# Weighted Composition Operators on the Zygmund Space

Authors: Shanli Ye; Qingxiao Hu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We characterize the boundedness and compactness of the weighted composition operator on the Zygmund space𝒵={f∈H(D):supz∈D(1-|z|2)|f″(z)|<∞} and the little Zygmund space 𝒵0.


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## Body

## 1. Introduction

LetD={z:|z|<1} be the open unit disk in the complex plane C, let T={z:|z|=1} be its boundary, and let H(D) denote the set of all analytic functions by D. For f∈H(D), let(1.1)‖f‖Z=sup{(1-|z|2)|f′′(z)|:z∈D}.An analytic functionf∈H(D) is said to belong to the Zygmund space 𝒵 if ∥f∥𝒵<+∞, and the little Zygmund space 𝒵0 consists of all f∈𝒵 satisfying lim|z|→1-(1-|z|2)|f′′(z)|=0. From a theorem of Zygmund (see [1, vol. I, page 263] or [2, Theorem 5.3]), we see that f∈𝒵 if and only if f is continuous in the close unit disk D¯={z:|z|≤1} and the boundary function f(eiθ) such that(1.2)|f(ei(θ+h))+f(ei(θ-h))-2f(eiθ)|h<∞,for alleiθ∈T and all h>0. It can easily proved that 𝒵 is a Banach space under the norm:(1.3)‖f‖*=|f(0)|+|f′(0)|+‖f‖Z
and that 𝒵0 is a closed subspace of 𝒵. It is easily obtained that(1.4)|f′(z)-f′(0)|≤12‖f‖Zlog1+|z|1-|z|forf∈Z,(1.5)lim|z|→1-|f′(z)|log(1/(1-|z|))=0forf∈Z0.
For some other information on this space and some operators on it, see, for example, [3–5].An analytic self-mapφ:D→D induces the composition operator Cφ on H(D), defined by Cφ(f)=f(φ(z)) for f analytic on D. It is a well-known consequence of Littlewood’s subordination principle that the composition operator Cφ is bounded on the classical Hardy, Bergman, and Bloch spaces (see, e.g., [6–9]).Recall that a linear operator is said to be bounded if the image of a bounded set is a bounded set, while a linear operator is compact if it takes bounded sets to sets with compact closure. It is interesting to provide a function theoretic characterization of whenφ induces a bounded or compact composition operator on various spaces. The book [10] contains plenty of information on this topic.Letu be a fixed analytic function on the open unit disk. Define a linear operator uCφ on the space of analytic functions on D, called a weighted composition operator, by uCφf=u·(f∘φ), where f is an analytic function on D. We can regard this operator as a generalization of a multiplication operator and a composition operator. In recent years, the weighted composition operator has been received much attention and appears in various settings in the literature. For example, it is known that isometries of many analytic function spaces are weighted composition operators (e.g., see [11]). The boundedness and compactness of it has been studied on various Banach spaces of analytic functions, such as Hardy, Bergman, BMOA, Bloch-type spaces, see, for example, [12–16]. Also, it has been studied from one Banach space of analytic functions to another, one may see in [17–26].The purpose of this paper is to consider the weighted composition operators on the Zygmund space𝒵 and the little Zygmund space 𝒵0. Our main goal is to characterize boundedness and compactness of the operators uCφ on 𝒵 in terms of function theoretic properties of the symbols u and φ. We also characterize boundedness and compactness of uCφ on 𝒵0.Throughout this paper, constants are denoted byC, they are positive and may differ from one occurrence to the other.


## 2. Auxiliary Results

In order to prove the main results of this paper, we need some auxiliary results.Lemma 2.1.
Iff∈𝒵, then(i)
|f(z)|≤∥f∥* for every z∈D;(ii)
|f′(z)|≤log(e/(1-|z|2))∥f∥* for every z∈D.Proof.
