VECTOR-VALUED HOLOMORPHIC FUNCTIONS ON THE COMPLEX BALL AND THE ANALYTIC RADON-NIKODYM PROPERTY

2000,20B(2):278-288

VECTOR-VALUED HOLOMORPHIC FUNCTIONS ON THE COMPLEX BALL AND THE ANALYTIC RADON-NIKODYM PROPERTY 1 t

Chen Zeqian ( ~ it.ft) Ouyang Caiheng ( it fa If ) Wuhan Institute of Physics and Mathematics, Chinese Academy of 'Sciences, Wuhan 430071, China Abstract

The complex Banach spaces X with values in which every bounded holomor-

phic function in the unit ball B of Cd(d

> 1) has boundary limits almost surely are exactly

the spaces with the analytic Radon-Nikodym property.The proof is based on inner Hardy martingales introduced here. The inner Hardy martingales are constructed in terms of inner functions in B and are reasonable discrete approximations for the image processes of the holomorphic Brownian motion under X -valued holomorphic functions in B.

Key words Complex Banach spaces, the analytic Radon-Nikodym property, inner functions, inner Hardy martingales 1991 MR Subject Classification 46B20,32F05,60G46

1

Introduction

A complex Banach space with values in which every bounded analytic function on the open unit disk has radial limits at almost all boundary points is said to have the analytic Radon-Nikodym property. A paper of Bukhvalov and Danilevich[5] examines this topic.The class of Banach spaces with the analytic Radon-Nikodym property is slightly larger than that with the Radon-Nikodym property.Indeed,it is shown in [5] that L 1 and more generally all Banach lattices not containing Co also verify the analytic Radon-Nikodym property. After their work,several characterizations of the analytic Radon-Nikodym property have been obtained ([2,4,10,11]),among which Edgar[10] first gives a martingale characterization of the analytic Radon-Nikodym property by means of analytic martingales.(The analytic martingales are defined by Davis,Garling and Tomczak[8] in connection with their study of complex uniform convexity.) Edgar's proof involves techniques not used in the proof of the martingale characterization of the Radon-Nikodym property.One of the items appearing in the proof is Brownian motion.On the other hand,a geometrical characterization of the analytic Radon-Nikodym property is presented in [11]. This paper deals with the characterization of a complex "Banach space X in which the following property holds:Every bounded X -holomorphic function on the unit ball B of C d ( d > 1) has radial limits at almost all boundary points.Evidently,this problem of the complex ball B 1 Received

(No:19771082)

May 4,1999.

Project supported by the National. Natural Science Foundation of China

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is not trivially a slice problem, although the slice method is useful that allows us to apply facts from the function theory of the disc D to questions in the complex ball B.Slice functions are not relevant in this problem.Indeed,if we apply the slice method to this case in terms of results in [5],we will encounter a question how to describe the measurable structure of the uncountable union of null sets. One main result of this paper is that the complex Banach spaces on which the above property holds are exactly the spaces with the analytic Radon-Nikodym property.The proof relies on another main result which asserts that every holomorphic image process of Brownian motion in B can be approximated by inner Hardy martingales introduced here.The construction of the inner Hardy martingales is dependent on inner functions in B. Note that,in the case of the disk D,Bourgain[3] observe the fact that the analytic image processes of Brownian motion in D has a discretization, to an analytic martingale can be followed directly from both the Ito's formula and the Cauchy-Riemann equation.That one involves the analytic martingales to study some problems on the disk D is natural[8,lO,11]. However, the existence of inner functions in B had been a long-standing open problem of Function Theory in the Complex Ball B.It was said to be the inner function conjecture[15,p403] and affirmatively proved by Aleksandrov[l] and Low[12]. Then,the fact that the inner Hardy martingales can be used to approximate the holomorphic image processes of the Brownian motion in B is no obvious.It seems to us that without the inner functions of B one could not prove the main result mentioned above (the details will be presented in Section 5).It is the authors' belief that the inner Hardy martingales could be used to study some other problems of the complex ball as have been done by the analytic martingales in the disk[3,S]. The remainder of the paper contains three sections.Section 2 presents the basic definitions and preliminaries.In Section 3,we introduce the inner Hardy martingales in complex Banach spaces and its some properties are exhibited.Section 4 is devoted to the connection between the inner Hardy martingales and Brownian motion in B. Finally,in Section 5,we shall prove the main result .

