': V ~ S"M × N

~ ~/~ S~'~4

f : ~-f~ ~ M X N

and

M

induced from

> (f~ S~'N

M

T ' M × N --+T*N. M

Theorem 2.2.6'.

We have the following homomorphism of substitution: -I~

M -"~e

I". ~ Theorem 2.3.1'.

We have the following homomorphism of

integration

alon$ fibers :

~ , ~-l(^

~N ~

Theorem 2.4.1'. denote by

pp

induced from

Let

MI, M 2

--,

I~N)

~ i~M

~M

be two real analytic manifolds.

the canonical projection T*(MIK M 2) -->T*M~

CM

for

~f~*(MI~ ~ = i, 2

We

M2) --~(f~S*M~

respectively.

Then we

have the bilinear homomorphism of product : -i ^ -I Pl ~ M I × p2 ~ M 2 2.5. on

0-M,

Micro-local operators.

> ~MI~M 2 '

As stated in

~ 2.1,

But a larger operation ring operates on

Kawai [I]).

We sketch about this.

the precise discussion.

Let

~ M

~M

(cf. Kashiwara-

It is left to the reader to give

f : N --->M

be a real analytic map.

first, we construct the sheaf of operators which transform ~N" N

f

decomposes into

with the submanifold of

N --->N ~ M --->M by the graph map. N × M.

The natural map

yields the isomorphism

by which we identify

~f~SN(NXM )

operates

and

N~S~'~.

~M

At into

We identify

T*(N × M) --~T*M

300 Definition 2.5.1. f)

on

N × S'M- S ~

We define the sheaf

~N~M

(also denoted by

by

M

(2.5.1)

N×M ~M 2kM)alN×~-i S ~ ' / I ~ S ~ i ' M

where

~-M

is the sheaf of densities on

M

with real analytic func-

tions as coefficients. We call a section of denote by

~M

~N~M

f.

M.

We remark that in Kashiwara-Kawai pseudo-differential

[I] we used the name of

operator instead of micro-local operator.

paper, we reserve "pseudo-differential

In this

operator" for the more restricted

type of micro-local operator, which we called pseudo-differential ator of finite type in Kashiwara-Kawai theory of pseudo-differential

[I].

This is obtained as follows.

oper-

We will present the detailed

operators in Chapter II.

has a canonical element, written IN9 M.

We

~ M i_~dM , whose section is called a micro-

the sheaf

local operator on

a micro-local operator over

~(x - f(y))dx,

Note that

~N~M

which we denote by

By the integration along fibers,

we can define

Since

~ N x M = ~-N ~ ~M'

by tensoring

( ~ N )~(-I) , we get

By checking the singular support, we find that this image is contained in

0~,(~ N_>M).

IN.~M is the image of

By the following proposition, Proposition 2.5.2. (2.5.2)

i E ~N

~ N~M

by this homomorphism.

transforms

We can define f'.(~N~M ® Z Y - I ~ M )

---> d N '

C M

into

~ N"

301 a

The homomorphisms induced by

1N~M ~ ~2~N~M and the above homomorphisms

c o i n c i d e w i t h t h e homomorphism of homomorphism of

integration

Roughly s p e a k i n g ,

substitution

along fibers

( 2 . 2 . 1 1 ) and w i t h t h e

(2.3.2).

( 2 . 5 . 2 ) and ( 2 . 5 . 3 ) a r e given by

(K(y, x ) d x , u ( x ) ) ~

fK(y,

x)u(x)dx

(v(y)dy, K(y, x)dx)~-* ( ~ v ( y ) K ( y , x ) d y ) d x

o

We leave the proof to the reader. Proposition 2.5.3. f' : N × L --->M × L

Let

be

L

be the third real analytic manifold and

f X iL.

Let

p

be the projection

(NXL) >(~2~ S*(MxL) - ~-~SNM × e - N ~ 2 ~ S*L --+ N x ~ MxL M

S*M

Then we can define a canonical homomorphism -i P

~ N->M --->~ N× L-~M×L

given by

K(y, x)dx ~->K(y, x) f ( £

- £')dxd~ '

By P r o p o s i t i o n 2 . 5 . 2 and P r o p o s i t i o n 2 . 5 . 3 , we o b t a i n c o m p o s i t e s of micro-local operators. Proposition 2.5.4. a n a l y t i c maps.

Let

f : N--~M

Pl : N ~ - I S * L L

and

- N ~ 2 ~ S M L --+ Nx~2~S*M M M

P2 : N X ~ f ~ S * L - N L ~ I ~ S M L

-->M x ~ S * L - ~ f ~ S ~ L

deduced from

S'L-~

2B': M ~ ~ L

g : M--~ L

be two r e a l

and

are the projections

SML - - ~ 2 ~ S~',~ and

f.

Then we can

define the bilinear homomorphism of composite of micro-local operators Pl-i~ N ~ M and we have

X p;l~M,e

--~ ~ N ~ f L

IN~ M o IM~ L = IN_~L .

By these propositions

~M

has the structure of a ring,

~M

302

has the structure

of a left

a ~M-MOdule.

right

2.6. f(x)

~M-MOdule

~N~M

is a

sional manifold and

X

f(x)

~ M

has that of a

We can define (cf. Sato

the complex conjugate

[i]).

Let

M

be an n-dimen-

be its complex neighborhood.

as we need, we can define

By shrinking

the (unique) anti-holomorphic

which is equal to the identity on the C-antilinear

~M

(~ - I ~ N , 2 ~ - l ~ M )-bi-Module.

Complex conju~ation.

of hyperfunction

and

M.

c

2

= id.

map

X

c : X --+X

We also denote by

c

homomorphism -i

(2.6.1)

c : ~X-->c

f(z) e-~ f-(z) = f(z c)

O- X,

By this we can define n

c

aM(~x) _ ~ M < c-1eX) ~ c-la~(OX)~(*X ) By tensoring

tOM,

we obtain the

(2.6.2)

c : ~M

The map

c

¢-antilinear --> ~ M

c : MX~\~ --* ME-A-

can be lifted to the map

is equal to the antipodal map

a

on

homomorphism

S~.

This map

Hence, we can define

~ ( ~ - I ~ x) -~ ~ ~(~-lc-l~ x) n ,Xs~(c'l~-lex

By t e n s o r i n g

~M'

we o b t a i n

)

n ~ c-l~s~x(~-l~x ).

the ¢-antilinear

homomorphism

a

(2.6.3) This

implies

c : £M - ~ £ M that,

for

every hyperfunction

" f,

we h a v e

S - S(~) = S - S(f) a

3.

Techniques

for construction

of hyperfunctions

We have defined hyperfunctions abstract method. construction

and microfunctions

and microfunctions

by a rather

In this section, we give several technical methods

and their examples.

We refer to Gel'fand-Shilov

of

303

[I].

In Chapter II,

3.1.

we will give more techniques.

Real analytic

fundamental methods of holomorphic

functions

of positive

type.

is to construct hyperfunctions

functions,

or more precisely,

One of the most

as the boundary values

the method using the homo-

morphism

The following

theorems are useful for this purpose.

Theorem 3.1.1.

Let

x0

be a point in

(Cf. Komatsu

M = ~n

and

zero real tangent vector.

Assume that a holomorphic

defined on an open set

in the complex neighborhood

contains

x+~Tt~

U

for every point

every sufficiently

small positive

tinued to a holomorphic such that

U' U SMX

Proof.

For

is a neighborhood

F = {(z I ,z2) ~ ¢2.,

t.

Then,

of

Xo+~f~

rl, r 2

j = 1,2

Then we have,

and

,

- 2~ < x j <

sinh r I

for every holomorphically H ~ × U ( C 2 ~ U;

Proof.

x0

U' in

in

that

sinh r I [i], we have

X -M

n > i.

~ ,

Set 0~yj~r.j

cos x 2 sinh(r2-Y 2) +

~C2XU ) = 0

for

sinh r 2

~ i,

> 11

sinh r 2 U

in

k ~ I.

0 ~ v. (j =1,''',4), J

v4 +

and

~.

Suppose

convex open set

w. = u. + ~ f ~ v j , J J

v3

which

can be con-

We put G = ~w e C4;

By Morimoto

f(z)

numbers.

cos x I sinh(rl-Y I) for

near

is

two lemmas.

be positive

z J" = x.J + ~ f ~ y j

f(z)

X = Cn

~0

the theorem is trivial.

At first, we prepare the following Let

sufficiently

number

be a non

function

function defined on an open set

n = i,

Lemma 3.1.2.

x

~

[2].)

v I ~ r I,

v 2 ~ r2~-

£m,

304

k 4 HG×U(C × U; O-C4×U) = 0 We embed

k # 4.

{(z, u) E ¢2× U I -TE < Re z. < ~ J

into

¢4X U

by

f : z = (Zl, z2) ~-+w = (Wl, w2, w3, w 4) , W l = Z I, Then

w 2 = z 2,

F = f-l(G).

w3=sin(~i-~r I -Zl),

w4=sin(~fir 2 -z2).

Since there is the exact sequence

0 <--- f.

~--- (~ 4 4--- ~ 2 4-- ~ ~-- O, ¢ xU ¢4xU ~E4~U

2× U

we obtain HF×uk (C 2~U;

O

for

2×U ) = 0

k < 2 q.

Lemma 3.1.3.

Let

f(zl, z2)

e.

d.

be a holomorphic function of two

variables defined in a neighborhood of I z 6 ¢ 2", {xjl
j=l,2

U ~zEC2; Then,

f

Y2=0,

0
Ixjl
Y l = r l ~ ly2~< re1 .

can be extended to a holomorphic function defined in a neigh-

borhood of Iz E C2;

Proof. subset

U

Xl=X2=0'

Yl
We can suppose

a2tanh 2-~2r 2 IY2~< ~t Yl ~ " alsinh 2-~ir I

a I = ~/2,

a 2 = ~/2.

There is an open

such that

F = {z 6 C 2;

Ix I] < 5'

Ix21 < ~

cos x I sinh Yl sinh

U

preceding

; 0

H Fl(u

0~ Yl ~ rl'

sinh

and that 2)

04-Y24_r 2 ,

cos x 2 sinh(r2-Y2)

r 1

is a closed subset in lemma,

+

,

= O.

f

r 2

is defined on

It follows that

U-F. f(z)

By the is continued

¢

on

U.

It

is

trivial

that

U

contains tanh

~ze C2;

x I =x 2 =0,

0 4 Y l ~ r I,

r2

0 ~ Y 2 < sinh r I Yl } " q.

e.

d.

305

We can suppose that

Now, let us prove Theorem 3.1.1. = (i, 0, "'', 0),

and that

U

x 0 = 0,

contains the subset

n

F = I z ~ cn;

~ x.2 < 2a 2, j=l J

D Iz ~ £n;

~ x.2 < 2a 2, j=l J

Y2 . . . . . Yn =0,

n

Let

V =

z E U;

2 2 y2+'''+Yn<S

Yl=r,

be t h e e n v e l o p e o f h o l o m o r p h y o f

U

U.

z3 . . . .

0
Then

Zn =0

contains i(Zl,Z2) ~ ~2, iXll
V

~Xl~
X l = X 2 =0'

Since

transformations

"--,

in

Since

U'

Yl=r,

0
U

F

JY21<sl V

contains

is invariant under real orthogonal

z ), n

< r,

Y2=0~

tanh(~s/a) IY2~< sinh(~r/s) YI~

U

contains

~ 2

Y2+ " ' ' + Y n

2

tanh(~s/a)

< sinh(Tcr/a)

1

Yl "

contains

x~ +'''+x2
is simply connected, f(z)

function defined on 0+~(I,

(z2,

x=0,

This implies that U' ={z ~ on;

~x21
0
by the preceding lemma.

;

0
is also a holomorphically convex subset,

~z ~ C2'

~z ~ Cn

~x21
U'

0, "'', 0)0.

U' U ~ S M

~Y~+'''+y2n
can be continued to a holomorphic is obviously a neighborhood of

This completes the proof of Theorem 3.1.1. q.

e.

d.

When we construct elementary solutions of differential operators, we encounter hyperfunctions of the type F(x) = ~ aj(x) ~ j ( ~ (x)) jE~ with suitable growth conditions on

~ a~(x)~,

where

306

1 2~,q~i

~j(-C ) = (See Example 1.4.1 in # 0.

j~ ( _ T ) J +1 " F(z)

#1.4 of Chapter II.)

~(z)

as the boundary

F(x)

We give a meaning to the hyperfunction

value of the holomorphic function

converges if

F(z).

The following lemmas are powerful in this case. Definition 3.1.4.

A real analytic function

real analytic manifold

M

if

~(x0) # 0,

Re d ~ (x0)

= 0

implies

or if

Im

defined on a

is said to be of positive type at

~ (x) ~ 0

Note that if

~(x)

~(x)

x 0 6 M,

is a non zero covector and

for every

x

sufficiently near

is of positive type at

x0,

of positive type at any point sufficiently near

Re ~(x)

x 0.

then

~(x)

is

x O.

In the sequel, we set D£ = ~ Lemma 3.1.5. type such that Then every

Let

~ > 0

Im~

?(x)

~ (x0) = 0

( ~ - l ( o & ) -M) U ~ S M

Proof. that

~£;

> - £1Re~l 1

be a real analytic function of positive

and

~0 = d ~(x0)

is a real covector.

is a neighborhood of

and for every tangent vector

~

such that

We choose a local coordinate system

Re ~(x) = Xl,

x

0

= O,

~ =

~ / ~ x I.

x 0+~

~ 0

for

< ~ , ~ 0 ) > 0.

(Xl, --., Xn)

such

By the preparation theorem

of Weierstrass, we have ~(x) = g(x)(x I - f(x')) where have

x'

signifies

Re ~ = x I R e g - R e

(x2, "-., Xn) , gf.

,

g(0) = i

Therefore

and

f(0) = 0.

Re gf = x I ( R e g - I).

We From the

assumption, we have Im ~ = -Im gf ~ 0 Since

Re gf = 0,

function

h(x').

when

xI = 0 .

g(0, x')f(x') = - ~ h ( x ' ) For sufficiently small

for some positive valued x',

f(x') ~ Dg.

Hence

307

follows the lemma,

q.e.d.

Conversely, we have Lemma 3.1.6. ~ ( x 0) = 0

and that

the closure of x0+ ~

T0

~(x)

Let

?(x)

be a real analytic function such that

7 0 = d ~ (x0)

~z ~ X-M;

~(z)=

01

for some tangent vector

is of positive type at Proof.

such that

is a non zero real covector. in

~

~

does not contain

such that

~,

~0 > ~0,

then

X 0 •

We can choose a local coordinate system x 0 = 0,

If

Re ?(x) = x I

and

~=

~ / D x I.

(x I, "'"

x ) n

We write

?(x) = g(x)(x I - f(x')) with

g(0) = 1

and

f(0) = 0.

for sufficiently small

x'

By assumption,

we have

Im f(x') ~ 0

As shown in the proof of the preceding

i emma, g(0, x')f(x') = - ~ h ( x ' ) for some real valued function near

i,

we have

h(x').

h(x') ~ 0.

Since

g(0, x ')

is sufficiently

Therefore

Im ?(0, x') = -Is g(0, x')f(x') = h(x') ~ 0 • q. e. d. Boundary values of hyperfunctions with holomorphic parameters

3.2.

and examples.

The second method is to construct hyperfunctions as a

boundary value of hyperfunctions with holomorphic parameters. Let

X

be a complex neighborhood of a real analytic manifold

of dimension codimension (X, Y, M)

n d

and

Y

be a real analytic submanifold of

containing

M

as its submanifold.

is locally isomorphic to

(¢n

X

M

of

Suppose that

cn-d ~ [Rd, ~n). d

We have

~{ ( ~ X )

-- 0

for

k # d.

Set

~ X , Y = ~ { Y ( ~ X ) ® mY/X"

308

~X,Y

is isomorphic to the solution sheaf of

Cauchy-Riemann

system

(cf.

My

~2.2 in Chapter III).

Now, we can apply the theory of microfunctions to the situation

where

(X, M, ~X )

a section of

~M

developed in

is replaced by

Then the argument goes on in the same way. 0OM/Y = A M ,

of the partial

~i

(Y, M, O-X,y).

Since

~M

n-d

(O"X,Y)

is represented as the boundary

value of a section of ~X,Y" We denote by ~ X , Y the sheaf n-d -i a ~[S~f(~M/Y~X,y) ~tOM/y, where -feN/Y is the projection MY* -->Y. There exist a canonical projection

p : S~-S~X

tive canonical sheaf homomorphism

p

X,Y

--* SMY

and a surjec-

CMls_sx.

We give several examples. Example 3.2 1 Y(Y) • " x+iy Y(Y) x+iO+iy i x+iy

can be considered as the boundary value

of the hyperfunction is,

by definition 21 ( 9 i , ) -- ~x +

'

Y(Y) z+iy

Im z > O.

Y(Y) + Y(-Y) x+iy x + iy

Y(y) = i ~(y) = --i ~(y) x+iO+iy 2 (x+iO+iy) 2 x + i0

. ~

2 ~ ~x

which is defined on

Y(-y) 9yJx-i0+iy

-i~(y) = 2(x-i0+iy)

=

_

_

i ~'(y) 2 x-i0

Therefore I ~(~+i~)x+iy

i

= ~f(x)

~ (y)

In the same way, we have (x

+y

+I) x+iy =0'

and

(y~-X~y

Now, let us calculate the integral

~x+--~ydX

This is obtained by x+i----~dx

for

This is independent of the choice of

a ~ 0 < b. a

and

b°

I (x+iy) x+i----~ ~y = I.

as microfunction.

309

~ab

Y(y) ~),+ Y(y) f dz x+i0+iy dx = z+i--~ dz = Y(y) z+iy

= Y(y) (log(b+iy) - log I (a+iy)) where we take branches such that log b i b

Y(-Y) dx x-i0+iy

~

Y(SY)dz ¥- z+iy

=

,

log b,

log I a

=

log(-a)+

~i.

Y(-y)Ilog(b+iy) -log2(a+iy)~

where we take the branch such that

log 2 a = log(-a) - ~ i

= logla - 2~i.

Therefore ~

dx = log(b+iy) - logl(a+iy)+ 2miY(-y) =- 71i(sgny). Example 3.2.2.

Consider the differential 2n

c+

P(x, D) = ~x--~+ ~ x I ~x 2 discussed in Mizohata A fundamental

solution of

I

P(x, D)

~

~

Note that

Y(x I - xi) z - __S!-- , 2n+l ,2n+l 2n+l ~Xl -Xl )

parameter

z

Im z < 0.

!

and

is given by

Im z < O.

, 2n+l ...... kx I -x I,2n+l )

Y(x i - x l)

x 2-X~+i0-

x1

>o})

Y(x I - x{)

+

x 1~

=

i , 4 , 2n+l 2n+l) x2-x2- 2n+l kXl -xl

- 2~-~--~x2-x'-i0-2

when

R2

[I].

I E(x, x') - 2TgqTT

when

operator on

o~

, 2n+l

~x I

,2n+l. I "

-x I

)

is a hyperfunction with holomorphic

In fact, the denominator never vanishes We can easily

verify that

310 P(x, D )E(x,x')

= ~(x-x')

and

P*(x',D

X

,)E(x,x')

= f(x-x')

.

X

By the fundamental theorem (cf. Sato [3] and Chapter III) we can conclude that

S -S(E(x, x')) ~a

Hence, E.

coincides with the anti-diagonal

= ~ (x, x'; V~f~(<~ , dx> + < ~ ' , dx'>) ~), E(x, x')dx'

We have

x = x',

~+

~' = 0}.

is a kernel function of micro-local operator,

EP = PE = i.

This shows that

P : ~ ---> ~

say

is an isomor-

phism. Of course, the operations on hyperfunctions and microfunctions defined in

~2 help us to construct functions together with the tech-

niques given in this section. Example 3.2.3.

(~-hyperbolic

(cf. Kawai [7], Kashiwara Let

P(D)

We will give the examples of this sort. operators with constant coefficients

[4] ).)

be a linear differential operator with constant coef-

ficients, whose principal symbol is denoted by

P (~).

We decompose

m

P(~) so that (3.2.1)

where

Q(~ )

+ Q(~),

is a polynomial of order

< m.

It is evident that

(-)kQ(Dx)k I [i (-2~f~) n-I Jk--0= Pm(~) k+l ~n-l-mk-m(<X'~>+i0)~(~)'

K(x)

~(~)

= Pm(~)

=

n__ >~ ( - ) J - l ~ j d ~ i A j=l

an elementary solution of

P(D).

A " . " A.d ~.j . A

,

is formally

-i A' = (2~_l)(,'T)'k+ 1

(~&(T)

the precise meaning of this function see

Ad~n

~2 in Chapter II).

;

about

The

integrand of (3.2.1) is a (multivalued) holomorphic functions on # 0

and

<x, ~

# 0.

We fix a non zero real covector G-hyperbolic

Pm ( ~ )

in the direction

~

~

.

We say that

(in a neighborhood of

P(D) 70 )

is if

311 Pm(~+~f~t~)

# 0

for

(sufficiently near

0 < Itl << i and every non zero real covector

~0 ).

Then, it is easy to see that the integrand

of (3.2.1) can be considered as a hyperfunction from the direction

~+~0'

really a fundamental the whole

~n,

<x, ~ > + ~ 0 .

solution of

P(D).

as the boundary value In this way,

Moreover,

if

~0

K(x)

is

may vary in

we can easily check that the singular support of

is in the proper cone contained in the singularity of the solution of

~x; < x , ~ # P(D)

~ 0 I.

propagates

K(x)

It shows that in finite velocity.

The ideas of construction presented in the preceding examples will be developed by means of pseudo-differential

operators in ~ 2.3 of

Chapter III. Example 3.2.4 of Hans Lewy [2]. Chapter III).

P=~-~z~,

(See also Sato [3] and

~2.3 in

Set

~

~ ~ t Q = -P = ~-~-

( ~, ~ ) 6 (C × ~) -~0~

(°( ( C+) , where

signifies the point

C+ = ~ ¢ I I m ~ ) O

I•

~Z~ (Re< ~, dz>+ ~dt)~o .

We define i I E (z,z,t, z' ,z' ,t') . . . . . . . . . . . 2~2~-I~-- z-z F(z,~,t,z',~',t')

=

K(z,~,t z' ') = , ,z',t Note that Then and

t-t'+~[zl

!

t-t'+~z']2-~z'12-(o~-

~)zz'

i I I 2K2~II ~-~' t-t'+~Iz~2-~Iz'12-(~-~)z~ ' , ~-~ I 2 2~_~ (t_t,+~lz~2_~iz,~2_(~_~)z~,) 2 2 - ~ z ' ] 2- ( ~ - ~ ) z z '

is of positive type.

E(z, -z, t, z', -z', t) l~dz'dE'dt'l , F(z,~, t, z', ~', t')lldz'dz'dt'~ K(z, ~, t, z', ~', t') ~I ~dz 'dz' dt'l

operators, which are denoted by

E, F, K.

are kernels of micro-local They are defined on

~L=

312

PE = i,

EP=I-K

,

FQ = i,

QF = I - K

.

This shows that 0 ---> £ M

J~

K --> £ M

£M

is an exact sequence on Example 3.2.5. such that

2

aD..

(See

= ~ + ~.

D I + 2 ~ X l D 2.

P ---. dM --~ 0

~2.3 in Chapter III.) M = ~Rn.

Let

~,

P(x, D) = D 1 - 2~ XlD2,

~ , ~< ( ¢+ Q(x, D) =

As is shown in Schapira [i], they are not solvable.

order to analyse the structure of kernels and cokernels of and

In

P(x, D)

Q(x, D), we construct microfunctions, which are almost elementary

solutions of

P(x, D)

and

Q(x, D).

Set

E(Xl,X2,X{,X ~) dt -_ 4 ~12 i (t-o[(Xl-X{)) (x 2 -x~+25 -2 (~Xl+~X ~) t+ o(~ -2 (x I -x{)2- r- 2t 2+i0)

=

,

12

,2

47Ei~x2-x2+~Xl+~X I +i0

-

{/x (xl-xl) ,+ 2 - ,2

-xi)

]

- . ], 2 ,2 , ' 2-x20tXl+{~x I +iO+~x I ~x2-x2+~Xl+~X I +i0-~x I

F(xI,x2,xi,x ~) at -4~i 2 ~ (t+~(Xl-X{))' (x'2'x~+2~-2(~l+~X{)t+~'2(Xl-X{)2-~-2t2+io)

=

, 2.........,2 ...... 4Tti~Xm-X2+~Xl+~X I +i0 !

K(xI,x2,x3,x¼) =

Note that !

,----2 - ~ ~2--7~,2 -' x2-x2+RxI+#Xl +i0+gx I ~XE-Xx+~xl+~Xl +i0 -~x I"

~" "x 4~i

i ,2+.^.3/2 " ,+~ 2+~ 2-x2 x I I)Xl iUp

x 2 -x~+ 2~-2(~Xl+~xi)t+~-2(Xl-Xi)

x2 -x2+~x~+~xi

2

2 - ~'-2t2 and

are real analytic functions of positive type.

Y(Xl-X{)~t+ax~+~xi2+iO of the hyperfunction

+~Xl~ -I,

for example, means the boundary value

Y(Xl-X{){~+~x~+~xi2+~Xll -I

which is defined

313

on

T~£+

with holomorphic parameter

Xl,X2,t)

. If we denote by

A(Xl,X 2,

the integrand

I 4 2 (t -~ (Xl-X {) ) (x2 -x~+2[ -2 (~Xl+~X {) t+~# ~-2 (Xl_X{) 2_~-2t2 ) ' then this satisfies the following equations: (Dxl-2~EIDE2+~Dt)A (Xl ,x2 ,x I' ,x~, t) = - 2Tti i X2-x~+i0 I ~(Xl-X{)~(t)' (-Dx{ +2~X{Dx~+ ~Dt)A (Xl'X2'X{'X2't) =

-

I

I

2~i x2-x~+i0 ~(Xl-X{)~(t) + 27g~(x2_x~+25-2 (~Xl+~X{ ) t+~F-2 (x l_x{) 2_ Z-2t2 )2 " n

n

E <Xl 'x2'x{ 'x2) j=3 - ~ ~ (xj -x'. ' j)dx ' F <Xl'X2'X{'X2) -~-j=3~ (xj -x ''.j)dx

and

n ' ' j]~3 are kernel functions of micro-local K(Xl,X2,Xl,X2) ~(xj -x'.)dx J operators defined on we d e n o t e them by

~ = ~(x,~i-~< ~, dx>~o) 6~i~S*M;

E, F

and

K,

then the preceding

~2)0}.

If

calculations

yield the following relations: PE = i,

EP = I - K

,

FQ = I,

QF = I - K

.

Therefore the sequence 0 --~ ~ M is exact. 31=0'

Set

N = ~x e M;

~2 ~ 0~.

we identify

Q-~ ~ M

EL• ~ M

x I = 01,

By the projection,

Z with

~S*N

defined on

~

--~0 M

z =

$:iS*M;

xl=0,

M

(~2 > 0).

We define the sheaf homomorphisms --->~N

P --~£

~:

in the following way:

eN --~ ~ M

and

~:

~M

314

2 ,xn) ~ u ( x 2 + ~ X l , X

: u(x 2,

]

3,.'',xn) =

u(x ,x3, 'xn)dx 2+~Xl-X2

: V(Xl,'-',Xn) I > KVlxl= 0 . Then, it is easy to see that ~i

= i:

C N

i~

=

~

K:

and

This shows that

~

>~N

'

--> g M

M

"

define the isomorphism

: ~ N ~ - Ker ~ M P . Note that

~v

= vlxl= 0

for

v & KerCMP.

We define the sheaf homomorphisms ~ : ~N

--~ £ M

and

~ : CM

--~£N

as follows: : u(x2,

4

.

.

•

"

: v(x I ,

", Xn ) ~

K(u(x2'

•

•

•

' Xn) ~ (Xl)) '

l Iv(xl,x',x3, .,x)

"

2

, Xn) ~

- 2~ ~

[l

x2_x~+~xim+~f~0

!

!

dXldX 2 •

Then, it is easy to see that ~=

I :

~'~= K : T h i s shows t h a t

~

and Z

£N

N

~M --'-~ ~M

g~/ d e f i n e the isomorphism :

CN-%CokerCMQ

Note that £u

= f (Xl)U(X 2, "'', Xn) -QF( ~ (Xl)U(X2, -- ~ (Xl)U(X2,

"'"

'

xn )

mod

Q

M"

"'', Xn))

CHAPTER II,

FOUNDATION OF THE THEORY OF PSEUDODIFFERENTIAL EQUATIONS

i.

Definition of

i.I.

pseudo-differential operators

.Definition of holomorphic microfunctions.

The introduction

of microfunctions has brought a new aspect to the theory of linear differential equations

In Kashiwara-Kawai [i], the authors

introduced the notion of pseudo-differential operator (of finite ordeO which operates on the sheaf of microfunctions.

Originally pseudo-

differential operators are defined on the cosphere bundle of the real analytic manifold and the calculation of pseudo-differential operators is algebraic.

It turns out that the pseudo-differential operator can

be also constructed on the cotangential projective bundle of the complex manifold, just in the same way as a real analytic function is the restriction of a holomorphic function. In this section we will give a definition of pseudo-differential operators. Let

X

be a complex manifold of dimension

submanifold of

X

with codimension d.

n

Regarding

and X

Y

be a complex

and

Y

analytic manifolds, we construct a real monoidal transform a real comonoidal transform : YX --~X We denote by Y

in

X~ -+X

with center

be the usual monoidal transform of PyX, PyX

<~,~=0,

X

T~:Yx~-+X,._.

Y.

Let

with center

Let

EyX

where

be the submanifold of

PyX × P~X Y EyX instead

( ~ , ~ ) ~ P y X X PSX ° Using Y of DyX, the method employed in §1.2 of Chapter I gives a topological space

Y.

the normal and conormal projective bundles of

X : ~-l(y) = PyX.

defined by

~:

as real

YX* = (X-Y) U P~X

316

Proposition

i,i,i.

-i 6~ kyx(T~{ (>X) = 0

Jgpkyx(Z-16~ x) = 0 Proposition

for

k ~ i

for

k ~ d

112.

{k [~-i S~X~ ~{ 0-X) = 0

Since we can prove these propositions in the same way as the corresponding Theorems

1.3. I

in Chapter I , we omit the

and

i. 3.2

for

k ~ d-l,

proofs. Proposition

i.i.3.

~{p~x( ~ k -I(~x) = 0 Proof. of

X

We fix a local coordinate system

such that

Y={x=0}.

1.2.3

(x I, • .-,x d ,y i ,...,y n-d)

Let x~ = (x0,Y0;~ 0) ,

g0=(l,0,...,0), be a point of tion

d ~ i

P~X.

x0=0 , Y0=0 ,

The same argument as in Proposi-

of Chapter I proves that

o { kP~X"(,g-i ~ X )x~ = li__~Hks(u £ ; ~ X ) = ~ Z. g ( ~kX ) 0 where

U£-Zg

Z£=

t(x,y) eX; lxll >_6lx~l,

U&=

{(x,y)~X;

Ix I ~ 6

is covered by the

d-i

V=2,''',d} ,

, [Yl < £~

holomorphically convex sets

~i ~) = {(x,y)&U 6 ; Ixll < E~xvl I which implies that

(~=2,...,d) ,

,

317

k HZ

g

(Ug; (~X) = 0

for

k > d

Let us show .

k

for

i_~mJ4z~ (6~X) 0 = 0

k < d-i

To prove it, it suffices to show

j
where ,

Zg = {(x,y) E X ; ?£(x,y) - g(~x2~2+ .... +~Xd~2)-IXl;

2

< 0} .

Since the hermitian matrix

l h'as

(d-I)

j

positive eigenvalues, we have (Andreotti-Grauert [i]).

=0

q.e.d.

In the same way as in the theory of microfunctions, we give Definition

1.1.4.

We define the sheaf

~YIX ~

C~IX

on

S~X

f

by

X = 04-[ SyX (

g~l

d X = ~

By Proposition

1.2.2

-i

a

S~X (I~fR ~ X )

in Chapter I,

we have

on

SyX

and

318

eyl X Recall that

=

R d-

~yl X

Ra Ir~.~ml~ yix .

is the sheaf on

d ~yl x = ~ y ( ~ x Proposition

)

Y

defined by

(ef. Sato

[i])

I.i.5.

R k ~ *g~YIX = ~ YIX

for

k=0, d#l,

= O'Xl Y

for

k=d-l#O,

= 0

for

kfi0, d-i

If d=l, then we have the exact sequence

(iii) Proo f.

0 --~ ~yix "-+Tr]R.(~iX ) --~ ~XIY -+0 We have R k % * ~R = R k + d - l ~ * ~ R

"

Since

k

I xlY

for

k=2d

0

for

k~2d ,

for

k=d

for

k~d ,

~-~R (rex) =

k I ~YIX ~{y (~X) = 0

319

the exact sequence "'"

j~k( ~X )

Rk-I

-~

-~

k+l

~. ~

-~ ~ t ~ R

(~x ) -+

~. y k+l

(~x) --~

proves the proposition. q .e .d. Theorem 1.1.6. onto

P~.

Then

Let

~

Rk ~,(~]X)

canonical exact sequence, (1.1.2)

be the canonical projection of = 0

for

Y,(CyIx )

and call a section of

PyX.

and there is the

__~d-l( PyX ~-l~>x)

We define the sheaf

Proof of Theorem 1.1.6. onto

k # 0

d # I,

0 --+ 7~-l~yix --~ ¥.(C~YiX)

Definition 1.1.7.

S~

for

Let

Cyl X t

Cyl X

on

--~ 0 P~

(~-I~X)

be the canonical projection of

= ~-l((~X{y)[-2 ] .

This induces a triangle

i ( 0 X iy) [-2]

|Rrp~(~-l~x

)

> ~Rt,U R r S ~ ( T ~ I ~ X) ,

which implies that Rkt,~

= 0

to be

a holomorphic microfunction.

We have ~ t

SF

for

k # 0 ,

320

and gives the exact sequence 0---~ ~ - - ~ t . ~ , - - - >

---> 0 ,

"~-l(Oxly)

i ~ =~pyX(~-l~x).

where

Ey_X C PyX ~ P~X Y defined Dy

< ~ , ~ > = 0,

be the hyperplane

6 PyX,

where

~ ~ Py~K. We have the

following diagram :

~x-

SyX ~ ~5

, s~x

SyX ~ EyX C-~-"S~ X PyX PyX j Y

l PyX ~

TC

EyX

>

ql

P~

•

The fiber of s is homeomorphic to the oriented one dimensional i sphere S over SyX × EyX and to an interval outside it. Hence PyX we have ;R~g~ts(7rll~LR) = j. T C - I ~ [ - 2 ] . The canonical triangle concerning

s

reduces to

IRT I j.]l -i ~

(1.1.3)

[-2]

/ ~ ~1~s * ~ 1 ¢ ~

321

The cohomologies in this triangle can be calculated as follows:

~TI, j * ~ I ¢ ~ R

= IRT*Rt*~I¢~

- 1 ~-

-i

-i

Triangle (1.1.3) is therefore reduced to

-l~. C

/

(1.1.4)

~,.C a

~RT,]I"I ~[d-2]

. . . . .

We will calculate

RkT.7~-I~

.

Put

G = (X-Y) U S ~

We give a topology on

G

× EyX PyX

in the same way as on

G

~2

be the canonical projections,

>

y~

q72

,

SyX

PvX

~2

in Chapter I.

Let

~'~

> YX*

is proper and separated.

same way as in the proof of Proposition

~r

G

YX +

1.2.2.

In the

of Chapter I, we have

322

and -1

-i~

Since

• ~T2 (~-i&x ) = ~-l(~xIy)[2-2d] the triangle

~P~X~ T 2 ( ;RIFp~x(~-I (~X)

-i 6%X)

"~ IRT2~R~ _i (~21~-i~X ) 1~2 (P~X)

reduces to the triangle

~i-i(~XIy) [2-2d]

~{p~x(d-i7r-16~x)[l_d]

> ~a.]~-l~[-l]

This triangle yields the identity -i •d-l. p~x ~Ir ~X ) Rk1~,]r-I ~

= l~-l(~xly) 0

and the exact sequence

for k=d-2~0 ,

for k=2(d-2)#0

for k~d-2,2(d-2)

.

323 (I.l.5)

I -i -i ~'~ 0 ---+0~p~x(~ e X) .... > T.~ ¢

c~-l(6~XIy)

for

~

0

d=2

Triangle (1.1.4) yields the exact sequence 0--~

~-I ~YIX

60÷ K - I R ~ ,

-~

CR

a0 > r. al+

~-IR2~ *~ R

C(R --÷bo b

RIy, CR

i

Rd-2T*~-I~

>

dhE1 ¢

~0

Since

Rd-I~.TC-I~ = 0

and

Rd'l~.%-l¢ -

61) T6-1R2~.¢ R

(both sides are isomorphic to If

for

or

=2

is an isomorphism for

?[-l(~xIy)) , ~I

d=3

is always isomorphic.

l

d > 2, then RI~.~R

d >3

= 0

and

d-l. -i ~X ) RN-2]~.T[-I~ = ~f p~XiTC

,

and the theorem is obvious. If

d=2, RI~R,O R = ~XIY

sequence (1.1.5).

and

60

is equal to

c

in the exact

This proves the theorem in the case d=2. q .e .d.

324 1.2.

Operations on holomorphic microfunctions.

holomorphic microfunctions

We define the sheaf

Px

We have defined of pseudo-differ-

ential operators to be ~ (0, dim X) the (dim X)-form on X×X with XlX~X respect to the second variable with holomorphic microfunetions as coefficients.

In order to define the ring structure on ~X' we must

explain several

operations on

the holomorphic microfunctions.

The

argument goes on in the same fashion as in the case of (real) microfunctions. Proposition

1 2 i,

Let

be a complex submanifold of

X~ X~

be a complex manifold and for ~=1,2

respectively.

we can canonically define the bilinear homomorphisms of

Pl

( YI XI) × P2

Y~ Then

product

(~ Y2[X2 )

~OY~lx Y21XIXX2 t SYI~Y2 * "" X 2- S ~YI ~" X ix Y2 (XI~ X2 ) -yl ~ S"Y2 -i Pl (CYII ~) ~ p21(~Y21X2 )

"~Y#

where

p~R

Y2 XlX X2 IP~I~Y2 (XI× X2) -Yf P~2X2-P~IXIXY2

is the projection

of

S~tY2(XlXX2)-Y ~ S~2X 2 - S~IX I XY2 is the projection of onto

P* X~

Y.,

'

P*YlXY2 (XlX X2)

onto

S~X~

Y~P~EX2

and p~

- P~IX~Y2

for ~=1,2 respectively.

This is proved as in Proposition Proposition

1.2.2,

Let

2o4~Io in Chapter I.

f : X'-+X

be a holomorphic map

:

325

transversal

P~X ~ X'~ P~,Xa - ~) • X Then we can canonically define homomorphisms of

to a submanifold

Set

Y' = Y X X' X substitution :

where

p

Y

of

X

(that is,

f* : p-I(CyIx)----*Cy,Ix,

,

f. : p~l( ~ R R yIx)---~ Cy, IX,

,

is the canonical projection

P~,X ' ~

""

PEX × Y'---~ P~X Y

PR : S~,X' ~ SEX K Y' --~ SEX . Y Proof. Let d=codim x(Y) = codim x,(Y').

and

We have the diagram of

continuous maps

X• ~R

~

>

i

?c' X'

and

(~)-I (S~)

= SE,X'

Y

'[t

-> X

f

Hence we have the sequence of canonical

h omomor ph isms ~-lj~d

-i

d

s~x(~ ex) -+ ~s~,x'

=

([-i -i

~x )

ex)

*x,)

This gives the homomorphism

-I ~ PR ~Y~X -+~Y'IX'" Then denoting by

~

the natural projection of

and operating the functor

~.

S~,X'

on this homomorphism,

onto

P~,X'

we obtain

326

P'I Cy~x -~Cy,j x, q.e.d. Corollary

1 2.3. Let

in a complex manifold

X.

YI' Y2

be two transversal submanifolds

Then we can define the canonical

homomorphisms of product : -I Pl ~YIJ X Xp21gy21X --~ j-I ~yla y21X

~-i ~YIJXKP2 ~ ~ - i [~Y2 ~ ~x __+j"i ~vYI ~ Y21X Pl where Y1 × P~2 X X

P*YIn Y2 X - P~I X X× Y2

~

P~X

P~I • Y2 X P~ S*Yl~Y2 X - S~IX Xx Y2 - YI xS~2X

,

J S~ 1 f~ Y2 X

Note that T*YI X IYI ~Y2 ~ T*Y2 XIy I ~Y2 = T~I ~ y2 X

holds.

This is a trivial corollary the preceding two lemmas. Remark.

We define several operations in this section.

These

operations do not betray our common sense except for the following fact:

If

Ul~ ~YIJX

and

u 2~ ~ Y2~X

, then

327 Ul(X)U2(X)

Proposition n'-dimensional

= (_i) c°dim YI codim Y2 u2(x)ul(x)

1.2 4.

Let

manifold

X'

such that the composite

We denote by p

p : P~Xp

f* : P

p~

Y d-~X'

Y f~ X

the canonical maps

~,X--+ P~'

of

integration

-i£ (n') YIX'

____+ ~ (n) --YX

-1 p~{(n ') f* : PU{ ~'YIX

mean

and

Let

: SEX - S~,X -+S~X'

Then the canonical homomorphisms

can be defined.

be a holomorphic map of an

X' to an n-dimensional manifold X.

be a complex submanifold of is a closed embedding.

f: X'--> X

~R (n) ~ ~Y~X

In this notation the suffixes

(n') (resp.

(n))

tensoring the sheaf of n' forms (resp. n forms).

Moreover if

f

is smooth,

i.e. df

has maximal rank, then the

integration of exact forms (with respect to the exterior derivative along the fiber) is zero. Proof.

Consider the following diagram P~ S~X-

S~X'

Y~

YX~*

S~,X

Y¥. x,- s ,x X

1+

f X

"t'C

S~X

328

As shown in Chapter I, we have the canonical morphism ~{f""~'X'f~(n')[n']-~ O'X(n)[n]

Pulling back this morphism by ~

we obtain

~f,p-~-iz,-ivx ,(~'(n')[n']-+-m -l(~(n)[n ]

This gives

s~x-s£, ~- ~. x(~- ~, -1 ~x' '* (n')) [~, ]

-~r s~x(~-~ O-x~n))En]i S~X - S~,x By composing this homomorphism with

--l~_ p

-I

l, S~X,(~'

(n')

] x,v n' O-X, )[n']-->~ S~X,.S~,X(~-I~'-I'~(n'))[

and by taking the O-th cohomology, we obtain the desired homomorphism

If

f

is smooth, then the composite -n'-n-lrn, ] dx,/ X~ n'-nr '-n] ~f! ~X'/X L -n ~ ~f,~x,/xLn

is zero. This proves the last statement,

q.e d.

329

1.3. Sheaf of pseudo-differential the sheaf of pseudo-differential

operators.

Now we can define

operators and endow it with the ring

structure. Definition l.b i. identify

Y

Let f: Y-~X

with the graph in

We define the sheaf

_~ Y ~ X

be a holomorphic map.

Y • X

and

and

~X~Y

P$(Y × X) on

Y X P*X X

We

with

Y × P*X. X

by

dim X (1.3.1)

~Y--~ X = £ Y I Y × X

~X

A~LX

(1.3.2)

~)X ~- Y = ~ Y I Y ~ X

~)y

dim Y ~Ay

The sheaf

~X

(1

on

%

P*X

= am

We call a section of

is defined

by

x

~ a pseudo-differential

the canonical element

operator.

~Y~X

~(x-f(y))dx, which will be denoted by

We can prove the following lemmas in the same way as in

has Iy~ X~ 2.5 of

Chapter I Lemma 1,3,2. a submanifold

Z.

Let

f : Y~X

Set

Z' = Y XZ, X

be a holomorphic map transversal to

P : PZ' Y "~z'~< PZ X--+PZ X Z

* Y -~Y × P*X . q : PZ' X Then there are canonical bilinear homomorphisms

330

-I

(1.3.4)

q

(1.3.5)

P

-1 ~Y -~X × p

-i

~X ~Z~X

Lemma 1.3.3.

Let

£ZIX

-+ ~Z'IY

'

dim X -1~ --~Z' ~ ~X ×q X+Y ~Y

f:Z~Y,

pl:Z ~ P*X - Z ~ P~XOZ~ x P*Y X Y Y

~dim Y Y

g:Y~X

be two holomorphic maps,

and

p2:Z × P*X - Z × P~X~~ Y ~ P*X X Y X

Then we have the following canonical bilinear homomorphisms

(1.3.6)

(1.3.7)

-+ Pl-i ~Z-~Y × P2 i ~ -Y->X

~)Z-~XIZ × P*X - Z ~< P Xf~ X Y

'

P2-i~X~-Y × Pl-i~Y+Z -->~X+Z IZ × P*X - Z × P*X y Y

The bilinear homomorphism

(1.3.6) endows

structure. Note that the bilinear homomorphism Ring structure

to

~X~X

and that

~X~X

~X

with a Ring

(1.3.7) gives

a

is isomorphic to ? X

by

~X~-X 9 K(x,x')dx ~-+ K(x', x)dx e ~X The bilinear homomorphisms the structure of left structure of right

(1.3.4) and (1.3.5) endow _dimX ~x-MOdule and ~YIX ~xILX

CyIx

with

with the

~x-MOdule.

Recall that we denote by ~ X the sheaf of differential operators on

X,

~dim X (0,dimX) which is defined to be ~L X (~XxX )"

By Proposition 1.1.5, we obtain a ring homomorphism

~ X --~ ~r, ~ X

( ~: P*X-->X),

which is an isomorphism when dimX > i and injective when dimX = i.

331

Lemma 1.3.4. P*(X K Z) - X~P*Z

Let onto

p

be the canonical projection of P*X.

Then there is a canonical homomor-

phism of Rings P -1 e x -'~x,z This homomorphism is obtained by K(x,x')dx'

J

> K(x,x')~(z-z')dx'dz'

~dimX K(x,x')dx' 6 £XI X × X ~ x ~ X

where

dimZ

~(~-~')dz' ~ ~ZtZ×Z ~z~Z Remark.

and

= ~z

In the calculus of holomorphic microfunctions, we must

pay attention to the signature.

