Differential forms orthogonal to holomorphic functions or forms, and their properties

Translations of

MATHEMATICAL MONOGRAPHS Volume 56

Differential Forms orthogonal to Holomorphic Functions or Forms, and Their Properties L. A. Ai zenberg

Sh. A. Dautov

,lll1414tEPEHl.lHAJII.HLIE 4lOPMI.I, OPTOrOHAJlI.HbIE rOJlOMOPIltHI.IM IItYHKUKJlM l1JIH 4lOPMAM, Ii HX cBOAcrBA

JI. A. AIDEHliEPr H m. A• .lJ,AYTOB KlllATEJJl>CTBO .HAYKA. outKPCKOE (JT,ItEJlEHHE HotocwiMf'ClC 1m

T.o!Ub1M hom tho: Ru.oial\ by R. R. SlmM Tmubtion .diled by l.ro J. Lolrman 2lXlO MalMtn.a!.ia S,J,j.cl ~ Prim.,y JZA2S, 32FIS: Secoadory JZA35. 3ZAt5. AunAcr. This boot is de-.d 10 tho deKriplioo 01 •• teriof dilf"rrntal rom» OfthoIonal to bolomorpllio rorms of C."lif>nl if , .. 0) ~1h rapm to iJltqf._ oo'lldod domain D i" C·. Tho Maninoll;' Bochno.. Koppda\atl f_lIla. _hiell is ... iJll 1, do not possess. Chapter I is devoted to tools that will be used and results related to the main problem. In particular, we present an elementary derivation of the MartinelliBochner-Koppelman integral representation of forms, generalizing the Martinelli-Bochner integral representation for holomorphic functions on the

one hand, and the Cauchy-Green formula for smooth functions in the case n = I on the other. Also we give a characterization of the trace of a holomorphic function on the boundary of a domain in C". In Chapter IV, some applications are indicated: we generalize the classical theorems of Hartogs and the Rieszes, describe the general form of integral representations of holomorphic functions, and construct the Martinelli-Bochner representation of a distribution lying on 6D'(R2n" ), by means of which we introduce a new definition for the product of distributions. In the text itself, we shall refer only to such original works as are necessary for understanding the book. All references are listed at the end of the book. After Chapters IV and VII the reader will also find a brief historical survey; in

INTRODUCTION

3

the former some unresolved questions (as of 1975) are indicated (a few of which are subsequently solved in Chapters V---VII, written in 1981 for the American edition).

We shall employ the following notation. C" is the space of n complex variables, whose points are denoted by z, , z°, ' °, etc. If z = (z1,. .. , zn) then +zj, and I = ,, ... ,1" j. For z, E C" we define e). Lemma 1.7 shows that the second integral on the right in (1.8) tends to zero as e -+ 0, so that (1.5) and (1.18) yield I

f Y A X0,9-I = Dr

j

aDr

A gVq-

-

j

D,

A

gVq_,.

(i.19)

In both the integrals on the right side of (1.19), we may differentiate under

the integral sign; the first is a proper integral, while the second has a singularity of the form I z

after differentiating the integrand and

16

I. INTEGRAL REPRESENTATION OF FORMS

hence is absolutely convergent. Hence, the left side of (1.19) is likewise differentiable, and we get

A kg A V9_I -

f

aD

D

ay(e) A azg A Vq-1-a

D

y A Uo,9_, = 0. (1.20)

Adding (1.17) and (1.20), and using the definition of Uo, 9 and (1.16), we obtain

i(z) - fany

A Uo,9

-,l aY A U0.a - af ntY AUo.q- I v,

We shall show that i(z) = y(z ). From (1.17) and Stokes' theorem applied to DE we get

i(z)=lim F-'O

Let us compute this limit. To do this, we go over from Oq to the form

Oq= (-1)QargA Vq- azgA Vq_1. From the invariance of the first differential, it follows that dig IK = 0, where K = ( E C": I at.g IK = arg IK; hence (1.5) and the definition of 94 show that e9 1K = 9q IK-

- z 1or

We set

sk=

a =-ag

=

k

kz

?n

Then A Oq

X

ly (n 1) 1. A0 (2ri )n

(_1)9df, ^ ... ndf9 n 74 Skdfk n k=1

I jq_l

(I,j)(-l)'d[I,j] n

n4Il Jdzi - df, n ... AdfQ n 2 (-g)dik 0

A

j=1

Q(J, k)(_l)k-i X df [J k] A dt[kjdzr

J kV

§1. THE MARTINELLI-BOCHNER-KOPPELMAN FORMULA

(-1)"(n

n

2 8kdf. A ...

(21ri)n

17

n df [1,...,q, k]

k=9+I 9

A[

2 G((19,

d zQ

a

aI

[j],t*.*.)q-3k)-)

A dF, A

n da, A .. ndaq

n d z k + a((1 ,...,q), k)(-I) 9

n

A ...

n

da

n ... [k] ... ndzQ

i=1

Observe also that

k}dfj A ... A

A

A

(df, I1 dfz A ... Adfq) I1 df [1,...,C k

--- df, a((I,..., [klv... q), k)df, A ... Ad-4 11d ' [1,.. ,,q]dzk A di1 A ... [kJAd1q q}) k)d

k

A

A ... AdYq

Using these equalities, we get n

n

t..

(2iri)7'

Y n 84

A

k=}

A ... 11dzq + X.

Here the form X is a sum of terms of the type

const

A

j] A day,

j k, I= (.,..

, 'q).

I. INTEGRAL REPRESENTATION OF FORMS

18

Thus finally we get by Lemma 1.7

i(z),

n 69 = Jim f

lim f

((2n1Tk

t0

n di, n

(_l)n+k_If

A 99

ZIE

ndzQ + lim f

!l i X

F-+O

(n - 1)! (2771)n

(2iiYf(Z)dI1 n ...

k_1

y(z).

n

§2. Theorems on the saltus of forms

V. In the theory of functions of one complex variable, the following results are well known (see, for example, Muskelishvili [11, §22). PROPOSITION 2.1. Let D be a domain in C' with smooth boundary, and let

qE

Then the functions I

(Z)

2ri

d z

(1rp+ is defined in D, and T- in CD) can be extended continuously to the closure of the open set on which they are defined, and

T' G) - T- G) = (W),

E aD.

(2.1)

To formulate the second proposition, we need some notation. Let .a be a domain with smooth boundary, and let ip E C(aD) and z0 E aD. On a line _V passing through zo and not tangential to aD, choose points z E D and z' (4 D at equal distances from zo, and consider the limit ilm Z,Z -)Zo

((p

[q(z)-q(z')]

(2.2)

are defined as in Proposition 2.1).

PROPOSITION 2.2. The limit (2.2) exists and equals 4p(zo ). The limit is approached uniformly if the nonobtuse angle between the line V and the tangent to 8D at z0 is not less than afixed $o>0. It must be noted that Proposition 2.1 solves the so-called additive Riemann problem:

Let D be a domain with smooth boundary, and let yv E Co.x(aD). Find functions c + E A C(D) and ( p -

A C(CD) such that qv + - qr = qv on D.

§2. THEOREMS ON THE SALTUS OF FORMS

19

We shall need analogues of the Riemann problem and Propositions 2.1 and 2.2 for forms of several complex variables. 2°. Before proceeding to consider the analogues mentioned above, we state the following result without proof. THEOREM 2.3. Suppose aD E Cm+',A and f E 0 < A < 1. Then the functions defined by the integral

f

where m >.-t 0 and

f(')d[k]Ad pt

I

(2.3a)

y-ZI2n-2

inside and outside D can be extended to the closure of the respective open sets as Cm+'u", and the functions defined by the integral functions of class

dAd as functions of class

(2.3b)

-ZI2n-Z

fDr

0 < A' < A.

The proof of this theorem can be carried out along the lines of Gyunter's proof ([1], Chapter II, §19) for the case of real dimension three. But the computations are much more complicated in our case. Since the coefficients of the kernels Up, q are derivatives of the function Z12-2n , and we may differentiate under the integral sign in (2.3a) and CI (2.3b), we obtain

-

and Y E Cl (D), then the forms defined by the integral V(, q)(D, Y) inside and outside D extend to the closure of the COROLLARY 2.4. If aD E

C'"+',a

p'

respective open sets as forms of class C( 'P,)"), and the forms defined by J2 extend as forms of class C(', Q1, Xj, with 0 < A' < A.

1)(D, Y)

3°. We observed that (1.1) is the analogue of the Cauchy integral formula for

forms of several complex variables. Therefore it is natural to consider an analogue of the integral of Cauchy type an integral of the form I p, q(D, y)(z). Here D is a domain with piecewise smooth boundary, and y E C(p, q)(aD ). Let Y+ denote the form defined in D by the above integral and Y- the one defined in CD. THEOREM 2.5. If aD E C2 and Y E fl,q)(aD), 0 < q < n - 1, then

Y=Y+1aD-Y 6 (by Corollary 2.4, Y+ 18Dandy-laDare defined).

I . INTEGRAL REPRESENTATION OF FORMS

20

PROOF. It suffices to prove that

f Y AT- f (7IaD - Y Ian

AT

an

an

for any p E q_ 1(8D ). We extend p to C" as an (n - q - 1)-form of class C' . Then T can be represented in the form

n-q-l

(P _i=0 I The forms 1,

-y±

1)(Cn).

Pi E

and -y are of type (n, q); therefore -y A pi = y ± A (pi = 0 for 1. Hence we need prove (2.4) only for T E C(o, n _ q _ 1)(C" ).

... , n - q -

Let p be a function defining D; thus, D = (z E C": p(z) < 0), p E C2(C"), grad p 0 0 on aD. Set D, = (z E C": p(z) + 1-' < 0) and D1= D i. Then for sufficiently large 1 we have

D, CD ED' CC",

aD' E C2,

0D1 E C2,

and also, for any X E C2,,_ , (C" ),

X = lim

'aD

1-i00

X= faD'

aD,

X

Thus

f

D

Y -Y J A 9 = li

n(p

_

fa D,Y+

f

DAY-n(P

-,llm

OO

Z)

Z) A f

-

f

y() ^I

AUn,qG1 Z)

(Z) A (Uq(, Z))

L

- f(3Dl)z T(z) A

Z))] .

The reversal of the order of integration is permissible since the integrands are

jointly continuous. By Lemma 1.5, Un, qG, z) = - Uo, n _ q _ I (Z' h' ). Hence by

§2. THEOREMS ON THE SALTUS OF FORMS

21

.1) and Stokes' theorem we get

f (y-yA qo = lim = urn f aD)r

(z)AU0(z,fl

(aD),y(Y) 11

l-' Qp

D

+j

A

(D'\Dr)t

1-1-00

a(p(z) n Uo.n-4-j(z,

+a

(z) A UO.n-q-2(Zt f(D'\Dl)z

= J y n qg± lim J

Jy()Af

+

01

(z)AUo,nq1(z,) (p(z) A UO,n-q-2(Z

A f(D'\Dj)z

From the form of the kernels Ua,,, we see that the last limit above is zero as a consequence of lim

f df [j] A

iff Adz= 0,

(2.5)

z) E C(aD X Df\Df). We show that the integral l

z 12n- I

aDx(D \D,) i

dsdv

(2.6)

exists. Here dv is the 2n-dimensional volume element, and ds the area element on aD. Since the integrand is positive, (2.6) exists if and only if the following limit exists: l

lim

e-.o FaDx(D'\D,)l\(t '-zj<e)

--- z [Zn-i

duds.

Now (2.7) is a proper integral; hence we may go over to the iterated integal, and the existence of the limit (2.7) is equivalent to that of lim

ds

where (D 1D,)\(It-Z <E)

--'O aD

l dv.. z 12n- I

I

But lime-.o g(C) exists, and

0 < &,-Q)
0, and choose a (2n - 1)-ball B in a such that 1) B C U;

-zl and Uand 3)1w -Z-J< a suitable constant C. Condition 2) is satisfied because y is continuous, and condition 3) because of the relations H 1 w - aiiI . Thus from (2.12) we get

1Iz'zn k

k

d depends only on a, b, C, C, and ft.

IZI

(2.13)

24

I. INTEGRAL REPRESENTATION OF FORMS

Similarly,

'k

-

+

z

zk

zf

IZI2n

where d l depends only on a, b, C, Finally,

'Q Z

I

k

di z


0, the assertions analogous to those of Propositions

2.1 and 2.2 and Theorems 2.5 and 2.6 are false, as the following example shows.

EXAMPLE 2.9. Let D be the same domain as in Example 2.8, a -== dz" _ q+ 1

A ... A dz,_ j A dz l n ... A dzF and y = dz A a. Then y E CtP*, q)(a D ). For

26

I. INTEGRAL REPRESENTATION OF FORMS

z E B,

Y+Ian - 7_6 = Yi Ian - 71

laD'

where

Yi = Since

A Up,q(J z) 1

A Up, q is of type (n, n - 2) in , we have Y(')AUp,q(J,Z)I8=0,.

i.e. y, = 0; hence for z E B we have y+ IaD - 7_6 = 0, but y {aD 0. For such forms, the jump theorem can be obtained in the following way: Let D be a domain with a piecewise smooth boundary, and y E pq)(aD). Let f denote any form in C( p, q)(D) such that y IaD = y, and let

Y±(z±) = Ip,q(D, Y)(z±) - 5J2 z+ E D, z'E CD.

Y)(z±),

THEOREM 2.10. Y.. extend continuously to 8D, and 'Y = Y+IaD - *Y-IaD*

(2.18)

PROOF. By (1.1), we have for z E D

+(z) = Y(z) + ID.v(D.

(2.19)

and for z ii D

y- (z) = I,,,(D,

) (z).

The integral that occurs in the definition of Ip,q is absolutely convergent for all z E C"; hence I p, q(D, al) E C(p, q)(C"). Hence the forms y ± can be extended continuously to aD preserving (2.19) and (2.10). Substracting (2.20) from (2.19), we get (2.18).

6°. We now proceed to the additive problem of Riemann for forms. We formulate it as follows: Can every y E C(p,q)(lD) be represented in the form (2.21) Y = Yilan - 7261, with y E Z(p, q)(D) and y2 E Z(p, q)(CD ), and it not, what are the conditions on y of the validity of such a decomposition?

THEOREM 2.11. I. If aD E C' and y E C(p,q)(lD), then the decomposition (2.21) holds only if there exists j E C( p,q)(C")such that Y IaD = y and al = 0 on aD.

27

§2. THEOREMS ON THE SALTUS OF FORMS

m > 1, then a sufficient condition for II. If aD E Cm 1,A and y E (2.21) to hold is the existence of a Y, E Cp, 9(C") such that 11 IaD =-y and aYl A dz, IaD = 0 for any J= (jp+ ... ,j"). The forms yl and y2 can then be chosen inCp;qj,0COG) -p(z))+C2I

for EV\D,zED, 1'--z

Z

12-

<e.

In the rest of this section, we shall denote by u the vector-valued function

P

PI

Pn

We recall that a domain D C C" with boundary of class C2 is said to be strictly pseudoconvex if D = (z E C": p(z) < 0), where p E C 2(G" ), grad p 0 on aD, and the Levi form "

a 3p

2 r

j=

is positive for all z E aD and all w Op z azI

1

az'azJ

( z ) Wj 1wj

0 such that ap

W+.,.+

(6) We state the result only in the form we need.

az'

z . w -- = o.

I. INTEGRAL REPRESENTATION OF FORMS

32

3°. Let us consider integrals of the form

f Y(z) A D1,1,q,r,n-q-r-2(t, u, 5zt,

apt) A Dp,n-p(az, aJ ),

D.

A D1,l,q,r,n-q-r-2(t, u, azt, ku, apt) A Dp,n-p(az, aJ ),

D

where t = (f - i)/1 ' - z

12

as usual, and y is a form of the appropriate

dimension. Our interest is in the following question: is it possible, by imposing suitable conditions on y, to extend the forms defined by these integrals to forms on V, differentiable as many times as we need? Let us first explicitly write down the singularity of the integrals: LEMMA 4.1.

DI ,I,q.r,n_q_r_2(u, t, alt, ruq att) _

z)

" 21

I ( - Z) 12(n-r-

1

where the µqr are forms of type (0, q) in z and (0, n - q - 2) in

with

coefficients in Cm-2((V \D) X V). The proof proceeds exactly like that of (1.4). We can now answer the question asked above. LEMMA 4.2. Let m > 3 and y E Cp,Q

l.If y-0on V\D, then

f Y(z) A Di,l,n_q,r-2(t,

u,

5't, ku, apt)

D.

ADn_p,p(az, a') E

2)(V\D).

2. If y = 0 on D, then

f(v\D)rY(') A Dl,l,q-r,r,n-g_2(t, u, at, n Dp n-p(az,

art) E C(p,g2 2)(D )

PROOF. We shall prove the first part; the proof of the second part is entirely similar. It is sufficient to prove the lemma for m < cc.

From Lemma 4.1, we see that our assertion is a consequence of the following: Cm_3'A(V),h

Ifh E

0 on V\D,andcp E Cm-2((V\D) X V), then

Izl

z)(fk

DZ

r+1

zk)

m-2 (V 2(n-r-1) dz n dz E C

\D).

(4.2)

33

§4. CASES OF SOLVABILITY OF THE 3-PROBLEM

For ' E V\ D, (4.2) is a proper integral, so we may differentiate under the integral sign m -- 2 times. Hence I(') E C"r2(V \ D ). The derivatives of the integrand are linear combinations of expressions of the form f I, I' h(Z)D

Zk

fk

r+1

where III + I' + J + Jill < m -- 2 and X

QX9 9 z,

z) is continuous on (V \

D) X V.

And we have

Df,I'

M1

1

+1 1

I4D(p' z)

Z))r+l

I

DJ,Jtk2k - 12(n--r--1) Z

M1 - Z 12(n-r--l)+IIJ±J'V--l I

z) E (V \ D) X D.

where M1 does not depend on

We have h = 0 on aD = (z: p(z) = 0); hence by Hadamard's lemma Cm-4,? (V)

(Arnol'd [I ], §2,6.4) there exists h E such that h = ph 1; in turn, we 0 on aD. Repeating have h 1 = h p-1 = 0 on V \ D, and so, by continuity, h 1

1

.

this procedure m --- 3 times, we get h = p'n- 3h 2, where h 2 E C °" (V) and h2=- 0 on V \D. For A' such that 0 < A' < the function h h z lies in C((V \ D) X U); hence M21p(Z)jn-3+X'

I'M C I (W

z) 1111+I'11+r+1 i +r11

P(Z)

P(z)

oq Z)

I

J - ZI

. - Z 12(n--r-1)+1}J+J`11--i M3

Ili + Jill - I 1 4PG9 Z) 1'r+ I

I

- z 12n-V-2(r+ 1) "

where MZ and M3 are suitable constants.

Now, from property 4) of z) and the fact that p(z) < 0 for z e D and 0 for E V\D, we have

z)l,

lp(z)l
2. PROOF. The necessity of (4.3) is a simple consequence of Stokes' formula. Indeed, for any IL E ID

,A-y= d(jLna)-aD

D

ttAa=O.

Let us prove I and the remaining part of II. Let V be the domain of holomorphy introduced in 2°. Since supp y C D and ay = 0, for z E V we have by (1.1)

y(z)

`

a12

12

(v,

Y).

(4.4)

y) E AP(CD ), since supp y C D. By Hartogs' theoIf = 1, then I2 'p-qrem, Ip, q_ 1(D, y) can be extended to all of Cn as a form a, E AP(C" ). It now follows from (4.4) and Corollary 2.4 that a = fp,q- t (D, y) --- a t is a form with the desired properties. '(D,

We now consider the case q > I. Then, by Lemmas 1.3 and 1.4, for z E V \ D we have

Irv-1 (D

f 7(Z) A :µy- A

P' C

-fa f y(z) A µq_2 A

n- P)

Dp'n-p(az,

1

pn - p) , Dp.,,_p(az,

(7) If q = n, it will be supposed that y satisfies (4.3).

1

(4.5)

§4. CASES OF SOLVABILITY OF THE a-PROBLEM

35

for q < n, and 12

f 4y(z) A W,,O(u, , z)

, - J(D

Pn

+ 5f -y(z) Auo A -

1

P

n1

v (al,

(4.6)

(for convenience, we have deviated from our usual notation and interchanged the places of and z).

Using Stokes' formula and the equality y laD= 0, we see that the first integral in (4.5) vanishes. Since z), and consequently WP,o(u, ', z), are holomorphic in z on D, the first integral in (4.6) also vanishes by virtue of (4.3). By Lemmas 1.3 and 1.4 we see that the second integrals in (4.5) and (4.6) are linear combinations of the integrals considered in Lemma 4.2; hence 2

on V\D

p,q- I(D,

and $ E Cm+1 (P , q _ 2)(V ). Corollary 2.4 and (4.4) now show that a = Ip,q_ 1(D, ' the desired properties. It remains to prove III. By (1.1), we can represent a as a

IP,q_1(V, a)

(4.7)

is a form with

IP,q_1(V, Y) - alp,q-2(V, a).

As in the proof of Theorem 2.10, we can show that

IP,q-1(V, a) E Z(p,q-1)(V)Since V is a domain of holomorphy, I1 P,q- (V' 1

with f E 1

C(Pp, q _

2)(V ). The proof is completed by applying (4.7) and Corollary

2.4.

THEOREM 4.4. Let D be a bounded strictly pseudoconvex domain with C'" + 2

boundary and y E Z(., q)(5 ), m > 1. Then there exists a E CCp, q-1)(D) such

that aa=y. PROOF. Consider the domain of holomorphy V introduced in 2°, and a

y E p,q)(C) such that

y on D and supp Y C V. By (1.1),

-Ip,q(V,

alp,q_ 1(V, y ).

For q = n we have J2 p,n(V, a y) = 0; hence the theorem follows in this case from Corollary 2.4.

36

I. INTEGRAL REPRESENTATION OF FORMS

Now let q < n - 1. Then, since ay = 0 on D, Lemma 1.4 yields for z E D

(z)

= a-

a

A

A

(v\D)t

-f

A

atµ2

v

p! (n

1

p)!

1

n P (n - P)

p,pI OZ

D

a- I2

p,q- 1( v

.1

)

Dp,n-p(az,

Since ay = 0 on a(V\ D ), the second integral above vanishes by Stokes' theorem. The first integral is a linear combination of integrals considered in Lemma 4.2; hence the proof is by appealing to Lemma 4.2 and Corollary 2.4. COROLLARY 4.5. If D is a strictly pseudoconvex domain in C", n > 2, with boundary of class Cm+ 1, m > 2, and g E and if ag can be extended to CD as a form of class C(o, i 1 '(CD ), then g can be extended to C" as a function in Cm(C" ).