2

Definitions and Preliminaries We will make use of materials on Banach spaces and several complex variables. Throughout

this paper, (X, II II) denotes a complex Banach space,X* the dual space of X.C will denote the complex field,and c- will be the Cartesian product of d copies of C;here d is any positive integer> I.Equipped with the usual inner product <, > and the associated norm I I, c- is an d-dimensional Hilbert space whose open unit ball will be denoted by Bd,that is,the complex ball Bd

= {b E c- : Ibl

< 1},or simply by B when it seems unnecessary to mention the dimension

d.The boundary of B is the sphere S, and the normalized surface Lebesgue measure on S is denoted by G'. Also,P(b,s) = (1-lbI 2 )d/ ll - < b,s > 2d (b E B,s E S) is the invariant Poisson kernel in B = Bd. 1

Recall that F : B ~ X is said to be holomorphic if for each x* E X* , x* F is holornorphic in B.It is known (for example see [13]) that if F is holomorphic in this weak sense,then itIs

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holomorphic in the stronger sense that F is the sum of a power series

F(b) =

LL 00

xab

a

n=Olal=n

where X a E X, b E B and the series converges unifornly on every compact subset of B. The class of all such functions is denoted by H(B, X). If F : B

--+

X is any function,and 0 < r < 1,then F; denotes the dilated function defined

=

for Ibl < 1/r by Fr(b) F(rb). If 0 < p < oo,we obtain the Hardy class HP(B, X) of holomorphic functions F : B satisfying

--+

X

We shall denote by HOO(B,X) the space of X-valued bounded holomorphic functions on the complex ball B and its norm is given by IIFlloo =: sUPbEB IIF(b)ll. We say that F : B --+ X has radial limits almost surelyif for almost all s E S, lim r -+ 1 F(rs) exists in X. Proposition 2.1 Suppose
[z" F(b) I

< exp

(is

log [z" F(b + r s)Ida(

>0

s)) ::; exp (is log IIF(b + r s)lIda( s)) ,

consequently,

IIF(b)1I

=

sup Ix* F(b)1 IIx*lI=l

:s exp

( flog IIF(b+ rs)lIda(s)) .

Js

This shows that log IIFII is subharmonic in B. Thus,by the Jensen inequality,
u = P[h] for some h E LP(a) with Ilhllp = I p • Remark 2.1 It follows obviously from Proposition 2.2 that a function F E H(B,X) is in H(B, X) if and only if
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M-invariant,if F o'ljJ E M for every F E M and 'ljJ E Aut(B);an immediate consequence of the result is: Every H¢>(B, X) is M-invariant. Before making the statement of the next corollary,which happens to be useful in the

>

sequel,let us recall some notations.For a

1 and s E S,we let Da(s) be the set of all z E

c-

such that 11- < z, s > I < a(l - IzI2)/2. It is clear that Da(s) C B.lf F : B - t X and a > 1, the maximal function MaF : S - t [0,00] is defined by (MaF)(s) = sup{IIF(z)II : z E Da(s)},

=

and the radial maximal function MradF is defined on S by (MradF)(S) sUPo~r<11IF(rs)ll. An immediate consequence of Proposition 2.2 is (e.g., Theorem 5.6.5 in [15]) Corollary 2.1

> 1 corresponds a constant C(a) < 00 such that

To every a

llMoF1Pda S C(a)llFlI~ for every F E HP(B, X), 0 < P < 00. Remark 2.2 In the case of the unit disc,the above result was first observed by Bourgain [3].This is a surprising result since it holds for any complex Banach space. Now,let us examine some facts about K-limits as follows. We say that a function F : B - t X has K-limit z E X at 8 E S, and write (K - lim F)( 8) = z if the following is true:For every a > 1 and for every sequence (bn ) in D a (8) n B that converges to s, F(b n ) - t X in the norm of X.Also,for a, b E B,we define d(a, b) = 11- < a, b > 11/ 2 and Q(8,6) = {z E S : d(8, z) < b} for

E S,6 > O. The letter Q will always denote a "ball" Q(s,6). Occasionally,when the "center"s is of no particular importance,we shall write Qb in place of Q(8, b). 8