In this note, we use the following

concordant system of calculation rules.

i)

I~(x-y)f(y)dy- f(x), I~(x)dx=(-l)n

2)

ff(x,y)g(y)dx = (If(x,y)dx)g(y) ,

3)

If(x,y)dxdy =

f (If(x,y)dx)dy •

where

x=(xl,...,x ~ ,

332

1.4.

Concrete expression of holomorphic microfunctions.

If we

fix a coordinate system of a manifold, then we can describe a holomorphic microfunction concretely by the aid of defining function.

This

is analogous to the fact that holomorphic functions can be developed into power series.

Before discussing the general case, we mention

a simple example. Example 1.4.1.

Let

In this case ~:P~X--~Y P~X

with

Y.

Let

?(x)=0

¢.

choice of

A xl

-->

of codimension i.

S~X

on

o,

-linear.

be a defining equation of

R a section of ~Y~X

~Y~X

X

is an isomorphism, by which we identify

0 --~ ~gy~X --+ C yix

which is, of course, ~ X

of

be a submanifold of

We obtained the exact sequence

(i 4.1)

modulo

Y

Y.

Log y(x)

In fact, log ?(x)

is defined on

(More precise arguments are given later.) corresponding to ?(x).

-~log

We denote it by

~(x) Y(?(x))(or

defines X-Y

The section

is independent of the Yyix(X)).

Since

~(Y( ~ (x))):l, £yix = ~yIx~'xY(?(x)) (as

(~x-M°dule' not as

~ x-MOdule) .

Now, we define the holomorphic function % with parameter

(1.4.2)

~

][ ~-(~)

by l ~(l+Ai 27~'~ (-T)~+I

~(T)

of one variable

333

Then we see that the following remark holds.

Remark.

If

~

is an integer

i

< 0, ~x(-c) stands for

-~-i ~

'U-~'-l~l°g'l-(

2 ~Ci'~ (- X- i) !

where

" 1~ - ~ ) I

~=i

~ is the Euler constant.

~(~)

satisfies the following

properties :

i)

~

= ~x+l(~)

2)

n ix(z )

(-)nr(X+l) = r(~-n+l) ~X-n(~)

Moreover, Let

X=C 2= ~ (A,'r) 1

set

D Y=

(modulo holomorphic

{ (~k,'r)

~ X~T=0

p:V~S~X be the universal covering of

defines a section of sections of

p

-i

p-i

Y~X

functions).

}

S~X.

Then

, and equations i) and

~(T)

2) hold as

~yl X

Every section of

CyIx

on

U CY

is

a

cohomology class of the

type (1.4.3) where

~a.(x)~j(? j~ J ~aj(x) I j ~

neighborhood (1.4.4) I)

V

(x))

is a series of holomorphic of

U

in

X

C > 0

for every

K in

V,

such that

laj(x) l -< (-j)! C-j 2)

satisfying the following estimate:

For every compact set there is

functions defined on a

~ > 0,

for there is

j< 0 C~ > 0

and

x~K

such that

,

334

Cg Ej laj(x) l <_ ~--

Estimate

j > 0

and

xeK.

(1.4.4) is equivalent to the fact that the series (1.4.3)

converges normally Definition

(that is, absolutely converges

1 4 2.

aj (x) ~ (j) (?(x))

~ j~

the section of

~Y[X

with the cohomology class (1.4.3).

Suppose, moreover, where

locally uniformly).

We denote by

(1.4.5)

roj=idy

for

that a contraction

j:Y~X).

r:X~Y

Then every section

u

is given (that is,

of

~ Y[X

on

UCY

is uniquely expressed in the form u =

(1.4.6)

where {bjl estimate X-Y

~ j~

bj(r(x))~(J)(?(x))

is a series of ho!omorphic

(1.4.4).

,

functions on

In fact, every holomorphic

can be developed

function

(1.4.6) when the codimension

codimension X

be a submanifold d.

such that

be coordinates homogeneous We set

of

Y

is larger than i.

P~X ,

coordinates.

where

manifold

X

with

system (Xl,...,xd,Yl,...,yn_ d) of

is defined by the equation of

on

by a series similar to

of an n-dimensional

We fix a coordinate

Y

f(x)

~ aj(r(x)) ?(x) j jE~

We can express a holomorphic microfunction

Y

satisfying

into the Laurent series

f(x) =

Let

U

xl=...=Xd=0.

Let (y, ~

~ =(~l,...,~d ) is cotangential

335

Ixl
(1.4.7)

Ug= ~(x,y) E X;

(1.4.8)

ZA, g = ~(x,y) e Ua ;

<

£},

(1.4.9)

j=2 ..... d~ ,

for

ReOkXl)>-~Im(Axl)l,~Xll_>6Ixjl

GA = ~(0,~0) E S~X; -1

~2 =. ..=~d =0, for a complex number

X

Re(A-I~I) > llm(X ~1)! ~" We can define the

with absolute value i.

homomorphism (1.4 i0) •

Hd (U£ ; eX) b------+r(Gx; ~yix ) ZX,£ A,~

(see Proposition

1.2.4 and the proof of Proposition

1.2.3 in

Chapter I) Ug - ZX, ~ is covered by holomorphically V k,~ (~) (~=i, "" "' d)

convex sets

defined by

V X,~ (I)= {(x,y) ~U&" Re(Xxl)
vk,e=

=

iXli, L ., 1 for

Set =

d ~ v (~)

,

~(~) =~

p =2, .... d

v~) X,6 1nu~

We have the exact sequence

d ~ (~(V)

9=11" A,~'" ~X) -~ F(v~,~,• ~X)

~'~,,~.

H dzx,~(u~,~x)-~0

336

Set ve

=

t(x,y) eUe;

xj~0

""

V e = ~(x,y) eU&;

Let

f(x,y)

ux

IXll
for

be a (multivalued)

~[(V£;~X )

of

j=l . . . . .

d}

,

_^ (1)

j=2 ..... d}-Vx, &

analytic function of the type

1 f(x,y) = ?(x,y) + ~ ~(x,y)log x 1 2K~2~

(1.4.11)

where

for

f(x,y)

branch of

by

log Xl.

and

~6~(~¢;DX).

bx,&og~,g If

Gx~

Since

~)k, (~)=0 ,

the image

is independent of the choice of the G~ ~

,

(which is equivalent to

Re Xflt-l>0), then the diagram

F(Vk,g

LIV , f . ; (#X) -~'x, ~ :. H dz),,a

Hd

(ug; 0 x )

d

z,~(u~; ~x )

; Hzx,~uz,~ (U;Ox)

is commutative, which implies that

Therefore there is a section UIG x = u x We say that We put

If we denote

u E F( lXl U =I Gk; C ~1 y~X) x~=(0,dxl~0) ~P~X ,

f(x,y) is a defining function of Z&= {(x,y) ~Ug; IXl[ ~£1xjl

for

then

such that u ~CylX, x ~

u. j=2 ..... d 1

Then, we have the exact sequence d ].,(~ ( ~ )

~=2

"~

; 69X) --+ ~(V~; 69X) ~~

H z~ d-I

(u~;6~x)~ o,

337

where

~)

= ~

V (P) ~ U g

2_<~d We denote by

bg

the homomorphism d-i d-i HZ& (Ug; O X ) - - ~ P ~ X (~-l~x)x~

Note that, if

f

is given by (1.4.11), then

~ (Nx,~ o~X,~ (f)) = ~o~(~)

(1.4.12)

,

where

Proposition defining function ('1.4.13)

Every germ

i~4.3.

of

CYIX

at

x~

has a

of the type

f(x,y)

f(x,y) =

u

> i ,a~(y) ~ ( x ) =(~ i'''" "~d )~d" ~2 ' " " " '~d~-0

which converges locally uniformly on for

W~ = i(x,y) ~ X; Ix~<~, {y1
for sufficient small

j=2 ..... d }

g, where we set d 9=1

(see (14.2)). In order that the series (1.4.13) converges locally uniformly on WE

for sufficient small

there exist r > l

C~

g>0, it is necessary and sufficient that

and a neighborhood

U

of

y=0

such that

338

(1)

sup la~(y)I=~ Cl~ir yeU

(2)

lira j ~ j-~m

(3)

lira j_>_~

= 0

-JJ77

Proof. proof.

Let

-J

J

<0¢.

Since the last statement is easily verified we omit its E

be the subspace of

~yix _ ' ~. consisting of the

microfunctions with defining functions of the type (1.4.13). ~d

a~(y)~(x)'s

constitute

~yix, K(x~ ) , which implies that

E D ~YIX,~(x~)

By exact sequence (l.l.2),it is sufficient to show

that the image of

E

equal to J{ ~

under

~: Cyix,x ~ _ _ ~ pXi~d-l" ~ -i ~.X)x~

(~'l~x)x~ .

is

If we set

Xl~l-i ~(x,y) =

~_~

~#0

a (y)

(-~l-1)'.

~ 2 (x2)

.

.

.

.

~d

(xd) '

then

Now, let we can develop

g(x,y) E F(V~; ~X ).

Since

is a Reinhardt domain,

~g

g(x,y) into the Laurent series D~(Y) x~ ~ d , ~i>_0

~, b~ (y)x ~

is holomorphic on

~fV~j )

for

j=2 ..... d, which

~.>_0 J

implies that

_y~(g)

is equal to the image of

339

Xl~i-i (1.4.14)

b<(y) x ~

=

~

~ ~ d , ~ 2 < 0 , . .. ,~d<0

a ,(y)

~'E

1>o

iZ0,

d -~-~ ,(x~) (-~i-i). ' f W=2 ~

.....

by setting ( -~l- 1) ! a ,(y) = ~2!...~d ! b~(y) If (1.4.14)

converges

the proposition.

This completes

on

,

V£, then

where ~a~}

~' = -~ .

satisfies

the estimate

of

This shows that

the proof of Proposition

14.3.

We put W = ~(y,~,p);$Ecd-~01, Z = ~(y,~,p); Let

u(x, y)

p~¢,

yEY I ,

p=01CW"

be a section of

~yl X

on an open set

fL of

P~X

Then v(y,~,p)

=

I ~(p-<x, ~>)u(x,y)dx

can be defined by Proposition 1.2.4

to be a section of

~' =

I(Y,~,P)

then

u F--> v

Example

s

Corollary

1.2.3 and Proposition

on

= Z ; p = 0, ( y , ~ ) 6

is the map

~} .

Z ~ (y,~)|)

turns out to be the sheaf homomorphism

1.4.1 asserts v(y,~,p)

where

~ZI W

6 P~

This means that if

1,2.2

a. (y ~) I '

that =

v(y,~,p)

is uniquely

7~ aj(y,~)~(J)(p) j~

satisfies

(y,<~,dy>~)~ s

-i

expressed

~Y~X

-+

as

,

the growth condition

given in (1.4.4).

P~X ,

zlw"

340

Since

v(x,~,p)

is homogeneous of degree (-1) with respect to

a.(yj,~) is homogeneous of degree Proposition 1.4,4. in a neighborhood of (1.4.15)

Let

j

(~,p),

with respect to ~.

u(x,y) be a section of Cy~ X

defined

x~=~,dXl~ ) with the defining function f(x,y) = ~ d

a~(y)~(x).

~2 ..... ~d ~0 Then v(y, ~,p) = I ~(p-<x,~>)u(x,y)dx reduces to v(y,~,p) =

(1.4.16)

~ aj (y, ~) ;(J) (p) j&2

where (i 4.17)

a.(y,~) =

•

J

Proof.

Z

a~(y)~

I~ t =j

At first, we calculate (~0=(i,0 .... 0)) v(Y,~0, p) = I~(P-Xl)U(x,y)dx = I u(p,x',y)dx'

where

x'=(x 2 ..... Xd).

It follows that

, v(Y,~0,p)

function defined by the defining function (_)d-i I f(p,x,,y)dx, " Here

is the cycle of dimension

(d-l)

~(p,x',y); Ixj~= g/2 for j=2 ..... d I We have

is the micro-

341

I~- f(p,x' y)dx' = I~a~(Y)~l(P)~,(x')dx'

= (_)d-i

~ a~(y)~l(p) ~' =0

where

~'

signifies (~2 ..... ~d )"

(1 4 is)

This shows that

v(Y,~o,p) = ~ a~(y)~(J~)(p) ~'=0

Moreover, for

~£ ~

, we have

D ~ v(y, ~,p)

= Ix~(~i)(P-<X' ~>) U(x,y) dx , = (_~)'~i ~ x~(p_(x,~>) u(x,y) dx . Since

(_)1(~1~ ! (4-(~)!

a ~ (y)~_(5(x)

is a defining function of x~u(x,y), we have by (1.4.18)

D~v(y,~,p)I~0 _(_~)IOL Z (_)1~J! ~'-(~'=o.

=

a~(y)~ (l~-~)(p)

(~-~)!

Z ~! a=((y)~(~) (p). ~'=~'(~-~)!

This implies that

D where

aj(y,~) I~=~0

~ is given by

(o(-~) '. a~(y)

(J- l~'l , ~'). Since

degree j, we obtain the desired result

aj (y,~)

is homogeneous of

342

a.(y,~) = J

~ a (y)~ I~ I=J

q.e.d.

It is easy to verify that the growth condition of in Proposition

Theorem 1.4.5.

manifold

Let

(1.4.4).

Let

U

Y X

(x I ..... xd,Yl,...,Yn_d) Xl=...=Xd=0.

given

1.4.3 is equivalent to that of the convergence of (1.4.17)

and the growth condition

n-dimensional

~a~(y)l

This proves the following theorem.

be a d-codimensional

We fix a local coordinate system of

X

such that

be an open set in

U' = ~ ( y , ~ ) ~ Y ~ d - ~ o ~ )

Then the set of series

submanifold of an

Y

is defined by

P~X

Set

; (y,~)E U ]

of holomorphic

I aj(y,~)IjEZ

functions satis-

fying: i) j 2)

aj(y,~)

is a holomorphic

function on

U'

homogeneous

of degree

with respect to ~; j J~laj(y,~)i

tends to

0

when

j-~,

uniformly on every

compact subset in U' ; 3)

i _--~ - ~ a j ( y , ~ ) l

compact subset in

is uniformly bounded when

~(U;

~yix)

by

5~(p -<x, ~>)u(x, y)dx = ~_, aj(y, ~ ) ~(J)(p). jeZ

(1.4.18) X

be a complex manifold with a local coordinate system

(x I ..... Xn). P*X

on every

U',

is in i-i correspondence with

Let

j~O

Let (~I ..... ~n )

is expressed as

case, by taking

X

(x,~m). and X~X as

be a cotangent v e c t o r

The point of

We will apply Theorem 1.4.5 in this Y

and

X

respectively.

343

Definition

1.4,6.

Let

~'=~(x,~)~X~(¢n-{01); of holomorphic i)

pj(x,~)

degree 2)

j

~

(x,~)e~l.

with respect to

(1.4.19)

j

P*X

and

~pj(x,~)~ jEZ

be a sequence

:

function defined on

~'

homogeneous

of

~;

satisfies the estimate ~Ipj(x,~)~

when (i .4.20)

Let

functions satisfying

is a holomorphic

~pj(x,~)}

be an open set of

:

tends to

0

locally uniformly in

~L'

is locally uniformly bounded in

~'

j--~.

1

Ipj

when

j < 0.

Then we denote by

(1.4.21)

P(X,Dx) =

~ pj(x,Dx) j~2

the pseudo-differential (1.4.22)

P(X, Dx)S(<x,~>-p)

Note that In fact, if

operator on

characterized

is a kernel function of

is a kernel function of

P(X,Dx)~(x-x' ) is equal to

IK(x,x")~(x"-x')dx"

This is evidently equal to

K(x,x').

Remark.

In Kashiwara-Kawai

the consideration Pj(X,Dx)

by

= ~p.(x,~)~(J)(<x,~>-p). j~Z j

P(X, Dx)~(x-x')dx'

K(x,x')dx'

~

P(X,Dx) , then by the definition.

[i] we have restricted ourselves

of pseudo-differential

for some integer

P(X,Dx).

m

to

operators of the type

and have called them pseudo-

j<m differential

operators of finite type.

But in this paper and also in

our forthcoming papers we call them pseudo-differential finite order, since we must use

pseudo-differential

operators of

operators of

344

the form (1.4.21) in later sections.

We sometimes call pseudo-

differential operators of the form (1.4.21) pseudo-differential operators of infinite order

We also remark that the sign of the parameter

j has been changed in this paper in order to avoid confusions. used

P_j(X,Dx) in place of

(We

Pj(X,Dx) in our preceding note Kashiwara-

Kawai [i]). 1.5.

Adjoints, composites and coordinate transformations.

Now we establish some calculation rules concerning pseudo-differential operators, which are expressed in terms of Pj(x,~). We first investigate the formal adjoint operator pseudo-differential operator

~ Pj.(x,Dx). P(X,Dx) = j=~j

R(X,Dx) of Note that R(x,D x)

satisfies by definition

(1.5.1)

P(X, Dx) ~ (x-x')=R(x',Dx,) ~ (x-x').

Without loss of generality we may consider that near

(x;~)=(0;(l,0 ..... 0)), i.e.,

~2 ....~'~n~0 a~(x)D~x

P(X,Dx)

is defined

P(X,Dx) has the form

Then by the definition the left hand side of

(1.5.1) is the holomorphic microfunction with defining function

(1.5.2)

~_~ a~(x) ~[~(x-x' )= ~. ~2 .....~n >-0 ~j>_0

-~i~x# a ~(x' ) (x-x')~ ~(x-x' ).

~2 ..... ~n >0 By the definition of

(1.5.3)

~(x-x') we have

(x-x')~l~(x-x') = (-1)I~I~ (~_~)! i~_~(x-x'), n

where

i~[ signifies

~ ~ j=l j

On the other hand the right hand

345 side of (1.5.1) is equal to (-l)nR(x ',Dx, ) ~(x'-x).

(1.5.4) Hence if we denote

(1.5.5)

R(X,Dx)= ~b,(x)D: ,

br(x) =

then we have

(-I) ~j~!

~.

~2 ' .... Wn>-0

By these calculations we have the following ~o

Theorem 1.5.1.

The formal adjoint operator

R(X,Dx)=

Rk(X, D x) k=-~

~o

of a pseudo-differential operator (1.5.6)

~(x,q)=

Proof. (1.5.7)

P(X, Dx)= ~Pj(X, Dx)

is given by

(-l)J D~D~P.(x,~) ~ ~, '~ x J k=j -~i

By the definition we have

(-~)Pj(x,~)=(~)i~=

j a~(x)q ~= ,a~=j(~---~. a~(x) q ~-~

Combining (1.5.5) and (1.5.7) we have --~ ~¥= Rk(X'~)= Ir~k b~(x)

-~ 2~ k=j -J~i

This is the relation which we wanted. In the sequel we denote by

qe.d.

P*(X, Dx) the formal adjoint operator

of the pseudo-differential operator

P(X,Dx) . Note that the operator

P*(X,Dx) depends only on the choice of the section n of ~]'X"

dx=dxl, ""

"

,dxn

We next investigate the composition rule of two pseudo-differential operators

P(x,D x) = j__2~_fj (x' Dx) --

and

Q(X,Dx)= k=-~ ~ Qk(X,Dx)

346

By the definition the kernel function of the composite of P(x,Dx) and Q(X,Dx) (1.5.8)

is I P(X'Dx) ~(x-x')Q(x"Dx') ~ (x'-x")dx'

On the other hand the integrand of (1.5.8) is equal to P(X, Dx) Q*(x",Dx,) ~(x'-x) S(x"-x'). Denoting

P*(x, Dx)

by

~ j

homogeneous of order

P~(x,Dx) , where

P~(x, ~)

is

=-~

j with respect to ~, we have

[P(x, Dx) Q*(x" ,Dx,,) ] *=Q (x", Dx,,)P*(x,Dx) =

2_. Qj(x",Dx,,)P~(x,Dx) j,k

=

~ (~ ~(x"-x)YDx ~ Qj(X,Dx,,))P~(x,Dx) j,k ~ " n

~. t"x" j,k Qj(X,Dx,,)P~(X,Dx)+/=i 4-x~)" S ~(x' x"' Dx' Dx")

for some pseudo-differential operators

S£(x,x",Dx,Dx,,) (~=i..... n).

That is, we have proved that

(P(x,Dx)Q* (x",Dx,,))*=T (x,Dx,Dx,,)+ ~ (x~'-x£)S J(x,x", Dx,Dx,,) ~=i

holds, where

T (x,Dx,Dx,,)= .~ Qk(X,Dx,,)P~(x,Dx). J,K

This immediately implies that P (x, Dx) Q*(x", Dx,,)=T*(x, Dx, Dx,,)+ ~ S~*(x, x",Dx, Dx,,)(x~-xA) £=1 Since

(x~-x~)~(x'-x)~(x"-x') vanishes, we have obtained the following

347

(1.5.9)

P (x, Dx) Q* (x", Dx,,)~(x-x' )~ (x' -x") = T*(x, Dx,Dx,, ) ~ ( x - x ' ) ; ( x '

-x")

Let R(X,Dx,Dx,,)= ~- R~(X,Dx,Dx,,) denote

T*(X,Dx,Dx,,).

Then applying

(1.5.6) in Theorem 1.5.1 and Leibniz' formula for differential operators successively we have R£ (x,~, ~") =

(1.5.10)

=

~ (-i) j+k D~D~ (Qk (x, ~") p~ (x, ~) ) ]=j +k-I~t o~ !

~ (-i) j+k D~D~(Q~(x, ~") ~=j+k-l~l ~I• x ~ K (j=i-I~I

O"

"I

(T!)2i-1~1+k(71)~ ! ~! Dx~IQk(X,~,,)Dx~2+~D~+~Pi(x,~ ) i=i+k- I~t -I~I =~i+~2

=

~

(-1) k+I~I

~ 1

£=i+k-t~i-I~l ~!~i!~2! Dx

~+~1

qk(x,~")D~D~

Pi(x,~)

d= dl+~2

r=~2+~ (-1) k ~ i Wl ~1 ! Dx Qk(X,~")D~Pi(x,~),

>,

=

£=i+k- I~iI since --

(-1) l#~

i (l-l)

~-~ ~2 ! ! = "~'i ¥=~2+~ ' ~. • holds for l~i > 0 and _

= 0

(_l)IP I

2, , =l ~2+~=0 ~2" ~" holds clearly.

Now we substitute (1.5.10) for (1.5.8) and obtain

348

I

P(X,Dx)~(x-x')Q(x',Dx,)~(x'-x")dx'

= I R(x, Dx, Dx,,)$( x-x' ) S (x' -x") dx'

=R(X'Dx'Dx") f °¢(x' -x) ~'(x"-x' )dx' = 7Z.~R~(x, Dx, Dx,,) f (x- x" ) = ~ Rz(x,D x,-Dx)~(x-x'') (-l)k (D x Qk(X,-Dx) )(D~ Pi )(x,Dx)~(x-x''))

= ~(~=i+~k-'~i'~I! I(

=

~

2=i+k- M1

~ t D~Qk(D~Pi)(X,Dx)~(x-x")). "

"l

Therefore we have proved the following theorem. Theorem 1.5 2. Qk(X, Dx)

Let

P(X,Dx)= ~ Pj(X,Dx) j=-~

be pseudo-differential operators.

and

Q(X, Dx)=

If we denote the

k = -~

composite

P(X,Dx)Q(X,Dx) by R(X,Dx)= ~ R£(X,Dx) ,

term R~(x,~) (i.5.11)

each homogeneous

is calculated by Rz(x, ~)=

~-- ~V i Da p. (x, ~) D~xQk(X,~) L=j+k_t~i " 'l J

In the next place, we will study the transformation law of the expression

P(X, Dx) = >-.Pj(X,Dx) under coordinate transformations.

Lemma 1.5.3. (xI ..... Xn)

Let f(x)

be a holomorphic function of

defined in a neighborhood of

dxf(0)=0, and

P(D t,Dx)=j6 ~P.(Djt'Dx )

tor defined on a neighborhood

(1.5.12) where

of

P(Dt,Dx)~(t+f(x))Ix=0 =

a i is given by

x=0

such that

n variables f (0)=0

and

be a p seudo-differential opera

dt00.

Then

(t) jE~__. Za.~(J) J

349

(1.5.1~)

ek (~) (1, O) Proof.

a.

-

J

~-~ =4'~' P j=k- I~ i+P " ~ , ~Z+, l~ ~_2~

signifies

(I'0)~Dx (f(x))~ =0

D~ Pk(l, ~)I ~=0

By the definition, we have P(Dt,Dx) ~(t+f(x)) I x= 0 = I{ P (Dr,Dx) ~(t-t' ) ~(x-x' ) }~ (t '+f (x') )at' dx' Ix=0 = I{P (Dt, -Dx,) ~ (t-t') g(x-x' )} ~(t'+f (x')) dr' dx' Ix__0 = I~P (Dt, -Dx,) ~(t-t')~ (-x')} ~ (t'+f(x')) dt' dx' =

(_)n I p (Dt, _Dx, )~(t_t, )~ (_x,) it,=_ f(x, )dx'

: IP(Dt,-Dx)~(-x )~(t-t')l

dx t'=-f(x)

We can easily show, using Proposition 1.4.4, that P(Dt,-Dx)~(x)~(t-t') is defined by the defining function E ~, n c F,~=((x) ~(t-t' ) ,

where

c ~,~ is determined by

(1.5.14)

Pj (~,~)=

Z c ~ ~+ i~| =j ~'~

In other words, c ,~=

i ~(6) ~,. ~+I~I (i,0)

Therefore the integrand by the defining function

P(Dt'-Dx)~(-x)S(t-t')It'=-f(x)

is defined

350

F(x, t) = (-I)n

(1.5 .].5) Let ~

~.,

c

be the cycle of dimension <xecn;

Ixj[ = ~,j=l . . . . .

for s u f f i c i e n t l y small &.

~(-x)~(t+f(x))

n: n

P(Dt,Dx)3(t+f(x))lx=O i s , t h e r e f o r e ,

defined by the defining function G(t) = ~0_F(x,t)dx We calculate

G(t).

(-)nG(t) = ~0" ~ c ,~(-x)~(t+f(x))dx ~,d = ~ c ~(-x) ~,~ ~,~

]~p(t+f(x))dx ,

because the series (1.5.15) is normally convergent. (_)n I~- ~(-x) ~v(t+f (x))dx

X

6o

= ~~--~'-I (D~xf(X~)Ix =0{~+e(t)" /~=0 It follows that G(t) = ~

C V ~ ~'

i

(D~xf(X~ix=0)~

C

-- ~

{Dx~f(x~

This is the desired result,

qe.d.

351 From this lemma, we can prove the following theorem Theorem 1.5,4,

Let

f(x)

defined in a neighborhood P(X,Dx)=

~ P.(X,Dx) j~2 j

neighborhood of

x0.

(1.5.16) where

such that

(x0,df(x0)~).

Put

function of

df(x 0) # 0,

x 6X

and

operator defined in a

Y= ~xE X; f(x) = 0 ~ .

be a section of ~YIX

defined in

a

Let

neighbor-

Then ~_~ cj(x) ~(J) (f(x)) j e~

P(X,Dx)U(X ) =

cj(x)

(1.5.16)

x0

be a pseudo-differential

u(x)=~aj(x)~(J)(f(x)) hood of

of

be a holomorphic

is given by i

cj(x) =

~!pl

(x, dxf (x)) X

j =k- I~ I+~+

k~, ~Z,W~+,~2+,]<1 >2V

X{Dx~(~(x,z)ga~(x))Iz=x} where

~(x,z)

is given by

(1.5.17) Here

P~)

,

f(x) = f(z)+<x-z,d f(z)>+f(x,z) signifies the derivative with respect to the second

arguments. proof.

Set

P(x,D )~(t+f(x))=~bj(x)~ (j)(t+f(x)) X

Then

b.(x) ]

near

x0 .

is uniquely determined. Set

We fix a point

s=t+f(xl)+<X-Xl,df(Xl)>

P(X,Dx)~(t+f(x))

xI

sufficiently

Then

= P(X, Dx)~(t+f(xl)+<X-Xl,df(Xl)>+~(x ,xl)) = P (x, Dx+d f (Xl) Ds) ~ (s+~(x, Xl) ) .

352

Hence if we put

Q(Ds,Dx) = ~,Pj(Xl,Dx+df(Xl)Ds) ~ then, by the

preceding lemma, P(x, Dx) ~ (t+f(x)) IX=Xl =Q(Ds'D x) ~ (s+f(x' Xl)) IX=Xl i p(a) = ~f---V~.J j (x I , df(Xl))(D~ ~(x, Xlfl

X=Xl )~(j+~-l~1) (t+f(xl)) ,

which implies (1.5.18)

bj(x) =

-i Pk(~)(x,df(x))(Dax~(X,Z)V Iz_x). 2-~ ~!~' j=k- ~) +~ "

Now let us prove the formula (1.4.16). R(x,Dt) = ~aj(x)D~

Set ,

v(t,x) = R(x, Dt)~(t+f(x)) Then U(X) = v(t,x)It=0 We denote by

S(x, Dt,Dx)= ~S%(x,Dt,D x) the composite

P(X, Dx)R(x,Dt) . By the formula (1.5.12), =

~

~=j+k-I~l

p!~) (x, ~)%,~ (x, ~) ' J

1 ~-v

~k ~(~) =

~ ~--F ~'j (x,~)Dxak(X) ~=j+k- i~I "

•

Formula (1.5.18) yields S(x,Dt,Dx) f(t+f(x)) I ~(~) (x, i, df (x))(Dx~(x, z)~ iz=x) ~ (j - I~D +~) (t+f(x)) = ~ ~-~-V-.V! ~j Hence C. (X)

J

=

j=k-l~l + ~ ! 1~" Sk(~)(x, i, df (x)) (Dx~~(x' z)~ ]z=x )

On the other hand

353

Sk(~)(x, i, df(x))

(~) #i' P~ (x'~)Dx~ak-1+'#'(x)~l~=l,~=df(x)

.k-~+t~I

= ~D~ ~, "~ 1

(a+~) (x, df(x))D~xak_f+l~ i (x)

1,¢ This implies

1

c (~)= Z o~!~!~[ P J j=k+p- lal -I~l +~

(~+~)

(x, df(x)) ×

(x)

j=k+p- I~I +£

I z--x)

i ~(~) (x, df(x))D¥(~(x,z)qai(x)),x ~ ~! ~! ~k z=x

completing the proof.

q e.d.

We can calculate the transformation of the expression P(x,D)=~Pj(x,D)

of the pseudo-differential operator under coordinate

transformation, by the aid of Theorem Theorem 1.5.5. x~=(Xl..... xL)

Let

X

1.5.4

be a manifold,

x=(x I ..... Xn) ,

be the two local coordinate systems of X,

and

~=(~I' ....~n )' ~=(~I ..... ~n ) coordinates of

P'X, i.e.

be the corresponding homogeneous fiber ~xk ~J= ~k ~ ~k" Let P be a pseudo-

differential operator on P*X. coordinate system, and Then

Suppose

P=~j(x,D~)

P =~Pj(X,Dx) by the first by the second coordinate system.

P. (x,~) and ~. (~,~) are related by the formula J J ~ ~

(_)~+j~l

(~+~)(~,

354

where (x~,<~,d~>~) = (x, <~,dx>~) Proof.

It is easily obtained by applying Theorem

1.5.4

to

P(X, Dx)~(<~,~>-p) " We leave the precise calculation to the reader. q.e.d. Let

~X,m

be the subsheaf of ~X

consisting of the

pseudo-

differential operators P(x, Dx) = ~Pj(x, D x) such that

Pj(x,~)=0 for j>m.

definition of ~X,m system.

Let

is independent of the choice of local coordinate

O-p,x(m )

m with respect to ~° Theorem

1.5.5

~X,m 9P =

Then, Theorem 1.5.5 states that the

be the sheaf of homogeneous functions of degree ~p,x(m)

is an invertible

~P*X-Module.

implies that ~ Pj(X'Dx) ~ jim

Pm (x'~) & ~ P * X (m)

is a homomorphism independent of the choice of local coordinate which we will denote by

system,

~" m

Definition 1.5.6. tor.

If

Let

P(x, Dx) be a pseudo-differential opera-

P(x, Dx) E ~X,m ' then we say that

P

call 0"m(P)

the principal symbol of order m of P.

some

then we say that

f ~X

mEZ,

P

is of order ~ m If

is of finite order.

P ~ ~X ,m

and for

We denote by

the sheaf of pseudo-differential operators of finite order. propositio n 1.5.7.

~

=

k2 ~x m mEX

is a sheaf of rings with

filtration and its graduation f

=

(~ ( ~ X , m /~X,m-I )

355

is a sheaf of rings isomorphic to Proof.

~) G~p,x(m ) m~2

This is a direct consequence of Theorem 1.5.2 and Theorem

1.5.5.

q.e.d.

Let

Y

be a submanifold of

X

~Y/X = ~ / ~ 2

be the normal sheaf of

defining

We denote by

Y.

~Y/X

with codimension d.

Let

Y,

where ~ is the coherent sheaf d the sheaf det ~Y/X, = A ~ y / x •

Then we have a canonical homomorphism (f(y)(dx) ® - I ~

( ~ Y / X ) ® - I --~ ~ Y I X defined by Sate [i] subsheaf of

~6.4, by which we consider

~YIX " "71 is the projection

f(y)I(x)) -i ®-i 71 ( ~ Y / X )

P~X-~Y

as a

We define the

i

sheaf

~yiX,m

~-I ~yiX, m = ~X,m(~ l ~ y i x ) = ~ X , m ~ Y i x ( X )

by

Proposition 1.5.8. Module , that is 2)

i)

f ~ y ~ x = U ~YIX, m

~ X , m ~YIX,£

~-i ~YIX

m~Z.

is a filtered

C CyiX,m + ~ .

f X = ~m ( £ y i X , m / ~ Y I X , m _ l ) gr £YI

~ P * X (m) ~ mEZ Y ~y

for

as a graded

is isomorphic to

f gr ~X-- ~ p . x ( m ) - M o d u l e . m

This is easily proved by the following lemma. Lemma 1.5.9. X

such that

phism

Y

~X ~YIX

We fix a local coordinate system is defined by

x I = ...= x d = 0.

(x I ..... Xn) of Then the homomor-

defined by

~ X 9 P (x, Dx) e-~ p (x, Dx) ~ (x I ..... Xd) is surjective

and its kernel

Dd+l,...,D n.

Moreover, we have

~

is generated by

x I .... ,x d

and

356

d

3m= 2.

J ~ ~m

n

= j=l ~ ~ m Jx. + j --=d~+ i ~m_iDj

Fundamental properties of pseudo-differential

operators

In this section we establish fundamental theorems on pseudodifferential operators of finite order, i.e., the invertibility elliptic pseudo-differential

operators,

of

the equivalence of two pseudo-

differential operators of finite order under suitable conditions and the possibility of "operational calculus"

(The meaning of "the

equivalence" will be clarified in Theorem 2 1 2 ) Note that the theorems given below play an essential role when we investigate the structures of (systems of) pseudo-differential equations in later sections. 2.1.

Theorems on ellipticity @nd the equivalence

differential operators

of pseudo-

We first establish the following theorem

which shows the invertibility

of elliptic pseudo-differential

operators° Theorem 2 i.i~ order at most

m.

Let

P(x,D)

be a pseudo-differential

Assume that its principal symbol

vanishes in an open set

&CP*X

Pm(X,~) never

Then there exists a unique pseudo-

differential operator of finite order (2.1.I)

E(x,D) such that

PE = EP = i

holds in £g, where I denotes the identity operator,

i e., the pseudo-

differential operator whose kernel function is given by Proof

operator of

S-function.

We first construct right and left inverses of

micro-locally,

that is, for any

(x0,~0)

in

P(x,D)

~. we find pseudo-

357

differential that

PE=FP=I

E=F E2

operators

holds.

F

in a neighborhood

Then we have

El

is defined in

Then we have

and

and

PE2=E2P=Iu2

P(EI-E2)=0

0 = EI(P(EI-E2))

U IC~

in

E2

in

P

= (EIP)(EI-E2)

E 1 and

U2 ~

satis-

U1 ~ U 2 # ~ .

= EI-E 2 right and left

is sufficient for the proof of the theorem.

prove the existence of micro-local

hence

hence

Therefore the existence proof of micro-local

inverses of

(x 0,~0) such

operators

Assume that

Uln U2,

of

F=F(PE)=(FP)E=E

Now consider two pseudo-differential

PEI=EIP=Iu1

holds.

and

holds there.

such that

fying

E

left inverse of

P.

We first

Then it is clear

from the method of the proof given below that we can construct the right inverse of

P

also.

Now we begin the proof of micro-local

existence of the left

inverse.

We first define a pseudo-differential

Q_m(X,~)

is given by

tor of finite order pseudo-differential

i/Pm(X,~). R(x,D) by

operator

operator

Q_m(X,D)

Define a pseudo-differential

I-QP. I-R

where

opera-

We want to prove that the

is invertible and whose

(right and

left) inverse is given in the form S =

Z

R

,

I=0 where R.

R£

denotes the

~-th power of the pseudo-differential

Note that the pseudo-differential

(-I), hence

S

operator

defines a pseudo-differential

R

operator

is of order at most

operator of finite order

if it exists. To prove the existence of

S

we use the following estimate

concerning the composition of two symbols, which is due to Boutet de Monvel and Kr~e [i].

To state the estimate we introduce the following

auxiliary norm of the symbol of pseudo-differential

operator

358

P(x,D) =

Z P_k(X, D): k=0 -2-

N(P;t) =

2(2n) -k k! ~~k~t2k+l~+~l ((1~l+k),(l~l+k))ID~D

k , ~ ,~ where

t denotes

a parameter.

Then

Monvel and Kr~e [I] shows that N(PQ;t).

the

N(P;t) N(Q;t)

(Definition 1.4.6)

Remark I

de

is a majorant series of

~_~ R £ ~=0 operator by this majoration. Therefore

Thus we have obtained the micro-local left inverse

P by defining Let

pseudo-differential Denote by

of Boutet

(Boutet de Monvel and Kr@e [i] Lemma 1.2).

clearly defines a pseudo-differential

F of

calculations

Pij0(x,D)

F=SQ.

This completes the proof of the theorem.

P(x,D)=(Pij(x,D))l~i,j~ ~ operators.

be an

~x~

Define

m. by max ord P..(x,D). J I~i~£ lj the homogeneous part of Pij(x,D) of order m.j

(which may be identically zero).

Assume that

0 det(Pij(x,~)) ~ 0

in

~ CP*X

.

Then it is easily verified that there exists a unique pseudo-differential

matrix of

operators

E(x,D)

~x£

matrix of

such that

PE = EP = I (£) holds in

~ , where

I (~)

denotes the diagonal matrix of size

whose components are the identity operators. The proof of this statement can be given by the same reasoning as above and we leave the detailed proof to the reader. Remark 2

The above proof of Theorem 2.1.1 shows at the same

time the following proposition on "operational calculus". proposition 2 1 2.

Let

P(x,D)

operator of order at most 0 with

be a pseudo-differential

Po(xO ~)=0_

Assume that

f(s) is

359

a holomorphic

function of

s e@

near the origin,

i.e.,

f(s)= ~ a . s j. j=0 J Then the series >-, a.P(x,D) j defines a pseudo-differential j=0 j operator of finite order (in fact, of order at most 0) near (x0, u).n The proof is clear by the aid of the above introduced N(P,t) of pseudo-differential

formal norm

operator.

In the sequel we denote

~

ajP(x,D) j

by

f(P).

j=0 Now we prove the following striking theorem on the equivalence of two pseudo-differential that

P(x,D)

and Q(x,D)

their principal

operators

and

Q(x,D).

are operator-theoretically

It asserts

equivalent if

symbols coincide and satisfy a suitable condition on

the non-degeneracy,

or in other words it asserts that two pseudo-

differential

equations

equivalent.

Here the equations

9~: P(x,D)u = 0

denote the left

~f-modules

where

~Q

~p

P(x,D)

and

~

and

~: Q(x,D)u -- 0

: P(x,D)u = 0

~f/~p

and

and

~f/~Q

are left ideals generated by

,

are

91:Q(x,D)u -- 0 respectively,

P(x,D) and Q(x,D),

respectively. Theorem 2.1.2 and

Q(x,D)

( Kashiwara

- Kawai

be pseudo-differential

that their principal grad(x ' ~)Pm(X, ~)

symbols

[i] Theorem 7.).

operators of order

Pm(X,~) and

is not parallel to

Qm(X, 7)

m.

S(x,D)

(~ ,0) whenever

of finite order so that

P(x,~

Assume

coincide and that Pm(X, ~)=0.

Then we can find locally two invertible pseudo-differential R(x, D) and

Let

operators

P(x,D)R(x,D)=S(x,D)Q(x,D)

holds. Proof. (x 0, ~0) ing

At first consider the problem in a neighborhood

where

theorem.

Pm(X 0, ~0)#0. Now assume that

of

Then the theorem is clear by the precedPm(X 0, 0 )

= 0.

Then by the assumption

of the theorem we may assume that grad(x,~)Pm(X , ~) is not parallel to

360

(~,0) near

(x0,~ 0)

(2.1.2)

We also assume for the sake of simplicity that 0 ~

P(x,D) =

j 0 x =0, ~0=(0, i,0 ..... 0)

Pj(x,D), Q(x,D) = P0(x,D)

=-~

and ( ~ / ~ X l ) P 0 ~ 0.

differential operator of finite order (2.1.3) holds near

We want to find pseudo0 R(x,D) = ~ Rk(X,D ) so that k= -~

P(x,D)R(x,D) = R(x,D)Q(x,D) (x0, 0 ) .

Using Theorem 1.5.2 following equality for any

we see that

R£(x, ~) should satisfy the

~:

i ~ P. D ~ R k = ~ io~! D ax P0 D~ ~ Rk ~_~ d--FD~ 2_~ £=j+k-I~; " j x ~=k-lx;

Therefore, if we denote by with

P0' i.e.,
~0

the Hamiltonian operator associated

gradx>- IgradxP0,

grad~>,

Rk'S should

satisfy 2~ ~i.J D x~ P0 D~~ Rk ( ~ 0 + P - I)R~ = ~=k+l-l~J

(2.1.4)

k_>~+l _

~

1

D~ P. D ~ R k

~=j+k+l-t~i ~ ! 2

J x

for any By our choice of local coordinate system (2.1.2) the hypersurface i ~I=01

is non-characteristic with respect to

the Cauchy datum

0

for

R£

(~

-I) on

HP0.

{~i=0}

Hence we give

and find a holo-

mprhic solution of (2.1.4) which is homogeneous of order respect to

~

(since the Cauchy datum for

defined in a neighborhood of

{~i=0}

R~

is

independent of

defined to be a solution of (2.1.4) with Cauchy data initial surface

~ ~i=01.

0)

~

and is ~.

I

with

R0

is

on the

Therefore what remains to prove is that

361 Rk'S

satisfy the growth order condition with respect to

k

imposed for the symbols of pseudo-differential operators.

which is (See Defi-

nition 1.4.6.) To prove this fact we use the method of majorant. tion that tor

P

Pj(x, ~)'s

By the assump-

are the symbol of a pseudo-differential opera-

we have (-j)~ I~,I~c -j+l x j f D~ P ~(

~fl~l(l_s)l+l~l '

(2.1.5)

D~ c, ~

for suitable constants IN ~ where

s

and the notation

f>>g

a majorant one of "''' ~n"

~ s=0~

relations

~ > I, for sufficiently large

g

n

+ ~2+'''+

~n-l+ ~i

denotes the relation that the series

as a power series on

f

is

x I, "''' Xn' ~i' "''' ~n -I'

Now it is clear by the estimate (2.1.5) that the proof

is completed if we can find of

and

denotes Xl+'''+x

~3'

Pj <<

(-j)~ I~I'.~ i c-J+l ..... ~i~t(l_s)l+l~l

independent of

~£= *~(s) ~

defined in a neighborhood

so that it satisfies the following

(2.1.6) and ( 2 1 7 ) ,

since

R£'s

are defined successively

by the differential equation (21.4). (2.1.6)

~£< (- ~)! C ~ ly large I ~i

(l-s)

>> ~

C

for sufficient-

Ill .

M (2.1.7)

for some positive constant

=

+ (~-~+ "''+~n ~ -I~t ~ "

+'""+~n

1)I L(s)

~I~I(i_s)~l

k>_~+l (-j)'. I~'. ~ i c - J + l D~< +

2~

~=j+k+l-l~l kk~+l

~,

~,~I (l_s),~L

x ~k(S)l '

362 M

where

is a positive constant depending only on the estimate (2.1.5).

Therefore it is sufficient to find

(2n-l)

d ~g

(2 1.7'){~ - (l-s)}" M }

2M >> l-s

satisfying (2.1.6) and

~(s)

M

ds - l - s ~I

-j+l ~i l (-j)! l~t! c p ~-~ ~--~ 2 I~I ~=j+k+l-l~l (l-s) l=tl

d ~a

ds I~I ~k

ka~+l Y,ote that the right side of (2.1.7') 2M ~ l-s ~=j+k+l-~ kZf+l

is equal to

(-j):c-J+l(~+2n-l) p d~ (l_s)~ K2v ds 9 ~ k

since

--~_~i~l!~.~i = (~+2n_l)l~l I~I=> holds. Therefore, fixing the constant sufficient to find

(2.1.7")

~%'s

(1- 1_--77) ~ s

>> -2b-

a
and

sufficiently large, it is

satisfying

1-s

~ (-j)!c -j+l (~+2n-l)~ d ~ *=j+k+l-v (1-s)V ~ d--~ ~k k_>~+l

l-s

where

~

b

are some positive constants.

To facilitste the calculations we define

~(t)

by

~£((l-a) t)

and rewrite (2.1.7") as follows: dR (2.1.8) dt

b (l-t) ~ >>2b

~_~ &j+k+l-w k~+l

(-j)! c -j÷l ~+2n-l~ dV~k (l_t)~a+l ( (l-a)~ 2~ ) dt~

T h e r e f o r e i t i s s u f f i c i e n t t o f i n d y1's

<- !