Also, for every extension of ag to a form a E Z , (C" ), there exists an extension f e Cm(C") of g such that a f = a.

PROOF. Suppose ag is extended to a, E C(o,1)1.A(C"). Then aa1 E C(o,2) 2, `(C" ), and supp a a, C D. Further, if n = 2, for any µ E A 2(D) we have

fAaa1=f&Aaa1=f pAa1 = D

D,

3D,

µnag=0. aD,

Here D 1 is a domain with smooth boundary such that D C D, and µ E A 2(D 1). Now Theorem 4.3, III shows that there exists h E Cm(C") such that ah = a,

on CD. The function p = h - g is holomorphic on CD; hence by Hartogs' theorem it can be extended to an entire function. Therefore g = h - p can be extended to C" as a function of class Cm(C").

To prove the second part of Corollary 4.5, we note that, by Theorem 4.4, a = 8f, with f, E C "'(C" ). Now g - f 1 is holomorphic in CD; let f2 be the entire function extending it to U. Then f = f 1 + f2 is the desired extension. This corollary is a generalization of Hartogs' theorem to smooth functions.

For domains which are not domains of holomorphy, an assertion similar to Corollary 2.5 is false. An example of a function g not extendible continuously to CD, for which ag admits a C00 extension to CD, will be constructed in §8 (see (8.1)).

CHAPTER II

FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

§5. Polynomials orthogonal to holomorphic functions

1 °. Let D be a bounded domain in C", with smooth boundary, and let 0 E D. Let wk(z ), k = 1,.. . , n, be C' functions on aD such that, for all

zEaD, (W(Z), Z)

(5.1)

We set(') n

w(w, z) =

:: (-l)"wdw[k] A dz,

and [ k ] signifies that dwk is to be omitted. If w (w, z) is nondegenerate on aD, then every a E 1(aD) can be written as a w qv(z )w,

where w

with 'p E C(aD ). Instead of discussing forms orthogonal (with respect to integration over aD) to holomorphic functions, we shall discuss functions 'p that are orthogonal to holomorphic functions in the sense that

faDU(z)w(w,

z) = 0

for all f E A(U). We shall denote the subspace of such functions 'p E C(aD ) byO(aD). In certain situations, it is useful to have a description of the polynomials P(z, w) which lie in O(aD). Observe that, for n = 1, P(z, w) and w(w, z) are respectively P(z, l 1z) and dz/z. (1)Note that w(w, z) = w(w, z, 0) of

§1.2°.

37

fT1

H. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

38

THEOREM 5.1. P(z, W) E O(8D) if and only if

z}+ .-. +,t_t

TI-- ... fin-- i

Tn- I

T1

P

zn

Z n-1

dT1 A ...11dTn- I

1

z"

zI

I

where the Qj are polynomials such that Q1 (o, z 1, ... ,1 /zn) - = 0, j = I , ... , n.

We divide the proof of this theorem into several lemmas. LEMMA 5.2.1f f E A(D) and P(z, w) has the form m

P(z, w) = Y aQ

(5.4)

r#* Z AW#A,

k=1

where ak = (at ,... ak) and f3k

then (z)Zak

Qax Sk

PIZ, wIlIZ>W

DPk,O

f(I/3k1I+n

This lemma follows easily from the Cauchy-Fantappi8 formula (see (13.2)), written (for sufficiently close to 0) in the form

A0 _ (n - 1)! f8D (]Az)w(w,w)z))n (217i)"

LEMma 5.3. In order that a polynomial P(z, w) of the form (5.4) lie in 0(8D), it is necessary and sufficient that, for all n-tuples y = of nonnegative integers, Z ap`+y

Oak .8k D#(11 =1

k 11

4-

r..`.' o.

)

(5.6)

z= O

PROOF. It follows from (5.5) that (5.6) signifies orthogonality of P(z, w) to

all monomials z''; hence the necessity is obvious. The right side of (5.5) depends neither on the form of D, nor on the concrete choice of the w; (z ), i = 1, ... , n . Thus, whether P(z, w) lies in o(aD) or not depends only on the

properties of P(z, w) itself. In particular, P(z, w) E o(aD) if and only if P(z, w) E O(OB(0, r)), where B(0, r) = (z: I Z I 12 + . + I Z. 12 r 1). We have seen in Examples 1 and 2 of 3 G how the w, (z) can be conveniently chosen for a

ball. Each f E A (B (o, r)) can be uniformly approximated on aB(o, r) by polynomials in z. Hence the orthogonality of P(z, w) to all monomials z *t already implies that P(z, w) E O(3B(o, r)).

.

§5. POLYNOMIALS ORTHOGONAL TO HOLOMORPHIC FUNCTIONS

39

Finally, from (5.6) we deduce LEMMA 5.4. A polynomial P(z, w) of the form (5.4) lies in O(aD) if and only if, for all n-tuples y of nonnegative integers,

pil...gk!

k_ k_

a`k

(liii+ n-1! = 0.

(5.7)

The proof of Theorem 5.1 now reduces to comparing (5.7) with the equality

P z,

TI

, ... , Z1

Ti.f ... "Tn-1s 1

T"-1

] - Tj- ... -Tn-1

Z"-I

z"

dT1 n ... n dz" _ ,

#kj ... #nk

`

aak.ok

(1113k11 ' It

z

COROLLARY 5.5. Every polynomial P(z, w) can be represented as the sum of a polynomial Q(z, w) E O(aD) and a polynomial R (w) depending only on w.

PROOF. It suffices to consider a polynomial P(z, w) of the form (5.4) for

which 8k - ak =-y is a constant vector. If - at least one coordinate of y is negative, then P E O(aD ). If all the coordinates of y are nonnegative and (5.7) is satisfied, then again P E O(aD ). It remains to consider the case when all the components of y are nonnegative and (5.7) does not hold. In this case, we can add a monomial ao,.,w? to P, choosing ao,., so that (5.7) is satisfied for the polynomial P + a0 7w'', i.e. that this "augmented" polynomial lies in O(aD). Note the inclusions A o(aD) C O(aD) C C(aD ), where A o(aD) is the space of traces on aD of functions f(z) holomorphic in D and continuous on D with f (O) = 0. From Theorem 5.1 it f olows that O(aD) is not a ring for n > 1. We stress that the characterization of the polynomials in O(aD) given by Theorem 5.1 and the representation in Corollary 5.5 do not depend on the concrete form of D.

2°. For large classes of domains, the functions w1(z ), ... , w"(z) can be chosen so that

polynomials P(z, w) are dense in C(aD). Then the forms P(z, w)w are dense in

(5.8)

The following theorem may

be considered as a new step in the characterization of forms orthogonal to holomorphic functions. THEOREM 5.6. If w; E CZ(aD), i = 1,... n, then a polynomial P(z, w) belongs to O(8D) if and only if the form Pw can be extended into D as a a-exact form of

type (n, n - 1).

II. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

40

For the proof, we need LEMMA 5.7. P(z, w) E O(aD) if and only if P(z, w) can be represented as a linear combination of polynomials of the following two kinds.

with ak > fJk for some k,

zawfl,

z a w a+Y(

(5.9)

Ca+Y,l - zjwr ) ,

(5.10)

here a, /3 and y are nonnegative integral vectors, and C.,I = (/B, + 1)(11 /3 I I + n)-'

.

PROOF. It follows from Lemma 5.4 that polynomials of the form (5.9) or (5.10) lie in 0(WD). To prove the necessity, it is enough in view of the same lemma, to show that a polynomial m

wY 2 aak Zakwak, k=1

(5.11)

lying in 0(aD) is a linear combination of polynomials of the form (5.10).

Without loss of generality, we may suppose that a1 = 0 and a k k = 2,.. ., m. Then (5.11) can be written as m

2

k=2

0,

zakwak+Y

aak

(z,w,k z,wfk

m

+ 2 aakCak +YJk k=2

Cak+Y.Ik

Z akwak+Y

(5.12)

+ a0wY.

z,wfk

The first term in (5.12) is a linear combination of the polynomials (5.10); the remaining polynomial has lower degree in z than the initial one and also lies in O(aD). Repeating this argument, we will have (after finitely many steps) a representation of (5.11) as a sum of a linear combination of the polynomials (5.10) and a polynomial bwY; by Lemma 5.4, bwY E 0(aD) only if b = 0. PROOF OF THEOREM 5.6. Sufficiency follows immediately from Stokes' theorem. We shall prove the necessity for polynomials (5.9) and (5.10) (see Lemma 5.7).

Suppose the polynomial has the form (5.9). We may suppose that k = 1. Then Zaw.8 = (Z1W1)'Z2 ... Znnw2 2 ... wI4nZ1 t-ftl

Z -flt(1 - z2w2 - ...

-Znwn)plZ22 ... Znnw2 2 ... w!

and so it is enough to consider the polynomial P = z" ,

a1 >0.

. zan wf2 .

.

w,n with

§5. POLYNOMIALS ORTHOGONAL TO HOLOMORPHIC FUNCTIONS

41

By Lemma 1.6, for zk :#- 0 we have 1)! (_1)k_1dw[k] Adz. (2iri)n

w(w, z) _ (n

(5.13)

Zk

0

Therefore, for z 1

Pw

_ (n

n]

1)! Za,-1Z a2 ... ZanW2" 2 ... wRndw 1 A dz

(2ffz)

n

(n (217i

n

2

1

1)!

)n(,82

+ 1)

zi -1222 ... zn^ W2# 2 + ... w!n dw[1, 2] A dz

This equality can be extended to all of aD by continuity. It remains to consider the case when P E O(aD) has the form (5.10). In this case n

n

k=1

k=1

P(z, w) _.ZaWa+Y Ca+Y,! I ZkWk - Z1W` = ZaWa+Y

For z,

akZkwk.

zn =-71= 0 we get from (5.13)

,

Pw = ((27r[))' zuwa+Y I (_1)m_lamwmdwlm] A dz.

(5.14)

0 may be dropped since both sides of (5.14) are z,, The condition z, continuous on aD. If m > 1, then

(WiWmZ"WdW[l, m] A dz)

_ (a, + yj +

1)zawa+Ywmdh[m] A dz

+ (-l)m(am + ym + 1)zawa+Yw,dw[1] A dz, and so (5.14) implies

Pw=awlzawa+Ydw[l] A dz + aµ,

where a

is

a

constant, and p a linear combination of the forms

(2ri)-

W 1 wm z a w a + Ydw [ 1, ml A dz. Further, in view of (5.13),

Pw=

_

-

1az1wlzaWa+ow+aµ - P1w+all.

(n - 1).

Hence P1(z, w) E O(aD), and it follows from (5.7) that a = 0. 30. We now consider examples of classes of domains for which there exists a vector-valued function w(z ), z E aD, such that (5.1) and (5.8) hold and the form w(w, z) is nondegenerate.

42

11. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

EXAMPLE

1.

Let D = (z: p (z) < 0) be a strictly pseudoconvex bounded

domain with C2 boundary and containing zero; suppose that D is also linearly convex (Aizenberg [11), which in this case means that, for every z E aD, the z l )pz, + ... + Gn -- zn )pz" = 0) does not inanalytic tangent plane tersect D. We set p`

...

ipZ'}

(5.15)

np=

For this choice of w,, (5.1) holds. Let us consider the form w formed from the wi as in 1 °.

LEMMA 5.8. w(w, z) is nondegenerate on D.

PROOF. We represent aD as the union of disjoint sets F1,.. . , Fn such that 0 on F,,,. Then we see by an easy computation that, on rm,

pi'.

---

(_l)m_Ie(p)di[mL p m(zIpzI +

A dz ,

+znpzn Yn

where C (p) is the generalization of the Levi determinant (see Fuks [1], Chapter 11, §12.3, or Vladimirov [11, Chapter 111, §18.5) to the case of n variables (see Rizza [ fl):

p

0

. f ' pPZn

r

e (P) = -

Pi',

pz,,i,

PZJfZJ

pzn

pzIZX

It remains to show that B(p) 0 on aD for strictly pseudoconvex domains D. But this can be easily checked by making a nonsingular linear change of variables such that the analytic tangent plane to D at a given point of aD is parallel to a coordinate hyperplane, and using the following property of B(p): if z = z( ) is a biholomorphic map, and p,(r) = p(z(r )), then

e(p1) = e(p)

,,..., n

2

(5.16)

The proof of (5.16) in the general case is the same as in the case n -= 2 (see Fuks [ 11, Chapter 11, § 12.3).

Let h be the locus of (z, w) in C2' as z ranges over aD. As in the proof of Lemma 5.8 (by means of the same linear change of variables), it is easy to prove LEMMA 5.9. h is a smooth manifold without complex tangent vectors.

§5. POLYNOMIALS ORTHOGONAL TO HOLOMORPHIC FUNCTIONS

43

Let D = {w: (w, z } I for all z E D) be the compact set dual to D, and let D be the dual of D. D is an open set (see Aizenberg [I and [2]). Let us also require that D be connected.() Then D is linearly convex in the sense of Martineau [1). LEMMA 5.10.

h = (iYxL5) n {(z,w): (z,w)= 1),

(5.17)

where h is a polynomially convex compact set.

PROOF. D may be interpreted as the set of complex direction vectors w of

analytic hyperplanes of the form a,,, = (z: (w, z ) = 1) that do not meet D. Hence the right side of (5.17) is the set of pairs (z, w) in C 2" such that z E ZID

and a,,, passes through z but does not meet D. From the uniqueness of the analytic tangent plane, it follows that w has the form (5.15). Thus (5.17) holds.

D and D are polynomially convex, since D and D are connected (see, for example, Makarova, Kudaiberganov and Cherkashin [I]); hence (5.17) implies the polynomial convexity of h. In view of Lemma 5.9, continuous functions on h are uniformly approximable by holomorphic functions (Harvey and Wells These in turn can be approximated by polynomials P(z, w), because of Lemma 5.10. Thus condition [1]).

(5.8) is satisfied. EXAMPLE 2. Now, in contrast to (5.15), we set

wj = ;/! z 12,

j -- l,...,n.

(5.18)

For this choice of the wJ, the vector-valued function w(z) does not depend on D. and always satisfies (5.1). Let us consider conditions under which (5.8) holds. Let D be a bounded domain with 0 E D. Let B denote the uniformly closed algebra of functions on 8D generated by z 1, ... , z1z and w,,. . . , w", and let

B. be its restriction to the (complex) one-dimensional analytic plane a, 0 E a.

Clearly, B separates points on aD and contains 1. Let a = {z: z = at (a t) ... ,ant ), t E C' }, a E C ". Then w1 - a j I a I2t, so that B. is generated 1

by t and

r'.

LEMMA 5.11. For (5.8) to hold, it is necessary and sufficient that, for every a, the set OD n a have no interior points and that its complement in a consist of two connected components.

For the proof, we need the following results of Cirka [1). Let x be a compact set in C", a subalgebra of C(X) and F

{fa}a E r a family

of real-valued functions in QX). Then a continuous function f on x can be (2) It is not clear whether the connectedness of ' is a consequence of the hypotheses of Example 1.

44

H. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

approximated uniformly on X by polynomials in elements of A and F if and only if f can be approximated on every Xa = (x E X : fa(X) = a a) , a,,,, E R, by elements of A. PROOF OF LEMMA 5.11. Consider the functions f k = Zj Zk I z 2k = 1, ... , n,

f k= lkjl fjk E B. Then f k+ fkj = 2 Re f k E B. These functions do not separate points lying in the same plane. By taking the restriction of f k to two planes a1 and a2 and considering the functions f k f j;', it is easy to see that Re fk and Im f k separate points of different planes. Thus the set (z E aD: Re fjk = bjk, Im fk = cjk) is either empty or coincides with aD n a for some a. Sufficiency. If aD n a satisfies the conditions of the lemma, then, by a theorem of Mergelj an [11, polynomials in t and t` are dense in C(aD n a). Hence tirka's result implies that B = C(aD ). Necessity. If B = C(a D ), then

Ba=C(aDna).

(5.19)

In the interior of aD n a, each function in Ba, being a uniform limit of holomorphic functions, is holomorphic; hence (5.19) can hold only if aD n a

has no interior points. If the complement of aD n a had a component w containing neither 0 nor oo, then for to E w we would have (t - to )-' E C(aD n a); but (t - to )-' E Ba by the maximum modulus principle. It remains to consider the question of the nondegeneracy of the form w for the choice (5.18) of w(z ). LEMMA 5.12. Let D be a bounded domain with smooth boundary. Then w(w, z) is nondegenerate on aD if and only if, for all z E aD

zl pii + ... +zn pi?

0,

(5.20)

i.e. no analytic tangent plane passes through 0 E D.

The proof reduces to simple computation. For instance, if at z E aD we have pin 0, then

w(w, z) =

"

IZ12n k=1

(-1)n

Izi 2npi,

k-

ik di [k] A dz

(z1pz, + ... +Znppn)dZ[n] A dz.

Thus the nondegeneracy of w is equivalent to (5.20).

Thus all the required conditions of 1 ° and 2° are satisfied by the function w(z) of (5.18), provided D satisfies the conditions of Lemmas 5.11 and 5.12. EXAMPLE 3. If the class of domains D is the same as in Examples 1 or 2, but p E C"+2, then, by Theorem 5.4 of Hbrmander and Wermer [1], (5.8) is valid

§6. THE CASE OF STRICTLY PSEUDOCONVEX DOMAINS

45

for polynomials in z and v, where v = (v 1 , ... , v"), each vi being a C" +' function sufficiently close to wi in the C2 topology, i = I,.. . , n. Thus it remains only to choose the vi such that the analogue of (5.1) holds and w(v, z) is nondegenerate on 3D. EXAMPLE 4. Let D be a domain of the class of Example 1, and let w; E C"+' as in Example 3. We extend the w; to some open neighborhood V of aD so that

(5.1) is preserved. Consider homeomorphisms pt C C"+' of aD onto the boundary 3D, C V of a domain D, depending continuously in the C2 topology on the parameter t, 0 < t < 1, such that cp, in the C2 topology is the identity map of 3D. Again applying Theorem 5.4 of Hormander and Wermer,

we see that, for e > 0 sufficiently small, the polynomials P(z, w) have the property (5.8) on 3D,, 0 < t < E. Thus the w; of (5.15) work not only for the domains of Example 1, but also for domains which are close to them in the above sense.

§6. Forms orthogonal to holomorphic forms: the case of strictly pseudoconvex domains

1 °. We now proceed to the characterizatioi) of forms orthogonal to holomorphic functions, i.e. we shall answer the following question: Let D be a bounded domain with smooth boundary in C". Then which forms a E Cu,, _ j )(8D) are orthogonal to functions in An- p(5) in the sense that

faD

for all j E A"_ p(D )? Let us denote the space of these orthogonal forms by An p(D ).

As already observed, in the case n = 1, for any domain D, we have a E A p(D) if and only if a E Z(p,o)(D) fl Cp,O(D ), p = 0, 1. It is easy to give a sufficient condition for a form a to belong to An p(D). Indeed, if a = y I a D for some y E Z( p,"_ J

faD

µAa

D

P), then a E A p(D

), since

=0

by Stokes' theorem, for any µ E An -P(5). By a standard application of the Hahn-Banach theorem, it can be seen from Theorem 2.1 that Z°°, "_ (D) is weakly dense in the space of forms (arid even measures) from A C1 (D) in the topology of C*(aD ), the dual space of C(0D ).

2°. For strictly pseudoconvex domains, we can show that the condition stated in 1 0 is not only sufficient, but also necessary. More precisely, we have

46

II. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

THEOREM 6.1. If D is a strictly pseudoconvex domain in C", n > 1, with aD E C m + LX, m > 1, and if a E C(p:n_ I)(aD), then a E

if and only if

there exists y E C,"_2(D) such that 87 IaD = a. Theorem 6.1, Part II of Theorem 4.3 and Part III of Theorem 4.3 for q = n are essentially equivalent. PROOF. Sufficiency was proved in 1 °. Necessity. By Theorem 2.12, (6.1) a2 1L' a = al where a, E Z p n'_ 1 )(D) and a2 E Z(p; n'_ 1 )(CD ). If we extend a2 to a form 1aD-

1)(C"), then -y= a a2 satisfies the conditions of Theorem 4.3:

a 2 E C(p'

n-

fAAY==

µ/\a2=1D2

aD

since a 2 IaD = a - a 1 IaD, and a, a IaD E A p(D ). By Part III of Theorem CD,and ft EC(p,"_2)(C"). By Theorem 4.4, a = 5y, and y, E C( p,"_ 2)(D ). Now (6.1) yields that Y = y j - 8 has the desired properties. Theorem 6.1 cannot be carried over to arbitrary domains because it is not true that all a-closed forms (which, as shown in 10, are orthogonal to holomorphic forms) are a-exact for every domain. Thus the following result seems natural: THEOREM 6.2. Let D = O\ U k (Sj j), where 2 and the Ski are strictly pseudoconvex domains with C'"+2 boundary, m > 1, such that 2i cc Sl and SZi n a j = 0

1)(aD). Then a E An p(D) if and only if there exists for i = j; let a E 1)(D) such that IaD . a. aE The "exotic" form of the domain in which such a characterization is given reflects to a certain extent (and for n = 2 almost completely) the true nature of

i

things, as Theorem 8.1 shows.

PROOF. Let a E A n p(D ). By Theorem 2.12, a = a1 IaD - a2 1aD-) where

ai E

Z

a2 breaks up into forms a°, a1, ... , ak, a° E Zp;n'_ 1)(CSO), ai E Zp;n'_ 1)(2). By Theorem 4.4, ai = aY,, Yi E q,-2)(j). Extend the yi to C" so that

Q\ supp Yi CE

k

U UP,

j=1

j96i

and denote the extended forms also by y,. Consider /3 = a2 - (ay1 + . +a). Clearly, l3 = 0 on SZi, i = 1,... , k, and l3 = a° on CO. Further,

47

§7. THE GENERAL CASE

IaD E An p(D), since ay, 11D and a2 18D belong to An p(D). And since

An-P(Q) = A,-P(D), we have ao Ia, E A_(2). By Theorem 6.1, a°IaD = 3)' 61 where yo E Ctp,n_ 1)(SZ). Let q' E C°°(Cn), (P = 1 on CSC and q9 = 0 on U; Sl,. Then # IaD = a(gvy0) IaD. Set k

ay,- a(9rYo)j= 1

Then a E Z p n1 1 )(5), and k

a I8D = Cl 13D

i=1

IaD + (PYo) 18D

= a.

