THE BERGMAN NUMBER OF A PLANE DOMAIN CHRISTINA KARAFYLLIA Abstract. LetDbe a domain in the complex plane C. The Hardy

2025-05-06 0 0 386.43KB 13 页 10玖币

侵权投诉

THE BERGMAN NUMBER OF A PLANE DOMAIN

CHRISTINA KARAFYLLIA

Abstract. Let Dbe a domain in the complex plane C. The Hardy

number of D, which ﬁrst introduced by Hansen, is the maximal number

h(D) in [0,+∞] such that fbelongs to the classical Hardy space Hp(D)

whenever 0 <p<h(D) and fis holomorphic on the unit disk Dwith

values in D. As an analogue notion to the Hardy number of a domain D

in C, we introduce the Bergman number of Dand we denote it by b(D).

Our main result is that, if Dis regular, then h(D) = b(D). This gener-

alizes earlier work by the author and Karamanlis for simply connected

domains. The Bergman number b(D) is the maximal number in [0,+∞]

such that fbelongs to the weighted Bergman space Ap

α(D) whenever

p > 0 and α > −1 satisfy 0 <p

α+2 < b(D) and fis holomorphic on D

with values in D. We also establish several results about Hardy spaces

and weighted Bergman spaces and we give a new characterization of the

Hardy number and thus of the Bergman number of a regular domain

with respect to the harmonic measure.

1. Introduction

The Hardy space with exponent p > 0 is denoted by Hp(D) and is deﬁned

to be the set of all holomorphic functions fon the unit disk Dsuch that

sup

0<r<1Z2π

|f(reiθ)|pdθ < +∞.

The fact that a function belongs to Hp(D) imposes a restriction on its growth

and this restriction is stronger when pincreases, that is, if 0 < q < p then

Hp(D)⊂Hq(D). For the theory of Hardy spaces see [3].

In [8] Hansen studied the problem of determining the numbers p > 0 for

which a holomorphic function fon Dbelongs to Hp(D) by studying f(D).

For this purpose, he introduced a number which he called the Hardy number

of a domain. The Hardy number of a domain Din the complex plane Cis

deﬁned as

h(D) = sup {p > 0 : |z|phas a harmonic majorant on D}.

2010 Mathematics Subject Classiﬁcation. Primary 30H10, 30H20; Secondary 42B30,

30C85.

Key words and phrases. Bergman number, weighted Bergman spaces, Hardy number,

Hardy spaces.

I would like to thank Gregory Markowsky for the valuable comments and the referee

for the corrections and especially for providing the idea for the proof of Corollary 4.1.

arXiv:2210.12190v2 [math.CV] 28 Feb 2023

2 CHRISTINA KARAFYLLIA

Later, in [13] Kim and Sugawa proved some equivalent deﬁnitions for h(D).

Let fbe a holomorphic function on Dand set

h(f) = sup{p > 0 : f∈Hp(D)} ∈ [0,+∞].

Since Hp(D)⊂Hq(D) for 0 < q < p, it follows that f∈Hp(D) for p<h(f)

and f /∈Hp(D) for p>h(f). In case p=h(f), then both can happen (see

[10]). Let Dbe a domain in C. The Hardy number of Dcan equivalently

be characterized (see [13, Lemma 2.1]) as

h(D) = inf{h(f) : f∈H(D, D)} ∈ [0,+∞]

h(D) = sup{p > 0 : H(D, D)⊂Hp(D)},

where H(D, D) denotes the set of all holomorphic functions on Dwith values

in D. Combining the deﬁnitions above with the monotonicity of Hardy

spaces, we infer that for a given plane domain D, the Hardy number of D,

h(D), is the maximal number in [0,+∞] such that f∈Hp(D) whenever

0<p<h(D) and fis holomorphic on Dwith values in D. Moreover, Kim

and Sugawa [13, Lemma 2.1] proved that h(D) = h(f) for a holomorphic

universal covering map of Donto D.

The Hardy number has been studied extensively over the years. A clas-

sical problem is to ﬁnd estimates or exact descriptions for it. What we

know, so far, about the Hardy number of an arbitrary plane domain is some

estimates proved by Hansen in [8] and an exact formula for it involving har-

monic measure proved by Kim and Sugawa in [13]. Their proof is based on

Ess´en’s main lemma in [5]. However, we know more on how to estimate it

for certain types of domains such as starlike [8] and spiral-like domains [9],

comb domains [12] and, more generally, simply connected domains [10], [11].

In this paper, we prove one more way to compute the Hardy number of a

regular domain with the aid of harmonic measure (see Theorem 1.2).