If / E L 1(S, X) then

Lemma 2.1

lim

b~O U for almost all

s E

(Q1 ) s

f

lQ(s,b)

IIf -

S.Hence

f(s)llda = 0

f

/(s) = lim (Q1 ) b~O a s

lQ(s,b)

fda

for almost all s E S. Proof Since

1 )

I a (Qs

f

lQ(s,b)

fda /

J(S)II ~

U

(Q1 ) s

f

lQ(s,b)

IIf -

f(s)llda,

the second assertion follows from the first statement.To prove the first statement, assume without loss of generality that / is separally valued.Let (x n ) be a countable dense subset of /(S).By Theorem 5.3.1 of [15],one has lim

b~O

U

(Q1 ) s

f

lQ(s,b)

Ilf - xnllda = Ilf(s) - xnll

for almost all s E S and for all n.For any s E S such that this holds for all n,one obtains limsup

b~O a

(Q1 ) s

f

1

JQ(s,b)

. 1 ) ~ hmsup -(Q b~O a s Q(s,b)

= 211/(8) - xnll

Ilf -

f(s)llda

II/ -

xnllda

+ IIX n -

/(s)11

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for all n.If e > 0 is given,a choice of n such that II/(s) - xnll < €/2 completes the proof. Lemma 2.2 If / E L 1(S,X) then (K -limP[f])(s) /(s) at every Lebesgue point s of

=

J.

Proof The proof is as the same as that of [15.Theorem 5.4.8] and omitted. Proposition 2.3 Let X be a complex Banach space, 1 ~ p ~ oo,and let F E HP(B, X). Then the following assertions are equivalent: (a) F has radial limits almost surely on S.

= Is

(b) There is / E LP(S, X) such that F(b) P(b,s)/(s)d{}"(s), for all bE B. (c) F has K -limits at almost every point of S. Moreover,if 0

<

P

Is IIFr

<

00,

F E HP(B, X) and /(s) =: lim r --. l Fr(s) exists for almost all

=

/IIPd{}" O. Proof (a) =>(b). Let / : S -+ X be ·/(s) = lim r --. l F(rs). Then Corollary 2.1 implies that / E LP(S, X).For any z" E X*, z" F E HP(B) and hence s E S,then lim r --. l

x*F(b)

-

=

L

P(b,z)x*!(z)da(z)

= z"

(L

P(b,z)!(z)da(z))

Is

for all b E B. Thus, F(b) = P(b, s)/(s)d{}"(s) for all b E B. (b) => (c) follows from Lemma 2.2,while (c)=>(a) is obvious. The proof of the last assertion is as the same as that of Theorem 5.6.6 of [15] with the help of Corollary 2.1.The proof is complete. Remark 2.3 A more detail of boundary properties of Hardy and BMOA classes of vector-valued holomorphic functions on the complex ball is presented in [6].

3

Inner Hardy Martingales Let U be an open set in a complex Banach space X.An upper-semicontinuous function

¢; : U

-+

[-00, (0) is said to be plurisubharmonic if for every z, y

U,one has

4J( x) <

1 2

7<

E X and

{x

+ ,8y : /,81

~

I} C

4J( X + ei9 y )d8 /211".

We denote by PSH(U) the space of all such functions. We are interested here in the case where the function ¢; is Lipschitz continuous.We denote

by LIP(U) the set of such functions ¢; on U,that is, I¢;(x) - ¢;(y)1 ~ Kllx - yll for some K > 0 and for all x,y E U.We shall write PSH 1(U) for PSH(U)nLIP(U). Definition 3.1([9]) Let (0, f, P) be a probability space and let (fn)n~O be an increasing" sequence of sub-rr-fields of J=.Suppose U is an open set in a complex Banach space X and Mn : -+ U is an f n -measurable random variable,for all n == 0,1,2,···. The sequence (Mn)n is said to be an P S H -martingale in U with respect to (0, F, (fn)n, P) if M n is Bochner

°

=

integrable for each n 0,1,2··· and,for every ¢; E PSH 1(U),the sequence (¢;(Mn ) ) is a realvalued submartingale with respect to (0, F, (J=n)n, P),that is,

for every n

= 0,1,2,· .. and all E E J='n.