(2.1.6')

C

satisfying

L

and (2.1.8')

--

(1-t)(41UC

f o r some c o n s t a n t s b , c satisfy

-1

b -

(2.1.6')

and

z

k+l-l-v)!cZ-v lik+l-v:l-t)p+lg~

(3.

and ( 2 . 1 - 8 ' ) ,

We now s e e k f o r

k dv9k ) (C d t v I .

yL(tO1s which

assuming t h a t t h e y have t h e f o l l o w i n g

form:

where

R

and

N

a r e suitable constants.

t h e following estimate (2.1.10),

I n f a c t , i f we can prove

t h e n we have t h e r e q u i r e d

Y e w 's

i n t h e form (2 .1.9) :

Note t h a t t h e t r i v i a l f a c t t h a t

holds i f (2.1.11)

prq.

By t h i s f a c t we s e e t h a t t h e f o l l o w i n g e s t i m a t e

i s s u f f i c i e n t t o prove ( 2 . 1 . 1 0 )

-Nt + holds.

2

-> -Nk + V +

i f we t a k e N 1

so t h a t

364

In the sequel we take

N=5

for example.

To prove (2.1.11)

with

N=5, it is sufficient to prove

(2.1.12)

(k+l-£- ~) I,(-k) I (_5k+>)~ .......R£_ k (-~)(-~)! (-5k)! f!k+l-v £+l~k!O

A~ =

is dominated by a constant independent of ~.

To prove this fact

it is sufficient to prove the sequence

(2.1.13)

Bg =

~

(-4j-5~-5) ! (-j-L-I)~

j=0

(- ~-i) ! (-5j-5f-5)! R j

is dominated by a constant independent of

~, since we have the follow-

ing inequalities: (2.1.14)

A£ = O!v_<j+2 0<j <- ~-i

(,~+2-~) ! (-j-~-l)! (-~) (-~)!

(-5J -5~-5+~! (-5j-5~-5)! R

l+j

(-4j-5 £-3)'. (-j-~-l) ! (-~) (-~)! (-5j-5~-5)! R l+j

!3 Z j=0

Z 7--5 R - i--1

(-4j-5~-5)! (-j-£-l)! (-~-i)[ (-5j-5~-5)!R j

= 75 B£ R

On the other hand we have the following inequality:

(2.1.15)

B~_ I = i+ I -~=~0 I p (_4j_5~_4) 1.(_j_f_l).~ j (-@)(-~-i)! (-5j-B~-B)!R j

Therefore combining (2 1.12)~(2 is bounded as far as

R

i.~5 R -<

B~

1 15) we see that the sequence

B%

is sufficiently large and this completes the

proof of the theorem. Remark i.

It is clear by the method of the proof that we can

365

take

R(x,D)=S(x,D)

in the statement of Theorem 2.1.2

at least at

"generic points", i.e., the points where grad(x,~)Pm(X,~ ) parallel to (7,0).

Moreover note that grad~PmCX, ~) ~ 0

under the assumption that (2.1.16)

is not

m#0.

if

Pm(X, ~) # 0

In fact we have

= m Pm(X, 7)

by Euler's identity for homogeneous functions Remark 2

If we perform the contact transformation on P*X

first and assume that the principal symbol of the operator

at

P(x,D)

is

n' then we can prove Theorem 2.1.2 in an analogous way to the proof of Theorem 2.1.1.

The only difference from the proof of Theorem 2.1.1

is that the matrix

R(x,D')

of pseudo-differential operators

used

there consists of pseudo-differential operators of finite order in the present case since the matrix tors

M(x,D') of pseudo-differential opera-

used there reduces to a single pseudo-differential operator of

order at most

0

in the present case.

But here we have preferred

the proof given above which is more elementary in its nature, as we have taken into account the fundamental nature of Theorem

2.1.2

in our reasonings in the later sections.

2.2

Theorems on division of pseudo-differential operators.

In this section we prepare some lemmas concerning the division of pseudo-differential operators.

They play an essential role when we

investigate the algebraic structure of the sheaf of pseudo-differential operators.

(Cf. any textbook of the theory of functions of

several complex variables.) We first prove a Sp~th-type theorem for pseudo-differential operators of finite order.

366 Theorem 2.2.1. of finite order

Let

m

(0;i,0,--',0)

order

S(x,D)

of

(x; ~) =

Denote its principal symbol by

Pm(X; ~)

~n = 0.

defined in

~'

~'

operator of finite

can be uniquely written in the form

S(x,D) = Q(x,D)P(x,D) + R(x,D), Q(x,D)

and

order defined in

R(x,D) ~'

are pseudo-differential

and

R(x,D)

operators of finite

has the form

p-I (j) ~ R (x,D')D~ j=0

(2.2.2) Here

and

Then we can find a neighborhood

such that any pseudo-differential

(2.2.1) where

~

operator

Pm(0"l,,0,.--,0, ~n)/ ~ pn is holomorphic and never vanishes

in a neighborhood of of

be a pseudo-differential

defined in a neighborhood

(0; I, 0, "'', 0)E P ~ . assume that

P(x,D)

D'

denotes

(DI,-..,Dn_I).

In other words

[ ?l(Xn),[ ~2(Xn),...,[~p(Xn),R(x,D)]...]] real analytic functions Proof.

(x n)}j=l P

~j

The assumption on

P (x

vanishes for any set of

depending only on 7)

x n.

implies that the pseudo-

m

differential operator

P(x,D)

(2.2.3) where

has the form

Pl(x,D) + P2(x,D)D~ PI(X,D)

and

P2(x,D)

are pSeudo-differential

finite order in a neighborhood of

(0;I,0,''',0),

tic there and the principal symbol of of degree smaller than

p,

PI(X,D)

which vanishes for

Since an elliptic operator is invertible from the beginning that

,

P2(x,D) m 1

Np (Pl,t)

P2(x,D)

is ellip-

is a polynomial in

~n

(x; ~') = (0;i,0,''',0).

(Theorem 2.1.1), we may assume

and that

Then it is clear that the principal symbol of identically for

operators of

(x; 7) = (0;i,0,''',0, ~n )

order PI(X,D) ~ p. PI(X,D)

vanishes

and it implies that

becomes as small as we want for suffieintly small

~'

and

367

Itl, where we fix the domain in which

Here

2 (2n) -kk! ID~D ~p ~ (x, D) I t2k+~+~i ( (l~+k)! (l~i+k)!)sup ( x , ~ ) ~ " x ~ p-K

N~'(PI ;t) = ~ P k~O

~,~z0 where

~n varies.

I~l=l

Pp_k(X,D)

denotes the homogeneous part of order

the pseudo-differential operator assumed to be equal to

0

and

PI(X,D). ~'

(Cf. # 2 . 1

in

There

D p

of is

is not written explicitly.)

After this normalization of the operator pseudo-differential operators

p-k

~(x,D)

P(x,D)

and R(x,D)

we seek for

so that they

satisfy (2.2.4) If such and

S(x,D) =~(x,D)(D~ + PI(X,D)) + R(x,D) Q(x,D)

R(x,D)

Q(x,D)

to

and

satisfy Q(x,D)

Now we find sive

(2.2.1)

are obtained, then clearly.

~(x,D)

(2.2.5) where D

Q(x,D)=~(x,D)P21(x,D)

In the sequel we abbreviate

for simplicity. Q(x,D)

and

R(x,D)

approximation, that is , we set

and

in

R(x,D)

satisfying

(2.2.4) by succes-

Q0(x,D) = 0

and define

Qk(X,D)

(k ~ i) by the following recursion formula S(x,D) = Qk(X,D)D~ + Qk_l(X,D)e~(x,D) + Rk(X,D),

Rk(X,D)

has the form (2 2.2), i.e.,

~(x,D)

of degree smaller than p. n In the sequel we use a rather rough notation

for a pseudo-differential operator

T(x,D)

~JT(x~D) I Dn=0 ~DnJ

of the form

in order to designate the pseudo-differential operator Clearly we can take as

is a polynomial

Rk(X,D )

P~I i_~_ ~J (S(x,D) (x,D)PI(X,D)) I DJ j-~-0 j! ~D j -Qk-i Dn=0 n n

~T£(x,D')D~

~0

j!Tj(x,D').

368

and as

Qk(X,D) (S(x,D)-Qk_I(X,D)PI(X,D)+Rk(X,D))DnP

Here the above definition of

Qk(X,D)

makes sense, since

S(x,D) - Qk.I(X,D)PI(X,D) + Rk(X,D ) has the form Sk(X,D )

Qk

Sk(X,D )

{Qk(X,D)} (2.2.6)

as

Rk(X,D), that is, the above definition means Qk(X,D).

Moreover, denoting by

S(x,D), we easily see by induction on

is at most

order of

for a pseudo-differential operator

by the choice of

that we take order of

Sk(X,D)D ~

~ - p

PI(X,D) and

and that of

is at most

IRk(X,D)}

p.

Rk(X,D )

k

~

the

that the order of

is at most ~, since

the

Thus we have two sequences

such that

(Qk+I(X,D)-Qk(X,D))D~ + (Rk+l(X,D)-Rk(X,D)) = -(Qk(X,D) - Qk_I(X,D))PI(X,D).

We denote by respectively.

qk

and

Clearly

rk

(Qk+l(X,D)-Qk(X,D)) and (Rk+l(X,D)-Rk(X,D))

qkD~+rk = -qk_iPl

holds.

On the other hand

Cauchy's inequality for holomorphic functions immediately (2.2.7)

implies that

N~(rk;t)<
for a suitable choice of ~'. (2.2.7) (2.2.8)

obviously means that ; ~' N£~' (qkDnP t)~(p+l)N L (qk_iPl (x,D) ;t)

On the other hand, in virtue of the Schwarz lemma for holomorphic functions in one variable we easily see that (2.2.9)

N~_p(qk;t)<
holds for a polynomial Cp(t)

in t beginning with

l~nlt

and being of

369 degree at most

p, which depends only on

p and ~n~ " Therefore,

combining (2.2.8) and (22.9), we conclude &O v

(2.2.10)

~O v

~0 w

N~_p (qk; t) << (p+l)Nl (qk_iPl (x, D) ;t)+Cp (t)Nl_p (qk; t) v

~t

~v

<< (p+l) N ~ (PI (x, D) ;t)Nf_p (qk-I ;t)+Cp (t) N£_p (qk; t) As remarked above, making

l~n~<~

for fixed ~>0, we can take ~0' and

~v

so

t= £ > 0

small that

(p+l)Np (Pl(X,D);g) (l-Cp(£)) < 1/2.

Then

we have (2.2.11)

N~_p(qk ;~' £) ~ 2-kN ~'£_p(q0;g) = 2

Therefore it is clear that

-k ~' N£_p(QI(X,D),'£).

~ qk defines a pseudo-differential k=0 Moreover (2.2.7) proves that

operator of finite order.

N~ (rk, g) ~ CpN£_p(qk, e) for a suitable constant (2.2.11)

c P

depending only on

p

and ~'

Therefore

implies that 2-~ k=0

rk

defines a pseudo-differential operator of finite order R(x,D). Obviously

R(x,D)

has the form (2 2.2).

Moreover the uniqueness of

the above defined pseudo-differential operators

Q(x,D) and

R(x,D)

follows from the uniqueness assertion of the Sp~th theorem for holomorphic functions. In fact, if there exist

Q(x,D) and

R(x,D)

satisfying

Q(x,D)P(x,D) + R(x,D) = 0, the application of the Spgth theorem for holomorphic functions to the principal symbol obviously proves that Q(x,D) = R(x,D) = 0.

Thus the proof of the theorem is completed.

As in the case of holomorphic functions, Theorem 2.2.1 immediately proves a Weierstrass-type preparation theorem for pseudodifferential operators of finite order, that is,

370 Theorem 2.2.2

Assume that a pseudo-differential

satisfies the assumption of Theorem 2 2.1.

operator

P(x,D)

Then we can write

P(x,D)

in a unique way in the form (2.2.12)

where

P(x,D) = Q(x,D)W(x,D),

Q(x,D)

and

W(x,D)

are pseudo-differential

order defined in a neighborhood tic there, and

W(x,D)

Dp +

"

n

W (j)(x,D')

defined in Proof. 2.2.1 and

of

(0;i,0 .... 0),

p-i ~

w(J)(x,D')D j n

j=0

is a pseudo-differential

operator of order ~p-j

~o whose principal symbol vanishes for (x;~')=(0;l,0 ..... 0). Take

Dp n

as

S(x,D)

in Theorem 2.2.1.

asserts the existence of pseudo-differential p-i R(x,D) = ~ R(J)(x,D')D j such that j=0 n

(2.2 14)

Then Theorem operators

n

"

The method of the proof of Theorem 2.2.1 is at most

p-m

also shows that the order of

and that of R(x,D)

is at most

p.

by

q(x,~) the homogeneous part of Q'(x,~) of order p-m p-i and by ~ r(J)(x'~')~ jn the homogeneous part of R(x,~) order

p j--0 in 7'

respect to

~'.

i.e., r (j) (x, 7' )

We denote in of

is homogeneous of order p-j

with

Then, comparing the homogeneous parts of order

of the both sides of (2.2.14),

(2.2.15)

Q'(x,D)

D p = Q'(x,D)P(x,D) + R(x,D)

"

Q'(x,D)

Q(x,D) is ellip-

has the form

(2.2 13)

Here

~

operators of finite

~ P = q(x,~)Pm(X , ~ ) +

p

we have

p-i r(j)

(x,7'

j=0 Then the Taylor expansion of the right hand side of (2.2.15)

in a

in

371

neighborhood of

(x;~)=(0;l,0,..~0) shows that

(2.2.16)

q(0;l,0 ..... 0) = I/Pm(0;l,0 ..... 0)

and that

r(J)(0;l,0 ..... 0) = 0 (j=0 ..... p-l).

Q'(x,D)

is elliptic by the definition

that there exists the inverse operator (2.2.14)

(2.2.16)

implies that

Hence Theorem 2.2.1 Q(x,D)

of

Q'(x,D).

asserts Then

proves that P(x,D) = Q(x,D)(D~ - R(x,D)).

Therefore, if we define

W(x,D)

by

D p - R(x,D), then all the n

requirements concerning

Q(x,D)

and

uniqueness of the decomposition of

W(x,D)

are satisfied.

The

P(x,D) in the form (2.2.12) is

clear. Now we extend Theorem 2 2 1 tothe case of pseudo-differential operators of infinite order assuming that the divisor finite order, that is, Theorem 2 2 3. finite order 2 2 1.

P(x,D)

we have the following. Assume that a pseudo-differential operator of

P(x,D)

satisfies the assumptions posed in Theorem

Then we can find a neighborhood

~'

of

(x; ~)=(0;I,0 ..... 0)

such that any pseudo-differential operator (of infinite order) defined in

~'

S(x,D) = Q(x,D)P(x,D) + R(x,D),

Q(x,D)

and

R(x,D)

infinite order) defined in

"

Proof. ~

are pseudo-differential operators (of ~'

p-i ~ j=0

(2 2 2')

order

S(x,D)

can be written uniquely in the form

(2.2.1') where

is of

in

Denote by D.

and

R(x,D)

has the form

R(J)(x,D')D j

S6(x,D)

n

the homogeneous part of

Then we can apply Theorem 2.2.1

to

S(x,D) of S%(x,D)

and

372

find

Q~(x, D)

and

R~(x, D)

such that

S£(x,D) = Q£(x,D)P(x,D) + R~(x,D) , where R~(x,D)

has the form ( 2 2 2 ) .

If

~Q)(x,D)

and

R~(x, D)

~= -~

define pseudo-differential operators Q(x,D) then they clearly satisfy (2.2.1') gence of the series

Qf(x,D )

~=-~

and R(x,D)

and (2.2.2').

and

~

~=-~

respectively,

To prove the conver-

R~(x, D), we use the follow-

~=-~

ing lemma. Lemma 22.4.

Let

~A~ (x, D)I7=_~

and

IB ~(x, D)I~=_~

sequences of pseudo-differential operators defined in Assume that

A~(x,D)

the order of &O >0

B(x,D)

is homogeneous of order is at most

~

~

be two

~ ~ P*X.

with respect to

D,

and that there exists a constant

such that

(22.17)

N£Ot (A£;t) =>N~f.,O' (B~;t)

holds for any polydisc

~'C ~

Itl < gO'

Suppose that

and for any

~

and

t

satisfying

~-~ Ai(x,D) defines a pseudo-differential ~=-~

operator

A(x,D) (of infinite order).

Then

~- B~(x,D)

also defines

~= -~

a pseudo-differential operator (of infinite order). Admitting this lemma for a while, we will show how to apply it to our situation. we use

Q~(x,D)

Following the notation of the proof of Theorem 2 2.1, as A£(x,D)

in the lemma.

That is, A£(x,D)

is

defined by (2.2.5') where

SI(x,D) = A~(x,D)D p + R~(x,D) ,

Rl~(X,D)

has the form

(2 2 2).

p-i aj (S~(x,D) - ~ _~i j=0 j! aD j n

Then

A~(x,D)

S~(x,D) ID =0 n

is given by

Dj ) D- p n n

373

~_~ A %(x, D)

hence

is given by

p-i i ~J (S(x, D) - 7_. S(x,D) I D j ) D -p j=0 j! ~D j D =0 n n n

Therefore

~

A£(x,D)

H

obviously defines a pseudo-differential

=-~

operator, since

S(x,D)

is a pseudo-differential operator.

Moreover, it is clear by the above definition that homogeneous of order

~-p.

(Here the order of

A£(x,D)

A~(x,D) is

is

~-p, not

as required in the lemma, but this translation of the suffix does not matter).

On the other hand the method of the proof of (2.2.10)

shows that N~

~-p holds for

(Q~;t) <__ ~_~ N ~' (Qf;t)/2 k = N w' (AL;t) k=l ~-p ~-p

Itl < ~

for a suitable choice of

ciently small polydisc

~'

(2.2.7)

E > 0 for any suffi-

also shows that

N~ (R~'t), <= N~_p(A~;t) for

It I < ~.. Therefore

2~ Q2(x, D)

L =-~ ential operators Lemma

2.2.4.

Q(x,D)

and

~

RI(x,D) define pseudo-differ-

J=--

and

R(x,D)

respectively in virtue of

Thus the proof of the theorem will be completed except

for the uniqueness of the division in the form (2.2.1') Lemma 2.2 4.

Now

if we prove

we go on to the proof of the lemma.

Proof of Lemma 2 2 4.

Denote by a£

sup

[A~(x,~)~ where

(x,~)~' ~'f~0.

Since

A(x,D)=

_~# --~£(x'D)

is a pseudo-differential

operator defined in ~, the following estimates hold by the definition:

374

(2.2.18)

For any

~ ~ 0

(2.2.19)

There exists a constant ~

there exists

C1 > 0

Ci*(-i) !

If we choose a ploydisc

~$ such that

for ~ $ O.

~"~'

, then (2.2.18) combined with

Cauchy's inequality proves that for any ~f

such that

g >0

we can find

£>0

and

such that

N~co"(A~;t) <

(2.2.20)

~--

~ '. ~I~

= ~,~>__0

I~l!t~l!~!

_ (i-~) "2n ~f~

Then assumption (2.2.17)

i~ - I~+~ i

for ~_> *S"

implies that

N~ (Bi;t) __< (i-~) -2n jPT

(2.2.21)

holds for ~__>~ Denote by k in D

t I~+~

of

for sufficiently small B~(Xk",D)

B~(x,D)

I t~

and Bk(X,D ) the homogeneous part of order

and ~__2~_~B~(x, D)

makes sense at least formally).

respectively.

We also denote

(The latter

sup

I BT~(x,~)I

(x,~)~" l~l--i and

sup

(x,~)~"

IBk(X,~)l

by

b~

and

bk

respectively.

Then (2.2.21)

proves by the definition that

t.-2n ~t-2( I -k) ( i-k)~ b~ __< (~-F~ ~ ! holds for

~.

Hence for any

~

we can find

~E

such that

375

(2.2.22)

~ t -2(~ -k)k!(~-k)!

k!b k <__ (i-~) -2n ~>k

~ !

~ k (i-7) ~ ~.-1 = (l-~)-2nt2k(--) t for

k > ~

Now for any

> 0

we firstly fix

so that

t=t 0

It01 = min(~ , ~) , secondly fix

~

so that 2

to = min(--~-- , ~) and lastly choose the corresponding constant

~

Then (2.2.22) immediately proves that for any exist

C2

and

kg

(2.2.23)

~ > 0

there

such that

k!b k ~ C2 Kk

for

k ~ k~.

On the other hand (2.2.19) proves in the same way as above that

~,,(B ;t)~ (l_~)-2n(_~)!Cl~t-

(2.2.24) holds for

~$ 0.

Taking (2.2.18)

into account we may also find

C2 > 0

~' ~ .. t.-2n C~ N~ (B ; t ) ~ ( l - ~ ) ~--~

(2.2.25) for any ~ > 0 . Now for any (2.2.26)

k < 0

we have

. O k C_~t_2 be <-- (l-~)-2n(~=~ i (~-k)(- ~)[(~-k)!

2o

~=0

such that

376

Denoting by

II

and

12

the first and the second series in the

parenthesis, we want to prove that for a suitable constant

t=t 0 < ~

and a

C3 Ii, 12 ~ (-k)! C3 k

hold.

We first investigate

Ii.

Clearly

I I ~ (-k)Clk(-k)! ~ c4k(-k)! for sufficiently large

C4

As for

12, we fix

t = to = rain(8 , --~) , 2 to S .... 2

and choose the corresponding constant C5 >

0

~

as before.

Then taking

so that i i C5t---~ - < ~-

we find by (2.2.21) and (2.2.25) that

(2.2.27)

~ C2~(~ -k)! t~ k 12 (-k)! <= ~=0 to (-k) !

~(~ -k) ~(C5t0)2k + C; 2k ~ 2~ ~=~f to ~! (-k)!

Since ~-~ ~i,~2~0 converges absolutely for

( ~i + ~2 )!

~I £2 x I x2

~i [ ~2 !

IXll < 1/2

and

Ix2~ < 1/2, the second term

in the right hand side of (2.2.27) is smaller than C6C52k for

C 6 > 0.

The first term there is clearly smaller than C7 (t~) -2k

377

All summing up, we have proved that there (2.2 2 8 )

exists

C3 > 0

such that

b k ~ (-k)!C 3

holds for

k ~ 0

The estimates

(2.2.23)

and (2.2.28)

imply that

B£(x, D)

~=-~ defines a pseudo-differential

operator (of infinite order).

completes the proof of Lemma 2.2.4 existence of pseudo-differential

This

and at the same time proves the

operators

Q(x,D)

and

R(x,D)

Finally we prove the uniqueness of the division of

S(x,D)

satisfying (2.2.1') and (2.22').

P(x,D)

in the form (2 2 1').

In order to diminish the technical

difficulties we employ here the matrix notation, that

P(x,D)

by

that is, we assume

has the form (DnI(P)- A(x,D')),

where

I (p) denotes the identity matrix of size

>-.A.(x,D')

is a matrix of pseudo-differential

p

and

A(x,D') =

operators such that

j~l j Al(X, ~')l(x;~')=(0;l,0 ..... 0) A.(x,D') j

is a nilpotent matrix.

is homogeneous of order

P(x,D)

j

in

D'

is obviously possible by Theorem 2 2 2

Here

Such representation of Moreover also by

some technical reasons we prove the uniqueness by dividing P(x,D)

from the left.

S(x,D) by

Clearly this does not present any trouble,

since by taking the adjoint of both sides of (2.2.1') we can reduce the problem to the present situation. Q(x,D) =

~QJ(x,D')D j j$0 n

and

R(x,D')

Now assume that satisfy

378

(2.2.29)

P(x,D)Q(x,D) = R(x,D')

Then clearly we have ~ QJ(x,D ') = ('~x

- A(x, D')) QJ+I (x, D'),

j>l =

n

by c o m p a r i n g t h e c o e f f i c i e n t s h a v e f o r any

(2.2.30)

in

(2.2

Therefore, we

29).

A(x,D,))kQj+k(x,D ,) .

QJ(x,D') = ( ~x n

the homogeneous part of p.

Dj + l n

j,k>l

Now we prove by letting

any

of

k

tend to infinity that

QJ(x,D')

of order

p

in

D',

QJ (x,D') p vanishes for

For that purpose we use the following notation:

Let

P(x,D)=(P(J'k)(x,D))

be a

ential operators of order at most ~. simplicity that norm-matrix

matrix of pseudo-differ-

(In the sequel we say for

P(x,D) is of order at most ~).

N~(P; t)

formal norm of

p×p

of

Then we define the

by the matrix consisting of the

P(x, D)

P(J'k)(x,D),i e., (NL(P(J'k);t)).

N)(P;t ) >>N£(Q;t)

if any component of

of the corresponding component of

N£(P;t)

We also denote is a majorant series

N£(Q;t).

Using these notations we easily see that (2.2 BI)

N~I+~ 2 (PIP2,t). <~ N~I(PI;t)N~2(P2;t )

holds for any pseudo-differential operator most

~j (j=l,2).

m

of order at

In fact

N11+~2((PlP2)(P'q);t)

<< r=l ~'

P.(x,D) J

_ . ~ _(p,r)_(r,q) = N~l+~2(r~=l~l ~2 ;t) m

N~I+L2(P P' r)P2(r'q) ;t)<
379

= (N£1(PI;t)N~2(P2;t))(P'q) In passing,(2 2 30) proves that (2.2.32) QJ(x,D')=(-A(x,D'))kQj+k(x,D')+ where

f

is a monomial of length

3k_l ~ fs(A(x,D'),QJ+k(x,D'),Dn), s=l

(k+l)

in

A(x,D')

QJ+k(x,D')

s

and

Dn, which contains

D

and contains QJ+k(x,D') once On the n other hand , Cauchy's inequality and the assumption that ~ QJ(x,D')D~ is a pseudo-differential operator imply that (2.2.33)

N~ (QJ+k; t) <<(l-(~))-2n sup IQj~+k.ix, ?7 )I C ~+k+j+l -2n) <
for

£+k+j ~ 0

((_~.k.j)!C-( ~ +k+J)(l_(!))-2n ) holds for constants

C

and

6>0.

full use of the assumption that operator.) fixed

p

is a pseudo-differential

and (2.2.33) prove that for any

j

(2 . 2.34)

sup ~ _~

I QJCx p.,~)i

'

Nk(Ak;t)N ~ (QjTk;t) t-2 ( ~ +k-p) ( ~ +k-p) !

2~

_zP-k 3k_l +

~+k+j < 0

(Note that here we need not make

Q(x,D)

Now (2.2.31),(2.2.32) and

for

Z

s=l

fs(NI(A;t)'N£ (QJ~+k;t) , Nl(Dn,t)))

380 ~_~ 3kCl(~,t)kc ~+k+j+l -2 ~+k-p)(~+k-p)I ~>__max(p-k, -j-k) (l-(t))2nt ( (£+k+J)'t

+

~_ 3kCl (t~,t)kc - (£+k+j) (i_ (t)) -2n(.j[-k-j) !(~+k-p).1 p-kK~_<-j -k-i

Here the symbol

A < B

means that any component of the matrix

smaller than the corresponding component of Al(X , ~')

B.

A

is

Since the matrix

is nilpotent at (x; ~')=(0;i,0 ..... 0,0), by the assumption

and since we consider the problem near take, for any fixed shrinking and take

~

CI(~;t)P

sufficiently small

~

by letting

t, constant

sufficiently small k

(x; ~)=(0;i,0 ..... 0,0), we may as small as we please by

Therefore, if we fix so that

(3CIC)m ~ i,

tend to infinity in (2.2.34)

P sup

{ Q!(x,

t < E/2

we conclude

that for any fixed j

and

~') ~ = O.

Note that the summation

3kc~c- (2

+k+j) (I-~) -2n(~-k-j) ! (%+k-p)!

p-k~ ~-j -k-i contains only

(-j -p)

terms.

Thus we have proved that any pseudo-differential operator (of infinite order)

Q(x,D)= ~ Q J ( x , D ' ) D j j~0 n

satisfying (2.2.29)

identically zero in a neighborhood of (0;1,0, .... 0).

should be

This proves the

uniqueness of the division (2.2.1') and at the same time completes the proof of the theorem. Remark i.

Assume that

S(x,D)

and

P(x,D)

are

linear differ-

ential operators satisfying conditions in Theorem 2.2.3 (and that S(x,D)

is of finite order).

Then the pseudo-differential operator

381

R(x,D)

obtained in Theorem 2.2.3 (and also in Theorem 2.2.1

S(x,D)

is of finite order) turns out to be linear differential

operator

(of finite order if

S(x,D)

is so).

To prove this, we

consider the problem introducing an auxiliary variable denote

~/ ~ t

sequel.

For the sake of clarity we denote

On the other hand, since

S(X, Dx)

ential operators by the assumption, =(x,t;0,1).

Hence we can divide

way in the form of (2.2.1') =~,0;0,I). (2.2.1')

S(X,Dx) by

In fact, in

it follows from the uniqueness

nor

R

depends on

are defined even for ~=0 Q(X,Dx)

and

R(X,Dx)

Let

by

P(x,D x)

Dx

in the

are differ-

P(X,Dx) ~

Q

of

in the unique (x,t; 7' ~)

nor the residue

of

(t, ~)

R

in

~'={(x,t; ~ , ~ ) E ~ ; ~ # 0 1

of the division of the form

and the analyticity of the coefficients Q

D

t.

i.e., defined even for (x,t; 7' ~ )

Moreover neither the quotient

that neither

and

in a neighborhood

depends on (t, T).

if

Q

and in

~ .

and, by the definition,

are linear differential

R

(2.2.1')

clearly implies Thus

Q

and

R

this means that

operators which

satisfy S (x, Dx) = Q (x, Dx) P (x, Dx) +R (x, Dx) with

R(X, Dx)

satisfying

Remark 2.

(2.2.2').

Theorem 2.2.3 easily gives the following proposition

concerning the "operational calculus" which extends Proposition 2 1 . 2 . Note that Proposition 2.2.5 below is closely related to Theorem 5.2.3. In fact Theorem 5.2.3 was our first motivation

for obtaining the

following proposition. Proposition 2 , 2 5 . differential component of

Let

P(x,D)

operators defined near

be a

pxp

matrix of pseudo-

(x; ~)=(0; ~)

such that any

(P(x,D)) m, the m-th power of the matrix

P(x,D),

is of

382

order at most m-~ . Let variable

t

which satisfies

(2.2.35) where

f(t) be an entire function of one complex the following condition

li--~ r -P log M(r) = 0, r-~

M(r)= max If(t) I Itl=r Then

and

~=l+m_ ~

f(P(x,D)) defines a matrix of pseudo-differential

(of infinite order) in a neighborhood denotes,

(2 2.B5).

of

(0; ~).

Here

operators

f(P(x,D))

as usual, aj(P(x,D)) j

,

if

f(t) =

j~O Proof.

As is well-known,

the assumption on

~ ajt j j=O f(t)

is equivalent

to saying that (2.2.36)

iim j ~a~ ~/j = 0 J j-~

'holds. Now define entire functions

gj(s), j=O ..... m-l, by k

gj(s) = ~ amk+j s k=0 Clearly we have m-I f(t) = ~ j=O On the other hand condition j=O ..... m-l, Here

gj(tm) t j

(2.2.36) proves that

is a (pseudo-)differential

operator

gj(Dxo) , (of infinite order).

x 0 denotes an auxiliary variable which we have introduced for

convenience. differential

Then Theorem 2.2.3 operators

in a neighborhood

proves the existence of pseudo-

Qj(xo,X,Dxo, Dx)

and

of (x0,x ; 70' ~)=(0,0;0,~),

R.j(Xo,X,Dxo ,Dx) defined such that

383

J

J

'

where

R. is a polymial in D of degree at most m - ~ - i The J x0 choice of gj(s) obviously implies that R. depends neither on J Dx0 nor on

x0

and that

R.j

is nothing but

gj(P(X, Dx)m), which

is evident from the method of the proof of Theorem 2.2.3. clearly

Then

f(P(x,D )) is given by X

m-i ~- gj (P(X,Dx)m)p(x,Dx)j

j=0 This completes the proof of the proposition. Remark 3.

The method of the proof of Theorem 2.2.1

and Theorem

2.2.3 immediately shows the following Theorem 2 . 2 6 .

If

Pm(Xl,0; ~0)/x~

vanishes in a neighborhood pseudo-differential

of

operator

Xl=0 ,

is holomorphic

then we can divide by P(x,D) any

(of finite order)

ing form in the unique way in a neighborhood (2.2.1")

where

S(x,D) = Q(x,D)P(x,D)

Q(x,D)

order if

and

S(x,D)

R(x,D) is so)

~. x j R (j)(x',D).

j=0 Here

x' denotes

(x 2 ..... Xn) .

in the follow-

of (Xl,0;~0):

+ R(x,D)

has the form

p-1

(2.2.2")

S(x,D)

are pseudo-differential and R(x,D)

and never

, operators

(of finite

384

3.

Algebraic properties of the sheaf of pseudo-differential operators As we lay stress in Sato [2] and in Kashiwara [i],

a system of pseudo-differential equations as a

we understand

~x-MOdule.

(Cf.

Guillemin, Quillen and Sternberg [i] Quillen [I] and Spencer [I], [2].) In this section, we will prove the finiteness properties of 2X'

which

we need in the study of systems of pseudo-differential equations. 3.1. Let

Pseudo-differential operators with holomorphic parameters.

f : X ---~Y be a smooth holomorphic map, with fibre dimension

and

P*(X/Y) = Px(X ~ X) Y Then, in the same way as

n

be the relative cotangent projective bundle. ~X ,

the sheaf

~X /Y = ~ X ] ~ X ~xf~X/Y

has

the structure of sheaf of rings. There are a canonical projection P~

- P*Y × X ~+ P*(X/Y) Y

and a canonical injection -I P

~X/Y -'-> ~ X l P*X-P*YxX Y (by the integration of the holomorphic microfunction). It is easy to see that the image of this homomorphism is a subsheaf of

~X

consisting of the pseudo-differential operators

P(x, D x) ~ ~ X

such that

every holomorphic function

for

P(x, Dx) ~(f(x)) = ~(f(x))P(x, Dx) ~

on

Y. n

Set

~X/Y,m = OXIX~X,m ~xaX/Y

?X/Y,m = *---lira @X/Y,m/~X/Y,k k The "product" homomorphism transforms

,

~xf/Y = m~j ~ ~X/Y,m '

~X/Y = ~j ~ X / Y , m m

~X/Y,ml × ~ X/Y,m 2 --9 ~X/Y,ml+m2 /%

~X/Y,m I ~X/Y

and

~X/Y

A

'

'

A

@ X/Y ,m2 --~ @X/Y,ml+m 2

have structures of filtered Rings, and their

385

~ ~,O'P*X/y(m) : m~

graduations are both isomorphic to

m~Z

(~X/Y,m/~X/Y,m_l)

-~ O

mEZ

(~X/Y,m/~X/Y,m_l)

-"~ - * ~t~ ((~ p ,,,~'~/Y(m))" m~.Z

Note that

r~*X/y(m)

is the m-tiple of a natural relatively ample

line bundle. For every holomorphic map P*(X'/Y') = Y' X P*(X/Y) Y where

p

Y' -->Y,

if we set

and we can define the map

is the projection

P*(X'/Y') --+ P*(X/Y).

P*(X/pt) = P'X,

X' = Y'× X, then Y -I P --/~XIY--~ ~X'/Y'' We have

~X/pt = ~X"

We call a section of ~X/Y a pseudo-differential operator (on X/Y), f a section of ~ X / Y a pseudo-differential operator of finite order (on

X/Y)

tor (on

by

and a section of

~X/Y

X/Y).

We abbreviate in the sequel ^ f ~, ~, ~, ? , "'" 3.2.

toms.

a formal pseudo-differential opera-

O'p~/y,

^ ~X/Y'

~X/Y'

f ~ X/Y' "'"

Properties of the Ring of formal pseudo-differential opera-

Let

~i (resp. ~ )

be a ~

f_

(reap. ~-)Module.

The filtration

A

of

~(resp.

sheaves

~)

~k}k6~

is, by definition, the increasing sequence of subof

~

(reap. !~klkEZ

~k+m (reap. ~m~kCik+m ) by

~

(reap. ~ )

~k/~k_l

~ ~(k) = ~

eemma 3.2.1. of

the

Let

(~ f)g (reap. ~£)

~-Module

~/~-i

~m~k

such that

~m~k

C

= ~ k + m ) . We denote

(resp. % / ~ _ i

).

Note that

0 ~(k). ~

A

(reap. ~

)

be a sub-~ f- (resp.~-)Module

with filtration defined by

~/~k = #~?L Then

(afortiori,

of ~ )

(reap. ~/ik = ~ n

(~k) )"

386

i)

If

~

=

~,

2)

If

~

(resp.

then ~

is a coherent Proof.

)

~

--->~

is finitely generated,

~ =~fL

Q 6 ~f~ . Then, there is

the residue class of

Q.

~

(resp. ~

)

and

P

6 ~/l

j

.

It is clear that

^ ~ ~ 0

is

such that

Q

such that

A. -A. J J

Assume Q =

Aj E ~

then

~-Module.

At first, suppose that

Conversely, let

where

is an isomorphism.

~ Aj Pj , A_ & q9 J 3

There is

O"

f

A

A

~-I"

If we put Q=

~A.P. , J J

then Q - Q = ~(Aj -Aj)Pj e $~I_i. It implies that

Q e g~

.

By i), it is sufficient to show the state-

Now, let us prove 2). ment on

~

Assume r

.,~

A

A

Pj & ~ t 0 • j=l We d e n o t e

~O/~_m_l

by

~ m

the image in

~

of

7"/'l 0

~

j~__l~m P'j

Note that "

is a coherent Ring (since it has a filtration every gradua-

tion of which is a coherent

~-Module).

r

is a finitely generated

r

(~0/~_m_l)-Sub-Module " "

( ?"'m t " ~-i )'~ '

of

A

which is a coherent coherent ~m l~m

( ~ O / ~ _ m _ l ) - M o d u le

(~ol~_m_l)-Module. is a sub-Module of

~m (~m/~

•

Therefore

,

~ m

is a sub-Module of

~ m

a

and

0 )~ , which is a coherent

A

(~01~_m_l)-Module.

is

A

It implies that

~m

is a coherent

A

(~0/~_m_l)-

387 Module and, afortiori, an

a coherent

~-Module.

~>m~mE~+

is, therefore,

increasing sequence of coherent sub-Modules of a coherent Module

~.

It

is locally

stationary

and

77l = U ~m

is a coherent

~-Module.

First of all, we give several properties on Proposition filtration

3.2.2.

~k

= ~

Let

~L

be a

~-sub-Module

Suppose t h a t

~ (~k)£"

~

of

with

is generated

~

P.]

residue classes of the sections r

~k

:

^

(j = 1 ' " ' ' ,

r)

of

be a section of

~k

~'~i~.0.

Then

^

~ ~kPj " j=l Proof.

borhood

U

Let of

x ~ P*X/Y x.

and

Q U,

By shrinking

defined

(1¢ = 0 , ' ' ' )

on

U.

We d e f i n e

inductively %

Q~ & P ( U ; ~ k _ l ) )

on

=

U

over a neigh-

we can suppose that

and that there is an elliptic pseudo-differential 1

by t h e

A

operator and

U

is Stein, A

of order

R:0, j ~ ["(U;

k_u)

such that

,

Q)) = 2 . , R ~ , j P j . + Qu+l (v=0 ' "'" ) J They can be d e f i n e d s i n c e a l l Q e F(U; )~[k ) can be expressed as =

~ R . P . + S, j ] ]

where

~ - k ~ ~ ~(U; ~ 0 ).

R. & ~(U" ] '

and

The principal symbol

q

E F(U; ~ k _ l ). of

~-k~

is contained

~

in

~(U; ~ ) ,

which implies that there is

In fact,

A

R. 6 F(U; ~ 0 ) J

such

that n-k~-

Z R.P. E F(Uj ~FL 1 ) J J ,

and hence

- Z a

.P. ] 3 e p(u;

k i). ^

If we put

Rj = ~ R p , j ~ r(u; ~ 0 ),

Corollary 3.2.3. coherent

~-Module,

then

^

Q = ~RjPj ]

.

Under the assumption as above, if then

~[

is a locally

finitely

q.e.d. ~i

generated

is a ~]~-Module.

388

Proposition 3.2.4. Proof.

Let

~

~

is a coherent Ring.

be a finitely generated Ideal of

~ .

It is

A

sufficient to show that Pj E J0

~

(j = i, ..-, r)

generate

~.

Let

be sections whose principal symbols

(~j ) ~ ~ r 0 be such that ~j

Let P. ^J P) ~

be the kernel of the homomorphism ~ r

Let

^Pj..

defined by

~

is locally of finite presentation.

AQjPj A 6 ~-I"

Then

,%

by Proposition 3.2.2, there is

Rj 6 ~ _ i

__

A

~

A

such that A

QjPj = ~RjPj. Hence A

A

~.

(Qj -Rj) ¢

A

, and

A

A

A

Qj m Qj -Rj

^

mod

~ -i"

_

This implies that 9;% is the kernel of ~ : ~ r --+ coherent over

~

By Corollary 3.2.3,

.

, and, therefore,

~rL is locally finitely generq. e. d.

ated. A

Let

Proposition 3.2.5. "I~kl.

977.0

Suppose that

921 be a ~ -Module with filtration is a locally finitely generated

~0-Module

A

is a coherent ~-Module.

and that

Then

~.

is a coherent

^

~ -Module,

and

)Y[m

=

lim

e--k

~

Proof. ~'L

defined

Let by

the kernel of ~.

r

Wi 0 =

A

~ 0uj.

~Uj~l_<j<_r.

~

Let

be a homomorphism

~ r __+

Let ~

is obviously surjective.

be

The sequence 0--+

--* ~r---*~'~---~

7g

is evidently exact, which implies

~

0

is a coherent ~-Module.

By

A

Corollary 2.2.3,

9%

is locally finitely generated over

~

, and

A

hence

7;L is coherent. o

~

o --,

"

Consider the diagram with exact rows

m

/Zi k) __+

~

( ~^m )r (~m/~k)

~

--~

^

~m

......~

0

(~m/Dlk) ---> 0

389

The middle arrow is an isomorphism, which implies the surjectivity of the right arrow. condition as

Since the filtration

{~kl'

{~kl

satisfies the same

the left arrow is surjective.

Therefore,

right arrow is an isomorphism,

the

q.e.d.

We say that a filtration satisfying the condition given in Proposition 3.2.5 is a good filtration. Proposition 3.2.6. Proof.

Let

the kernel

F :

~

~0

of

F0

is locally finitely gener-

be an extension of

F " We have

kenel of

is a coherent Ring.

It is sufficient to show that for every homomorphism

F0 : ~ r0 --~ ~0' ^ ated.

~0

~'0 = ~ r 0 ~ 7 ~ .

F0

and

~g

be the

By the preceding proposition, A

is coherent.

By Proposition 3.2.2,

~

is finitely generated.

If

A

PI' ..-, P~ ~,

then

are sections of PI' "''' P£

~0

generate

whose residue classes in ~D

over

?0

~

by Proposition 3.2.2. q.

Proposition 3.2.7.

generate

e. d.

Let

A

be a sequence of coherent that

~f~', ~fL and

~0'

and

G(~fl0) C ~

~-Modules

~,,

satisfying

G oF = 0.

have good filtrations such that

0"

Suppose F(~'0 ) C

Assume, moreover, that

is an exact sequence of coherent

~-Modules.

Then the sequence

(3.2.1)

is exact and so is (3.2.3)

~ ' 0 ---~~ 0

Proof. ~0

= ~

of

~ : ~

Let

& 20 --~

~

be the kernel of

is a coherent ".

0

Therefore

G.

~0-Module. ' --~ ~

Since

~N

is coherent,

is evidently the kernel is surjective.

390

Let ~0

~

by

--+ ~

be the cokernel of

~--~

~0

--~ Z

--+0

~'

--->~

for every

Therefore,

(3.2.1)

~fO

defines a good filtration of

is exact, we have

~{0 = ~ k

and

k < 0.

and

~

= 0.

By Proposition

(3.2.3)

be the image of ~[

Since

This implies that

3.2.5, we have

are exact,

~

= 0.

q.e.d.

A

Proposition 3.2.8.

The stalk of

Proof.

an

Let

I

be

ideal of

~

is a noetherian ring.

~^x "

Let

~I'

"''' ~ r

be a

A

system of generators of

I = I0/I_i,

is the residue class of

Pj ~ I O.

Suppose that # = ~ J

+

A

A

Q,

P. J

~ j

P

~.j

is generated by

Let us prove

is a finitely generated Ideal of

J

~x = ~ = ~ ~ -P j .

on some neighborhood of Pj,

which implies that

Definition X

3.2.9.

~

A sheaf of rings

, ~

Let

of ~

x.

j Q e I. Then

defined on that

#

U. is

By Proposition 3.2.2,

Q E E

Now, to state one more property of

U

This implies

x.

and

I = >-~pP.. j J

are defined on a neighborhood

is a c°herent Ideal and

generated by

Ik = I ~ ~ k , x

where

j

~ P-. 3

q.e.d.

we give the following on a t o p o l o g i c a l

space

is said to be noetherian if the following equivalent conditions are

satisfied. a)

For every open set

U

in

X,

of locally finitely generated generated

~-Module

for every

x E U,

such that a')

a")

there is an integer

~

~

~9~ij]j~+

-sub-Modules of a locally finitely

defined on

~ j l V = ~Iml V

a) is true for If

7i

~

every increasing sequence

for every

U

is locally stationary m

and a neighborhood

(that is, V

of

j ~ m).

=~£.

is coherent, the above conditions are equivalent to

a) is true for

~ = ~

.

It is well known that the sheaf of holomorphic functions on an

x

391

analytic space is noetherian.

of

Proposition 3.2.10.

~

Proof.

be an increasing sequence of coherent Ideals

~

.

Let

Then

~jlj ~ j

is noetherian.

is an increasing sequence of coherent Ideals of i%

~.

It is locally stationary.

Therefore,

by Proposition 3.2.2,

is locally stationary, We can show that

~}j}

q.e.d. f

~

inherits the properties

A ~

of

proved

above. Before showing this, we must introduce a special but sufficiently general kind of "quantized"

contact transforms.