For k = 1, Theorem 6.2 admits an alternative formulation: COROLLARY 6.3. Let D1 CK D2 be strictly pseudoconvex domains with C m + 2

boundaries, m > .1, and let a j E _i)(aD), i =_ 1, 2,. . .. Then the a, are restrictions to the aDj of the same form a E Z(p p 1)(D2\ DI) if and only if

faD,µ/a1= µ/a2 aD2

for any u EE An-p(52)'

30. Just as Parts II and III of Theorem 4.3 for q = n yield Theorem 6.1, Parts I and III for q < n yield THEOREM 6.4. If D is a strictly pseudoconvex domain in C", n > 2, with C "12

boundary, m > 1, and if a E Z(mP; )(CD ), 1 < q < n - 2, then there exists y E C(p.9_ 1)(Cn) such that ay = a on CD.

This theorem is an analogue of Hartogs' theorem on the extension of holomorphic functions from the exterior of a compact set. §7. The general case

For the characterization of forms orthogonal to holomorphic forms in the general case, we need the notion of the envelope of holomorphy of a closed bounded domain. If a compact set K can be written as rl m Dm, Dm+ i a Dm where each domain D. has a schlicht (i.e. univalent) envelope of holomorphy H(Dm ), then we shall say that K has a schlicht envelope of holomorphy, and define H(K) as (1 m H(Dm). The envelope of holomorphy H(K) so obtained does not depend on the choice of the sequence Dm, m = 1, 2,.. ., and preserves

a number of properties of the envelopes of holomorphy of domains (see Aizenberg [2]).

48

II. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

m > 1, a compact set Dhas a schlicht THEOREM 7.1. If aD E envelope of holomorphy, and y e C(p.n , (aD ), then y E An p(D) if and only if (D) and Y2 E Bp,"- ,(CD) n q p;n- ,B(CD), 0 < A' < A, there exist y, E Z 11-2n ) as I z I -* oo and is a-exact in CHID), such that where y2 = O(j z (7.1)

Y-Y1 IaD-Y2IaD If p = n, then y, E B",n_,(D).

PROOF. SUFFICIENCY. Consider a E An _ p(D ). There exists a bounded domain Q with smooth boundary such that H(D) C Q and a E A n _p(Q ). By assumption, Y2 = a/i in CHID ). Now, by Stokes' theorem,

IDaQ

a Ay= aAYI - 1aD2 'aDJtD

anaf6

d(aAl)=0.

_(_i)" p-'

aQ

Necessity. Using Theorem 2.12, we write y in the form (7.1). Obviously,

y, E An P(D); hence y2 E An p(D). We must show that y2 is a-exact in CHID). Let H(D) = n Qm, where Qm + 1 C Qn, and the Qm are strictly pseudoconvex with C°° boundary. If a E An _ p(Qm ), then, since y2 is a-closed, we have by Stokes' theorem 1aQm

A y2Ay2=0, aD

for all m. As in the proof of Theorem 6.1, we get that for

i.e. Y2 E

every m there exists 6,,, E Cp,"_ 2(C") such that -y2= a S. in CQm. By induction we shall now construct 6,, E C p," _ 2)(CQm ), m = 1, 2,. . ., such that a 6, , = Y2 and Sam, = 8,, on CQm.

We put S' = S. Suppose 8; is already defined, and 0,,, = 6m+ I - S;". Then a em = a Sm+ I - a 5;,, = 0 on CQm. By Theorem 6.4, 0,,, can be extended to all of Cn

as a form in ZZ p,,, _ 2 (C" ); denote this extension also by 8m and set

Sm+I = 8m+I 6m+I

em-

e,n

Then S;"+1 = aSm+1

aem = Y2 on CQm+1 and S;"+1

8"=0onCQm.

I f we define S as equal to Sm on CQm, then S E C p,"_2)(CH(D )) and

as=y2. The question arises whether the second term on the right of (7.1) is necessary, or whether in the general case, as in the strictly pseudoconvex case, the characterizations of forms in may be confined to that of forms with a-closed extensions into D. From the results of §8, it follovWs that the second term in (7.1) is in general necessary.

49

§8. CONVERSE THEOREMS

§8. Converse theorems

It is natural to ask whether there exist domains for which it is impossible to characterize forms in A n p(D) in a manner similar to Theorem 6.2, and if such

domains do exist, then to describe the class of domains for which the characterization is possible. For domains in C2 , the answer is provided by the following result. THEOREM 8.1. Let D be a domain in C2 with C2 boundary, and suppose that,

for every a E A 1(D) fl C(2,,)(8D), there exists it E

)(D) such that a I a D =

a. Then

D=w\ Oj, i=1

where 7 and the SZi are domains of holomorphy, S2, i

9, and Sz; fl SZ, = 0 for

j.

PROOF. Since D is a bounded domain with boundary of class C2, we have D = S2\ U i SZ,, where Sz and the 2, are domains with C2 boundary and connected complements, with SZl C fI and SZ, fl SZ _ 0 . 1. Suppose Sz is not a domain of holomorphy. Then there exists z ° E aD at which the Levi form is negative definite; hence there is a neighborhood U of z°

such that (z E U: F(z, z°) = 0) \ z ° C Sz, where 2

2

F(=, Z°)

aP. (zo)(Zi -

Z°) +

Y'

as aZ. (z°)(z-z)(z-z)

(see, for example, Gunning and Rossi [1], Chapter IX, §B). Let T E C°°(C2), and supp 9) C U and p = 1 in a neighborhood of z°. Put

g(z) = q)(z)[F(z, z')]-'

(8.1)

for z E CSZ. Then ag E Z I)(CSZ), since ag = 0 at z 64 U U S2 and ag = ap(z)[F(z,z°)] for z E U\0. Further, if a =

Jag A dz lay 0

on aSZ,

onaSz,,i= 1,...,k,

then a E A" (5). Indeed, a satisfies the conditions of Theorem 7.1, since

a=aIIaD-a2laD,where al0, and a2=5gAdz onCSZand0ontheI,. By assumption there is an it E Z'(2,1)(D) such that a 1

= a.

II. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

50

By (1.1) applied to B \12, where B is any ball such that U U 2 C B, we have

forz EB\SZ

AdA

ag A dz

-a

ag(e) A a A

z)

z).

(8.2)

f(B\fl)r

By the same formula applied to D, we have for z E B\St

z) - a f a() n

0

z).

(8.3)

The first integral of (8.2) and that of (8.3) are negatives of each other, since ag IaD = a lag; = 0 and a Ian = ag A dz W. Hence by adding (8.2) and (8.3) we get ag A dz = ah A dz, where h is defined by

h(z) A as = -

5g(e) A a A

J«A

Z)

z).

Dt

h (z) is continuous in C 2, since the integrands in (8.4) have integrable singularities (see (1.4)). Now g - h is holomorphic in B\Q, since a(g - h) A dz = 0, so that g - h extends holomorphically into B. Therefore g extends continuously to zo. On the other hand, lim 1g(z) I = 00 Z--+ZO

This contradiction shows that the assumption that SZ is not a domain of holomorphy is false. 2. Suppose now that some Ski is not a domain of holomorphy. Then there is a ball B such that all functions of A(S21) extend holomorphically to B, and B\SZ, is not empty. Let ° E B\5,. Without loss of generality, we may suppose that

°=0and BnSZ,n{zEC2:z2=0} a=

0.

U2,1(0, z) Ian;

on 80,,

0

on asp and S2 , j

i.

§8. CONVERSE THEOREMS

51

Exactly as in step I above, we see that a E A 1(D ). Hence there is an such that a IaD= a. By (1.1) applied to SZi, for z E SZ; we get aE U211(0,') A

U2,1(0, Z) = f(a

z)

1 j) t

-5

U2,1(09 a) A U2,&, z).

(8.5)

By the same formula applied to D, for z E fI, we get

O=f a(s")A

(8.6)

aDt

D

Again as in step 1, we see that the first integrals of (8.5) and (8.6) are negatives of each other. Hence, if we add them together, (8.5) and (8.6) yield U211(0, z) = ah A dz, with h defined by

h(z) A dz = -

(j )t

z) -

U2,1(0, ) A

Dt

a(') A

z). (8.7)

By Corollary 2.4, the first integral on the rigit above is of class C(12 0(S21) and

the second integral is proper for z E 0,, so that it also lies in C(12 ,0)(2j), i.e. h E C°°(2,). On the other hand, U2,1 (0, z ) =

i1dz2 - zZda1

1

2711

)Z

1z14

dz = a

?idz

(2ri)2IzI2z2

for z2 0; hence h - a,((21ri)2 1 z 1ZZ2)-' is holomorphic in D\{z: z2 = 0}. Then g = zah - al((2,wi)Z 1 z 12)-I is holomorphic in D, and so it extends

a-' and B fl (a2 = 0) is connected; holomorphically to B. But hence g cannot be extended to 0 E B. The theorem is proved in full. COROLLARY H.Z. Let D be a domain in C2 with C2 boundary, and suppose that,

for each a E A1(D) n C(2 1)(8D), there exists y E q20)(D) such that ay E Z('2,,)(D) and ay IaD= a. Then D is a domain of holomorphy. PROOF. By Theorem 8.1, k

D=sZ

UsZ; 1=1

is a domain of holomorphy and k is the number of bounded components of CD. The corollary will be proved if we show that k = 0. where S2

Suppose that k 0, and let ° lie in a bounded component of CD, and let r be the boundary of this component. Then U2,1(s' °, ) E Z(12 ,j)(5) by Lemma 1.2;

52

II. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

hence U2, G°, f 18D E A 1(D ). By hypothesis, we therefore have a Y E 20)(D) such that U2,1 U °, z) = a Y 18D' But then by the Martinelli-Bochner formula ((1.1) for p = q = 0) and Stokes' theorem we get fr U2,1(01 0 = fr 5-Y = 0-

This contradiction proves the corollary.

CHAPTER III

PROPERTIES OF a-CLOSED FORMS

OF TYPE (p, n - 1)

§9. The theorems of Runge and Morera

The classical theorem of Runge asserts that a domain in C' is polynomially

convex if and only if it is simply connected. Wermer (see Wermer [1] or Gunning and Rossi [1]) has constructed an example of a domain which is biholomorphically equivalent to the tricylinder but not polynomially convex. Thus there are no necessary and sufficient topological conditions on a pair of domains in C" which ensure that homolorphic functions in the smaller domain

can be approximated by functions which are holomorphic in the larger domain. For a-closed forms of type (p, n --- 1), the situation is exactly the same as that for holomorphic functions of one complex variable. THEOREM 9.1 (Runge's theorem for forms). Let 2 1 C 07 be open sets in C". Then the following conditions are equivalent: 1. Every form lying in Z('P* n ,1)(S2 3) can be approximated in C p,, _ 3(S ; ) by .forms lying in Z(',.n_ 1)(02).

2. If 22\fl1 = K fl F with K compact and F closed in 02, and if K fl F = 0,

then K = 0.(') PROOF. I

--), 2. Suppose 92\0I = K U F, with F closed in f 2, K compact, Pick ° E K. Then A .. , A D1,n_ E Zrp,,l_,. (fzs )

K rl F = 0 but K

a=

0.

By condition I we can approximate a in fl f)(21) by the ak E Z(I*P,n--1)(02 )_ Let D be an open set, consisting of finitely many connected components with (')Condition 2 means precisely that each bounded component of CuI meets Ctt2. 53

III. a-CLOSED FORMS OF TYPE (p, n - 1)

54

smooth boundary, such that K C D C I2\ F. Then aD C S2, and by Stokes' theorem

f a A 4p+ 1 A ... Ad'n = lim

ak A

l

k-Woo faD

aD

A ... Adin

= klim f8akAd+IA...Ad=O. - oo D On the other hand, by the definition of U00 0 and the Martinelli-Bochner formula (see (1.1)), we have (277i

a n dip+, A ...

(n

aD

)n 1).1

)n

faD

)

U0,0\J J

- (n(27ri- 1). 1

This contradiction proves that 1 - 2. We shall prove the implication 2 - * 1 first only for m = oo ; the case of an arbitrary m will be handled later, using Theorem 9.3. By the Hahn-Banach theorem, it is sufficient to prove that any continuous linear function T on C(70, n _ 1)(521) which vanishes on Z(p, n _ )(02)vanishes on 00

Z(p,n-1)(

1)-

N ow, since T is linear and continuous on C (,P* n-1)(21), which is the proj ecC(p,n-1)(L), T can be extended to a continuous linear tive limit of the spaces

functional on .C(p,n_ 1)(L) for some compact set L C fl (see, for example, Gel'fand and Silov [11, Chapter I, §4.1). Here we may assume that CL has no bounded connected components contained in 2; otherwise we need only adjoin to L all the bounded components of CL contained in SZ to obtain a new 1

compact set which does satisfy the required condition. Put

E CL. (9.1) 7 (U,,n-1(', Z)), For a C°° function g, Og/Oxk tends to gzk uniformly on all compact sets; hence we may differentiate (9.1) with respect to and use Lemma 1.2 to get a>p

-T

qJ

-1)"Ti(azUn-21 Z))

= (-1)nTs\az`Ti'(Z)UP,n-2Gr Z))J = O where (p E C°°(C") is equal to 1 in a neighborhood of L, and equal to 0 in a neighborhood of E CL. This means that J(') is holomorphic on CL.

For' E"21 D"°4 = Tz(Df'°Up,n-l(J z)) =

0,

z), and hence all its derivatives, belong to Z(p, n -1)(522) (for 0 in each component of CL E C12 2 ). By the uniqueness theorem,

since U,,,,

§9. THE THEOREMS OF RUNGE AND MORERA

55

meeting CO 2.Also, = 0 in the unbounded component of CL as well. This is proved for n = 1 in Hormander [11 (§1.3). For n > 1,' can be holomorphically

extended from the unbounded component of CL to all of C" by Hartogs' theorem, and lim M -'oo

From Liouville's theorem it follows that 4 = 0 in the unbounded component of CL. Condition 2 of the theorem and the choice of L guarantee that CL has no other components, i.e. 4 = 0 on CL. Now let a E Z(p."_ and let D be an open set with smooth boundary consisting of finitely many connected components L C D C SZ 1. Let T E C°°(C"), (p = 1 in a neighborhood U of L, and supp 9) C D. Then, for z E U, (1.1) gives

a(z) = Ip,n-1(D, a) - alp,n-2(D, a) A Up,n-1(JI Z) - 5(g7Ip,n-2(D, a)). Since pI p, n _ 2(D, a) E C(p, n _ 2)(C") and aD C CL, we deduce that

TafDz(=A aD

z))

- 7'la(9,Ip.-2(D, a)))

A 4'(') = 0.

Thus 2 -- 1 is proved (for m = oo).

As in the case of one variable (see Hormander [ 1 ], § 1.3), we have, in particular, for SZ2 = C" COROLLARY 9.2. Z(p,"_ 1 )(C") is dense in Z(p,"_ 1)(S2) if and only if 0 has connected complement in the one point compactification of C".

THEOREM 9.3. Let SZ be an open set in C" and a E C(p, n _ 1)(9), m > 0. Then the following conditions are equivalent:

1. For every ball B (E SZ and every I = (i1)in_p), 1 < ii < . . . < ip < n, laB a n dal = 0.

(9.2)

2. a can be approximated in C(p,n-1)(S1) by forms from Z(p,n_ 1)(14 3. a E Z(mp,n_1)( )

Weierstrass' theorem on the limit of a sequence of holomorphic functions shows that, for n = 1, Theorem 9.3 is precisely Morera's theorem. For n > 1, a

tii. a-CLOSED FORMS OF TYPE (fir, n --- 1)

56

form satisfying the equivalent conditions of Theorem 9.3 can be nondifferentiable. As an example of such a form we may take 4p(zi)dz[n], where qp(z,) is a

continuous but not differentiable function on C'. Thus it is impossible to obtain a theorem in the same formulation as Morera's theorem for n = 1. PROOF OF THEOREM 9.3. We first introduce some notation.

If g is locally integrable in 2 and j E C°°(C") with compact support L, then for all z E Sl such that z - L C St (2) we set

(g*f)(z)f 8awz - )df A d f g(z -

A

s

2

PT(z)

eXp

(r2_Iz) r

for Iz 1< r,

forI z1> r

0 (K

1Z

is chosen so that Jp,.dAd = 1).

If 91

akJdi[k] n dz, E C(P,R-1)(SZ)v

J k=1 then

ajz} J k=1

(akj*p,.)(z)dff[k]AdzJ.

It is known (see, for example, Bremermann [1], §3.2, Lemma 2) that ar E ,, 1)(S r where fl '' = (z E 0: p(z, 89) > r), and that a,, converges to a

a in C4'.n_ 1)(l) as r -- , 0o. We first prove that I ---3 2. Let B C 0, and let the multi-index I' be such that dz1. A dz f = dz. Then

J,.Aj=f,.1Zi n

=1 Bt I' f ciakr(z -

=f

CS

(f8B z«(Z

f f it

n d n di[k] n dz

-JA1

n dt

a(z) A dzj

A d = 0.

(2) z -- L denotes the vector difference, i.e. the set ( E C": = z -- w, w E L).

§10. THE FIRST COUSIN PROBLEM

57

Hence aar A dz, = 0 in W. Since the multi-index I is arbitrary, it follows that aar = 0. Now it is easy to see that the open sets Ur, Sl satisfy condition 2 of Theorem 9.1. Thus we may apply Theorem 9.1 with m = oo to conclude that

1->2. For m > 0, the implication 3 -- I is an immediate consequence of Stokes' theorem. So, let m = 0 and B C Sl. Then B C Rr for sufficiently small r, since B is compact, and so

am dzl= lim faarAdzl

f a Adze= lim

r-'0

aB

= lim r

r-'0

aB

JB :k=1

B

(_i)kf aki() C

P.(z - )d/ dJ di / dz,

t

where I' is as before. Using the equality [ pr(z ik = -[ pr(z definition of a derivative in the space of distributions, we get

f a n dzl = h B

r

m,

f eda n dz fc :

k

and the

y (_1)k_1akJ,()pr(z )dA d -

s k=1

=1im J df A

fk

0.

This proves 3 -+ 1. The implication 2

3 is trivial for m > 1, and follows for m = 0 from the fact that differentiation is a continuous operator in the space of distributions. CONCLUSIONS OF THE PROOF OF THEOREM 9.1. By Theorem 9.3, every

a E Z_ 1 )(0 J) can be approximated in

C( p,,, _ 1 )(S21)

by forms from

Hence Theorem 9.1 for the case m = oo can be applied to deduce the theorem in the general case. Z(I*P, n

_ 1)(0 1).

§10. The f ilst Cousin problem, separation of singularities and domains of existence .

For forms of type (p, n - 1), we can pose the following problem, analogous to the Mittag-Leffler problem: Let 2 be an open set in C" and (z') a discrete sequence of distinct points in 2. Suppose forms a; E Z(p,,, _ 1 )(S2 \ (z')) are given. Does there exist an a E Z(I*P, n _ 1)(2\{z)) such that a - a . E Z(p, n _ 1 )(U) for some neighborhood U of

z`

III. 3-CLOSED FORMS OF TYPE (p, n - 1)

58

As in the case of functions, this problem can be generalized to the following one, which it is natural to call the first Cousin problem for forms: Let 2 be an open set in C", and {fl,) an open covering of Q. Suppose forms a, , E Z,,,_ 1)(Sl f n 0j) are given, satisfying the conditions a.

+ aj, = 0 on SZ; n UP onSZInSZjnQk

a.J+aik+aki=0

for all i, j and k. Can we find forms a, E Z(p,n_ )(00 such that a. = aj - a; on 2, n SZj?

The forms a,j are called the Cousin data, and the a, (if they exist) a solution of the first Cousin problem.

LEMMA 10.1. For any /3 E ')(2), the equation a y = /3 has a solution Y E C(

PROOF. Let {rk) be a sequence of positive numbers decreasing to 0, such that

Gk = ark n B(O, k) 0 , k = 1, 2, ... , and let k E C°°(C") be equal to 1 on Gk +1 and supp Pk C SZ. Put p 1 = 41, and Pk = 'k - Pk for k> 1. Then 1on SZ.By (1.1)we have aYk=Tkl8, Tk=0on -1

where

Yk-

S2Pk$n

By Corollary 2.4, Yk E C(p. n _

). Since 4)k = 0 on Gk, k > 1, we get 1)(0).

Yk E Z(,nP, n _ 1)(Gk ). As we observed in the proof of Theorem 9.3, f 2 rk and 2

satisfy condition 2 of Theorem 9.1; hence so do Gk and U. Thus we can choose X k E Z,,,_ 1)(2 ), k > I, such that the differences between the corresponding coefficients of yk and X k, and all their derivatives of order up to min(m, k), are less (in absolute value) than 2-k on Gk- I since Gk- I a Gk. Then the series 00

Y

Y1 + k=2

(-)

together with all its derivatives of order up to m converges uniformly in the interior of SZ to the form Y, so that Y E C(,,,_ 1)(0). By termwise differentiation, we get 00

aY = TIP + I Tkft = /3. k=2

THEOREM 10.2. If the Cousin data are of class C', m > 0, then thefirst Cousin problem has a solution which is of class C"'.

§10. THE FIRST COUSIN PROBLEM

59

Note that for the solvability, with any data, of the Cousin problem for holomorphic functions of several complex variables, it is necessary and sufficient that the first Dolbeault cohomology group be trivial. Therefore Theorem 10.2 is not valid as it stands in the case of holomorphic functions of several variables. PROOF OF THEOREM 10.2. Let {p,} be a partition of unity subordinate to the

covering (w,), i.e. the sequences {cp,) of functions and {i,} of integers satisfy the following conditions: 1. (P, E C°°(C") and supp p, C Q;,. 2. On any compact subset of 0, all but finitely many of the p, vanish.

3.11 p,= l on Q. Put hk =

1° qv, a;,k; then h k E C(p,"_ 1)(0k ). Furthermore, 00

00

hk-hi=

T1(a;,k-aij) _ Ya P1ajk=ajk, 1=1

1=1

and so the form f3 which is equal to ahk on 11k is well-defined. And a E C(p,")(si ), since 00

41 /\

ahk r= i

By Lemma 10.1, there exists y E Cc p,"_ 1 )(I) such that ay = /3. The forms

ak = hk _y are the desired ones, since a ak = ahk -ay = 0 and ak - aj = hk

- y - hi + y=ahk.

In the special case when the covering consists of two sets, we have COROLLARY 10.3. Let Q. Q, and SZ2 be open sets in C", with 2 = a, n 22. Then, for any a E Z(1nP,,_ 1)(2), there exist a; E Z,_1)(21), i = 1, 2, such that

a=a1 -a2onS2. That is, for a-closed (p, n - 1)-forms, the theorem on separation of singularities is valid for any triple of domains, whereas for holomorphic functions of several complex variables this is not always the case. THEOREM 10.4. For any domain D C C" and any p, 0 < p < n, there exists a form a E Bp,"_ 1(D) which cannot be continued across aD as a a-closed form. PROOF. Let

be a countable dense subset of 8D. Consider the function °°

1

G(Z) - 1 kz

1

IzI2n2'

(10.1)

If K C D is compact, then for E 8D, z E K and an r independent of and z. Therefore, the series (10.1) is majorized by the

Iii. a-CLOSED FORMS of TYPE (p, n - 1

60

convergent series 1

°',

1

2 r 2n-2 k=1 k

Hence (10.1) converges uniformly in the interior of D to a harmonic function, and urn

G(z) = co.

Clearly, the form

2

J k=1

- dF [ k I A dzj aZk

has the desired properties (here J = (j1,. .. , jp )). Indeed, suppose that a can be

extended continuously to a neighborhood of a point z ° E 8D. Then all the functions Gzk, and hence also all the G1k = = (Gzk),(3) can be extended continu-

ously to a neighborhood of z °. But then G itself can be extended smoothly to this neighborhood, which is impossible in view of (10.2). §11. Theorems of approximation on compact sets

a-closed (p, n --- 1)-forms enjoy approximation properties analogous to those of holomorphic functions of one complex variable on compact sets just as well as on open sets: THEOREM 11.1. Let K be a compact set in C", and a E C p, n _ 1)(K )g m ' 1. Suppose that a has an extension to a neighborhood of K such that 8a and all derivatives of order up to m --- I vanish on K. Then a can be appoximated in C( p,n - 1)(K) by forms which are C00 and 8-closed in a neighborhood of K.

If in addition CK is connected, then the approximating forms may be chosen in Z(,n.- 1)(C ?I ).

If formulated in the same way as above for holomorphic functions of several complex variables, Theorem 11.1 would no longer be true (Fuks 21, Chapter III3§16.3). PRooF. Let D be a bounded open set, consisting of finitely many connected components with smooth boundary, such that K C D and a E C(p, n M 1)(D ). We represent a as an integral by means of (1.1). By Lemma 1.2,

-jp,n_1(D, a) E Z(p,n--)(D) (3) This equality holds because G is real-valued.

61

§11. APPROXIMATION ON COMPACT SETS

By Corollary 2.4,

Pplvn-2(D, a} EH C(F,,,--,)(D), so we can approximate I p, n _ 2(D, a) in the topology of Cp, ,, _ 2(D) by forms lying in C(F.,, _ 2)Then a ,n _ . 2(a) is approximated in C(F,n 1)(D ) by forms lying in Z{p,n- 1)(D). To prove the theorem, it remains to approximate I p,"w 1(D, O a). Now a a has the form

f=

' h1di A dz1 ,

011...

,jp}, h j

e C"(5).

J Here the hr vanish with all derivatives up to order m - 1 on K; hence they can

be approximated in C'-1(D) by functions h' E C°°(D) which vanish in a neighborhood of K (see, for example, Malgrange [1], Chapter 1, Lemma 4.2). Then

a'.=

Dh'df A d ) C- C(0,0

J2

,,,---1

The a' are a-closed in a neighborhood of K (since the h' vanish in a neighborhood of K), and in this neighborhood we may differentiate the under the integral sign.

integral defining the operator 1p

We shall prove by induction on m that the sequence (a') converges in m-1 1)(K) to Qa = Ip,n_ 2 1(D, aa). C(pfn_ First let m = = 1. For each J, (h,) converges uniformly on D to h., i.e., given e > 0, we can find an N such that I h' -- hi I< e on D for I > N. Let ak1 denote the coefficient of dz [ k ] A dz1 in a', I = 0, 1,. . .. Then (see (1.14))

k-1

(ff-

akJlz

D

Fk

n

-Z

(11.1)

for l = 0,1,... (h,° = hj). Hence for z E K and 1 > N

«kJtZ) - akJ(Z)l -I ((2n1ri)n'

k- I

j

e(n - 1)!.

din CEO

D

n

fk_

where C

(n - 1)! V

n

dv

fa

and dv denotes Lebesgue measure.

2n I

I

U (z + D) zEK

z Izn d

Ad

III. a-CLOSED FORMS OF TYPE (p, n - 1)

62

Suppose now that our assertion has been proved for m = s. We proceed to prove it for m = s + 1. Note that hj = 0 in a neighborhood of K, and that the kernel in (11.1) is a function of - z. Hence, by Stokes' theorem, for z E K we have

az;

"kJ(Z)

= fo; fa

fk-I

FhJ)_l)

D

h'

fkZn n a I'-zI

)(_)k+i_ Jk - Zk d n dp[i].

(11.2)

I-zIZ"

A similar formula is also valid for the derivatives with respect to the a;. The

sequence formed by the second integral in (11.2) converges, with all its derivatives, uniformly on K; hence we see by means of the induction hypothe-

sis that the sequence {(akj)z.} converges in CS-'(K). Therefore {akj} converges in CS(K) to akU, and the first assertion of the theorem is proved. Now, if CK is connected, then we may also suppose that the neighborhoods

of K in which the approximating forms are defined also have connected complements. Then the second assertion follows from Theorem 9.1. We shall call a domain D in C' admissible if, for every z E OD, there exists a neighborhood UZ of z and an a E C" such that 5n UZ + Ea Cc D

(11.3)

for 0 0: there exists a E C" such that (11.3) holds for a and UZ = B(z, r)). By the definition of an admissible domain, r(z) > 0. It is not hard to show that r(z) is lower semicontinuous.

THEOREM 11.2. Let D be an admissible bounded domain, and let a E Then a can be approximated in C(p, n, ,)(D) by infinitely differentiaZ(p, n _ ble 5-closed forms defined in a neighborhood of D. )(5).

Further, if f a E Z(p, n _ 1 )(D ), then the approximating sequence converges to a in Ck(p,n_ 1)(D).

For the proof, we need

LEMMA 11.3. Let D = D, n D2 (D, D 1 and D2 are domains in C') and (D,\ D) n (D2\ D) = 0. Then any a E Z(p,n _ 1)(D) can be represented in the form a = a, - a2, where a, E Z(p,n_ 1)(A), i = 1, 2. If, moreover, a E C(p,n_ 1), then a, E C(p,n_1)(D1), i = 1, 2.

63

§11. APPROXIMATION ON COMPACT SETS

PROOF. The hypothesis of the lemma implies that D = D, n D2. Choose T, and P2 in C00(C" ) so that 1 - P2 = 1 and supp T, n D,\ D = 0. (q), and P2 exist. Indeed, choose neighborhoods U, and U2 of D,\ D and D2\ D such that

u, n U2 = 0, and functions 4, such that 4i, = 0 on U, and J, > 0 on CU, Ji, E C°°(C" ). The p, = p(4' + 02)-' and P2 = -'P2('P 1 +'2)-' will serve.) We define two forms aye 1 A a

fl(z)

on D1,

a 9p2 A a on D2,

-f(z) = f(DiUD2) fl(J) A Up.n-I(J' Z) /8

is well-defined since a 971 = a P2 onD 1 n D2 = D, and we have a E

C(p,n)(D1 U D2) and y E Cu,,,. 1)(D1 U D2 ). We shall prove that ,e = aY.

Here, the differentiation is to be understood in the sense of distributions. Approximate /3 in C(p,n)(D1 U D2) by forms fl, E C( pp,n)(DI U D2 ), and set

Ti(z) -

(DIUD2)

/3,() A

Z).

By (1.1) we have 6, = 5-f,, and since the y, approximate y in ,_1)('I U D2 ), (11.4) follows from the continuity of the differentiation operator in the space of distributions. We set a, =- cp,a - Y, i = 1,2. Then a, E Z(p,n_))(D,), since aa, acp1 A a

-ay=ft

Further, a1 -a2=TIC' -Y-pea+y=(p1 -gv2)a

a on D. If moreover a E C(p,n_ 1)(D), then # E )(D1 U D2) by Corollary 2.4. Then y E ,_))(DI U D2), and so a, E C(p,n-1)(D,). PROOF OF THEOREM 11.2. Let a E Z(p, n _ 1 )(D ). Then for every S > 0 there exist domains D,, D C D,, and forms a, E Z(p,n_ 1)(D, ), i = 1, ... ,1, such that

a1+---+a,=a

(11.5)

on D and diam(aD n aDj) < S. We prove this as follows. Choose N such that (2.3-')Ndiam D < 8(2n )1/2. In the first step, put b = infZEaD Re z1, c = SUPZ E a D Re z 1 and R = supZ E D I z 1+ 1. Then the domains D,

D' =DU (z: IzI 0 in (9o as lzl--+ oo and l Im zn 1

c > 0. Hence we may take limits in (14.1). The rest of the proof proceeds as in the case n = 1 (Bremermann [1], §§5.6, 6.5); the necessary uniform estimates at the corresponding places are valid.

We shall say that a distribution T is of class A, if there exists an aT E Bn, n -1(C

n \R2 n -1) representing T in the sense of Theorem 14.3 and a» (see (14.4)) is bounded on the sets l Im zn I>c for all c > 0. Clearly, 00 C A. 3°. We give below some examples of distributions and their representations. 1) 6- the Dirac distribution. Then

as(z) = (&, (n - 1)!

Z)) = Un,n-1(0, Z)

(-l)"ikdi[k] A dz. (27x1) IZI

k=1

2) Define the distribution 6 + by .

e--0 +0

9p(x)(xn - ie)dx

(n - 1)! Rzn-1

2i7r

(Ix 12 + E2

for p C -6D. For n = 1, this gives the usual S+ (Bremermann [11, §7.2). Let us show that 6 + E 6 '. The expression

(n-l)!ej 2 1,n

Txdx f*2n- I

(I X 12 + .2)n

80

IV. SOME APPLICATIONS

differs from the Poisson integral for the half-space only by a constant factor; hence its limit as E -- 0 exists and equals z Ep(O).

Consider the integral

(n - 1)! 2ia "

X.99(x)dx fR2n-) (I X 12 + 62)n

Since fp E 6D, we may suppose that the integration extends over a ball B, = {x: x E W` 1, 1 x I< r}. Further we replace q)(x) by [qv(x) - cp(0)] + (p(0), and observe that X,lIX

f6. (1x12 + e2)

since the integrand is odd and the domain of integration is symmetric about 0. Now

2Ix;f ve(x)-T(o) The functions

0

j=1

` ax w(tx)ar

j=1

x 1-Z" are integrable in By, and Z + EZl-n CI'XkxnTklxll lxL2Z;

I

hence we may take the limit under the integral sign. Thus S+ is defined. It is obviously linear, and its continuity follows from the weak completeness of 6D' (see Shilov [lJ, Chapter II, §9.3). Suppose cp E 6D has support outside some ball; then x12 + fR'n(P((IX)(

'621

fU,17- 1p(x)Iixfl_iId x

II2

l

Taking limits as e -+ +0, we get (p(x) I dx

1(3+ 01 < C fR2X-1 ixi2n-I

Hence S+ E 0,,', for any a < 0 (see Bremermann

i.e. 5.,. = O( x I §7.2).

Finally, we write down the representation of S+

as+(a)

(n - 1)! x (_l)kzdi[k]Adz, (2i1zI2) k=1 0,

I1T1 Zn < 0.

Im z > 0,

81

§14. REPRESENTATION OF DISTRIBUTIONS

3) S_ = 6 - S+ ; hence the representation of S_ has the form Im zn > 0,

0,

(n -

k- -

1),

n

(2iIzI2)flkI

1- 1 / Z,.Ul I K l / \ UZ,

lIR Z_.. --, V.

4) The principal part P(x x I_2?) is defined by ( )

`P\xn I X 1-2n),

IlITl

JR zn

- (1x12+e

(n - 1)! P

lnE

= S_-S+ .

l7l n

(14.6)

It follows that P(xn x-2n) E C9a for any a < 0. As in the case n = 1 (Bremermann [1], §7.6), it is easy to show that P(xn x1-2n) is equivalent to the Cauchy principal value of the integral Xnlxl-2xcp

( X) [1X.

fR 2n-1

The representation of this distribution is obvious, taking into account (14.6) and the representations for 8+ and 8-. 5) Consider the distributions 8k, k = 1, ... , n - 1, defined by (Xn + lXn+k)(p(X)dX

(1x12+e2)n

As in Example 2, it is not hard to show that 8k E 04 for any a < 0. As a representing form for Bk, we may take

(i/2)_hIzI_2P1[(_1)h1_Ic_1zdi[k] / a 9k(z) = -

IM Zn > 09,

0,

Im z < 0.

6) Finally, we define the distribution 00 by

(oo,qi)f =

qp(X)dx I x 12n-2

IV. SOME APPLICATIONS

82

Clearly, 0o E ®. for any a < -1, and

(i/2IZI2)'[d1[n] / dz + (l) kIZnZ1 Xdi[k] Adz],

«eo( z) = j

Im zZ > 0,

Imzn0,TEA.

an(z,, Z2,.. . ,Zn)

BA contains differences of all representations of the TEA satisfying the conditions in the definition of the class A.

The elements of BA are harmonic functions, bounded on all the sets C > 0). Consider the map

(Im Zn

f E BA -* T E A such that for every T E 6D lim cpfdi[n] A dz = (T, (p). e-. +o y=e

We shall show below that ( is an isomorphism of BA onto A. We denote by BA the algebra generated by BA. We introduce the following equivalence relation in BA*: f I f2, if, for any p E 6D,

lim

e-'+o y- U. Let A be the quotient space of BA by this equivalence relation, and f the 1%

image of f E B4 under the natural map B4 -+ A.4 We have BA C A. If *_'

T EA -*f EE BA ,

j= 11, 23P

then by the product Ti T2 we shall mean the element (f, f2) of A. We define the derivative in .4 by

aXk f

- (af/ax.

It is defined unambiguously; indeed, if f, -- f2, then

lim f

J

y=e

(f1-f2)--dx=0 k

83

§15. MULTIPLICATION OF DISTRIBUTIONS

for any qq E ®-2n, i.e.

afl/axk-aft/axk. THEOREM 15.1. 1) 4 is an isomorphism of BA onto A.

2) If S, T E A and the supports of S and T are disjoint, then S T = 0. 3) If S and T are defined by bounded continuous functions S(x) and T(x), then

S- T = S(x)T(x). PROOF. 1) By definition, 4 is surjective, A and BA are vector spaces, and 4 is

linear; hence we need only show that the kernel of 4, is trivial. If f E Ker 4, then, for any q) E 6D,

E)dx = 0.

lim e-' +0 JR2-1

(15.1)

But since the left side above is a distribution lying in o'2,, and 6D is dense in n2 n,

(15.1) holds for any p E= -0-2 i as well (Bremermann [1], §6.7).

Put

qlt(x) = K(x, t, tn) =

(n -

1).

?In

'

t2n

(Ix

't12+t2n)n

where it = (t 1, ... ,t2,,_ The function f(x, y + E) is harmonic and bounded for y > 0; hence by Poisson's formula 1).

f('t, t2n + E) =

R201-1

pt(x)f(x, E)dx.

Letting e - + 0, we get f(t) = 0 for t2n > 02) Set u = 4 -' T' and v = 4 -'S. We must show that, for any qv E GD,

lim j

e-' +0 R2n-

u(x, E)v(x, E)p(x)dx = 0.

(15.2)

Ou tside the support of S, we have v (x, 0) = 0; hence v (x, y) = yv 1(x, y) in a neighborhood of any such point. If we now choose p with support in such a neighborhood, then qD(x) v(x, e) -- 0 in 6D as e --o- +0; hence (15.2) holds (see

Shilov [1], Chapter II, §9.3). (15.2) also holds for functions with support outside that of T. Since every p E 6D can be represented as the sum of functions pj with supports disjoint from that of either S or T (see Bremermann [11, §3.6), it follows that (15.2) also holds in the general case. 3) Let w = 4 -'(TS)- Then u, v and w are harmonic extensions of S(x), T(x) and S(x)T(x) to the half-space (x: Im zn > 0), given by the Poisson integral formula. Also, u --* S, v - T, and w -+ ST uniformly on any compact set (see the proof of Theorem 14.1). Hence uv -+ ST uniformly on any compact set, so that uv - w. Q.E.D.

84

IV. SOME APPLICATIONS

We remark that, for n = 1, several definitions of the product of distributions have been introduced by means of analytic representations (see, for example, Bremermann [ 11, Russian pp. 245--259)* Ivanov [ l ], or Itano [ 11), but this kind

of definitions leads to difficulties for n > 1, basically for the reason that the product of distributions with disjoint supports need not be zero; even the product of the zero distribution with a nonzero one need not be zero (I tano [11),(6) Theorem 15.1 shows that the definition of the product given here does not lead to these difficulties. In conclusion, we present some examples. 1) Y'-1Bk

- (1/2) H ,ZA

Z

(i/2fzI2)'.-t

and 41 '180 =

I2",

It follows from the identity Izi2-2n(a1aZk)(Zk1I

12n) Z

_ -YlZk11 Z Ion

that eo o (a1azk)Bk = -n8k 0 ek2)

(n- 0! (-l)"(ZR - Zn)

(2iIz 12Y and Ix

1-2n1 1

in-I

(2+z). n

(21z12) '1

Bp o (a/a)o = -2n8 o P(xnIx

1-2n),

I a zn --- 'Fn 12n-2 3xn 12n

Zn - Fn

(15.3)

since Zn --- Z wn

12n

z 11n

For n = 1, (15.3) goes over into the known identity

8' = -28,o P(x -1) (see Ivanov [11, p. 18). * Editor's note. The citation is to an appendix added to the Russian translation. (6) other methods of defining the product of distributions have been developed by Vladimirov [21-[4].

BRIEF HISTORICAL SURVEY AND OPEN PROBLEMS FOR CHAPTERS I-IV

1°. The formula (1.1) was first proved for (0, q)-forms by Koppelman [1], and then generalized to (p, q)-forms in Aronov and Dautov [1]. Lemmas 1.1 to 1.4 and 1.6 were formulated by Koppelman [1], [2]. The elementary proof of (1.1) given in §1 is a detailed exposition of the proof in Aronov [1]. The results of §§2, 4, 6, 7 and 13 had been noted previously under stronger

smoothness assumptions. The use of Holder conditions has enabled us to sharpen these assertions in this book. Theorem 2.5 was formulated in the case q = n - 1 by Aizenberg [6], [7] and

Serbin [1], but the Tatter's proof is faulty. For arbitrary q, the theorem was proved by Dautov. Theorem 2.6 is due to Dautov and Kytmanov [1]. Corollary 2.7 was observed for p = 0 by Lu Qi-keng and Zhong Tong-de [1]. Example 2.9 and Theorem 2.10 were suggested by Dautov. Theorem 2.11 is also due to Dautov; the solvability of this boundary value problem had been established by Andreotti [ 1 ] and Andreotti and Hill [ 1 ], but their solution was not explicit. Theorem 2.12 is contained in Aizenberg [6], [7]. Corollary 2.13 is taken from Aizenberg [7].

Theorem 3.1 was proved by Weinstock [1] for domains with smooth boundary, but his proof works only with a stronger smoothness assumption than 8D E C', e.g. if 8D is a Lyapunov surface. The proof presented here is due to Aronov and Dautov [21, and is based essentially on the same ideas as in

the case n = 1 (see Muskhelishvili [1], §29.3). Apart from the tangential Cauchy-Riemann equations (3.2), there are other local differential conditions which may be imposed to secure holomorphic extension of f(z) into D (Hormander [1], Theorem 2.6.13). The possibility of replacing local differential conditions by global integral ones seems to have been first observed by Fichera 85

86

HISTORICAL SURVEY AND OPEN PROBLEMS

[ 1 ] (see also Martinelli [ 1 ] and Kohn and Rossi [ 11). Theorem 3.2 is taken from

the work of Weinstock, but the proof presented here is due to Aronov and Dautov [2].

The construction of a "barrier function" (§4.2°) was given by Khenkin [I ] (see also Ramirez de Arellano [1] and Ovrelid [1]). Lemma 4.2 was implicitly present in Dautov [1]. This lemma is basic for the proofs of Theorems 4.3, 4.4, 6.1 and 6.4. Theorem 4.3 is equivalent to Theorems 6.1 and 6.4 except for

minor details (this form of the assertions of Theorems 6.1 and 6.4 was proposed to the authors by V. P. Palamodov). Theorem 4.4 is due in the C°° case to Kohn [ 1 ], whose work is based on L2-estimates for the a-Neumann problem (see also Hormander [1], Chapter IV). We remark that a solution of

the a-Neumann problem by integral formulas, as well as estimates in the uniform metric, were first obtained by Khenkin [2] and Grauert and Lieb [I ] (see also Khenkin [3], Lieb [ 1 ], Ovrelid [ 11, Romanov and Khenkin [ 1 ] , Kerzman [1] and Polyakov [ 11, [2]).

The results of §5, excluding Theorem 5.6 and Lemma 5.11, are proved in Aizenberg [8], [9]. Theorem 5.6 was published in Dautov [1]; for n = 2, it has

been obtained jointly by Aizenberg and Dautov. Lemma 5.11 is due to Kytmanov and Preobrazhenskii [1].

The results of §6 are due to Dautov [11, [2]. An assertion equivalent to Theorem 6.4 in the C°° case had been proved earlier by Kohn and Rossi [1]. In Andreotti and Hill [2], there is a local variant of the Hartogs theorem for C°° forms, from which one can also deduce a global result (of the type of Theorem 6.4). Theorem 7.1 is taken from Aizenberg [6], [7].

The propositions of §8 are due to Dautov [4].

Theorem 9.1 is a precise analogue (both in formulation and method of proof) of Runge's theorem in one variable (see Harmander [1], Theorem 1.3.1). The results of §9 and Theorem 10.2 occur in Dautov [3]. The possibility in principle of obtaining such assertions had been observed by Malgrange [2]. The results can also be deduced from Corollary 2.13 and analogous assertions for harmonic functions (see Palamadov [11, p. 377). Corollary 10.3 and Theorem 10.4 are contained in Aizenberg's note [7].