Supposef∈𝒵, z∈D and 0<t<1, then
(2.1)|f′(zt)|≤|f′(0)|+12‖f‖Zlog1+|zt|1-|zt|,by (1.4). It follows that
(2.2)|f(z)-f(0)|=|z∫01f′(zt)dt|≤|z|∫01(|f′(0)|+12‖f‖Zlog1+|zt|1-|zt|)dt≤|z||f′(0)|+12‖f‖Z∫0|z|log1+s1-sds≤|z||f′(0)|+log(1+|z|)‖f‖Z,hence
(2.3)|f(z)|≤|f(0)|+|f′(0)|+‖f‖Zlog2≤‖f‖*.
One may easily prove (ii) by (1.4). The details are omitted here.Lemma 2.2.
Supposef∈𝒵, then ∥ft∥*≤∥f∥*,0<t<1, where ft(z)=f(tz).One may easily obtain it by a calculation.Lemma 2.3.
SupposeuCφ:𝒵0→𝒵0 is a bounded operator. Then uCφ:𝒵→𝒵 is a bounded operator.Proof.
SupposeuCφ is bounded in 𝒵0. It is clear that for any f∈𝒵, we have ft∈𝒵0 for every 0<t<1. According to Lemma 2.2, we obtain that
(2.4)‖uCφ(ft)‖*≤‖uCφ‖‖ft‖*≤‖uCφ‖‖f‖*<+∞.Then
(2.5)‖uCφ(f)‖*=limt→1-‖uCφ(ft)‖*≤sup0<t<1‖uCφ(ft)‖*≤‖uCφ‖‖f‖*<+∞.Hence, uCφ:𝒵→𝒵 is a bounded operator.


## 3. Boundedness ofuCφ

In this section, we characterize bounded weighted composition operators on the Zygmund space𝒵 and the little Zygmund space 𝒵0.Theorem 3.1.
Letu be an analytic function on the unit disc D and φ an analytic self-map of D. Then uCφ is bounded on the Zygmund space 𝒵 if and only if u∈𝒵 and the following are satisfied:
(3.1)supz∈D(1-|z|2)|u(z)(φ′(z))2|1-|φ(z)|2<∞;(3.2)supz∈D(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|log11-|φ(z)|2<∞.Proof.
SupposeuCφ is bounded on the Zygmund space 𝒵. Then we can easily obtain the following results by taking f(z)=1 and f(z)=z in 𝒵, respectively:
(3.3)u∈Z;(3.4)supz∈D(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)+φ(z)u′′(z)|<+∞.
By (3.3), (3.4), and the boundedness of the function φ(z), we get
(3.5)K1=supz∈D(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|<+∞.
Let f(z)=z2 in 𝒵 again, in the same way we have
(3.6)supz∈D(1-|z|2)|4φ(z)φ′(z)u′(z)+φ2(z)u′′(z)+2u(z)(φ(z)φ′′(z)+(φ′(z))2)|<∞.Using these facts and the boundedness of the function φ(z) again, we get
(3.7)K2=supz∈D(1-|z|2)|(φ′(z))2u(z)|<+∞.
Fixa∈D with |a|>1/2, we take the test functions:(3.8)fa(z)=h(a¯z)a¯(log11-|a|2)-1-∫0zlog11-a¯ωdω,
for z∈D, where
(3.9)h(z)=(z-1)((1+log11-z)2+1).Then we have
(3.10)fa′(z)=(log11-a¯z)2(log11-|a|2)-1-log11-a¯z,fa′′(z)=2a¯1-a¯zlog11-a¯z(log11-|a|2)-1-a¯1-a¯z,and sup1/2<|a|<1∥fa∥*≤C by [3], where C is not dependent on a. Therefore, for all λ∈D with |φ(λ)|>1/2, we have
(3.11)C‖fa‖*≥‖uCφfa‖*≥supz∈D(1-|z|2)|(uCφfa)′′(z)|=supz∈D(1-|z|2)|(φ(z))2(φ′(z))2(2φ′(z)u′(z)+φ′′(z)u(z))fa′(φ(z))+fa′′(φ(z))(φ′(z))2u(z)+u′′(z)fa(φ(z))|.Let a=φ(λ), it follows that
(3.12)C‖fa‖*≥(1-|λ|2)|(2φ′(λ)u′(λ)+φ′′(λ)u(λ))fφ(λ)′(φ(λ))+fφ(λ)′′(φ(λ))(φ′(λ))2u(λ)+u′′(λ)fφ(λ)(φ(λ))|=(1-|λ|2)|(φ′(λ))2u(λ)φ(λ)¯1-|φ(λ)|2+u′′(λ)fφ(λ)(φ(λ))|≥(1-|λ|2)|(φ′(λ))2u(λ)φ(λ)¯1-|φ(λ)|2|-(1-|λ|2)|u′′(λ)fφ(λ)(φ(λ))|.Then, by Lemma 2.1 and (3.3), we have
(3.13)(1-|λ|2)|(φ′(λ))2u(λ)φ(λ)¯1-|φ(λ)|2|≤(1-|λ|2)|u′′(λ)fφ(λ)(φ(λ))|+C‖fa‖*≤‖u‖Z‖fa‖*+C‖fa‖*.Hence
(3.14)sup|φ(λ)|>1/2(1-|λ|2)|(φ′(λ))2u(λ)|1-|φ(λ)|2≤2sup|φ(λ)|>1/2(1-|λ|2)|(φ′(λ))2u(λ)φ(λ)¯1-|φ(λ)|2|≤C‖fa‖*<∞.