Its more generalized

exposition will be treated in later sections. 3.3.

Contact structure and quantized contact transforms. We review

on contact manifolds.

Let

X

be a complex manifold and

subbundle of the cotangent bundle orthogonal L*

subbundle of

L,

is the dual bundle of

~d~,

v I A v 2> e ¢

L.

Li × L ~ × L

X

TX --~L*, where

9 (v I, v2, d ~ )

with a specified line subbundle

such that (3.3.1) is non degenerate. X

the half of

l+dim

is a fundamental X,

X.

L

It is equivalent to

is odd and that

nowhere for every nowhere vanishing section

(3.3.2)

the

L ~ x L ~ --~ L ~(-I)

saying that the dimension of

manifold

L~

This gives the alternating bilinear homomorphism

A contact manifold is a manifold

L

We denote by

that is, the kernel of

(3.3.1)

T*X

X.

be a line

gives a multilinear homomorphism of vector bundles

L ~ x L~ ~ L ---+C × X.

of

T~"]< of

L

~

~ A(d~)n-I of

L,

where

We say that a nowhere vanishing section

1-form of the contact manifold

X.

vanishes n

is ~

of

For a contact

(3.3.1) gives the exact sequence of vector bundles 0 --~ ¢ × X - - + T * X ® L ®(-I) -~H TX --+L* ---> 0,

392

where

H

is given by

Set there

L× = L-X.

Then

is a canonical

symmetric --+T*L ~

bilinear defines

i H ( e ~ 0)~ ( _ l_ ) ) ( d ~

closed

a 1-form

is a non-degenerate

L×

is an associated

A map

X

and

is a fundamental

projective of

manifold

X

typical

X

of

and its exterior bilinear

manifold

form on

of

a local

differential

TL x.

We say that

of the same dimension.

transformation

if

f*~

1-form

~y

Y of

Y.

isomorphism.

of a contact

T*Y -Y.

as a skew-

X.

for every fundamental

of a manifold

is locally

Pn_l )

X

example

P~

P~'~Y is

there

L~

be two contact manifolds

1-form of

Therefore,

on

that is,

L x --+L x x L x --+L x X T*X X L

is said to be a contact

bundle

manifold

@

structure,

is non-degenerate

In fact,

alternating

Such a map is, a f o r t i o r i , The most

TL x.

symplectic

Y

f : X --+Y

has a symplectic

2-form which

form on

d@

Let

Lx

= 8 .

Y.

manifold

The associated

It is well-known

isomorphic

symplectic

that every contact

to a cotangential

is a local coordinate

is the cotangential

system

(Xl'

projective

bundle.

"''' Xn' PI'

"'''

such that = dx n - P l d X l . . . . . Pn_idXn_l .

In this case, coordinate

(Xl'

system.

"''' Xn' PI' Corresponding

we can choose a local coordinate of the associated

symplectic

and

--. +

of

~ = L ~.

~IdXl + Frequently

for that of in

X;

X.

.'., Pn_l )

is said to be a canonical

to each canonical system

manifold

~ n d× n = ~ n ~

(Xl, L×

For example,

in this case,

defines

c.

of

we use sometimes

(x, ~ )

for every non zero number

"''' Xn'

such that

we use the terminologies

and

(x, c ~ )

We say that

coordinate ~i'

~n )

structure

as a substitute

(x, ~ ) indicate

(x, ~ )

"'''

PP = - ~ / ~ n

a symplectic L×

system,

as coordinates the same point

is a canonical

393

homogeneous coordinate system of be a homogeneous function on

X

X.

A section of

of degree

L ®(-r)

is said to

r.

Now, we explain several notions concerning contact structure. f

and

g

are homogeneous functions on

X

of degree

r

spectively, then we can define the homogeneous function degree

(r+s-l),

called Lagrangean bracket. f = ~ ~ ~(-r)

a fundamental 1-form and [f, g] = { r ?

;x

[f, g]

g =

~ ~ ~®(-s) ,

~ ..... n-1

~

An analytic subset

Y

is said to be regular if

form

~

of

Let

and

vanishes on g

A

for

vanishing on

A.

does not exceed the half

Y

TY.

~Iy

L ~ T~X

nowhere vanishes on

where

H

It is clear that

a bicharacteristic

Y

Y

is given in (3.2.2). ~

for a fundamental l-

~

X.

Let

~

be

is a subbundle

defines a completely integrable system

We call a maximal integral submanifold of

(or sometimes a bicharacteristic manifold).

dimension of a bicharacteristic of X.

consists of the zero section, in

be a regular involutory submanifold of

of Pfaffian equations.

in

X.

X.

H(T~X ® L®(-I)),

if

A

~®(1-r-s)

be an involutory submanifold of a contact manifold

other words, if

Y

then

i + dim X. Let

of

is

of a complex contact

[f, g] f

It is known that the codimension of

Y

A

is said to be involutory, if

every homogeneous holomorphic functions

of

of

n

~

Definition 3.3.1. X

re-

~ = dXn " ~pjdxj

+_f_~._+ ~-~f-) _ ~ {~ - ~ + p j ~---~)~pj = ( ~xj ~x n ~ ~ jP~--I PJ ~Xn ~Pj j=l " ~x.J

manifold

s,

~x n

n-1

,

If

and

If

Y

The

is equal to the codimension of

By the classical theory (cf. Jacobi [I], Carath@odory [i]),

is a regular involutory submanifold, then we can choose a

394

canonical coordinate is defined by

system

Pl . . . . .

Pr = 0.

• "" = dPn_l = dXr+ I . . . . Definition

(x I, ..., Xn, PI'

3.3.2.

n.

If

Y

dPr+l =

= dXn = 0. An analytic

A

such that

is defined by

Then

set

of

A

(2n-l)-dimensional

if

contact manifold is said to be Lagrangean codimension

"''' Pn-i )

A

is involutory and of

this condition is equivalent

is non-singular,

to

It is known that if

a)

~ IA = 0

b)

dim A = n - I.

is a non-singular Lagrangean

Y

then there exists a canonical • .., Pn_l )

such that

Y

local coordinate

is defined by

The above explanation

system

x I .....

is applicable

submanifold of

(Xl,''' , Xn, PI'

Xn = 0.

to real analytic manifolds as

well as complex manifolds.

But, for real analytic manifolds,

that the line subbundle

of

L

T*X

is oriented,

S*M

of a real analytic manifold

example of signed contact manifold. locally isomorphic Let

X

and

to Y

~X

[I] and Egorov

[i].

M

The cotangential is the most familiar

Every signed contact manifold

is

S~"~. be manifolds with the same dimension.

every contact transformation of isomorphism of

we require

and then we call such

a real analytic manifold a signed contact manifold. sphere bundle

X,

and

~y.

P*X

to

P'Y,

Then,

for

we can define an

This approach is initiated by Maslov

See also HUrmander

[2].

To show this, we prove

the following T h e o r e m 3.3.3. same dimension

n

Let and

defined by the equation morphic

function on

X ~

and

Y

be two complex manifolds with the

be a non-singular hypersurface ~(x,

X X Y.

y) = 0,

Suppose that

where

~.(x, y)

of

X x Y,

is a holo-

395 P~(X ×Y) ~ (P~"IK)~ Y = P~(X xY) n X X (P~"~f) = ~. We denote by --~ P*Y. i)

p

p

and

q

the projection

PI(XxY)-->

P*X

and

P~(XxY)

Then is a local isomorphism if and only if the determinant

of

dy~'a 0 dx~

1

nowhere vanishes on

A . If

p

is a local isomorphism,

dxdya'3- )

t h e n so i s ii)

q.

Suppose that

gf ~XxY

p

is a local isomorphism.

Let

whose principal symbol never vanishes.

Then

u

be a section of P(x, Dx) F-~ P(X,Dx)U

gives isomorphisms

-1 P -i P P iii)

Suppose that

p

~£~

P X

IX×Y '

f f ~ X -~/~IXxY

X,m -+ XxY,m

Dy)

Then

(= (Q*(y, D ))udy) Y

of sheaves of rings -I P

q-l~y ~ X -~

-1 P Moreover,

u.

is a local isomorphism.

P(x, Dx)(udy ) = (udy)Q(y, gives isomorphisms

'

-I '

P

@X

f ~ q -l~f Y '

~ -l~ -~ q Y,m

~X,m

the isomorphism P

-I

O_p, X _~ q-l~p~,~f

induced by the last homomorphism,

coincides with the composite

-i P Proof.

The first statement i) is evident.

an obvious consequence preparation case when

GL P~'~ ~ 6~p~(XxY ) ~_ q-l(> P~f

of ii).

We will prove ii) by using Weierstrass'

theorem (Theorem 2.2.2). u = Y(~(x,

y)),

The last statement is

If we prove this theorem in the

then there is an elliptic pseudo-differ-

396

ential operator

A(x, D x)

such that

u = A(x, Dx)Y(~.(x, y)).

fore we may assume without loss of generality that Let

~i'

= 0,

that

"''' ~ 2 n

The Ideal = 0,

~

of

for

~X×Y

is generated by

P•*(X ~ Y)

~I'

and

Y,

y.

"''' ~2n-I

and

1

degree

0.

Let

P*(X×Y),

Suppose that

degree

).

~ , ~

yj = qj(x, ~ ).

P

and

~2n~L = I.

such that ~2n

(x, y; ~ , ~ ) where

PY(~(x, y))

(see Lemma 1.5.9).

P~(X × Y)

pj(x, ~ )

with respect to

and

~j-Pj

Let

are coordinates of

~j = pj(x, ~ ),

~ , and

qj(x, ~ ) ajk,

are homogeneous of bjk

on a suffi-

such that

Bjk

0

and

bjk

are of degree

be pseudo-differential operators of order

(-i) whose principal symbols are ajk, bjk, respectively. 2n-i ~ Ajk]6k+Aj,2n~)-~2n , Rj = k=l

det(sj,k, bj,k )

(Ajk , Bjk ) implies that

never vanishes on

p* (X ~Y),

the matrix

is invertible by Remark i after Theorem 2.1.2.

0,

We set

2n-I = ~ B j k ~ k + B j , 2 n ~'L~2n " Sj k=l Since

x

are holomorphic functions homogeneous of

are homogeneous of degree

Ajk,

be a homogeneous

= ~ ajk ~ ( ~ k ) + a j , 2 n 6"(0-~2n ) , k=l 2n -i = ~ bjk6"(~k)+bj 2n0"(~g~2n ) k=l

Yj-qj ajk

y

is defined by

There are holomorphic functions

I

and that

x

6" (~i)

are cotangent vectors corresponding to

ciently small open set in P*(X × Y) 2n-I

(-i).

and that

[J~i' ~j]

coincides with the zeros of the principal symbols

coordinate system of and

i = i, "'', 2n-I

consisting of the

''', 6"(~2n_i) , ~ ' ( ~ 2 n

X

u = Y(fL (x, y)).

be a base of vector fields such that

~ l • D. = 0

There-

This

397

2n-I

n

6

k=l

j =I

j =i

(3.3.3) Moreover,

and

S. J

are of order

R.

J

are of order

0

i

]

with principal symbol

with principal symbol

~j -pj(x, ~ )

yj -qj(x, ~ ).

Set

I Rj(x, y, Dx, Dy) = Dyj - P.(x, y, D x , Dy) j Sj.(x, y, Dx, Dy) = yj -Qj(x, y, D X, Dy) By induction on such that If

k,

we will show that we can take

[Yi' Pj] = [Yi' Qj] = 0 k = 0,

then this is obvious.

[Yi' Pj] = [Yi' Qj] = 0 Weierstrass

for

i < k.

and

k ~ i

and

S. 3

i = i, "'', k. Suppose that

By the preparation theorem of

(Theorem 2.2.2), there are

Pj, Qj, Tj, Gj

such that

i p.J = TjR k + P.J , Qj = GjR k + Qj ,

(3.3.4)

and

[P%, yk ] = [Q0 , yk ] = 0.

= 0

for

i < k,

we have

By replacing

= ~fRj

+

Zf

Rj,

Sj

by

This implies

Since

[Pj, yi ] = [R k, yi ] = [Qj, yi ]

[Pj, yi ] = [Qj, yi ] = 0

the uniqueness of division.

results.

for

R. J

for

i ~ k,

by

We have

S O = ~ - ~ (Dy o - P%) + ~

Dyj - P.,j

yj -Qj,

that we Can assume

[Yi' Pj] = [Yi' Qj] = 0

(Yj - Qj)-

we have the desired

that

for

i = I, .-., n.

In the same way as above, by using Theorem 2.2.2, we can assume, moreover, that

[Dyj, Pj] = [Dyj, Qj] = 0 .

Rj

=D

YJ

-P

for j

(x,

i = I, ''-, n.

Hence

D x) ,

(3.3.5) Sj = yj - Qj (x, Dx) Since the principal symbols of vanish on

PI(XxY),

[R j, Rk],

[Rj, Ski,

RI, "'', Rn, SI, ''-, S n

[Sj, S k]

must

commute each other.

Let

398

A(x, y, Dx, Dy)

be a pseudo-differential operator.

2.2.3 and Theorem 2.2.6, (3.3.6)

A

can be written in the form

A(x, y, Dx, Dy) = ~Gj(x, + ~Hj(x,

(If

A

is of order

~_ m,

y, D X, Dy)Sj(x, y, Dx, Dy)

y, Dx, Dy)Rj(x, y, Dx, Dy)+~(x,

then

~

is of order

-i P are surjective. Let =

A(x, Dx)

~ m).

Dx )"

It follows that

-I ~X

--~ ~/[I X×Y

and

p

~ X , m --+ ~X~Y,m Y(~-)

If we show the injectivity, then the proof completes. be a pseudo-differential operator such that A(x, Dx)Y(~L)

By induction on

0.

By using Theorem

k,

we show that

A

is of the form

n A(x, Dx) = j=k ~ Gj(x, y, Dx, Dy)Rj (x , y, D x , Dy) n

+ j~=iHj(x, y, Dx, Dy)Sj(x, y, Dx, Dy) and

[Gj, yi ] = [Hj, yi ] = 0

for

Suppose that this is true for T. J

and

K. J

k.

i < k.

If

k = I,

this is obvious.

By Theorem 2.2.3, there are

~j, H j,

such that Gj = TjR k + Gj , Hj = K Rjk + H

and that

[Gj, yk ] = [Hj, yk ] = 0.

[Gj, yi ] = [Hj, yi ] = 0

for

i ~ k.

n

j By the same argument as before, We have

n

n

n

A(x, D x) = ~_~ G.R.+ ~-~.S.+ (Gk+ ~__J.R.+ j=k+l j j j=l j 3 j 3 3 n

Since

n

~ G.R ~ H.S. A(x, Dx) -j=k+l j j - j=l J j

by T h e o r e m 2 . 2 . 3 .

In this

Kj Sj)R k.

way,

commutes with

the induction

completes.

we can suppose n

A(x, Dx) = ~-~ HjSj , j=l

[Hj, yi] = 0 .

Yk'

it must vanish Therefore,

399 In the same way as above, by using Theorem 2.2.6, we can show =0.

A(x, Dx)

q.e.d. The isomorphism of an open set

U

in

P~X

to an open set

P~'~f, given in this theorem is a contact transformation.

V

~(x,

in

y)

is said to be the senerating function of this contact transformation. Let tion. of H

F : P~K ~ U --~ V C P*Y Then,

P~'Z< D U

(by shrinking G ~ W C P*Z

U

and

be an arbitrary contact transformaand

V),

P*Z O W

F

is obtained as the composite

H> V C P~"-f, such that

are contact transforms defined by generating functions.

G

and

By Theorem

3.3.3, there are isomorphisms

H-I~P ~ : F -1 ~ y

Composing these isomorphisms we obtain an isomorphism

9x[u.

This isomorphism

and the isomorphism

~

transforms

F-l(~p~,9{ ~ >

with the isomorphism induced by

F'I~ Y,m

to

In other words,

~X,mlU' coincides

~P~"~flU induced by F.

_%

the following

diagram

F-I~y,m F -l (> p~,~f(m)

' ~X,mlU

~ (Yp~.,7~(m)Iu .

is commutative. Note that the isomorphism contact transform

F.

is not uniquely determined by the

This is similar to the relation between the

quantum mechanics and the classical mechanics.

So, we call

~

a

quantized contact transform. The "quantized" investigate

contact transforms play an important role when we

the structure of systems of pseudo-differential

We treat more precisely "quantized"

contact transforms

section, by using the theory of maximally overdetermined

equations. in a later

systems.

400

Example .3.:3..4 function

(Legendre transform).

Sg(x, y) ~(x

(x, ~ )

and

n

(y, ~ )

i xj = - ~j ~ n I for x n = ~ n I ~j = Yj ~n =

n-i -Yn + ~ xjyj

y) = x '

Then

We take as the generating

j=l

are in correspondence by the relation: YJ

j < n

=

i

for

j ~ n

for

j < n

Yn = <x, ~ >

for

~j = -xj ~n

j < n

'

Pseudo-differential

operators

P(x, D )Y(~(x, x

and

P(×, Dx)

Q(y, Dy)

are related as

y)) = Q*(y, D )Y(f~(x, y)). y

Explicitly, -i x. = -D D for J Yj Yn x

n

=~D

y

, y>D -I Yn

Dx. = yjDy n J D

= D Xn

-i yj = D .Dx xj n

j < n

for

for

j < n

Yn = D~ l<x' D > n x j < n

Dyj = -x.D]x

for

j < n

n ,

D

Yn

= D Yn

Xn

are fundamental relations. 3.4.

Faithful flatness.

for pseudo-differential and

^ ~

By using Weierstrass' preparation theorem

operators, we can show the faithful flatness of

f

over

Theorem 3.4.1.

Let

~ : X --+Y

be a smooth map.

Then

A

a)~

~X/Y

is faithfully flat over

~f/y •

b)~

~X/Y

is faithfully flat over

f ~X/Y "

c)~

~Xf/y

is coherent and noetherian.

d)? Every stalk of

~X/Y,0

e) 9 Let

~f/y-SUb-Module

~f

be a

is a noetherian ring. of

f r (~X/y)

defined in an

401 open set

U

of

P~'~X/Y. Set

If

PI' "'', P£

= )r f, -~k (~X/Y,k ~ ~ ~ = ~0/~-i " are sections of ~ 0 whose residue classes

~£

generate

, then we have

~

r = j=l ~ ~X/Y,0 P"J "

~0 Proof.

PI' "'''

We prove this theorem by induction on the dimension of

P~/Y. We abbreviate

~X/Y'

~X/Y'

by

~P*X/Y' "'"

~, ~,

@

in

the sequel. A

a)~ implies evidently c)~, since a)? implies e)~. morphism ^

^

~=

y ~

pf

In fact, let

P : (~f)£ ---~~if

is coherent and noetherian.

~f

defined by

~If . By a)~, 0

>~

be the kernel of the homoP.. J

,~

~=?®

Set

~f

,

?f

>~.

is exact. Moreover, the sequence

0 is

exact, where

~_~

~-97

~

=

~'~

'-

%o1~7-1'

k = ~ n P k " Since

~-~>~,

by ~emma 3.2.1,

0 is an exact sequence. (3.4.1)

~

>~

;i

for every

( ~ ~n~IPj) n ~ m m.

Since

,0

This implies that

J

Pj

~,0

~[

C ~ ~mPj

J

is generated by

Pj,

~f

is generated by

in virtue of a)~. Hence

(3.4.2)

~f

C ~ m

By (3.4.1)and (3.4.2), we have We will show that

( ~_ ~mPj). j

~ 0 C ~ ?0Pj. J

a)? implies d)~.

Let

p

be a point in

P*X/Y

402

and

I

Since

be an ideal of ~I m(m) l

~0,p"

Set

Tk = Ik/Ik-l"

is an increasing sequence of ideals of

is a noetherian ring, there exists for every

Ik = I ~ ~ k,p'

m ~ m 0.

~ = ~ p fl.

Set

m0

such that

I = ~T

and

~p

P

I m(m) = l_m0(m 0)

m(m),

which implies that

m

I = l_m0(m0).

Let

symbols generate

PI' "''' Pr l_m 0

be elements of

By e)~

we have

r

I m0

l_m 0

whose principal

I ~

~_m0,P

r

j=l ~ ~0Pj C l_m0.

Therefore,

is finitely generated and

I m 0 = j~=I~0Pj .

I

Since every

is finitely generated,

l_m/l_m_l

I

is finitely

-m 0 generated.

This shows

d)~.

Now, we will prove a)~ Suppose that system of of

X

Y

and

coordinate Let

p =

and

and b)~ from the hypothesis

Y = £L ' X = Y ~ C n '

'', y£)

(YI' "'', y~ , Xl, .-., xn)

(YI' "''' YL' El' "''' Xn; system of

(y0

(YI'

P*X/Y.

0 0) , x , ~

~I'

be a point of

is a coordinate

is a coordinate

"''' ~ n )

~(y, x) = y.

of induction.

D. J

P*X/Y.

system

is a homogeneous

signifies

~ /~x.. J

It is sufficient to show

that, for every finitely generated proper ideal

I

of

f ~p,

~f

f/ I) = 0 ,

T o r l P ( ~ p , yp

(3.4.3)

~f T o r l P ( ~ p, ~f/i)p = 0 , A

ypl #

~p

Let that

and

~pl # ~p.

P(y, x, Dx) ~ I

P(y, x, D x)

be a non-zero element of order

is defined in a neighborhood

there exists a finitely generated Ideal whose stalk at symbol of

p

coincides with

P(y, x, Dx).

(Case i)

I.

~

of

U

of

~ f

Pm(y , x, ~ )

p

m.

Suppose

and that

defined on

U

is a principal

We decompose the proof into several cases.

The case where

Pm

(y0 , x 0 , ~) ~ 0

as a function of

403

~0 = (0, ''', 0, i) = (0, q,0)

We can assume that pm(y0, x 0 Set

,0) Y' = C x Y.

and

m We define

~' : X --~Y'

by

~'(x, y) = (x I, y).

There is a canonical projection g : P*X/Y - X × P*Y'/Y --->P~](/Y' Set

p' = g(p),

g -i~

, C~,

~'=

f ~'f = ~ X/Y''

~' = ~X/Y''

~'

We have

x/Y'"

g-i p ,f C y f and g-i ~' C # .

By Weierstrass' preparation theorem (Theorem 2.2.2), we have

(~,f,)m~p f/ fp P P ~p ,

, )m (~p,

Therefore, if we denote by (3.4.4)

A

_%~p/~pP

and

the module

~ / ~-fP, p

Lf

A

(~p,)m ~ ~p/~pp. then we have

~ p % e f ~_-?p, ' ~ ef , ~,f,

Let RQ & ~ fP

ef ' fL f ^ ~P ~f ~---~p' ~ " ~p P p' f Then, there is R & ~0,p such that

Q ~ I 0 = I ~ ~0,p" and

6_0(R)(y0 , x 0 , ~I'

the residue class of

I

in

L f.

~

,%

~ 0.

In fact, let

u

be

Note that, by Weierstrass' prepara-

tion theorem (Theorem 2.2.2) m-1

~0,P u = ~ ~'0,p'(Dl/Dn)Ju = ~ j =0 j =0 This implies that !

~0,pU

!

!

0,p (DI/Dn)j u.

is a finitely generated module over

!

0,p'"

~0,p' is noetherian by the hypothesis of induction. N Therefore, {--~0 ~ ' = 0,p'(DI/Dn~QUIN is stationary; hence, we obtain the relation N-I (DI/Dn)NQu = ~ Sj(y, x, D')(DI/Dn)JQu ,

j=0

where

! Sj(y, x, D ' ) 6 ~ 0,p'"

Let

QI' "''' Qr 6 10

principal symbols generate such that

6"(Pj)(y0, x~,

be a set of generators of T = I0/I_I. ql'

~,0)#

There exist 0

and

I,

whose

f Pj(y, x, Dx) E ~ p

PjQj e yfP.

Set

404

L[3 = ~ fP/ ~pPj. f

Qj

be the kernel of

r ~) L[ j=l J

Gf = 0

Set

is exact.

0 --+~'

qj : L[ --+ L f. J

induces a homomorphism q > L f.

~ Nf

~,f N f

The sequence

Gf

M f = ~f/I. P

Nf

Let

> Lf

~ 0

Consider the diagram ^'

Gf

~'

Lf

~'

Mf

M f --+0 . f ~p

f Pp

f Pp

f *p

The middle two vertical arrows in (3.4.5) are isomorphisms and the upper sequence is exact by the hypothesis implies the exactness

of induction.

This

of the lower sequence.

~f

Since

by (3.4.4),

Tot.P( ^ J ~p'

L f) = 0

for

j # 0,

we obtain by (3.4.4)

f Tor_1'P(~p,

M f) = 0.

In the same way we have f Tor P ( ~ p , M f) = 0.

,~ ~~ M f = 0. ~p

Suppose

tive.

~^p ,'

Since

is faithfully

surjective, which implies In the same way,

(Case 2)

Then

~p,

~'

?'f

Lf

is surjec-

G f --+ L f

is

M f = 0. ~p

® Mf = 0 f ~p

X' = Y' X C n,

h(y) = (Y2'

= ~X'/Y''

~ 'f p,,

~

implies

M f = 0.

The case where Pm(y , x 0, ~0) ~ 0.

y, = ¢ £ - I

defined by

~p

flat over

In this case, we can assume that setting

(9,fG f --~ ~^p ,,

, y£),

we can prove

pm(Yl, y,0, x O, ~0) # 0. ' : X' --+Y'

X = X ~,X',

(3.4.3)

By

h : y = CL__+y,

g : P*X/Y --~ P*X'/Y',

in the same way as (Case i) by

using Theorem 2.2.6 instead of Theorem 2.2.3.

,We omit the precise

405

argument. (Case 3)

The case where dim Y ~ I.

By a coordinate (Case 4)

transformation,

The case where

By a "quantized" (Case I). manifold

i.

denote by

~

we can reduce this case to submanifold

a submanifold vanishes

~i . . . . .

Y = pt.

such that the restriction

identically)

~ n-k = 01

in a contact

can be taken in

by a contact transformation.

The last case.

of dimension

~'f

1-form

Ix = 0,

What remains

Then

and

contact transform,

(that is, by definition

(Case 5)

operators

dim X > 1

Note that every under-Lagrangean

of the fundamental the form

we can reduce this case to (Case 2).

to investigate Suppose

~'

(reap.

that ~',

(reap. formal,

is the case when X = ¢,

~'f)

Y = pt

P~"~/Y ~ P ~

(Case I) by using Theorem

2.2.6.

X

is

~ X 9 p = ~0}.

We

the Ring of pseudo-differential

of finite order) with constant

is evidently a field.

and

coefficients.

The proof goes on in the same way as We omit the detailed discussions. q. e. d.

Although we do not repeat,

the propositions

also shown to be valid for the sheaf Definition

3.4.2.

to be admissible, of

x

A

~x-MOdule

if for every

and a coherent

~ f ~£g

x 6 fL,

f

~x-Module

~f

stated in

I 3.2

are

by using this theorem. defined

in

~ C P*X

is said

there exist a neighborhood defined on

U

U

such that

@ F~Lf"

f Px Remark I. to manipulate

Since

~f

algebraically.

is a coherent Ring,

~f-Modules

are easy

But, as is seen in later sections

§ 5.2 and ~5.3 of this chapter and of

~f-Modules

52

of Chapter III),

(e.g.

the structure

cannot be fully investigated without extending

the

406 operation ring to

~.

This is the reason why we have introduced the

artificial notion of admissible systems.

At the same time we must

confess that we do not know at present how to manipulate general

~-

Modules without assumption of admissibleness. Remark 2. (resp. ~ )

Let

X

be a complex manifold.

the sheaf of differential operators on

tial operators of finite order on Algebra

(resp.

-l~_Algebra).

X).

f

i)

~X

2)

~X

3) ~xf ~f

~X

X

(resp.

~X

(resp. differen)

is a

~

-i

~X-

By using the method employed in this

section, we can prove (see Kashiwara

4)

We denote by

[3]) that

is a coherent and noetherian Ring. f ~X"

is faithfully flat over is flat over

In order

that

~ q~-l~/~= 0 -i f

v~-l~ f

x" ~ Xf

a coherent

all over

P'X,

Module

satisfies

it is necessary and sufficient that

;~x is locally isomorphic to the direct sum of copies of 3.5.

X

of a finite number

~X'

Operations on systems of pseudo-differential

will explain how we can manipulate

~-Modules

equations.

We

in the same fashion as

microfunctions. Let As in

? : Y ~X

be a holomorphic map of complex manifolds

X, Y.

~ 1.2 of Chapter I, we denote by

the projections The sheaf

= ~

: P~"IK~Y- Py'IK --+ P*Y ,

= ~

: P * X x* Y - PyX --> P~'D{

i n d u c e d by

~ .

~Y-~X = C Y ~ Y * X ~ X

~-i ~ X ) - bi-Modules,

on which

defined in ~1.3 ~-i ~ y

is a

(f-l~y,

operates from the left snd

407

-i ~ X

~ X = ~o Xd i m X

from the right, where

Lemma 3.5.1.

Let

~:

Z --+ Y,

? : Y --> X

be two holomorphic

map s. a)

Assume

~ : Y--~X

~f-Module,

is smooth.

flat over

~ Xf

i

~f~x

is a coherent

f y ~Y~X ~Y~X = ~ Y ~~f

and

.~-I f

Then,

f

-I f

Z~Y'

~

f

) =

f ~z~x ~

0

~-i JOO~ i b)

Assume

(~ Z~Y'

~ : Z --+Y

f ~y-MOdule,

0

is an embedding. f ~Z'

flat over

~ ~Y~X ) =

and

~Z~Y

for

i # 0,

for

i = 0

for

i # 0 is a coherent

f ~ ~y= PZC.~Y f

-1

$0~

i = 0

f ~Z~Y

Then

-I f f ~? ~Y -l~f , f ~ ~Z-~X i (~ ~ZC-~Y ~ Y-~X) = [ 0

~

for

~

Y( ~ ? @ZC~Y' ~ Y ~ X ) =

for

i = 0

for

i # O,

for

i # 0 .

Z~X

0

Since this is proved easily, we omit the proof. Lemma 3.5.2. (Xl, "'', Xn) defined by

Let

be a submanifold of

X

of codimension

be a local coordinate system of

X

such that

xI = 0

Y

and

q

be a point in

D I + P I ( X , D')DI-I+ "''+Pro(x, D') of order signify f = ~ X u'

m

~/~= ~ X

~

f

and ~f

P(x, D)u = 0.

~-l(q).

(D 2, "'', Dn )"

= ~ X / ~ X P = ~ X u"

fx equation

Let

is

P(x, D) =

be a pseudo-differential operator

defined in a neighborhood of

(x2, ..', Xn )

P*Y.

Y

I,

Then we have

We set u

Here ~f

x', D' = ~X/~

is defined by the

P

408

?f l)

J~

(~ Y~X' ~ )

= J ~ i X( ~Y-~X' ~f) f

= 0

for

i # 0 .

2) f

?x is a free

~y

(resp.

~f)-Module with basis I y , x ~ U ,

i y ~ x ® D l U , "'',

m-I 1y.~x© D1 u Proof. superfix

f

We only give the proof for

we can suppose that m

.

Only by adding the

to the following proof, we obtain a proof for

At first, let us prove i).

= ~i °

~

Since

The question being local in

q = (0; 0, 0, "'', 0, i)

~Y~X

=

Px/Xl~x

and

Pm(0;

Y x P'X, X

~i' 0, "'', 0, I)

as a right Px-Module, in order to xI ~---~ is injective. Assume that

prove I) it suffices to show that XlQ(X , Dx)U = 0.

~f.

By Sp~'th's theorem (Theorem 2.2.3),

Q

is of the

form m-i Q(x, D) = G(x, D)P(x, D) + ~, Sj(x, D , )D JI

j=0 Since

XlQ(X , D) = 0 rood ~ P,

Spg'th's theorem.

Therefore all

we have

~ XlS j(x, D , )D JI = 0

Sj(x, D')

vanish,

by

which implies

Q(x, D)u = 0. Now, let us prove 2). Y x P*X. X have

Let

Z

Note that this is not of local

be the intersection of

Y × P*X X

and

nature in

Supp ~

. We

Z = ~x* ~ Y x PCq~; Pro(x*) = 0}. X We prove 2) by the induction on the number of elements in the finite

set

Z ~ ~-l(q).

one point, say

Suppose, at first, that ~q01"

Z ~ f-l(q)

consists of only

In this case, 2) is a direct consequence of

Sp~th's theorem. In fact,

Q(x, D)

Q ( x , D)

is of the following form in the unique fashion

XlG(X , D ) + H ( x ' ,

m-1 D)P(x, D ) + ~ S j ( x ' ,

j=0

4

D')DI.

409 Now, suppose that the number of We decompose and

Z2

Z

Z ~ ~-l(q)

into the disjoint union of

are closed

(and hence open) in

Z

is greater

ZI

and

Z2

and that

than

i.

such t h a t

ZI n ~

-i

(q)

Z1 con-

sists of only one point. By Weierstrass'

preparation

unique pseudo-differential

theorem (Theorem 2.2.2), there is a

operator

Pl(X, D)

of order

mI

of the

form

D~I+

ml-1

Pl(X, D) =

BI(X, D')D I

+ ''. +Bml(x , D')

(for brevity, we say such a pseudo-differential strass' type of degree vanishes on P = P2PI

Z I n ~-l(q)

P mp

such that the principal

and never vanishes on

for some elliptic pseudo-differential

in a neighborhood ~-l(q)

ml)

of

Z I a ~-l(q).

and of Weierstrass'

is also of the form and elliptic on Set

Supp ~ p C We have Let

~i Z~. ~Y

~

P2

~-l(q) _ Z~

= ~i~2

Sp

operator

m 2 = m - m I.

~ = i, 2

' ~2

PI

~-l(q) _ ZI and that P2

In the same way, type of order

respectively.

= ~X/~xP2

by the isomorphism

defined

defined on

is of Weierstrass'

for

=~xul

symbol of

is, afortiori,

type of order

P = SIS 2.

= ~X/~xSI

operator is of Weier-

= ~xU2"

We have

u = S21Ul~Pllu 2.

= ~/~IY ~ ~ 2 Y "

w = w l~w 2

be a section of ~ y .

By the hypothesis

of induc-

tion there are unique m~-I

', D) Q(~)(x',

Q(P)(x such that

wp

Then we have = SIIP2u2 = 0, generated by

= Iy~ X ~

=

~=0 ~j ~(P)(x', D')D~

D)u~

for

P = I, 2

w = Iy~x~(Q(1)S 2+Q(2)Pl)u. PIS21Ul = P21SIU I = 0. Iy~ X ~ D~u

(0 ~ j ~ m)

Now, let us prove that

Iy~ X ~ D~u

respectively.

In fact, we have

This shows that over

~y

S2PIIu 2 is

JOy.

are independent over ~y.

410

Set

= ?X/~xP1

~I

= 7xvl "

Then

~=

~i~2

by

u = Vl~)Pllu2 ,

Assume that Q(x, D) = satisfies

iy~X ~ Qu = O.

m-i ~. Qj(x, D ,)D JI

j=0

We divide

Q

Q(x, D) = G(x, D ) + G in

DI

is a polynomial in of order

We have

DI

by

PI"

H(x, D)P I.

of order < m I

and

H is a polynomial

< m 2.

iy~ X ~ Qv I = Iy~X ~ Gv I = O.

induction, we have

G 6 XlCP X.

= iy~X ~ Hu 2 = 0.

By the assumption of the -I Iy~ X ~ QPI u2

On the other hand,

This implies that

H £ x I ~X"

Therefore

Q E xl~X. q. e. d.

Reducing to this lemma, we can prove the following theorem, which allows us to pull back ~-Modules. Theorem 3.5.3.

~X

~ : Y --~X

be a holomorphic map and

~ Xf-Module defined on an open set

be a coherent ~=

Let

~f 971f "

Let

V

be an open set in

P*Y

U

of

P*X.

~L Set

such that

~x Supp~L f ~ f-l(v) --~V

0O

is proper (and hence finite).

m-l~ f $~0"i X ( ~ E . ~-l~f) Y~X' -i (V) for i # 0 .

l) on

=

2)

(

Then we have

-I~x ~ = $~i

(~Y~X;

f

~ -i~ ~ ) = 0

-i ~fC) ~-L,,f Lo ifX

is a coherent

~yf-Module on

then

•

~-l~x

V,

and, if we set

?y

~*~f

~* ~= is canonically isomorphic

?Y

to ?*~. Proof.

We prove Theorem 3.5.3 by decomposing into several cases.

411

(Case I) sion

The case when

Y

is a submanifold of

X

with codimen-

i. Without loss of generality, we can assume that

n

¢ ,

and that

Y

is defined by

We can suppose that

V

x I = 0.

Set

is a neighborhood of

X

is a domain in

Z = Supp ~t = Supp $fif. q = (0

dx ~ ) ~ P ~

and

n

Z ~ ~-l(q) u~

consists of only one point

be a system of generators of

exist pseudo-differential

q0 = (0, dx n ~).

~f

operators

defined in Pi(x, D)

U.

Let

Ul' "'" '

Then, there

of order

m.l

such that

m.

Pi(x, D)u.l = 0 We set

~ if = ~

denote by by

Ul,

and

~m.(Pi )(0,

"'', u~.

Let

and 2) are true for (3.5.i) Since

~*~f ?*~f

Therefore

? *~f--

be the kernel of

~f.

(x' D).

We

the homomorphism defined F f.

By Lemma 3.5.2, I)

Consider the following exact sequence --+ ~ * Z f ---> y * ~

is coherent,

~*~f

i = i, ''',

£ i = ~ X ~xf~i = ~ X / ~ x P i

"'" @ ~ gf --->~ f

~f

for

f

/ ~ XfP i (x, D),

F f : ~ f = ~If ~

~i' q) = ~ I l

~*~;Lf

--+ 0 .

is locally finitely generated.

is locally finitely generated, whiah implies that

is coherent.

Next consider the following diagram

% (3.5.2)

~y

~y

.>

~Y

--->

In this diagram, both rows are exact and the middle arrow is an isomorphism.

Therefore the right arrow is surjective, which shows that the

left arrow is surjective.

Hence the right arrow is an isomorphism.

This completes the proof of 2) in (Case i). Now, let us prove

412

-l_f

(3.5.3.) Since

j,,,,.~ ~x(~$.,. x, ~f> =0 is

~ Y"~X

to

it suffices to show that Assume that a

~/~f Xl~ ~r~f

u &~

pseudo-differential

+ Am(X , D')

for i ~ O.

satisfies operator

such that

is injective. XlU = 0.

P(x, D) = DI+AI(X,

P(x, D)u = 0. ..

Hence

Xl]

[...[[P,

Since

Here,

D'

u ~ ~r~f,

there is

D')DI-I+--signifies

(D2,''',Dn).

_ m-times , x I

],

...,

-~X l ] U

=

m

~u

=

0

.

This shows (3.5.3). From (3.5.1) we have the exact sequence f ~X

-

~* ~ ' X ' I

This shows that

f (~Y~X'

~*~f

--~ ? * ~ f

---~ ? * ~ f

~i) = 0.

___> ~, ~f

~, --~

f ~y,

faithfully flat over y~,, ~ X ( ~ y . ~ x ;

~f)

?

*~

is injective. --+ ~ * ~

Since

~ ~y

is injective

--->O. is

Therefore

This completes the proof of Theorem 3.5.3

in (Case I). (Case 2) Let

Z

The case where ~ : Y --~X be a submanifold of

of codimension --+ P*Y

I.

Let

X

which contains

Y

~' : P*X × Z- PZ X --~ P'Z, X

be two canonical projections.

consequence

is an embedding.

f"

of (Case i) and the fact

YY~Z'

Pz is

easily

( C a s e 3) Since

: P*Z x Y - P y Z Z

Then the theorem is a trivial

z(

which

as a submanifold

for

?z~x ) = o i > 0 ,

verified. The c a s e

f @Y~X

is

where

flat

over

: Y --~X X

is smooth.

and coherent over

~

,

the

413

theorem is evident. (Case 4)

The general case.

Setting

Z = Y x X,

we decompose

~

j P Y f--~Z-->X.

into

Since

60

and

3

Z(?y z ,

for

) = 0

i > 0,

This completes the

the theorem is a consequence of (Cases 2 and 3). proof of Theorem 3.5.3.

The condition in Theorem 3.5.3 is a generalization of the notion of "non-characteristic" of the initial surface with respect to a differential operator.

Therefore, we give the following

Definition 3.5.4. (reap. ~ )

Let

be an admissible

defined on an open set If

~ : Y --~ X

U

~ - I s u p p ~L ~ ~-I(v)

over

V.

of

~

~f~

~x-MOdule (reap. a coherent pxf-Module) PRO(, and

V

be an open set in

~--~V (reap. 59"-1 Supp ~ f

proper, then we say that (reap. ~tf)

be a holomorphic map,

P~'~f.

~ ~-I(v) --~ V)

is

is non-characteristic with respect to

~,(~y_~x _l~v~Y-191 )

(reap. ~*(~f-~x ~

~L ~m'-l~f)) x

is denoted by

~* ~

~f~ (reap. ~ f ) b y breviated to

~y

(reap. ?'91 f) ~ .

~*~

and called the induced system of

(reap. ~ * ~ f )

is frequently ab-

f (reap. ~ y ) .

In the same way as in the proof of Theorem 3.5.3, we can prove the following theorem. Theorem 3.5.5.

Let

? : Y --~ X

be a holomorphic map, and

be a right coherent

~f-Module defined in an open set

Set ~ =

Let

~gf ~

~y.

U

be an open set in

P*X

V

of

~f P#~f.

such that

f

Yy

-i Supp ~ f ~ %IY-I(u) --+ U

is finite.

(This means, by definition,

414

that it is proper and with finite fiber.) Then we have

I)

~

on

(~

"~'-I(u) for

2)

~ Y~X ) =

( ~-I~z,

~#~'i

~y~x ) = 0

i # 0.

?,~f

= IDa,( ~-l~If

f ~ ~y.~x) -i f

f is a right

f ~ x - M o d u l e on

coherent

U,

~* ~ f ~ ~X

and,

~.

i f we s e t

~i f =

is canonically isomorphic to

f% We o m i t t h e p r o o f o f t h i s

theorem.

The reader can easily verify

that this theorem is a direct consequence of Theorem 3.3.3 by using "quantized" contact transforms. Therefore, the "integration along fibers" and "substitution" are the same operations in this point of view. If

~

is a left ~-Module,

then

~ ~ ,

~ )

is a right

]

-Module.

This corresponds to the dual system of a system

following theorem gives

the relation of

Theorem 3.5.6.

: Y --~X

~if)

Let

be an admissible

respect to which P*Y.

?

~x-MOdule

~*

and

~

~ (

, ~ ).

be a holomorphic map,

(resp. coherent

. The

@~_ (resp.

~-Module)

is non-characteristic over an open set

with V

of

Then we have canonical isomorphisms :

(3.5.4)

?~'~R~ X ( i ~ i ,

~x)[dim X] ¢~ ~ ? y (

~*~,

~y)[dim Y]

of ~y-MOdules, and (3.5.5) f

~'~R~

of ~ v-Modules.

f(~f,

~ Xf ) [dim X] <~-~ ~ , t f ( ~ * ~ i f , ~f)[dim Y]

415

Proof.

By decomposing

to show the theorem when

~ ~

into

y c_+ y x X --~ X,

it is sufficient

is an imbedding and when

is smooth.

In the sequel, we give the proof only on ~ . (Case i) The case where ~ IR~y(

~e,

is smooth.

~y) = ~ R ~ y ( ~ Y ~ X

~

~-l OJ ~/'L; ~y)

~l? x = ~R~-I~x(~-I~'

~J~y(~Y-~X'

~ Y))"

It is easily verified that ~R~g~y(~y_~ X, ~y) = ~X+y[dim X-dim Y] Therefore IRj~fy(?*~,

=

AR~

fy) = ~ _ i f x ( ~ - l g ~ _ ,

_I~x(~'I~, ~-I~x)

~

pX~y)[dim X -dim Y]

~ X~y[dim X-dim Y]

~-l~x = ~*IR~x(~, (Case 2)

~ X) [dim X - dim Y]

The case where

~

is an embedding.

At first, we prove the following Lemma 3.5.7. Let

Z ¢+Y c_~X be a triple of manifolds.

Then

there exists a canonical homomorphims £yl X ~ Proof.

CZix[codimyZl

We have N~y(~y,

~ZIy) = Cz[-codimyZ ].

In fact, m~y(~y,

~Ziy) = ~ @ y ( ~ y , =~FZN~y(6~y,

pR~z (~Y))[codimYZl Oy)[codimyZ]

= IR~z(Cy)[codimyZ] = Cz[-codimyZ].

416

Czlx-- x*

y ZjY'

woa o

¢ --+ HOm~y(Oy, ~Ziy[codimyZ]) --->HOm~x(~YjX, CZ~x[codimyZ ]). The desired homomorphism is the image of I~ ¢

under this homomorphism.

This completes the proof of the lemma. Now, return to the proof of (Case 2). non-characteristic with respect to We denote by (3.5.6)

~w X

the invertible

~

Suppose that

. Set

~ : YC~ X

n = dim X,

~x-M0dule ~ d i m X

d -- codimxY.

We have

(Cyiy× x ~ ~y×x ) 71 ~*~[

= (£YIY×x @ q>'Y×X) ~Y×X @ (~y~x ~Y ® 9 *~) = ( (~YIYY,X @) °O'Y~X) (9

(~y×xc_,X×X ® 9/_.)

:~Y~X

~x

= (£YIX~X ® l)'xxx) ~ 9/_.

%

By applying the preceding lemma to the situation

Y c+X c-~XxX,

obtain

73X

=

dXlX×X~ lYx ~ CYlXxX ¢) 'l)-x[d I .

Hence, we have,together with (3.5.6),

~----~CjDX~.y~I

?*~i[dl = (eyixxx~)~y)

® ~Y

~*~_[d].

By using this homomorphism, we obtain

IR.M,C~py( ?*TN.; ~ y)[-d] x.Y

--, ~ . ~ , ? x ( ~ ;

= ~*IR~x(~; Hence we obtain the homomorphism

~x+y) ~x )

?*~; F x.y )

is

we

4t7

(3.5.7)

~J~y(~*~;

~y)[-d] --+ y * ~ X ( ~ ;

~k).

Now, we will prove that this homomorphism is an isomorphism. can assume without loss of generality that

d = codimyX = I

induction on

Y = ~x I = 0).

d.

Assume that

X = cn

and

ing the resolution of ~fL , we can suppose that p = Dlm+Al(X ' D,)D~-I+ ...+Am(X, D'), differential operator of order ~ X

j.

m

by the

By consider-

~X/~X P

A.(x,j D')

In this case,

Therefore

(~; 9 X) = ~x/P~x[-I].

= ~ y[-l].

where

~=

and

is a pseudo-

~*~

~ * ~ X

We

m = ~ y

and

(~, ~ X )

This shows that

m : ]* * m ~ .

jC~y is an isomorphism. Remark.

(~, Fx ) [1]

Y

This completes the proof of Theorem 3.5.5.

Theorem 3.5.5 is valid if we replace

~

by ~

Then,

this theorem gives the following interesting consequence, which is proved in Komatsu-Kawai [I] and Kashiwara [3]. Corollary 3.5.8. manifold N, M.

M

Let

Let

N

with codimension ~7~ be a coherent

non-characteristic (that is not meet with

be a submanifold of a real analytic d,

Y, X

be a complex neighborhood of

~ x-MOdule f with respect to which S -S(~)

= S u p p ( ~ x ~ %X ~ ) C P*X

Y

is

does

P~).

Then, we have an isomorphism ~rN~ where ~ y

f (~i, ~ M ) ~ ~X is a coherent

~N/M <'~'IR~0~ f($~y, ~N)[H] ~Y f-Module ~ f ~y Y~X ~~ ~ ' ~ N / M = ~ ~(ZM) "

(Cf. Schapira [3].) The following theorem permits us to consider the "product" of systems.

418 Theorem 3.5.9. Let X~ be a complex manifold and ~f)

~i~ (resp.

be an admissible (resp. coherent) ~X~- (resp. ~fxv-)M°dule

defined in an open set ~

of P*Xv (~= i, 2 respectively).

P*(X 1 X X2) -X 1 ~. P*X2 -P*X 1 ~ X2 ~

denote by p~ the projection P*X~. Then, ~

= ~XIXX2

We

-i

( p l I ~ ~¢ p21~2 )

-

el ~)XI~ p21~X2 C

(resp. ~ f

f = ~XIXX 2

-i f -I f (el ~ i ~ P2 ~12)) -l~f -l~f Pl TX I ~P2ffX 2

is an admissible (resp. coherent) Moreover, if ~ p =

@X ~ ~ f ~ ' Uf

f )Module. ~XI×X2- (resp. ~XI~X2then

~=

~XIX X2

f

Xl, X 2

by

Since this theorem is easily proved, we omit the proof. We denote A f~ ~ f f ~il~ ~ 2 (resp. @fgl 2) the ~XI×X 2 - (resp. 2 )

Module that

~

(resp. ~f)

(~f, ~f2 ) ~

defined in the statement of the theorem. Note ~fl $ ~ 2f is an exact functor.

419

4.

Maximally overdetermined systems 4.1.

Definition of maximally overdetermined systems.

We can

grasp a (hyper-, micro-)function by the knowledge of the (pseudo-) differential equations which the function satisfies.

In this point of

view, the maximally overdetermined systems play an important role, because these are systems whose solutions constitute a finite dimensional vector space. of the

Roughly speaking, the codimension of the support

~-Module indicates the lack of the numbers of the parameters

of the solutions of this system.

A maximally overdetermined system is,

by definition, the system whose support has the largest possible codimension.

More precisely,

Definition 4.1.1.

f ~x-MOdule

A coherent

maximally 0verdetermined if the support of A section

u

J

non-singular analytic subspace of

~'P*X

is said to be

is of dimension ~_ d i m X - i .

of a maximally overdetermined system is said to be non-

degenerate, if the symbol Ideal

P(x, D)u = 01,

~

~

J

of

u

is reduced and defines a

P~IK. Setting

~ k = {P(x, D) 6 ~ k ;

is, by definition, the coherent Ideal

$0/~-i

of

"

In the sequel, we study maximally overdetermined systems, and "quantized" contact transforms. 4.2.

Invariants of maximally overdetermined systems.

We prepare

some auxiliary lemmas in order to analyze the structure of maximally overdetermined systems. Lemma 4.2.1. system

Let

(x I, "'', x n).

coordinate system of P(x, D) = ~ Pj(x, D)

j~m

X

be a complex manifold with a local coordinate n

(x I, "'', x " ~I' P*X.

"'' ~n )

is the homogeneous

For every pseudo-differential operator

of order

~ m

defined on an open set

~

of

420

P*X , we define the first order differential operator of degree

(m-l) defined on the inverse image

~(m) Lp

homogeneous

~-I~ under ~::T*X --->P~"-X

by In LP(m) = HPm+ ( P m - I - ~ = I

(4.2.1)

n

~P

L (m)

_i

m ~x ~

~

~

)+ (Pm-i

n

~2p

2 ~ ax~ ~=I

)"

satisfies the following properties:

(dx l-'-dxn)-I/2L(m)(dx l.-,dxn) I/2 -p

l)

)

~P

= ~ ( ____~m ~ ~=I ~ ~x~ Then,

~2P m ~x~

is independent of the choice

of local coordinate system. 2)

If

of order

P(x, D)

~ m

and

and ~ i ,

Q(x, D)

are pseudo-differential operators

respectively, then we have

a)

LpQ(rmP~) = PmL~) + Q~L~m) + ~[Pm'l Q~] ,

b)

L =(rmF~-l) [P,Q]

Proof. proof.

[L~m), L~ ~)].

Since i) is easily proved using Theorem 1.5.5, we omit the

The following formula is useful:

for every vector field

~

~ (l og det F) = Tr(F-I~F)

and every matrix valued function

F.

q. e. d.

2) is trivial. By using this lemma, we prove the following theorem. Theorem 4.2.2. u

Let

~f[ be a maximally overdetermined system and

be a non-degenerate section of

Lagrangean submanifold J =

~

~

with a non-void connected

as its support.

Set

~k =I P e ~ k ; Pu=01'

~ 0 / ~ _ I . Then, there exists a unique complex number

fying the following: and every (4.2.2)

P(x, D) E

for every local coordinate system m

1 n (Pm-l-~ ~

such that

dPm E j ~ l + ~

~

satis-

(xI, "'', x n)

' we have

~2P m )~O _-- ( ~ + ~ ) d P m

rood J

~=i ax~ ~ v This means that the difference belongs to

J~9. I.

~

signifies the

421 n

fundamental 1-form Proof. order

If

~ ~vdx~ ~=I

P ~ ~m'

differential

P 6 ~m-l'

then

operator

then

Lp(m)

on

Lp(m) lYt

]~-

can be considered as the first

homogeneous of degree

vanishes on Y~ . Therefore,

depends only on the principal symbol of

P.

If

(m-l).

Lp(m) 171

p E J(m) = ~ m / ~ m _ l ,

we denote by on

~(m) the corresponding first order operator P ~ , where P 6 with principal symbol p. m There e x i s t

homogeneous functions

homogeneous functions

Pl'

"'''

Pn

ql'

"'''

of degree

If

qn

1

ep(m) I/~

of degree

such that

0

and

~ =

n

j~=lpj dqj

and that

/h

is defined by

be a pseudo-differential

n j~__~lqjpj. Then

Then

operator

dP I -~ ~

~(i) = _ ~ P u ~ P1

rood J.

+ ~

ql . . . . .

whose p r i n c i p a l

Set

qn = 0.

P 6 ~i

symbol coincides

I ~ P0-~__l

~=

Let

~2PI ~x~ $ ~

with

_ i(i) el (I).

We have

"~P~

~(0) ~(0)~(i) L~(1)~(0) ~(0) qj (~) = qj PI (i) = rl qJ (i) = ~[Qj,Pl](1) , and qj' where

Qj E J 0

and

~-(Qj) = qj.

i(0) ( qJ

(~) = 0. qj It follows that ~

Q 6 ~

= - qj '

Therefore

)~(0) (i) - i (°) ~) = (i+ ~

Therefore,

~Ppq~

qJ

qj (I)

H

satisfy

operator of order

is constant on ~.

dQ£ -= ~ (~-i)

mod J

and let

Q-RP ~_i

+ ~0~£

We set R

' we have

N = ~.

Let

be a pseudo-differential

with principal symbol

i[?, Since

# ~(0)

qJ (i) =

=

? . Then, we have

422

It remains to prove the coordinate invariance of

~

But this

is evident by the property I) of the preceding lemma, Definition 4.2.3. We call of

u,

and denote it by

~

,

q.e.d.

determined by (4.2.2), the order

ord(u).

"order" is an important invariant of sections satisfying maximally overdetermined systems. symbol.

One more important invariant is the principal

Since we do not use it in this paper, we omit its explanation.

We will give several properties of the order.

Since the proof is

easy but spends a lot of pages, we leave the proof to the reader. Propositi9 n 4.2.4. I)

If

u

system, and order

m,

"ord" satisfies the following properties:

is a non-degenerate section of a maximally overdetermined

P(x, D)

is an elliptic pseudo-differential operator of

then ord(P(x, D)u) = m + o r d ( u ) .

2)

Let

~ : Y --+X

overdetermined system and support

/h

an embedding.

ord(?*u) 3)

u

~ : X ~ P~X flY: y X×

Then, the section ~ -l~i)

e -if

~/i be a maximally

be a non-degenerate section of

Assume that

and the restriction of

f fl]Y*(~ Y~X

be a holomorphic map,

P~X ---~P*X

p~-P~

---+P~'~

~ * u = iyo X ® u

9~

with

is transversal to to

~

-i

(/L )

is

of ~-i(~)

is non -degenerate with support

and

= ord(u). Let

overdetermined with support restriction of

?

: Y --->X be a holomorphic map, ~$-Module, and

u

~ . Assume that ~

to

~-I(A

is a non-degenerate section of

be a right maximally

be a non-degenerate section of

~Y )

$2

is transversal to

is an embedding. ~,~

=

~, (fU-I ~

/~

Then

and that the ?.(u) = u ~ iy_~x

f ~ ? y_+x) with -I f

423

f w-I (A

support dim X. dim X

~1.x

and

dy ,

4)

systems on

of

~i

dx

are nowhere vanishing sections of

_ dim Y dIy

and

respectively.

Let

support

and ord((dy) -I ~ u) - ~I dim Y = ord((dx) -I e ?,u) -

),

X v be complex manifolds Xp

~_~

and

for

% ~2

up

9~u be maximally overdetermined

be non-degenerate

~ = i, 2.

Then

with support

UlU 2

A I ~ ~2

Pl-I (AI) N P2 I(A2) , where

p~

P*(XIX X 2) -XIX(P*X2)-(P*XI)×

sections of

~)i u

is a non-degenerate

with section

(which is, by definition,

is the natural projection from

X2

to

P*Xu),

and

ord(UlU 2) = ord(u I)

+ ord(u2). We will give several examples. Example i.

Ord(x ~) = - (~ + ½

the residue class of

i

in

). Here

~/~(xD

x~

is, by definition,

- ~ ).

We employ analogous

interpretations

in Example 2 and Example 3.

Example 2.

n ord(xll'- -x~n) n = -( ~ - ~ u P=l

+n

2 ).

n

Example 3.

ord(~ (Xl)''" ~(Xn)) = [ .

Example 4.

ord((x 3 _y2)

1

u = (x 3 - y2)

i 6

6) =

.!3

is defined by the equations (3yDy + 2XDx + l)u = 0, (3x2D

y

+ 2yDx)U = 0

(9xD 2 - 4D2)u = 0 y x If

(x, y; ~ , ~ )

is

the homogeneous coordinate

gential projective bundle, then the support 3y~ Therefore

7~

is non-singular.

+ 2x~

= 0,

We have

]h

system

of

u

9x~ 2 = 4~ 2

of the

cotan-

is defined by

424 I v = ~udx In fact,

(3yD

w = u Ix= 0 = y

-

= 0.

l+2DxX)U

~

= y

ord(v)

3

= - ~5 = ord(u) - ~i .

(3yDy + 2XDx + l ) u

because

This does not contradict with Proposition condition

= 0.

ord(w)

4.2.4,

= - ~I # o r d u.

for the transversality

of support is not fulfilled.

a+~ Example If we put = 0.

5.

I ( x + y 2 ) a dy

u = (x+y2)a

This implies

and

v = ~udy

(xD x _ ~ - ~i) v

By the following

theorem,

: 0.

,

operator

ord v = - a - i = ord u - ~

a non-degenerate

Theorem 4.2.5. P*X

and I)

x* Let

of order Let

~

and

~ ,

/~

$~_f and

be maximally

of an elliptic pseudo-

submanifold

overdetermined

Set

~

=

~X

~xf ~ f ,

~=

of

systems

sections with the same support

respectively.

.

0.

be a point of ~f

i

section is determined

be a connected Lagrangean

generated by non-degenerate o(

I I (xD X - ~ - ~+ ~ D y y ) U

then

by the order and the support up to multiple differential

2

= x

~_

®Xf

~X

of order

~f.

Then we have

j 0 ,

if

~ ~ ~

mod Z ,

locally isomorphic Therefore, 2)

if

at -= ~

Suppose

mod Z,

that

ated by a non-degenerate Then,

if

m = ~-~

v

~if

then

~f

and

is a maximally

section

is a non-degenerate

u

?if

Cjh, if o~---~ mod 7. are locally isomorphic.

overdetermined

of order

section of

is an integer and there exists

to

~ ~f

system gener-

with support of order

~ ,

locally an elliptic

~

. then

425

pseudo-differential P(x, D)u.

( ~

We say that

3)

mod ~

~f

P(x, D)

~

~ 3.3,

applying

,

U

of

x

with support

Y~ ~ U,

the "quantized"

contact

~)

E Pe~;

x I = 0,

~f

= ~u,

where

system of order

overdetermined U.

transform,

l+dim

contact transform,

X°

defined

Therefore,

we can suppose that

~2 = "" " =

u.

such that, for every

defined on

the order by the half of

v =

of the choice of

overdetermined

We note that the "quantized"

and ~L = {(x ,

such that

there exists a simple maximally

~

varies

m

independent

is a simple maxim@lly

system of order Proof.

of order

is, therefore,

There exists a neighborhood

complex number

in

operator

~n = 0}.

by X = Cn

Then 3) is

evident. Now, let order

~

with support J L .

u

is a non-degenerate

By using the method employed

section of in the proof

of Theorem 3.3.3, we can suppose that P(Xl, DI)V = 0 , D.v = 0 J

for

j = 2, "--, n ,

v = Q(x, D)u for some elliptic pseudo-differential P(Xl, DI) symbol

is a pseudo-differential

XlD I.

Before continuing

Lemma 4.2.6.

If

pseudo-differential a neighborhood differential

t = 0,

operator

~

This is proved

operator of order

the proof, we prepare

of order

i

of order 1

Q(t, D)

of order

the following "'"

is a

with one variable

-I

0

defined

such that

(tD+ ~ ) Q ( t ,

D).

= a0(0). in the following way:

0.

with principal

then there exists an elliptic pseudo-

P(t, D) = Q(t, D) is given by

Q(x, D)

P(t, D) = t D + a 0 ( t ) + a l ( t ) D - l +

operator

of

operator

by developing

Q(t, D) =

in

426

bo(t) +bl(t)D -I+ "'" , we determine

bj(t)

successively by the follow-

ing formula : (tD+ ~ -ao(t)+j)bj(t) where

~j(t)

is determined by

bk(t )

=

Yj(t) ,

(0 $ k ¢ j).

We leave the

precise argument to the reader. Now, we return to the proof of Theorem 4.2.5.

By using the above

lemma, we can assume that '~4.2.3)

(xID I

-

~)v

=

D.v = 0 for J v = Q(x, D)u Q(x, D)

is a pseudo-differential

Now, let us prove i).

0

,

j = 2,...,n

operator of order

,

0.

By the preceding argument, we can assume

tha t ~f/~

f(XlD I + ~ + ~i)

and ~f

= ~f/~f(XlDl

+ ~ + ~i) +

+ ~n ~fD. = ~fu j=2 J

• j=2

fD

fv J = ~

"

Then, ~-~X(~,

9g) = {A(x', Dl)V ; DjA(x', Dl)V = 0 I (XlDl + ~ + ~ ) A ( x ' ,

Since Set

DjA(x', Dl)V = 0, A(x', DI) = ~ ajD j~Z

A(x', DI) . Then, Therefore,

implies that

3)

= 0.

is independent of

if If

O(~ ~ mod ~, 0( - /9 = m

A(x', DI) = aD I , which implies that ~ o ~ f ( ~

j = 2,''',n,

Dl)V = O} . x' = (x2,''',Xn).

(XlDl + ~ + ~ ) A ( x ' ,

( ~ (~ - ~ - j) ajD~)v . jeZ 0~(~,

for

Dl)V = A = 0,

which

is an integer,

f, ~ f )

=~o~,~,

then ~)

=¢ A . Let us prove 2).

By i),

m

is evidently an integer.

Multiplying

427 an elliptic pseudo-differential

operator, we can assume that

v

satisfy the same pseudo-differential

v

is equal to

u

u

and

equation, which implies that

up to constant multiple.

This completes the proof

of Theorem 4.2.6. 4.3.

~ 3.3,

Quantized contact transforms

general case - -

we defined a special kind of "quantized"

In

contact transforms.

In this section, we give more general kind of them. Let

X

and

Y

be two complex manifolds of the same dimension

and ~

be a Lagrangean submanifold of

Jh ~X×

P*Y =7~ ~ (P*X) x Y = ~

p : J[--+P*X

and

P*(XXY).

n

Assume that

and that the natural projections

q : Yl--~ P*Y

are embeddings.

Set

U = p(Y[),

V = q(7~). Assume moreover that the maximally overdetermined system ~f

with support

.are given.

~ry

and a non-degenerate section signifies the invertible

~y-MOdule

K

of

~y

~ y .n

~y ~ f

~y

~f Y

is a

(p-l~fx, q - l ~ $ )-bi-M°dule"

We set

~=

~Xxy

O f

~Z f.

Then

~X~Y the integral transformation with kernel function framework.

P(x

(4.3.1)

makes sense in our

We have the following theorem.

Theorem 4.3.1. where

K

,

i) By the correspondence

Dx) E p - l ~ X

Q(y, D y ) 6

,

P

-ij0

P(x, Dx)K = KQ(y, D y ) '

q- IjD y , we have the isomorphisms

-1 X --~q JaY '

P

-19oX,m

P

-i .~ -I~ ~X,mq Y,m

~ q-l~y,m

Moreover,

-I P

_~q-i ~ P¢'~X(m)

~ pelf(m)

428

is commutative. If we denote by by

~

7. the diagonal embedding

the projection

of ~y-Modules

X×y

--+X,

to the category of

#°x-Modules, defined by the isomor-

Let

be a Lagrangean submanifold in

ql : ~ i - - ~ and that

Jt 1

P~'

W = q2(~2). X×Y

(reap. K2)

7)"Z ® ~ f ) . XxY yyx Z

P*(Y× Z).

P2 : /~2--> P ~

p2(J~2) = ql(Y~l).

system on K1

to a ~ x - M o d u l e

~)).

be a Lagrangean submanifold in

and

~

® L*( ~

2)

Let

and

Set

Y×Z)

P*(XxY)

Assume that

)

and

~2

Pl : Y~I --->P~TK,

q2 : Y~2--~ P*Z

U = pl(Jil),

~/~I (reap. ~

(reap.

and

then the functor from the category

phism (4.3.1), transforms a ~y-Module y,(Lry

X ~ Y c_~ X × Y xY

are embeddings

V = p2(~2)

= ql(Al)

be a maximally overdetermined

with support

~I

(reap. Y~2 ),

and

f ~ Y O@) ~ i (reap. Y the diagonal embedding X x y x Z -->

be a non-degenerate section of

We denote by and by

~

~

the projection

the Lagrangean submanifold of

P*(Xx Z)

XX y × Z -->Xx Z.

Let

~

be

determined by the contact

transform

q2P21qlPl I : U -->W.

P*(Xx Z);

(x, y; J , ~ ) 6 /~I

and

Then

= ~,(gry £

~ ~f))

is a maximally overdetermined

~

K = ~ , Z * (KI× K2)

~f

g*(~

In other words,

7~ ={(x, z; ~ , ~ )

(y, z;-~, ~ ) E Y~2

for some

y, ~} .

Y system on

Xx Z

with support

degenerate section of phisms K2

q2P2-i qlPl-i~ X

and

is a non-

~'Z ~Z ~ f " Moreover, the composite of isomor-~ q2P2-I~ Y

and

q2P21~y

~o Z

as in i) coincides with the isomorphism defined by This theorem being obvious, we give no proof.

special case of this theorem.

defined by

KI

K.

Theorem 3.3.2 is a

We say that isomorphism (4.3.1) is the

"quantized" contact transformation with kernel function

K.

As seen in

429

T h e o r e m 4.2.5, a non-degenerate determined

section satisfying a maximally over-

system is determined by its singular support and its order

up to multiplication by elliptic pseudo-differential O.

Therefore,

operator of order

if the contact transformation and the order are given,

then the "quantized"

contact transformation

is determined up to the

inner automorphism by the elliptic pseudo-differential order

0.

But, we can never canonically

transformation

5.

operator of

specify the "quantized"

contact

for the given contact transformation.

Structure theorem for systems of pseudo-differentia.l

equations

in the

complex domain In this section we establish the fundamental the structure of a system of pseudo-differential order in complex domain at generic points,

theorem concerning equations of finite

i.e., we will firstly prove

in T h e o r e m 5.1.2 as the simplest case that any system differential

~

of pseudo-

equations of finite order with one unknown function and

simple characteristics

can be transformed micro-locally

into the partial

de Rham system :

~x~ u = 0 ,

i = I,

"'', d

l 1!

.

11

by a suitable quantlzed contact transformation. means "locally on "micro-locally" on

S~'°M, not on

P~"~, not on

in this sense M"

X"

Here "micro-locally"

In the sequel we use the word

(and sometimes in the sense that "locally

when we consider the problems in the real domain).

Later we extend Theorem 5.1.2 to more general systems by the aid of pseudo-differential 5.1.

Structure

operators

of infinite order.

theorem for systems of pseudo-differential

tions with simple characteristics.

equa-

The idea of the proof of this

430

section is as follows:

To begin with, we "straighten out" the charac-

teristic variety of the system P'X,

~

by a contact transformation on

which is nothing but the classical

ential equations of the first order. the "geometrical

optics".

integration theorem of differ-

In other words, we first treat

Secondly we apply to the system

invertible quantized contact transformation a system

~'

of pseudo-differential

P*X

~

= 05

the system

~_'

(of finite order) and obtain

out", i.e., has the form

in the new coordinate

after the contact transformation.

tible pseudo-differential

an

equations of finite order whose

characteristic variety is "straightened {(x', ~') ~ P*X ; ~i . . . . .

~i

system on

Finally we find suitable inver-

operators of finite order which transforms

into the partial de Rham system

treat the "wave optics".

~

,

that is, we

In the later sections we use pseudo-differen-

tial operators of infinite order and we should be more careful in some details,

though the essential idea is the same.

We first recall the following classical integration theorem of Jacobi

(Cf. ~ 3.3.).

Lemma 5.1.1. codimension

d

Let

V

in a contact manifold

can find ~ neighborhood system

(x, p)

(5.1.1)

be an involutory and regular submanifold of

of

U

U

of

y

Then for any

0

in

V

we

and a canonical local coordinate

U ~ V = Ipl . . . . .

Pd = 0 I "

~4.3 we can always find micro-locally

contact transformation

a contact transformation,

(of finite order) associated with

hence Lemma 5.1.1 allows us to assume without

loss of generality that the characteristic variety satisfying the following conditions form

y

so that

By the aid of the results of a "quantized"

0

Y.

V

of the system

(5.1.2) and (5.1.3) has the

431

(x, ~) E P ~

; ~I . . . . .

~d = 0~ ,

as far as we restrict ourselves to the consideration of the micro-local property of the system

~

Therefore we next investigate the

structure of the system

~

whose characteristic variety

V

has the

form (5.1.1). Theorem 5.1.2.

Let

~

be a system of pseudo-differential

equations satisfying the following conditions (5.1.2)

~

is a coherent

denotes

a left

(5.1.2) and (5.1.3).

~f-Module of the form

ideal

of

~'~f,

of

~f~

i.e.

~f/J

the

,

equation

where ~'~

has

one

unknown function. (5.1.3)

The symbol ideal

J

is simple at

(x 0, ~0)

and that

the variety of its zeros is regular with codimension in

P'X,

i.e.

~i

is simple characteristic at

Assume further that its characteristic variety near

(x0,

7 0 ) = (0; 0, "'', I).

V

(x 0,

d (~ n-l) ~0).

has the form (5.1.1) is micro-

Then the system

locally equivalent to the partial de Rham system (5.1.4)

~

: ~x'. ~u - 0,

i = i, "'', d ,

l

that is,

~i

is isomorphic to

~L

near

(x 0,

~0)

as a left

~f-

Module. Proof. ators

By the assumptions on

Pl(X, D),

..., Pd(X, D)

principal symbol of

Pj(x, D)

loss of generality that Theorem 2.1.2.

$~I and

of the left is

~j.

V,

we can find the gener-

~f-ldeal

~

so that the

Moreover we may assume without

Pl(X, D) = D I (= ~Xl

by definition) by

In fact we can find an elliptic pseudo-differential

operator of finite order

Q(x, D)

fore we can choose the generators

so that

QPI Q

IPj(x, D)]~=I

If we can prove that the generators

IPj(x, D)I

-i

= DI

so that

holds.

There-

Pl(X, D) = D I.

can be chosen so that

432

(5.1.5)

Pj(x, D) = Dj,

j = I, -'', k+l

under the assumption that we have already chosen the generators d {Pj(x, D)Ij= I

so that

(5.1.6)

Pj(x, D) = D j,

j = I, "'', k ,

then the theorem is proved by induction on assumption (5.1.6) we first prove that it depends only on

To prove (5.1.5) under

Pk+l(X, D)

(Xk+l,.--, Xn, Dk+l,--- , Dn).

we can apply Theorem 2.1.2 to

QPk+iQ-I = Dk+ I.

If this is proved, then

Q(Xk+l,.-.,Xn,Dk+l,''',Dn)

Since the pseudo-differential operator

mutes with the operators (5.1.7)

can be chosen so that

Pk+l(X, D) and find elliptic pseudo-

differential operators of finite order that

k.

DI, "'', D k

QDjQ

-I

= Dj,

Q

so com-

by the definition, we have j = I, "'', k

This proves (5.1.5) under assumption (5.1.6). Now we prove the fact that has the form

Pk+l(Xk+l,

Pk+l(X, D)

can be chosen so that it

-.-, Xn, Dk+l, "--, Dn).

To begin with, we

prove that it can be chosen so that it has the form (5.1.8)

Pk+l(X, D) = Dk+l+Rk+l(X,

Dd+ I, "'', D n)

If it is proved, then we have (5.1.9) since

[Dj, Rk+l(X, Dd+l, IPj (x' D)~dj=l

clearly implies that fact, if

"'', Dn)] E ~

generate the ideal Pk+l(X, D)

holds for j =I, "'', k, ~

has obtained the required form.

[Dj, Rk+l(X, Dd+ I, ---, Dn)]

has order just

then (5.1.9) implies the principal symbol of Dn) ]

vanishes on

{71 . . . . .

homogeneous part of order Dn)]

m

~d = 01"

[Dj, Rk+ l(x, Dd+l,

m-l.

(>

-~0),

[Dj, Rk+l(X, Dd+ I, ''',

[Dj, Rk+l(X, Dd+ I, -'',

[Dj, Rk+l(X, Dd+ I, ..., Dn)]

This is a contradiction.

--', D n)] = 0

m

In

It immediately means that the

of the operator

vanishes identically, that is,

of order at most

The relation (5.1.9)

holds.

Therefore

is

433

Therefore what remains to be proved is the fact that can be chosen so that it has the form (5.1.8)•

Pk+l(X, D)

We first note that we

can assume without loss of generality that the pseudo-differential d ~PN(X, D)}j=k+ I

operators since

Pj(x, D) = D.

for

has the form j = I

J

-.-

~Pj(x ' Dk+l,

k

... ' Dn)}dj =k+l'

by assumption

(5.1.6).

On

'

the other hand applying to theorem of Weierstrass

Pd(X, Dk+l,

"-', Dn )

for pseudo-differential

we find an elliptic pseudo-differential

the preparation

Operators

operator

(Theorem 2.2.2),

Rd(X , Dk+ I, "'', Dn )

of finite order for which (5.1.10)

Pd(X, Dk+l,

-'', Dn) .

"''' Dn)(Dd+Qd(X'

= R d(x, Dk+l'

holds.

Here

Qd(X, Dk+l,

operator of order at most "'', Dn)

does not contain

-.., Dd, 0

Dk+ I,

"'', Dn)

.

~d

means

(We use this notation

sense in the sequel for the simplicity of notations). assume that (5.1.11)

{Pj(x, Dk+l,

d -'', Dn)~j=k+l

---, Dd,

"'', Dn) ,

Pd(X' Dk+l,

--., Dn) = D d+Qd(x,

j = k+l,

A

in this

Therefore we may

..., d-l,

Dk+ I, "'', D d, "'', D n)

by the use of Sp~'th's theorem for pseudo-differential ~Pj(x ' Dk+l,

Qd(X, Dk+ I,

has the form

Pj(x, Dk+l,

2 •2. i) . Applying to

"•, Dn ) )

is a pseudo-differential

and the symbol D d.

A

"', D d,

''', Dn)]dj=k+l

operators

(Theorem

the preparation

theorem of Weierstrass and the theorem of Spa'th for the division of pseudo-differential

operators in this way, we may assume without loss

of generality that the generators

I Pj(x ' D)} dj=l

of the system

$~6

are chosen from the beginning so that (5.1.12)

Pj(x, D) = Dj,

j = i, "'', k ,

Pk+l(X, D) = Dk+l+Rk+l(X, This proves the theorem.

Dd+ I, ..-, Dn).

(5.1.10) and at the same time it completes the proof of See the proof of Theorem 3.3.3.

434

Thus Theorem 5.1.2 combined the remark before it proves the "wave optics".

Lemma 5.1.1 on the "geometrical optics" and Theorem 5.1.2

will play their essential role in later sections. 5.2.

Equivalence of pseudo-differential operators with constant

multiple characteristics.

In

consideration of the system simple.

~5.1 we have restricted ourselves to the $~= ~ f/~

This assumption, however, is

whose symbol ideal rather restrictive

J

is

when we

consider the general system of pseudo-differential equations.

It would

be natural to desire to weaken the assumption of the simple characteristic condition in the above sense.

But in the framework of distributions

such a trial soon meets great difficulties. of distribution solutions of the equation

and that of the equation

For examples, the structure 22 ~i I : PI (D)u 2 u = 0 xI

22 ~ ~12 : P2 (D)u = (-~x~ - ~

)u = 0

are

X2

completely different, though their characteristic varieties coincide even to the degree of multiplicity. that the equations

~I

and

~12

On the contrary one can verify are equivalent in the framework of

hyperfunctions by the explicit construction of linear differential operators of infinite order

A(x, D)

and

B(x, D)

which are, roughly

speaking, invertible such that Pl(X, D)A(x, D) = B(x, D)P2(x, D) holds.

(See the example after Theorem 5.2.1 for the precise meaning

of "'invertlbll" " ity").

Note that such operators of infinite order cannot

operate in the space of distributions.

The reader will clearly find

here one of the advantages of the employment of hyperfunctions in the theory of linear differential equations even of finite order.

Motivated

by this observation, we will now extend Theorem 5.1.2 to the case where the symbol ideal

J

of the system

~i

has constant multiplicity.

Of

435

course even in this case, "geometrical for the system

~.

i.e., Lemma 5.1.1 holds

Hence applying a suitable invertible "quantized"

contact transformation

of finite order associated with the contact

transformation which "straightens ~,

optics",

out" the characteristic variety of

we may assume without loss of generality that the characteristic

variety has the form . . . . .

=

0

as a set, as far as we are concerned with the micro-local property of the system

~g

.

But as for "wave optics",

encounter a difficulty, situation,

i.e., Theorem 5.1.2 we

that is, we cannot apply Theorem 2.1.2 to our

since Theorem 2.1.2 concerns only with the case where the

symbol ideal

J

is simple.

The direct extension of the method of the

proof of Theorem 2.1.2 would, of course, be possible, but here we prefer a little more easy-going way, that is, we extend Theorem 2.1.2 by the aid of pseudo-differential the principal

symbol of the operators under consideration has the

simplest form of "geometrical

operators of infinite order assuming that

m ~ i'

Clearly it does not cause any troubles by

optics" explained above.

To begin with we prove the following theorem as a substitute

for

Theorem 2.1.2 in our new situation. Theorem 5.2.1. order

m

Let

P(x, D)

be a pseudo-differential

defined in a neighborhood of

(x 0,

Assume further that its principal symbol is differential

operator of

~0) = (0; 0, "'', 0, I). m ~ i"

Then the pseudo-

equation : P(x, D)u = 0

is micro-locally,

i.e., in a neighborhood of

to the pseudo-differential

equation

(x O,

~0),

equivalent

436

m

: D1 u = 0 as a left

~-module

can transform

the equation

pseudo-differential Proof.

(, not as a

~f-module,

~N_

operator

tial operator (5.2.1)

into the equation

preparation

differential

~

by the aid of

theorem for pseudo-differen-

(Theorem 2.2.2) we can find an elliptic pseudo-differenof finite order

Q(x, D)

for which

P(x, D) = Q(x, D ) ( D ~ - P 0 ( x ,

holds, where

That is, we

of infinite order.

By the Weierstrass

tial operators

of course).

D'

denotes

operator

(D 2

'

-'-

D')D~ -I . . . . . Pm_l(X, D'))

D ) '

and

P.(x, D')

n

is a pseudo-

j

of order at most

j,

which does not contain

D I.

Therefore we may assume that P(x, D) = D m1 _ P0(x ' D,)DI-I . . . . . Pm-i (x ' D')

(5.2.2)

from the beginning, invertible ~

(Theorem 2.1.1).

can be rewritten

(5.2.3) where mKm by

since elliptic pseudo-differential

Now, using the matrix notation,

are

the equation

in the form DIU = M(x, D')U,

U

is an unknown vector with

matrix whose components (5.2.2),

m

components

and

are pseudo-differential

M(x, D') operators

is an induced

i.e.,

0

l

0

0 (5.2.4)

operators

M(x, D')

\

1 . ""

""

° .

°

"0

Pm_l (x,D')

Pm_2(x,D')

Then it is clear that the operator

D n,

...

hence

Pl(X,D')

1

P0(x,D')j

the matrix of operators

437

If 01 Dm-2 n .

C =

"I

is invertible in

~_ .

of pseudo-differential (5.2.5) where

Using the operator operators

A(x, D')

C,

we define the matrix

by

A(x, D') = DiI(m) - C(DII(m) -M(x, D'))C -I, I (m)

denotes the identity matrix of size

account the commutativity find that the matrix

of the operators

A(x, D')

D1

m. and

Taking into Dn,

we easily

has the form

0

Dn

\

0

Dn

"'''°

"°*° ..

(5.2.6) "° •

m_I(X,D,)D~ re+l, Pm_2(x,D')D n-m+2 , We decompose

A(x, D')

Dn

0

, PI(X,D ,)Dn I , P0 (x ,D' )/

into the form A0(x , D') + N ,

(5.2.7) where

N

=

O

Dn

0 I

Then it is clear by the definition that any component of the matrix of A0(x , D')

is a pseudo-differential

operator of order at most

the other hand any component of the matrix differential

operator of order at most

~

N£ and

(i ~ ~ ~ m-l) N~ = 0

for

0.

On

is a ~ ~ m.

438

Therefore

the product of the matrices

consists of pseudo-differential x

, -.-, x

A(x ~I) , x ' , D' ) ...A(x ~m) , x ' , D' )

operators of order at most

denote different arguments.

m-l, where

Making full use of this

fact ~e will now prove the existence of the invertible matrix of pseudo-differential (5.2.8)

R(x, D')

operators of infinite order for which

(D I -A(x, D'))R(x, D') = R(x, D')D I

holds.

The matrix

R(x, D')

is given, at least formally, by the fol-

lowing series of matrices of pseudo-differential

operators

(of finite

order) R k (x, D')

,

k=O

where

R k(X, D')

equations

is defined by solving the following differential

successively: R 0(x, D') = Id(D')(m)

where the right hand side signifies the diagonal matrix consisting of identity operator in (5.2.9)

(x', D'),

k _~ I

~ Rk(x , D') = A(x, D' )R k-I (x, D') ~x I

with initial condition on equation

(5.2.9)

Rk(x, D')

xI = 0

of order

Aj(x, D')

of order

j

in

given by

0.

Here the differential

means that the homogeneous part j

in

D'

;Xl R (x, where

and for

Rk(x, D') of J satisfies the differential equation

~ ) = A N(X

)

denotes the homogeneous part of

A(x, D')Rk-l(x, D')

D'

It is easy to verify that

Rk(x, D')

is given by

(5.2 .I0)

i

x0 IA(tk'X''D')(

I

It2

0tk A(tk_l,X',D')( J0tk.l... ( 0 A(tl'X' ,D' )dtl) ...dtk_2)dtk.l )d ~ .

439

Since the coefficients of

A(x, D')

without loss of generality that

are holomorphic, we may assume

Rk(x, D')

is given by

(5.2.11)

f'" "~

A (tk,x ',D')A(tk_l,X' ,D') •• "A(t l,x' ,D')dt I- • -dtk, Vk

where

denotes a real k-dimensional simplex whose volume is

IXll Vk

k!

As we have proved before, any component of the matrix A(tk, x', D')A(tk_l, x', D') "'" A(tl, x', D') i s o f o r d e r a t most

integer such that

(k-ms) + (m- 1)s,

where

s

is the greatest

ms ~ k.

Now we will prove that the series

~ Rk(x, D') k~0

defines a matrix of pseudo-differential operator order).

converges and

R(x, D')

(of infinite

By the method of the convergence proof given below, it will

suffice to show that the series

~ RmS(x, D') s~0

converges.

Denote for

simplicity A(tms,X',D')...A(tm(s_l)+l,X',D'), by

B s, "'', B 1

As usual we denote in the sequel by

the formal norm of the pseudo-differential operator

N£(P; t)

= ~

respectively.

..., A(tm,x',D')..'A(tl,X',D')

Pj(x, D),

P(x, D)

i.e.,

j~=~ =

2 (2n)-kk!

D~p

k(X,

t2k+i~+~

k~_0 If

P(x, D)

is an

m×m

tial operators of order at most definition fjk > 0

(pjk(x' D))l~_j,k_~m of pseudo-differen-

matrix

maj-max N£(P jk,• t). l~j, k~m

£

,

then

Here for

N£(P; t)

fJk(t) = ~~0 f~kt ~=0

we define maj-max f J k ( t ) l~j,k~m

= ~ max ( f ik) t l . ~=0 l~j,k~m

Under this agreement it is clear that

denotes by with

440

N~I+£2(AIA2," t) << m N£1(AI; t)N~2 (A2; t)

(5.2.12) holds for at most

m×m

matrices

~..j

A. 3

We denote by

7_ RmS(x, D')

of order

of pseudo-differential operators of order Rj(x, D')

j.

the homogeneous part of

Then for a constant

CO > 0

and a

s~0 suitable neighborhood

~'

of

(x; ~') = (0; 0, ''', 0, i)

we have by

(5.2.11) and (5.2.12)

(5.2.13)

sup IRj( x, (x, ~' )ELO'

~')I ~_

~

((m-1)s-i)' s~

(m-l) s_2j

(ms) ~

CO

I~'I =i sup

Nm_l(BS ; ~,)...Nm_I(BI ; ~,) & j-(m-l)s

sup

(x, ~')(~0' (tl,''',tms)eVms I~'I=1

for any

g > 0.

Here we may take

C O __<m(

sup

IXl Im)

and

g '2

(x,~' )~ ~' = 2ng .

Clearly we can find

C1 > 0

such that

sup sup N (B s. ~,) (BI; g,) _< s (x, ~')e~' ...,tk)eV k m-i ' '''Nm-i - C1 I~'I=1 (tl' for &'<<

i.

Therefore (5.2.13) implies that (5.2.14)

sup ~Rj(x, ~')I <= ~ ...((m-l)s-j)~ .. (x, ~' ) ~ ' (m-l) s2j (ms)'.

JC2 g

I~'|=i holds for constants

g, C 2 > O.

Denoting by

r. 2

the left hand side

of (5.2.14), we will now prove (5.2.15)

lim ~j!rl = 0 j~ 3

and (5.2.16)

r.J =~ (-j)~C -j

for a constant

C > 0

Clearly (5.2.15) and (5.2.16) prove that the series

if ~

j <__ 0. RmS(x, D')

se_0 defines a pseudo-differential operator (of infinite order). We first prove (5.2.15).

Let

~(j)

denote

441

((m-l)s-j)~ s (ms)! C2

In virtue of Stirling's formula, i.e.,

(m-l)(s-l)~j xXe'X(2Ttx) I/2 ~ P(x+I) ~ x x e -x(2~x)i/2el/12x we easily see that for sufficiently large

for

x > 0

j

ff ~O

?(J) ~J m-i

F((m-l)x-J+!,) x r(mx+l ) C 2 dx (C2e)XeJ ((m-l)x-j) (m-l)x-j

~2~ J~° Let

a = m-i ~ i

and

dE

(~)mx

m-i --Y m-i

x

°

Then

e-J ~(J) < ;]~Cy(y-j)y-j(my)my(m_l )dy ~ 2 = holds for a suitable constant >I,

since we may make

taking

~0'

C3, C4.

;j~(Y-J)Y-J dy (C4y)ay

Here clearly we may assume

C2 = CoCI g-(m-l)

sufficiently small.

.

as small as we please by

Obviously we have

(y-j)Y-J lj

dy ~ Jj~YY-J dy ~ f 7 y-ay+Y-Jdy (C4y)ay = (C4Y)ay j_j -

since

a > i

and

j_(a_l)Ydy ~ j_j

;7

C4 > i. Clearly j-(a-l)j lira (a-l)log j- 0 j->m

-

as

j-(a-l)j (a-l)log

j

'

a > I ,

hence, using Stirling's formula again, we have lim ¢ j~ T(j) = 0 . j--~ Clearly lim J~i~ ~_~ ((m-l)s-j) ~ C2S = 0 J-~ ..J... +l>s>" j (ms) m-i = m-i hence

,

C4

442

lim j-~

J~j !r .

= 0

.

J

This proves (5.2.15). Now we go on to the proof of (5.2.16). gives for any

For

j < 0

(5.2.14) clearly

A > i r. g-JA j j (-j)~ ~

s~O

((m-l)s-j)~ s A j (ms)'(-j)' C2 "

"

and the right hand side obviously tends to zero for sufficiently large A,

because we may assume

C 2 < i.

This proves (5.2.16).

Thus we have shown that the series

~. Rk(x, D') k~0

defines a matrix of pseudo-differential operators

converges and

R(x, D').

recursion formula (5.2.9) immediately implies that

Then the

R(x, D')

satisfies

the following: (5.2.17)

I ( Z x I - A(x, D'))R(x, D') = 0 R(x, D')I,Xl=0 = Id(D')(m)

Therefore (5.2.18)

R(x, D')

satisfies

(DI -A(x, D'))R(x, D') = R(x, D')D I

by the Leibniz formula for pseudo-differential operators. Moreover, considering the adjoint operator can find

R(x, D')

tD I

t(A(x, D')),

we

such that

(tDl _ t(A(x, D')))R(x, D') = ~(x, D')tD I

(5.2.19) Note that

{ R(x, D')IXl=0 = tld(D')(m) t(A(x, D'))

prove the existence of Now we define clearly satisfies

has all the properties of

A(x, D')

used to

R(x, D').

S(x, D')

by

tR(x, D').

Then by (5.2.19)

S(x, D')

443

(5.2.20)

since

R(x, D')

~

S(x, D')(D I -A(x, D')) = DIS(X , D')

[

S(x, D')IxI= 0 = Id(D')(m)

does not contain

(5.2.21)

D I.

Therefore

, S(x, D')

satisfies

DIS(X , D') = S(x, D')(D I -A(x, D')) .

Now we want to show that (5.2.22) holds.

R(x, D')S(x, D') = S(x, D')R(x, D') = Id(D') (m) If (5.2.22) is proved then

R(x, D')

is clearly invertible in

a neighborhood of

(x; ~ )

invertible there.

This is nothing but the required result, that is the

systems

9t

~[

and

By the way,

= (0; 0, 0, "'', 0, I)

are equivalent as left

as

Id(D') (m)

is

p-Modules.

(5.2.22) can be easily verified as follows.

We first

note that (5.2.23)

DIS(X , D')R(x, D') = S(x, D')R(x, D')D I

and (5.2.24)

(D I -A(x, D'))R(x, D')S(x, D') = R(x, D')S(x,D')(DI-A(x , D'))

follow from (5.2.18) and (5.2.21). for pseudo-differential (5.2.25) Moreover

In virtue of the Leibniz formula

operator we deduce from (5.2.23) ~ x I (S(x, D')R(x, D')) = 0 .

(5.2.17) and (5.2.20) imply that

(5.2.26)

S(x, D')R(x, D')IXl= 0 = Id(D') (m)

Therefore,

if the solution of the Cauchy problem (5.2.25) and (5.2.26)

is unique, then S(x, D')R(x, D') = Id(D') (m) holds.

On the other hand the uniqueness property of solutions of the

above Cauchy problem is clear. operator

In fact, if a pseudo-differential

444

F(x, D') = ~_~ Fj(x, D') j=-~ satisfies

(5.2.25)

and t h a t

D ' ) l x l = o = O,

F(x,

I

~Fj(x, 7' )

then we have for any

j

-0

~x I Fj (x, ~')IXl=0 = 0 . This immediately proves that j

Fj(x, 7')

and it amounts to saying that

identically vanishes for any

F(x, D')

vanishes identically.

