ハーディー空間に関する研究

九州大学学術情報リポジトリ

Kyushu University Institutional Repository

ハーディー空間に関する研究

真次, 康夫

https://doi.org/10.11501/3063846

出版情報：Kyushu University, 1992, 博士（理学）, 論文博士 バージョン：

権利関係：

STUDIES ON IIARVY SPACES

YASUO MATSUGU

Department of Mathematics Faculty of Science Shinshu University

Matsumoto, Nagano Pref., Japan

CONTENTS

Preface

Part 1 The Strict Inclusion Relation between the Spaces

1-1 <p <U) on the Open Unit Disc in C

(Jour. Fac. Sci. Shinshu Univ. 18(1983), 1-10.)

1

Part 2

The Inclusion Relation between the Bergman Spaces ₁₂ AP<U), the Hardy Spaces HP<U) and the Nevanlinna

Space N<U) on the Open Unit Disc in C

(Jour. Fac. Sci. Shinshu Univ. 18(1983), 11-25.) Part 3

Determining Sets for N<Bn> and HP<Bn>

(Jour. Fac. Sci. Shinshu Uni v. 18 ( 1983), 55-62.)

28

Part 4 The Zero Sets of Functions in the Bergman Spaces 38 and the Hardy Spaces

(Jour. Fac. Sci. Shinshu Univ. 18(1983), 63-73.)

Part 5 Determining Sets for HP<Bn) II

(Jour. Fac. Sci. Shinshu Univ. 18(1983), 27-29.) Part 6 On a Conjecture of Z.Jianzhong

(Bull. Austral. Math. Soc. 45(1992), 163-170.) References

51

55

64

PREFACE

Let n�1 be an integer. Let H(Bn) denote the space of all holomorphic functions in the open unit ball Bn of the n-dimensional Euclidean space Cn. Every function f in H(B n ) has its nonnegative order Ord f of the Nevanlinna characteristic Tr(f), O�r<1. All the functions in H(Bn) are ranged into uncountable classes according to their order. The set 00(Bn) of those functions in H(Bn) ^which

are of zero order is thus a very 'thin' subclass of the space H(B ), n nevertheless it is a very significant object of study in the analytic function theory in Bn. The Hardy spaces and the

Nevanlinna space N(Bn) are extremely important subspaces of o0(Bn).

The inclusion relations between these subspaces are as follows:

if O<p<q<(X).

The origin of the theory of Hardy spaces (of course in the case of the dimension n=1) is commonly associated with the publications of G.H.Hardy [5] and F.Riesz [17] in 1915 and 1923, respectively.

A.Zygmund [29], I.I.Privalov [16], K.lloffman [9], P.L.Duren [2], P.Koosis [13] and J.D.Garnett [3] are basic references for the theory of Hardy spaces on the unit disc in the complex plane. Around 1970, the theory of Ilardy spaces in the multidimensional case started to develop abruptly. W.Rudin [18], [21], [22], E.M.Stein [26] and S.V.Shvedenko [24] contain many information on this field.

The present thesis is devoted to mainly on the zero sets of the Hardy spaces HP(Bn), n�1, O<p�(X)' and the inclusion relation between them. We call a nonnegative real-valued function � defined on [O,(X))

a moduLus Junction if it is a nondecreasing and nonconstant function

such that �(t) = �(et) is a convex function on (-oo,oo) . For each modulus function �. we can define a Ilardy-Orlicz space H�(Bn). The Hardy spaces Hp(Bn), O<p<oo, and the Ncvanlinna space N(Bn) are special Hardy-Orlicz spaces. In this thesis, we also treat general Hardy-Orlicz spaces H (B ) .

cp n

It is well known that (in the case of the dimension n=1) all the spaces Hp(B1), O<p�oo, and the Nevanlinna space N(B1) admit the same zero sets which are completely characterized by the Blaschke condition. When n�2, the situation is considerably more complicated.

In [14], R.O.Kujala proved the following:

To

(1) The zero sets of N(B ) satisfy a generalized Blaschke

n

condition ( P) .

(2) The zero sets of H (Bn) 00 have a property (Q) of the Blaschke

( 3)

each (a) (b)

condition type.

There exists a non-constant function /EN(B ) whose zero set

n Z(f) ^{is a} determining set for Hoo ( Bn) , that vanishes on Z(f), then g_:::O in B lienee

N(Bn) are different from those of these three results, Kujala Is the necessary condition ( p)

Is the necessary condition ( Q) n

of H 00 ( B ) .

n

raised the in ( 1 ) also in ( 2) also

is, if gEH (Bn) ⁰⁰

the zero sets of

following problems:

sufficient ?

sufficient ? (c) Can determining sets for H (Bn) 00 or N(Bn) be reasonably

characterized ?

The problem (a) was solved affirmatively by G.M.Ilenkin [8] and H.Skoda [25], independently. The zero sets of the Nevanlinna space

N(B ) are thus characterized completely by the (generalized) Blaschke n

condition (P). The second problem (b) was solved negatively by the

author in [15]. One of our purposes l1ere ls to discuss the last problem (c). This thesis consists of six parts. Two parts (Part 3 and Part 5) of them are concerned with the problem (c). In Part 3, we characterize completely the determining sets for N(Bn), by using the Ilenkin-Skoda theorem. (See Theorem 1 in §3.3.) The

characterization of the determining sets for Hp(B ) , O<p�w, is much n

more complicated. We shall only show the existence of various determining sets and non-determining sets for Hp(Bn), O<p�w. (See Theorem 2 � Theorem 5 in §3.4 and §3.5.) One of these results is as follows:

Theorem. Let n � 3 be an integer. Then there exists a Junction f _E n Hp(Bn) satisfying the foLLouing tuo conditions:

O<p<w

(i) The zero set of f is not a determining set for H00(8n).

(ii) The zero-divisor vf does not equaL v9

In Part 5, we prove that the above theorem is still valid when n=2.

In [20], W.Rudin proved that, if n�2 and O<p<q<oo, then the zero sets of Hp(B ) are different from those of Hq(B ). C.Horowitz [10]

n n

and J.II.Shapiro [23] proved analogous results regarding the zero sets of the Bergman spaces Ap(B ) , n�l. O<p<oo. In Part 1, we shall

n

establish the following !-dimensional version of the Rudin's theorem:

Theorem. _(Theorem 1 in §1.1) Assume � and � are tuo modu�us

Junctions. If lim �(t)/�(t) t�oo

such that 2/ � H�(B1).

then there exists an f _E H�(B1)

By applying this theorem and the Rudin's theorem, we shall describe the strict inclusion relation between the Hardy spaces Hp(Bn)' n�1. O<p�oo. For example, Theorem 3 in §1.3 is as follows:

U _Hq(Bn)

�

HP(Bn)

�

n _Hq(Bn)

p<q<oo O<q<p

( O<p<oo) ,

For a given modulus function �. the (generalized) Bergman space A�(Bn) is defined as

A (8 ) = {/EH(8n):

� n

f

8 � ( I f I ) dA. < oo} ,

n

where _A. is the Lebesgue measure on en. If �(t) = tP , O<p<oo, then A (8 ) are the ordinary Bergman spaces AP(8n). In Part 2, we prove

� n

the following generalization of the Shapiro's theorem:

Theorem. _(Theorem 1 in §2.1) Let � and� be moduLus Junctions.

Assume that lim �(t)/�(t+1) = oo and that there exists a reaL number t�oo

T such that �(T) > 0 and sup �(t+1)/�(t) < oo t':1:.T

an f _e A�(81) �ith the JoLLo�ing property:

Then there exists

If m is a positive integer, bE H (81), g E H(81), g�O, and 00

h = (fm+b)g, then some constant muLtipLe of h faiLs to be in A� (81),

'lit

�here �m(t) = �(t/m).

By making use of this theorem, we shall study the inclusion relation between Ap(81), O<p<oo, N(81) and Hp(81), O<p�oo. (See §2.4 and §2.5.)

In Part 4, we first prove two other generalizations of the Shapiro's theorem , to "weighted" Bergman spaces on the one hand, and to the case of the dimension n _� 2 on the other hand. (See Theorem 1 and Theorem 2 in §4.3.) By using them, we shall amplify the Rudin's theorem and the Horowitz-Shapiro's theorem. For example, Theorem 3 in §4.4 is as follows:

Theorem.

(a) For any p E(O,oo) and any integer n � 1,

v( U _Aq(8n))

�

v(AP(8n))

�

v( n _{Aq(8 )).}

p<q<oo O<q<p n

(b) For any p E(O,oo) and any integer n � 2,

v( U Hq(B

)) �

v(HP(B ))

�

^v( n Hq(B )).

p<q<oo n n O<q<p n

Here v denotes the divisor map from H(Bn) into D + (Bn) (see §5.2).

Each Hardy-Orlicz space H (B ) contains a subclass II + (B ) of

cp n _cp n

the Smirnov type. In 1990, Z.Jianzhong conjectured that, for given

two modulus functions cp and �. if and only if

Hcp(B1) = H�(B1). In Part 6, we prove that this conjecture is true not only on the unit disc B1 of C but also on the unit ball Bn ^of

en for any dimension n � 1.

The author wishes to express his sincere gratitude to Professor Joji Kajiwara for his valuable suggestions.

Part ₁

The Strict Inclusion Re lation between the Spaces Bcp(U) on the Open Unit Disc U in C

- 1 -

1.1. Introduction.

Let n�l be an integer. Let H(B ) denote the space of all n

holomorphic functions in the open unit ball Bn of the complex n-dimensional Euclidean space en. Let �:(-oo,oo) � [O,oo) be a non- decreasing convex function, not identically 0, and let H�(Bn) be the class of all /EH(B ) whose growth is restricted by the requirement

n

sup

!88

^�(loglf(ru) l)da(u) < oo, O<r<l n

where

8

Bn is the boundary of Bn and a is the Euclidean volume element

on the unit sphere 8Bn in en normalized so that the volume of the sphere is 1. If �(x)= max{O,x}, then H (B ) is called to be the

� n

NevanLinna space N(Bn). If �(x)=epx, O<p<w, then H�(Bn) are called to

be the Hardy spaces HP(Bn). By H00(Bn) we shall denote the space of all bounded holomorphic functions in B .

n

In [20], W.Rudin proved the following theorem:

Theorem A (Rudin [20], p.58). ^Fix n�2. Assume that � and� are nonconstant, nondecreasing, nonnegative convex Junctions defined on (-oo,oo), and that

lim �(t)/�(t) = w.

�00

Then there exists an !EH�(Bn) uith the fo��ouing propert y:

If beH (Bn), gEH(Bn), g�O, and 00

h = (f+b)g,

then some constant muLtipLe of h faiLs to be in H�(Bn).

In the case n=l, this theorem is not valid. Indeed, if �=epx, O<p<w, and �=(2+p2x2)epx, then Theorem ^A implies that the zero sets of functions in HP(B1) differ from the zero sets of functions in Hq(B1), for any q>p. But this is false when n=l.

The main purpose of Part 1 is to prove the following analogue of Theorem A in the case of n=1:

(The open unit disc in C and the unit circle will be denoted by U and J, in place of 81 and 881, respectively.)

Theorem 1. Assume that� and� are as in Theorem A. Then there exists an fEH�(U) such that 2f�H�(U).

Applying Theorem A and Theorem 1, in §1.3 we shall describe the strict inclusion relation between the Hardy spaces HP(B ), O<p�w,

n and the Nevanlinna space N(Bn).

1.2. Proof of Theorem 1 ⁱⁿ §1.1.

We need the lemma which was used to prove the Rudin's theorem (Theorem A in §1.1) in [20].

Lemma (Rudin [20], pp.59-60) . Suppose

(a) � is a finite positive measure on a set Q;

(b) v is a reaL measurabLe Junction on ^n, �ith o�v<1 a.e., uhose essentiaL supremum is 1;

(c) A is a continuous nondecreasing reaL function on [O,w) ,

�ith A(O)=O and A(t)�w as t�w;

(d) O<o<oo.

Then there exist constants ckE(O,w), for k=1,2,3, ... , such that fnA(ckv )d� = k ^o.

These ck aLso satisfy

uhenever 1�1<1.

If O<t<w and if Yk=Yk(t) is the set of aLL xEQ at uhich

lim

k-.oo fy k A(okvk)d�

⁼ b.

that

Proof of Tl1eorem 1.

Without loss of generality, we cun assume

(1) �(t)

= o if t�o.

j

J=1,2,3, ...