For allλ∈D with |φ(λ)|≤1/2, by (3.7), we have(3.15)supλ∈D(1-|λ|2)|u(λ)(φ′(λ))2|1-|φ(λ)|2≤43supλ∈D(1-|λ|2)|u(λ)(φ′(λ))2|<+∞.Hence (3.1) holds.
Next, we will show that (3.2) holds. Fix a∈D with |a|>1/2, we take another test functions:(3.16)ga(z)=h(a¯z)a¯(log11-|a|2)-1
for z∈D. It is proved that sup1/2<|a|<1∥ga∥*≤C above, where C is not dependent on a. Therefore, for all λ∈D with |φ(λ)|>1/2, we have
(3.17)C‖ga‖*≥‖uCφga‖*≥supz∈D(1-|z|2)|(uCφga)′′(z)|=supz∈D(1-|z|2)|(φ′(z))2(2φ′(z)u′(z)+φ′′(z)u(z))ga′(φ(z))+ga′′(φ(z))(φ′(z))2u(z)+u′′(z)ga(φ(z))|=supz∈D(1-|z|2)|(2φ′(z)u′(z)+φ′′(z)u(z))(log11-a¯φ(z))2(log11-|a|2)-1+2a¯1-a¯φ(z)log11-a¯φ(z)(log11-|a|2)-1(φ′(z))2u(z)+ga(φ(z))u′′(z)(log11-a¯φ(z))2|.Let a=φ(λ), it follows that
(3.18)C‖ga‖*≥(1-|λ|2)|(2φ′(λ)u′(λ)+φ′′(λ)u(λ))(log11-|φ(λ)|2)2(log11-|φ(λ)|2)-1+2φ(λ)¯1-|φ(λ)|2(φ′(λ))2u(λ)+u′′(λ)gφ(λ)(φ(λ))(log11-|φ(λ)|2)2|≥(1-|λ|2)|(2φ′(λ)u′(λ)+φ′′(λ)u(λ))(log11-|φ(λ)|2)|-(1-|λ|2)2|φ(λ)|1-|φ(λ)|2|(φ′(λ))2u(λ)|-(1-|λ|2)|u′′(λ)gφ(λ)(φ(λ))|.Hence
(3.19)(1-|λ|2)|2φ′(λ)u′(λ)+φ′′(λ)u(λ)|log11-|φ(λ)|2≤(1-|λ|2)2|φ(λ)|1-|φ(λ)|2|(φ′(λ))2u(λ)|+(1-|λ|2)|u′′(λ)gφ(λ)(φ(λ))|+C‖ga‖*.By (3.1), Lemma 2.1, and the boundedness of the function φ(z), we get
(3.20)sup|φ(λ)|>1/2(1-|λ|2)|2φ′(λ)u′(λ)+φ′′(λ)u(λ)|log11-|φ(λ)|2≤sup|φ(λ)|>1/2(1-|λ|2)21-|φ(λ)|2|(φ′(λ))2u(λ)|+sup|φ(λ)|>1/2‖u‖Z‖gφ(λ)‖*+C‖ga‖*<∞.
For allλ∈D with |φ(λ)|≤1/2, by (3.5), we have(3.21)sup|φ(λ)|≤1/2(1-|λ|2)|2φ′(λ)u′(λ)+φ′′(λ)u(λ)|log11-|φ(λ)|2≤log43sup|φ(λ)|≤1/2(1-|λ|2)|2φ′(λ)u′(λ)+φ′′(λ)u(λ)|<∞.Hence (3.2) holds.
Conversely, suppose thatu∈𝒵, (3.1) and (3.2) hold. For f∈𝒵, by Lemma 2.1, we have the following inequality:(3.22)(1-|z|2)|(uCφf)′′(z)|=(1-|z|2)|(φ′(z))2(φ′(z))2(2φ′(z)u′(z)+φ′′(z)u(z))f′(φ(z))+f′′(φ(z))(φ′(z))2u(z)+u′′(z)f(φ(z))|≤(1-|z|2)|(2φ′(z)u′(z)+φ′′(z)u(z))f′(φ(z))|+(1-|z|2)|f′′(φ(z))(φ′(z))2u(z)|+(1-|z|2)|u′′(z)f(φ(z))|≤(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|loge1-|φ(z)|2‖f‖*+(1-|z|2)|(φ′(z))2u(z)|1-|φ(z)|2(1-|φ(z)|2)|f′′(φ(z))|+(1-|z|2)|u′′(z)|‖f‖*≤(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|loge1-|φ(z)|2‖f‖*+(1-|z|2)|(φ′(z))2u(z)|1-|φ(z)|2‖f‖Z+‖u‖Z‖f‖*≤C‖f‖*.This shows that uCφ is bounded. This completes the proof of Theorem 3.1.Theorem 3.2.
Letu be an analytic function on the unit disc D and φ an analytic self-map of D. Then uCφ is bounded on the little Zygmund space 𝒵0 if and only if u∈𝒵0, (3.1) and (3.2) hold, and the following are satisfied:
(3.23)lim|z|→1-(1-|z|2)|u(z)(φ′(z))2|=0;(3.24)lim|z|→1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|=0.Proof.
Suppose thatuCφ is bounded on the little Zygmund space 𝒵0. Then u=uCφ1∈𝒵0. Also uφ=uCφz∈𝒵0, thus
(3.25)(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)+φ(z)u′′(z)|⟶0(|z|⟶1-).Since |φ|≤1 and u∈𝒵0, we have lim|z|→1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|=0. Hence (3.24) holds.
Similarly,uCφz2∈𝒵0, then(3.26)(1-|z|2)|4φ(z)φ′(z)u′(z)+φ2(z)u′′(z)+2u(z)(φ(z)φ′′(z)+(φ′(z))2)|⟶0(|z|⟶1-).By (3.24), |φ|≤1 and u∈𝒵0, we get that lim|z|→1-(1-|z|2)|u(z)(φ′(z))2|=0, that is, (3.23) holds.
On the other hand, by Lemma2.3 and Theorem 3.1, we obtain that (3.1) and (3.2) hold.
Conversely, let(3.27)M1=supz∈D(1-|z|2)|u(z)(φ′(z))2|1-|φ(z)|2<∞;M2=supz∈D(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|log11-|φ(z)|2<∞.For all f∈𝒵0, we have both (1-|z|2)|f′′(z)|→0 and |f′(z)|/log(1/(1-|z|2))→0 as |z|→1- by (1.5). Since u∈𝒵0, given that ϵ>0, there is a 0<δ<1 such that (1-|z|2)|u′′(z)|<ϵ/3∥f∥*, (1-|z|2)|f′′(z)|<ϵ/3M1 and |f(z)|/log(1/(1-|z|2))<ϵ/3M2 for all z with δ<|z|<1.
If|φ(z)|>δ, it follows that(3.28)(1-|z|2)|(uCφf)′′(z)|=(1-|z|2)|(φ′(z))2(2φ′(z)u′(z)+φ′′(z)u(z))f′(φ(z))+f′′(φ(z))(φ′(z))2u(z)+u′′(z)f(φ(z))|≤(1-|z|2)|(2φ′(z)u′(z)+φ′′(z)u(z))f′(φ(z))|+(1-|z|2)|f′′(φ(z))(φ′(z))2u(z)|+(1-|z|2)|u′′(z)f(φ(z))|≤M2|f(φ(z))|log(1/(1-|φ(z)|2))+M1(1-|φ(z)|2)|f′′(φ(z))|+(1-|z|2)|u′′(z)|‖f‖*≤ϵ3+ϵ3+ϵ3=ϵ.
We know that there exists a constantM3 such that |f(z)|≤M3, |f′(z)|≤M3 and |f′′(z)|≤M3 for all |z|≤δ.
If|φ(z)|≤δ, it follows that(3.29)(1-|z|2)|(uCφf)′′(z)|=(1-|z|2)|(φ′(z))2(2φ′(z)u′(z)+φ′′(z)u(z))f′(φ(z))+f′′(φ(z))(φ′(z))2u(z)+u′′(z)f(φ(z))|≤M3(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|+M3(1-|z|2)|(φ′(z))2u(z)|+M3(1-|z|2)|u′′(z)|.Thus, we conclude that (1-|z|2)|(uCφ(f))′′(z)|→0 as |z|→1-. Hence uCφf∈𝒵0 for all f∈𝒵0. On the other hand, uCφ is bounded on 𝒵 by Theorem 3.1. Hence uCφ is a bounded operator on the little Zygmund space 𝒵0.The following corollary is just as Theorem  2.2 in [27].Corollary 3.3.
Letφ be an analytic self-map of D. Then Cφ is a bounded operator on 𝒵 if and only if
(3.30)supz∈D(1-|z|2)|(φ′(z))2|1-|φ(z)|2<∞,(3.31)supz∈D(1-|z|2)|φ′′(z)|log11-|φ(z)|2<∞.Corollary 3.4.
Letφ be an analytic self-map of D. Then Cφ is a bounded operator on 𝒵0 if and only if φ∈𝒵0, (3.30) and (3.31) hold.Proof.
By Theorem3.2, Cφ is a bounded operator on 𝒵0 if and only if φ∈𝒵0, lim|z|→1-(1-|z|2)|(φ′(z))2|=0, (3.30) and (3.31) hold. However, by (1.5), φ∈𝒵0 implies that lim|z|→1-(1-|z|2)|(φ′(z))2|=0. Then, Cφ is a bounded operator on 𝒵0 if and only if φ∈𝒵0, (3.30) and (3.31) hold.


## 4. Compactness ofuCφ

In order to prove the compactness ofuCφ on the Zygmund space 𝒵, we require the following lemmas.Lemma 4.1.
Suppose thatuCφ be a bounded operator on 𝒵. Then uCφ is compact if and only if for any bounded sequence {fn} in 𝒵 which converges to 0 uniformly on compact subsets of D, we have ∥uCφ(fn)∥*→0 as n→∞.The proof is similar to that of Proposition 3.11 in [10]. The details are omitted.Lemma 4.2.
Let{fn} be a bounded sequence in 𝒵 which converges to 0 uniformly on compact subsets of D. Then limn→∞supz∈D|fn(z)|=0.Proof.
LetK=supn∥fn∥*<∞. Given any ɛ>0, there exist 0<t<1 such that (1-t)1/2<ɛ. If t<|z|<1, by Lemma 2.1, it follows that
(4.1)|fn(z)-fn(t|z|z)|=|∫t/|z|1zfn′(zt)dt|≤K∫t/|z|1|z|loge1-|zt|2dt≤2e-1/2K∫t/|z|1|z|(1-|zt|2)1/2dt≤Ke-1/2(1-t)1/2<Ke-1/2ɛ,where we use the fact that x1/2log(e/x)≤2e-1/2 for all x∈(0,1]. Then
(4.2)supt<|z|<1|fn(z)|≤Ke-1/2ɛ+sup|z|=t|fn(z)|.Noting that {fn} converges to 0 uniformly on compact subsets of D, we get
(4.3)limn→∞supz∈D|fn(z)|≤limn→∞supz∈D(Ke-1/2ɛ+sup|z|≤t|fn(z)|)=Ke-1/2ɛ.Hence, limn→∞supz∈D|fn(z)|=0.Theorem 4.3.
Letu be an analytic function on the unit disc D and φ an analytic self-map of D. Suppose that uCφ be a bounded operator on 𝒵. Then uCφ is compact if and only if the following are satisfied:
(4.4)(i)lim|φ(z)|⟶1-(1-|z|2)|u(z)(φ′(z))2|1-|φ(z)|2=0;(ii)lim|φ(z)|⟶1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|log11-|φ(z)|2=0.Proof.
Suppose thatuCφ is compact on the Zygmund space 𝒵. Let {zn} be a sequence in D such that |φ(zn)|→1 as n→∞. Without loss of generality, we may suppose that |φ(zn)|>1/2 for all n. We take the test functions:
(4.5)fn(z)=φ(zn)¯z-1φ(zn)¯((1+log11-φ(zn)¯z)2+1)(log11-|φ(zn)|2)-1-an,where
(4.6)an=|φ(zn)|2-1φ(zn)¯((1+log11-|φ(zn)|2)2+1)(log11-|φ(zn)|2)-1such that limn→∞an=0. By a direct calculation, we may easily prove that {fn} converges to 0 uniformly on compact subsets of D. From the proof of Theorem 3.1, we see that supn∥fn∥*<∞. Then {fn} is a bounded sequence in 𝒵 which converges to 0 uniformly on compact subsets of D. Then limn→∞∥uCφ(fn)∥*=0 by Lemma 4.1. Note that
(4.7)fn(φ(zn))=0,fn′(φ(zn))=log11-|φ(zn)|2,fn′′(φ(zn))=2φ(zn)¯1-|φ(zn)|2,it follows that
(4.8)‖uCφfn‖*≥‖uCφfn‖Z≥(1-|zn|2)|(φ′(zn))2(2u′(zn)φ′(zn)+φ′′(zn)u(zn))fn′(φ(zn))+u(zn)fn′′(φ(zn))(φ′(zn))2+u′′(zn)fn(φ(zn))|=(1-|zn|2)|(2u′(zn)φ′(zn)+φ′′(zn)u(zn))log11-|φ(zn)|2+(φ′′(zn))2u(zn)2φ(zn)¯1-|φ(zn)|2|≥(1-|zn|2)|(2u′(zn)φ′(zn)+φ′′(zn)u(zn))log11-|φ(zn)|2|-2(1-|zn|2)|φ(zn)¯u(zn)(φ′′(zn))2|1-|φ(zn)|2.Then
(4.9)limn→∞(1-|zn|2)|(2u′(zn)φ′(zn)+φ′′(zn)u(zn))log11-|φ(zn)|2|=limn→∞(1-|zn|2)2(1-|zn|2)|φ(zn)¯u(zn)(φ′′(zn))2|1-|φ(zn)|2,if one of these two limits exits.
On the other hand, let(4.10)hn(z)=h(φ(zn)¯z)φ(zn)¯(log11-|φ(zn)|2)-1-∫0zlog311-φ(zn)¯ωdω(log11-|φ(zn)|2)-2,so
(4.11)hn′(z)=(log11-φ(zn)¯z)2(log11-|φ(zn)|2)-1-log311-φ(zn)¯z(log11-|φ(zn)|2)-2,hn′′(z)=2φ(zn)¯1-φ(zn)¯zlog11-φ(zn)¯z(log11-|φ(zn)|2)-1-3φ(zn)¯1-φ(zn)¯zlog211-φ(zn)¯z(log11-|φ(zn)|2)-2.One may obtain that hn⇉0(n→∞) on compact subsets of D by a direct calculation and supn∥hn∥*≤C<∞ by the proof of Theorem 3.1. Consequently, {hn} is a bounded sequence in 𝒵 which converges to 0 uniformly on compact subsets of D. Then limn→∞∥uCφ(hn)∥*=0 by Lemma 4.1. Note that u∈𝒵,hn′(φ(zn))≡0 and limn→∞supz∈D|hn(z)|=0 by Lemma 4.2, it follows that
(4.12)0⟵‖uCφhn‖*≥‖uCφhn‖Z≥(1-|zn|2)|u(zn)hn′′(φ(zn))(φ′(zn))2+u′′(zn)hn(φ(zn))|≥(1-|zn|2)|u(zn)(φ′(zn))2|φ(zn)|1-|φ(zn)|2|-(1-|zn|2)|u′′(zn)hn(zn)|⟶(1-|zn|2)|u(zn)(φ′(zn))2|1-|φ(zn)|2,as n→∞. Then limn→∞(1-|zn|2)(|u(zn)(φ′(zn))2|/(1-|φ(zn)|2))=0. The proof of the necessary is completed.
Conversely, Suppose that (i) and (ii) hold. Let{fn} be a bounded sequence in 𝒵 which converges to 0 uniformly on compact subsets of D. Let M=supn∥fn∥*<+∞. We only prove limn→∞∥uCφ(fn)∥*=0 by Lemma 4.1. This amounts to showing that(4.13)supw∈D(1-|w|2)|(2φ′(w)u′(w)+φ′′(w)u(w))fn′(φ(w))|⟶0,supw∈D(1-|w|2)|u(w)(φ′(w))2fn′′(φ(w))|⟶0,supw∈D(1-|w|2)|u′′(w)fn(φ(w))|⟶0.By Lemma 4.2 and uCφ bounded on 𝒵, which implies that u∈𝒵, then
(4.14)supw∈D(1-|w|2)|u′′(w)fn(φ(w))|≤‖u‖Zsupz∈D|fn(z)|⟶0.If |φ(w)|≤r<1, by (3.5), then
(4.15)(1-|w|2)|(2φ′(w)u′(w)+φ′′(w)u(w))fn′(φ(w))|≤K1max|z|≤r|fn′(z)|.If |φ(w)|>r, by Lemma 2.1, then
(4.16)(1-|w|2)|(2φ′(w)u′(w)+φ′′(w)u(w))fn′(φ(w))|≤M(1-|w|2)|(2φ′(w)u′(w)+φ′′(w)u(w))|loge1-|φ(w)|2.Thus,
(4.17)supw∈D(1-|w|2)|(2φ′(w)u′(w)+φ′′(w)u(w))fn′(φ(w))|≤K1max|w|≤r|fn′(w)|+Msup|φ(w)|>r(1-|w|2)|(2φ′(w)u′(w)+φ′′(w)u(w))|loge1-|φ(w)|2.First, letting n tend to infinity and subsequently r increase to 1, one obtains that
(4.18)supw∈D(1-|w|2)|(2φ′(w)u′(w)+φ′′(w)u(w))fn′(φ(w))|⟶0,as n→∞. The third statement is proved similarly.
If|φ(w)|≤r<1, by (3.7), then(4.19)(1-|w|2)|u(w)(φ′(w))2fn′′(φ(w))|≤K2max|z|≤r|fn′′(z)|.If |φ(w)|>r, then
(4.20)(1-|w|2)|u(w)(φ′(w))2fn′′(φ(w))|≤M(1-|w|2)|u(w)(φ′(w))2|1-|φ(w)|2.Thus,
(4.21)supw∈D(1-|w|2)|u(w)(φ′(w))2fn′′(φ(w))|≤K2max|z|≤r|fn′′(z)|+Msup|φ(w)|>r(1-|w|2)|u(w)(φ′(w))2|1-|φ(w)|2,which also implies that
(4.22)supw∈D(1-|w|2)|u(w)(φ′(w))2fn′′(φ(w))|⟶0,as n→∞. This completes the proof of Theorem 4.3.In order to prove the compactness ofuCφ on the little Zygmund space 𝒵0, we require the following lemma.Lemma 4.4.
LetU⊂𝒵0. Then U is compact if and only if it is closed, bounded, and satisfies
(4.23)lim|z|⟶1supf∈U(1-|z|2)|f′′(z)|=0.The proof is similar to that of Lemma  1 in [6], we omit it.Theorem 4.5.
Letu be an analytic function on the unit disc D and φ an analytic self-map of D. Then uCφ is compact on the little Zygmund space 𝒵0 if and only if u∈𝒵0 and the following are satisfied:
(4.24)(i)lim|z|⟶1-(1-|z|2)|u(z)(φ′(z))2|1-|φ(z)|2=0;(ii)lim|z|→1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|log11-|φ(z)|2=0.Proof.