Thus

we have proved that S(x, D')R(x, D') = Id(D') (m) holds. Concerning

(5.2.27)

R(x, D')S(x, D'),

it also follows from (5.2.24) that

~X-~l(R (x,D')S (x,D')) -A (x,D ')R (x,D') S (x,D')+R (x,D')S (x,D')A (x,D') =0

Clearly we have by (5.2.17) and (5.2.20) (5.2.28)

R(x, D')S(x, D')IXl= 0 = Id(D') (m)

We will now show that, if a pseudo-differential operator =

~ Fj(x, D') j =-~

(5.2.27 ')

F(x, D')

satisfies

--F(x, ~x I

D')-A(x,

D')F(x, D')+F(x,

D')A(x, D') = 0

and (5.2.29) then form

F(x, D')

F(x, D')Ix =0 = 0 1 is identically zero. If we expand

~_~ x J F (J) (x' , D' ) ,

then

F(x, D')

(5.2.29) is nothing but

j=0 F(0)(x', D') = 0 . Moreover (5.2.27') and (5.2.29) imply that

in the

445

~-~-F(x, ~x I that is,

F(1)(x ', D') = 0.

D')I

~x~F(x' i

that

F (j)(X',D') = 0

F(x, D') = 0

Since both

if

'

Differentiating

the same way using the induction on

that is,

= 0 Xl=0

j

that

D')IxI=0 = 0,

for any

F(x, D')

R(x, D')S(x, D')

(5.2.27') we can verify in

j ~ 0 ,

j ~ O.

Therefore we have proved

satisfies

(5.2.27') and (5.2.29).

and

Id(D') (m)

satisfy (5.2.27) and

(5.2.29), the above assertion proves that R(x, D')S(x, D') = Id(D') (m) holds.

Thus we have proved

(5.2.22) and, at the same time, completed

the proof of the theorem. Remark.

The proof above shows immediately that determined systems

of pseudo-differential

equations ~f~: DIU = 0

and : (D 1 -A(x, where

U

is an unknown

matrix

consisting

differential -modules

of

vector D1

operators if

A(x,

(5.2.30)

any

(5.2.31)

If we

not

D')

denote

by

I,

D')

containing

of

A(x,

the D')

there

is an DI,

Al(X , D')

then

= 0

,

m components,

A(x,

satisfies

component

of order

and

of

D'))U

are

D1 m×m

~I)

,

is of order at most

the

A I (x I

~0 -1.

of pseudoas

at most

I.

is of order

(~o)

'" "

matrix

left

following:

exists

,

, x , D )

diagonal

equivalent

homogeneous

part

an integer

~0

component of Al(X

is the

, x' , D')

of

A(x,

D')

such

that

any

446

Before extending Theorem 5.2.1 to more general systems we give an example, which was our first motivation. Example.

Let

n = 2

and consider

22 2 ~x I

P1 (D) -

and

P2(D) =

~2 ~x~

Then we can find linear differential operators

~x 2

in an explicit form so that

IAj( x' D)I~= I

I P2Al = A2PI,

A3P 2 = PIA4 ,

(5.2.32) A4A 1 m 1 hold.

Here

mod

AIA4 ~ 1

mod

~ P2

denotes the sheaf of linear differential operators

(of infinite order).

Then the equations

equivalent by the correspondence Proof.

~PI'

First note that

u = A4u ,

cosh(Xl~2)

sense as differential operators. 2n .~ xI ~n sinh(Xl~-2) ~-~ (2n)' ~x n and n=0 2

P1 u = 0

In fact

and

P2 v = 0

are

v = AlU. and

s i n h ( X l ~ 2) ~2

make

cosh(x1~o)

is by definition

is by definition 2n+l Xl ~n n=0~ (2n+l)'. ~x 2 "

Now we set sinh (Xl~-~2 ) AI(X,D ) = (cosh(Xl~-D2))(l-xlD I) +

D1 , S inh (xi$~2)

A2(x,D ) = (cosh(Xl~D2))(l-xlD I)+

(DI-2XlD 2) , sinh ( X l ~ 2 )

A3(x,D ) = (cosh(Xl~2))(l+xlD I) +

(-DI+XlD 2)

and sinh (Xl~2) A4(x,D ) = (cosh(XlV~2))(l+XlD I) + Then one can verify (5.2.32) by a direct calculation.

(-DI-XlD2). We leave it to

the reader. Remark i.

If we employ the matrix notation as in (5.2.3), then

we easily find instead of (5.2.32)

447 (5.2.32')

\~sinh<-x:~) cosh(-Xl~) /~-~2 D/"~sinh(Xl~)

c°sh<Xl~2)J

:(" 0) 0

DI

holds. Remark 2.

This example shows that the structure of the hyperfunc-

tion solution sheaf, the equation of because

not merely the microfunction

PI(D)u = 0

the operators

pseudo-differential

Aj(x, D)

operators.

is a linear differential of the origin of ential equation -Modules.

and that of

¢n

P2(D)u = 0

are differential

whose principal and

m

D~u = 0

and

if

~,

D'

P(x, D )u = 0

respectively. as in (5.2.16).

P(x, D)

then the differ-

To prove this we introduce an auxiliary variable W

not merely

are equivalent as left

of the origin of

the sake of clarity of the notations we write D

operators,

defined in a neighborhood

symbol is

consider the problem in a neighborhood

of

are the same,

Note that, more generally,

operator of order

P(x, D)u = 0

solution sheaf, of

D

x

and

Dx,

t

and

£n+l.

For

instead

To begin with we rewrite the equation Then we easily see that

for

~=

2 m ,

X

any component of the matrix of differential

operators

M(~) - M<x~), x',~,) ". M(x~1), x',~x,) is of order at most

f-i

is of order

and any component of the matrix of pseudo-differen-

m(m-l)

tial operators respectively.

C(Dn)

and

since any component of

C(Dn)-I

is of order at most

Of course near the point where

other pseudo-differential

operators

not matter as far as the order of

C M (g)

and

~n = 0 C "l,

(m-l) and

0

we must use

but clearly it does

is concerned.

Therefore,

448

replacing

m

by

~ ,

we can proceed as in the proof of Theorem 5.2.1

without using the matrices

C

and

C -I.

Though

A(x, D ,) and

C -I

X

are matrices of pseudo-differential operators, not differential operators,

M(K, D ,) is a matrix of differential operators.

Hence in a

X

neighborhood find

~

of

R(x, t, Dx, , Dt)

(x, t; ~, ~) = (0, 0; 0, I) and

S(x, t, Dx,,Dt)

(5.2.17) and (5.2.20) respectively. that

R

(t, Dt)

and

S

in

~'

R(x, Dx ') and

we can

so that t~ey satisfy

R

nor

= {(x, t; 7' ~) & ~ ; ~ # 0~.

S(x, Dx,)

P*W

The proof of the theorem shows

are unique, hence neither

assures the unique existence in

in

~'

S

depends on

In fact the theorem

of pseudo-differential operators

satisfying (5.2.17) and (5.2.20) respectively

and there they must coincide with

R(x, t, Dx,, Dt)

and

S(x, t, Dx,, Dt)

respectively.

On the other hand, the analyticity of the coefficients

of

clearly proves by the unique continuation property that

R

and

neither

R

S nor

S

depends on

are defined even when definition that

R

and

~= S

0,

(t, D t)

OO . Therefore

R

and

S

and it immediately implies by the

are linear differential operators, not

merely pseudo-differential operators. differential equation

in

P(x, Dx)U = 0

Thus we have proved that the and

m Dlu = 0

are equivalent as

left ~-Modules. Lastly we remark that the equivalence as left ~ -Modules trivially implies that of holomorphic solution sheaves of these equations, since operates on 5.3. equations.

~

.

Structure theorem for regular systems of pseudo-differential Now we extend Theorem 5o2.1 to more general systems.

the course of the proof Theorem 5.2.1 itself is very powerful.

In To

449

begin with we reformulate Theorem 5.3.1. every fiber of

f

projection from variety of

admissible

Let

f : Y --~X

be a smooth map.

is simply connected.

Y m P~ X

P~.

of admissible

it in the following form.

Let

to ~,

We denote by

We identify

be an open set in

~x-Modules

~y-MOdules

P*X.

$I

~

defined on

~-i(~)

~Y the canonical

Y X P~ X

P*X.

with a sub-

Then the category

3)_ and the category of

defined on a neighborhood

with support contained in

Suppose that

~-l(fa)

of

is equivalent by the following

correspondence

= PY~X

(5.3.1)

(5.3.2)

~ ~-l~i m-lpx

~-i ~ = ~ p y ( ~ Y ~ X ' ~)

hence

= ~Y*J~2y(~Y~X' Proof.

~)"

At first note that the problem is of local character.

the induction on the fiber dimension of fiber dimension is equal to

I.

and that

the homogeneous

f(t, x) = x.

fiber coordinates

we may suppose that the

Hence we assume in the sequel that

X = cn = ix = (Xl, -'-, x n) E ~n~, 6¢n+i}

f,

By

y = ~n+l = ~(t, x) = (t, Xl, -.-, Xn )

Let

(~'

of

P*Y.

~) = (< ' ~I'

"''' ~n )

Since

~-1%) = ~-l A J~y(

Ty_~x, T Y->X ~ % X

is evident,

it is sufficient to show that, for every admissible

Module

with support contained in

97%

admissible

~x-Modul e

~

~f

be a coherent

there exists an

such that = ~Y-~X

Let

~y-l~)~

~y-

~f-Module

~ ~y-i ~ - l~x such that

~f~= ~ y

~f ~[f.

9y

Since

be

450

Supp ~i f C { ( = 0 ~ ,

f

is a quotient of f m (~y)

where

~t'

f

which is the cokernel of

Dt-A(t'x'Dx)~ ( ~ f ) m ......... is an

A(t, x, Dx) = j~l~-~Aj(t, x, Dx)

differential operators of order Al(t , x, ~ )

$ I,

(mxm)

matrix of pseudo-

such that the principal symbol

is nilpotent and independent of

t.

By Theorem 5.2.1,

there is an invertible matrix of pseudo-differential operator such that

R(t, x, D X) (5.3.3)

~D t -A(t, x, Dx)}R(t , x, Dx) = R(t, x, Dx)D t ,

and (5.3.4)

R(t, x, Dx) It=0 = Id.

(Here we consider in the neighborhood of the origin Ul! '

. • •

' U m!

be the images in

~'f

t = 0.)

Let

of the basis of ( ~ f ) m

lul

U

!

_--

\u'/ m

We have {D t -A(t, x, Dx)~U' = 0 . Set

v' = R(t, x, Dx)-lu ' e ~dL' = ~ y

Dtv' = 0, Let

and the components of ~"f

be the kernel of

v'

~f ~'f. Py

Then by (5.3.3)

generate

~['.

--~ ~ f .

Set

~'f

= Py Since I (~Y~X' we have

~' ) = 0

Set

451 ~X~ I ~y(~Y~X' ~ )

: 0

In fact, as D 0 ---> ~ y

t ~ ~ y ___>~ Y ~ X --+ 0

holds by the definition, ~z~t2 (~Y-+X' ~ " ) ?Y vanishes. Then this reasoning holds without any changes if we take

$~L" as

hence ~>~y(~Y~X'

~")

= 0

holds. Therefore we have the following diagram with exact rows:

% 0

"-

1

W/."

....

~'

%

~-'

~

9"d.

.------~

0 .

Clearly ~"~y(~Y-~X'

m iiF_l~ xV = j=l ~

~')

Set

: Oq~o~py (@ Y~X' ~ ) '

~'

These are

~Y-l~x-Modules.

?y( : ~"

~ Y~X'

c]F£'),

Consider the following diagram with

exact rows:

(5.3.5)

~Y~X

~

~l~x ?rL"

£"

--~' ~Y'-~X

.....

>

~)

£'

~TY~X

~zL'

The middle arrow is clearly an isomorphism.

® Z -~0 ~l~ x

~x

~,

~

>

0

.

452

This hand

implies

satisfies

97["

the same condition as

Therefore instead of

~i ,

On the other

that the right arrow is surjective.

by the application

we conclude

~_ ,

i.e.

Supp ~ '

of the same reasoning

to

C ~["

that

~

~Y~X

,~ " ~

~'Zi,_"

~-~x is surjective,

Hence

that is, the left arrow in (5.3.5) is surjective.

the right arrow is an isomorphism. We embed X ¢-+Y.

X

into

Y

by

t = 0.

We denote by

j

the embedding

Set @?- = j*$??. ,

These are admissible

@i' = j * ~ '

and

~2." = J*~i"

Consider

~x-Modules.

the following diagram with

exact rows: 0 --~ £ " I t = 0 0

Since

the

middle

Therefore, arrow

is

the

)

arrow left

TL"

is

arrow

an isomorphism.

presentation,

---> ~ ' I t = 0 ---> £ 1 t = 0 ~

TL'

isomorphic, is

>

the

surjective,

Since

~

right which

is

a

9l

--+ 0

-

>

arrow

0

is

implies

~x-MOdule

.

surjective.

that

the

locally

of

right finite

we h a v e

J6.~_ _~x(~-l%, Z )It=0 =J~?x (~' ~1 t=0 )" Therefore,

in a neighborhood

o--~

of

~"

t = 0,

>

Z'

with exact rows.

~

£

~y-I

~/, ' __._~.~,-i~ - ' ~ 0

The middle arrow is isomorphic.

arrow is surjective,

and so is the left one.

~tpX(~)y.+x,

>o

T

t 0 --+ ~¢-i ~ "

we have the diagram

~)

Therefore

This implies

: "~r'l ~F~ ,

the right

that

453

completing the proof. We give several direct applications

of Theorem 5.3.1.

More

profound applications are given in later sections. The shape of the support of the If

P(x, D)u = Q(x, D)u = 0,

ideal

I

of

u

(I =

then

4/~_1

~x-MOdule [P, Q]u = 0.

if we put

~m

closed under the Poisson bracket operation. result.

(See also Guillemin

Theorem 5.3.2.

is of special kind. Therefore

= {P6 ~m;

the symbol Pu = 0})

is

We have more decisive

[i].)

The support of an admissible

~x-Module

is an

involutory analytic set. Proof.

Let

Suppose that functions

~

~

is not involutory.

f, g

homogeneous

in a neighborhood f

and

g

be an admissible

vanish on

may suppose that if we put

~

f =

then there is a C ~ x I = 01,

g = Xl'

? : X--+Y,

Ty-Module

~[

x* e ~

1

and

0

such that

respectively, [f, g] = 1

defined

and that

By a suitable contact transformation,

~i'

Y = C n-I ,

A

Then, there are holomorphic

of degree

of a point

~x-MOdule with support

~

must vanish.

x* = (0, dx 2 ~ ) .

By Theorem 5.3.2,

~(x I, ..., x n) = (x 2, such that

~

= ~*~

we

.

"', Xn), Since

Supp

This leads to a contradiction.

In order to proceed further, we should prepare some more propositions on the cohomological properties of the sheaf has proved several cohomological properties linear differential sheaf

~

f

,

in the case of

~f.

of the sheaf

Kashiwara ~ f

[3]

of

operators of finite order, and concerning the

we can also prove the same results in the same manner as ~f.

ties of the sheaf Theorem 5.3.3.

Hence, ~ f Let

in this report, we only mention the proper-

without proof. ~

be a coherent

f ~x-Module.

In order that

454

f ~X)

~xf(~If,

= 0

for

i < d ,

it is necessary and sufficient that the support of set of codimension

~_ d

Theorem 5.3.4.

in

Let

~t

is an analytic

P~. ~ Xf -Module.

~//~f be a (left) coherent

Then

f ~x-MOdule

the support of a coherent right

if( f ~x is an analytic set of codimension

~_ i.

Since •

_

F

f

~x~t if(~/~- , ~ X ) = 0

for

i > d

Px is equivalent to saying that ~_ d,

$~f

is locally of projective dimension

and that the codimension of non empty involutory set does not

exceed the dimension of Corollary 5.3.5•

X,

we obtain, f ~X

The global dimension of the stalk of

cides with the dimension of

coin-

X.

Now, let us define the good family of ~-Modules. Definition 5.3.6.

We say that a coherent

~f-Module

~_f

(respec-

A

tively an admissible

~x-MOdule

i f f ~x~ f( ~ , ~ X ) vanishes except for o~ly one

~)

is purely dimensional,

(respectively

~X(9~,

if

~ X ))

i.

By Theorem 5.3.3, this is equivalent to saying that the local projective dimension of

~i f

codimension of the support.

(respectively

@II )

is equal to the

This is the notion corresponding to that

of Cohen-Macaulay module in the theory of commutative algebra. Moreover,

if

~f

(respectively

its support is a regular submanifold,

~)

is purely dimensional and

then we say that the system

~fLf

455

(respectively ~i)

is regular.

This notion is a natural generaliza-

tion of the so-called constant multiplicity condition on a single differential equation.

The following theorem is a direct consequence

of Theorem 5.3.1 and it shows the usefulness of the condition that the system

~i

is regular.

It will be clear that Theorem 5.3.7 gives a

natural generalization of Theorem 5.1.2. Theorem 5.3.7. let

~0

and

~.

Then

~

~

Let

~

be a regular submanifold of

be admissible and regular

By a suitable "quantized"

subset of

~(x, ~) ~ P*X ; ~i . . . . .

Y = Cn-d.

The map

Y,

set of

~x-Module with support

~0" contact transformation, we may

assume without loss of generality that

to

and

is locally isomorphic to a direct summand of a

direct sum of finite copies of Proof.

P*X

i.e.,

~

X = Cn ~d = 0}

and

~

with

is an open

d~

n-l.

Set

denotes by definition the projection from

~(Xl, ---, Xn) = (Xd+l, ---, Xn )"

Then

~

X

is an open

X K P*Y. Y

The question being of local character, we can assume that each fiber of

~

--~ ~(A)

is simply connected.

exist admissible

~y-MOdules

and

hold.

~

=

~*~

~0

and

~y(~, ~

In the same way by replacing

~

and

such that

~i0 = ~ * ~ 0

On the other hand Theorem 3.5.6 proves that

Since the left hand side vanishes for

This implies that

~

By Theorem 5.3.1 there

i # d

~y)

= 0

by the assumption, for

i # 0.

is locally projective. ~0 ~0

is seen to be locally projective. by

suppose from the beginning that

~ ~

and

~0

Therefore,

respectively, we may

is an open set of

P*X.

456

Now we prepare the following: Lemma 5.3.8.

Every stalk of

~ Xf

is a simple module as

~x-bi-

Module. Proof. P(x, D)

Let

I

be a non-zero element of

symbol of

P(x, D) 0

assumption

P £ I

symbol of

[P, xl]

~x I Pm (x' ~ ) indices

(~, ~)

the other hand, Therefore

Since

implies is

I

Io

Pm(X'

[P, D I] ~ I.

~ )

Let

denotes the principal f ~X,x*'

is a two-sided ideal of

[P, Xl] ,

~I

Pm(X, ~ )

the

Since the principal

and that of

[P, DI]

is

on one hand and since there exists a pair of multisuch that I

DxD~P re(x, ~)

does not vanish at

x*

contains an elliptic pseudo-differential

f I = ~X,x*

"

f ~x-MOdules

on

operator.

This completes the proof of Lemma 5.3.8.

Proof of Theorem 5.3.7 continued. projective

~ X f, x * "

be a non-zero two-sided ideal of

Let

~0

f

and

~-

be locally

such that f

%=

, f

Px

2x

There exists a non-zero homomorphism f Then by Lemma 5.3.8, there exist f ~X such that

f

Tx" Ul, "'', u r E ~i~

and

RI

..., R '

r

r 1 = ~ ~(uj)Rj j=l Therefore,

there is a surjective map

(

)r --~ ~ Xf"

Since

~if is a

quotient of a free Module, ~ f is a quotient of the direct sum of finite _f f copies of ~ 0 " Since $F6- is locally projective, ~ is locally a direct automata of

(~it0)N. This completes the proof of Theorem 5.3.7.

CHAPTER III,

STRUCTURE OF SYSTEMS OF

PSEUDO-DIFFERENTIAL EQUATIONS I.

Realification of holomorphic microfunctions We have thus far discussed the theory of pseudo-differential equa-

tions etc. in the complex domain

To apply the theory developed in

this paper to the hyperfunction theory, we must give a relationship between holomorphic hyperfunctions and (ordinary) hyperfunctions. This resembles the relation between holomorphic functions and real analytic functions. i.i

Realification of holomorphic hyperfunctions.

n-dimensional manifold, and Y

X

and

Y

N

be a submanifold of

be complex neighborhoods of

is a submanifold of

X

and

M

M and

Let

M

be an

of codimension N

respectively.

N=M~Y

Recall that ~y(~x

) = ~y~x[-d]

By Proposition 1.1.2 in Chapter I we have the canonical homomorphism (i.i.i)

m~y(~X)

IN " - ~ P N a ~ Y ( ~ X ) [ n - d ] @ W N

"

The right term of (i.i.i) is equal to ~N(~X)[n-d]~

N =

a N N a r M ( ~x )[ n- d] ~N

n

= a~N~M(@X

=

)[-d]~N

0 ~ N ( ~ M ) I-d] ~0]N/M

where d ~N/M = ~{N(~M ) = ~ N ~ M

= ,

aPN~M

['d]~N/M

d,

458

It gives a homomorphism 0 X : ~yix ® w N/M-~N(6M

(1.1.2)

)

This is expressed explicitly using the local coordinate system as follows. such that

Let N

(x I .... ,Xn)

is defined by

The section

u =

be a local coordinate system of Xl= ..=Xd=0.

a~(x")~(~)(x ')

~

M

GIx

of

is expressed by

the defining function

(1.1.i)

f(x) =

~

a,(x")~(x')

,

d

where II.)

x'=(x I ..... Xd) The image of

and

u(x)

x"=(Xd+ 1 ..... Xn).

of Chapter

(Cf. ~ 1.4

under the homomorphism

X

in (1.1.2) is

given by

O.1.4)

~(x) = ~ a =

(x")D~, ~ ( x ' )

(_)d

£1...£df(Xl+£1i0,x2+E2 i0 ..... gl ..... ~=±i Xd+ ~i0,xd+ 1 ..... Xn).

Now we can prove the following Proposition

i.i.i.

The homomorphism

~ in (1.1.2)

and its image is the sheaf of germs of hyperfunctions support in

N

M

with

Suppose

~(x)

~S~M given by (1.1.4) vanishes.

a~(x") = all

on

with singular support contained in the pure imaginary

conormal spherical bundle Proof.

is injective

a (x") v a n i s h .

Since

(_) L~l f xt~(x)dx ' al J

T h i s i m p l i e s t h a t t h e homomorphism X i s i n j e c t i v e ,

459 The image is evidently contained in the sheaf of germs of hyperfunctions with support contained in in

N

and singular support contained

~E-~S~. Let us prove the converse in two s t e p s (1-st Step) Let

v(x)

S~

S~=I_ ~

d=l.

be a hyperfunction with support in

support in ~

holomorphic

The case where

, defined in

dXl} , v(x) functions

a neighborhood

of

N

and singular

x=0.

Since

is the sum of boundary values of two

:

v(x) = f+(xl+i0,x 2 ..... Xn)-f_(xl-i0,x2 ..... x n) where

f+ (resp. f_) is a holomorphic

function on

~+ (resp~_).

%

contains IzeX

= C n" Izl = maxlz.l
Since Supp v ~ N , function

f(z)

i

there are an open set defined on

~9_D{xEM and that such that

f(z)i~ f(z)

1

n ImZl~ ~ ~ llmzjl}. j=2

= f±(z).

fi

~

and a holomorphic

such that

= Rn ; Ix~0

is holomorphic on ;

By Theorem 3.1.1 in Chapter I, ~ contains [z6X for sufficiently following

n ; Izi<6, ~IZll > ( ~ I I m z j l ) } j=2

small

~ , ~ > 0.

To proceed further we prepare the

Lemma 1.1.2. The holomorphic function f(zl,z2)

defined on a

neighborhood of

can be continued to a holomorphic function defined on a neighborhood of

Proof. By Lemma 3.1.3 in Chapter I, f

for every

r 1 and

ax. Letting

L r ,

rl

is holomorphic on

+ 00,

we obtain the

I.

desired result. q.e.d. N

Now, let r~

2

+

where

c,

fi

by

be the envelope of holomorphy of - n w (w,z) w (e z , 0 , * -,0). Then

p

satisfy

-

By Lemma 1.1.2, w

z

a. Set 7 :

€

2

;

2


and

(a) contains

p<&.

yl(a)contains Re z

=

0,

1I.m z

1 < $pa,

sor some positive number a. Therefore,

Im w

<

contains

log

461 Since

is invariant under rotations, ~ contains n

Iz~g ; 0< IZll<~,Rezj= 0

for

j=2 ..... n~

i imz21 2 +...+[imZn~2 < ( g ~ a ) 2 ~ and therefore contains I z ~ ¢n ;0
Imz212+...+ llmznl 2

<

4

m

This completes the proof of (1-st Step). (2-nd Step)

In the general case, we set

v(x", ~ ,p) =

where and Then

U(X) (d-l): i (-2~i) d (p-<x', g>)d

dx'

x' = (xI ..... Xd)'X" = (Xd+l ..... Xn)' ~ =(~i ..... ~d )' u(x)

is a hyperfunction with singular support in ~7~ S*M N

v(x", ~,p)

is a real analytic function in

~ E £ d, p~C-{01, holomorphic in (~ ,p) with respect to (~ ,p). -v(x", ~,p-i0)

x"

'

when

and homogeneous of degree (-d)

Now, let ~EIRn,

p~IR.

is a hyperfunction with support

Then in

v(x", ~ ,p+i0)

e=~(x", ~,p);p=0}

and with singular support contained in the pure imaginary conormal bundle of L

This implies, by (1-st step), that v(x", ~,p)

holomorphic on

I(x", ~ ,p) E Cn-d x C d× C; ~x"~ < ~ , J~l<~, 0
for sufficiently small

g

Since

v(x", ~,p) is homogeneous in

(~ ,p), v(x", ~ ,p) is holomorphic on

{(x", ~),p); Therefore

is

lx"~
v(x", ~ ,p) can be developed as

O+p} .

462

a~(x") ~ ~p -d-lv.I ~6~d + which is normally convergent

Therefore

~(x) = ~ (-)i~l ( _ 2 v E ~ ) d (d+ ~ i - l ) ~ is in the image of the homomorphism

v(x", ~,p+iO) =

a~ (x'')D~x' $ ( x ' )

A

and

(d-l) : f U(X) ( - 2 ~ ) d J(p-<x', ~>+iO) d

dx'

Since

(d-l),. (12~) d

]"

1

=

(<x',~) +iO) d ~0(~)

g

(x'),

we have Iv(x", ~ ,<x', ~>+i0)~0(~ ) =

(d-l)'' I( (-2m~) d

=

u(y'x")

dy ~ ( ~ )

<x'-y',[> +i0) d

I ~ (x'-y')u(y',x)dy' I!

= u(x',x").

In the same way, we have Iv(x", 5 , <x', ~>+i0) ~ ( ~ ) = ~(x). This implies that

u(x)=~(x)

and at the same time completes the proof

of the theorem. Remark.

In the proof of (1-st Step) we have not made the full

use of the assumption.

Concerning this point we refer the reader to

Morimoto [3],[4]. 1.2.

Realification of holomorphic microfunctions.

same notations as in §i.i.

We denote

by

j

We use the

the canonical embedding

463 of

~S~

(--~-S~X ~ S~X)

into

S~X.

In this section, we define the canonical homomorphism j-I

(1.2.1)

~ Cyl X

0 > ~[ ~ f ~ S ~ ( ~ M) ~ ~N/M

At first we prove the following Proposition 1.2.1. submanifold of We denote by

X, p

and

Let N

X

be a real analytic manifold,

be a submanifold of

and ~I the canonical maps

Let ~ be a sheaf on

X.

M

M

be a

of codimension

MX"--#Nx"

and

d.

NX*--~ X.

Then, we can define two canonical

homomorphisms

(1.2.2)

'RiP S,MX

(~ -i~

) [d],

'R~ S~X (~r-1~ ) i S~X × N --~'~ s~x x N (p-l~-l~)M M

(1.2.3)

Proof.

The first homomorphism is nothing but the homomorphism

defined in Lemma 2.2.5 in Chapter I. The second homomorphism is easily defined, since

p -I(s ~ X)=S ~X

N. M We will apply this proposition to the situation

q e.d. XDYDN

use the notations:

y~y, ~Y/X: ~ - - ~

X,

XN/x:NX * --+ X,

in order to avoid confusion.

~M/X: "2 -~ X

Homomorphism (1.2.2) reduces to

We

464

-I × N (~IN/X~X) y

~S*X Y

[n-d]

Next, we apply the proposition to the situation

XDMDN

Then homomorphism (1.2.3) reduces to (i .2.5)

-i

mrS~X(~N/X6}X)I S~X ~

,Rr

N

M

M

As the composite of (1.2.4) and (1.2.5), we obtain (1.2.6)

-i (>x)IS~X x nRrS~X (~Y/X Y

N ® ~N/M

-i IR~"S~X ~S~X (TEM/X ('9"X) [n-d]~g°M/X " Taking the d-th cohomology of (1.2.6), we obtain ~

j{0

s~x ~ s~x (C M)

Y J X ~ ~ N/M

Thus we get the following Proposition 1.2.2. (1.2.7)

j-i

If we denote by

p

We have the canonical homomorphism

IR __+~ ~yi X ~ ~N/M

0

i

(~M)

the two-fold embedding of

~S~M

into

P~X ,

we have -i (1.2.8)

P

__~ gYl X O ~ N / M

0 ~fT S~M (~M)

This homomorphism is compatible with (1.1.2). Remark.

The cohomology

~ X

definition of homomorphism (1.2.7)

-i (~N/X~'X)

encountered in the

is an important one to formulate

the boundary value problem in the hyperfunction theory. Corollary 1.2.3.

Let

M

be a real analytic manifold,

X

be a

465

complex neighborhood of P*X.

M, and

p

of

be the projection

~f~S~X

to

Then we have a homomorphism of sheaf of rings

(1.2.9)

P-I~x

Therefore

is a left

~M

--~ ~ M

p-i ~x_Modul e and

~M ~M~M

is a right

p-l~x-Module.

(~M

coefficients.)

This is compatible with the operation of the sheaf ~ X operators.

of differential of Chapter I,

is a sheaf

~M

of densities with real analytic

As is seen by Examples 3.2.4

is not flat over

p-l~x

"

3.2.5

This shows how delicate

and interesting the (local) theory of (pseudo-)differential the real domain is. (See

and

equations in

52 of Chapter III.)

Now, let us prove the injectivity of (1.2.8). At first, suppose that system of M

d=codim M N =I. Let so that

N

be a section of

~YIX "

the image

u

~

of

is defined by

Xl.0.

If we set

under

~=f(xl+~-~0,x' ) at dXl.

(x I ..... Xn)=(Xl,X' ) Let

be a coordinate

u= ~aj(x')~(J)(x I)

f(x)= ~aj(x')~j(Xl) ,

homomorphism (1.2.8) is given by If

u

vanishes at x

then

f(x)

holomorphic in a neighborhood of x, which implies that Now, suppose so that

N

d >i.

~Y|X " We denote by

~

(x,y) ~ (<x, ~>-p)dx=0.

lu(x,y) ~

Let (x,y)

is defined by

of

x=0.

is

u=0.

be a coordinate system of

Let

u=~aj(y,D x) ~(x)

u(x,y) the image of u. By the result in

(<x, >-p)=0, which implies

proposition 1 2 4.

then

u=0.

Homomorphisms

d=l,

M

be a section

If u=0, we have

Thus we have obtained

(1.2.8) and (1.2.9) are

injective. Until now, we have assumed that a real analytic submanifold

N

of

Y M.

is a complex neighborhood of But, there are cases where

this condition is not necessary, as is shown in the following

466

Example 1.2.5.

Let

?(x,~,t) be a real analytic function of

(x'~'t)=(Xl ..... Xn'71 .... '~n' tl ..... t£) positively homogeneous of degree 1 with respect to 7, defined in a neighborhood of (x0, ~0,t0) =(0,~0 ..... ~0n, t~ ..... t~)(~0#0) i) 2)

?(x,~,t)

and satisfying

is of positive type ,

~(x,~,t)Ix= 0 = 0 and

grad x ?(x,~,t)l x=0=~.

We determine an n-vector of real analytic functions ~ (x'~' t)=(~j (x' ~' t))j=l ..... n

by

t(x,~,t) = <x, ~((x,~,t)> ~(x,~,t) lx=0

=

,

"

~-

Then we have

=

(x)

(n-l) ',

[ det(grad~ ~(x,~, t)) ~(~)

(-2~i) n

(?(x,~, t)+i0) n

as a microfunction in a neighborhood of (0,t0,~f~0dx ~o).

If

~

is

defined for every ~#0 , then this formula is valid as a hyperfunction. This is proved by showing that det (grad~(x ,~, t)) t(x,~, t)n

{

1

10)(~)

<x,~> n

is an exact form. Generally, -I (-2/E~) n-I

I

~' aJ (x' t' ~) ~ n-l+J ( ? (x' ~' t)+i0) °° (~) j~_m

is a (realification of) holomorphic microfuction of order principal symbol is

am(0 ' t, ~)

functions homogeneous of degree ~aj(x, t,Dx)

when

a.(x, t, ~) j

j with respect to

is a pseudo-differential operator.

m

whose

are real analytic ~

such that

467

1.3.

Real "quantized" contact transforms.

we defined "quantized" contact transforms

In

~4.3

of Chapter I~

In this section, we "real"-

ify them. Let n, and

M X

and and

We denote by P*(X×Y),

N Y

be complex neighborhoods of

P' PM

and

~S*M--~P*X

submanifold of

PN and

P*(X~Y)

open embeddings F:V~U

be real analytic manifolds of the same dimension M

the two-fold embeddings ~S*N-~P*Y.

U,V

F is)

real,i.e:,

and

F~R:V~->U ~

~P*X,

~=~

.

We assume that

Set ~_ =p-l(~),

are

~

is

U =pMI(U)

is the contact transform with graph

Suppose that a simple maximally overdetermined with support

~-~P*Y

the images of these projections.

is the contact transform with graph ~_.

(and, afortiori,

~S*(M~N)-->

Let A. be a Lagrangean

such that the projections

We denote by

V~=pNI(V).

and N, respectively.

/~, a non-degenerate

section

K

system ~i on

n of ~ X

~ ~

h R. X×Y

and a

ex homomorphism

~:p-l~i

~

~M×N@OOM

are given.

By Theorem 4.3.1 of

Chapter , II, we have the isomorphism (1.3.1)

T:F-I~x ~yIv

If we write

K = ~(K),

(1.3.2)

then

T:F~I~M

u(x) ~ - ~ K u

gives the homomorphism

--~ ~NIV~R

It is evident that (1.3,1) is consistent with (l.3.2);that is, we have

T(Pu) = T(P)T(u)

We will show that if an isomorphism. case where

~

for every ~

P e ~x

and

u E ~M

vanishes nowhere on ~ ,

"

then (1.3.2) is

To see this, it is sufficient to investigate in the is the conormal bundle of a submanifold

of codimension 1.

Z

of

X×Y

In fact, every contact transform is obtained as

the composite of contact transforms defined by generating functions.

468 Therefore, we can suppose that ~ = If

Z

Since

is defined by ~

-i P

~(x,y)=0,

~ZjX×y

with no loss of generality.

then we can suppose

(p-l~ziXxY , £MxN )

K= ~(~(x,y)).

is locally isomorphic to

CA~

~XxY

we can suppose that

~= ~(~(x,y))

If we prove that

with no loss of generality.

I~(?(x,y)) ~ (~(x',y))dy

is the kernel

function of an elliptic pseudo-differential operator, then the composite

is isomorphic.

This leads to the desired result.

Now, let us calculate the integral l(x,x') =

I~(~(x,y)) ~ ( ~ ( x ' , y ) ) d y

.

By Weierstrass' preparation theorem, we can assume that

?(x,y)

is of

the form ?(x,y) = Yl

f(x'Y2 ..... Yn )

Therefore, we have l(x,x') =

~ g(f(x,y 2 ..... yn) - f(x',y 2 ..... yn))dY2"''dy n.

There exists a vector of real analytic functions

G(x,x',Y2,...,yn)

such that f(x,y 2 ..... yn) - f(x',y 2 ..... yn) = <x - x', G(x,x'~y 2 ..... yn)~ We introduce new parameters

t 6 ~ +, ~ = tG (x,x',y 2 .... ,yn )

Then, we have

I(x,x')

= I ~ (<x - x ' ,

G ( x , x ' 'Y2

Yn)>)dY2" " "dY n

= I A(x,x',~) g (<x -x', where

A(x,x', ~)

~ ))-i ~Yn nowhere vanishes.

is the absolute value of (det(~, gY2

'

(See Theorem 3.3.3, i) of Chapter II.)

A(x,x', ~ )

"

"

"

'

,

469

By Example 1.2.5,

l(x,x')

differential operator. Theorem 1.3.1.

is a kernel of an elliptic pesudo-

Thus, we obtain the following

If

Supp(~) = ~ ,

then homomorphism (1.3.2) is

an isomorphism. Remark.

Note that the integral

maximally overdetermined system.

l(x,x')

clearly satisfies the

This implies that

l(x,x')dx'

is a

kernel function of an elliptic pseudo-differential operator up to a multiplicative constant.

Hence

l(x,x')dx'

defines either an inver-

tible micro-local operator or an identically zero operator.

In order

to remove the last possibility we have needed the above argument.

2.

Structure theorems for systems of pseudo-differential equations

in the real domain By the results of the preceding section the word "solutions of pseudo-differential equations" acquires the substantial meaning in the sense that the sheaf of pseudo-differential operators operates on the sheaf

~

of germs of microfunctions.

of j-th extension groups

~(~,

pseudo-differential equations.

C)

Hence we may consider the sheaf of a given system

@N. of

Moreover, we may sometimes obtain

results concerning the hyperfunction solutions, or even real analytic solutions, of linear differential equations by the aid of the microlocal results concerning the microfunction solutions of linear differential equations, which are clearly pseudo-differential equations by the definition.

(See Kawai [4]-[7] for example.)

Note that the

employment of pseudo-differential operators facilitates the treatment of linear differential equations, as is shown in this section.

(In

fact almost all procedures employed in this section cannot be performed in the framework of linear differential equations.)

At the same time

470

the theory of pseudo-differential

equations becomes much more subtle

and interesting, as is shown by the celebrated example of Lewy concerning the non-existence of local solutions, for example

(cf. Lewy [2]).

Such subtle phenomena are best illustrated by the aid of microfunctions (see Sato [3], Kawai [2]).

The main purpose of this section is to

decide the structure of solution sheaves of a general system pseudo-differential

equations of finite order.

result on the structure of

~tJf(~,

~)

~

of

It gives a decisive

(or

~(?~-'r

£

))"

T (See also Guillemin

[I], where the announces a result very close to ours

under the assumption of simple characteristics obtained by the method of the so-called sub-elliptic estimate.) flatness of

~

over

Here we note that the faithful

( ~3.4 of Chapter II) immediately implies

that for any admissible system

~=

~

~f

of pseudo-differential

pf equations 2.1.

~(~,

~ )=

~X2~f(9~f,

~ )holds

Structure theorem I - - p a r t i a l

for any

j.

de Rham type -- . We begin

our discussions by investigating the case where we can directly apply the results we obtained concerning the system of pseudo-differential equations in the complex domain (~ 2 and ~5 of Chapter II). Theorem 2.1.1. operator of order for any the sheaf

(x, ~ ~

Let m

q) on

P(x, D)

defined on in

be an elliptic pseudo-differential ~C~/~S~'~M,

i.e.,

Pm(X, ~ / ~ )

# 0

~i . Then it defines a sheaf isomorphism of

~g .

The proof is immediate from Theorem 2.1.1 of Chapter II. Corollary 2.1.2 (Sato [2]-~[5]).

Let

u(x)

solution of the linear differential equation Then

S.S.u = supp sp(u)

be a hyperfunction

P(x, D)u = 0

or order

is contained in the characteristic variety

of the differential operator

P(x, D),

i.e.,

~(x, ~ / ~ )

E~S*M;

m.

471

P m

Remark. then

u(x)

This

corollary

depends

real

shows

that

analytically

if

on

Pm(X O, xI

(t,

near

x

O, " ' " , 0 ) )

0

.

(See Sato

[1],

8 about the notion of real analytic dependence on a parameter.) is pointed out by Kawai [i],

Let

sp(u) l ~ n Proof.

constant

-Iv = 0,

map

6 > 0,

from

V C U

and

= (0; ! / ~ ,

of

0

such that

Supp u C {x E V~ x I > 0~

which will be fixed later, we set ~. ,

we can find a

2 2 V = {× e IRn~ - ~ + g(x 2 + "'" + x n) ~ x I <

is contained in

= (x, ~---~-gradxt(X))

t(x)

(x;~)

For sufficiently small

.so that

2 2 - g(x2+ '''+Xn) ~

such that

O E M = ~n

vanishes identically.

For a constant

~ > 0

(x, ~ f ~ )

which contains

u(x) ~ ~(V),

6 ( x 22+ "'" + x n2 ) .

t = x I+

be a neighborhood of

Then there exists a neighborhood

any hyperfunction and

U

~-]_S~'~

be an open set in

0, "-', 0).

(See also Kawai [2], H~rmander

In fact, we have the following

[2].)

Proposition 2.1.3. ~.

It

~5 that this remark is a key to the proof

of the uniqueness theorem of Holngren. [I] and Schapira

# O,

U

and that, for any

is contained in

V n ~x ~ x I _~ 0 1

to

~t 6 ~R ~

~

x

in

V,

and that the

It~ < ~ I

is proper.

Then the hyperfunction under consideration depends real analytically on

t

and

Supp u

is proper with respect to

function of one variable

~(t)

t.

with support in

Then for any hyper-

it;

for any volume element with real analytic coefficients in

V,

Itl<~

and

(x)dx

the integral IV u(x) %(t(x)) ? ( x ) d x

is well-defined by the conditions on

I while the integral

-~

S

t-I

u.

Moreover it is equal to

defined

472

ft_l u(x) ?(x)dx , which is clearly a real analytic function of for

-~ < t < ~

function.

u(x)~(t(x)),

must be zero, since ~,(V),

~(x)

V,

which has compact support in

the space of all hyperfunctions with compact ~(V),

the space of all real analytic

under the natural pairing (Sato [2]).

is an arbitrary hyperfunction with compact support in u(x)

must be identically zero. Remark.

V,

is an arbitrary real analytic function.

support, is the dual space of functions on

vanishes identically

by the unique continuation property of a real analytic

Therefore

Note that

t,

Since

%(t)

{t; -~ < t < S},

This proves the proposition.

Kashiwara has pointed out that the condition on

~L

can

be weakened to the following: ~h

does not contain

(0 ; ~'~ , 0, ''', 0)

(See Sato [5] p.56).

See also Morimoto's exposition in these lecture notes concerning the related topics. We also note that the above proposition combined with Theorem 2.1.8 or Theorem 2.2.9 below immediately proves the following unique continuation property of hyperfunction solutions of a system

~

of

linear differential equations across some characteristic boundary points.

Precisely speaking we easily verify the following:

H~rmander

[i] for a rather special case.)

Theorem 2.1.4.

Let

~L=~x~M;

Assume that a hyperfunction

u

have smooth boundary.

Let

x0

be in

~.

Assume further that there exists a neighborhood

such that, for any ~( ~ ( x , ± ~

?(x)<01

is a solution of the system

fying the conditions in Theorem 2.1.8.

?<x)= 0 ] .

(Cf.

x

in

0o

grad~(x) ) ) ~ fL # ~

satisfying ( x , ~ g r a d holds, where

~

satis-

= { x ~ M; ~

of

x

x~(x)) ~ S u p p ~ ,

~(x,±~-~ grad ~(x) )

0

473 denotes the bicharacteristic manifold of and

u

V

of

vanishes in

x

0

in

M

such that

(x,~f~ grad ?(x))

through

~I~ denotes the projection from ~ S * M

a neighborhood as

~

to u

M.

Then there exists

vanishes in

V

as far

V N ~.

The analogous statement for the system of linear differential equations

~l

satisfying the assumptions in Theorem 2.2.9 will be

clear and we omit the details.

Clearly it is also possible for the

case treated in Theorem 2.3.10 or more generally in Theorem 2.4.1. these cases the above remark of Kashiwara may play its own role.

In We

leave the details to the reader. Corollary 2.1~5 (Kashiwara [2]). The sheaf Proof.

~

is flabby.

It is not difficult to verify by a direct calculation

the following decomposition of n-dimensional

~-function to curvi-

linear waves : (2.1.1)

~(x) =

~

(l-~x' ~>)n-l+ ~2(l-~x'~'>)n-2(~x12-~x'~'>) ~0(~ ), ~=I

where S n-I

(n-l) ~ (-2~) n

(<x,~>+~(~x~ 2 - < x , ~ > 2 ) + ~ 0 ) n

Im ~ > 0, and

W(~)

a~(~)

belongs to the (n-l)-dimensional unit (co-)sphere denotes the volume element of the unit sphere, i.e.,

n = ~(-I)J-I~jd~IA'''Ad~j_IAd~o+IA'''Ad~ j=l

n"

Direct calculations also show that the following identity (2.1.2) of microfunctions holds. (2.1.2)

I(<x-y, ~ > + ~(~x-yl 2 -<x-y, ~ > 2 + ~ 0 ) X X X(
,

~>+~(IY

12 - ~ Y , ~ > 2 ) + ~ f ~ 0

~dY

474

n+l n+l ~ n- i = -27~ 2 ~/2[ ~(-~-~--~--) (~+~) - 2

×

F(-~) V(-~) X ×

for

Im ~) 0, Defining

(<x, ~>

-~ ~ +- ~

+

n+l +/~+ 2

(ixi2_<x, ~) 2)+ ~ 0 )

Im ~ > 0. ~(x, f )

~ (~, q)

by 1

=

(<x,~) +~i-f(Ix]2-<x , ~ )2)+/i~o)n for

7 ~ Sn-I , we find by (2.1.1)

(2.1.3)

and (2.1.2) that

/ !n ~ (x-y, ?) ~(y, ?)dyu)(~ ) ye ?~S n-I

defines a kernel f u n c t i o n (of order at most 0). (2.1.4)

o f an e l l i p t i c

pseudo-differential

Moreover

S.S. ~(x, ~ ) C i(x, ~ ;/21( ~ i' ~2 ))~S*(gRn×Sn-l); X = O,

holds by the definition of

~n

by

M

and

~i = ~

~(x, ~), where

homogeneous fiber coordinates denote

operator

of

sn-lx Nn

defined by (2.1.5)

sheaf homomorphisms.