The results of §11 are due to Dautov, and were published (except for Theorem 11.4) in [3]. Theorem 11.1 had been proved, in the case when the. compact set is the closure of a domain with smooth boundary and the form is C°°, by Weinstock [2]. For n = 1, the similar assertion is due to Browder (see Gamelin [1], Chapter II, Corollary 1.2). Theorem 11.2 for n = 1 is a known theorem of Keldysh [1] with a supplementary condition. Theorem 11.4 is a

87

HISTORICAL SURVEY AND OPEN PROBLEMS

generalization of the theorem of Hartogs and Rosenthal (see Gamelin [ 1 ], Chapter II, Theorem 8.5).

The results of §12 were obtained by Aizenberg. In less general form, they have appeared in his papers [8] and [9].

The Cauchy-Fantappie formula (13.2) was proved for convex domains by Leray [11, and for arbitrary domains by Aizenberg [101, [5], Koppelman [2], and Kenkin [1]. In the last article, the proof is close to Leray's. The footnote to formula (13.2) in fact Koppelman's proof [2]. Theorem 13.1 was noted by Aizenberg [5]; the remaining results of § 13 are due to Dautov [ 14].

The propositions of §§14 and 15 are taken from Aizenberg and Kytmanov The definition of the product of two distributions considered in §15 was first introduced for n = 1 by Ivanov [11, [2]. [1].

2°. To conclude, we indicate some open problems which we think are of some interest.

1) The Martinelli-Bochner-Koppelman formula is a generalization of the familiar Martinelli-Bochner representation for holomorphic functions to exterior differential forms. Is there a similar generalization of the Cauchy-Fantappie formula?

,

2) Can one replace "all 3-closed forms" in Theorem 3.1 by "all the forms U0 0(, z), z E CD"'? In other words: Is the representability of a function in a given domain by the Martinelli-Bochner formula sufficient for it to be holomorphic?(;) This question can be generalized as follows: 3) If a E Z(P, g)(D ), then a is given by (1.1) without the second summand.

and suppose it can be represented by (1. 1) Conversely, let a E C(, r 9)( without a second summand. Does it follow that a is a-closed in D? The same question can be asked under weaker smoothness conditions (Holder condition, etc.).

4) Can one assert in Theorem 2.12 that Yt C Bp, 1(D) when 0 < p < n --- I? 5) Every harmonic function generates a form of class B,,,,- whose coefficients are derivatives of the function. Describe the harmonic functions generating forms lying in Bn," _ (CD) f A y (D }. A simple solution to this problem may enable one to describe the dual of the space of all holomorphic functions in an arbitrary domain D C C" (or a compact set K C C") in a simple way, as some quotient space of the space of functions harmonic in the exterior of the domain (or compact set). 6) Does there exist for every domain D C C" a vector-valued function w(z), z e 8D, such that (w, z } - I on aD and the polynomials P(w, z) are dense in C(aD)? 1

(')For the ball in C", Aronov [21 has answered the question positively.

HISTORICAL SURVEY AND OPEN PROBLEMS

88

7) Are the orthogonal polynomials P(w, z) E O(aD) dense in O(aD )? A positive answer to this question would enable us to state the analogue of the Hartogs-Riesz theorem of §12 in a way similar to the classical theorem: one would be considering measures orthogonal to the polynomials in O(aD ) described in §5.

8) Elucidate for which domains the second summand in Theorem 7.1 is necessary. The results of §§6-8 solve the problem only in certain cases (the case of strictly pseudoconvex domains from which a finite number of strictly pseudoconvex domains have been removed; the case of domains not covered by Theorem 8.1, when n = 2).

9) Characterize the class of domains for which the Cauchy-Fantappie formula is the general form of the integral representations for holomorphic functions (analogues of the Cauchy integral). The theorems of §13 show that for n = 2 this class contains the strictly pseudoconvex domains and consists only of domains of holomorphy. This problem is related to the next two problems. 10) Is the Cauchy-Fantappie formula invariant under biholomorphic maps?

11) Find the general form of the integral representations of holomorphic functions for an arbitrary bounded domain with smooth boundary.

12) Prove approximation theorems for forms in Z(,_ I ), similar to the well-known results of Vitushkin [1] and Mergelyan [1].

13) Extend the definition of the product of distributions of class A in R2r-' to arbitrary distributions. 14) The representations of distributions in §14 are closely related to their Fourier transforms when n = I (see Bremermann [11, Chapter III). Is there a similar convenient connection when n > I? A positive answer to this question would lend further support to our view that forms in B,_1 (or Z(n, n _ I)) are good analogues of the holomorphic functions of one variable. Note. After this book went to press, solutions have emerged for some of the problems listed above. Aronov and Kytmanov [1) have obtained an affirmative answer to question 2) for functions of class C'(D). Also, Kytmanov [3] has

settled question 3) positively for forms in ,q)(D). In another paper [5] by the same author, problem 13) is solved in arbitrary dimension.

CHAPTER V

INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED DIFFERENTIAL FORMS AND HOLOMORPHIC FUNCTIONS

§16. A characteristic property of a-closed forms and forms of class B

In this section, we answer question 3) in the " Brief historical survey and ' open problems" section. THEOREM 16.1. Suppose a (D) E C 2" and y E C( P Q)(D ), X > 0. Then y is a a-closed in D if and only if, for all z E D,

Y(z) = Ip.9(D, Y) - Blp.a-i(D, Y)

(16.1)

If the coefficients of y are harmonic in D, then the conditions on aD and the behavior of y on aD can be relaxed. THEOREM 16.2. Let D be a bounded admissible domain with piecewise smooth boundary, and suppose y E C(p,q)(D) has coefficients harmonic in D. Then Y is 8-closed in D if and only if (16.1) holds.

10, and I.,,

Note that, if q = 0, then I p, q 9(D, y) has coefficients which are harmonic in D. Hence from Theorem 16.2 we deduce COROLLARY 16.3. Suppose D is as in Theorem 16.2, and -y E q,O)(D). Then,

for z E D, -y:= Ip,o(D, -y),

if and only if the coefficients of y are holomorphic in D. 89

(16.2)

90

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

For the proof, we need to use the double form

z) =

((2n1Ti)n'

karU(I'

I

[I, k]dz,

k)

(a part of UP.Q), as well as the following lemma. LEMMA 16.4. Let D be a bounded admissible domain in C", n > 1, and suppose

f E C '(D) is harmonic in D. Then f can be approximated in the topology of zk (Z-2", zk (4 D,

CA(D) by linear combinations of fractions of the form I

PROOF. Since D is admissible and bounded, the space of functions harmonic

in a neighborhood of D is dense (in the topology of CA(D)) in the space of functions of class CA(D) harmonic in D (see Weinstock [3]). Hence we need only prove the lemma for functions harmonic in a neighborhood of D.

Now let f be harmonic in the closure of a domain G D D with smooth boundary. Then, by Green's formula, for z E G

.t(Z) _ (2n 12)Q2n

a(n) I [g,z ) a

an

Gt

Z)I

do, (16.3)

z) = I - z I22n; also, where Q2" is the area of the unit sphere in C", and a f(')/an denotes the derivative of fin the direction of the outward normal to aG at , and do the surface element of aG. By replacing the integral in (16.3) z)/an by difference quotients, by the approximating Riemann sums and we obtain the assertion of the lemma. PROOF OF THEOREMS 16.1 AND 16.2. The necessity follows immediately from

the Martinelli-Bochner-Koppelman formula (1.1). Let us prove the sufficiency. If the form

2'

A dfJ

J

I

satisfies (16.1), then each of the forms

d

Yf

I

also satisfies (16.1), Indeed,

IP,9`Y) = I D

I'

A If

df, n

n (l)'wq(, Z)

A dZk

k

= f 11 2' (,j()df, A wq A aD

A a(J)d [J] A dz,)

§16.5-CLOSED FORMS AND FORMS OF CLASS B

_ I I fo _

A w4 /

91

A dzj

A dzj. J

Similarly,

Y) _ 2 aloqJD1 Y) A dzJ.

J Hence it is enough to consider forms of the type (0, q)

y(z) = 2'yr(z)dY,. I

Suppose now that the conditions of Theorem 16.2 are satisfied. If (16.1) holds for y, then y1(z) = II,q(D, ay) = 0 for z E D by (1.1). But the coefficients of y, are harmonic outside D, and tend to 0 as I z I -+ oo; also, being convolutions of compactly supported bounded measurable functions with the locally integrable functions I Ik, they are continuous in C". Hence by the uniqueness theorem for harmonic functions we have y, = 0 on C". Writing out this identity, we see that, for every multi-index I'(i,< i 2 < . < '0,

ay, D 11

p-

k4Z Il

aJk

-

A

-

ag m6ZI

L

d-

A d = 0.

(16.4)

By Lemma 16.4, the y, ( ) can be approximated in C' (D) by linear combinations of the g(', zs), zs (4 D, s = 1, 2,.... Hence (16.4) yields D 1,

7 k6-11

rr-.

aak

A 7 a(I,m)-a df [I,m}Add--0.

(16.5)

m(41

Summing over I in (16.5), we get

fDh1d= 0, i.e. 8y = 0.

Suppose next that we are in the situation of Theorem 16.1. The coefficients of the form y2 = Io, q(D, y) are harmonic in D, and y2 E Coo, q)(D) (see Corollary 2.4). Further, a -y2= a y by (16.1), and hence

,02, JD, aye) = 10,q(D, ay) = 0. Thus, (16.1) holds for y2, and it remains to apply Theorem 16.2. We have just considered the properties of forms representable by (16.1), i.e. the Martinelh-Bochner-Koppelman formula without the second summand. It is

92

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

interesting to discuss the class of forms to which the first summand belongs. If we consider (1.1) with p = n and q = n -- 1, we see that the first summand has the form II,n._.,(D, Y)

Z).

YlJ) Un,n--1

aD

It is clear that this form is of class B1(D). Forms of this important class have already been encountered before in this book (see Theorems 2.12, 7.1 and 14.1-14.3). Let us present some new properties of this class of forms. They can be written as

k-. a-dF[k] ac A dz,

YG(Z)

a Zk

with G harmonic in D.

THEOREM 16.5 (UNIQUENESS). Let aD E C', and suppose G E OD) is harmonic in D. If the restriction of Yy to aD is zero, then Yo =- 0. PROOF. n O=f

GYG(Z)= fdU^YG=1

8D

D k=1

D

aG azk

2

di A dz.

Hence aG/azk = 0, k = 1,... , n, i.e. G is antiholomorphic in D, and so YG 0. We remark that Theorem 16.5 remains valid even for unbounded domains, if we assume G = 0(1 z 12 _ 2 n) as I z 1 --9, oc. THEOREM 16.6. If n > I and aD is connected, and Y E Co n, Y(Z) `== In,n-1(D, Y

a

z

1)

is such that

D,

(16.6)

then Y -w 0.

PROOF. Consider the functions

(n - 2) fD (27ri

)n

z)

G+(z),

z E Dt

G-(Z),

Z q5 5.

These functions are harmonic (outside 8D). It follows from the properties of the potential of a simple layer that G'' = G- on D. By Theorem 2.5, (16.7) YG) Ia = Yc+I an - TG-I ao9 so that (16.6) implies Yc-6 = 0. By Theorem 16.5, yG-= 0 outside D, i.e. G' is

antiholomorphic outside D and vanishes at infinity; hence G-= 0. Hence G+ = 0 as well. Now the assertion of the theorem follows from (16.7).

93

§16. a-CLOSED FORMS AND FORMS OF CLASS B

This theorem shows that there are no nontrivial forms representable in D by (16.6) (i.e. by (1.1) with only the first summand). The next result describes forms representable by (16.6) outside D.

THEOREM 16.7. Let a(D) be connected and of class C, and suppose G CG 1 (CD) is harmonic in CD, with G = 0(1 z 12-2") as I z I -* oo. Then

Z a D, YG(z) = -In,"-1(D, YG), if and only if G can be continued antiholomorphically into D. PROOF. Necessity. By considerations similar to those in the proof of Theorem 16.6, we see that G- is antiholomorphic in D and that Yc = Yc- Hence G is antiholomorphic in CD and tends to zero as I z I -* oo, so that G = G-. Sufficiency. Suppose G can be continued antiholomorphically into D: denote the continuation by G,. Then G, E C'(8D) and G, is antiholomorphic in D; hence G, E C'(D) (Shabat [1], p. 334). Let

n- - 2)! (2,.Ti

)n

Z) =

aD

G'(z), G-(z),

z E D, z (4 D;

then 1'c' Ian - Yc-Ian = -Yc laD = 1'c, Ian - Yc laD'

so that

(-fG+ -yc,) ID = (YG7G) IDLet D = {z E C": p(z) < 0), where p E C'(C") and grad p 18D 11 G

0. Set

a

}" aG ap grad pr-1 ; k=1 34 azk

NG is defined near aD. It is easy to see that NG IaDa'ff = YG Ian'

where as is the surface element of W. Now we see from (16.8) that NG+_G, = NG--G for z E 1D, and G+ -G, _ G - - G on 1D. It follows from this and the complex form of Green's theorem that the function G+ (z)

G,(z),

,.G-(z) - G(z), is harmonic in C" with

0; hence

z Ei

z (4 D,

= 0. Consequently G+ (z)

= G, (z) and G -(z) = G(z). Hence YG+ = 0 and YG- = YG

94

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

§17. Holomorphy of continuous functions representable by the Martinelli-Bochner integral; criteria for the holomorphy of integrals of the Martinelli-Bochner type

If p = 0, the kernel in (16.2) becomes the Martinelli-Bochner kernel UO.OIJ. Z) _ -Un.n-1(Zr

0 - `n - 1)!

nk

Zk)d [kJ A d

I-zI

(2irz)"

nd the formula itself reduces to the Martinelli-Bochner formula for holomorphic functions:

rcZ) = f o

ZED,

z),

(17.1)

which is valid if D is a bounded domain with 8D piecewise smooth, and f E AC(D). The Martinelli-Bochner kernel z) is not holomorphic in z for n > I. Nevertheless, the following sharpening of Corollary 16.3 holds. THEOREM 17.1. If D is a bounded domain with piecewise smooth boundary, and

the representation (17.1) is valid for an f E CA(D), then f E A(D).

We may drop the condition f E C'(D), but we must then impose additional conditions on 8D: THEOREM 17.2. If f E C(D) is representable in a bounded domain D by the Martinelli-Bochner integral (17.1), then j is holomorphic in D in each of the following cases:

a) 8D (a C2; b) n = 2, and 8D is connected and of class

If f E AC(D), then

iDo,0(r, Z, = o,

Z

D.

(17.2)

As it turns out, (17.2) completely characterizes the traces on 8D of functions holomorphic in D: COROLLARY 17.3. In each of the two cases considered in Theorem 17.2, the condition (17.2) holds if and only if for f E C(8D) there exists F E AC(D) such that F lao = f. .

Let Pk,l be spherical (i.e. homogeneous and harmonic) polynomials forming an orthonormal basis for LZ(8B(0,1)), where 8B(0,1) is the unit sphere in C". Here,1 is the degree of the polynomial, and k its index among the polynomials

95

§17. HOLOMORPHY OF CONTINUOUS FUNCTIONS

of degree I occurring in the basis, so that 1 = 0, 1 , 2 ... , and k = 1, ... , a(1), where 2(n + k - 1)(2n + k - 3)!

k! (2n_2)! COROLLARY 17.4. Suppose D is bounded, and 8D is connected and of class C2.

Then an f E C(8D) is the trace on 8D of a function from AC(D) if and only if "

aPk,r f(f) I (_1)j_' aD j=1

k = 19 21 ...,x(1). For n = 1, this corollary is the well-known classical criterion, namely

1= 0,1,2,...;

orthogonality to all the monomials i",1= 0, 1, 21 ... .

We shall not present the proof of the rather easy Theorem 17.1 here, but refer the reader to Aronov and Kytmanov [11 (see also Corollary 16.3), or to the book by Aizenberg and Yuzhakov [1], §1. We proceed to the proof of Theorem 17.2, which is more difficult.

Let f E C(aD ), aD E C'; consider the following integral of MartinelliBochner type:

f()Uo,o(z)f f+(z), f (z),

z E D, z (4 D.

aDD

Consider lim Z, Z, t E a D [ f + (z) - f -(z')], where z, z' -- in the same manner as in (2.9). By Theorem 2.6, this limit is equal to f(), so that Theorem 17.2

and Corollary 17.3 are equivalent. Therefore it suffices to prove Corollary 17.3.

We introduce some notation. Let z0 E 3D, z+ E D and z - E CD, with the last two lying on the normal to aD at z ° and I z

=

P

- ZEaD inf I'-zI, inf 1-zI, zEaD

z

z

z0

.

Define

QED,

then D = (z E C": p(a) < 0). If 8D E C2, then (see Volkov [1], §2) the following assertions hold: a) There exists a neighborhood U of 8D such that p E C2(U). 1 in U.

b) I grad p C)

'P (Z+ k

and z$EU.

ap Z'

(z

ap

a(Zk)

(zo),

k = 19 2,... n,

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

96

Let

ap aF±

NF -k=I" I

a_

aZk

.

k

LEMMA 17.5. Suppose aD E C 2 and F E C(aD ); then lim Z+, Z

(NF (z+) - NF (z-)) = 0

uniformly in z °. If NF can be extended continuously to D, then NF can be continuously extended to CD, and conversely.

PROOF. Let F E C(aD). It is clear that the second part of the lemma follows from the first. Since NF = NF+ c for any constant C, we may assume F(z °) = 0. Observe that

df [k] A d IaD =(2i

)n(_I)k-I

a

, k

where do is the surface element of W. Therefore

z) Ian -

(n-i)!

f_kI'

I

,k_ Iznda

hence

NF (2+) - Ni (z-)

(n 0! J

F(J) ko)

D

+n n' aD f '

k21 a a k=1

J ak

Z+

I2"

I_z_I2n] J

Zk F+m=1

C10

(aplafm) (fm - Zm

I r . Z+2n+2

Ik_1

Zk )2m=1

(ap/am)(m

- Zm))

z-12n+2

Let us denote the first integral above by I,, and the second by I2. By performing a unitary transformation and a translation, we may take z° to

0, and the tangent plane to aD at z ° to the plane a = (w E C": Im w" = 01. Then aD will be defined in a neighborhood of the origin by the system of equations

'p='w, where

"=u"+h (w), ),'w=(wl,...,w,_1),and w=CM

be of class C2 in a neighborhood W of 0 in a, and z ± will be of the form (01000909 iyn ).

97

§ 17. HOLOMORPHY OF CONTINUOUS FUNCTIONS

aD is a Lyapunov surface with Holder exponent 1. Hence

w E W,

14'(w)I < cIw12,

a

_

0, such that I) B C w,

EB,

2) 1

3) B X [-a, a] C U,

+cI wI2)6d1.

Hence luopo( 0) = (_l)"?1 A

dv[1] A dv[1]

Here, 'o = (v2, ... , vn ). Under our change of variables, the lines {': ; = t, 2 - = c2 t, ... cn t } go ov r to the lines (v: v2 = c2) ... , v,, = cn }, and the boundary of D to a certai cycle F. If f is holomorphically continuable in dimension one, then, after th change of variables, the new function f(u) will be continuable in dimension one along all the lines (u: vz = c2, ... , vn = Furthermore,

rvv A f Df(flU00(,O) = ff(v)

A('v) = f (r)x('v)f ,(U)nrf(v) vv'

where 17 is the projection v -+ 'v. Note that, by Sard's theorem, the .7r-'(,v) n r are smooth curves for almost all 'v E ir(I'). The origin lies outside the domain bounded by r; hence the inside integral above vanishes for almost all 'v, i.e.

(17.2) holds at points z sufficiently far from D. Since an integral of the Martinelli-Bochner type is _harmonic, it follows that (17.2) holds on the unbounded component of CD. Suppose now that z belongs to a bounded component Q of CD, and suppose z = 0 (we can always achieve this by performing a translation). Consider a cone H. with vertex 0, formed by complex lines through 0, where E denotes the area of the portion of 3Q lying inside H E. By the same change of variables as

before, this cone goes over into the cylinder formed by the lines (v: va c 2 , ... , vn = cn } . And II, can be chosen so that: 1) as e -} 0, the area of a(Q fl H,) fl 13D tends to 0, therefore

f

f(')Uoo(,O)1imfoa(Qnn.)f)U o,o',0),

105

§17. HOLOMORPHY OF CONTINUOUS FUNCTIONS

and 2) aD \a(Q fl r1,) maps to a compact set I', under the change of variables. Then it can be shown, as before, that

ff(v) v re

AX'v

C

)

= 0.

1

Hence (17.2) also holds for z E Q.

Corollary 17.3 is closely related to the criterion for the holomorphy of integrals of Martinelli-Bochner type. THEOREM 17.9. Suppose n > 1, and D is bounded with aD connected and of

class C'; and let F E C1(0). If F+ is holomorphic in D, then F+ extends continuously to D and its restriction to aD coincides with F.

THEOREM 17.10. Suppose n > 1, and D is a bounded domain with aD connected and of class C 2. Let F E C(aD ). If F+ is holomorphic in D and extends continuously to D, then its restriction to aD coincides with F.

The content of the above two theorems is that, in contrast to the onedimensional case (integrals of Cauchy type), an integral of Martinelli-Bochner type is holomorphic in the case n > 1 if and only if it is a Martinelli-Bochner integral, i.e. F+ = F on aD (or equivalently, F 0). The condition that aD be connected cannot be dropped, as the example of the spherical shell D = (z: 1 < I z I < 2) and the function

_

1 1= 2,

1,

F E C'(aD), but F+ = 1, i.e. F+ does not coincide with F on aB(0,1) C 3D. For the proof of Theorems 17.9 and 17.10, we need the following assertion on the saltus of derivatives of an integral of Martinelli-Bochner type (Aronov [3l).

LEA 17.11. Let D be a bounded domain with 3D E Cl: D = {z E C": p(z) < 0}, where p E C'(C") and grade IaD =# O. If F E 0(8D), then Z

aF4- (z +

lim o

(

aF-(z-)

ask

azk

1

-

aF(Z°) aZk

I

aF(Z°)

Pk m=1

aZm

Pm+

uniformly in a°. Here z° E 8D, Z": lie on the normal to 8D at a°, z+ E D, a-(4D, I z+ -z°1=1 z--r° I ,and

Pk(z°) =

a a

ko)

Pk - Pk

106

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

PROOF OF THEOREM 17.9. We extend F to a function F of class C' in a neighborhood of D. For F, the Martinelli-Bochner formula for smooth functions (the special case p = q = 0 of (1.1)) is valid:

f

f

U0,0( z)

8D

f

A

D

z) (17.14)

z).

