Hyperfunctions on Hypo-Analytic Manifolds (AM-136), Volume 136 9781400882564

Table of contents :
CONTENTS
Preface
0.1 Background on sheaves of vector spaces over a manifold
0.2 Background on sheaf cohomology
CHAPTER I HYPERFUNCTIONS IN A MAXIMAL HYPO—ANALYTIC STRUCTURE
Introduction
1.1 Analytic functionals in open subsets of ℂ^m
1.2 Analytic functionals in ℂ^m
1.3 Integral representations of analytic functionals carried by small compact subsets of a maximally real submanifold
1.4 Support of an analytic functional carried by small compact subsets of a maximally real submanifold
1.5 Hyperfunctions on a maximally real submanifold of complex space. Hyperfunctions on a manifold equipped with a maximal hypo—analytic structure
1.6 Hyperfunction theory, polynomial convexity and Serre duality
Appendix to Section 1.6: Homomorphisms of Fréchet spaces
CHAPTER II MICROLOCAL THEORY OF HYPERFUNCTIONS ON A MAXIMALLY REAL SUBMANIFOLD OF COMPLEX SPACE
Introduction
II.1 Boundary values of holomorphic functions in wedges
II.2 FBI transform of a hyperfunction. Inversion of the FBI transform. Hypo—analyticity charaterized by the exponential decay of the FBI transform
II.3 Local representation of a hyperfunction as a sum of boundary values of holomorphic functions in wedges. Hypo—analytic wave front set of a hyperfunction
II.4 Delimitation of the hypo—analytic wave—front set of a hyperfunction by the decay of its FBI transform
II.5 Edge of the wedge wedge
II.6 The sheaf of microfunctions
CHAPTER III HYPERFUNCTION SOLUTIONS IN A HYPO—ANALYTIC MANIFOLD
Introduction
III.1 Definition of hyperfunction solutions in a hypo—analytic manifold
III.2 Statement of the main theorem on invariance of the sheaves SA^(q)
III.3 Proof of the Main Theorem
CHAPTER IV TRANSVERSAL SMOOTHNESS OF HYPERFUNCTION SOLUTIONS
Introduction
IV.1 Analytic functionals depending smoothly on t
IV.2 Hyperfunctions depending smoothly on t
IV.3 Hyperfunction solutions depending smoothly on t
IV.4 Partial regularity of all hyperfunction solutions
IV.5 Consequences and additional remarks
IV.6 FBI transform of hyperfunctions depending smoothly on t
IV.7 Use of the fine embedding. CR hyperfunctions
IV.8 FBI minitransform. Jump Theorem in hypo—analytic structures of the hypersurface type
IV.9 Nonvanishing of the local cohomology in hypo—analytic structures of the hypersurface type
Historical Notes
Bibliographical References
Index of Terms

Citation preview

Annals of Mathematics Studies

N um ber 136

Hyperfunctions on Hypo-Analytic Manifolds

by

Paulo D. Cordaro and Frangois Treves

P R IN C E T O N U N IV ER SITY PRESS

P R IN C E T O N , N E W JERSEY 1994

Copyright © 1994 by Princeton U niversity Press ALL R IG H T S R E SE R V E D

T h e Annals o f M athem atics Studies are edited by Luis A. Caffarelli, John N . M ather, and Elias M . Stein

Princeton U niversity Press books are printed on acid-free paper and m eet the guidelines for perm anence and durability o f the C om m ittee on Production G uidelines for Book Longevity o f the C ouncil on Library Resources

Printed in the U nited States o f America 10 9 8 7 6 5 4 3 2 1

Library o f C o n g r e ss C a ta lo g in g -in -P u b lic a tio n D ata Cordaro, Paulo D . Hyperfunctions on hypo-analytic manifolds / by Paulo D . Cordaro and Frangois T reves. p. cm. — (Annals o f mathematics studies ; no. 136) Includes bibliographical references and index. IS B N 0-6 91-02993-8 (cloth) IS B N 0 -691-0299 2-X (pbk.) 1. H yper functions. 2. Submanifolds. I. Treves, Frangois, 1930 -. II. T itle. III. Series. Q A 324.C 67 1994 515'.782— dc20 94-29391

T h e publisher would like to acknowledge the authors o f this volum e for providing the camera-ready copy from which this book was printed

To Andre Martineau, a friend, in m em oriam

CONTENTS Preface

xi

0.1

Background on sheaves of vector spaces over a manifold

3

0.2

Background on sheaf cohomology

12

CH A PTER I HYPERFUNCTIONS IN A MAXIMAL H Y PO -A N A LY TIC STRUCTURE

Introduction

25

1.1

Analytic functionals in open subsets of iCm

29

1.2

Analytic functionals in Cm

38

1.3

Integral representations of analytic functionals carried by small compact subsets of a maximally realsubmanifold

1.4

Support of an analytic functional carried by small compact subsets of a maximally real submanifold

1.5

52

66

Hyperfunctions on a maximally real submanifold of complex space. Hyperfunctions on a manifold equipped with a maximal hypo—analytic structure

1.6

77

Hyperfunction theory, polynomial convexity and Serre duality

91

Appendix to Section 1.6: Homomorphisms of Frechet spaces

107

C H A PTER II MICROLOCAL THEORY OF HYPERFUNCTIONS ON A MAXIMALLY REAL SUBMANIFOLD OF COM PLEX SPACE

Introduction

113

viii

Contents

II. 1

Boundary values of holomorphic functions in wedges

11.2

FBI transform of a hyperfunction. Inversion of

115

the FBI transform. Hypo—analyticity charaterized by the exponential decay of the FBI transform 11.3

125

Local representation of a hyperfunction as a sumn of of boundary values of holomorphic functions in wedges. Hypo—analytic wave front set of a hyperfunction

11.4

141

Delimitation of the hypo—analytic wave—front sett of of a hyperfunction by the decay of its FBI transform

150

11.5

Edge of the wedge wedge

161

11.6

microfunctions The sheaf of 'microfunctions

170

CH A PTER III H YPERFUNCTION SOLUTIONS IN A H Y PO -A N A LY TIC MANIFOLD

Introduction 111.1

179

onsinin Definition of hyperfunction solutions a hypo—analytic manifold

111.2

n theinvariance invariance Statement of the main theorem on of the sheaves

111.3

181

195

Proof of the Main Theorem

211

C H A PTER IV TR A N SV ER SA L SM OOTHNESS OF HYPERFUNCTION SOLUTIONS

Introduction

235

IV. 1 1

AnAnalytic functionals depending smoothly on t

239

IV.2

Hyperfunctions depending smoothly on t

259

C ontents

ix

IV.3

Hyperfunction solutions depending smoothly on t

272

IV.4

P artial regularity of all hyperfunction solutions

295

IV.5

Consequences and additional remarks

306

IV. 6

FBI transform of hyperfunctions depending smoothly on t

316

IV.7

Use of the fine embedding. CR hyperfunctions

327

IV.8

FBI m initransform . Jum p Theorem in hypo—analytic structures of the hypersurface type

IV.9

337

Nonvanishing of the local cohomology in hypo—analytic structures of the hypersurface type

344

Historical Notes

357

Bibliographical References

365

Index of Term s

371

PREFACE The m aterial in the present monograph represents, in a sense, the next logical step in the development of the local (and microlocal) theory of

hypo—analytic structures described in the book [TREVES, 1992]. Roughly speaking, a hypo—analytic structure on a C00 manifold is w hat makes possible the holomorphic extension of functions, distributions, and more generally, as we show in the pages th a t follow, hyperfunctions. The theory developed here relies on concepts, m ethods and results found in the book cited above. However, we have made a rule of referring to it only for technical assistance, and of repeating every definition needed for the comprehension of the present text. Distributions are the objects of study in an involutive structure: as solutions, rig h t-h a n d sides, initial data, etc. An involutive structure on a C°° manifold M is the datum of a vector subundle V of the complexified tangent bundle CTM th a t satisfies the Frobenius condition [V,V] C V (ie., the com m utation bracket of two smooth sections is a section). For a distribution u in an open subset Q of M to be a solution means th a t Lu = 0 for all e

sections L of V over Q, ie., L E (^(O jV ).

An im portant subclass of involutive structures are the locally

integrable structures: the structure on il defined by the vector bundle V (whose rank is equal to n) is locally integrable if every point p has an open neighborhood

grals", ie., C® solutions

in which there are m = dim M — n "first inte­ such th a t dZ\!\u • - AdZm ^ 0 at p. Real

structures (ie., V is locally spanned by real vector fields) and real—analytic structures (ie., M and V are of class CW), complex structures (ie., V®V = (.Tjt) and more generally elliptic structures (ie., V+V = CTM) are locally integrable; and so are strongly pseudoconvex CR structures of the h y p e r surface type if dim M > 7. However there exist involutive structures th at

xii Preface

are not locally integrable. On this and on related topics we refer the reader to C hapters VI, VII of [TREVES, 1992]. Deeding with a fixed set of first integrals Z \v ..,Zm in an open subset Cl of M leads to novel questions. The approxim ation form ula in [BAOUENDI-TREVES, 1981] (see also Section II.2, [TREVES, 1992]) asserts th a t each point p E ft has a neighborhood

in which every C° solution

-1 h is constant on the pre—image Z (zq) of every point z0 E Cra in the range of Z = (Z i,...,Z m). This allows us to pushforward the solution h under the m ap Z, to get a kind of CR function h on the, generally highly singular, set Z(U ). In turn this leads to the question of the possibility, or impossi— P bility, of extending the pushforward A as a holomorphic function in a do­ m ain of and the boundary value of a similar function on the opposite side (Theorem IV.8.2). We close C hapter IV by showing th a t a num ber of nonsolvabity

xx Preface

results valid for distributions, which include the classical ones for the Lewy and M izohata vector fields, remain valid for hyperfunctions. The

m ain

definitions

and

results about

sheaves

and

sheaf

cohomology needed in the text are gathered in two short introductory sections, 0.1 and 0.2. We have om itted most of the (mostly semitrivial) proofs. We have thought it proper to add a postscript, briefly surveying the rather rich history of hyperfunction theory and highlighting the contributions of various authors to the presentation in this m onograph. Much of the research and the writing for the monograph was done while Cordaro was a visiting member at the Institute for Advanced Study in Princeton, N. J. The visit of Cordaro was funded in part by G rant No 92/1402—7 from FA PESP (Sao Paulo, Brazil) and in part by a grant C N Pq/N SF under the US—Brazil Cooperative Research program. The work of F. Treves was supported by NSF G rant DMS—9201980, as well as by NSF G rant INT—9103833 (US—Brazil Cooperative Research).

Paulo D. Cordaro, Instituto de M atem atica e Estatfstica, Universidade de Sao Paulo, C.P. 20570, 01452-990 Sao Paulo, S.P. (Brazil). Francois Treves, D epartm ent of M athem atics, Rutgers University, New Brunswick, N. J. 08903 (USA).