A: ~_M ~ N / ~ N and (2.1.6)

Here we denote by

germs of real analytic functions on N

In the sequel we

N.

By (2.1.4) the integral operators B:~N/ ~N ~ ~ M

~2 = 0 ]

(~i' ~2 ) denotes the

S*(~nxsn-l). by

and

~N

and

are obviously

and ~N

the sheaf of

and that of hyperfunctions on

475

N, respectively and by

~M

that of microfunctions on

~-~ S*M ~ ~ N

(~2.1.5)

A(~(x)) = /~(x-y, ~) ~(y)dy

,

(2.1.6)

B(~ (x, ? )) = ~esn_ I ~(x-y, f) ~(y, ? )dy ~ ( ~ ).

Since BA( ~(x))=

J~sn_l ~(x-y', ,)J~(y'-y, ~)~(y)dydy' ~o(~)

=

J ( ~ s n _ I ~(x-y-y", ~) ~(y", ~)dy"~(~)) ~(y)dy

holds, the integral operator BA : C M -> ~ M defines an elliptic pseudo-differential operator, hence it gives rise to a sheaf isomorphism from ~ M

to

~M

other hand the quotient sheaf ~ N / ~N is flabby and

HI(~, aN)= 0

Euclidean space.

C M

in the real

is flabby.

asserts that everything is micro-locally trivial

outside the characteristic variety, i e., outside the Supp admissible system

~

of pseudo-differential equations.

~

? This is a very important question.

~

of the

Then, how

does the microfunction solution sheaf of the system ~ Supp

On the

is flabby, since the sheaf ~ N

holds for any open set ~

Therefore the sheaf

Theorem 2.1.1

by Theorem 2.1 1

behave on

In the real simple

characteristic case~ it can be answered by the results in ~ 5 of Chapter II , that is, we have the following Theorem 2.1.7. stating the theorem

Before

we recall in an intuitive and analytic fashion

the notion of bicharacteristic manifold of the system

~

= ~f/~

of

pseudo-differential equations with one unknown function of finite order defined in a neighborhood

V

of

(z0, ~0) 6 P * X

Clearly the

476

definition given below is nothing but the analytical version of the definition in not by

~ 3.3

of Chapter II. (In this section we denote by

x, a point in a complex manifold

X.)

bicharacteristic manifold of the system ~ Vc P'X, is, of course,

Definition 2,1,6

(Bicharacteristic manifold

=...=Pd(Z, ~ ) = 0} (din-l)

V

, which is a submanifold of

Let

~

be the characteristic variety of the

is a regular submanifold of codimension

through

d

#=~z0,k

is by definition the unique

integral manifold through

(z0, ~0)

Assume that

in a neighborgood

Then the bicharacteristic manifold (z0, ~0)

of a system

V = {(z, ~) e P*X~PI(Z , ~ )

, i.e., the zeros of the symbol ideal of ~ .

(z0, ~0) 6V.

strip of

P(x,D).

with one unknown function).

system ~

The notion of

a generalization of bicharacteristic

a linear differential operator

z,

~0)

~

of

of

d-dimensional

of the system of Hamilton opera-

toTs n

~P.

~ ~ Remark

- ~

"~~l )"

If the characteristic variety

in a neighborhood in ~ S * M

of

(x0, ~

V

of the system is real

~0)e~E~S*M, then we obtain a

unique d-dimensional real submanifold of ~ S * M manifold of the Hamilton operators. defined

We sometimes

which is the integral call the above

real bicharacteristic manifold only the bicharacteristic

manifold for short. Now Lemma 5.1.1

and Theorem 5.1.2 of Chapter II combined with the

above Remark yield the following theorem Theorem 2.1.7.

Let

~=~f/y

be a system of pseudo-differential

equations of finite order whose characteristic variety

V

in the

pure imaginary cosphere bundle is real and regular and its symbol ideal

477

J is simple in a suitable neighborhood in

of

U

( x 0 , ~ i ~ 0).

Then,

U

(2.1.8)

~-fJf(~,

£)

= 0

for

j # 0

Y holds, while for ~0~f(~,

j=0

the solution sheaf of the system

C) = ~ f ( ~ ,

C)

is a sheaf supported on

V

which is

locally constant along each bicharacteristic manifold and flabby in the transversal direction. a flabby sheaf C U 0

More precisely we have a manifold

and a smooth morphism

the bicharacteristic manifolds lying in and that the solution sheaf

~-I~u

isomorphic to

Proof

5.1.1

of the "geometrical

Note t h a t

U nV ~ U 0 ,

so that

are just the fibers of

£) restricted to

U~V

is

0

the contact

the characteristic variety

V

variety

optics

can be p e r f o r m e d i n t h e r e a l

change.

UoV

~f(~, j~

Since the characteristic

the procedure

y:

U0,

V

",i.e.,

analytic

is real

and r e g u l a r ,

t h e p r o o f o f Lemma

category without

transformation

any

used to straighten

to the form ( ~i, = "''= ~d, = 0

}

out is

obtained by solving the classical Hamilton-Jacobi equation, which is possible in the category of real valued real analytic functions under the additional assumption that the characteristic variety Therefore in virtue

of Theorem 5.1.2

~

: ~x~ u = 0, i=l, .... d.

is real.

and the above Remark

we may assume from the beginning that the system de Rham system

V

~

is the partial

On the other hand the

i

assertion of the theorem is clear for the system

~ , hence we have

proved the theorem. Remark.

Theorem 2.1.7

obtained for a single equation

is a generalization of the results P(x,D)u=0

in Kawai [2].

478

If

~

variety

V

is an admissible and regular system whose characteristic is real in a neighborhood

employment of

~o

instead of

~f

U

of (x0, ~2~ ~0), then by the

we can prove the following improve-

ment of Theorem 2.1.7. Theorem 2 i 8.

Let

~

be a system of pseudo-differential

equations satisfying the above conditions, regular and

V

be r e a l

~)

~

be admissible and

Then, in U

~(~, while~(~,

i.e.,

C)

0

=

for

is a sheaf supported on

V

j#0,

which has the unique

continuation property along each bicharacteristic manifold and flabby in the transversal direction. UO,

~ flabby sheaf

CUo

and a smooth morphism ~ : U ~ U 0

bicharacteristic manifolds that to

More precisely we have a manifold

lying in U

the solution sheaf ~ ( ~ ,

£)

so that the

are just the fibers of restricted to

U

is

~

and

isomorphic

y-l¢ U

Proof.

0 As in the proof of the preceding theorem, we may assume

from the beginning that

V = [~l = "" "= ~d = 01.

Let

~

be the

partial de Rham system ~x

u = 0, i = i ..... d. i

Then Theorem 5.3.7 system

~

immediately implies that for an integer

is a direct summand of

structure of ~ ( ~ , C )

5.3.7.)

i.e , r copies of ~

the .

The

is given in the preceding theorem and the

assertion of the theorem is c l e a r 5.3.1

~r,

r

directly to investigate

Clearly one may also use Theorem

~z~j~(~,C).

(cf. the proof of Theorem

479

2,2 In

Structure theorem II

-

EaFtial Cauchy Riemann type

~2 1 we have restricted ourselves to the investigation of the case

where the

microfunction

solution sheaf

only by the aid of the results in

~5

Y~(~,

£)

can be studied

of Chapter II.

study the subtler feature of the microfunction

Now we will

solution sheaf.

section we treat one of two extreme cases where the results in Chapter II cannot be applied directly. studied in

~2 3.

between

~

~5

of

The other extreme case will be

The result concerning the general case follows

immediately from the results in that the subtle

In this

~ 5 . 1 ~ §5.3.

(See

~ 5.4.)

We note

features are due to the intricacy of the relation

S'M, where the sheaf of microfunction

characteristic variety

V C P*X

is defined, and the

of the system ~ .

We call the

reader's attention to the fact that we conventionally treat the "geometrical optics"

on

S*M

by the technical r e a s o n s

It is easily

performed to transfer the geometrical results obtained on ~S*M

if we use

f(x, ~ )

fC(x, ~)

which is homogeneous

>-~.f~(x-----~( - i ~ ) ~

if

~S*M

in ~ .

f(x, ~) =

designates the point of in

instead of

~S~.

S*M

to

T(x, ? ) for analytic function Here

fC(x, ? ) denotes

5-.f~(x)(i~f

where

(x, i ~ )

fc

at the point

The value of

is equal to the complex conjugate of

f

at

x*

x*

In this section we treat the case where the real characteristic variety

V~ = V n ~ f ~ S * M

has the complex structure

that is, where the characteristic variety

V

(by Theorem 2.2.1),

of

~

= ~f/~

satis-

fies the following conditions: (2.2.1)

V

is regular in a complex neighborhood

(2.2.2)

V

and ~

Here

~

(2.2.3)

~.

(x0,i~0)~f~S~.

intersect transversally at any point of

denotes the variety complex conjugate to V~

of

V~ = V ~ f ~ S * M .

V.

is involutory and regular (, hence (2.2.1) follows from

480

(2.2.2) and (2.2.3)). Remark.

In the sequel we use the notation

bracket of homogeneous analytic functions ~

I

~

~ ( ~f j=l ~ j S*M.

~$ ~xj

~f ~xj

25 ~j

)

where

~f, g}

f(x, ~) (x,~)

and

as the Poisson g(x, ~),

i.e.,

is a coordinate of

We do not prefer the more usual notation

(f, g)

in this

report since it seems a little more confusing in complicated formulas. As in ~5 of Chapter II we first treat the "geometrical optics" of the system

~,

that is, we seek for the real contact transformation

which "straightens out" the characteristic variety above conditions.

P*X

Assume that the regular involutory submanifold

satisfies conditions (2.2.1),

can find a contact transformation on

(2.2.2) and (2.2.3). S*M

near

naturally extends to a complex neighborhood of that the characteristic variety

i (z'' ~ ') E P'~'~ ' ~ i + ~ ' f in a neighborhood of Proof. functions (x0, ~0)

V

(x0, q0)

Then we which

in

P'X,

so

~2 = O' " ' ' '

~2d-l+~'i~

'2d = O}

(x0, ~0).

In virtue of condition (2.2.1) we can find holomorphic Pl(Z, ~), o'-, Pd(Z, ~),

and homogeneous in

dPl, .-., dP d

linearly independent on

defined in a neighborhood of

~ , so that

.....

Pd(Z, ) : 01

and the canonical 1-form V.

0~ = ~ ~jdzj

are

Moreover condition (2.2.3) implies the

existence of holomorphic functions defined in a complex neighborhood of

A~,k(Z , ~ ) (x0, ~0)

and satisfies (2.2.4)

(x0, 70),

V

has the form

l(Z, and that

satisfying the

More precisely we have the following theorem.

Theorem 2.2.1. of

V

{Pj, Pk}: 5-~

£=i

-

)

of

(z, ~ ) which are

and homogeneous in

481

there, since holds.

{Pj'~k} (z, ~ ) = {Pj'Pk~ (z, { ) = -{Pk,Pj] (z,])

The relation (2.2.4), however, is still insufficient in order

that we should perform a contact transformation on S*M out"

V

to

"straighten

into the required form.

For this purpose we solve a suitable differential equation given below and use the induction on dim X. denote the defining ideal of

V

and

In the sequel V

~

respectively.

and

f

We first prove

the following Lemma 2 2.2, which is a key in the induction process. To begin with, we note that there exists a holomorphic function homogeneous in

G(z,~),

~, such that it satisfies (nod ~ ),

{ G ' Pi] ------0 d belongs to y , i.e., G = ~ g J P . j=l J

for a suitable

gJ, and

dG , dPl,.. . , dPd, are linearly independent.

In fact the existence of such

G clearly

follows from the classical Jacobi's theory on integration of first order differential equations. function

q(z, ~)

so that

neighborhood of (x0, ~0)

Using such

G

we choose an auxiliary

{ ~'ql (x0' ?0) # 0,

q(x, ~) = q(x, 7) in a

in S*M

and that q(x 0, N 0) = 0. d Lemma 2.2.2. Define ~(t) = Z FJ(z, ~ ,t)Pj(z, ~), j=l t E•. Then by a suitable choice of F j(z, ~ ,t) we have (2.2.5)

{ ~(q)

(2.2.6)

{ Pj , ~(q))~_0

Proof.

' ~(q) I = 0

To find the suitable

Kowalevsky theorem.

where

, mod ~ .

F j(z, ~ ,t) we use the Cauchy-

For this purpose we prepare the following lemma

on the calculation of Poisson brackets.

482

Lemma 2 2.3. Let F(z, ~,t)

and G(z, ~,t)

be holomorphic

functions in (z, ~ ,t) near (x0, ~0,0). Then for any holomorphic functions with

f(z, ~), g(z, ~)

and q(z, ~)

defined near

(x0, 0)

q(x 0, ?0)=0, the following relation (2.2.7) holds.

(2.2.7)

{F(z, ~,q(z, ~))f(z, ~),G(z, ~,q(z, ~))g(z, ~ )}

= ~F(z, ~,t) Dt +

(g(z, ~ ){ q(z, ~),G(z, ~ ,t)}I t=q(z, ~ )

t=q(z, ~)

G(z, ~ ,q(z, ~ ))lq(z, ~ ),g(z, ~ )} )

+ G(z, ~ ,q(z, ~)){F(z, ~ ,t),g(z, ~ )I[ t=q(z, ~ ) ~(z, ~ )IF(z, ~ 't),G(z, ~ 't)l I t=q(z, ~ ))f(z, ~)

+~G(~' t~t) IIt=q(z' ~)(f(z' ~ )IF(z' ~ 't)'q(z'~ )}It=q(z, ~ ) + F(z,~ ,q(z, ~)){f(z, ~),q(z, {)} ) + F(z, ~ ,q(z, ~))If(z, ¢),G(z, ~ ,t)} It=q(z, ~)

i

j

)

+ ~-f(z, ~ ){F(z, {,t),G(z, { ,t)} t=q(z, { ) g(z, {) + F(z, ~ ,q(z, ~ ))G(z, ~ ,q(z, ~ )){ f(z, ~ ),g(z, ~ )J

Here the symbol { q~z ~

G~z ~ t~}i~:q~z ~

for example me~s

that we first compute the Poisson bracket of q(z, ~ ) and G(z, ~ ,t) regarding Proof.

t as a parameter and next substitute

q(z, ~ ) for t.

A direct calculation will yield (2.2.7)

{q,ql=0 . We leave it to the reader.

if we note that

483

Proof of Lemma 2.2.2 continued.

Applying the above lemma we

easily obtain d { $(q), ~(q) } = Z (HJ(q)P. j=l J where

HJ(t)

(2.2.8)

HJ(q)Pj),

is given by ~ Fj

d

{ ~ ( t ) ' q]-~--~t + z~ (}{Fk(t), FJ(t)]P k + {Pk, FJ(t)] ~k(t) k=l d + Fk(t) ~ AJgF£(t)). f=l

If we define by gJ the Cauchy data for FJ(t) for = { ~, q) ~ 0 near (x0, ~0).

t=0, ~(0),q}

Hence the Cauchy-Kowalevsky theorem

applies to the non-linear differential equations 1 } {~(t),q

(2.2.9) Then the solution

FJ(t)

HJ(t) = 0.

of these equations

clearly satisfies

(2.2.5). Thus what remains to prove is relation (2.2.6). To prove this, we set Yj(t) ={ Pj, ~(t)j (22 .i0)

and prove

~j(t) ~ 0 mod ~.

It is clear by the choice of G

that

~j(O) = {Pj, G ] -~ 0 mod ~ . Hence, in order to prove (2.2.10) it is sufficient to show that

d pk (2.2.11)

~9t J

(t) ~k(t) k=l I=-0 + {Pj' {~(t), q ]

To prove (2.2.11) we first set

rood ~.

484

d d (t) = ~ ( ~ (l{Fk(t),FJ(t) } Pk +{Pk,FJ(t)} ~k(t) j=ik=l d ~ AJk~Ff(t)))P.. ~=i J

+ Fk(t)

Then clearly ~(t) - ~(t) = {~(t), ~(t)} hold.

and ~ (t) =_ 0

rood j

Moreover we have _

d

d

{ @(t), ~(t) I = { ~ FJ(t)Pj,~(t)}-- Z FJ(t) ~j(t) j=1 j =I

rood ~.

Hence d ~(t) -- ~ FJ (t)~j(t) j=l

(2.2.12)

mod ~ .

On the other hand (2.2.9) immediately implies ~ ~ (t) ~--~+ I~(t)'q I

(2.2.13)

= 0.

Hence we have 9@ 1 d D-~ + {~(t) ql ~ FJ(t)?j(t)' j=l

0

mod ~.

This immediately proves (2.2.11) . Thus we have proved (2.2 10). Now we prove (2.2.6) by the aid of (2.2.10). (2.2.10) implies

and (2.2 12) show that ~(t)- 0 ~t

{Pj

-- 0

mod

~(q)} '

IPj =

Therefore

using

~(t)}]t=q + {Pj,q} ~ '

(~)

In fact relations mod ,14., hence (2.2.13) (2 2.10) again, we have

I t=q

=0

This is the relation which we have wanted to prove.

mod j

This completes the

proof of Lemma 2.2.2. Proof of Theorem (2,2,1.) continued. 2.2.2 and the induction on dim X

Now using the above Lemma

we prove the theorem.

Assume that

485 for any

V

manifold V

of codimension at most Y

of dimension

n-2

d-I

in

P*Y

for any complex

satisfying conditions (2.2.1)~(2.2.3)

has locally the form !

~(z', ~') E P'Y;

!

~l+~f~ ~

= 0, "'', ~ d _ B + ~ f ~

by a suitable real contact transformation. (q)

given in Lemma 2.2.2.

g(z, ~ ),

- g=l HPd

We first replace

Pd

by

We want to prove the existence of

which is homogeneous in

(2.2.14)

~2d_2 = 0~

,

and satisfies =0

HPdg

(2.2.15) and (2.2.16) Here

Hpglv, = 0 . J

Hp. J

denotes as usual the Hamilton operator attached to Hp. j

n

~ pj

~

~ Pj

k=l

~ k

~Zk

~Zk

Admitting the existence of such how the induction on V0 = V~

d

proceeds.

since

J

~k

g(z, ~)

for the moment, we show

In fact

~(z, 4) 6 e~"~; ed(z, ~) = Pd(Z, ~ ) = g ( z ,

~) =g(z, ~) =01

satisfies conditions (2.2.1)~(2.2.3) when we replace Y0=P~

e. :

P*X

by

a~(z, ~); Pd(Z, ~) = Pd(Z, ~) = g(z, ~) = ~(z, ~) = 0},

HPd

and

H~d

are inner derivatives along

VnV.

Hence the

induction hypothesis allows us to find a homogeneous local coordinate system

(x2,-..,X2n ; 73,''' , ~2n )

~2j_l+~2n=0,

j =2,'--,d I.

on

YO

We extend

so that xj.

V0

and

is given by ~j

to

by solving the following first order differential equations: HPd ~j = H~d ~j = H g ~ J = H-g ~j = 0 , (2.2.17) HPd x.j = H~d xj = Hg ix" = H-X.g J = 0

P*X

486

by giving their Cauchy data on

x.. 3 (2.2.17) clearly constitutes an inte-

Then by (2.2.14)~(2.2.16), grable non-characteristic = ~j,

~k~y 0

{Pd = P d = g = ~ = 0 }

Cauchy problem and

holds, where

I~j,

~Y0

local coordinate system

SO that

is given by

set

V

Pd =

~I+~

~2

Y0"

(x; ~)

{~2j_l+V~f~2j and

~j,

and

~k}l{pd=~d=g=~=0 }

Hence we can find a on

= 0,

P*X

near

(x O, ~0)

j = i, "'', d 1

if we

g = Xl+ ~f~x 2 .

Now what remains to prove is the existence of (2.2.14) ~(2.2.16).

~j

denotes the Poisson bracket

associated with the contact structure of homogeneous

by

g(z, ~)

At first we overlooked condition

satisfying

(2.2.15).

We

are much obliged to Mr. Tetsuji Miwa, who has kindly pointed it out and suggested us the way of correcting it. By (2.2.5) we can take the canonical local coordinate (z, 7, x; ~ , sects

~,

~)

on

T = ~z = z = 0}

a complex coordinate analytic manifoold only on

¢.

have

= ~ dz +

~

can find

~ ,

g'

defined on

P*X

so that

transversally.

and that

¢, ~

so that

~

inter-

(z, ~)

¢2

~j dx.j .

z, 7

By (2.2.5)

of degree

0

is

of the real

is the complex conjugate of

are cotangent vectors of

~d~ + ~

V

In this context,

system of the complexification

which is homogeneous V

Pd = ~

system

z

Hence we

and (2.2.6)

we

in

and

(~,

~ )

so that I H~dg' = i Hp.g' = 0 , J

Let

g"

be an extension of

homogeneous

in

tial equation

( ~, ~ ).

g'IVnT

j = I, "'', d to

{z = ~ = ~ = 0}

which is

Then we claim that the following differen-

(2.2.18) admits a unique solution

h

on

T.

487

~ (2.2.18)

2

~j

~x.

~h I I Zh ~[ $ [ =-[>-~ ~ ~j ~x.j

~x.

~

~h

~xj

~[ )

~j--

h - g" rood { . Admitting the unique existence of such a solution h

for the

moment, we show how we can prove the existence of the required g(z, ~ ) by the use of h. Condition (2.2.15) assures the existence of

g homogeneous in

(~, ~, ~ ) which satisfies

i (2.2.19)

H~dg = i HPdg = 0 glT = h .

Since glVnT = hlVNT = g"l V~T = g 'IVNT hold

and since V ~ T

is

non-characteristic with respect to the equation (2.2.19) restricted on V,

the uniqueness assertion of the Cauchy-Kowalevsky theorem

implies glv = g'Iv" On the other hand (2.2.19) implies that HPd{g' g } = { H P d g' g I - { g' HPdg} = 0 holds. In the same way H~d{g, g }= 0 holds. Moreover (2.2.18) and (2.2.19) show that {g' ~IIT = =

~

+

~z

~

~ ~

~Z~ ~j

~z ~g

~~

~z" Sg ~ + ~" I ~~gj Zg

~x.j - ~x.j -~O

~h ~ (~h ~ ~ - ~ + ~ ~ j ~xj

~x.j ~xj Z~ Zg

T

=h ~ )=0

~x.j ~ j

Hence, applying the uniqueness assertion of the Cauchy-Kowalevsky

~j

488

theorem again, we conclude that required

g

is obtained.

existence of

h

holds.

Thus the

Therefore what remains to prove is the

satisfying

existence of the solution differential

{g, g } = 0

(2.2.18). r(z, w, x,

It is reduced to the unique ~ ),

s(z, w, x, ~ )

of the

equation

s}

9r-l~r,

9z

~W

2

s-1~ Ir,

s

(2.2.18') rlz=0 = r0(w , x, ~ )

slw=0 = s0(z, x, $) where

{r, s~

r0(w, x,

~ )

and

s =

denotes the Poisson bracket in and

a neighborhood

,

s0(z , x, ~ )

of the origin.

are given holomorphic If we set

~_~ Sjk(X , ~)zJw k, j, k~_0

(x, ~ ),

r =

and

functions in

~ r._(x ~)zJw k j ,k_~0 jK '

(2.2.18') implies the following recur-

rence formula:

.

jrjk

= 12

~__~

Ir.,k,, J

sj,,k,, }

(j ~ i)

{ r j , k , , Sj,,k,, }

(k _~ 1)

j-l=j '+j" k=k '+k"

1 ~ kSjk = _~j=j,+j,,

k-l=k '+k" It is clear by the induction on

~ = j+k

that the above recurrence

formula defines uniquely the formal solution of (2.2.18'). to prove the convergence method.

In order

of the formal solution we use the majorant

Obviously the solution of (2.2.18") with positive coeffi-

cients in its power series expansion gives the majorant series of the solution

(2.2.18') if

respectively.

~Iz=0

and

~lw=0

majorize

r0

and

so

489

= _

~z

> ( ~

(2.2.18")

J

Moreover we can assume that have the forms

J

C.+ J

~j

~'(z, w, x, ~ )

~(z, w, ~ x j + ~ j )

spectively, that 0, x, ~ )

~J

+

2

~w

~(z,

+ --

2

and

and

~J

~j ~(z, w, x, ~ )

~(z, w, ~ x j +

~_.~j)

~(0, w, x, ~ ) = l_a2w 1 l-a(~_xj+Y_~j) 1

- l_a2z i l - a ( ~ x jI+ ~ j )

If the solution (2.2.20)

~(z, t)

~z

?(z, t ) =

and that

for a positive constant

(+t

?(z, t)

= ~(w, t)

majorates

~(0, w, t) =

I i l_a2w 1-at

= ~(z, t)

majorates

~(z, 0, t) -

I I l_a2z 1-at '

and

~(z, w, t)

a.

of the equation

has a Taylor expansion with positive coefficients and if

majorates

re-

~(z, w, t).

and

~(z+w,t) Iz= 0

? (z+w, t)l w=0

then

?(z+w, t)

Therefore, the proof is

finished when we prove the existence of the holomorphic solution ? (z, t)

of (2.2.20) with positive coefficients in the Taylor expan-

sion which majorates

i i l_a2 z 1-at " We find

(2.2.20) with the initial condition

if(z, t)

1 if(0, t) = i-at"

by solving The Cauchy-

Kowalevsky theorem asserts the existence of the unique holomorphic solution of (2.2.20), while the positivity of its coefficients is obvious.

Therefore what remains to prove is that the solution

is a majorant series of

1 1 l_a2 z 1-at

Expanding

?(z, t) =

?j(t)z j , we clearly see from (2.2.20) that

~?~(t) =

j~-O~ d ~ j ~_~ j+k=£-I dt that

~(z,t)

d?k dt ~L(t)

holds

Now we will first show by the induction on

is a majorant series of

2~ a l-at

In fact

490

a2j+l

a2k+l

(l-at) 2

(1-at)

9~(t) >> j+k=~-i

a 2 (j+k+l) j+k=£-I

(l-at) 4

~a 2~

£a 2£ > > ~ (l_at)4 1-at Therefore (Z, t) = ~

~j (t)z j

j~_0

1 >>l-at Thus

~(z, t)

~ j~0

(a2z)j

1 = 1-at

has all the required properties.

convergence of the formal solution of the equation

1 2

l-a z

This shows the (2.2.18') and at

the same time it has completed the proof of Theorem 2.2.1 at long last. In virtue of Theorem 2.2.1 we may assume by the use of an invertible "quantized" contact transform that the characteristic variety V

under consideration has the form

(2.2.21>

,~(~,g> ~ p~; ~1+,/:~j2=0, -.-, g2d_l+4":-f~'2d--OJ

in a neighborhood of

(xO, V < T ~ O ) .

On the other hand Theorem 5.1.2 of Chapter II immediately yields the following Theorem 2.2.4.

Let

~=

Ff/~

be a system of pseudo-differen-

tial equations with one unknown function of finite order with simple

491

characteristics.

Assume further that its characteristic variety

has the form (2.2 21).

Then the system

~i

V

is micro-locally equiva-

lent to the partial Cauchy-Riemann system ~ : ~ju = ~i (

(2.2.22)

That is, ~i and ~

~

+ v~

of

~Qo (x,D)l j=l d Qj(x,D) is

can take

X

V

~

so that the principal symbol

as the generators of (z~...,Z')n

~f-modules.

we can find the genera-

Then Theorem 5.2.2

new local coordinate system in

and

of the left ideal

~2j_l+~2j

j j=l

j =i ..... d.

are micro-locally isomorphic as left

Proof. By the assumptions on ~i tors

~--~2j ) u = 0,

~x2j -i

.

implies that we

In fact if we choose a

in a neighborhood of

x

0

so that

I

z~j_l

= z 2 j ' l2- ~

z~j =

z2-_l+~fT z 2. ~ '] 2

z~ = zk Theorem 5.2.2

for

M,

j=l,

' '

j=l,

"'",d '

"'"

,d

'

k ~ 2d ,

is immediately applicable.

formation is not permissible on nate s y s t e m

z2J

As this coordinate trans-

we return to the original coordi-

Then the conclusion is c l e a r

Theorem 2.2.1

combined with Theorem 2.2.4

implies that the

structure of the microfunction solution sheaf of the system whose symbol ideal satisfies conditions

J

~=

~f/~,

is simple and whose characteristic variety (2.2.1),(2.2.2)

and (2.2.3), can be completely

investigated by the study of the partial Cauchy-Riemann On the other hand, as is clearly perceived, function) solution sheaf of

~

system

~

.

the structure of (micro-

is intimately related to the sheaf of

holomorphic functions, that is, we have the following

492

Theorem 2.2.5. ~i u = "'" = manifold

Let

~d u = 0

N,

@i

be the partial Cauchy-Riemann system

defined on an

n =n~2d

dimensional real analytic

whose complexification will be denoted by

exist a locally uniquely determined complex manifold (m+d)

containing

such that

pl N

induced to

N

N

and a canonical projection

is the identity and that

injection from

N

to

X.

j

holds. and

7[ 0

O-X

denotes

Proof. dinate system

(vl

from

Y

to

~i

where

on

X.

the

~

denotes the

~O ~'

j # 0,

~N ) = [~ms~(7 ~01~X )

canonical

map f r o m

NX*

to

j = 0

(Zl,.-- , Zm, Wl,... , Wd, Vl,-.. , Vd)

(2.2.24)

X.

Zm, t I, ''', td) .

on

Y

N=IImz I ..... Imv d=0}

so that the

is given by and that

(Wk- V ~ V k )

= ~jk'

j, k = i, "'-, d ,

~j(We+~T~Vk)

= 0 ,

j, k = i, "'', d

We also define a local coordinate system on Then the projection

p

X

by

(z I, "'" ,

is, by definition,

given by

p(z, w, v) = (z, w + ~ v ) . Under these assumptions it is clear by the definition that

holds.

X

The problem being of local character, we fix a local coor-

(Re Zl, ..., Rezm, ReWl, ..-,ReWd, R e v I, ''', ReVd)

(2.2.25)

X

Here the

denotes the sheaf of holomorphic functions on

local coordinate system on

hold.

p

of dimension

Then -i

~-l~f

Here

~*~,

X

Then there

is the tangential system

from the Cauchy-Riemann system

tangential system means the system

(2.2.23)

~

Y.

~ f ( ~ ,

0-y) --~p-l~ x

493

On the other hand if we denote by ~ from

N~

(2.2.26)

to

Y

the canonical projection

the definition of sheaf rR ~ - l ~ y

~

gives

f (3z-l~, £ N )

= ~-l~f

(@i, [R~S~ y (Tz-l~y))[n] ~y

= ~SV

(IR~_l~yf

(7~-I~' Va-I~Y ))In]

= ~ S V (-I ~ ~ f ( ~ , ~y))[n] = IRrs~ (7E-Ip-I~'x)[n]. In passing, Lemma 2.2.3 of Chapter I proves that (2 2 27)

~ ~S~f(~-ip =i 0_X) [codimyN] = N r S ~ X ( ~ 01~x)[codimxN].

Moreover it is shown in § 2 (2.2.28)

~X(~01~X

of Chapter I that ) = 0,

j#m.

Therefore, combining (2.2.26),(2.2.27) and (2.2.28),

we clearly see

t,.at (2.2.23) holds, since codimyN=n

This completes

and codimxN=m.

the proof of the theorem. Now Theorem 2.2.5 reduces the problem of the investigating how microfunction solutions of the system

~

under consideration

behave,

i.e., the problem of propagation of regularity of the solutions, to that of local cohomology groups with the sheaf of holomorphic functions as its coefficients, while the latter has already been investigated (Kashiwara[l]), i.e., the following lemma holds. Lemma 2.2 6 (Kashiwara[l],Theorem 6.2). mentary set of an open polydisc in

Cd

and

Let ~Ujl

A

be the comple-

be a fundamental

494 system of open neighborhoods of Denote by

X

(x0 ~0) E S*M

a complex neighborhood of

M.

li__m ~.KA (MX*~cd., 3

whose fibers are convex.

Then

%~£d )

vanishes for k
~

by the aid of the above lemma, we give the

definition of the virtual bicharacteristic manifold of the system under considerations. Definition 2,2,7 (Virtual bicharacteristic manifold). be a system of pseudo-differential characteristic variety (2.2 3).

V

equations of finite order whose

satisfies conditions(2 2.1),(2.2.2)

The virtual bicharacteristic manifold

(x 0, ~0) ~ V ~S*M

of the system

~

V =V~V,

~

and

through

is by definition the real (2d-

dimensional) bicharacteristic manifold through involutory manifold

Let

(x 0 , ~ 0 )

of real

which surely exists under the conditions

above Since the notion of bicharacteristic manifold is invariant under contact transformations,

that of the virtual bicharacteristic manifold

is invariant under real

contact transformation.

is the partial Cauchy-Riemann

1 j.u =

- -

x2j_l

Moreover,

if

system

+ ~-l~x~j)u. = 0,

its virtual bicharacteristic manifold

A-

j=l ..... d, through

(x0, 7 0)

is

clearly given by ~(x, ~)~ 7 = ~0

and

x.=x 0 3 J

for

j=2d+l,

n "'"

.

Therefore combining Theorem 2.2.5 and Lemma 2.2.6 we immedlately find

495

the following Let

Theorem 2.2.8.

u

be a microfunction

whose symbol ideal

J

variety

V

that

vanishes on an open set

u

satisfies conditions

is simple and whose characteristic (2.2 1), (2.2.2) and (2.2.3). U

The special case of this theorem, single equation, As in

solution of the system

in ~,

then

u

vanishes on ~.

i.e., the case where

is announced in Kawai[3]p.424.

Assume

~

is s

See also Andersson[l].

~ 2.1, combining Theorem 2.2.1 and Theorem 5.3.7 we can

prove analogues of Theorem 2.2.4 and Theorem 2.2.8 by the aid of pseudo-differential

operators of infinite order, that is, we have the

following Theorem 2 2 9. pseudo-differential fies conditions (x0 , ~ 0 )

Let

~i

equations whose characterlstic variety

Then in a real neighborhood

~$(~,

the system

u

in

~,

V

satis-

(2 2 1), (2.2.2) and (2.2.3) in a neighborhood of

~ VN~f~S,M"

while any

be an admissible and regular system of

0~p(~,

£) £),

= 0

for

i.e., any

U

of

(x0,~-T~0),

j~0

microfunction

solution u of

has the unique continuation property along the virtual

bicharacteristic manifold stated in Theorem 2 2 8. Proof.

In virtue of Theorem 2.2.1 we may assume by the aid of

suitable "quantized" contact transformation has the form (2.2.21). ~ju=0,

j=l ..... d.

Let

~0

from the beginning that

be the partial Cauchy-Riemann system

Then Theorem 5.3.7 immediately implies that

a direct summand of the direct sum

V

(~)r

of

r

copies of

~

is

~0"

Thus the assertion of the theorem trivially follows from Theorem 2.2.5 and Lemma 2.2.6.

This completes the proof of the theorem.

496

2.3.

Structure theorem III --Lewy-Mizohata type -- .

Now

we go on to the study of another extreme case where Theorem 5.2.1 cannot be applied directly.

That is, we will discuss the case where

the structure of microfunction solution sheaf of a system

~=

~f/~

is described by the aid of "generalized Levi form". The definition of "generalized Levi form" will be given in Definition 2.3.1.

We note that our work around Theorem 2.3.6

is deeply

affected by instructive articles of Lewy [I], [2] and recent works of Naruki [i], [2] on tangential Cauchy-Riemann systems.

We also note

that Guillemin [I] has announced results close to ours in by the aid of the so-called sub-elliptic estimates. [I].

C~-category

See also Kuranishi

We call the reader's attention to the striking fact that our final

result (Theorem 2.3.10) does not require any regularity condition on the characteristic variety, which is one of the greatest advantages of employing (pseudo-) differential operstors of infinite order. To begin with we give the definition of generalized Levi form of an involutory variety in Definition 2.3.1. where V

X

Let

V

be an involutory variety in

is a complexification of the real manifold

is defined locally by

borhood of V

P~"~K.

(x0, ~ f ~ 0 )

pl(z, ~) . . . . . E ~f~S~i.

pd(z,

P'X,

M.

Assume that

~ ) = 0

in a neigh-

Then the seneralized Levi form of

is by definition the hermitian form whose coefficients are the

Poisson brackets Ipj, p~ }(x, ~ f ~ ) Remark. invariant on

.

Clearly the signature of the generalized Levi form i s V

under any real contact transformation and well-

defined independently of the choice of the defining functions of

V.

pj(z, ~ )

497

Using the notion of generalized Levi form we obtain the following Theorem 2.3.2.

Assume that the generalized Levi form of an

involutory variety

V

(i.e.,

positive eigenvalues and

at

has

(d-p)

(x0, ~ f ~ 0 ) .

is non-degenerate and has the signature p

negative eigenvalues)

Then by a suitable real contact transformation, which

naturally and uniquely extends to a complex neighborhood, V (C P'X)

%

q

and

the variety

takes the form of the characteristic variety of a system

of the following form, considered in a neighborhood of

x' = 0

~' = (0, -.., 0, I): ~q

where near

(d-p, p)

d

: ( ~x[ j

2

~x[ j

~x' ) u = 0, n

j =l,''',d,

is the codimension of the characteristic variety

(x,0; ~,0) = (0; 0,---, 0, i)

and

ate real-valued quadratic form of

(x{,-'',

ture of the generalized Levi form of Remark.

Since the equation

the quadratic form

q(x')

q(x')

~

~ q

x~)

V

in

P~

denotes a non-degenerwith the same signa-

. has a covariant expression,

can be brought to the form

, I) 2 + "''+ (x~) 2 -(xi)2 . . . . . (Xp) 2 + (Xp+

so that

$T[q has the form:

+ ~ i x'.

( j

J

=0,

j = i, "'', p ,

= 0,

j = p+l,

n

(2.3.1)

? -,]'2"fx,. ( "~-x2-x'. j .1

In the sequel we call the system Mizohata system" (of signature

T)u

"'', d.

n

~q

of the form (2.3.1) "Lewy-

(d-p, p)).

(Cf. Lewy [2] and Mizohata

[i].) Note that the Lewy-Mizohata

system, which, at first sight, seems

to be a little queer, can be naturally obtained by restricting the

498

following very simple system

~

defined on

u = 0

~

: { 7+~f~ (~x n

~Rn+l

j = i, ''', d ,

~ _)u ~Xn+ I

= 0

(d {,n-l)

to the hypersurface P d 1 2 2 Xn+ I + ~ (~. xj - ~ _ xj ) = 0 . j=l j=p+l In the same way we can obtain the "mixture" of the Lewy operator and the Mizohara operator in the following way: Let N.

N = ¢ X ¢~×

~m ( ~Rr

and denote by

(•,

Consider the system of linear differential U

=

..~u

~-

....

~E I

-

~u

~u

~£-

~x I

_

z, x, y)

equations ~u

=

a point in

~

defined by

0

~ xm

and a quadratic form q(z, ~-, x) = >-~ajkz j z~ + ~'bjkX j z k + ~ b j k X Then the system hypersurface

~

defined by restricting the system

~Re("a-q) =0} C N

Im(q - ~ ), ~

j z~ + ~ C j k X j x k.

is the required one.

~

to the

Setting

t =

is

)~. J

~xj

~t

u = 0

j = i, "'', m

~"

~t

u = 0

j = i, "'', ~ .

J

Proof of Theorem 2.3.2.

In order to find the required contact

transformation we prepare the following two lemmas on the calculation of Poisson brackets. Lemma 2.3.3. neighborhood 1/2

f(x, ~)

(x 0, ~0) E T ~

with respect to

(2.3.2) and

of

Let

~

be an analytic function defined in a - M,

which is homogeneous

and satisfies f(x 0,

0)

= 0

of order

499

2 ~1

(2.3.3)

(x o,

{f,

Then we can choose an analytic function and a complex variable

t

~(x, ~; t, ~)

and its complex conjugate

real valued in a neighborhood of i(x, ~ ; f(x, ~), ~(x, ~))

(xO, ~0; 0, 0)

~,

in

(x, ~ )

defined and

and such that

is homogeneous with respect to

~ , so

that we have (2.3.4) I {f(x,~)i(x,~;f(x,~ ) '~(x,~)),~(x '~)~(x,~;f(x '~),~(x,~))~ = I " 2~f~ Proof.

We will prove the lemma in a little more general situation,

that is, we will prove the existence of

~(x, ~

t, ~)

with the

properties described above, so that we have (2.3.5) ~f (x,~)~(x,~; f (x,~),~(x,~)) ,~(x,~)~(x,~; f (x,~) ,~(x,~))~

= {f(x,~), ~(x,~)~F(x,~; f(x,~),~(x,~)) for any given

F(x, ~; t, ~)

valued in a neighborhood of ~(x, ~ ))

that is defined and strictly positive (xO, ~0; O, O)

is homogeneous with respect to

Defining

~(x, ~,• t, ~)

by

so that

F(x,~; f(x,~),

~.

( l(x,~, • t, ~))2 , we easily see

that the left hand side of (2.3.5) is rewritten as (2.3.6)

~f (x, ~),f (x ,~)~(x,~; f (x,~),F(x,~)) +~f(x,7) {~(x,~; f(x,~), f(x,~)), f(x,~)l l-+~f(x,~){f(x,~),~(×,~, f(x,~), ~(x,~))~ .

Now we introduce the following derivations functions

g(x, ~)

(2.3.7)

in a neighborhood of

~

and

acting on

(x0, ~0) :

{g, 7 1 ,

(2.3.8) Moreover we define

/k~(x, ~; t, i)

and

~(x,

~; t, ~)

by

500

and

tf~E v

~respectively, where we have defined ~,~(x, ~) by _ ~ t~ (2.3.9) ~(x, ~; t, t) =~, ~m-O~-~~ p ( x , ~) .-TT..--~>x.w. Then we see that the requirements imposed on certainly satisfied if

~(x, ~; t, ~)

~(x, ~; t, -t) are

satisfies the following degener-

ate differential equation of the first order : 1 ~(x, ~; t, i ) + ~ t (

(2.3.10)

1+~t(

~ ( x , ~; t, i) ~t + fL~(x, ~; t,i))

~ ( x , 9; t ~) ~[ .....'.. + % ~ ( x , ~ ;

t,~)) = F(x,~ ; t,t).

Hence we will now prove the existence of an analytic solution of (2.3.11)

%~(x, ~; t,~) = ~

with

~(x,

and

--

t~E ~,

~) ~,

00(0,

> O.

For that purpose we use the method of majorant. we set

F(x, ~; t,~) = ~,~>__0 ~_~ c~(x, ~) t/~~ ~ ~

~ 7~(x, ~)

and

Coo(X, ~) > 0

and decide

with

To begin with,

c~(x, ~) =

~/~v(x, ~)

by the follow-

ing recurrence formula (2.3.12), which follows from (2.3.6) by the t~ v comparison of the coefficients of #¢~ v~ on both sides. (2.3.12)

( l + ~1+ ~ ) ~ 1/ ~ p =

=-~ i

{f, [~ ..........,.

/~~ - I , V ....

_ I2~ , ~ _ i +

Since

&~-I,V

= cv/~

by their definitions, we can clearly determine

C~W"

~ f, [}

and

c/~u

~v(x, ~)

recursively through (2.3.12) in a unique and consistent way, so that ~00(x 0, ~0) = c 0 0 ( x 0 0 )

> 0

and

~y~x, q) = ~u~(x, ~)

hold.

Therefore what remains to be proved is the convergence proof of the

tJ~EV

series #,~>=0 ~cv(x'~)~-~"~-:

501

Now we set

n 0 2 n )1/2 r = (j__~l(Xj-xj) + ~ (~j _ ~ ) 2 "= j=l

and prove that

(2~+2~- 2)~! g~+~(l - ~) 2#t+2u

(2.3.13) holds for some

~,

a majorant series of

Here the symbol

g>0. f

(2~)~

and

f<< g

= ~ - (2k), k=l

means that as usual.

g

is

We first

note that C0~ ~ ~

(2.3.14)

r

) <<

holds for some

CO' ~-'

(i -

f0 > 0,

since co

(2.3.15)

F(x, ~; t, ~) << (I- r-E-)~0 (I- ~t)(l_t) ~0' 0- > 0,

holds for some by induction on

~+V=A.

C O > i.

We go on to the proof of (2.3.13)

2 ~1

As

~f, ~}(x 0, ~ 0 ) > 0

tion, we can majorize the coefficients of the derivation

/~

and

~

~/~x. J

and

by the assu~;/ ~ j

in

by C1 (I - i )

for some

C I,

~I > 0.

(2.3.16)

Now assume that

*x~w(x, ~) <<

( 2 ~ + 2 w - 2)! g ~ + ~ (I- r-l-)2~+2~

fl Then we have

holds for /~+~ ~_

(2.3.17) for ~ + ~ (2.3.18) for

~+Y=

A-~v ~l~b~v '

<<

2nCl (2A) ~ r )2~+2

= ~ . Therefore (2.3.12), (2.3.14) and (2.3.17) prove that ~/Rp(x, ~ )<4 ~ + i,

since

4nCl (2X) "'2 C 0 ~ ~" r_E_)-I r )2A+2_ + )~2A+2 (I~%(I- ~I ~i (2+~+~ ~0 ~,

P ~ 0.