A

D

The second integral in (17.14) is the convolution of a bounded compactly supported measurable function with the locally integrable functions I fk - Zk I - Z I-In , and hence is continuous in C". Consequently the first integral, i.e. F+ , extends continuously to D. Now F+ is holomorphic in D, and therefore I

aF+/aik = 0; hence it follows from Lemma 17.11 that the aF-/aik extend continuously to CD and their restriction to aD has the form aF -

aF

aZk

aZk

aF

" m=1

(17.15)

1,2,...,n.

k

pm,

Zm

Multiplying (17.15) by Pk and summing over k, we then get "

k=1

aF -

- Pk =

aZk

"

2

aF

k=1 aZ k

" " aF Y Pk Pk I al_ pm m

Pk

k=1

m=1

since n

2 PkPk = 1. k=1

Next, consider the form n

µF-

()k -I ai- dz[k] n aF.

k=1

k

Recall that

dz[k] A Lil 13D =(2i

)n(_I)n+k-

lPkda

where da is the surface element of D. Hence

AF-6

-

)n y

V

da = 0.

k=1

By Stokes' formula, we deduce that OF-

0 = J DF N'F-= k=

azk

2

dv;

Q,

§ 18. TRACES ON THE SHILOV BOUNDARY

107

this is permissible, since F - = 0(1 z 11-2" ), and aF -/aZk = 0(l z I2-2n ) as zI

oo. (17.16) means that F - is holomorphic outside D. Since aD is

connected, Hartogs' theorem now shows that F- extends holomorphically into D. However, F - - 0 as I z I--* oo ; hence F - = 0. By Theorem 2.6, F+ extends continuously to D and F+ 18D = F. PROOF OF THEOREM 17.10. Let z o and z ± be as in Lemma 17.11. Apply Lemma 17.5. In our case NF = 0, and so NF extends continuously to CD and Ni 6 = 0. The proof proceeds from this point exactly in the same way as that of Theorem 17.9. Note that neither of the Theorems 17.9 or 17.10 contains the other. They can

be regarded as results concerning the possibility of extending F from aD holomorphically into D. §18. The traces of holomorphic functions on the Shilov boundary of a circular domain

Let D be a bounded strongly starlike circular domain in C", i.e. a domain invariant under all rotations z -- a Fez, 0 < 0 < 217, such that XD = (Az: z e D) C D for any A, 0 < A < 1. Let S denote the Shilov boundary of D, i.e. the smallest closed set lying in D and such that

sup{If(z)I : z ED} = sup{If(z)I : z E S} for all f EAC(D). If D is strictly pseudoconvex, then S = aD; in general S is a subset of aD. Before proceeding to the description of the traces on S of functions from A C(D ), we give a characterization of polynomials orthogonal to A C(D ). Let it

be a nonnegative measure on S, invariant under the rotations z -+ eiez, 0 < 0 < 27r, and massive on S, i.e. S \ E = S for every E C S with µ(E) = 0. Denote by [A C(D)] the space of functions 9p E C(S) such that f S cp fdµ = 0 for all f E A C(D). If D is the unit ball and µ = w(i, z), then [A D)]µ = O(D). Let P 2(µ) be the space of polynomials in z,,.- , ", with the inner product

(f,g) = ffd&. If the multi-indices I and J are such that 11111 µ under rotations we have (Z19 Zj) =

II J II, then by the invariance of

f

Z1i.rdµ = s

= e i801 111 -

Of 11111),

(ZI

)

Zj

).

108

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

Hence W, 2J) = 0; thus, if f and g are any homogeneous polynomials of different degrees, then (18.1) U9 9) = 0. If follows that we can choose a complete orthonormal system of polynomials (p} in P2(µ), where I is the degree of the homogeneous polynomial, and k its

index among those of degree 1. For multi-indices I and J with 111 II < II J Il> we

put

PT.J(Z) = JalJkTflJll-Il/ll0). k

where

ark =

!-J k

df

SZ z

LEMMA ]8.1. A polynomial P(z, z) lies in [AC(D)]µ if and only ijit is a linear combination of polynomials of the form

Zl.P,

11111 > IIJII,

ZIIJ - PIJ(z) I

Hill < 11A.

(18.2)

(18.3)

PROOF. Sufficiency. D is strongly star-like, so functions from A C(D) can be approximated uniformly on D by polynomials. Since (q) is complete, we need only prove that, for a polynomial P(z, z) of the form (18.2) or (18.3) and for all k and 1,

f SP(Z' Z )4pidµ = 0.

(18.4)

If P(z, i) is of the form (8.2), then by (18.1)

f

S

z

Idli = (zlq),k , Z J) = O

since a'(pl is a homogeneous polynomial of degree 11 1 II + 1 > 11 J II. Next, if P(z, i) is of the form (18.3) and I 1& 11 J II - 11111, we have (using (18.1) again)

fP(z. Z)Tidµ = (z'q', zJ)

- (q, Pir(z)) = Q.

Finally, if 1= 11 J II - 11111, then (18.4) holds by the choice of the aljk.

Necessity. Let P(z, F) = El,j blfz,iJ belong to [AC(D)]µ ; then

+

P(z, Z) Illti>IITII

bIjPlj(z) II11111II

bl,(z'.F-' - Plj(z)')

§18. TRACES ON THE SHILOV BOUNDARY

109

i.e. P(z, Z-) = P1(z, i) + P(z), where P1 is a linear combination of polynomials

of the form (18.2) or (18.3), and P(z) is a holomorphic polynomial. Since P(z, i) E [ A C(D )] µ by assumption, and P1(z, i) E [ A C(D )] µ by what we have already proved, we see that P (z) E [A C(D )] µ . Hence

f1P(z)I2c1µ = js P(z)' P(z)dµ = 0. Since u is a nonnegative and massive on S, it follows that P(z) = 0 on S, and hence that P(z) = 0 on D (S is the Shilov boundary). Thus P(z, i) = P1(z, z ). Q.E.D. If D is n-circular, i.e. invariant under all n-rotations z -> (eie,z1, ... , eienz ), 0 < 01, ... , 0 < 21r, and if u is also invariant under n-rotations, then as a

complete orthonormal system we an choose {ajz'}, consisting of monomials

with al = (fSZJiJdµ)-'

.

In this case Lemma 18.1 becomes

LEMMA 18.2. The polynomial P(z, z) lies in [AC(D)Iµ if and only if it is a linear combination of polynomials of the form

z,ij,

with 'k > jk jor some k, 2

ZIZI+J

al+J

(18.5)

ek1z

where ek is the multi-index with 1 at the kth place an 0 elsewhere.

Lemmas 18.1 and 18.2 are analogues of Lemma 5.7. When D is the unit ball, w = I and ,4 = w (z, z ), Lemma 18.2 and Lemma 5.7 are identical.

For the proof of the main result, we need the Poisson kernel. First we construct the Szegd kernel h (r, z) of the circular domain D by the formula

h(f, z) =

I q)k(Z) (Pk

k,!

z Ei D, E

I

It can be shown that: 1) h (f, z) is holomorphic in z on D and continuous in 2) h (f, z) = h (z , ) on D X D; and

on D;

3) forallfEAC(D),

f(Z) = ff()h(, z)

dµt.

The Poisson kernel can be constructed from the Szegb kernel:

z)

Ih(z)I2 h(i, z)

(18.6)

110

V. INTEGRAL PROPERTIES CHARACTERIZING 2-CLOSED FORMS

For z E D we have h(a, z) > 0, since (18.6) and 2) above imply

h(z, z) = fh(z, )h(f, z)dµt = fs Ih(, z)12

dµ > 0.

Further, if f E AC(D), then

fez)

h(z

z)f(z)h(z>

z) - h(i, z) f f(flh(i, )h(, z)dµt

z)dµt

=

(18.7)

THEOREM 18.3. Let D be a bounded strongly star-like circular domain in C"

such that every point of its Shilov boundary S is a peak point for the algebra AC(D), and let µ be a nonnegative massive measure on S invariant under the rotations z -> e'BZ, 0 < 0 < 2r. Then a function qD E C(S) is of the form f IS for some f E AC(D) if and only if

=0

(18.8)

fSTPdj%

for every polynomial P of the form (18.2) or (18.3).

If D is n-circular and u is invariant under all

the n-rotations z -30 (eia, z,, ... , e`0nzn ), 0 < 0s, ... , On < 21T, then we may replace (18.2) and (18.3) above by (18.5). PROOF. The necessity follows from Lemma 18.1.

Sufficiency. Since each point of S is a peak-point for AC(D ), we have for each z 0 E S and any neighborhood W of z 0

P(', z)dµq = 0.

lim z-'z zED

(18.9)

Sew

The argument for this can be found in Koran [1]. Further, since P(', z) > 0 and

fP(',z)d,.i.-l,

z ED,

S

it follows from (18.9) (see, for example, Hoffman [11, p. 18) that if p E C(S), then the function

Z fp()P(, z)dµr,

z E D,

§19. COMPUTATION OF MARTINELLI-BOCHNER INTEGRAL

111

is continuous on D U S. Since S is the Shilov boundary of D, {f(rz)) therefore

converges uniformly on D as r -4 1; hence f extends continuously to D. It remains to show that, if p satisfies (18.8), then f E A(D). From (18.7), we see

that, for any g E A C(D) and any differential operator T with constant coefficients,

fg(')7(P(, z))dµ = 0. It follows in particular that

z)) L=o E[AC(b)] From the form of the Poisson and Szego kernels, we see that the function on the left above is a polynomial in and . By Lemma 18.1, it is a linear

combination of polynomials of the form (18.2) and (18.3). This and the hypothesis of the theorem imply that

f (pa) [TzmPal z))] 1z=0 dji = 0, i.e. all the coefficients of the form

af(z)

f

z)dp

(18.10)

and all their derivatives vanish at z = 0. Since the Poisson kernel is realanalytic in z, the form (18.10) has real-analytic coefficients; hence 5f = 0 on D, i.e. f E A(D). §19. Computation of an integral of Martinell i-Bochner type for the case of the ball

We consider on the unit sphere aB(0,1) the space L2(aB(0,1)) with the usual inner product

(f, g) =

aB(o, 1)

fds,

where do is the surface element of the sphere. We shall identify this space with

the space G 2(B(0,1)) of harmonic functions f in B(0,1) represented by a Poisson integral from L2(aB(0,1)). Let 6Y,,t be the finite-dimensional vector space consisting of homogeneous harmonic polynomials of the form Ps,t . aIJZ J-J II111=s IIJII=t

112

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

The union of all the P3 , is dense in L2(aB(O, 1)). As we know, (17.1) is valid for f E G2(B(O, I)) n A(B(O, I)):

(Mf)(z) =

f

B(0.1)

zEB(O,1).

A0

LEMMA 19.1 Let PS, E s,r. Then the MartinelliiBochner type integral of PS:t is of the form

n+s-wI

=

n+s+t-I P

MP

s'

s,r

This follows from the fact that the harmonic extension of k P,,x from aB(O, 1) into B(O, 1) is given by

Iifl2

kPs,t+ n+s+t --I

a

k

's,t,

and from the easily verified formula Z) JaB(O'l)

here

=

I

4' L

z)da;

I - I Z 12

z) is the Poisson kernel for the ball: 1)!

PQ> z) -

t - 1z12 ZI2n

COROLLARY 19.2. If QS'

t " yj

Y.

V a,, IV

11111=S 11J11Wt

then

MQS,t = 2 RS-P, t-P 9 p.4 where the R. _p, #_I, are the harmonic polynomials defined by s+p+n---I

R S--p't--p

;I-(+ t + - p--. l)r (_1)f(S+t _f ---2p+n--2)!

A being the Laplace operator:

AR =

I n

a2R

2j i+p

§19. COMPUTATION OF MARTINELII-BOCHNER INTEGRAL

113

This corollary follows from Lemma 19.1 and the Gauss representation of an

arbitrary polynomial on the sphere as a sum of homogeneous harmonic polynomials (see, for example, Sobolev I I], Chapter 13). We introduce the differential operators b and b by setting n

=2 b

n

a

a

_

h

=k aZk k=1

ik 6Z= k=l

k

THEOREM 19.3. Suppose f E G2(B(O,1)). Then the Martinelli-Bochner type integral off has the form n

I 44'0

(19.1)

k=l where 1z

dl j

I' -nf "' m aak

lZlm"-

`Yk

IZI-'If

f(')d,'J'

k= 1,...,n.

J

PROOF. Since f E G 2(B(0,1)), in the interior of B(O, 1) an expansion

f()= 2P(),

(19.2)

s, 1>0

converging uniformly on compact subsets of B(O,1), in particular on each sphere MR r), 0 < r < 1. From the definition of the class G z (B(0, 1)), it is not hard to show that (19.2) is also valid on aB(o, 1), but in the sense of convergence in L2 (aB(0, I)). By Lemma 19.1, we deduce that, in the interior of B(0, 1),

n+s

M_

1

P

--

---

2

t

P

To prove the first part of (19.1), we observe that ---n

0z) =1z1

s,r>a

flzlIn_2+:+s P1, f

d1fl

o

+s+ nt-I zi r+s

Iz _

I

n+s+t

1

k=

)=Yj n+s+I -P "

apX, 4 = aZk

I

2;

S,1>0

t

n+s+t

-P 1

114

V, INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

The proof of the second part of (19.1) is similar. The above theorem enables us to compute Mf for any f E G'(B(O, 1)). The functions V and &k are harmonic in B(O, 1). To compute them, it is convenient

to change over to polar coordinates. If f is polyharmonic or real-analytic in

B(O,1), then it may be convenient to compute Mf by using the Almani representation of f (see Sobolev [1], Chapter 14): k>O

with the fk harmonic in B(0,1). We present some examples to illustrate the applications of Theorem 19.3. EXAMPLE 1. Let f E GZ(B(0, l)), and

(n-2)! r Tf

(2ir)"

8B(0, l)

.f

I

dad' Z 12n-2

ZEB(0,1)

the potential of a simple layer. Then

7j, _ IZI1-nflZI II'2f()dI1. 0

In particular, if Tf = F, then f = (n - 1)F+ bF + bF. Consider the integral equation

f + ATf = q),

(p E GZ(B(0,1))

(19.3)

on 3B(0,1). Then its solution is of the form

f _ (P -

AIZII -n-,N Jlzl

112p(fldII,

JAI< n - 1.

(19.4)

ExnWLE 2. Let f E LZ(aB(0, I)), and

Sf = 2f

z),

z E aB(O, 1),

aa(o. >)

be the singular integral of the Martinelli-Bochner type. Consider the singular equation

f + XSf

E L2 (aB(09 1)).

If ip has an expansion T _ 2p=0 PpS.ys in B(0,1), where p and q are nonnegative integers, then

f_

p+q (1 - A) q + (1 + X) p 1P ICI c 1

2Xq(p + 4)(n - 1) IzI [(1 - a)q + (1 + J1)p] Z " o

f

_

(n - 1)(1 + A)(P + 4)(1

- A)q + (1 + A)p

IZI

§ 19. COMPUTATION OF MARTINELLI-BOCHNER INTEGRAL

115

If we know only the values of p on aB(O, 1), we must replace qv in (19.4) and (19.6) by its Poisson integral. Then we obtain integral representations for the solutions of (19.3) and (19.5), i.e. resolvents of the operators T and S.

To compute integrals similar to Mf in more general domains than the ball, we may use the ideas of §5. We will have to require that (5.8) holds, or that the polynomials P(z, w) are dense in L2(aD ). We indicate other applications of Lemma 19.1. The following theorem is an immediate consequence of the lemma. THEOREM 19.4. For n > I the operator M maps Let OB(O,1)) into itself, is bounded, and has norm 1. All positive rational numbers in [0, 1] are eigenvalues of infinite multiplicity for M. The spectrum of M is [0, 1].

Denote by Pr11 the orthogonal projection of G 2(B(0,1)) into the Hardy space H 2(B(0,1)) consisting of elements of A(B(O,1)) satisfying the condition that faB(o, r) If 12 do is uniformly bounded for 0 < r < 1. T, his operator is called Szego's operator, and has the integral representation

(Pr11f)(z)

f[) 8B(O,I)

k=1

(-1)df [k] A d .

z))

For n = 1, integrals of Martinelli-Bochner type are integrals of Cauchy type, and M = Pr1,. For n > 1, the kernel U0,0 is not holomorphic, and M and PrH are not the same, as the following example shows. EXAMPLE 3. Let f =zk . Then J' is not holomorphic in B(0,1). Indeed,

(n-i)!

n

(2i)n JaB(o,

Zm/Gk - Zk) dy m7

1

YJ nm=1

I_zI2n+2

(-0

z 12n

since

_ Uo,o( ', z) IaBto,ly =

(n- I)! n

n

(m`"`Zm)'m do L-zI2n

we have a

aZk

+

(n-I)! (n 2 -r'

and the integral vanishes only if n = 1.

- I)f

3B(0111)

1tk12

da,

1

/

df [k] A

J

dt,

116

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

However, the following theorem is true: THEOREM 19.5. For every f E L2(aB(0,1)),

lim Mkf = PrH f,

(19.7)

k -oo

where Mk denotes the kth iterate of M.

PROOF. For each homogeneous harmonic Polynomial P,,t, (19.7) follows immediately from Lemma 19.1, since PrH Ps,t =

fP,1,

if t=

to,

if t > p,

But such polynomials are dense in G 2(B(0,1)), and the family of iterates of

M is uniformly bounded in norm; hence (see Dunford and Schwartz [1], Chapter II, §3, Theorem 6) (19.7) holds on all of L2(B(0,1)). §20. Differential boundary conditions for the holomorphy of functions

Let D = (z E C": p(z) < 0) be a bounded domain with aD E C'. We consider the following problem: suppose we are given a function f E C'(D) harmonic in D, and a continuous vector field w = (w, (z),. . . , wn(z )) on aD, with

(w,gradp)=

W 8z

'

Z E an,

(20.1)

i.e. the vector w never lies in the complex tangent plane TTD(z) at z to aD for any z E aD. Then does the condition " k = =1

al = 0, -wk azk

z E aD,

(20.2)

imply that f is holomorphic in D? This problem is an analogue of the oblique derivative problem for real-valued harmonic functions. If we do not require (20.1), then there does exist an f e A(D) satisfying (20.2), e.g. D = B(0,1) C

C2,w=(0,1)andf=z,. When w = grad p, (20.2) becomes the Neumann normal a-condition for functions harmonic in D. and the answer to our question is affirmative: THEOREM 20.1. Let D be a bounded domain in C", and let f E C(D) be harmonic in D. Then f E A(D) if and only if the a f/azk, k = 1,. .. , n, can be extended continuously to D and satisfy the Neumann normal a-condition n

k =1

af

azk

ap

azk

-

01

z E aD.

(20.3)

117

§20. DIFFERENTIAL BOUNDARY CONDITIONS FOR HOLOMORPHY

PROOF. If f e A(D), then all a f/aik = 0, and (20.3) is trivial. Conversely, if 8 fl azk E C(D), k = 1, ... , n, and (20.3), holds, then n

of

k= , azk

ap

azk

where do is the surface element of aD. As in the proof of Theorem 17.3, we now conclude easily that f E A(D ).

Let us reformulate our problem differently. Decompose w into its normal and tangential components to aD at a point z E aD:

w(z) = a(z)grad p+ T(z); here a never vanishes on aD, and T(z) is orthogonal to grad p,

i.e.

(grad p, T)=0 on 8D. Then (20.2) becomes

of aP

of , ?k(z) a(Z)

k=I azk

k_, ask aZk Consider again the form n

kI of d; [ k ] A dz-.

k=1

Using (20.4), we can write it on 8D as

µI JaD = I ak,m(z)djn dz[k, m] A dz IaD,

(20.5)

k>m

where the ak,m E C(8D). Then the new formulation of our problem is: suppose f E C'(D) is harmonic in D, and (20.5) holds; does it follow that f is holomorphic in D? (20.5) reduces to the Neumann normal 8-condition when all the ak,M are zero. Condition (20.5) can also be written in integral form:

AZ) = fan with

z) + (-1)n 2 d(gak.m) A

[k, m] n dal = 0

k>m

z) = -(n - 2)!(21ri)-" I

z 12-Z". It is then clear that the problem

of the holomorphy of the functions defined by a Martinelli-Bochner integral (see Theorems 17.1 and 17.2) is a special case of the present problem (the case

[1km=4 We now present the solution to our problem for certain classes of domains D and functions THEOREM 20.2. If n = 2, and 8D is connected and of class C', and a harmonic function f E C'(D) satisfies the condition

µJ1an=adfA where a is a constant, then f E A(D).

(20.6)

118

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

PROOF. If we show that, for each Ps,1 E 6Ys,t (see § 19 for the notation),

S,t=0,1,2,...,

f

2

an

I ) k- I OPS,t d

k=1

[k] A d = 01

(20.7)

holds, then our assertion will follow from Corollary 17.4. By (20.6) and Stokes' theorem, Z

(-l)'d[k]Ad=fakDPs,1µf k

=d= '3D'"

,Ad. aD

If t = 0, i.e. Ps,, is a holomorphic polynomial, (20.7) holds. Let t > 0. We transform dPs,t A d' into the form

nd _

aPd-

,

aP

Ad

.

This can be achieved by setting

- 4 dr

P

ar,

2

(formal integration with respect to 2 ). Then a2Ps,t

aP

a2Ps,t

=

d _ aps,t 2

aft

so that AP =0. Also, P E 33+,,t_ 1 Thus 2

aD

f(U)

(_l)k_td[k]Ad

k=1

4k

2

an

k-(_l)hff.d

k=1

Now we apply induction on t, using the fact that (20.7) holds for t = 0, and conclude from (20.8) that (20.7) is valid for all t. THEOREM 20.3. Let D = B(0,1) and suppose that the lie in AC(D). Then, if f E CA(D) is harmonic in D and satisfies (20.5), it follows that f E A(D). PROOF. Consider first the case µ113a(0,1> =

where Qi.o E 'Yio

n df [k,m] A d IaB(o,1)'

(20.9)

119

§20. DIFFERENTIAL BOUNDARY CONDITIONS FOR HOLOMORPHY

Writing out (20.9) for the case of the ball, we have

of j= 1

aj

j=

a.

of

!(

Q1,d` 0

aB{0, 1).

k

m

(20.10)

aim

ask

We shall look for the solution of (20.10) in the form of a series

f

I PSTt s,t;;0

where Ps,t E'Psrt. Then (20.10) yields i

apslr

j

+ --- Q 1, ` )

s*r

_

am

kaPst -

(20.11)

.

s,tD ask We now extend the function on the right above harmonically into B(0,1) using Gauss' formula (see, for example, Sobolev [1], Chapter 11): if Pk is any homogeneous polynomial of degree k, then its harmonic extension Pr from aB(0,1) into B(0, .1) is given by

Pr = 2 Zk-2s 0

S

where

(_,)j(k-j-2 +n-2)! k-2s+n-1 Zk-ZSsI(k+nsJI zI js1

ZjDj+,Pk.

In our case a

Pk

-r

Q1;0 a m

k

-si t

and k = I + s + t. Hence (20.11) leads to the system of equations 2j tPs, t

djt

j

aP5-1- l,t+ ask

j;;o: 0

bj2jj+

+

+ I cj j>o

l,t+

1,0

2J

(ai_12

Q

j+?Q1

k +

BPS--r r+

ar

1,4mk

aP3-14+1+1 o

ask

M

M ---1 t+f+ l aim

k

(20.13)

where the a j, bj, ... , cj are constants. For s = 0, we have tP4,, = 0 for any t by (20.13); hence P.,t = 0 for t 0. Next we have tP1,, = 0, ... , tPs, f = 0, so that the series for f consists only of holomorphic summands, i.e. f is holomorphic in B(0, 1).

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

12 0

If the ak,m are arbitrary holomorphic functions, we can decompose them into sums of homogeneous polynomials and apply the earlier considerations. Then we get a system of equations of the type of (20.13), with each equation involving only finitely many summands. In the case of B(0, r), the Neumann normal a-condition has the form _ of In zka_ =0 zk

on aB (O, r).

k= 1

Suppose now that this condition is satisfied everywhere in B(0, r). Then it turns out that the requirement that f be harmonic can be dropped, and f will still be holomorphic in B(0, r) as before: THEOREM 20.4. Let f E C'(B(O, r)) fl C°°(0), and suppose that 91

ik

of aZ-k

=0

z EB (O, r )

(20.14)

k

Then f is holomorphic in B(0, r). If f E C2 (B(, r)) fl C°°(0) and a 2f

n ZkZm azkazm

09

z E B(0, r),

(20.15)

k, m=,

then f is pluriharmonic in B(0, r). PROOF. Observe that (20.14) implies that f is holomorphic on each complex line I through 0, and (20.15) implies that f is harmonic on each such line. Thus we need only appeal to a result of Forelli [1], which asserts that an f which is harmonic on each complex line 1 through 0 and C°° at 0 is pluriharmonic in B(0, r).

Rudin [1] has shown that, if a function f in B(0, r) satisfies the Laplace equation A f = 0 and the Laplace-Beltrami equation d' f = 0, where a2

n

A

A k I

m=

ZkZmaZkaZm -

then f is pluriharmonic in B(0, r). Theorem 20.4 shows that, if f E C2 (B) ft C°°(0), then the single equation (20.15) is sufficient. The single condition A,f = 0 is not sufficient. For example, let

f-

(IzI2

Iz2I)

+11)I IZlzk.

ko (n'+ k

(20.16)

Then f is real-analytic in B(0,1), since the radius of convergence of (20.16) is 1. It is easy to verify that A' f = 0, but f is not pluriharmonic, since

(IZ112 _Iz212) ; (j+ 1)! (j + l)(n + 1)! k.m

ZkZm

aZ a

(n +j +

IZI2j.

§20. DIFFERENTIAL BOUNDARY CONDITIONS FOR HOLOMORPHY

121

If f E Ck(B(0, r)), Theorem 20.4 is not true; e.g. let

- zj=1 I zj

f(Z) Since

I z j I4

2. Then the restriction of Pk to 8B(0, r) can be extended holomorphically into B(0, r) if and only if ek Pk = 0 on 0B(0, r), where

(n+k-2l--3)!(2l-1)!!2,1 (n+k-2)!(l+1)! !0

n

k

2 Zj j_1

j

PROOF. Let Pk = Pm, t, i.e. a1JZIZ-J .

Pk 11111=m II.JII=t

We extend Pk from aB(0, r) into B(0, r) by Gauss' formula; the extension, call it Pr, can be written as Pr = 7sz0 Zk _ 2s, where Zk _ 2s is defined by (20.12). P. is holomorphic if and only if

apr

n

2 zj azj_ = 0

j=1

in B(0, r) (see Theorem 20.4). But n

j= 1

ij aZaZ-2s - (t - S)Zk-2s j k

Then

2 j=1

i 8Pr a1i

1)(k-j-2s+n-2)!A+sP

I

S!j! (k + n

j, S

s

AP(-1)(k-2s+n-1)(k-l-s+n-2)!(t-s) !>0

s:kt - s): kk -r n - s - i):

j+s=1

(-1)5(k-2s+n-1)(k-l--s+n-2)!(t-s)

(_')/Q!Pk I;;,__0

s=0

s:kt - sJ:kK -rn - s - iJ:

The following lemma now concludes the proof of the theorem.

k

12 2

V. INTEGRAL PROPERTIES CHARACTERIZING a-CLOSED FORMS

LEMMA 20.6. If k > 21, then

5=o

(-1)3(k---2s+n- 1)(k-l-s+n-2)! s!(1-s)!(k+n-s- 1)!

JO,

ifl > 0,

ii, ifl=0,

and

(-1)3(k - 2s + n - 1)(k-l-s+n-2)!s

Sl.(I-s)!(k+n-s- 1)!

(-1)'(n+k- 21 - 1)! (21 - 2)! (n + k - 2)!1! (1 - 1)!

ifl=0. PROOF. Let us compute for instance the first sum, which we denote by S. We have

S_ (-1)I (k-21+n- 1)(k-21+n-2)! 1! (k + n - I - 1)!

+(-1)1-1(k-2I+n+1)(k-21+n-1)!+ (_1)'(k-21+n-1)![' k-21+n+ Il 1+ k+n-1 + (1- 1)! (k+n-l- 1. 1

(-l)'-'(k-21+n-1)!(1_ 1)(k+n-21)+ 1! (k + n - 1)! 1)'-'(k - 21 + n)! (I - 1) 1! (k + n - 1)!

+

(_1)1+2

(k - 21 + n + 3)(k - 21 + n)! (I - 2)!2!(k + n - 1 + 1)!

+...

_ (-1)'2(k-2l+n+ 1)!(1- 2) + .. . 2!1(1- 2)!(k + n - 1 + 1)!

(k-l+n-2)! (1- 1)!l(k + n - 2)!

+

(k+n- 1)(k-l+n-2)! 1!(k + n - 1)!

The second sum can be computed in the same way.

§20. DIFFERENTIAL BOUNDARY CONDITIONS FOR HOLOMORPHY

123

COROLLARY 20.7. Let P = 20 Pk be an arbitrary polynomial, where the Pk are homogeneous polynomials of degree k. Then P can be extended holomorphically from aB(0, r) into B(0, r) if and only if

k=0

ek pk IaB(0, r)

0.

We note that the condition EkPk = O on 8B(0, r) may be replaced by the condition ek Pk = 0 in B, where

_ k

n

_

a

2 Zi a

j=1

-

2 1;190

(n+k-21-3)!(2l- 1)!!21 1

21+2 !+ IZI

CHAPTER VI

FORMS ORTHOGONAL TO HOLOMORPHIC FORMS. WEIGHTED FORMULA FOR SOLVING THE a-EQUATION, AND APPLICATIONS §21. Forms orthogonal to holomorphic forms

In this section, we sharpen the results of §6. First some ancillary results: LEMMA 21.1. Let D be a bounded strictly pseudoconvex domain in C". Then there exist positive constants r, 8 and c such that

I) For any

with I p(') I< r in

8), we can perform a smooth change of

variables T = T(Z)with the properties T2 = Im z); 1) T1 = P(z) -

3)1/c >I aT/aZ I > c; z E

8). II) For any z such that I p(z) I < r in B(z, 8). we can perform a smooth change of variables T = r() with the properties 1) T1 = p(J) P(Z); T2 = Im `W, Z);

EB(z,6); 3) 1 /C > 18 T/af I> C;

E B(z, 8).

z) was introduced in §4.2'. The function PROOF. We need one more property of Khenkin's barrier function not noted in §4.2°: there exist r > 0 and a nowhere-vanishing continuous function h on (z: I p(z) I < r) such that, for z e Vwith I p(z) I < r, d14 (', z) lz=r = d3' I

z) I= hdp

(see ovrelid [1]). From this we conclude that the maps

Ti = p(z), T2 = Im'DG, Z),

and 125'

Ti = pM =

lr2Im(,z),

126

VI. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

and grad p(z)

have rank 2 at points where z =

0. The lemma follows

easily from this. LEMMA 21.2. Let

('ri+ Ior,k,ll

fTIrs+r, we get 1a.k.2n_ 1(E) = 0(I) El-k+a

I

f

dl

f46-1

l

ok

f (vu +

CiU

+ 1)

f1

kI

(VU + 1)"du

.

Assertion a) follows. In the case b), we have

la.k,2n-3-2m(E) _

J/'2rZ mdr J('2

rds

(TS + +

(rsu

a

E) rsdu r2mdr

= 0(1) f Zrz"'dr J(2 = O(1) JZ

(E+r2)'12 dr

(+r)

2(k

-a-m-2)

Assertion b) now follows by direct integration.

§21. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

127

LEmmn 21.3. Let D C R" be a bounded domain with smooth boundary and let f E C '(D). S u p p o s e I grad f(z) (= O(1) I P(z) I' Co,aGD).

for 0 < X < 1. Then f E

This assertion is a slight generalization of a result of Hardy and Littlewood (see, for example, Goluzin [1], Chapter IX, §5, Theorem 3).

We now improve the estimate for the solution of the a-problem given by Theorem 4.4. THEOREM 21.4. Let D be a bounded strictly pseudoconvex domain in C" with boundary of class C m+ 2, and let y E Z p; 9)(D ), m > 0, A > 0. Then there exists

a form a satisfying as =yin D, with a E C(p,q± lj 2(D ), C"'+1,a-1/2(D)

a E (p9-1)

0 < X ' < J1, if 0 < 71

2;

2

1.

(21.1) '

2

PROOF. We proceed as in Theorem 4.4. We need only show that the form

A u'q-, A Dp,nr-p(aZ, 3 ) (V\D),

satisfies (21.1). Lemmas 1.4 and 4.1 show that it is enough to verify that (21.1) is true for the function

H(Z) _ f (V\D)r

4Dr+

Z 1-r (,z)I-zl(" Z) I

1>d n

(21.2)

C'"-1,11(C") and h = 0 on D. If m = 0, we may suppose that with h E h E C(V\ D) and I for this, we have to extend y to I= 0(1) all of C" by means of Theorem 3.2 and Supplement 3.3 of Malgrange [11,

Chapter 1. For arbitrary m, by working out the argument in Lemma 4.2 more carefully, we see that 0(1) I Im1 . We shall carry out the rest of the proof in the case X > Z. The case A < Z can be handled similarly.

Differentiating under the integral sign and using the argument in Lemma 4.2, we conclude that, for z E D, 13jIJ

0(1) f V\D)s

1

(

Z)

CItJZ

12n-3+j'

Choose r and S according to Lemma 21.1 and split the integrals over V \ D

into integrals over (V\D)\B(z, S) and (V\D) Q B(z, S). The first integrals

128

VI. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

can be estimated above by a constant independent of z. Thus 2

%=0

=o(i)l 1 +

fl(V\D)f1B(z.S)]; I(W' Z)I3-,IJ -

ZI2n-3+j

2

j=0

p -'dvt °[(V\D)f1B(z.6)]t

rC1(P(0

+IIm4D(J.

P(Z)) + C2IJ - ZIZ

Z)I)3-JIJ

-

ZI2n-3+j

Here, we have used inequality 4 from §4.2°. Now make the change of variables

occurring in part II of Lemma 21.1. Using the definition of the Ia,k,1, the obvious inequalities

IA-1,2,2"-2\ 0 =

0(1)Ix-1,12,2"-1(e)'

4-1,1,2n-I(e) = 0(1)Ia-1,12,2n-1(e) and Lemma 21.2, we get for 11111 + IIJ II = m + 2 2

ID''JH(Z)l = 0(l)l 1 +2 4_1,3_J,2fl_3+f(IP(z)I) j=o

= O(1)11 + Ia-i.3.zn-s0P(z)D + 0(1)IP(z)I

A-12

Hence we conclude by Lemma 21.3 that all derivatives of H of order m + 1 lie in C °"X -1 /2(D ). Thus H satisfies (21.1).

We next give the sharpened version of Theorem 6.1.

THEOREM 21.5. Let D be a bounded strictly pseudoconvex domain in C", n > 1, with aD E C"`+2, m > 0, and let a E C(P,n_1)(aD), A > 0. Then a E

A ,n p(D) if and only if there exists a form y satisfying (21.1) such that

--a.

PROOF. The sufficiency was proved in §6. We prove necessity. By Theorem 2.12,

a=a,1an-az1an9 where a1 E Z(_ 1)(D) and a2 E Z_ 1)(CD). By Theorem 21.4, a1 = ay1 and y1 satisfies (21.1). Extend a2 to a form a2 E cp:n_1(C ). By (1.1), for

129

§21. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

z E (V n G )\ D we have 2

a2 = a2 = Ip,n-1(G, a2) - II2 ,n_ 1(G, aa2) - alp,n_2(G, a2

).

(21.4)

Here G is any domain of holomorphy with boundary of class C,+2 such that D C G. Using Lemma 1.3 and the fact that aa2 = aa2 = 0 on CD, we get for

zEG\D Ip.n_1(G, aa2)() = I p.n_1)(D, aa2)(')

-a D aa2 A µo -

4

aa2 A WW,0(u,', z). (21.5)

Here µo is the form defined in Lemma 1.3, for the choice

wI =t=(f -z)/l -z12 and The coefficients of WP.o(u, , z) are holomorphic in z, and

a2Ian-02 Ian-a1 Ian - a E so that the last integral on the right side of (21.5) vanishes by Stokes' theorem. The integral

72(r) =

aa2 A µo DZ

and its derivatives can be estimated as in the proof of Theorem 21.4 to conclude that y2 satisfies (21.1) with G \ D in the place of D. Extend y2 to all of G with the same smoothness, and denote the extension also by y2. Next, since

Ip,n- I(G, a2) E Z(1P,,,_,)(G), we have ay3 = Ip,n_ t(G, a2), where y3 E Then it follows from (21.3)-(21.5) that a = a y IaD, where y = yl C(pp, n _ 2)(G).

12 lp,n _ 2(G,

a2). Corollary 2.4 now finishes the proof. If we use Theorem 21.5 in place of Theorem 6.1 in the proofs of Theorems 13.2 and 13.4, then we can weaken their hypotheses: instead of the condition µ Z E C2» PD), A > 0, we need only require µ z E C20 - 1(aD ), A> 2 . We state the next theorem without proof. + Y2 - Y3 +

THEOREM 21.6. Let D be a bounded strictly pseudoconvex domain in C", and

suppose a is a (p, n - 1) form on aD with coefficients in L'(aD ). Then a E A n , (D) if and only if there exists $ E Zp,,,_ 1(D) such that lim E-o {p(Z)=-E)

$(z)Ap(z)f = a(z) A (p(z) aD

(21.6)

for any p E q,O)(D). Also, N(Z) A 5P(z) +IP(Z)I'/2I$(z)I dvZ = O(1)jjajjL'. (21.7) (p(z)=-E)

130

VI. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

Here, as before, p is a function defining D; by the absolute value of a form we mean the sum of the absolute values of its coefficients, and II II V denotes

the L' norm of the absolute value of the form. (21.6) says that, in some (generalized) sense, the restriction of P to aD coincides with a. (21.7) says that the tangential part of P has the same properties as a. For forms of type (p, n - 1), the a-problem is solvable not only for domains of holomorphy but also for domains D with aD E C2 for which the restriction of the Levi form (cf. §4.2°) to the complex tangent space to aD at a point has at least one positive eigenvalue for each E 3D. In the C°° case, this was done by Kohn and Rossi [1]. By combining the methods of proof of Fischer and Lieb [I ] with those of Theorem 21.4, it is possible to show that Theorem 21.4 is

valid for q = n - 1 and for domains of the above kind. After making the corresponding changes in the proof of Theorem 6.2, we thus obtain the following improved version:

'

THEOREM 21.7. Let D = S2\ U Sz;, where Sa is a strictly pseudoconvex domain

with al E Crn+2, m > 1, and the Ii C Sz are domains with boundaries of class Crn+2 such that S z; n SZ, =0 f o r j 1, j, I = 1, ... , k, and the restriction of the Levi form to the complex tangent space to afli at ' has at least one positive eigenvalue at each ' E ale , j = 1, ... , k. Let a E Cep: n_ 1)(8D ). Then a E An p(D) if and only if there exists a form it such that it IaD = a and

a E Z(p,n'' 1) '/2(D ),

j

a E Ztp;n'= 2(D),

0 < X' < X , if 0 < A < 1/2;

1/2 < A' < A, if 1/2 < A < 1.

§22. Generalization of Theorem 8.1 THEOREM 22.1. Let D be a bounded domain in C n with boundary of class C2, and suppose that for each a E A 1(D) n C(n,n _ 1)(D) there is an a e Z(1,, n _ 1)(D) such that a 18D= a. Then D = SZ \ U 0i, where the SZi a Si are domains such that ii, n Ii, =0 for j 1, j, 1 = 1,. . ,- , k, and the restriction of the Levi form of a 2i to the complex tangent space to a SZj at ' has at least one nonnegative eigenvalue at each E a Sz;, j = 1, ... , k.

Theorems 21.7 and 22.1 give similar necessary and sufficient conditions on the bounded components of the complement of D in order that all forms in have a-closed extensions into D. However, the most interesting part of the boundary, namely that of the unbounded component of the complement, remains unexplored.

131

§22. GENERALIZATION OF THEOREM 8.1

PROOF. Suppose that, for some j, 11 does not satisfy the conclusion of the theorem, i.e. there exists ° E a0i at which all the eigenvalues of the Levi form

are negative. Then there exists a neighborhood U of ° such that for any E U n SZj the function

F(z) =

n

ap

aZ (NZ, - fir) +

a2P

n

7, 1,m =

aZI

M

(NZ, ` 0(zm '" m )

E aIi U U (see, for example,

vanishes in u n CSC; if and only if z

Gunning and Rossi [11, Chapter IX, §B). A similar assertion was used in the first part of the proof of Theorem 8.1. It follows that there exist ' E -=u f, Si f, z ° E U n CS and r > 0 such that the function f (z) = l /F (z) satisfies

i>If(z)1,

z E CSI1 n 8B(! °, r),

(22.1)

and CSI J n B(' °, r) has piecewise smooth boundary. The form

Iuo,o(,z0),

E auk E aD\aSZj,

o,

lies in A l (D) n Cfw, n _ )(5); hence there exists a E Z(,,,,-,)(D) such that a 8D = a. As in the proof of Theorem 8.1, we see that U0,0(t, z °) = 5y in S2 f, where

Y( )

-f

U0,0(w, z4)U.,n--2(wq

} + fD* a(W) 11 U,,,,,-2(W, }.

By Corollary 2.4, Y E C (n, n _ 2)(s2 j ). By the Martinelli-Bochner formula (i.e. AFCQj n B(°, r j we have

(1.1) for p = q = 0) and Stokes' theorem, for any g

g(Z°) = J

fr

Z°

r)) A (ca; n

,)>r

Here IF = aS j n 8B(°, r). It follows that, for some constant c,

Is(Z°)) < c

max

Is(Z)I.

(22.2)

But it is clear from (22.1) that if g(z) = fm(z) with m sufficiently large, then (22.2) cannot hold. This contradiction proves the theorem.

132

VI. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

§23. Weighted formula for solving the a-problem in strictly convex domains and zeros of functions of the Nevanlinna-Dzhrbashyan class

To obtain certain precise results in strictly pseudoconvex domains, it became necessary to have the solutions of the a-equation with a right-hand side that grows near the boundary expressed by a formula. For simplicity, we present the formula here only for a strictly convex domain.

Let D be a strictly convex domain in C", i.e. a domain of the form D = (z E U, p (z) < 0), where U is a neighborhood of D, p E C 2 (U ), and the second differential (i.e. Hessian) of p is positive definite at all ' E U. Hence grad pIaD 0. For a strictly convex domain, the Khenkin barrier function (see §4.2°) may be introduced as follows:

PM = (P1,...,P) =

ap a

1

z) -

We set 1$(i, z) = of p, we have

,

...,

ap

Z) = (P, - Z).

By Taylor's theorem and the strict convexity zI2

for z,

2 Re z) + c, 1 p(z) > E D', D C D' C U, with a constant c1 > 0 depending only on Y.

Therefore 2 Re (D

z) -> p

2 Re

z) >

P(Z) + Cjj _ Z12,

p(z) + c, 1 - zj

2

(23.1)

Hence I41(3', Z)I =

{- P(z)I - Pq) - P(Z)

Z)k

+IzM $q, Z)I+I

(23.2)

-

0(1) depend only on D' and not on z, For a (p, q)-form 7; we define (Gp",qy)(z)

ZIZ

=

E Y.

`IP,4-I(D, akGI Z)7(0)(Z)

f 5takGl z) A y(J) A tJ'q

A p!

(n- p)! Dp,n-p(aZ+ (23.4)

§23. WEIGHTED FORMULA FOR SOLVING THE a-PROBLEM

Here, µ9

133

z 12 and u =

is given by Lemma 1.4, with w = t = ( -Z )/I z), and ak is defined by Z)

ak(

2n k

PG) Z)

Z)

L

k 2,

Z)

j=0

Z)

k

(23.5)

We have Z)

_ -k2nak-IM z)

2n- I

Z)

(W, zPP(0 - P(WeW, Z)

$Z( i)

4bq, =)1

(23.6)

From Lemma 4.1 and (23.6),.and the fact that .t , is a linear combination of forms of the type (1.10), we see that the kernels of the second term on the right in (23.4) have no singularitites. Also, ak contains a factor hence the

operator Gpq can be applied to forms y such that pkE Lt p,q)(D). THEOREM 23.1. Let q > 0, and suppose y E Z(' p,q)(D) and 0 > 0 are such that

pay E L'(p, JD) and pe-'i2ap n y E L(p,q+I)(D). Then for all k > 0 + 1 the forms I3k = Gp, qy satisfy aflk =yin D, and PP -'13k E L p q _ 1(D ).

PROOF. Assume first that y E Zp, q(D ). Substitute z)y(') in the Martinelli-Bochner-Koppelman formula (1.1). Since akG, z) = 0 for E aD, ak is holomorphic in z on D and ay = 0, we get

y(z) = -Ip,q(D,

z) A Y) - aIp,q-,(D,

z)y).

(23.7)

We transform the first term on the right above by Lemma 1.4 to

IP,9(D' ataklJl Z) A -y) =

=

ID;«kG,

8Z

Z) A

z) A

1DklJ

z) A Y\J 1 A Up,9Gl Z)

A (kjtiqI + afµa) A A µq-1 A

(_i)f d[ak(, z) A

P, (n'_ P),

p Au' A

DP.n-p(az,

p! (n-p)!

Dp

As already observed, the integrals have no singularities, so it is legitimate to bring O. outside the integral sign. Now, since arClk(t, z) = 0 for E aD and k > 1, we see by Stokes' theorem that the second integral above vanishes.

134

VI. FORMS ORTHOGONAL TO HOLOMORPHIC FORMS

Hence (23.7) yields y = aGp Qy. In the general case, we apply this equality to the form y,(r) = y(rz) E ZP,q(D) and pass to the limit as r -> 1.

We proceed to estimate the solution thus obtained. From (23.6) and the definition of Gp Q, we have fDz

lPe-i(Z)I (Gpkqy)(z)dvZ = 0(1) JD;(flIdvfDt

Ip°(z)I

UP.9-11J, Z)I IaklJr Z)I

Jr

+lak-IIJ

z) 12P11

I

z) Izn+ 1

I()I lk4D(J Z)I IiI}dv_ 1 4,j /yQZl Z) 12n

PB-1IZ)I

+O(l)fD (Y(J)n

I

l

lak-IIJ1Z)I

-

Izn+l lltlq-1jdVz-

Here, as before, dv denotes the volume element, and the absolute value of a form is the sum of the absolute values of its coefficients. Next, using (1.4), Lemma 4.1, (23.5), (23.2), (23.3), the relation q9 Z) I = I

- z ) I = 0(l) 1 - ZI 3,

9

and the definition of µ9_, (see Lemma 1.4), we get

fD 10'-'(Z)l i(Gpkq-Y)(Z)IdVz =Dn00) f I()I Ipk()IdvD I I°(z)I t

Z)

Ik

Z I2n-1 I

Z

+ I $(Jq Z) (k+21

+0(1) t fD IA ()I

I

Z

I

I''

Hence, by the same argument as in the proof of Theorem 21.4,

fDIPP-'(Z)l (G,'qy)(z)dvz t = 0(1){ 1 +1Ds I)I IPk( )- I dvS[

+fp jA ()I t

An application of Lemma 21.2 finishes the proof of the theorem.

) (dvi I2n-3

§23. WEIGHTED FORMULA FOR SOLVING THE a-PROBLEM

135

We state without proof an application of Theorem 23.1.

Let D = {z: p(z) < 0} be a bounded domain in C" with C2 boundary. We denote by Na(D) (a > 0) the class of holomorphic functions F in D such that

fDI()I

°`-' 1n+IF(Z)IdQ2jz)

< o0

(the Nevanlinna-Dzhrbashyan class). Here dak is the k-dimensional LebesgueHausdorff measure, and ln+ I = max{In t, 0}, t > 0.

THEOREM 23.2. Let D be a strictly convex domain in C". Then an (n - 1)dimensional analytic subset M of D is the set of zeros of a function from N.(D ), a > 0, if and only if

f

M

IP(Z)Ia+Ida2n-2(Z)

- >21'.,f I< 001, f Mj

where the Mi are the irreducible components of M, and yj is the multiplicity of Mj in M.

CHAPTER VII

REPRESENTATION AND MULITPLICATION OF DISTRIBUTIONS IN HIGHER DIMENSIONS

§24. Harmonic representation of distributions

Recently, there has been increased interest in the problem of multiplication

of distributions, and there has been a lot of work on the problem. In the International Conference on Generalized Functions held in Moscow (Novem-

ber 1980), different approaches to solving this,problem were discussed in several papers. Here we present a method based on representing the distributions by harmonic functions.

Let W" be the space of the variables (x,,.. . , x,,, y) = (x, y), and let R+ ' = {(x, y) E W+', y > 0), then R" = ((x, y): y = 0). We shall say that a function f defined in W++' has finite order of growth as y -+ + 0, if for every compact set Min R" there exist c > 0 and m > 0 such that If(x, y) I< cy -I for x E M and 0 < y < 1. Let H denote the set of harmonic functions f in W++' having finite order of growth as y --+ + 0, and K the set of harmonic functions

in R+ ' which can be extended continuously to R' U R" and vanish on W. Then K C H. Let K(x, y) be the Poisson kernel for the half-space R+ '

K(x, y) =

:

r(((n+

>>>z 2)

I X IZ

+ y2)cn+>>,z

.

If T E E'(R" ), then the harmonic function T*(x, y) = T K(x - t, y) is called the Poisson representation of T.

TlHouHt 24.1. If T E E', then T* E H and represents T in the following sense:

lim f

F-.f0 y =E T''`(x, Y)4)(X)dx = (T9 (P), 137

9P E Oo(R").

VII. REPRESENTATION AND MULTIPLICATION OF DISTRIBUTIONS

138

Here T* extends continuously to R+ ' U (R" \supp T), and vanishes on R" \supp T, and T *(x, e) = 4(1 x I -"-') as I x I -+ oo. The Poisson representation of a derivative of T is the corresponding derivative of the Poisson representation of T. This theorem generalizes Theorem 14.1, since (see § 14) C" = R2 n

aT(Z) ly=, - aT(Z) ly=-, = T*(x1,...,x2,,_1, E). The proof of Theorem 24.1 is the same as that of. Theorem 14.1. We need only show that T* E H. But since T E E', it is bounded with respect to some seminorm sup

I

xEK,IIIII<m

K

supp T,

and so

I (T, K(x - t, y)) I k cjjK(x - t, y)11 < c, y-s,

x E K.

THEOREM 24.2. For each T E D'(R" ), there exists an f E H which extends continuously by 0 to R+' U (R" \supp T) and which is a harmonic representation of T, i.e. lim E-i

+O

y=E

f(x, y)gv(x)dx = (T, q)),

qp E 6 (R").

(24.1)

If f E H represents 0, then f E K. Every f E H represents some Tf E 6D ' via (24.1). Thus 61 '

H/K.

THEOREM 24.2 generalizes Theorem 14.2 to the case of an arbitrary R" and an arbitrary T E 6D'. Thus, for distributions defined on hyperplanes, it is more convenient to construct their representations by means of the Poisson kernel rather than the Martinelli-Bochner kernel. The latter can be used for distributions defined on smooth hypersurfaces in C". The first part of Theorem 24.2 has in fact been proved already (Theorem 14.2). We proceed to the proof of the second part. LEMMA 24.3. Let f E H. Then, for any bounded domain a C R", there exist an index p and an F E H which can be extended continuously to R++' U SZ, such that

f = a 2pF/ay 2P.

For n = 1, as ' ' ar lemma for holomorphic functions is due to Tillmann [1]. PROOF. Let 0 be the La lacian in R"', and A that in W. Given f harmonic in R + ' , consider

Fj(x, y) = fF(x, 7J)dn + fi(x).

139

§24. HARMONIC REPRESENTATION

Then o F,(x, y) = a f (x, l)/ay + 04(x ). Choose T E C°° such that AT = -a f (x, 1)18y in R" (this is possible since the Poisson equation can be solved in

W). Then AF, = 0, and aF, /a y = f. If f E H, then there exist c > 0 and m > 0 such that cy_m,

lf(x, A I

x E S2, 0 +0, then f(x, y) -- Tf E 6D' in the sense of (24.1) as y -), + 0.

Thus, polyharmonic functions also represent distributions in 6D', and in this case Tf = Tfo

§25. MULTIPLICATION OF DISTRIBUTIONS

141

§25. Multiplication of distributions and its properties

Using the harmonic representations, we can define the product of two distributions by the method of Ivanov [1], [2]. Let H* be the algebra of functions generated by H, and K* C H* the subset consisting of functions representing 0 in the sense of (24.1). We shall call the quotient H*/K* =6D* the space of hyperdistributions, and denote by f* the image of f e H* in G *,

i.e.f*=f+K*. The map A : 6D' -> 6D * defined as Tf -* f * (f a harmonic representation of T) is an imbedding, since K C K*. Hence we shall suppose GD' C 6D*. Let T. S E

6D', and let f and g be their harmonic representations. Then by the product T o S of T and S we shall mean the hyperdistribution (fg)* = fg + K* E 6D*. In the sequel, we shall be interested in the question as to when T o S is a distribution, in the sense that the limit (24.1) exists for fg and an arbitrary ( E 6D. Note that if this limit exists for fg, then it exists for any function in fg + K*, and coincides with the limit for fg. We give some properties of this product. 1. S o T is well-defined, i.e. the class fg + K* is independent of the choice of the representatives f and g.

PROOF. It suffices to show that, if f E K and g E H, then fg E K*. Let p E 6D. It is easy to show that f = yf, in a neighborhood of the support of p, where f, is a real-analytic function. Hence ftp E 6 for fixed y, and ftp -+ 0 in GD as y --> + 0, while g tends to S in 6D' as y - + 0; consequently, by a lemma in Shilov [11 (Chapter II, § 9.3),

f(x,e)p(x)g(x,e)dx= (S,0)=0,

lim

e-++0

R"

i.e. fg E K*.

2. supp(T o S) C supp T fl supp S. In particular, T o S = 0 if supp T n supp S = 0 . Here, the support of a hyperdistribution is defined as follows. We say that f * E 6D* vanishes in an open set f in R" if, for any qv E GD with support in 0, lim+o e-.+

y=e

f(x,y)p(x)dx=0,

fef*.

Now the support supp f* of f* is by definition the complement of the largest open set in which f* vanishes. It is easy to verify by means of a partition of unity that this notion is well-defined. PROOF OF PROPERTY 2. Let qv E GD with supp qp C R" \(supp T fl

supp s) _

0. We may suppose that, for example, supp q) fl supp T = 0 (by means of a

VII. REPRESENTATION AND MULTIPLICATION OF DISTRIBUTIONS

142

partition of unity it is always possible to pass to functions 9) with arbitrarily small support). Then f can be written in the form yf 1 in a neighborhood of supp qn, and the rest of the proof proceeds as for Property 1.

3. If T E 6D' is defined by a function T(x) E L p(loc), and S by S(x) in LQ(loc), 1/p + l/q = 1, then To S = T(x)L(x) E L'(loc). PROOF. It can be shown that, corresponding to S(x), there exists u(x, y) E H such that u(x, y) -* S(x) almost everywhere as y - + 0, and for each compact set M

IIu(x, Y) - S(x)II as y -4 + 0, where q

IIrIIM

=

f

-M+ 0

If IfIgdxJ

q> 1.

To do this, we must use:

(a) Theorem 1 of Stein [2], (Chapter III, §2.1), asserting that, for every f e LQ(R" ), its Poisson integral defines a harmonic function J(x, y) such that f(x, y) converges to f(x) almost everywhere and 11 f(x, y) - f(x)II - 0 as y --- + 0; and (b) the procedure in Bremermann [11, §5.9, replacing power-series expansions by expansions in harmonic polynomials. Since S(x) E L9(loc) and T(x) E LP(loc), we will have S(x)T(x) E L' (loc). Let u(x, y), v(x, y) and w(x, y) be the respective harmonic extensions to R+ ' with the properties mentioned above. Then u, v and w are harmonic represen-

tations of T(x), S(x) and T(x)S(x). Furthermore uv - w E K*. Indeed, let 4p E 61)1 with supp qQ C M. Then

(uv - w)yv dx

c fMjuv

- wIdx < c(Ijuv

fRfl

-

+11w -

but II uv - ST 1) M --*0 as y - +0, since II u - T II°f and 11v- S II %f tend too. 4. The derivatives of hyperdistributions are defined as follows: a

f*

a

_1

_ of

This definition makes sense, since af/axk E K* for f E K*. For the derivatives of the product of two distributions, we have the Leibniz rule a

aXk

(To S =

aT aXk

oS+ To as

axk

143

§25. MULTIPLICATION OF DISTRIBUTIONS

This follows immediately from the fact that if f represents T, then a f/axk represents aT/axk. 5. The product of distributions is compatible with multiplication by multipliers,

i.e.ifaEC°°(R") andTE6',then a°T=aT. PROOF. Since a E C°°, its harmonic representation g(x, y) E H extends continuously to R+ ' U R" (solution of the Dirichlet problem for the half-space). Hence g E C°°(R. ++' U R") (Brelot [1], Chapter I, §8), so that p(x)g(x, y) -+

(p(x)a(x) in '! as y -* +0, for any T E'D. On the other hand, f(x, y) -> T in 6D' as y --> + 0, so that, by the lemma already mentioned (Shilov [ 1 ], Chapter II, §9.3),

j

Y

_ _e

f(x,y)p(x)g(x,y)dx -' (T, a(p) - (aT, 97) e- +0

6. If f E H is a homogeneous function of degree p, then its boundary value as y -+ + 0, i.e. Tf , is also homogeneous of degree p.

limo f_,)p(x)dx TJ, (P(tX))- elimo f f(x,y)p(tx)dx= Indeed,

y-et

\t

= tp+n (limo

If T E d' is homogeneous of degree p, then it is not hard to see that T*(x, y) is a homogeneous function of degree p, since

T*( tx t y) y)

(ii, K (u -- tx ,

)) =

t

n

Tu K

t

- x ,Y

= tp (Tu, K(u - x, y)) = tPT*(x, y). By means of the general procedure for constructing a harmonic representation, it can be shown that, if T E 6 ' is homogeneous of degree p, then T admits a harmonic representation which is homogeneous of degree p. It follows that, if T, S E 6D' are homogeneous of degrees p and q respectively, and T o S E 6D', then T o S is homogeneous of degree p + q. 7. Let x' _ (x l , ... , x m ), x" _ (Xm + 1, ... , x n ), and T E GL'(Rm ), S E 6D "(W - m ). Let f(x', y) and g(x ", y) be harmonic representations of T and S in R++' and R+ m+ 1 respectively. Then

(f +Kj-(g+Kx..) Ch+K* where h is a harmonic representation of the tensor product T X S. Thus multiplication of distributions is compatible with the formation of products.

VII. REPRESENTATION AND MULTIPLICATION OF DISTRIBUTIONS

144

PROOF. It suffices to show that the boundary value of f (x', y)g(x", y) in the sense of (24.1) is T X S. Consider the integral

fg(x", y)cp(x)dx" = (x', E),

(p e 6D.

y=E

We have (x', e) E 6 (Rm) and 4 (x', e) -> (S, cp(x)) = J(x') in 6D(Rm) as J(x') as c -- > + 0 (Lemma 24.3), and

er -* + 0. Since J(x', e)

DX, (x', e) = fg(x", y)DX-p(x)dx we have, by the same lemma, D x' (x, e) o D,' (Ji(x')) as e -4 +0. Now we apply the lemma from Shilov [1] (Chapter II, §9.3) once again and conclude

f

f(x',

_

y)s(x,,,

.v)'T(X)dx

E

=

f

_ .f(X', F

y)aX'F+0 -- (Tx,, Ox'))

= (Tx,, (Sx, T(x))) = (T X S, T(x)). Thus the definition of product which we have given has several natural properties; in particular, we will not have counterexamples of the Itano kind with our definition. §26. Examples of products of distributions Consider the vector space WI of functions, generated by functions of the form

Ax, y) =

Pk(x y)

(lxi2+y2)

k-m+(n+1)/2'

k> ,0 , m>

1,

(26.1)

where Pk(x, y) is a homogeneous polynomial of degree k and 1mPk = 0. These functions are polyharmonic of order m; actually they are the Kelvin transforms of the Pk(x, y). We first elucidate what kind of distributions they represent.

LEM&A 26.1. Suppose f is a function of the form (26.1) lying in

k+1-2m>0.Then

I

Pk(X, Y)xadx (I

=0

I

for all monomials x° with Ilall < k + 1 - 2m. PROOF. Let

AY) =1X,=1

PJx9 Y)xadx I2 +

2)k-m+(n+ 1)/2 '

y

Hall o

k-m+(n+ I)/2

Pk() ado

dt

-

0.

i11=1

The improper integral above is convergent, since II a II < k + I - 2m. It is easy to see that

Pk(x, y)xadx

llall-k-- 1--2m

R" (1x12 +Y 2)k-m+("+ 0/2

LEMMA 26.2. Suppose an f of the form (26. l) lies in V and k + I -- 2 m Then

' -- P.V.

f

Pf(x, y)xadx

------

0.

k + I ~~ 2m,

(1x12

where IF

n/2T(s

22(k+I -:-m)(k

-

1/2)Ak+I-S-M

(Pk

+ I - s - m)tI'(k - m + (n + 1)/2)

(26.3)

PROOF. Consider the integral Pk (x, y) x 2dx

I(Y)

(IX12+y2)k-pn+(n+l)/2*

fXl 0 (functions of the Nevanlinna-Dzhrbashy class) it was proved by Dautov and Khenkin [1], [2].

In these papers, there is also a generalization of Theorem 23.2 to strictly pseudoconvex domains in manifolds. 151

152

HISTORICAL SURVEY

Chapter VII is devoted to the solution of Problem 13. The results of this chapter are due to Kytmanov [5], [6]. We note that, for n = 1, Theorem 24.1 is proved in Bremermann [ 1 ] _(p. 49). That part of Theorem 24.2 which proves that the boundary value of a harmonic function of class H is a distribution can be deduced from a theorem of Roitberg [1] which states that, if L is an elliptic operator of order 2m in a bounded domain 2 C R" with sufficiently smooth

boundary, then every solution f of the equation Lf = 0 with finite-order singularity on aft defines a certain distribution on M. Theorem 24.2 gives a method for constructing this distribution. A different method of multiplying higher-dimensional distribution by means of analytic representations has been proposed by Ivanov and Verzhbalovich [1]. We rote that, with this method as with ours, it would not be possible to have counterexamples like those of Itano M.

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BIBLIOGRAPHY

161

Barnet M. Weinstock 1. Continuous boundary values of analytic functions of several complex variables, Proc. Amer. Math. Soc. 21 (1969), 463-466. 2. An approximation theorem for 3-closed forms of type (n, n - 1), Proc. Amer. Math. Soc. 26 (1970), 625-628. 3*. Uniform approximation by solutions of elliptic equations, Proc. Amer. Math. Soc. 41 (1973), 513-517. John Wermer 1. An example concerning polynomial convexity, Math. Ann. 139 (1959), 147-150; addendum, ibid. 140 (1960), 322-323. A. P. Yuzhakov and V. P. Krivokolesko 1. Certain properties of linearly convex domains with smooth boundaries in C", Sibirsk. Mat. Zh. 12 (1971), 452-458; English transl. in Siberian Math. J. 12 (1971).

SUBJECT INDEX a-problem, 31, 86

Formula, of Martinelli-Bochner for exterior differential forms, ((1.1)), 5, 7,..., 87 Functions orthogonal to holomorphic functions, 37

Cousin data, 58, 59 Derivatives of distributions, 82, 84 Determinant, Levi's, 42 Distributions of class A, 79, 81, 82, 88 Domain, admissible, 62 linearly convex, 41, 68, 74, 75 in the sense of Martineau,

Kernel, reproducing, 72-74 , Cauchy, 70, 71 , Cauchy-Fantappi6, 7, 73 , of Martinelli-Bochner, 71, 77

43

Problem, first Cousin, for forms, 58 of Mittag-Leffler for forms, 57 Riemann's additive, 18 for forms, 26 Product of distributions, 2, 76, 82, 88

, with smooth boundary, 1, 4, 18, 22, 26, ... , 86, 88 , with boundary of class C"n,\, 4, 19, 20,241..., 73, 74 , strictly pseudoconvex, 2, 31,..., 87, 88

Representation, integral, of holomorphic functions, 2, 71, 88

Envelope of holomorphy of a compact set,

Martinelli-Bochner, of a distribution, 77, 78 analytic, of a distribution, 76, 84

47

Equations, tangential Cauchy-Riemann, 30, 85

Solution, weak, of the tangential Cauchy-

Form, a-closed, 2, 27, 46,..., 87 , o-exact, 2, 39,..., 74 , double, 6, 7

Riemann equations, 30, 69, 70

- , of the first Cousin problem, 58, 59

Levi, 2, 31, 49 orthogonal to holomorphic forms or functions, 1, 2, ... , 71, 72 Formula, of Cauchy-Fantappie, ((13.2)), 38,

Theorem, of F. and M. Riess, 68-70, 71 Runge's, 53, 86 , for forms (Theorem 9.1), 53,

...,86

72,73,86-88

163

INDEX OF SYMBOLS A, 82

Hl, 67

lzI, 3

ACC(S2), 4

, 3

Ao (aD), 39

H(K)7 47 int M, 3

An (D), 45

11(P,,)(D, 'Y)(z), 8

ar, 56 an , 79

AP(K), 4 A(fl), 4 BA, 82 BA, 82 Bp,n-1, 27 B(z°, r), 3 Cn, 3

I(P,e)(D,'Y)(z), 8

7±, 20

11111, 3

'Y+, 26 6, 79

L (P); 42

M, 3 Mr, 3

6+, 79 &-, 81 °k, 81 0o, 81

MCN, 3 O(aD), 37 01(3D), 68 02(aD), 68

Cm'>`(F), 3 Cm,q (P) `F), 3

P0, 11

P(x-1), 81 P(xnI xI -2n), 81

C (P) ,q (11)' 4

diam M, 3 dz, 3

P(S, z), 31

dzj, 3 dz[4, 3

r(z), 62 supp a, 4

aa, 4

t(S, z), 3

D, 43

Up,q(S,z), 7 Vq, 9

Dijj 3 )em), 5

g*f, 56

Z p,q (1l)' 4

i;,11 µ, 11 Q

Pr(z), 56 P(z, M), 3 cr(I), 9 a(I, k), 9

>', 3 ±1 18

Wp,q(w, c, z), 7

I'(S,z), 31

z, 3

W (w , S, z), 7

ABCDEFGHIJ -EB-89876543 165