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Remark 3.1

283

It is known that L 1-b ounded analytic and Hardy martingales are both

PSH-martingales.As noted in [9],PSH-martingales are martingales in the usual sense. Definition 3.2

A function H : B

~

C is called an inner function in B if H E HOO(B)

=

is not constant such that its radial boundary values h verifying Ih(s)1 1 for almost all s E S. Remark 3.2 The existence of inner functions in B was provided by Aleksandrov [1] and Low [12] (see also [16]).This fact has been used to resolve several open problems of Function Theory in the Complex Ball B[1,16]. The inner functions also playa crucial role in this paper.We will use it to define inner Hardy martingales. . Lemma 3.1[16,Theorem 1.3] .If h is the boundary value of an inner function H in B with

H(O) = O,then the image measure dcr 0 h- 1 of a under h is the circle measure,that is,

for all I E L 1(dB/21r). Thus,h: S ~ T is a preserving-measure map from (S, a) to (T, dB/21r), where T C :

It I =

= {t

E

I}. Let SN be the sample space which is understood with the normalized Lebesgue product measure P = aN,where N is the set of all positive integers. If w E SN,we write w

=

(81,82,·· ·).We introduce the natural filtration (En) on SN so that En is the a-field generated

by the first n coordinates 81,···, 8 n of w = (81,82,···) E SN for n > O,and Eo = {0,-SN}. Definition 3.3 Let U be an open set in a complex Banach space X. An inner Hardy

martingale in U is a sequence (Mn)n~o of random variables M n : SN ~ U of the form:Mo = Xo and n

Mn(w) =

L: Ik(Sl'···' Sk-1)h(Sk), k=l

n

2:: 1,

for all w = (81,82,·· ·),where h is the boundary values of an inner function H in B with H(O) 0,11 = Xl E U and Ik : Sk-1 ~ X (k ~ 2) are all E k _ 1-measurable so that

=

for all 1f31 ~ 1 and n = 1, 2, . . .. . Lemma 3.2 Let U be an open set in a complex Banach space.Then,every Bochner integrable inner Hardy martingale in U is a P SH -martingale in U with respect to (SN, E, (En )n, aN). Proof Let (Mn)n be a Bochner integrable inner Hardy martingale in U. Suppose ¢ E P S H 1 (U) and E E En. Then, by Lemma 3.1 one has that

L~(Mn+l)dcrN L[1 =

2 ,..

~(Mn + f n+l eill )dO/ 21r] do"

:::::

L~(Mn)dcrN.

This completes the proof. Let U be an open set in a complex Banach space X and let ¢ : U ~ [-00, 00) be an upper-sernicontinuous function that is bounded above on bounded subsets of U.Set ¢o = ¢ and define for each integer n > O,the function

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then (¢n)n~O decreases pointwise to the largest plurisubharmonic function ¢ on U dominated by ¢ as shown in [4]. We shall call ¢ the plurisubharmornic envelope of ¢.([11]) Theorem 3.1

in LIP(U) such that

Let U be an open set in a complex Banach space and let ¢ be a function

¢ ,the plurisubharmonic envelope of ¢,is not identically -oo.Then for

each n ~ 0 one has ¢n(x) = inf E[¢(Mn)] for every x E U,where the infimum is taken over all Bochner integrable inner Hardy martingales (Mk)k=O in U and M o = z. Proof First note that all ¢n are also Lipschitz continuous on U with the same Lipschitz constant as ¢. Then we conclude the proof by induction on n. For n = 0, we have only M o = x, so

¢O(x) = ¢(x)

= E[¢(Mo)].

Suppose the formula is known for n - 1.Fix x E U and e > (l.Choose v E X so that {x

11'\ < 1} C

U and


~

L


+ f3v

:

f.