Hyperfunctions on Hypo-Analytic Manifolds

0.1

BACKGROUND

ON

SHEAVES

OF

V EC TO R

SPACES

OVER A MANIFOLD

Like most presentations of hyperfunction theory, ours makes m uch use of the language of sheaves and of sheaf cohomology. In these prelimi­ nary sections 0.1 and 0.2 we recall some of the basic statem ents of th a t language. For the proofs th a t are missing we refer the reader to the textbooks on sheaf theory (e.g. [CARTAN, 1950/51], [GODEMENT, 1958], [BREDON, 1968]). We shall be dealing exclusively with sheaves of complex vector spaces over a C00 manifold M, countable at infinity (and therefore p a r a com pact). For us a sheaf 7 will be prim arily defined through a presheaf: we are given a family 2^ of open subsets U of M which is a basis of the topology of M and: for each U E 2^, a vector space F(U); for each pair of elements U, V of 2C such th at V C U, a linear map ry : F(U) -> F(V) (thought of as a restriction m apping), with the standard transitivity property: if W C V C U, r ^ o r | | = r ^ ,. We list right away some of the main examples of presheaves (F (U ),r^ ) we shall be dealing with. For the sake of simplicity, in every one of these cases we take the family

2C to

comprise all open subsets of M (in the text we shall sometimes be forced to select smaller families). In every one of these examples the maps r y are truly defined by restriction from U to V C U. We content ourselves with indicating the selection of the linear space F(U).

Basic Examples: 1) Ck(U), the space of all Ck functions in U (0 < k < 4-ao); 2) ^ ( U ) , the space of distributions in U; 3) when M is real analytic, C°)(U), the space of real analytic

4

Sheaves of Vector Spaces

functions in U; 4)

when M = Cm, 0(U), the space of holomorphic functions in U. □

Presheaves o f spaces o f differential form s: 5) Returning to a real m ani­ fold M, we can also select F(U) = (^(U jA ^), the space of smooth differen­ tial q—forms in U. In the dom ain of local coordinates z^,..,z^. (N = dim M) such a form has an expression

(0.1.1)

/ =

£

/ dr.

|i|= q ‘

^

We use m ulti—index notation: I = {ip..,iq} with integers i^ such th a t 1 < i^ < • • • < iq < N, dzj = dz. A - • - Adz. ; and

G (^ (U ) = (^ (U jA 0).

W ith these spaces one can define the De Rham differential com­ plex, ie. the sequence of differential operators

(0.1.2)

d:

-* ( ^ ( U j A ^ 1) q = 0,1,...,

where the exterior derivative d acts in the usual fashion on a form (0.1.1):

df = | E

(0.1.3)

N S (a/j/dz.JdzA dZ j;

of course, d^ = 0. We shall use the standard terminology: / is said to be closed if d f E 0, exact if there is u G C ^ U jA ^ ”*) such th a t du = / . 6)

Now, once again, suppose M = Cm (where the coordinates are

denoted by z^,..,zm) and let p be an integer > 1. We can take F(U) = 0 ( p ) ( U ) , the space of holomorphic differential p —form s in the open

Sheaves of Vector Spaces

5

subset U of 1.

Other properties concern changes in the base manifold. We shall content ourselves with a few words about the restriction

from M to an

open subset 0 of M. If the vertical arrows stand for

we get the

com m utative diagram

20

Sheaf Cohomology

1

...

^v-1

->• • •,

) - » • • • - * r ( n , 7 < ‘' >)-* r ( f t , 7 0, since

(j)

h(z)dz =

|*|=1

(j)

31

h(z)dzy

\z\=r

for all h E 0(C\{O}). By moving the origin to an arbitrary point taking r sufficiently small, the circles \z\ = r can be brought inside

of U and U. □

Proposition 1.1.1 below shows th a t an analytic functional in U is carried by an open subset V of U if, and only if, it is carried by some compact subset of V. This allows us to extend the terminology of Definition 1.1.2 and to say th at an analytic functional in U is carried by an arbitrary subset S of U if it is carried by some com pact subset of U contained in S.

PROPOSITION 1.1.1.— For p E 0 ' (U) to be carried by the open subset V of U it is necessary and sufficient that there be a compact subset K of V and a constant C > 0 such that, for all h E 0(U):

(1.1.2)

| < M > | < C M ax \h \. K

Suppose there is p E O' (V) such th a t p = fyy/J. Then, given any rv U h E 0(U ), = . There is a compact set K C V and a constant C > 0 such th at | \ < C M ax | y | , K

V g E 0(V ), whence

(1.1.2). Conversely suppose (1.1.2) holds, with K CC V. First we note th at p^h -> is a well defined linear functional on the subspace />^0(U) of 0(V): by (1.1.2), p^h = 0 =} = 0. The rig h t-h a n d side of (1.1.2) defines a continuous seminorm not only on /^ 0 (U ) but on 0(V). By the Hahn—Banach theorem, the linear functional p^h -> extends from

32

Hyperfunctions in a maxim al structure

/> ^(U ) to 0 (V ) as a linear functional p th a t satisfies (1.1.2). Clearly p = t u " i-i V v /*- □

COROLLARY 1.1.1.— For p 6 0 ' ( U) to be carried by the compact subset K o f U it is necessary and sufficient that to each compact subset K ' o f U whose interior contains K there be a constant C > 0 such that, fo r all h 6 0( U):

| < M > | < C Ma x \ h \. K'

(1.1.3)

If K C Jfo/K .' C K ' C U the infimum of the constants C th a t can be used in (1.1.3) defines a norm | | / j | L/ . The space ^ ' ( Uj K) of the analytic functionals in U carried by K can be equipped with the locally convex topology defined by all these norms p -* ||/^|| ^ / - Since it suffices to let K ' range over a countable basis of compact neighborhoods of K this topology is metrizable. The strong dual 0 ' ( U) of 0(U) is complete. Since the subset of 0 ' ( U) consisting of the analytic functionals p th a t verify (1.1.3) for some C > 0 (depending on p) is a closed linear subspace of 0 ' (U) we conclude th at ^ / (U,K) is complete: it is a Frechet space.