Choose a se quence {X }

of nonempty connected open subsets of

T

^{so that}

(2) XJ c

{ei9er: O<S< � ^J

and XjnXka¢ if J*k. For each ja1,2,3, _... , pick

�JeXJ.

Define

( ZEC)

for each

j

^and

Then

D = {zEC: lz-�1=�

}.

and

(3) (4)

s1(U)

⁼ D,

s1(u1)

=

1,

max

ls1· (z) I

^{= max}

ls1(z) I

⁼

1,

zeT zeD

lsJ(z}l _<

1

^{for zeD,}

Z*uJ.

Moreover, the following inequalities hold:

.z(l+r} lsJ(z) 1

I

^{:>. lsJ(rz)}

I

^{:>. l}

s

^J(

z) I

00

for

O<r<l,

^ZE

i=1 U

^{Xi. In} fact, fix rE(0,1). _Put

( 5)

Then

( 6)

V(T)={lJEC; _{Re LJ=}

2}.

l+r ^Hence

min

IV(z) l

_{= l;r.}

ZET

(X)

u

eie1

Let ZE i=

u X

^{1 1, . •} Then, ^by ( 2) ' ^{we can} write Z= j for some

with ^{- rr rr}

2<01<2.

^{A simple} calculation shows that

( 7)

(

8)

2 1+T +2Tcos 2

OJ

<

IV(z) I =

2

+²c^os

e1

^1•

and

O<r<1. (5), (6)

^and

(7)

g^ive

(4).

Since t-.(X) lim •(t)/�(t)= •, there are numbers

t1>

3j such that

�(log(1+t)) ^> J3�(log(l+t)) if t> t ..

eJ

We now apply the lemma, for each positive integer J j, ^with (J,o)

in place of

(n,�),

^{and with}

Then O<cx _J .<1.

vj(z) ^c

l

s

3

(z) I,

A(

t) = �(log(1+t)), o1 = 21-2,

C( _• =

J max Is . ( z)

I .

ZET'XJ J

The lemma shows that there exist positive numbers

a1aok

(where kj is a sufficiently large positive integer) such that, setting J

( 9)

( ZEC) we have

(10) !T

�(log(1+1F11

))d(l

= ₂

; ^f

rr �(log(1+1fJ'ei8)

i))d8

⁼ 21-2,

-rc

(11) IF3(z) I

<

2-j ^on T'XJ ^{and for}

lzl<

1- ( 12)

f

y ^. � ^{(log ( 1} + IF J I

) ) da·

^>

J ,

-2

J

where Yj= {zeT: IF1(z)

l>tj}.

(13)

By (11),

_{Yjc Xj.}

Dy (8) and

(12),

fy

j �(log(1+1F J

.l))da

^> j.

- 5 -

J .-1

.we now define (14)

f(z) ⁼ :E ^F1(z)

j=1 ( ZEU) .

The series converges uniformly on compact subsets of U, ^by (11).

Hence /EH(U).

To prove that /E/lcp(U), for N=1,2,3, ... , define (15) HN (^{z) �} IF1 (z)+ ... + FN^(z) I

(16) H ( z) ^{= (X)} _L I FJ ( z) I

j=1 (zET).

Since the sets X. J are disjoint, (11) implies that

1 (X)

H(z) � {

It follows from (1) that

in r'3�1xJ

in XJ.

(X)

(zEC),

cp(log }f(z)) � { ^{0 in} T' U J=1 XJ

cp(log(1+ IF _J .(z) I))

Hence (10) ^implies

in xj.

(17) fr cp^(log H)da ^{� ()0} L 2J-2c 3 -¹ⁿ2 ^< 4.

j=1

Since F1+ ... + FN is a holomorphic function inC, log HN ^{is sub-}

harmonic in C. for each N, and so is cp(log liN), because cp is convex

and nondecreasing. Moreover, HN(z)�H(z) for zeT, by (15) and (16).

It follows from (17) that (18)

for 0<r<1. If we fix r and let N �oo, HN(rz) �lf(rz) I uniformly on T.

Hence (18) gives

Thus /E/lcp ( U) .

IT

cp(loglf(rz) I )do(z) � 4 (O<r<l).

rJ (1

9

⁾

We turn to proving that 2/�ll�(U). Fix jE{1,2,3, ...

}

and choose so that

o <rJ <1 ^an

d

where IIFJIIoo=max IFJ(z) I= aJ.

F

_{or zEYJ, by (1 4),}

_(9),

_(4), (11) and (19), zeT

I 2: F. ( rJz) I i=1 ^1,

� I FJ ( rJz) I - ^:E IF . ( rJz) I i=I=J ^1,

1+r ^k

�

(�)

2 JIF (z) 1-(1-2-J) J 1+r k

= IF.(z)l-{1-(

�

_{2 ) J}IF .(z)I-1+2-J}

J J

1+r. ^{k. .}

� IFJ(z) 1-{1-(

y

^{) 3}11FJ11oo-1+}

²

^-3

> IFJ(z) 1-1.

Since IFJ(z) l>tJ>3j�3 for zeYJ,

1

2

/(rJz) I > IFJ(z) 1+1 for zeY .. It follows from (13) that

J

fy

�(logl2f(rJz) l )da(z) > j.

J Thus

so that,

(j=1,2,3, ... ),

sup

J

�(logl2f(rz) I )da(z) ⁼ oo.

O<r<1 T

This means 21�/li{I(U). The proof is complete.

1.3. The strict inclusion relation between the Hardy spaces

HP<O)

_n ^an

d

the Nevanlinna space

N<O ).

_n

By Theorem A an

d

Theorem 1, we obtain the following

Theorem 2. Let n�1 be an integer. Assume that cp and �are

- 7 -

nonconstant, nondecreasing, nonnegative convex Junctions defined on (-co,co), and that

lim l/J( t) /cp( t) = co, t�co

Then there exists an fEH (B ) such that some constant muLtipLe cp n

of f faiLs to be in H"'(Bn).

Remark 1. If l/1 satisfies the growth condition

lim sup "'(t+l)/"'(t) < co, t�co

then H"'(Bn) is closed under scalar multiplication. (See Rudin [20], p.58.) In that case, the conclusion of Theorem 2 is simply

H"'(Bn)

�

^Hcp(Bn).