Let M 1 have the uniform distribution on the set {x + h(s)v : s E S}.Now for a fixed t E T, where T = {t E C : It I = 1},we can choose an inner Hardy martingale (Mk)k=l with values in U and M 1 = x + tv such that

By the above note,we can choose p > 0 so that if

It- t'l ~ p and t' E T

l¢n-l(X + tv) - ¢n-l(X

and I¢(z) - ¢(z')\ < e if liz - z'1/

~

then

+ t'v)1 < €

p.We then get for such t' E T,

¢n-l(X + t'v) > ¢n-l(X + tv) - e > E[¢(Mn)] - 2€ > E[¢(M~)] - 3€ where Mf = x + t'v, M~ = Mk(k = 2, ... , n).Since T is compact,thus we can choose measurably for each s E S,an inner Hardy martingale (M k)k=l with values in U and M1(s) = x + h(s)v such that

¢n-l(X + h(s)v)

~

E[¢(Mn)] - 3€, a.e. s E S.

Therefore we have obtained (Mk)k=l conditionally on M1.Putting them together,we get (Mk )k=O with values in U and M o = z so that

Since

€

is arbitrary,one has

where the infimum is over all inner Hardy martingales with values in U and M o = x.(Indeed,the details of the proof show that the coefficient of (Mk ) can be chosen to be step functions.) On the other hand,if (Mk )k=O is an inner Hardy martingale with values in U and M o = z , by E[¢(Mn)] ~ ¢n(x) and hence completes the proof.

Lemma 3.1 one easily concludes that

4

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The Approximation' of Brownian Motion in B Let (Wt)tE[O,oo) be the holomorphic Brownian motion in B starting at 0 (for example see

[7]).It is defined on a probability space (n,:F, P) and Wo == 0 almost surely.Let 0 < r < 1, and define a stopping time Tr : n -+ [0,(0) by Tr(W) == inf{t > 0 : IWt(w)1 ~ r}. Then Tr < 00 almost surely, and W T r is uniformly distributed on the sphere S; == {z E Cd : Izl == r },that is,

1 n

for all

f

f(Wrr)dP

=

1 s;

f(s)dO"r

=

r f(rs)do-(s)

is

E L 1 (a r ),where a; is the normalizied surface Lebesgue measure on Sr.Now set rk

==

1 - 2- k , and denote Trk == Tk, then W Tk -+ W Too ==: limt--+oo W t almost surely.We shall use the fact that if Brownian motion (Wtb)tE[O,oo) starts at b E B,then the distribution of W:oo on S will have density PCb, .). The close connection between the radial limits of holomorphic functions F in B and the convergence of the process (F(WTk))k is well known in the finite dimensional case (for example see [7]).In the sequel, we will prove that these connections also hold in infinite-dimensional complex Banach spaces. Proposition 4.1 Let X be a complex Banach space and let F E Hl(B, X). Then for every ¢ E PSHl(X), ¢ martingale.

0

F is plurisubharmonic on B. Consequently,(F(WTk))k is a PSH-

Proof Since F is holomorphic, by Theorem 35.7 of [13] ¢ 0 F is plurisubharmonic for every ¢ E PSH1(X),and hence subharmonic.Then by the scalar case,(¢ 0 F(WTk))k is a submartin-

gale.The proof is complete. Proposition 4.2

Let X be a complex Banach s,pace and let F E Hl(B, X). Then the

function F has radial limits almost surely if and only if (F(WTk))k converges almost surely. Proof If I(s) == limr--+l F(rs) exists for almost all s E S,then I E Ll(S,X) by Corollary 2.1 and F(b) == PCb, s)da( s) for every b E B so that F(WTk) == E[/(WToo) I:FTk ]. Such a closed martingale converges almost surely.

Is

Conversely,suppose (F(WTk))k converges almost surely to an X-valued random variable G. For each z" E X*, by the scalar case (x* F(WTk))k converges almost surely to Ix. (WToo)' where Ix· denotes the boundary limit of x.* F. Hence z" G == Ix. (WToo) almost surely. Consequently, G is weakly measurable for a-field generated by W Too. Evidently G has essentially separable range. Then G is measurable for the a-field generated by W Too and hence there exists a measurable

f :S

-+

X such that G == f(WToo). Moreover, f E Ll(S, X) by Corollary 2.1 and,for every

x* E X* one has

z" F(b) =

is

P(b, s)x* f(s)dO"(s) = z"

(is

P(b, s)f(S)dO"(S))

Is

for all b E B.Consequently, F(b) == PCb, s)/(s)da(s) for all b E B.This implies that F has radial limits f by Proposition 2.3 and completes the proof. Theorem 4.1