Let U be an open subset of Cm and let C(U) denote the space of continuous functions in U equipped

with

the

topology of uniform

convergence on the compact subsets of U. By the definition of the topology of 0(U) the latter can be regarded as a linear subspace of C(U), also in the topological sense; 0(U) is a closed subspace of C(U), since it is complete. We recall th a t the dual of C(U) is the space of the compactly

Analytic functionals in open sets

33

supported Radon measures on U, Mc(U). The transpose of the natural injection 0(U) -* C(U) (which is a homeomorphism onto its image) yields a

surjeciion MC{V)

0 ' ( U) . As a m atter of fact, O '{U )

£ Mc{U )/0{V)±. The

orthogonal of 0(\J) in Mc{U), 0(U )X, is clearly not empty: it contains any Radon measure of the form ( S duj/dz^)dxdy with TZj E C^(U). Thus we j=* may represent an arbitrary analytic functional /z in U by some compactly supported Radon measure /z* in U:

(1.1.4)

It is obvious th a t if K is the support of the Radon measure /z* then /z is carried by the com pact set K.

REMARK 1.1.1.— Actually the Radon measure /z* in (1.1.4) can be taken to be a density (z)dxdy with $ E C*(U). Indeed, 0(XJ) can be regarded as a closed linear subspace of ^ ' ( U) :

closed since it is defined by the

Cauchy—Riemann equations dh = 0. The topology induced by D ' (U) on 0{ U) is of course coarser than the one induced by C(U) (ie., the original topology of 0(U)). But actually these two topologies are the same because the closed graph theorem applies (see [SCHWARTZ, 1966], [DE WILDE, 1978]). It follows th at the natural injection 0(U) -» D '{U) is also an isomorphism onto its image, and therefore its transpose is a surjection of the dual of 2?'(U), C” (U), onto ^ ' ( U ) (we identify (j) E Cc(U) density {z)dxdy). For the readers uncomfortable with Functional Analysis let us recall a simple m anner of associating a function (j> E C*(U) to the measure

34

Hyperfunctions in a m aximal structure

/z*. If K CC U and 0 < rj < £ (j = l,...,m ) for a suitably small £ > 0, then z £ K 3 (z^-l-r^e

iO

10 zm-f rme m) E U and as a consequence, we have,

for all h E 0(1]),

/i 2?r •••/ 0

U

«

yj

A ^i+ ^e* 1,...,zm+ r me*

• -d0m.

Select ipQ € C°D([R) such th a t i>Q(r) = 0 if r


and

/.+ ®

i>0(r)rd r = 1; and set if>(z) = i>Q( | z j ) • • • i>Q( \ zm\ ). M ultiplying the

/ 0

preceding identity by r^ip0(r()' • • rm'4>0(rv^ and integrating with respect to rj from 0 to -fao for every j = l,...,n, yields h(z) = (27r)~mJ*

h(z-\-z' )ip(z' )d x ' d y ' ,

[R2m hence, if supp /z* = K, J * hdp* = (2n)~mj * [R2m

J*

h{zd-zf )i>{z/ ) d x 'd y ' d u* = z

[R2mL K 2m

J

h(z)(/)(z)dxdy, (R2m

where {z) =

^ {z —w )dp^ E C^(U). □ (R2m

Let P (z,() be a polynomial with respect to ( E Cm whose coefficients are holomorphic functions in U. We can view P{z,() as the symbol of the holomorphic differential operator P{z,djdz) — by the rule that, if / E 0(U) then / (^)Cj is the symbol of the operator / {z)d/ dzy Denote by operator u

t P (z,d /d z ) -»

its formal transpose [the formal transpose of the

f( z ) d u /d z j is the operator u

-»

-d /d z ^ (f

a )];

^P(z,d/dz) acts

continuously from 0{U) to itself and defines, by transposition, the contin­ uous linear operator P(z,d/dz): O' (U) -» ^ ( U ) . Thus

Analytic functionals in open sets

35

< P(z,dfdz)fi,h> = }

The spaces ff(K ) and 0 '( K )

Let K be a compact subset of Cm. We recall the definition of the space 0(K ) of germs at K of holomorphic functions. It is the inductive limit of the spaces 0(U), with U open, U D K, ie., the quotient of the disjoint union of the spaces 0(\J) modulo the following equivalence relation: if

E 0 (Uj) (j = 1,2), h\ 8 ^

3 V open, K C V C UiflU2,

such th a t h\ = h2 in V. This defines a restriction m ap />!!: 0(U) -* 0(K ) for any open set U D K; to i E 0(U) it assigns the equivalence class p h i of h K modulo the relation 8. The kernel of the map

K

is a closed subspace of

0(U): it consists exactly of those holomorphic functions in U th a t vanish in some neighborhood of K in U, hence in every connected com ponent of U which intersects K. Thus ptO{Ti) £ £7(U)/Ker pL carries a natural