Now we apply Theorem 2 to the description of the strict inclusion relation between the spaces Hp(B ), O<p�co, and the space N(B ).

n n

First we note that

Hco(Bn) c Hq(Bn) c HP(Bn) c N(Bn) if O<p<q<co. For each pE(O,co), we define

Then

Theorem 3.

Hp-(Bn)

�

^HP(Bn)

�

^Hp+(Bn)

Proof. (cf. Rudin [ 20], p.59, (c).) ( i) Put

(O<p<co),

(O<p<co),

cp ( t) (-co< t<co) ,

"'(t)

(t�O) ( t<O) .

Then cp and "' satisfy the assumptions in Theorem 2. ^{Moreover, "'}

satisfies the condition

lim l/l(t+1)/l/l(t) t-+oo

It follows from 1\emark 1 that

HI/I(Bn)

�

H�(Bn) = HP(Bn).

Since Hp-(Bn)c HI/I(Bn), this implies

( i i) Put

Hp-(Bn)

�

^HP(Bn).

� ( t) l/l(t)

{

-1 pt t e

Then Thoerem 2 and Remark 1 imply that

(t�p-1) (t<p-1), ( -oo< t<oo) ,

HP(Bn) = HI/I(Bn)

�

H�(Bn) c Hp+(Bn).

Remark 2. In the case n=l, some outer functions give another proof of Theorem 3. Let f be a positive measurable function on T

such that log fEL1(T). Define

it -1

n

^{e + z it}

Qf(z) = exp

{

⁽2

n

)

f_n

eit_ z log f(e )dt} ( ZEU) .

Then Qf(z) is called the outer Junction with respect to the function

J.

^We note the following theorem:

Theorem B (See e.g. Rudin [19], Theorem 17.16.). Fix pE(O,oo].

Let J be a positive measurabLe Junction on T such that log fEL1(T).

Then QfEHP(U) if and onLy if fELP(T).

Then

Now fix pE(O,oo). Put

f(eit)

{

t-1/p(-log t)-2/P

1

l t l

^-

l

^/p

(O<t<e -1)

(te[-n,n]'(O,e )),

-1

(te[-rc,rc]).

and

/ELP(T), f� U Lq(J), p<q<oo

g�LP(T), ge n _Lq�T), O<q<p

(Sec Ilardy-Littlewood-Polya [6],

§6.1.)

It follows from Theorem D that

Theorem 4.

Proof. _With

01eH (U), p

Q f$.HP ( U), g

Qff$. U Hq(U) p<q<oo

Q g _f$.. O<q<p n Hq(U)

<p ( t)

exp ( t2)

{ ^if t�o

1

exp ( t ³⁾

\{l(t) _{= {}

if t<O, if t�o

Hp- ( U) , Hp+ ( U) .

1 if t<O,

Theorem 2 establishes the existence of an /EH�(Bn) such that

lienee

for some constant _c. Note that

H00(Bn) _c H._� �(B)

n

_c H (B) _{<p n c} O<p<oo n _{HP(B ).} n

Remark _3. In the case

n=1,

as well as in Theorem 3, _some outer

functions give another proof of Theorem 4. _Put

(O< t<e -1)

1 ( tE[-n,n],(O,e-1) ) .

Then /E O<p<oo n Lp(J) ',Loo(T) _{and log} /EL1(T) . (See

[6]. §6.1.)

It follows

from Theorem B that

Q/E

n Hp(U)'H00(U).

O<p<oo Theorem 5.

Proof. Put

cp ( t) max{O,t}

exp

( /t)

t/t(t)

= {

e

( t�1)

(

t

<

1) .

Then cp and

�

satisfy the assumptions in Theorem

2.

In addition,

�

satisfies the growth condition

Hence

lim

t/t(t+1)/�(t)

t

^�oo

1

^<

00

U HP(Bn)

c

H,"(Bn ) � _{H (B)}

⁼

_N(Bn).

O<p<oo ^'f' cp

n

Remark 4. In the case

n=1,

a simple function gives another proof of Theorem 5. Put

Then /EN

( U) ,

f (

z) =

exp

( �: �)

If 1=1

* a. e., and

loglf(O) I

= 1

^{> 0}

( ZEU)

^•

* -rr

Here f denotes the radial limits of/. (See Rudin

[19], §17.19.)

If /E

U Hp(U),

then O<p<oo

1

^rr

·t

loglf(O) I _�

(2rr)-!

log

l

f^*(e t,

) ldt.

-Jt

(See

[19],

Theorem

17.17.)

Hence we conclude that

f� U HP(U).

O<p<oo

Part 2

The Inclusion Relation between the Bergman Spaces AP<U>,

the Hardy Spaces HP<U>

and the Nevanlinna Space N<U>

on the Open Unit Disc in C

2.1. Introduction.

By U and T we shall denote the open unit disc in the complex plane C and the unit circle, respectively. The space of all

holomorphic functions in U will be denoted by H(U). Let �: (-m,m) �

[O,m) be a nondecreasing convex function, not identically 0, and let H�(U) (resp. A�(U)) be the class of all /EH(U) whose growth are restricted by the requirement

1 n iS

sup (2n:)-

f

cp(loglf(re ) I )dS < m

O<r<1 ^-n:

-1 1 n iS

(resp. _7t:

f f

cp(loglf(re ) l)rdrdS < m ).

0 _-n:

If �(x)=max{_O,x}, Hcp(U) is said to be the NevanLinna space N(U) and

A�(U) will be denoted by BN(U). If �(x)=epx. O<p<m, then H�(U) (resp.

Acp(U)) are said to be the Hardy spaces Hp(U) (resp. the Bergman spaces AP(U)). By H00(U) we shall denote the space of all bounded holomorphic functions in U.

In [23], J.II.Shapiro proved the following theorem:

Theorem C ([23], Theorem 2.1.). Assume cp and � are strictLy positive, convex, increasing, unbounded Junctions defined on (-m,m), and that

sup cp(t+1)/cp(t) < m, sup �(t+1)/�(t) < m

-m<t<m -m<t<m

lim cp(t) = 0, lim �(t) = 0,

t�-m t�-m

lim �(t)/cp(t) = m, t�m

Then there exists an /EAcp(U) uith the JoLLouing property:

then

If n is a positive integer, bEH (U), gEH(U), gm

�

O, and

h4A� (U), n

uhere

h = (fn+b)g,

- 13 -

(In fact, Shapiro proved this theorem more generally on weighted Bergman spaces.)