Let X be a complex Banach space and let F E H(B, X) and €k

~

O.Then

there exists an inner Hardy martingale (Mn)~=o with Mo == F(O) and integers nl < n2 < ... < nk < ... so that E[II[Mnk - Mnk_J - [F(WTk) - F(WTk_1)]11J < €k for all k == 1,2,···. Proof By the arguments presented in [1o],it suffices to prove the following claim: If F is X-valued holomorphic in a neighborhood of B,then for any 0 < € < 1 and any b E B,there

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exists a bounded inner Hardy martingale (Mk=O)k=O with M o =:= F(b) and a random variable H distributed on S with the density P(b, .)such that EIIMn - F(H)II < €. Since F is holomorphic in a neighborhood of B,by the invariant Poisson integral formula for F it follows that there is M < 00 so that IIF(b 1) - F(b2 )11 ~ Mlb1 - b21 for all b1, b2 E B. The direct sum X Ea c- will be given the norm II(x,z)11 = max(llxll, Izl) for convenience.We define two subsets of X Ea Cd :

u = {(x,z) The function ¢ : U

-+

EX Ea Cd:

Izi < 1},

Q = {(F(s),s): s

E S}.

[0,00) defined by ¢(x, z) = dist((x, z), Q) is Lipschitz continuous. By

Lemma 3.2, ¢n defined above can be obtained as

¢n(x, z) = inf E[¢(Mn)] where the infimum is taken over all inner Hardy martingales (Mj)j=o with values in U and Mo (x, z). Since F is holomorphic, by the slice technique the function b -+ ¢(F(b), b) is plurisubharmonic in B and so ¢(F(b), b) ~ O. Therefore,there exists n such that ¢n(F(b), b) <

=

€'

[€/8(1

=:

+ M)]4d.

Consequently,there 'exists an inner Hardy martingale (Fj)j=o in U so that E[~(Fn)] <

€', Fo = (F(b),b).

We consider separately the components of Fj : Fj = Mj Ea Gj,

Mj E X,Gj E B.

Both (Mj)j=o and (Gj)j=o are inner Hardy martingales.We have

E[dist(Mn, F(S))] < €', Mo = F(b);

E[dist(G n, S)] < €', Go = b.

Now let H, = G n + Wt and r(w) = inf{t > 0 : IHt(w)1 2: 1}. Then H(w) Hr(w)(w) is a random variable and distributed on S with the density P(b, .). Since for every s E S,z E B, Is - zl2 ::; 211- < z, s > I,one has that

=

Consequently we have

The remaider is as the same as that of [10] and omitted.

5

The main results Theorem 5.1

Let X be a complex Banach space.Then the following assertions are equiv-

alent:

(1) X has the analytic Radon-Nikodym property. (2) All L1-bounded inner Hardy martingales in X converge almost surely. (3) Every function in H1(B, X) has radial limits almost surely. (4) For all 1 ::; p ~ 00, every function in HP(B,X) has K-limits almost surely.

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(5)

287

Every function in HOO(B,X) has radial limits almost surely.

Proof By a result due to Bu and Schachermayer [4] that say: If X has the analytic Radon-Nikodym property,then every PSH-martingale in X converges almost surely, it follows that (1) => (2) beca:use of that L 1 -bounded inner Hardy martingales are all PSH-martingale by Lemma 3.2.The fact that (3) => (4) is a consequence of Proposition 2.3,while the fact that (4) => (5) is obvious.To complete the proof,it remains to prove that (2) => (3) and (5) => (1). (2) => (3).Let F E H 1(B, X).Fix martingale (M n ) so that

for all k

€

> O.By Theorem 4.1, we can choose an inner

H~rdy

= 1,2,··· and some n1 < n2 < ... < nk < .... Then

L 00

E[I/MnJe - F(WTJe)//]::;

E[//[Mnj - M nj_ 1] - [F(WTj) - F(WTj_1)]II]

j=k+1

L 00

<

€/2j = €/2

k

j=k+l

for all k

= 1, 2, .....Consequently,

for all k; and the inner Hardy martingale (Mn)n is L 1 -bounded .In particular, (Mn)n is a bounded PSH-martingale by Lemma 3.2.Therefore,(Mn)n converges to some M E £1 almost surely by the assumption (2).Thus,for large k, k' we have E[IIMk - Mk' II] < € and consequently,