K

K

Frechet space structure; 0(K ) is the union of all these Frechet spaces, as U varies. We equip 0(K ) with the locally convex inductive limit topology of the Frechet spaces jKr0 (U): a convex subset of 0(K ) is open if its intersection with ptO {U) is open [An arbitrary open set in 0(K ) is the union of its convex open subsets. It is not true, in general, th a t a nonconvex subset of 0(K ) will be open if its intersection with each subspace pt0(}5) is open.] K In practice, it suffices to use a countable basis of open neigh­ borhoods Uj of K such th a t

CC Uj and such th a t every connected

component of Uj intersects K (j = 1,2,...). The latter ensures th a t the

36

m aps

Hyperfunctions in a maxim al structure

K.

are injective and therefore allows us to identify 0{U:) to J

p ^ 0 (U j) C 0(K ) for every j. Furtherm ore, instead of the Freshet spaces 0(U j) we can use the Banach spaces ^ ( U j ) consisting of the bounded holomorphic functions in Uj (equipped with the sup norm ). In this m anner we see th a t 0(K ) is a countable inductive limit of Banach spaces, the spaces ^ ( U j ) [identified to

A convex subset of 0(K ) is a

neighborhood of 0 if and only if, for each j = 1,2,..., its intersection with 0 b (Uj) contains an open ball centered at 0 in 0b(^j)* ^ linear m ap of 0 (K) into any locally convex topological vector space, in particular any linear functional on 0(K), is continuous if and only if, whatever j = 1,2,..., its restriction to ^ b ( ^ j) ^ continuous. Note, however, th a t the topology induced on ^ b ( ^ j) ^y ^ b ^ J +1) and therefore the one induced on 0 b (Uj) by 0(K ), is strictly coarser than the original Banach space topology. Indeed, the image of the closed unit ball in 0 b ( ^ j) ^ as a comPact closure in ^^(XJj

— by virtue of the

Cauchy inequalities, and by the same argum ent th a t proves the Montel theorem. One says th at 0(K ) is a nonstrict inductive limit of Banach (or of Frechet) spaces. Nevertheless, a subset A of 0(K ) is bounded if and only if it is contained and bounded in $ b (^ j) ^or some J- The bounded set A will be closed, and therefore compact, in 0(K ) if and only if it is a com pact subset of ^ b ( ^ j) ^or some j* We conclude th a t every bounded and closed subset of 0(K ) is compact: 0(K ) is a Montel space (in particular, it is reflexive). One says th a t K is a Runge compact set (or th a t K has the Runge property — in Cm) if every holomorphic function h in an open set U D K is the uniform limit of a sequence of entire functions, on a set Ke = { z 6

Analytic functionals in open sets

37

Cm; d ist( 2 ,K) < e } C U (for a suitably small e > 0). This means precisely th a t the restriction to K maps 0(Cm) onto a dense subspace of 0(K ). The transpose of the restriction map is an isomorphism of the strong dual of 0(K ) onto the space of analytic functionals in Cm carried by K — w hat we have been calling 0 ' ( K ) . The Frechet space structure of O' {K) is defined by the norms (cf. remark following Corollary 1.1.1) p -* sup | \< p th>\ ; h £ 0(

0).

1.2

ANALYTIC FUNCTIONALS IN Cm

Analytic functionals in the whole space Cm (or, for th a t m atter, in any open polydisk) have convenient representations. As the Taylor expan­ sion of an arbitrary entire function h converges to h in £J(Cm) we get, for any analytic functional p i n Cra,

=

(1.2.1)

E

h^

(0)/a!.

[We make system atic use of the m ulti—index notation.] Using the deriva­ tives of the Dirac distribution Form ula (1.2.1) can be rew ritten as

p =

(1.2.2)

E

( - 1 ) 1al ,

Form ula (1.2.1) can then also be restated as

=

E

£< (0)A(

(0)/a!.

THEOREM 1.2.1.— The Laplace—Borel transform p -» p defines an isomor­ phism o f O ' ( t m) onto the space E x p (Im) of entire functions o f exponential

Analytic functionals in whole space

39

type in Cm.

c L e t

p G O' ( I m). Then p(() is an entire function of £ in (Cm as one

sees by letting the Cauchy—Riemann operator act "under the duality bracket" at the right in (1.2.2). Suppose the open polydisk A^ = { z G Cm; | Zj j < rj } with “m ultiradius" r = (r^ ,...,^ ), rj > 0, carries p G ^ ( I 111). Then, for some constant C > 0, |/ l ( 0 |< C s u p

|e

| = C exp( E r j |C j |) .

r This shows th a t p G Exp(Cm). m Suppose g G 0( 0 (1 < j < m) and all ( G Cm. It follows at once from the Cauchy inequalities in the polydisk A

th at | rj (j l,...,m ), < C' E

• - r 4 a m |A (a ) ( 0 ) |/ a ! .

Applying once again Cauchy’s inequalities, now in the polydisk A „ with r r j ^ r j ^or eac^ ^ ==

Sets us

r *a l. . . r ^ “ m| At a) (0) |/ o ! < M ax | A |,

Vr whence

40

Hyperfunctions in a m aximal structure

E

m ff( a ) ( o ) A ( a ) ( o ) /Q! < C H (1 —r j / r j ) ' 1 M ax | h \ . j =1 A .