Applying Theorem 1 in Part 1, we studied there the inclusion relation between the Hardy spaces Hp(U), O<p��. The main purpose of Part 2 is to study the inclusion relation between tl1e Dergman spaces Ap(U), O<p<�, the Nevanlinna space N(U) and the Hardy spaces Hp(U), O<p��. To do so, we need the following generalization of Theorem C:

Theorem 1. Let � and � be nonconstant, nondecreasing, non- negative, convex functions on (-�,�). Assume that

lim �(t)/�(t+1) = �.

t-+�

and that there exists a number t0e(-�.�) suoh that �(t0)>0 and

sup �(t+1)/�(t) < �.

t�t0

Then there exists an /EA�(U) uith the !o��ouing property:

If n is a positive integer, beH (U), geH(U), g�

�

^{O, and}

h = (fn+b)g,

then some constant muLtip�e of h fai�s to be in A� (U).

n

2.2. Preliminaries.

It is easily shown that

H�(U) c Hq(U) c HP(U) c N(U), A�(U) c Aq(U) c AP(U) c BN(U)

� �

if O<p<q<�. Here we write A (U)=Il (U). For each pe(o,�), we define

Then

Hp-(U) U Hq(U), Hp+(U)= n Hq(U),

p<q<� O<q<p

Ap-(U) U _{Aq(U), Ap+(U)=} n _Aq(U).

p<q<� O<q<p

Hp-(U) c HP(U) c Hp+(U), Ap-(U) c AP(U) c Ap+(U).

Let /EH(U). Take a point aEU. Assume /�0 in U. Then a power series

f(z) =

�

^ck(z-a)k

k=m

converges in some neighborhood of a and represents I in this neigh- borhood. Here c � 0. The integer

m

v1(a) = m � o

is called the zero muLtipLicity of I ^at a. The integer-valued function v1 defined in U is called the zero-divisor of f.

Let � be a nonnegative integer-valued function defined in U.

Then � is called a positive divisor on U if and only if it is locally the zero-divisor of some holomorphic function, that is, for each

point aEU there exist a connected neighborhood V ^of a and a

holomorphic function I ⁱⁿ V such that /�0 and �=v1 in V.

We denote by D (U) + the set of all positive divisors on U. Then

we have the divisor map v from H(U)* into D+(U) defined by letting v(f) for I ⁱⁿ H(U) * be v1. Here, for any subspace X of H(U) we write

X = {/EX: /�0 * in U}.

We recall that �ED (U) + satisfies the BLaschke condition if and

only if

L �(z)(l-lzl) < m,

zEU

The set of positive divisors on U which satisfy the Blaschke condition will be denoted by D0. The following classical theorem will be used in §2.5:

Theorem _D (See e.g. Duren [2], §2.2.). For any pe(O,oo),

The following is an immediate consequence of Theorem 1:

Theorem 2. Assume that � and � are as in Theorem 1. In addition, assume that � satisfies the condition

Then

Hence

lim sup �(t+1)/�(t)<oo.

t-.oo

2.2. Proof of Theorem 1 in §2.1.

Our proof is a modification of the Shapiro's one of Theorem C.

(cf. Shapiro [23], pp.248-251.)

Step 1. Without of loss of generality, we can assume that

( 1) � ( t) 0 if t�o.

In fact, when �(0)>0, we put ( 2)

�(t)-�(a) cpa(t) = {

a

( t>a)

( t�o) .

Then cpa has the same properties as � does. In addition, cpa satisfies (1). Because of (2), cp-cpa is bounded, hence

For t'2:.0, define

A�(U)=Acp (U).

a

( 3) ct>(t) �(log t), ct>0( t) = �(log t+1),

'l'(t) �(log t).

Then �a is a continuous nondecreasing nonnegative function on [O,�) and cfl0(t)-.oo as

( 4)

t-.oo. By (1), cfl0(0)=0. Since

lim 'l'(t)/<IJ0(t) = oo t-+oo

lim �(t)/�(t+1)=oo, t-+oo

Put

}J ^{= sup} cp(t+l)/cp(t).

t^�t0

Since cp is nondecreasing, it follows from (3) that

( 5) �(s+t) ^� }J(�(s)+�(t))

where s0^=exp(^{t0). And} l<H<oo, by the hypothesis.

Step 2. By ^� we shall denote the Lebesgue measure on C=R2, so normalized that A.(U)=1. We now apply Hudin's lemma (in

§1.2),

for each positive integer k, with (U,A.) in place of (Q,g), ^{and with}

V(Z) ⁼ lzl, A(t) ⁼ 11>0(t),

c5 = (k2H)-l.

Then the following holds:

Lemma 1. There exist sequences {ckn}n=l,2,3, ^... of rea� numbe^rs such that

(a)

(c) n-too lim o kn yn = o uhenever IYI<l:

Q) • .

(d) lim!

n 11>0( lcknznl ^{)dA. = (k2ff)-l} for ^eac^h t>O.

n-too { ^lc^knz l>t}

Lemma 2. There exist four sequences {tk), {ak), {rk) and {pk) of rea� numbers, and one sequence {nk} of integers uith

s0 <

t1

^{< t}

2

_< ^{t3 < . . . , lim} tk = oo,

k�oo

0 < ^al < ^a2 < ^a3 < . ^{.. , lim ak Q)}

k^-too

0 < nl ^{< n2 <} n3 ^{< . .. ' lim} nk ^{= oo,}

k�oo 0 ^< r1 ^<

pl <

r2

_<

p^{2 < .}. ^{. ' lim} _k�oo rk ⁼ _k�oo ^lim pk ^{= 1'}

- 17 -

such _that if uk(z)= akz k n and

the foLLowing conditions hoLd:

(a)

(b)

( 6) ^(c)

{d) (e)

k-1 tk � _{4 :L: a}

j=1 J and _lJI( t) !ciJ0 ( t) > kH if J _{u Cll 0 ( I u k I ) d;..} ^:: _(k2H)-1;

J _R k _{� 0 ( I u k I )} dA. > ( 2k21f) -1;

luk(z) I _{� tk} t/ _lzl�rk;

luk(z) I ^s.: I u k-1 ( z) I Is if r1 S.:lzls.:pk_1.

t�tk;

Proof. We prove the lemma by induction. Choose any positive integer n1 and any positive numbers t1, a1, r1, p1, with s0<t1<a1, and O<r1<p1<1. Suppose k�2, and suppose the five sequences have been successfully chosen for all indices less than or equal to k-1. By

(4), there exists a positive number tk such that

tk ^> tk-1' tk ^> k, tk

�(t)/�0(t) ^> kH for

k-l

� 4 _2: ^a .,

j=1 _J t � tk.

