Bloch型空间上的Toeplitz 算子及分数阶导数刻画

贾策 曹广福 王晓峰 张艺渊

令μ为Cn中单位球Euclid Math TwoBA@n上的正Borel测度.本文主要刻画了Bloch型空间Bα（Euclid Math TwoBA@n）上以μ为符号的Toeplitz算子Tαμ的有界性和紧性，其中0<α<1. 当α>1时，本文利用分数阶导数给出了Bα（Euclid Math TwoBA@n）空间上的函数刻画的充要条件.

Toeplitz 算子；分数阶导数；Bloch 型空间

O177A2023.031001

收稿日期： 2021-01-23

作者简介： 贾策（1984-）， 天津人， 博士， 高级工程师， 主要研究方向为算子理论.E-mail： jiace@ibp.ac.cn

通讯作者： 曹广福.E-mail： guangfucao@163.com; 王晓峰.E-mail： wxf@gzhu.edu.cn; 张艺渊.E-mail： 1053296958@qq.com

On the characterization of Toeplitz operators and fractional derivatives  on Bloch-type space

JIA Ce1，2， CAO Guang-Fu1， WANG Xiao-Feng1， ZHANG Yi-Yuan1

（1. School of Mathematics and Information Science， Guangzhou University， Guangzhou 510006， China;

2. Institute of Biophysics， Chinese Academy of Sciences， Beijing 100101， China）

Let μ be the positive Borel measure on the unit ballEuclid Math TwoBA@n of Cn. We in this paper characterize the measure μ onEuclid Math TwoBA@n  for which the Toeplitz operator Tαμ is bounded or compact on the Bloch-type spaces Bα（Euclid Math TwoBA@n）， where 0<α<1. Additionlly， we also give a characterization for the functions on Bα（Euclid Math TwoBA@n） in terms of fractional derivatives， where α>1.

Toeplitz operator; Fractional derivative; Bloch-type space

（2010 MSC 30H30， 47B35， 26A33）

1 Introduction

Let Cn be the complex Euclidean space of dimension n and Euclid Math TwoBA@n the unit ball of Cn. For α>-1， let dvα（z）=Cα（1-|z|2）αdv（z） be the weighted volume measure， where cα=Γ（n+α+1）n！Γ（α+1） is a normalizing constant such that vα（Euclid Math TwoBA@n）=1. For α>-1 and 0

When the weight α=0， we simply write Ap（Euclid Math TwoBA@n） for Apα（Euclid Math TwoBA@n）.These are the standard Bergman spaces. When p=2， A2α（Euclid Math TwoBA@） is a Hilbert space. It is well known that the reproducing kernel of A2α（Euclid Math TwoBA@） is given byKα（z，w）=1/1-〈z，w〉n+1+α，where 〈z，w〉=∑ni=1ziw-i for z=（z1，…，zn）， w=（w1，…，wn）. The Bergman projection Pα is the orthogonal projection from L2（Euclid Math TwoBA@n，dvα） onto A2α（Euclid Math TwoBA@） defined byPα（f）（z）=fEuclid Math TwoBA@nKα（z，w）f（w）dvα（w）， f∈L1（Euclid Math TwoBA@n，dvα）.

The projection Pα naturally extends to an integral operator on L1（Euclid Math TwoBA@n，dvα）， see Ref. ［1， Theorem 2.11］.

贾 策， 等： Bloch型空间上的Toeplitz 算子及分数阶导数刻画

For α>-1， we also define the general Bergman projection of the measure μ as

Pα（μ）（z）=cα∫Euclid Math TwoBA@nKα（z，w）（1-|w|2）αdμ（w）.

For a measure μ onEuclid Math TwoBA@n and α>0， we define a Toeplitz operator as

Tαμ（f）（z）=cα-1∫Euclid Math TwoBA@nf（w）（1-|w|2）α-11-〈z，w〉n+αdμ（w），

f∈L1（Euclid Math TwoBA@n，dvα）.

Thus Tαμ（f）（z）=Pα-1（μf）（z）， where dμf（z）=f（z）dμ（z）. For α>0， the α-Bloch space Bα（Euclid Math TwoBA@n）， also known as the Bloch-type space， consists exactly of holomorphic functions f on Bn such that

‖f‖*Euclid Math TwoBA@α（Bn）=supz∈Bn（1-|z|2）α|SymbolQC@f（z）|<∞，

whereSymbolQC@f（z）=（fz1（z），...fzn（z））.

The Bloch-type space Bα（Euclid Math TwoBA@n） becomes a Banach space when equipped with the norm

‖f‖Bα（Euclid Math TwoBA@n）=|f（0）|+supz∈Euclid Math TwoBA@n（1-|z|2）α|SymbolQC@f（z）|.

It is well known that the above norm is equivalent to |f（0）|+supz∈Euclid Math TwoBA@n（1-|z|2）α|Rf（z）|， where Rf（z）=∑nk=1zkfzk（z） is the radial derivative of  f at z.