A more general class of holomorphic functions than Hardy spaces is

weighted Bergman spaces. The weighted Bergman space with exponent

p > 0 and weight α > −1 is denoted by Ap

α(D) and is deﬁned to be the set

of all holomorphic functions fon Dsuch that

|f(z)|p1− |z|2αdA (z)<+∞,

where dA denotes the Lebesgue area measure on D. The unweighted Bergman

space (α= 0) is simply denoted by Ap(D) and it is known as the Bergman

space with exponent p. Weighted Bergman spaces contain Hardy spaces,

that is, Hp(D)⊂Ap

α(D), for all α > −1 and p > 0 (see [19]). Actu-

ally, a more general result holds. By Corollary 4.4 in [17] it follows that

Hq(D)⊂Ap

α(D) provided that p

α+2 ≤q≤p. For the theory of Bergman

spaces see [4].

As an analogue notion to the Hardy number of a plane domain, here we

introduce the Bergman number of a domain in Cin the following way. If f

THE BERGMAN NUMBER OF A PLANE DOMAIN 3

is a holomorphic function on D, we set

b(f) = sup p

α+ 2 :p > 0, α > −1, f ∈Ap

α(D)∈[0,+∞].

Note that the number b(f) was ﬁrst introduced by the author and Karaman-

lis in [11] for conformal maps fon Dand it was proved that b(f) = h(f).

Let Dbe a domain in C. We deﬁne the Bergman number of Das

b(D) = inf{b(f) : f∈H(D, D)} ∈ [0,+∞].

Our ﬁrst observation is that the Bergman number satisﬁes the same basic

properties as the Hardy number (see [8, p. 236] or [13, Lemma 2.3]). That

is,

(1) if Dis bounded then b(D)=+∞,

(2) if C\Dis bounded then b(D) = 0,

(3) if D⊂D0then b(D)≥b(D0),

(4) b(D) = b(φ(D)) for a complex aﬃne map φ(z) = az +bwith a6= 0,

(5) if Dis simply connected then b(D)≥1/2.

Properties (1), (2) are proved in Section 2, properties (3), (4) are trivial

and (5) is proved in [11]. Moreover, we observe that since Hp(D)⊂Ap

α(D),

it is easy to show that h(D)≤b(D) (see Section 2). The question which

arises is whether the reverse inequality holds and thus h(D) = b(D), which

is not a direct consequence of their deﬁnitions. In this paper we prove that

if Dis regular then the answer is positive. We deﬁne a subdomain Dof

C∞=C∪{∞} to be regular if Dposseses a “barrier function” at each of

its boundary points [16, Chapter 4]. If Dis an unbounded subdomain of C,

then ∞is to be included among the boundary points of D.

Theorem 1.1. Let Dbe a regular domain and fbe a universal covering

map of Donto D. Then h(D) = b(D) = h(f) = b(f).

An immediate corollary (see Section 4) is that if p

α+2 < b(D) then f∈

α(D) for every holomorphic function fon Dwith values in D. In other

words, b(D) is the maximal number in [0,+∞] such that f∈Ap

α(D) when-

ever 0 <p

α+2 < b(D) and fis holomorphic on Dwith values in D. Another

consequence of Theorem 1.1 (see Section 4) is that, if Dis regular, then

b(D) = sup p

α+ 2 :p > 0, α > −1, H(D, D)⊂Ap

α(D).

We also establish the following result that gives a new description of

the Hardy number and thus of the Bergman number of a regular domain

involving harmonic measure. For the deﬁnition of harmonic measure see [7],

[16].

Theorem 1.2. Let Dbe a regular domain and let a∈D. If Er=∂D∩{z∈

C:|z|> r}for r > 0(see Fig.1), then

h(D) = b(D) = lim inf

r→+∞

log ωD(a, Er)−1

log r.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

THEBERGMANNUMBEROFAPLANEDOMAINCHRISTINAKARAFYLLIAAbstract.LetDbeadomaininthecomplexplaneC.TheHardynumberofD,whichrstintroducedbyHansen,isthemaximalnumberh(D)in[0;+1]suchthatfbelongstotheclassicalHardyspaceHp(D)whenever0

展开>> 收起<<

THE BERGMAN NUMBER OF A PLANE DOMAIN CHRISTINA KARAFYLLIA Abstract. LetDbe a domain in the complex plane C. The Hardy.pdf

共13页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

THE BERGMAN NUMBER OF A PLANE DOMAIN CHRISTINA KARAFYLLIA Abstract. LetDbe a domain in the complex plane C. The Hardy

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: