Guanghua He

College of International Finance and Trade,Zhejiang Yuexiu University of Foreign Languages,Shaoxing 312000,Zhejiang,China

Abstract.We define Bloch-type functions of C1(D)on the unit disk of complex plane C and characterize them in terms of weighted Lipschitz functions.We also discuss the boundedness of a composition operator Cφacting between two Bloch-type spaces.These obtained results generalize the corresponding known ones to the setting of C1(D).

Key Words:Bloch space,majorant,composition operator.

1 Introduction

Let D={z∈C:|z|＜1}be the unit disk of the complex plane C,and C1(D)be the set of all complex-valued functions having continuous partial derivatives on D.For α＞0,a function f∈C1(D)is called α-Bloch if

It is readily seen that the set of all α-Bloch functions on D is a Banach space Bαwith the normkfkBα=|f(0)|+kfkα.

Let ω:[0,+∞)→[0,+∞)be an increasing function with ω(0)=0,we say that ω is a majorant if ω(t)/t is non-increasing for t＞ 0(cf.[4]).Following[5],given a majorant ω and α ＞ 0,the ω-α-Bloch spaceconsists of all functions f∈ C1(D)such that

and the little ω-α-Bloch spaceconsists of the functions f∈such that

For 0＜α≤1,the weighted hyperbolic metric dsαof D,introduced in[1]is defined as

Suppose that γ(t)(0≤t≤1)is a continuous and piecewise smooth curve in D.Then the length of γ(t)with respect to the weighted hyperbolic metric dsαis equal to

Consequently,the associated distance between z and w in D is

where γ is a continuous and piecewise smooth curve in D.Note that h1(α =1)is the hyperbolic distance on D.

Let s,t≥0 and f be a continuous function in D.If there exists a constant C such that

for any z,w∈D,then we say that f is a weighted Euclidian (resp. hyperbolic) Lipschitz functionof indices (s,t).In particular,when s=t=0,we say that f is a Euclidian(resp.hyperbolic)Lipschitz function(cf.[12]).

In the theory of function spaces, the relationship between Bloch spaces and (weighted)Lipschitz functions has attracted much attention.For instance,in 1986,Holland and Walsh[7]established a classical criterion for analytic Bloch space in the unit disc D in terms of weighted Euclidian Lipschitz functions of indices.Ren and Tu[13]extended the criterion to the Bloch space in the unit ball of Cn,Li and Wulan[8],Zhao[15]characterized holomorphic α-Bloch space in terms of

In[16,17],Zhu investigated the relationship between Bloch spaces and Bergman Lipschitz functions and proved that a holomorphic function belongs to Bloch space if and only if it is Bergman Lipschitz.For the related results of harmonic functions,we refer to[2,3,5,6,12,14]and the references therein.

Motivated by the known results mentioned above,we consider the corresponding problems in the setting of C1(D)in this paper.In Section 2,we collect some known results that will be needed in the sequel.The main results and their proofs are presented in Sections 3 and 4.

Throughout this paper,constants are denoted by C,they are positive and may differ from one occurrence to the other.The notation A?B means that there is a positive constant C such that B/C≤A≤CB.

2 Several lemmas

In this section,we introduce some notations and recall some known results that we need later.

For each a∈D,the M?bius transformation ?a:D→D is defined by

If a,z∈D and r∈(0,1),we define the pseudo-hyperbolic disk with center a and radius r as

A straightforward calculation shows that E(a,r)is a Euclidean disk with center atand the radius

The following lemma is proved in[17].

Lemma 2.1.Let r∈(0,1),w∈E(a,r).Then we have

The following lemma is useful for us.

Lemma 2.2(see[5]).Let ω(t)be a majorant and s∈(0,1],v∈(1,∞).Then for t∈(0,∞),

As applications of Lemmas 2.1 and 2.2,we have

Lemma 2.3.Let r∈(0,1),w∈E(a,r)and ω(t)be a majorant.Then

3 Bloch functions

Let f be a harmonic Bloch mapping in the unit disc D.In[3],Colonna proved that the Bloch constant Bfof f equals to its Bloch semi-norm,i.e.,

where h1is the hyperbolic distance in D.

In this section,we first characterize the space Bαin terms of weighted hyperbolic Lipschitz condition and generalize Colonna's result to the setting of C1(D).

Theorem 3.1.Let f∈C1(D)and 0＜α≤1.Then f∈Bαif and only if there is a constant C＞0 such that

Moreover,we have

for all f∈Bα.

Proof.We first prove the sufficiency.For any z,w ∈ D,from the definition of hα(z,w),we assume that γ(s)is the geodesic between z and w(parametrized by arc-length)with respect to hα.Since hα(γ(0),γ(s))=s,we have

Dividing both sides by s and then letting s→0 in the above inequality gives

From the minimal length property of geodesics,

we obtain that

For the conversely,we assume that f∈Bα.Let z,w∈D and γ(t)(0≤t≤1)be a smooth curve from z to w.Then

Taking the in fimum over all piecewise continuous curves connecting z and w,we conclude that

for all z,w∈D.This completes the proof.

In the following,we characterize the spacesin terms of Euclidean weighted Lipschitz functions.

Theorem 3.2.Let r∈(0,1),f∈C1(D).Then f∈if and only if

Proof.Sufficiency.Let f∈C1(D).For z∈D,we have

By letting w→z,we obtain that

where the last inequality follows from Lemma 2.3.Thus,

The proof of Theorem 3.2 is completed.

A similar result is true for the little Bloch-type spaces.

The proof is almost the same as the one of Theorem 3.2 in[13].Thus we omit it here.

Remark 3.1.When ω(t)=t,Li and Wulan[8]obtained the analogues of Theorems 3.2 and 3.3 for holomorphic Bloch space on the unit ball of Cn.

4 Composition operators

Let φ be a holomorphic self-mapping of D.The composition operator Cφ,induced by φ is defined by Cφ(f)=f?φ for f∈C1(D).During the past few years,composition operators have been studied extensively on spaces of holomorphic functions on various domains in C and Cn,see e.g.,[9,10,18].In this section,we discuss the boundedness of composition operators between Bloch spaces of C1(D).

Theorem 4.1.Let α,β＞0 and φ be a holomorphic self-mapping of D.Then Cφ:Bα→Bβis bounded if and only if

Proof.First suppose that

For f∈Bαand z∈D,we have

and

Hence Cφ:Bα→Bβis bounded.

For the converse,assume that Cφ:Bα→Bβis a bounded operator with

for all f∈Bα.Fix a point z0∈D and let w=φ(z0).If α≠1,consider the function fw(z)=.Then it is easy to check that fw∈Bα.The boundedness of Cφimplies that

In particular,take z=z0,we get

Since z0is arbitrary,the result follows.

If α=1,we only need to consider the functionFollowing a discussion similar to the above,it can be proved that(1)holds.The proof of Theorem 4.1 is completed.

Recall that the classical Schwarz-Pick Lemma in the unit disk gives that for a holomorphic self-mapping φ of D,(1-|z|2)|φ′(z)|≤ 1-|φ(z)|2holds for all z∈ D.As an application of this result,it is easy to derive the following corollary.

Corollary 4.1.Let φ be a holomorphic self-mapping of D.Then Cφ:B1→B1is bounded.