Let H（Euclid Math TwoBA@n） be the holomorphic functions onEuclid Math TwoBA@n， for any two real parameters γ and t such that neither n+γ nor n+γ+t is a negative integer， we define an invertible fractional differential operator Rγ，t：H（Euclid Math TwoBA@n）→H（Euclid Math TwoBA@n） as follows. If f（z）=∑∞k=0fk（z） is the homogeneous expansion of f， then

Rγ，tf（z）=

∑∞k=0Γ（n+1+γ）Γ（n+1+k+γ+t）Γ（n+1+γ+t）Γ（n+1+k+γ）fk（z）.

The inverse of Rγ，t， denoted by Rγ，t is given by

Rγ，tf（z）=

∑∞k=0Γ（n+1+γ+t）Γ（n+1+k+γ）Γ（n+1+γ）Γ（n+1+k+γ+t）fk（z）.

Toeplitz operators have been extensively studied on many spaces of analytic functions， see，  for instance， Refs. ［1-18］. A fundamental problem is to determine conditions on the measure， necessary or sufficient， for the corresponding Toeplitz operator to be either bounded or compact. There is also some previous work on the characterization of bounded and compact Toeplitz operators Tαμ on α-Bloch spaces. In Ref. ［14］， the authors have completely characterized complex measure μ on the unit diskEuclid Math TwoDA@  under some restricted conditions for which Tαμ is bounded  or compact on Bloch-type spaces Bα（Euclid Math TwoDA@） with 0<α<∞. In Ref. ［13］， due to the limitation of technique in Ref. ［16， Theorem 2］， the  authors have only characterized the positive Borel measure μ onEuclid Math TwoBA@n such that Tαμ is bounded or compact on Bα（Euclid Math TwoBA@n） with 1≤α<2. In this paper， we will use another different technique to characterize the positive Borel measure μ onEuclid Math TwoBA@n for which the Toeplitz operator Tαμ is bounded or compact on Bα（Euclid Math TwoBA@n） with 0<α<1， which is an extension of Ref. ［13］. Besides， we also give a characterization of functions on  Bα（Euclid Math TwoBA@n） in terms of fractional derivatives and its module with α>1.

Our main results about the boundedness or compactness of Toeplitz operators Tαμ on Bα（Euclid Math TwoBA@n） with 0<α<1 are given in Sections 3 and 4， and the main results about characterization of functions on Bα（Euclid Math TwoBA@n） in terms of fractional derivatives and its module with α>1 are shown in Section 5.

2 Preliminaries

For w∈Euclid Math TwoBA@n＼{0}， the automorphism mapping φw：Euclid Math TwoBA@n→Cn is given by

φw（z）=w-Pw（z）-1-|w|2Qw（z）1-〈z，w〉，

where Pw is the orthogonal projection from Cn onto the one dimensional subspace ［w］ generated by w， and Qw is the orthogonal projection from Cn onto Cn-［w］ defined by Qw=I-Pw. More information about the mapping φw is described in section 2.2 of Ref. ［11］ or section 1.2 of Ref. ［1］， where we can find the following identity

1-〈φw（z），w〉=1-|w|21-〈z，w〉（1）

In Ref. ［3， Lemma 2.1］， we can find the inequality

|w-φw（z）|2≤2（1-|w|2）|1-〈z，w〉|（2）

Lemma 2.1［1］ For any  α>-1 and z∈Euclid Math TwoBA@n， we have

|w-φw（z）|2≤2（1-|w|2）|1-〈z，w〉|

if f is a holomorphic function on Euclid Math TwoBA@n with

∫Bn（1-|w|2）α|f（w）|dv（w）<+∞，

where dv is the normalized volume measure on Euclid Math TwoBA@n.

Lemma 2.2［1］ Suppose c is real and t>-1. Then the integrals

Ic（z）=∫Sndσ（ζ）|1-〈z，ζ〉|n+c，z∈Euclid Math TwoBA@n

and

Jc，t（z）=∫Bn（1-|w|2）tdv（w）|1-〈z，w〉|n+1+t+c， z∈Euclid Math TwoBA@n

have the following asymptotic properties：

（i） If c<0， then Ic and Jc，t are both bounded in Bn;

（ii） If c = 0， then Ic（z）～Jc，t（z）～log11-|z|2，|z|→1-;

（iii） If  c > 0， then Ic（z）～Jc，t（z）～（1-|z|2）-c，|z|→1-.

Lemma 2.3［16］ Let 0<α<2， β be any real number satisfying the following properties：

（i） 0≤β≤α if 0<α<1;

（ii） 0<β<1 if α=1;

（iii） α-1≤β≤1 if 1<α≤2.

Then a holomorphic function f∈Bα（Euclid Math TwoBA@n） if and only if

Fβ（f）=supz，w∈Bn（1-|z|2）β（1-|w|2）α-β

|f（z）-f（w）||z-w|<∞.

Moreover， for any α and β satisfying above conditions， the following two semi-norms supz∈Euclid Math TwoBA@n（1-|z|2）α|SymbolQC@f（z）| and Fβ（f） are equivalent.