Since e is arbitrary,( F(WTJe)) converges almost surely by the Chebyshev's inequality.An appeal to Proposition 4.2 yields that F has radial limits almost surely. (5) => (l).We shall use the inner function in B to prove this implication.Indeed,let H be an inner function in B with H(O) = O· and let h : S --+ T be the boundary values of H.Since h is a measure-preserving map from (S, a) to (T, dB/21r) by Lemma 3.1, there exist Borel sets M 1 C S and M 2 C T of measure one,and an invertible measure-preserving transformation I : M 1 --+ M 2 such that 0"( s E S : l( s) :/: h( s)) = O(See [14,Corollary 12 of p272]). Without loss of generality we set h = I on MI. Let f E HOO(D) with radial limits tp on T.Then f 0 H E HOO(B) with radiallimits,say ¢,on S.We claim that tp =

f ..

0

¢k

is the boundary values of

H(b) = lim

for all b E B.Therefore ¢ =

k

tp

rP(b, S)cpk

is 0

0

fk

0

H on S.Then

h(s)da(s) =

h almost surely and so

ip

rP(b, s)cp

is

= ¢

0

0

h(s)da(s)

h- 1 almost surely.

288

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~ATHEMATICA SCIENTIA

Vo1.20 Ser.B

Suppose now FE HOO(D,X).Then FoH E HOO(B,X).By the assumption (5),FoH has radial limits ,say 'ljJ,on S.Given x* E X*, then x* F E HOO(D) and x* F 0 H has radial limits x*'ljJ.By the previous claim,x* F has radial limits x*'ljJ 0 h-1.Then

for every z E D,where p(z, t) is the Poisson kernel in the disk and m = d8/21r. Consequently,

F(z)

=

£

p(z, t)1jJ 0 h- 1(t)dA(t)

for all zED and so F has radial limits ([5]).This completes the proof. References 1 Aleksandrov A B. The existence of inner functions in the ball. Mat Sb,1982,118:147-163;English transl: Math USSR Sb, 1983,46:147-159 2 Blasco O. Boundary values of functions in vector-valued Hardy spaces and geometry on Banach spaces. J Funct Anal, 1988,78 :346-364 3 ·Bourgain J. Vector-valued singular integrals and the H1-BMO duality. In: Chao J A, Woyczynski W A Eds. Probability theory and harmonic analysis. New York:Dekker,1986.1-19 4 Bu S, Schachermayer W. Approximation of Jensen measures by image measures under holomorphic functions andapplications.Trans Amer Math Soc, 1992,331: 585-608 5 Bukhvalov A V, Danilevich A A. Boundary properties of analytic and harmonic functions with values in Banach spaces. Mat Zametki, 1982,31 :203-214 6 Chen Z, Ouyang C. Mobius invariant vector-valued BMOA and the H1-BMOA duality of the complex ball. Preprint 7 Chen Z Q, Durret R, Ma G. Holomorphic diffusions and boundary behavior of harmonic functions. AlU1 Probabi, 1997,25(3):1103-1134 8 Davis W J, Garling J H, Tomczak-Jaegerniann N. The complex convexity of complex quasi-normed linear spaces. J Funct Anal, 1984,55:110-150 9 Edgar G A. Complex martingale convergence. LNM 1166. Springer-Verlag, 1985. 38-59 10 Edgar G A. Analytic martingale convergence. J Funct Anal, 1986,69(1):268-280 11 Ghoussoub N, Lindenstrauss J, Maurey B. Analytic martingales and Plurisubharmonic barriers in complex Banach spaces. Contemporary Math, 1989,85:111-130 12 Low E. A construction of inner functions on the unit ball of CPo Invent Math, 1982,67:294-298 13 Mujica J. Complex analysis in Banach spaces. North-Holland, 1986 14 Royden H L. Real analysis. New York: The Macmillan Company, 1963 15 Rudin W. Ftmction theory in the unit ball of C". Springer-Verlag, 1980 16 Rudin W. New constructions of functions holomorphic in the unit ball of cn. CBMS Regional Conference Series in Math 63. Amer Math Soc, Providence, 1986