By Le�na 1, there exists a positive integer nk with nk_1<nk' such that, letting

(7)

and

( 8)

( 9)

Put

nk nk-1

ak ^> tk' ak ^> ak-1' akpk-1 ^� ak-1r1 /S,

fu

¢0( l^akznkl)dA = (k2HJ-1•

Then pk_1<rk<l, by (7). Because of ^{{8) a}n^d (9), _{there exists a}

positive number pk ^with rk<pk<l such that

flrk<lzl�pk} �o<lakznkl)d� ^> (2k2H)-l.

This completes the proof of the lemma.

Step 3. We now define

f(z) k=1 _:E uk(z) ^{( ZEU) •}

The series converges uniformly on compact subsets of U, by (6-e).

Hence IE/I ( U) .

Lemma _3.

(10) (11)

Proof. By (12)

By (6-e), (13) (14)

Ill � Slukl/4 + 5luk+11/4 Ill ^� lukl/2

(6-a) and k-1

:E lu.fl s;:

j=l

co

:l: lu .1

j=k+1 J

co

:l: lu .1

j=k+1 J

( 6-d)' lukl/4

s;: 5luk+11/4

� lukl/4

on on

on

on

on

{ r k� I z I � p k + 1 } . Rk.

{lzl�rk}.

{r1s;:lzls;:pk+1},

{ r1 �I z I _�pk}.

(10) and (11) follow from (12), (13) ^and (14).

Lemma _4.

Proof.

f U <p (log

(t

_{I ) d).. = f} u c1> C If I ) d)..

!{ I z I �r } ^tP ( I I I ) dA. + ^:l: !{ I I

} tP ( I I _{I ) dA..}

1 k=1 T'k< z �T'k+1

Fix k€{1,2,3, ... }. ^By (10),

f{rk<lzl�rk+l} �� 1/l)dA � flrk<lzl�rk+l} �(SJukJ/4 +5Juk+ll/4}dA.

Put

- 19 -

{zEC: rk<lzl�rk+l' Sluk+l(z) 1/4 �s0}, {zeC: rk<lzl�rk+l' Sluk+l(z) 1/4 <s0}.

By (6-d),

Sluk(z) 1/4 � luk(z)

I

� tk ^> s0 It follows from (5) that

JE1�(5lukl/4+5luk+ll/4)dA � HJE1�(5lukl/4)dA + HJE1�(5Iuk+ll/4)dA H!E1�(loglukl+log(5/4))dA + H/E1�(logluk+ll+log(5/4))dA

� H!E1cp(loglukl+1)d;... + ff!E1cp(logluk+11+1)d;...

= H/E ct>0( lukl)d;... + H!E ¢0( luk+11)d;....·

1 1

On the other hand, by (6-d),

Thus

JE2�(51Ukl/4+51uk+ll/4)dA � JE2�(51uki/4+s0)dA

� JE �(5lukl/4+tk)d;... ^� JE �(9lukl/4)d;...

2 2 .

!E cp(loglukl+ log(9/4))d;... � !E cp(loglukl+ l)d;...

2 2

J E tt> 0

( I

u k I ) d;... . 2

! {rk<lzl�rk+1} ¢ (

If I

) d;...

� H

f

E cl> 1 0 (

I

u k

I

) d;... ⁺

N f

^{E lb} 1 0 (

I

u k + 1

I

) d;... +

f

E 2 ct> 0 (

I

u k

I

) d;...

�

Hfu

¢0( lukl)d;... +

Nfu

�0( luk+11)d;....

It follows from

(6-b)

that

Hence

f

{rk<lzl�rk+l} �( lfl)d;... ^� k-2 + (k+1)-2.

co

fu

�(loglfl)dA � J{lzl�r1}

�(

lfl)dA + r

{k-2+(k+l)-2}

< co k=l

This means /EA�(U).

Step 5. Let n be a positive integer, bell ^Q)

(U), gEI/(U)

• , and

h ^c (fn+b)g.

Put

ex =

8 ^{a sup}

I

b (

z) I .

ZEU

Then O�B<c:o, Since l^o

g

l^gl is subharmonic in U,

(15) -c:o < ex � (2n)-1J nl^o

gl

^g(r

e

i 9)

Ide

^{< Q)}

Choose a positive number -n o so that

(16) _lo

g a

^{+ ex - n}l^og 4 ^>

o.

Lemma

5 ^•

a h � A 1/1

( U) ^. n Proof. ^Define

( t�o) .

Then

(17) � Q) L

JR

�n(

lohl)dA

k=l k

(18)

Q) P n

= k=1 rk

L f

k 2rdr (2n)-1J_� �n(loh(rei9) l)dO. ^H

Fix rE(rk,pk]. By Jensen's convexity theorem and (15),

rc n

(2rcl-1! -n _'I' n (

loh(re£0)

I )dO = (2rc)-l

f '/t (loglah(re£9) I

)d9 -n n

TC

�

1/1

ⁿ _{((2n)-1J logl}

a

h(r

e

^£9)

IdS)

-rc

-n -n

�'It

ⁿ (log o + � +(2rcJ-1! rc logf(Jn+b)(re£9)

IdS).

-n

Since lim k-+c:o tk= ^c:o, t^here exists a positive integer K such that

(19) if k � K.

By (11), (19) and (6-d),

lfn+bl � lfln-6 > (lukl/2)n-(tk/4)n � ( luki/2)n-(luki/4)n � ( iuki/4)n for k':l:.K and zERk. lienee, for k':l:.K and rE(rk,pk],

(20) (2rc)-1J

lt ^log! (fn+b) (rei.EJ)

ldEJ

-rc

1 ^lt i9

�

(2rc)- ! n

l^og

lu

^k

(

^re

) I de -

nlog 4.·

-rc

By

(18), (20)

and (16), for k'L.K and rE(rk,pk],

(2n)-1J

TC '!In( lch(rei.9)

I )de

-rc

TC

� �n(log o + � - nlog 4 + (2n)-1J nlogluk(rei9) ld9)

-rc TC

�

�n((2n)-1J

nlogluk(rei.9)

I de)

-rc TC

c

t/t((2rc)-1J

logluk(rei.9)

Ide) -rc

n

= l[l(log(ae _k)

J

⁼ �[!(log uk( r)

J

⁼

'i'(uk(

r)

J

1

^{TC i9}

( 2rc) - !

