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)

=

~/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 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)


) = 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. 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 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[
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~.. 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 .

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 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 = t0)

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


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 ,~ >-~ 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-~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


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~Xx

~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)] £ ] , k ] + 2M[C~

I

lIB(A - A)-II~x_~x ~ ~ •

Therefore,

max (Kg, 2M~C~) ,

(] O, t = g

We may assume that

Tt

is an L(X)-

there is

~L > 0

converges on

R E=

0
(¢ - [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 -~ 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,

(~
---~-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~

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

=

=

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

~

(6)

Mj 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
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

~
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
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



>

•

and

It follows from Theorem 4 that

l n , ~ P ) "= H 0 (K, ~ P ' ) , HK(~ 4.

P(D)




>

~-...

~ £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 "'"

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
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 =

~(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)

= (-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 ~,(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

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 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
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 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 ~

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)
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


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

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A. Grothendieck,

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An Introduction to Complex Analysis in Several

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A. Martineau,

Les hyperfonctions de M. Sato,

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M. Sato,

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Math. Z., 96 (1967), 373-392,

main results are an-

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Linear partial differential operators and generalized

distributions,

Arkiv f. Mat.,

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Distributions, Complex Variables and Fourier

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[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-

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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.

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Sur les espaces de solutions d'une classe

g6n@rale d'@quations aux d@riv@es partielles,

J. d'Anal. Math.,

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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.

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Faisceaux sur des vari@t@s analytiques r@elles,

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Sur les fonctionnelles analytiques et la trans-

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Distributions et valeurs au bord des fonctions

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Completeness and the open mapping theorem,

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Ultra-distributions d@finies sur

~n

certaines classes de vari~t@s diff~rentiables,

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II,

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Un th@or@me de dualit@,

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Su certi classi di spazi localmente convessi

260

importanti per le applicazioni,

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Darstellung der Schwartzschen Distributionen

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Math. Z., 77

(1961), 106-124.

On the theory of general partial differential

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Differentiability properties of solutions of

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Linear Partial Differential Operators,

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The fundamental solution of linear elliptic differen-

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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

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 ~ ) )

~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:

0}.

'

-3

DM

~

M

/(1.3.1)

X

Theorem 1.3.1.

k J~SM


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

£)

~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