Assume that (i) and (ii) hold, andu∈𝒵0. By Theorem 3.2, we know that uCφ is bounded on the little Zygmund space 𝒵0. From (ii), we can show that
(4.25)lim|z|→1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|=0.
Suppose that f∈𝒵0 with ∥f∥*≤1. We obtain that
(4.26)(1-|z|2)|(uCφf)′′(z)|≤(1-|z|2)|(2φ′(z)u′(z)+φ′′(z)u(z))f′(φ(z))|+(1-|z|2)|f′′(φ(z))(φ′(z))2u(z)|+(1-|z|2)|u′′(z)f(φ(z))|≤(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|loge1-|φ(z)|2‖f‖*+(1-|z|2)|(φ′(z))2u(z)|1-|φ(z)|2(1-|φ(z)|2)|f′′(φ(z))|+(1-|z|2)|u′′(z)|‖f‖*≤(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|(1+log11-|φ(z)|2)+(1-|z|2)|(φ′(z))2u(z)|1-|φ(z)|2+(1-|z|2)|u′′(z)|,thus,
(4.27)sup{|(1-|z|2)(uCφf)′′(z)|:f∈Z0,‖f‖*≤1}≤(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|(1+log11-|φ(z)|2)+(1-|z|2)|(φ′(z))2u(z)|1-|φ(z)|2+(1-|z|2)|u′′(z)|,and it follows that
(4.28)lim|z|⟶1-sup{|(1-|z|2)(uCφf)′′(z)|:f∈Z0,‖f‖*≤1}=0,hence, uCφ is compact on 𝒵0 by Lemma 4.1.
Conversely, suppose thatuCφ is compact on 𝒵0.
First, it is obviousuCφ is bounded on 𝒵0, then by Theorem 3.2, we have u∈𝒵0 and that (3.24) holds. On the other hand, by Lemma 4.1 we have(4.29)lim|z|⟶1-sup{|(1-|z|2)(uCφf)′′(z)|:f∈Z0,‖f‖*≤M}=0,for some M>0.
Next, note that the proof of Theorem3.1 and the fact that the functions given in (3.8) are in 𝒵0 and have norms bounded independently of a, we obtain that(4.30)lim|z|⟶1-(1-|z|2)|u(z)(φ′(z))2|1-|φ(z)|2=0.
Similarly, note that the functions given in (3.16) are in 𝒵0 and have norms bounded independently of a, we obtain that(4.31)lim|z|⟶1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|log11-|φ(z)|2≤Clim|z|⟶1-(1-|z|2)|(uCφga)′′(z)|+lim|z|⟶1-(1-|z|2)|u′′(z)|‖ga‖*+lim|z|⟶1-(1-|z|2)2|φ(z)|1-|φ(z)|2|u(z)(φ′(z))2|,for |φ(z)|>1/2. So by (4.30) and u∈𝒵0, it follows that
(4.32)lim|z|⟶1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|log11-|φ(z)|2=0,for |φ(z)|>1/2. However, if |φ(z)|≤1/2, by (3.24), we easily have
(4.33)lim|z|→1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|log11-|φ(z)|2≤log43lim|z|⟶1-(1-|z|2)|2φ′(z)u′(z)+φ′′(z)u(z)|=0.This completes the proof of Theorem 4.5.Corollary 4.6.
Letφ be an analytic self-map of D. Then Cφ is a compact operator on 𝒵0 if and only if
(4.34)lim|z|⟶1-(1-|z|2)|(φ′(z))2|1-|φ(z)|2=0,lim|z|⟶1-(1-|z|2)|φ′′(z)|log11-|φ(z)|2=0.In the formulation of corollary, we use the notationMu on H(D) defined by Muf=uf for f∈H(D).Corollary 4.7.
Letu be an analytic function on the unit disc D. Then the pointwise multiplier Mu:𝒵(resp.𝒵0)→𝒵(resp.𝒵0) is a compact operator if and only if u≡0.



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*Source: 462482-2012-06-14.xml*