'!'

( I

^{u k ( re}

) I ) d

^{9 .} -n

It follows from (17) _that

co pk -1 lt 9

fu '!'n(

lchl)d� ':l:. _L

f

^2rdr(2n)

f

'!'( luk(rei) _l)d9

k=K rk -n

=

co

2: !

^{R tJ1}

( I

^{u k}

I

⁾

d;..

^. k=K k

By (6-a), (6-c) and (6-d),

(k=1,2,3, ...

)

^.

Thus

OJ

fu tJln(

^lc

h l

⁾

d

;.. � 2:

(2k)-1

= oo.

k=K

This means chftAl/1

(U).

The proof of Theorem 1 is now complete.

n

2.4. The inclusion relation between the spaces AP<U>.

Theorem 3. For any pE(O,w)

v(Ap-(U)*)

�

^v(AP(u)*)

�

v(Ap+(U)*).

Consequent�y,

(O<p<w).

Proof(cf.Shapiro [23] ,Corollary 2.2; Horowitz [10] ,Theorem 4.6.).

( 1) Theorem 2 with

<p ( t) ept (-w< t<w),

tept ( t':l:O) 1/t(t) {

0 ( t<O) ,

implies that

v(Ap-(U)*) c v (AI/I ( U) ) *

�

v(A (U)*) ^<p v(AP(U) *).

( i i ) Theorem 2 with

-1 pt

t e (t':l:p-1)

<p ( t) {

( t<p-1),

pe

1/t(t) ept (-w< t<w),

implies that

v(AP(u)*) v(AI/I(U) ) *

�

v(A<p(U) ) ^* c v (A p+ ( U) ) * . Theorem 4 .

Hw(U)

�

^{n AP(U).}

O<p<w

Proof. This is an immediate consequence of the following two facts:

(i) HP(U) c AP(U) (O<p<w),

(ii) H00(U)

�

^{n HP(U).}

O<p<oo

The latter is just the case of the dimension n=l of Part l,Theorem 4.

Theorem 5.

v( _U AP(u)*)

�

v(BN(U)*).

O<p<w ConsequentLy,

U AP(U)

�

^BN(U).

O<p<w Proof. Put

cp ( t) ⁼ max{O,t} (-w<t<w) ,

exp ( /t) _(t�l) t/l(t) ⁼ {

e ( t<l) .

Then <p ^and 1/J satisfy the assumptions in Theorem 2. ^lienee v( _U AP(u)*) _c v(AI/J(U)*)

�

v(A<p(U)*) ₌ v(BN(U)*).

O<p<w

2.5. The inclusion relation between the spaces AP(U), IIP(U) and N<U).

Theorem ( 1)

( i 1) (1i1) Proof.

so that,

6 ^• Suppose O<p<w.

liP ( _U)

�

^{AP ( U) ,}

1/p-( U)

�

^{Ap-( U) ,}

1/p+ ( U)

�

^{Ap+(U) .}

Then ZJe have

Choose q with p<q<w. ^Then

D0 ⁼ v(Hw(u)*) _c v(Aq(u)*) _c v(AP(u)*).

On the other hand, by Theorem 3,

v(Aq(u)*) _c v(AP-(u)*) � v(AP(u)*).

Hence

D0

�

V(Ap(U) *) . It follows from Theorem C that

v(HP(u)*)

�

v(AP(u)*).

Since Hp(U) c Ap(U), this implies (1). The same arguments prove ( i i ) and (iii ) .

But

Theorem 7. For any pe(O,oo) and any qe(O,oo), AP(U)

'\

^Hq(U).

Proof. ^If AP(U)cHq(U), then v(AP(u)*) _c v(Hq(u)*)

Do.

this is impossible.

Theorem 8 ^• Suppose 0<2p<q<oo. Then Hp(U)

�

^Aq(U).

Proof. ^For zEU, cE(-oo,oo), define

I (z) = (2IT)-1J ^IT 11-zeit,-1-c dt.

C -IT

If c<O, ^then I is bounded in U. If c>O, then there exists a positive

c

constant He such that

( ZEU) . (See Rudin [21], Proposition 1.4.10.)

Choose a ^with O<a<p-1-2q-1. ^Put b=p-1-a. Then 0 < 2q-1 < b < p-1

Define

Then, for re(0,1),

-IT (2IT)-1J IT

Since -ap<O,

Hence /EHP ( U) . -IT

f(z) (1-z) ^-b ( zEU) .

If( reit) I Pdt = (2IT) -1! ^IT 11-reit ,-bp dt -IT

l1-reitl-1+ap dt = I (r).

-ap

sup O<r<1 ^I

(r) < oo -ap

We turn to proving that fqAq(U). Put y=bq-1. Then y>l, since

2q-1<b. For rE(0,1)

( 2rc) -1 f TC ^{It (} rei t) ^{I q d t (2rc) -1} f ^TC 11-reit ^{1-bq dt}

-rc Since y>1,

Theorem

9.

-rc

U ^HP(U)

�

_U AP(U).

O<p<oo O<p<oo

Proof.

This follows from the fact

v( U HP(u)*) = D0

�

_{v( U} AP(u)*).

O<p<oo O<p<oo

Theorem

10.

(O<p<oo),

Proof.

This follows from the fact

Co�ollary.

U ^AP(U)

<\

N(U).

O<p<oo

Theorem

11.

Proof.

_Define

N(U)

<\=

f( z) exp ( 11+z) ( zeU).

-z

!Y(r).

Then fEN(U). (See Rudin [19], §17.19.) A simple computation shows lim (1-lzl )21Pif(z) I

lzl-+1 If !EAP(U), O<p<oo, ^then

lim (1-lz1)21Pif(z) I = 0.

lzl-+1

(O<p<oo),

(See l(uclin [ 21], Theorem 7. 2. 5.) lienee f�

Corollary.

(O<p<co).