On the other hand, it is obvious that

502

~'. u:

2+~+1~

<--

(2~+ ~ ) : - (/~+~-1): /&+V

_~ ( 2 ~ v + 2 2 ) - 2 ) : ' .

and -1

r

- (2~+2)

r

(1-To)

<< ( 1 - g ° )

hold. Now, choosing a smaller and take

~i

if necessary, we assume

~i = ~0

so that

L> 0

~i if2 £ < min (SnCl ,4C 0 ). Then (2.3.18) implies

( 2 ~ ) '. '. (2.3.19)

for any

~(x,

/¢k, V > 0

~) <<

gA+I (i

r )2~+2

/~ + V = ~ + I.

with

Thus the induction on

proceeds and (2.3.13) is proved by taking hb .. __it2~[W ~_~ I~'/~(x' 'l)/~!1;~ I--< ~-J

- -

2~

(2.3.20)

/~,~o

~ = ~0"

Hence we have

(2~+2 ~- 2) ['. Iti/al[lw _~u+l; r ~+2~ .

/., ~,~o ~ _

(1- --~- )

,m: v' ~

It is clear that the right hand side of (2.3.20) converges if I

~. (1- -~--) 2 This proves the convergence of the series a neighborhood of

(x 0, 70; 0, 0).

~_~ ~* ,~ ~_0

~(x, ~)~t'. P'.

in

Therefore the proof of the lemma

is completed. Remark. ~ ( x , ~;

t, ~)

In order to prove the existence of the analytic solution of the degenerate differential equation (2.3.10), we

have used the majorant series method. tence of

~

However we can prove the exis-

using the Cauchy-Kowalevsky

trick which we sketch here. ing up" with respect to We first define

theorem only by the following

The trick consists in the idea of "blow-

(t, ~). @ ( 4, t, ~) = ~ ( x , ~ ; t, 7)

by

503

A2~(x,~

; Xt, ~k~). Then we have ~A ~ ( ~ ' t ' i) = % ( t - ~

+t-~-~ +2~).

Hence (2.3.10) can be rewritten in the following form as an equation for

~

:

(2.3.10')

®+

~

+

/h(t~) = AF(x, ~, At, At).

of (2.3.10') by giving

Clearly we can obtain an analytic solution

@

0

and

as its Cauchy data on

concerning only

~ = 0.

Since

(x, 7), (2.3.10') ~X

~

proves that

X=0

It immediately implies that

holds under the above defined Cauchy data. ~(A,

t, ~ ) / ~ 2

is analytic.

Moreover the unique analytic solution

with Cauchy date

of (2.3.10')

~(A

0

clearly satisfies

c -it, c-l~) = c 2 @ ( ~ , t, ~).

~(c~, Thus we can take

are derivations

, t, ~ ) / ~ 2

as an analytic solution

~

of

(2.3.10). Let

Lemma 2.3.4.

f(x, 3)

defined in a neighborhood of respect to

~

and

g(x, 7)

(x0, ~0) E T'M-M,

f(x 0, ~0) = 0 ,

(2.3.22)

{f, ~l(x, ~) = 2 ~ ,

g(x 0, 70 ) = 0 , ~f, g~(x, ~) = 0 .

Then we can find another analytic function a neighborhood of ~

(x0, ~0),

g'(x, 7)

also defined in

homogeneous of the same order as

and satisfying the following :

(2.3.23)

g' ~ g ~f, g'}(x, ~) = 0

Proof.

homogeneous with

and satisfying the following:

(2.3.21)

respect to

be analytic functions

Define

gj(x, ~)

mod and

f, {g', fl(x, ~) = 0.

successively by

g

with

504

(2.3.24)

g0(x, ~) = g(x, ~), g0 (x' ~) = ~ g-ij - i

Then it is obvious that the series

(x, ~), f(x, ~)I ~

for

~., gj(x, ~)(f(x, ~))J

j 2- 1 " uniformly

j~_0 and absolutely converges in a complex neighborhood of

(x0, ~0),

since

(j-l) ~ ~ j~__l -2\I/2"J EJ(I - ( x ~ + n ~j) ) is a majorant series of

gj(x, ~)

proof of the preceding lemma. is easiler than that of Now we define

for some

g , ~ > O.

(See the

In this case the majoration of

*~(x,

~)

gj(x, ~ )

given there.)

g'(x, ~) = ~_~ 7., gj(x, 9)(f(x, ~))J

Then it is

j_~0 clear by the definition that

g' -= g

mod f.

On the other hand we can

easily verify that ~f(x, ~), gj(x, ~)} = 0 for any

j _~ 0.

In fact, since

{f' gj+l} =

-i If ~gj, ~}} = ~-~--~-({{f,gj],T}+{gj,{f,f}}) -I ,

holds by Jacobi's identity for Poisson brackets, we can verify by the induction on

j

that

{f, gj] = 0 holds for any

{f, g'~ = 0.

clear that

j ~ 0.

Moreover by the definition of

Hence it is g'(x, ~)

we

have -

j~_~O j~ Igjfj' ~} =j ~$0 ~I ({gj,~Ifj+gj.jfj-l{f,~})

= -2~ Therefore

--~ ~.,igj+if j + 2 ~ j _Z0

g'(x, ~)

>-, i !gjfj-I = 0 j _~i(j -I) "

has all the properties required.

This completes

the proof of the lemma. Proof of Theorem 2.3.2 the theorem.

continued.

By the induction on

d

Now we return to the proof of

as in the proof of Theorem 2.2.1,

505

Lemma 2.3.3 and Lemma 2.3.4 fj(x, ~)

(j = I, ..., d)

homogeneous of order

1/2

prove the existence of analytic functions defined in a neighborhood of with respect to

~

(x0, ~0)

and

and satisfying the fol-

lowing: (2.3.25)

~fj(x, ~), ~j(x, ~)~ = 2~f~

for

j = l,.-.,p ,

(2.3.26)

~fj(x, 7), f--j(x,~ ) ~ =-2~f~

for

j = p+l,-.-,d ,

(2.3.27)

~fj(x,~),

fk(x, ~)~ = ~fj(x, ~), ~k(X,~)l

= 0

for j # k,

and (2.3.28)

v = ~(z, ~) ~ P~"]~; fj(z, ~) = 0,

j = l,...,d}.

Note that the non-degeneracy of the generalized Levi form of implies that h., J

V

is non-singular.

respectively,

Re f. J

and

In the sequel we denote by

Im f. J

gj

and

(and also their analytic con-

tinuation to a complex neighborhood of

(x0, ~0)).

Then by the classical

integration theory of Hamilton and Jacobi we can find a real valued real analytic solution

gd+l(X, ~)

of the following system of linear

differential equations of the first order, so that it becomes homogeneous of order (2.3.29)

1/2

with respect to

~ "

I go' gd+l~ = {No, gd+l} = 0,

j = l,''',d .

In fact the consistency condition of (2.3.29) is easily verified in virtue of (2.3.25), (2.3.26) and (2.3.27) combined with Jacobi's identity for Poisson brackets, hence the Hamilton-Jacobi theory is applicable to the system (2.3.29). valued real analytic solution

Moreover we can also find a real

hd+l(X , ~)

of the following system

(2.3.30), so that it becomes homogeneous of order to

1/2

with respect

~ :

(2.3.30)

~gj, hd+l~ = ~hj, hd+ll = 0 ,

gd÷l' hd÷l -1

j = l,''',d ,

506 The consistency condition of the system (2.3.30) can be easily verified as in the case of the system (2.3.29). of the Hamilton-Jacobi theory prove independent solutions

G(x, ~)

gj(x, ~),

Thus the successive applications

the existence of functionally

hj(x, ~)

(j = I, ''-, n-l)

and

of the following system (2.3.31), so that they are homogeneous

of order

1/2

with respect to (i)

~ : j = l,''',p

{go, hjl = i,

for

(iii) Igj, hk~ = 0

Igj ' gkl = ~ hj, hk} = 0

(v)

{go' G I = ~hJ' G ] =

for any

j, k;

j = l,...,n-l.

0,

×,(x, j

~ 'n(X, ~)

j # k;

(iv)

Now we define

d+l,-..,n-l;

j = p+l,''',d;

(ii) ~gj, hi} = -i, (2.3.31)

and

(j = i, "'', n-l) '

and

3

by

J ~)-1 , x'.(x, ~) = gj(x, ~)G(x, ! ~j(x, ~) = hj(x, ~)G(x, ~) , j = l,''',p,d+l,''',n-l,

i (2.3.32)

!

j = p+l,''',d ,

~j(x, ~) = -hj(x, ~)G(x, ~) , ~n(X, ~) = C(x, ~)2

Then it is well-known that we can find a suitable real contact transformation which transforms

(x; 7)

into

(x'; ~')

(or Lagrange) bracket, if one prefers to do so). Carath~odory system

[I], ~ iii, §125

(x'; ~')

and ~127.)

the characteristic variety

form of the characteristic variety of ~ q , (2.3.33)

(z', ~') E P'X;

in a neighborhood of

, ~2 ~j-

(See for example

Under the coordinate V

clearly takes the

i.e., ~n' = 0 ' j=l,''',d}

J (x'; ~') = (0; 0, "'', 0, I).

denotes the complexification of of the theorem.

~q(z') ~zl

(by using the square

Here

(z', ~')

(x', 7'). This completes the proof

507 Now the result of optics" of the system

~1.3 and the above theorem of the "geometrical ~

with non-degenerate generalized Levi form

allow us to assume without loss of generality that the characteristic variety of

~

takes the form of (2.3.33) as far as we are concerned

only with micro-local properties of the system characteristic variety of the system

9~

~

.

Moreover, if the

has the form of (2.3.33),

then Theorem 5.1.2 of Chapter II asserts that the system system

~q

are equivalent as left

~f-modules.

~

and the

Therefore we first

investigate the structure of the microfunction solution sheaf of the system

~q,

which is easily deduced from the results concerning the

single equation

( ~Xl

t~

Xl ~--7-)nu = 0.

While the structure of the microfunction solution sheaf of the above 6quation can be directly investigated by the aid of the "good" elementary solution constructed in

~3.2 of Chapter I.

For reader's sake

we list again the lemma which we need in the sequel.

(See also Kawai

[3] Theorem 2.4.) Lemma 2.3.5. and

Q(x, D)

Denote

D 1 - 2~XlD 2

respectively.

Assume that

Then, in a neighborhood o0 of find a micro-local operator (2.3.34)

0---> C

is exact.

Then in M --->N

K

CO

D I + 2 ~XlD 2

by

P(x, D)

~ , ~ ~ ¢+ = ~z E ¢; Im z >0}.

( x ; ~ f ~ ) = (0; 0, ~f~, 0, "'', 0), we can such that

Q-~ C

Moreover, denoting

we may identify

and

~n

K

g

P_~ ~

by

M

and

~(x, ~) 6 S)'~M~ x I = 0

and

---~0 ~x E M j x I = 01

71 = 0 ) with

we can find micro-local operators (over

respectively)

CN -+ eM

N --~M

by

S*N. and

N,

508 and : CM--e

CN

such that (2.3.35)

~ =

Id.

and (2.3.36)

~=K

hold. The above lemma immediately implies that the chain maps associated with

K,

i.e., 0 ......' CM

P~ £ M - - - ~ 0

K ~ o

~0

>C -Ce

and 0 ' ~' ~ M 0

o

Q> £ M

> 0

~K

~£M Q~ ~M

are homotopic to the identity map.

>0

At the same time it is clear that

the cohomology groups of the above complexes vanish except for 0-th cohomology group

and 1-st cohomology group respectively.

remaining cohomology group is isomorphic to

Moreover the

~ N"

On the other hand the cohomology groups of the microfunction solution sheaf of the system

~q

can be directly calculated by

"tensoring" the above complexes suitably, that is, we obtain the following Theorem 2.3.6. signature

(d-p, p)

( x ; q'i-f~/) = ( o ; o , - - - ,

Let

~q

defined on

be the "Lewy-Mizohata" M.

system of

Then in a neighborhood

o , , f ' : l ) ~ f:is~,~

U)

of

509

4

~;f(~q,

~)

= 0

for

j # p ,

1

while the remaining flabby sheaf (Here

V

eW

cohomology

~:f(~q,

defined in

~

~)

is isomorphic

and supported on

denotes the characteristic

to a

W = V ~2~S*M.

variety of the system

~q

in

P~'X.)

Proof.

Denote

}x. +,[:7xj

~

j

(j

=

l,''',p)

(j

=

p+l,''',d)

n

and ~x.

by

Qj

and

F. J

in

60

-

~:~xj

j

-~

~x

n

P. respectively. Define micro-local operators E. and J J by those that are obtained in the preceding lemma as the

right inverse of

P. J

and the left inverse of

Qj.

K. = 1 -E.P. J J J

(j =p+l,.'.,d)

G.j = i-QjFj

(j = l,.'',p)

Setting

and ,

we find as in the proof of the preceding lemma that their kernel functions are given by I ]~ g(Xk-X;)dx' (x -x' +~_~_~I, 2, ,2. r-to)B/2 k#j,n n n 2 ~xjtxj )+4-1

4 ~ Here

(j=l,''',d).

= exp(-~). For the sake of simplicity of notations we denote

the sequel.

j -p

by

Then the preceding lemma asserts that the operators

Ei, Ki, Qj, Fj

and

G.]

Pi'

satisfy all the commutative properties which

we need, i.e., they satisfy the following: =

" P.E. i

i

1

l

FjQj

= 1

in

C)

0

,~

~n

II

•,+I

II

•~I

VII .r-3 Vll ,-4

VII • ,-4 "411

0

II

.,-4 I:~

C~I .~

0

.M

II

.,-.-+ II

~

+~-I C'd .P-~

0

"+-'+

II

";'-~

II

~

II

~ II

"~

I1

~ • ~t

"-~

~ .+-.+

II

MI

VII

~l

",,~II -;.-+

0

II

..~

II

~

0

.+-.-+

11

~

~

M

II

.+1

"--+ 0 ++

II

";"+; f.~

~

+t+7 CY

II

II

II

I

p+++

¢'4

o4

0 |I

II

II

F.~

II

c-4

@4

C~l

%

o4

.~ "v'l+

0 q-t

C~

04

,-4 "~-~

.;'-n

II

o"

~D

Oq

,--4 "++++

II

,-4

,r-+

II

~

~

•~ I

o.3

"~ 1"4 ~

,~

q4 °P4

01)

m

• ,4

+~ "q

t

£,j

0

~

bl) -~I

0 ,-4

0 q-I

0

z

F4

o

0

o

4~

o

.~4

eJ

4-+

~

O;

~ 0

~

'4,4 .,-I

0

u

0

t-' +

-

~

m

r,3

~ 4J

~

o3 ~4

0

V!

"If-.

¢~

+.+,

.r.,.+

0

+~J

•

°r,.~

ca

~)

II

II

o3

0

o3

,-~

,--4

~

0

0

~ ~

r-~

>

•

0o

--

• ,~

4-J

o

~

:

•

~

0 q-~

.u

,~

~

~

0 4J

qs

"

O"

0

+r.4 4-~

1_t

Jl

,--

o4

4J

o

0

511

where the differentials (2.3.38)

~(?~w)

~ are defined by d-p p = Z Pi(?)~eiAw+ ~Q'(~)~ i=l j=l j

In the sequel we identify the element denotes

? =

eilA...

abbreviate

~, ?IjeiAfj II1+ IJl=2 A eik

Kil

"Kik

i

if

(w)

for

to

the interior product of = v,

~ ~ AW

w = e I with

w

I = ( i 1,

KI

etc.

and

I ~ j,

by

ik)

?I J"

Here

and so f o r t h .

eI

We a l s o

In the sequel we denote by

e., J then

is defined by linearity.

wEAW.

and sometimes denote

C~AW ...,

for

e(1)

with of

fjAw

i.e., if i

w = ej A v

(w) = 0

ie . (w) J i e j (w)

then

and f o r g e n e r a l

w,

e.

J The map

if.

e.

is defined

i n t h e same

J

2 way.

Now we prove that the following chain map

~p

is homotopic to

the identity map: L

if

0 ~p(?~w)

~w

E ~)AW,

~#

p

=

(GI" " "GpKI" " "Kd_p ~ ij)eI A ej Z I=~ J={l'''" 'Pl if ~® w = Y_ ~ijeiA fj eC®Aw. In order to define the homotopy operators the i d e n t i t y (2.3.39)

we d e f i n e

the following

Ki~( ? ~ w )

operators

~s£1 between which operate

J~ on

P

C ~ Aw:

= Ki(T) ~ (w -eiAie.(W)), 1

(2.3.40)

G~(?~w)

(2.3.41)

Eil(T~ w) = Ei(?) ~ ie.(W),

= Gj(?) ~ (fj Aif.(w)), J i

(2.3.42)

Fj~(~w)

For s i m p l i c i t y Ej ~+ d - p

of notations

respectively.

= FO(?) ~ if.(w) J we d e n o t e

Then t h e r e q u i r e d

G.j~" and

F.j~

by

K~+d-p

homotopy o p e r a t o r s

!

s~ : ~ ® AW----s ~

1-1 AW

and

and

512 are given by (2.3.43)

d j-i ~ _ i E ~ ) ( ~ = (~ ~- K k J j=l k=l

sA(~w)

The proof that these operators

w).

surely give the homotopy between

P

and the identity is given by making full use of the commutativity the operators under consideration.

of

We leave the detailed proof to the

reader. On the other hand the kernel function of given by d r(l+ ~) (2.3.44)

2E~Z~

~ 3.2 of Chapter I shows as in the previous lemma

that we can find micro-local respectively

~

and

~

over

M --~N

and

= GI--'GpKI''-Kd_ p

~ N = Ix ~ M;

operators

such that

I ~

Here

is

n-i ~ ~ (Xk-X~)dx ' ~ k=d+ I ~f~ j=l d d 2 I+~ ~2d (Xn -Xn+ ( z + j=l r_ + , m o)

Hence the example in

N --+M

GI-..GpKI..-Kd_ p

=

Id

.

x I .....

x d = 0}

and we identify

W

with

S*N.

Thus the proof of the theorem is completed. The above proof of Theorem 2.3.6 is elementary in its nature but we can give more intrinsic proof of this theorem if we use of

~f.

Moreover i t

~n P~.

p~

instead

i s a p p l i c a b l e to more g e n e r a l s i t u a t i o n s ,

we now give the proof using pseudo-differential order.

~

hence

operators of infinite

The essential part of the proof is the following lenm~a.

Lemma 2.3.7.

Let

M, N

respectively.

Let

p

and

X

be

Z+

and

#x ~ M;

be the canonical map from

Consider an involutory submanifold and denote by

~n,

Z_

x I = 01 ~T~ S*M

Z = {~i-~x2~2

~(x, ~T~ ~ ) G p'I(z);

and to

= 0} ~2 > 01

of and

513

{(x,~f~)6p-l(z); ~2< 0} respectively. We also denote by f the projection from P * X X Y - P ~

X

to P ~

(and also the projection from

~ S * M ~ N - S.~'~ to 9C~S*M. Then for any admissible system of pseudoM differential equations ~ with support contained in Z we have the following isomorphisms: (2.3.45)

~*('R~p-l~x(P-l@f~' ~M)I Z+) --~]RJ~jP-l~y (p-l~y, ~N),

(2.3.46) JR~p_l~y(p-l~y, ~N)[-I]~k f,(~RJ~p_l~ X(p-l~, £M )I Z_)~N/M" Proof. We first prove (2.3.45). To begin with we construct a canonical homomorphism j. It is given by composing the following homomorphisms. There ~

denotes the imbedding map from ~T~S~ × N

to

,~ ~m~p_l~ (P-I~, e M ) --~ ~,[R ~

_ip_ipx(%-Ip-l~,

%-ICM )

-->~ ~,.R~_ip_l~x(Zilp-191,

I-I£M )

_ld -l~y,x ~ z-lp-l~, ~ Y~X ©L ~ M) --9RR ~ ~ ~ - i p - l ~ y (p p-l~x p -I~x L -iTy_~x ~ ~-ip-l~), ~,.(?y_~x ® ~ -i£ M) ) ---~[RJ~p-l~y ~'.(P p_l~x p -I~x --~Ji~ -I (p-19ty, e N ). L The last map is given in ~3.5 of Chapter II. Here ~

means the left

derived functor of Now what remains to prove is that the above defined map j is an isomorphism. Theorem 5.3.1 proves that we have the following resolution of ~i

by

£ = ~/~P,

where P(x, D)= ~x I

~/fx I

~ .. ~x2

514 ~r0

(2.3.47)

rI

r2

On the other hand Lemma 2.3.5 immediately implies that the map an isomorphism for

~

restricted to

tion (2.3.47), the map

j

Z+,

j

is

hence, in virtue of resolu-

is an isomorphism for the general case, that

is, (2.3.45) is proved. Now we go on to the proof of (2.3.46).

j,

the map

k

As in the case of the map

is defined by the composition of the following maps.

~R~

(p-i~y, £N) [_i] P -l~y L

--~R~

(p-l~y,p-l@y) ~ ~N[_I] P -i~Y

p-l~y = p

-I

N~>~y(~y,

~y)

L @

CN[-I]

P -l~y

L "v 0 ~P-I ' ~ ~ ~JR "~

~X (~};i, ~ X<_y)P _ % C . N

--~ f * P - l j R ~ x ( N '

L PX~-y ) ~ dN p -l~gy

L --~ ~ . f R ~

(p-iN, p-l? Xey) P-l~ X

)O.[R~

p-l?x

---> ~ . ~

(p-t~ , P

-i

(~ ~ N p -i~y L

~) X~-Y

~)

f-lp-l?y

(p-l~i, £ M ) ~ N / M

f

-ldN)

•

p-I~x Note that the above isomorphism Chapter II.

The above defined map

has been given in k

phism if the problem is restricted to

~3.5 of

is easily seen to be an isomorZ

by the aid of Lemma 2.3.5

and the resolution 0 4--- ~/g~-- £ r0 +- £ rl ~-- £ r2 ~-- "'" , where

~ = p/~P

with

P(x, D) =

~ -~f~x I ~ ~x I ~x 2

Thus the proof

515 of the lemma is completed. By the aid of this lemma we can obtain the following stronger version of Theorem 2.3.5 employing

~

instead of

~ f

Before stating the theorem we give the following Theorem 2.3.8 as a remark, since some of the reader may fear that the support Lemma 2.3.7 is so restricted that we can seldom use it.

Z

in

Though Theorem

2.3.8 is more general in its appearance than Lemma 2.3.7, it is equivalent to the lemma. Theorem 2.3.8.

Let

M

be a real analytic manifold,

manifold of

M

with codimension

borhoods of

M

and

from

~-~S~"~M to

codimension

I.

N

i,

respectively.

P~"IK. Let

Z

Z+

points

X

and

Denote by

Y

be a sub-

be complex neigh-

p

the canonical map

be a complex submanifold of

P~"N with

We assume that Z q Z ~Y

Let

and

N

(resp. Z_) (x, ~2~ ~)

K P~"XK. X

be the subset of ~ N ×~S*M

ZN = p

such that

-i

(Z)

consisting of the

~f, f C } ( x , ~ )

> 0

M

(resp. < 0),

if

(x, 4ri~)

Then we have the following canonical isomorphisms for any

admissible

Z

is defined by

~x-Module

(2.3.48) f ~ ( I R ~ w ~

"

(2.3.49) ~ , ( ~ R ~ P

i

P- ~X

~

f = 0

in a neighborhood of

with support contained in

(p-l~i ' CM) I

) _~>~

Z+

p-l~y

Z:

(p-l~y, ~N) I~(Z+ ) ,

-I (p-l~, ~M)I Z ~ _ I I ~ _ i (p-l~y, ~N)I~(z_)~ON/M[_I]. ~X P ~y

The proof of this theorem is immediately reduced to Lemma 2.3.7 by "quantized" contact transformation in virtue of the following lemma. Lemma 2.3.9. manifold of

X

Let

X

be a real contact manifold,

with codimension

i,

and

~

Y

be a sub-

be a complex submanifold

516

of the complex neighborhood that

~ ~ ~

that

~

X¢

of

X

if, ~ l ( y ) >

0

holds for a point

in a neighborhood of is defined by Proof.

holds on

~f = 0}.

is contained in the complex neighborhood

a canonical local coordinate system X

defined by

y

in

Y.

of

Y

and

Then there exists

is defined by

of

~x I = 0 } and

= 0} .

Clearly we may assume from the beginning that

Y.

Y

(Xl' "''' Xn' PI' ---, Pn_l )

such that

~ Pl+iXl

y

Y¢

Assume

f + ~ = 0

Then Lemma 2.3.3 asserts that there exists a non-vanishing

real valued function

F

such that

~Ff, ~ f } = - 2 ~ .

Hence we may

assume from the beginning that ~ hold on

~f

2

Y.

~}=

I

and

f + ~ = 0

Then clearly there exists a canonical local coordinate

system such that

f = pl+~f~Xl . This completes

the proof of the

i emma. Theorem 2.3.10.

Let

~

be an admissible system of pseudo-

differential equations defined on an open set Supp ~i

has the form

(Supp ~

of

P*X

such that

fl(z, ~) . . . . .

fd(z, ~) =0}.

may not be non-singular.)

Case (i)

Assume that the generalized Levi form of

L(~) has at least (x, ~ ' ~

q

=

negative eigenvalues in a neighborhood

has at least

p

,

i.e.,

d M) = 0

in

60

for

of

j < q.

Assume that the generalized Levi form

L(~)

positive eigenvalues in a neighborhood

~ ) e P-I(u)-

oO

Then

~(~,

Case (ii)

~

~ ~fj, fkl(X, ~ff~) ~ j ~ k l_~j ,k-Kd

) E P-I(U)

(2.3.50)

(x, ~

V = ~(z, ~) E U;

U

Then for any closed set

Z

in

00

~O of

of

517

(2.3.51) Here

~J

2,

Z(~i,~M)

dim prOJ(x,~)~g

tion of the system Remark.

~

= 0

~

j > q = sup dim p r O J ( x , ~ ) ~ - p.

(x,

~)~

denotes the length of the projective resoluat

(x,

If the system

then we say

for

~).

~fL

is q-convex at

satisfies the conditions of Case (i), (x, ~f~ ~ ).

In the same way if

satisfies the conditions of Case (ii), then we say at

(x, ~ ) .

~rL is q-concave

This naming is employed by taking account of the con-

nection of our results with those concerning tangential Cauchy-Riemann equations.

Moreover we may express (2.3.50) and (2.3.51) in other

ways as is well-known.

For the reader's convenience we recall the

following: (A)

Let

~"

We say that

3"

be a complex of sheaves on a topological space is of depth

~ d

X.

if one of the following equivalent

conditions is satisfied: (I)

o~J(~')

= 0

for any

j < d.

(2)

~(~')

= 0

for any

j < d

(3)

H~(X,

~') = 0

of (4)

for any

j < d

and any closed subset

Z

of

X.

and any locally closed subset

Z

X.

There is a complex

~"

quasi-isomorphic to

~J

are flabby and that

(B)

We say that

~"

~J

= 0

for

~"

such that all

j ~ d.

is of flabby dimension

~ d

if one of

the following equivalent conditions is satisfied: (I)

~(~')

= 0

(I') --d+l 01 Z ( ~ ' ) for any (2)

H~(X, of

X.

= 0

for any

j > d

and any closed subset

for any closed subset

Z

of

X

and

Z

of

X.

~J(~')

= 0

j > d+l.

~') = 0

for any

j > d

and any locally closed subset

Z

518

H d+l(x, Z

(2')

= 0 (3)

~') = 0

for any

for any closed subset

of

X

and

~J(~')

j > d+l.

There is a complex ~J

Z

~" quasi-isomorphic to

are flabby and that

~J

= 0

for

~"

such that all

j > d.

If we use these terminologies, (2.3.50) is equivalent to saying that the depth of

~ ( ~ ,

C M)

is equal to or greater than

(2.3.51) is equivalent to saying that flabby dimension of is equal to or less than

Case (i):

induction on

q.

M

its complexification

Y

~n,

~ x E M;

that

and

X

with

Cn

and denote by

respectively.

N

and

Assume

Then we can assume without loss of generality that

~

0

in

sufficiently small if necessary.

Then Theorem 2.3.2

proves that a suitable real contact transformation makes form

C M)

We prove (2.3.50) by the

~z ~ X~ z I = 0}

{fl' fl c ~ < by taking

~-~(~/~,

The problem being of local character, we may identify

x I = 0}

p ~ i.

and

q.

Proof of Theorem 2.3.10.

with

q

~l+~Xl

~2"

fl

have the

Then the isomorphism (2.3.48) proves that

~i(~,

~M)IW~f~S,N ~ ~o~i-l(~y,

~N).

On the other hand we have the relation Supp(~y) C (Here

~

denotes the canonical projection from

Therefore the system that

fj(x, ~ ) ( j

and since

~i

~iy

~ 2)

obvious.

is (q-l)-convex.

does not contain

is involutory.

fl' fcj I ( x , ~ holds.

F(Supp ~i). P*X Xx Y

to

P~.)

In fact we may assume

~i

since

fl = ~ l + ~ f ~ X l ~ 2

Hence

)I Xl=0 = If~ ' fj~(x'~f~)lxl=0

Hence the assertion that the system

~y

= 0

(j @ 2)

is (q-l)-convex is

519

Thus the induction on

q

proceeds and the proof of (2.3.50) is

completed. Case (ii).

In this case we use the induction on

p.

If

p = 0,

then it is clear that ~,Z(~f~,

£ M ) = J~J ( m p Z ~ m ~ ( ~ [

holds for any closed set since

~M

Z

in

~

, £ M) ) = 0 for

j > sup dim proj

(x,~)E~

(x,~)

'

is a flabby sheaf.

Now assume that

p ~ i.

Then as in the proof of Case (i) a

suitable real contact transformation permits us to assume from the beginning that fl (x, ~) =

~l-~Xl~2

•

Then we easily see that the generalized Levi form of

~y

has

(p-l)

positive eigenvalues. On the other hand it is clear that dim proj ~i = dim proj ~ y +

I.

Thus isomorphism (2.3.49) makes the induction on

p

proceed.

Therefore the proof of (2.3.51) is completed. Corollary

2.3.11.

Let

~fL be an admissible system of pseudo-

differential equations defined on an open set

~- of

that

(x, ~

point

~

is (q+l)-convex or (q-l)-concave at (x, ~

(2.3.52)

)

in

Supp~f~.

Ext~,z(~,@~i,

Let

~+

of the points at which

d M) = H ~ ( f L , ~ o ~ ( ~ ,

~+

and

~

implies that

(resp. ~ _ ) ~

are open in ~

= ~+

O ~_.

Z

for any

~ M )) = 0

of

be the subset of

is (q+l)-convex ~

)

Then

holds for any locally closed subset Proof.

~T~S~'~M. Suppose

~-

consisting

(resp. (q-l)-concave).

and the assumption of the corollary Now we consider the following exact

Then

520 sequence: (2.3.53) --~ Ext~,z~a+(gl +, ~ , C M) Then Theorem

2.3.10

hand side vanish.

asserts that the left hand side and the right

Hence

Ext~,z(~., ~

, ~M )

vanishes.

This is the

desired results. 2.4.

Structure theorem IV

--general

case -- .

Now we will

give our final theorem concerning the structure of (microfunction solution sheaf of) general system

~

,

which is, so to speak, the mixture

of the cases treated by Theorem 2.1.8, Theorem 2.2.9 and Theorem 2.3.2 respectively.

More precisely, the system

9~

which we now want to

investigate is assumed to satisfy only the following conditions in a complex neighborhood

~

of

(xO, ~ 0 )

denotes the characteristic variety of (2.4.1)

~

@ V ~ S ~ . ~

(in

There

V

P'X).

is an admissible and regular system of pseudo-differential

equations. (2.4.2)

V ~ V

is regular.

(2.4.3)

Tx,(V) O TE,(V) = Tx,(V ~ ~)

(2.4.4)

The generalized Levi form of the system signature

for any

x* £ V. ~i

is of constant

(q, p).

Concerning the system

satisfying above condition we have

the following decisive theorem. Theorem 2.4.1.

The system

is micro-locally isomorphic to

a direct summand of the direct sum of a finite number of copies of the system

(x;/~)

~0

which has the following form in a neighborhood of

= (0; 0, ..., 47f) e~is~'~:

521

U = 0,

j = i, "'"

+~

~

( ~ Xr+2k-i

r

)u = 0,

k = i, "'', s

r+2k

(2.4.5) (

(

~ +~T ~Xr+2 s+~

~ - ~f~ Xr+2s+~ ~Xr+2s+£

r = 2 codim V - codim (V f~ V)

Here

~ ) u = 0, ~x n

Xr+2 s+~

~x

) u = 0,

I-- 1,-'',q, 2 = q+l,''',p+q.

n

and

s = codim (V ~ V)

codim V

(p+q).

-

Remark.

Professor S. Hitotumatu reported in S~gaku Vol.9

(1956)

that Professor L. Bers had "conjectured" at the occasion of the symposium on analytic functions held at Princeton in 1956 that any "good" system of linear differential equations would consist only of (de Rham system and) Cauchy-Riemann system.

The above theorem has

realized the Bers conjecture with due modifications! Proof of Theorem 2.4.1.

In virtue of Theorem 5.3.7

ficient to do the "geometrical optics" for the system investigate the geometrical structure of defined by in

~

V.

{fl(z' ~) . . . . . fd(z ' ~) = 0 1 ,

and homogeneous of degree

(2.4.3) allow us to assume that

~ .

fl(z, ~),

~

Assumptions

(x 0, 0 ) .

i.e., to

V ~ ~

f j (z, ~)

"-', fr(Z, ~)

real coefficients in its Taylor expansion at

,

Assume that

where

1 in

it is suf-

is

is defined (2.4.2) and has only

Thus Lemma

5.1.1 of Chapter II proves that a suitable real contact transformation makes

f.j(z, ~) =

~j'

(j = I , .'' , r).

Then the successive applica-

tions of Theorem 2.2.21 and Theorem 2.3.2 show the existence of a real contact transformation which makes ,

!

fj(z, ~) =

~r+2 (j -r) -i +~-l ~r+2(j-r)'

fj(z, ~) =

~'j+s + ~ f ~ x ! J+s ~n' ,

j = r+l,''',r+s

j = r+s+l, "'" ,r+s+q ,

,

522

f.j(z, ~) =

~'j+s - ~f~xj+s' {n' '

Therefore we may take neighborhood of (x',

~')

(x', ~')

(x 0, 70).

becomes

j : r+s+q+l,

''', d (:r+s+p+q)

as the local coordinate system in a

Thus changing the notations so that

(x, ~),

we see that the integral transformation

associated with the above contact transformation allows us to assume from the beginning that Supp ~ =

V = ~(z, ~) ~ P'X;

~r+2(j-r)+l + ~

holds in

~r+2(j-r)

~j = 0 = 0

(j = l,...,r),

(j = r+l,''',r+s),

~j+s +$-f~Xj+s ~n = 0

(j = r+s+l,

~j+s - ~f~ Xj+s ~n = 0

(j = r+s+q+l ' ''' ' r+s+p+q)}

~

by taking

~

"'" ' r+s+q) '

sufficiently small if necessary.

Thus

Theorem 2.4.1 follows immediately from Theorem 5.3.7 of Chapter II. The above theorem gives us many informations concerning the structure of microfunction solution sheaf of the system vanishing of

~ ' ~J ( ~ ,

a direct summand of

~)

follows from that of

(~0)r.

Corollary 2.4.2.

Let

~

~(~0'

~

~x~(~L,

~ )

if ~YL is

be a system of pseudo-differential (2.4.1)~(2.4.4).

following structure in a neighborhood

Then

J #P,

£ M ) = O,

while the remaining cohomology group

=

U

~(~, of

~M)

(x O,

and a smooth map

~

from

V

~ U

has the

~'~ ~0):

There exist an s-dimensional complex manifold N

because the

In fact we have the following result.

equations satisfying conditions

manifold

,

to

Y,

a real analytic

Y ~f~S~

such

J

on

that there exists a sheaf summand of

~ r

where

of the solution sheaf ~ . u = 0, 3 holds.

~{

j = i, "'', s,

~r

Y x~f~S*N,

which is a direct

denotes the direct sum of

r

copies

of the partial Cauchy-Riemann system with respect to

Y,

and that

~ = ~-i~

523

In virtue of the method of the proof of the preceding

Proof.

theorem we can find complex manifolds

V. J

so that

(j = I, 2, 3)

(2.4.6)

V = VI ~ V 2 ~ V3 ;

(2.4.7)

V1

is the complexification of a real manifold ;

(2.4.8)

V2

satisfies conditions

(2.4.9)

The generalized Levi form attached to (q, p)

and

(2.2.1),

(2.2.2) and (2.2.3); V3

is of signature

codim V 3 = p + q,

Then by the methods of the proof of Theorem 2.1.8 and Theorem 2.3.10 and the assumption (2.4.1), we can find real manifolds and their complexifications

m~(~,

Y2

d M) ~ ~ ( ~

and

YI

YI

M2 ~ M I

C_~M

respectively so that

~ Ml)

~

(~Y2 ' £

M2 )[-P]

holds and that Supp(~y2) where

~

=

~((P*X ×X Y2 - P~2 X) ~ V2) '

denotes the projection P*X K Y2 X

-

P~ X --> P'Y2 " 2

Therefore in virtue of Theorem 2.2.9 ~(9[, Moreover if we set

~M ) = 0

= ~(~Y2

for

, C M 2 ),

j # p . the methods of the proof

of Theorem 2.1.8 and Theorem 2.2.9 immediately prove the unique continuation property stated in the corollary. find

Y

and

of the map

N ?

so that

Y X V~S*N

~ M 2.

In fact we can obviously We also note that the fiber

is nothing but the bicharacteristics

This completes the proof of the corollary.

attached to

V I.

524

BIBLIOGRAPHY

Andersson, K. G.: [i]

Analytic wave front sets for solutions of linear

differential equations of principal type, Andreotti, A. and H. Grauert:

[i]

to appear.

Th@or~mes de finititude pour la

cohomologie des espaces complexes,

Bull. Soc. Math. France, 9 0

(1962), 193-259. Boutet de Monvel, L. and P. Kr@e: and Gevrey classes, Bredon, G. E.: [I]

Pseudo-differential operators

Ann. Inst. Fourier, 17 (1967), 295-323.

Sheaf Theory,

Carath@odory, C.: [I]

[i]

McGraw-Hill, New York, 1966.

Calculus of Variations and Partial Differential

Equations of the First Order. Part i~ 1965.

Holden-Day, Amsterdam,

Translated from the German original, 1935.

Egorov, Yu. V.: [i] operators,

On canonical transformations of pseudo-differential

Uspehi Mat. Nauk, 25 (1969), 235-236 (Russian).

Gel'fand, I. M. and G. E. Shilov: Kyoritsu, Tokyo, 1963.

[i]

Generalized Functions. I and II,

Translated into Japanese from Vol. i of

the Russian original, 1959. Guillemin, V.: [i]

On subelliptic estimates for complexes,

Proc. Nice

Congress, ~, Gauthier-Villars, Paris, 1970, pp.227-230. Guillemin, V.W., D. Quillen and S. Sternberg:

[i]

The integrabili-

ty of characteristics, Comm. Pure Appl. Math. 23 (1970), 39-77. Hartshorne, R.:

[I]

Residues and Duality, Lecture Notes in Math.

No. 20, Springer, Berlin, 1966. H~rmander, L.: [I]

Uniqueness theorems and wave front sets for so-

lutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971), 671-704. : [2] 79-183.

Fourier integral operators, I, Acta Math. 127 (1971),

525

Jacobi, C. G. J.: [i]

Nova methodus, aequationes differentiales

partiales primi ordinis inter numerum variabilium quemcunque propositas integrandi, Werke V, Chelsea, pp. 1-189. Kashiwara, M.: [I] tions,

Algebraic foundation of the theory of hyperfunc-

S~rikaiseki-kenky~sho KSky~roku

Univ., Kyoto, 1969, pp.58-71 --

: [2]

No.108,

R.I.M.S. Kyoto

(Japanese).

On the flabbiness of the sheaf

~

,

Surikaiseki-kenkyusho

KSkyuroku No.l14, R.I.M.S., Kyoto Univ., 1970, pp.l-4 (Japanese). : [3] tions~ --- • [4]

Algebraic study of systems of partial differential equaMaster's thesis, Univ. of Tokyo, 1971 (Japanese). On e-hyperbolic

coefficients,

differential operator with constant

s@rikaiseki-kenky~sho

Kyoto Univ., 1972, pp.168-171 Kashiwara, M. and K. Kawai:

[i]

theory of hyperfunctions, Kawai, T.: [I]

KSky~roku No.145, R.I.M.S.,

(Japanese).

Pseudo-differential

operators in the

Proc. Japan Acad. 46 (1970), 1130-1134.

On the theory of Fourier transforms in the theory of

hyperfunctions and its applications,

S~rikaiseki-kenkyusho

KSky~roku, No.108, R.I.M.S., Kyoto Univ., 1969, pp.84-288 •

: [2]

(Japanese).

Construction of local elementary solutions for linear partial

differential operators with real analytic coefficients The case with real principal symbols -- ~ (1971), 363-397.

(I) --

Publ. RIMS, Kyoto Univ.

Its summary is given in

Construction of elementary solutions for I-hyperbolic operators and solutions with small singularities~

Proc. Japan Acad. 46

(1970), 912-916. and Construction of a local elementary solution for linear partial

526

differential operators. I, : [3]

Proc. Japan Acad.

4_7_7(1971), 19-23.

Construction of local elementary solutions for linear

partial differential operators with real analytic coefficients

(II) - - T h e case with complex principal symbols -- , Publ. RIMS, Kyoto Univ.

~ (1971), 399-426.

Its summary is given in

Construction of a local elementary solution for linear partial differential operators. II, : [4]

Proc. Japan Acad.

47 (1971), 147-152.

On the global existence of real analytic solutions of

linear differential equations, I and II,

Proc. Japan Acad.

4-7

(1971), 537-540 and 643-647. : [5]

On the global existence of real analytic solutions of

linear differential equations,

These Proceedings,

Part I,

pp.97-I19. : [6]

Theorems on the finite-dimensionality of cohomology groups,

I and II, : [7]

Proc. Japan Acad.

48 (1972), 70-72 and 287-289°

On the global existence of real analytic solutions of

linear differential equations (I), Japan

24 (1972).

Komatsu, H.: [I]

Cohomology of morphisms of sheafed

Fac. Sci. Univ. Tokyo, Sect. IA : [2]

to appear in J. Math. Soc.

spaces,

18 (1971), 287-327.

A local version of Bochner's tube theorem,

Univ. Tokyo, Sect. IA Komatsu, H. and T. Kawai:

Kuranishi, M.: [i]

J. Fac. Sci.

19 (1972), 201-214. [i]

Boundary values of hyperfunction

solutions of linear partial differential equations, Kyoto Univ.

J.

Publ. RIMS,

~ (1971), 95-104. Convexity conditions related to ½ estimate on

elliptic complexes,

Proc. Nice Congress,

Paris, 1970, pp.231-235.

~,

Gauthier-Villars

527

Lewy, H.: [i]

On the local character of an atypical linear differen-

tial equation in three variables and a related theorem of regular functions of two complex variables, Ann. of Math.

64 (1956),

514-522. --

: [2]

An example of a smooth linear partial differential equation

without solution, Ann. of Math. Martineau, A.: [I]

66 (1957), 155-158.

Le "edge of the wedge theorem" en th@orie des

hyperfonctions de Sato,

Proc. Intern. Conf. on Functional Analysis

and Related Topics, Univ. Tokyo Press, 1969, pp.95-I06. Maslov, V.: [I]

Theory of Perturbation and Asymptotic Method~

Moscow

State Univ., 1965 (Russian). Mizohata, S.: [i]

Solutions nulles et solutions non analytiques~

J. Math. Kyoto Univ., ! (1962), 271-302. Morimoto, M.: [I]

Sur les ultra-distributions cohomologiques~

Inst. Fourier, : [2]

Ann.

19 (1969), 129-153.

Sur la d@composition du faisceau des germes de singularit@s

d'hyperfonctions,

J. Fac. Sci. Univ. Tokyo, Sect. IA, 17 (1970),

215-239. -

: [3]

Support et support singulier de l'hyperfonctionj

Japan Acad. : [4]

Proc.

4_7_7(1971), 648-652.

Edge of the wedge theorem and hyperfunction,

These Proceed-

ings, Part I, pp.39-79. Naruki, I.: [I]

Holomorphic extension problem for standard real

submanifolds of second kind~

Publ. RIMS, Kyoto Univ.

~ (1970),

113-187. --

: [2]

Localization principle for differential complexes and its

application~

Publ. RIMS, Kyoto Univ., 8 (1972), 43-110.

528

Quillen, D. G.: [i]

Formal properties of over-determined systems of

linear partial differential equations, thesis presented to Harvard University. Sato, M.: [I]

Theory of hyperfunctions II, J. Fac. Sci. Univ. Tokyo,

(1960), 387-437. : [2]

Hyperfunctions and partial differential equations, Proc.

Intern. Conf. on Functional Analysis and Related Topics, Univ. of Tokyo Press, 1969, pp. 91-94. --

: [3]

Regularity of hyperfunction solutions of partial differen-

tial equations, Proc. Nice Congress, ~, Gauthier-Villars, Paris, 1970, pp. 785-794. : [4]

Structure of hyperfunctions.

Reports of the Katata sym-

posium on algebraic geometry and hyperfunctions, 1969, pp. 4-14.30 --

: [5] 9-27

(Notes by Kawai, in Japanese). Structure of hyperfunctions, SNgaku no Ayumi, 15 (1970), (Notes by Kashiwara, in Japanese).

Sato, M., T. Kawai and M. Kashiwara:

[I]

On pseudo-differential

equations in hyperfunction theory, to appear in Proc. A.M.S. Summer Institute on Partial Differential Equations held at Berkeley, 1971. Schapira, P.: [i] Une @quation aux d@riv@es partielles sans solutions dans l'espace des hyperfonctions, C.R. Acad. Sci. Paris

265

(1967), 665-667. : [2]

Theor@me d'unicit@ de Holmgren et operateur hyperboliques

dans l'espace des hyperfonctions, Annais Acad. Brasil. Sci. 43 (1971), 38-48. :

[3]

Hyperfonctions et probl@mes aux limites elliptiques,

529

Bull. Soc. Math. France Spencer, D.C.: [I]

99 (1971), 113-141.

Overdetermined systems of linear partial dif-

ferential equations, Bull. A.M.S. 75 (1969), 179-238. : [2]

Overdetermined operators:

some remarks on symbols, Proc.

Nice congress, ~, 1970, pp. 251-256.