We conclude th a t h -*

E g( a >(0)h^ a >(0)/a! is an analytic functional aGff?

carried by the closure of the polydisk A . Its Laplace—Borel transform is r obviously equal to g((). □

REMARK 1.2.1.— Given any bounded subset S of 0

there is C > 0 such th at e

(1.2 .10)

|cj < C flal,V a£

Let r > 0 be arbitrary and select e < r; the Cauchy inequalities entail

(1.2 .11)

< C ( l-e /r )-“

M ax

\h\

l zj l = r •

This shows th a t the analytic functional (1.2.9) is carried by {0}. On the other hand, it defines a distribution if and only if ca f 0 3 \ a \ < k Q < +oo. □

If a distribution u E £ / ((Rm) is identified to an analytic functional in Cm then it is easy to see, by the Cauchy inequalities, th a t the compact set supp u carries

Analytic functionals earned by a Runge compact subset o f the complex plane . The Cauchy transform

In the remainder of this section we assume m = 1. Let K be a compact subset of the complex plane and let 0 / (K) denote the space of analytic functionals in C carried by K, equipped with the Frechet space structure defined after the statem ent of Corollary 1.1.1. Then, given any w

44

Hyperfunctions in a m aximal structure

G C\K, the function z -» (z—it;)”1 is holomorphic in an open neighborhood of K. We shall hypothesize th a t C\K is connected. This is equivalent to the Runge property understood as follows: every holomorphic Junction in an open neighborhood V o f K is the uniform limit on K o f entire functions (see e. g., Theorem 1.3.1, [HORMANDER, 1967]). If K is a Runge compact subset of the plane, as thus defined, and if the analytic functional p in C is carried by K, ie., p G 0 / ( K), and if

w G C\K, we have the right to

consider

Tp(w) = (2tx)“l < ^ ( t w - z ) “1> .

(1.2.12)

The right hand side is the limit of < p ^P v (w;z)> for any sequence of polynomials with respect to z (depending on w) {Pv (w;z)}^_^ ^ converge uniformly to {2i 'k)~‘\

w —z)~l

th a t

in some compact neighborhood of K

on C\{iu}. The function Tp (defined in C\K) is called the Cauchy transform of p. Let x e C^(1R2), X = 1 in some neighborhood of K. Then

(1.2.13)

= j * h(w)dx{w)Tp(w)dwhdw, V h G 0(C).

S P b a c iL ^ (1.2.13): Since supp dx stays away from K there is an open neighborhood ft of K in which we have, thanks to the inhomogeneous Cauchy formula, h{z) = (2ix)"iy %A( w) 3%( iu) ( —z) ”*dwhdw. This entails

Analytic functionals in whole space

= (

2

45

h(w)dx{w){w—z)~^dwK&w> ,

where the duality bracket is the one between 0 (0 ) and O '( 0 ) (thus z stands for the variable point in 0 ). But we can interchange the integral and the duality bracket, whence (1.2.13). □

We shall denote by 0q(C\K) the space of holomorphic functions in C\K th a t tend to zero as \z\

H-ao. The space Oo(C\K) will be equipped

with the topology of uniform convergence on the closed subsets whose complements are bounded and contain K. The space 0 o(C\K) is a Frechet space; its topology is strictly finer than the one inherited from 0(C \K ). It is convenient to identify 0 o(C\K) to the space of holomorphic functions in the complement of K in the Riemann sphere S th a t vanish at infinity. Then the topology of 0 o(C\K) is simply the one inherited from the topol­ ogy of uniform convergence on compact subsets on 0 (S \K ); evidently

0 o ( C \ K ) is a closed vector subspace of 0(S\K). THEOREM 1.2.3.— Let K be a Runge compact subset o f C. The Cauchy transform defines an isomorphism o f 0 ' ( K) onto 0 o( t\K ) .

By letting d

act under the duality bracket at the right in

(1.2.12) we see right—away th at Tp G 0(C \K ). Set K£ = { z G t ; dist(z,K) < e } (e > 0). According to (1.2.12) there is a constant (7e > 0 such th a t

(1.2.14)

|r/j(w ) | < (7e M ax | w - z \ z GK£

V w G C \K £.

It follows at once from this th at | Tp(w) | -* 0 as | w\

-Hod; and th a t the

m ap O' (K) 3 p -* Tp G 0 O(^ \K ) *s continuous. T h at this m ap is injective

46

Hyperfunctions in a maxim al structure

follows at once from (1.2.13). Now consider g G 0 0(C\K) and let x be as in (1.2.13). For any h G 0(C) define

< p ^h > = - J g(z)dx(z)h(z)dzhdz.

(1.2.15)

Suppose we replace x by a different function Xi £ C^(IR2) with X\ - 1 in a neighborhood of K. Integration by parts shows th at

f g(z) d ( x - X i ) ^ ) K z) ^ dz = f (9g)iz) ( x - X i ) ( z)h(z)dzhdz = o and thus th a t the definition of p^ is independent of the choice of the cutoff xNow select h(z) = (2i*) ^ ( w - z ) ^ for a fixed w G C\K. Then select X so th a t xC™) = 0* We have, by (1.2.15), < P ^ {2 i'k Y \ v }- z Y ^ > — (2 i 'k Y ^ J g (z )d ( l-x ){ z ) (w -z )~ 1dzhdz = g( w),

again thanks to the inhomogeneous Cauchy formula and to the fact th a t g vanishes at infinity. We have thus proved th at p -» Tp is a continuous linear bijection of 0 ' { K) onto 0 o(C\K). Since both these spaces are Frechet spaces, the Cauchy transform defines a homeomorphism. □

COROLLARY 1.2.1.— Let

and K 2 be two Runge compact subsets of

such that K^UK2 is Runge, and let p be an analytic functional in

C

C. The

following is valid: I f p is carried by both

and K 2 then p is carried by the

intersection K jflK 2. I f p is carried by KjUK2 then there are analytic functionals p ^ and P2 in

C, carried by

and K 2 respectively, such that p = p^-py.

Analytic functionals in whole space

The intersection of two Runge compact subsets

47

and K 2 of C is

Runge. Indeed, the union (C \K 1)U(C\K2) = C \(K 1DK2) is a connected neighborhood of oo since each set C \K i (i = 1,2) is one.

: First suppose p is carried by

and by K 2. The Cauchy

transform Tp defines a holomorphic function gj in 0 o(C \K j) for each j = 1, 2. We have gt = g2 in (C X K ^ fl^ K j) = C \(K tUK2), since C ^ iq U K j) is a connected neighborhood of oo, according to the hypothesis. Therefore there exists a holomorphic function g G 0 o(C \(K ^nK 2)) th a t extends both g^ and g2. Theorem 1.2.3 implies th at it is the Cauchy transform of an analytic functional carried by K^flKg, necessarily equal to p. Now suppose p is carried by K^UKg. Let

and Q2 be two

arbitrary open subsets of C and h a holomorphic function in

Let x

G CGD(n^Uf22) be such th at supp x C ftj and su p p (l—x) C Cl2; (1 —x)h can be extended as a 0°° function in ft^, xh 33 a

function in Q2 and

h{dx/dz) as a Cm function in Q jU n2, by setting each one of these functions to be zero off h{dx/dz) and define h =

Select u G

= (1—x)^ + u m '

such th a t du jdz = ^2 = ^

~ u 'm ^2*

have

+ h2 in n ifin 2, Aj G 0(Qj), j = 1,2. Apply this to Qj = C\Kj and

h = Tp; we get Aj = 0(C \K j). Since | h(z) | -> 0 as \z\ nonnegative power series in the Laurent expansions of

-» -fro the

and h2 cancel

out, and we can eliminate them , ie. we may assume Aj G 0 o(C \K j). Then, for j = 1, 2, Aj is the Cauchy transform of an analytic functional pj G

Hyperfunctions in a m aximal structure

48

fl'C K j); and y = y t+ /t2. □

If the requirem ent th a t K 1UK2

be Runge is lifted

the

first

assertion in Corollary 1.2.1 is not any more valid, as shown by

EXAMPLE 1.2.2.— Consider the analytic functional fi in C defined by

= J A(*)dz, h G 0(C), 7ol

(1.2.16)

where

7 ol

is any smooth curve joining 0 to 1. It is clear th a t fi is carried

by all curves

7 0^

but th a t it is not carried by their intersection, the set

{ 0 }U{1 }: p defines a compactly supported distribution on R, the one defined by the characteristic function of the unit interval [0 , 1 ]. W hat fails in the proof of Corollary 1.2.1 when we lift the condition th a t KjUK 2 be Runge, is the extension to C \(K in K 2) of the Cauchy transform s g^ and #2. Make the following two choices for the path 7 ol 0

to

in (1.2.16): 1

first take

7

=

7

+, the arc of the circle | z —\ | = | joining

in the upper half—plane; second, take 70i =

7

', the arc of the same

circle joing 0 to 1 in the lower half—plane. But g^—g^ = 0 in the region 1z~ i I > 1 whereas g±—g^ — 1 in the disk | z —\ | < ^. This confirms th a t there is no common extension of g^ and g2 to C \( 7 +fi7 ~). □

Analytic functionals in whole space

49

Hyperjunctions on the real line

Let U be a bounded and open interval in R, t f / U its closure, d\J its boundary in R. Obviously

U is a Runge compact set since its

complement is connected. By definition a hyperfunction u in U will be an element of the quotient space #(U ) = 0 ' ( t f / u ) / 0 ' ( d U). According to Theorem 1.2.3 the Cauchy transform defines an isomorphism

(1.2.17)

fl(U) S 0 O(C\( t f / U ) ) / 0 0{ t \ 8 V ) .

The inclusion 0o(C \( $V tj)) C 0(C \( &/U)) induces an isomorphism

(1.2.18)

0 O(C \( t f / U ) ) / 0 o(i \ d \ J ) = 0(C\{ # / b ) ) / 0 { C \ 8 U ) ,

rsj as one sees by taking Laurent expansions. Now let U be a connected open subset of C such th a t U = UflR. By the reasoning in the proof of Corollary 1.2.1, given any g 6 0(U \U ) there are G € 0(C\( t? /U )) and F € 0(\J) IM\ such th a t g = F — G in U \U . It follows from this th a t the restriction m apping 0(C \( & /U)) -> 0(U \U ) induces a surjection

(1.2.19)

0 ( i \ ( &/U)) -+ 0 ( U \U )/0(U )

and consequently an isomorphism

(1.2.20)

0(C\( t f A l ) ) / 0 ( C\5U) * 0 (U \U )/0 (U ).

By combining the isomorphisms (1.2.17), (1.2.18) and (1.2.20) we obtain

( 1.2 .21 )

B(U) * 0 (U \U )/0 (U ).

50

Hyperfunctions in a maximal structure

Call n + (resp., n~) the open upper (resp. lower) half plane. Notice rv, + th at, since U is an interval, U \U has two connected components, U = Ufin* and U ” = U fin “. In other words, we may identify £7(U\U) to 0 (U +)x 0 (U ’). W ith this identification the restrictions 0 (C \( ^ / U ) ) 0(IL ) -* 0(U ) yield the factorization 0 (C \( f f / u ) ) -> 0 ( n +) x 0 ( n _) -4 0 ( u * ) x 0 ( u ~ ) / a ( u ) , of the m ap (1.2.19), where A(U) is the linear subspace of 0 (H +)x 0 (n ") consisting of the pairs ( 2 = Ci+* " ’ "Km-

Let then K be an arbitrary compact subset of 0 and h £ 0{U) with U an open neighborhood of K in Cm. Select a function x £

X -

1 in a open neighborhod W C U of K in Cm. We form (1.3.4) with u= x^. Let then ^ £ C ^ (# 7) , 0 < ^ < 1 everywhere, ip = 1 in a neighborhood U11 C U* of K. Let B denote the unit ball in Rm. If e > 0 is sufficiently small, supp Tp + ic 'B C U* for all €7, 0 < e7 < e. We can deform the dom ain of integration in the integral (1.3.4) from ft to the image of ft under the m ap z -» H(z,t) = z + np(z)tt with t £ Rm, 111 < e. We obtain

Hyperfunctions in a m aximal structure

54

£vv(B(z,t)) = (i//x)m '2 f e VJ '

with /

I,J

55

J

G C^(^4). We are using standard m ulti—index notation: I is a

p—tuple of integers ia : 1 < i^ < • • • < ip < m; dz^ = dz^ A - • • Adz^ ; like­ wise with J and dz . W e shall also deal with currents / G D* (A\hPy(*): this J

simply means th a t the coefficients f

I,J

are distributions in A. W hen all the

coefficients have compact support we write / G C^(^;AP,C^) or, in the case of distributions, /

G £ 7 (./4;AP,(*). The action of the Cauchy—Riemann

operator on a form or a current (1.3.5) is given by m (1.3.6)

3 /=

E

E

E

|l|=p | j | = q | = l

(3 /

/ 3z .)dz Adz Adz .

3

Of course 32 = 0 and D* (A\h?y(*) = 0 if either p > m or q > m. This gives rise to the various Dolbeault (or Cauchy—Riemann) complexes (for each p = 0,l,..,m ):

(1.3.7)

3: C V jAP’^) ^ C ^ A P ’^ 1), q = 0,1,...,m;

(1.3.8)

3: V ' (^;A p,q) -» V ' (.4;AP’q+1), q = 0,1,...,m.

The cohomology spaces of the differential complex (1.3.7) will be called HP,c*(.4). It does not m atter whether one uses forms with C00 or with dis­ tribution coefficients in the Dolbeault resolution of the sheaf m0

of

germs of holomorphic p—forms [ie., 3—closed (0,p)—forms (1.3.5)]. There­ fore the space H P ^ ^ ) is naturally isomorphic to the qf*1 cohomology space of the differential complex (1.3.8). We m ight also consider the sub—

56

Hyperfunctions in a maxim al structure

complexes of (1.3.7) and (1.3.8) defined by requiring th a t the forms or currents have com pact supports. Their cohomology spaces will be denoted by HP,(1(.4); HP,q(.4) is often interpreted as a 11homology11 space, of bide— gree (m—p,m —q). Consider m functions Asj G (^ ( A ^ A ) (j = l,...,m ) and define k(w,z) =

m E k:(w,z)(W:-Z:). j =1 J J J

In the open set H, — { (w,z) £ AXA', k(w,z) ^ 0 } we form the kernel k

N k(w,z) =

(1.3.9) (m- 1 M m ------ — X( — (2 ix * )m j = l

/N • - Adk i A - • • AdA;mAd(iu1- z 1)A- • •Ad(u;m- z m),

where d stands for the

"to tal11 differential,

ie., the differential in

(w,z)—space. The kernel N L is a smooth differential form of degree 2m—1 k in the region H.. According to Lemma VI.2.2 [TREVES, 1992], we also k have (always in

N k(w,z) =

(1.3.10)

/ _i \ | m /\ *— - X( —l J j ’^ifcdi^A* • -A diijA ' • • AdiimAd(u;1- z 1)A- • •Ad(it;m- z m), (2ixjfc)m j= l

where uj = k^/k.

LEMMA 1.3.2.— In the region

k

dAT, = 0. k

Integral representations of analytic functionals

We limit the variation of (iu,z) to d

=

Then, by (1.3.10),

m!(2ix)"md a 1A* • • A d A d ( u ^ - 2^)A - • • Ad(u;m- z m).

m But

57

m

E U:(w:—Z:) E 1 => j M J J J

m

E U:d(w:— Z:) = — E (w: —Z:)du:. M ultiplying j J J J J j «, J J J J

both sides of this equation by (-1 )

i l

t^di^A- • - Adi^A* • • Ad^mAd(tw1- z 1)A* • •Ad(ium- z m)

and summing with respect to i = l,...,m , shows th at d u fi* • • AdwmAd(i«1-2:1)A- • • Ad(u/m-2:m) E 0. □

We are going to apply Lemma 1.3.2 with a special choice of the functions ky As said the analysis will be carried out in the neighborhood of an arbitrary point of X taken to be the origin. We select the complex coordinates and the defining equations of X near 0 as described at the beginning of this section. In particular, (1.3.1), (1.3.2) and (1.3.3) will hold. Recall th a t 0 = Z(BQ) = /lT)B0 j. Now set p(w) = | v—$ ( t t ) |2 (w = u+iv) and, for each w E Bq1, consider the quadratic form in z—space, Q{w,z) =

S - ^ - 2 — (tw)2j i k. j,k=i

T aylor’s expansion yields

(1.3.11)

p(w) = p(z) + 2 $,