Part 3

Determining Sets for N<U ) n and IIP<O ) n

3.1. Introduction.

Let n�2 be an integer. Let N(B ) denote the Nevanlinna space n

on the open unit ball B of the complex n-dimensional Euclidean n

space Cn. Let Hp(Bn), O<p�oo, denote the Hardy spaces on Bn. H00(8n) is the space of all bounded holomorphlc functions in B .

n

In [14], R.O.Kujala proposed three problems. One of them is on

the complete chracterization of the zero sets of functions in N(Bn)' by the Blaschke condition. This problem was solved affirmatively by G.M.Henkin [8] and H. Skoda [25], independently. (See below Theorem E

in §3.2.) Secondly Kujala asked whether a certain necessary condition for the zero sets in H (B 00 ) (which is easily obtained through the

n

Jensen Formula) is also sufficient. This problem was solved

negatively in [15]. In this Part 3 we shall study the Kulala's last

problem. He asked ([14], p.260):

Can determining sets (or divisors) for H (B 00 ) or N(B ) be

n n

reasonably characterized ?

We shall consider when zero sets (of holomorphic functions) in Bn are determining sets for Hp(Bn), O<p�oo, or N(Bn). By using the Henkin-Skoda theorem (Thoerem E in §3.2), we can characterize completely the determining sets for N(B ) (Theorem 1 in §3.1). The

n

characterization of the determining sets for Hp(B ) , O<p�oo, n is much more complicated. We shall only show the existence of various

determining sets and non-determining sets for HP(B ) , O<p�oo. (See n

Theorem 2 � Theorem 5 in §3.4. and §3.5.)

3.2. Preliminaries.

Let H(B n ) denote the space of all holomorphic functions in B . n

Put

H(Bn)* = {/EH(Bn): /�0

ⁱⁿ

1/(Bn) ' * N(Bn)*

⁼

N(Bn)

n

Up(B )* = 1/P(B ) n n

ⁿ

1/(Bn)*

(O<p�oo).

We note that

H•(Bn) _c Hq(Bn) _c Hp(Bn) _c N(Bn) _c H(Bn) _if O<p<q<oo.

Let

/EI/(Bn)

* ^. In a neighborhood of each point

aEBn' f

can be expanded in a series of homogeneous polynomials:

The integer

f(z) =

^r

Pk(z-a).

k=O

v1(a) =

Min

{k

�O_:

Pk�O}

is called the zero multiplicity of

f

^at a. The integer-valued function

v1

defined in Bn is called the zero-divisor of f.

Let J.l be 11 nonneg11tive integer-v11lued

function defined in Bn.

Then _� is called a positive divisor on

B n

if and only if it is

locally the zero-divisor of some holomorphic function, thatis, for each point

aeB n

there exist _a connected neighborhood V of a and _a

holomorphic function

f

ⁱⁿ _V such th11t

Jf.O

11nd

Jl=Vf

ⁱⁿ _V.

We denote by D +

IBn)

the set of all positive divisors on

Bn.

^Then

we have the divisor map

v

^from

1/(Bn)

into D (

B

ⁿ

)

defined by letting

v(J)

^{for f} in

1/(Bn)

* ^be

v1.

* +

Let 1-l ^be 11 positive divisor on the open unit disc

B1

in the complex plane. Define

co

for O<r�l, and

n ( t) N�(r,s) =

!�

^{Mt dt}

for O<s<r�1, where rB1={.AEC: I.AI<r}.

Let n�2 be an integer. Let �ED (Bn). + Take a point �E8Bn' where

8Bn is the boundary of Bn. Define

�[�](.A) = �(.A�) for .AEB1.

Put E={�E8Bn: �[�]ED CB1)}. + Then a(8Bn'E)=O, where a is the rotation

invariant positive Borel measure on 88 for which o(8B )=1. (See

n n

e.g. Stoll [28], p.13.) We write

N�[�](r,s) if �EE, and we define

N � (r,s)

for O<s<r�1.

Let !EH(Bn) . * We denote by Z(/) the zero set of /:

Z(f) = {zeBn: /(z)=O}.

Then Z(f)={zeBn: v1Cz)>O}.

We shall say that a positive divisor �ED+(B ) (resp. a zero set n

Z(f) of some /EH(B ) ) satisfies the * BLaschke condition if and only n

if

N � (1, s) < w (resp. N v, (1, s) < w)

for some sE(0,1). (See Kujala [14], p.252 and Stoll [28], p.41.) Theorem F (Henkin [8]; Skoda [25]).

two conditions are equivalent:

(a) � e v(N(Bn) ). *

For �ED+(B ), the following n

(b) � satisfies the Blaschke condition.

Let X be a subspace of H(Bn). A zero setH in Bn (i.e. H=Z(g) for some gEH(B )*) is said to be a determining set for X _{if the}

n

assumptions /EX, HcZ(f) force f�O. llere the symbol HcZ(f) means the inclusion relation with multiplicity; i.e. for two zero sets

H(=Z(g))

and 2(/), we write HcZ(f) if and only if

v � v1

in B .

g n

We recall some results about the Hardy spaces Hp(Bn) and the Bergman spaces Ap(B ) . Assume O<p<w. For /EH(B ) , we define the

n n

HP-norm and the AP-norm as follows:

11/11 p H 11 /11 p

A

sup

{J88

1 /(r�) 1 Pda(�) } 11P,

O<r<1 n

{

f

B I I ( z) I P dA. ( z ) }

1 I

^P

n

Here A. is the Lebesgue measure on en normalized so that A.(B

)=1.

n Then

{/EH(Bn): 11/11 p< oo } , H

{ fEH(Bn): IIIII p< oo } . A

We note that Hp(B )cAP(B ) (O<p<w).

n n

Suppose n�2. Let f and g be functions defined in B and B 1,

n n-

respectively, and define a restriction operator p and an extension operator E by

(pf)(z') ='f(z',O) (Eg)(z',zn)

=

g(z')

( z' EBn-l), ( ( z' , zn) eBn) . We note that pE is the identity operator on B

1.

n-

onto

The following two theorems will be used in §3.5:

Theorem F (Rudin [21], p.127). Assume n�2, O<p<oo.

(a) The extension Eis a linear isometry of AP(Bn-l) into HP(Bn).

(b) The restriction p is a Linear norm-decreasing map of Hp(Bn) A p

(Bn-1).

Theorem G

(Rudin

[21], p.128).

then

If

(

z) I� 2

^{n/ P II f II HP}

( 1-

I z

I

) -n/ P

Assume

n�1,

O<p<w.

( zEB ) . n