Lemma 2.4［13］ Suppose that  0<α<1. If  f∈Bα（Euclid Math TwoBA@n）， then

|f（z）|≤11-α‖f‖Bα（Euclid Math TwoBA@n）， z∈Euclid Math TwoBA@n.

Lemma 2.5 For any z，w∈Euclid Math TwoBA@n， the following estimate holds：

|w-z|≤2|1-〈z，w〉|.

Proof According to inequality （2）， we have

|w-φw（u）|2≤2（1-|w|2）|1-〈w，u〉|.

The change of variable u=φw（z） yields

|w-z|2≤2（1-|w|2）|1-〈φw（z），w〉|.

This together with （1） gives the desired result.

3 Bounded Toeplitz operators

In this section， we are going to characterize bounded Toeplitz operators on Bα（Euclid Math TwoBA@n） for 0<α<1. To this end， for a positive measure μ onEuclid Math TwoBA@n and α-β>0， we call μ satisfies the condition Sα，β if

Sα，β（μ）（z）=（1-|z|2）β·

∫Bn（1-|w|2）α-β-11-〈w，z〉n+α+1/2dμ（w）<∞.

In fact， such a positive measure μ satisfying the condition Sα，β does exist and there are many. Next， we will give an example under the assumption that α-β>0.

Example 3.1 Let

dμ（w）=（1-|w|2）γdv（w），

where w∈Euclid Math TwoBA@n and γ>0. If γ=1/2 and β>0， or γ>β+1/2， then Sα，β（μ）（z）<∞ for all z∈Euclid Math TwoBA@n.

Proof we have

Sα，β（μ）（z）=（1-|z|2）β·

∫Euclid Math TwoBA@n（1-|w|2）α-β-1+γ1-〈w，z〉n+α+1/2dv（w）=

（1-|z|2）β·

∫Euclid Math TwoBA@n（1-|w|2）α-β-1+γ1-〈w，z〉n+1+α-β-1+γ+β-γ+1/2dv（w）.

If γ=1/2 and β>0， then by （iii） of Lemma 2.2， we have

Sα，β（μ）（z）=（1-|z|2）β ∫Euclid Math TwoBA@n（1-|w|2）α-β-1/21-〈w，z〉n+1+α-β-1/2+βdv（w）～1.

If β-γ+1/2<0， that is， γ>β+1/2， then by （i） of Lemma 2.2， we get

∫Euclid Math TwoBA@n（1-|w|2）α-β-1+γ1-〈w，z〉n+1+α-β-1+γ+β-γ+1/2dμ（w）<∞，

hence Sα，β（μ）（z）<∞ for all z∈Euclid Math TwoBA@n.

Theorem 3.2 Let 0<α<1 and μ be the positive Borel measure onEuclid Math TwoBA@n. If μ satisfies the condition Sα，β then

Sα，β（μ）（z）=（1-|z|2）β

∫Euclid Math TwoBA@n（1-|w|2）α-β-11-〈w，z〉n+α+1/2dμ（w）<∞

is bounded on Bα（Euclid Math TwoBA@n） if and only if Pα-1（μ）∈Bα（Euclid Math TwoBA@n）.

Proof It can be seen from Theorem 7.6 of Ref. ［1］ that （A1（Euclid Math TwoBA@n））*Bα（Euclid Math TwoBA@n） under the integral pairing

〈f，g〉α-1=∫Euclid Math TwoBA@nf（z）g（z）（1-|z|2）α-1dv（z），

f∈A1（Euclid Math TwoBA@n），g∈Bα（Euclid Math TwoBA@n）.

In order to prove the boundedness of Tαμ， we need to show

〈f，Tαμ（g）〉α-1≤CfA1（Euclid Math TwoBA@n）gBα（Euclid Math TwoBA@n）

for any f∈A1（Euclid Math TwoBA@n） and g∈Bα（Euclid Math TwoBA@n）.

Applying Fubinis Theorem and the reproducing property， we obtain

〈f，Taμ（g）〉α-1=cα-1∫Euclid Math TwoBA@nf（z）Tαμ（g）（z）（1-|z|2）α-1dv（z）=

cα-1∫Euclid Math TwoBA@nf（z）cα-1∫Euclid Math TwoBA@ng（w）（1-|w|2）α-1（1-〈w，z〉）n+αdμ（w）（1-|z|2）α-1dv（z）=

cα-1∫Euclid Math TwoBA@ncα-1∫Euclid Math TwoBA@nf（z）（1-|z|2）α-1（1-〈w，z〉）n+αdv（z）g（w）（1-|w|2）α-1dμ（w）=

cα-1∫Euclid Math TwoBA@nf（w）g（w）（1-|w|2）α-1dμ（w）=

cα-1∫Euclid Math TwoBA@nPα（fg-）（w）（1-|w|2）α-1dμ（w）+

cα-1∫Euclid Math TwoBA@n（I-Pα）（fg-）（w）（1-|w|2）α-1dμ（w）I1+I2，

where

（I-Pα）（fg-）（w）=f（w）g（w）-cα∫Euclid Math TwoBA@nf（z）g（z）（1-|z|2）α1-〈w，z〉n+1+αdv（z）=

cα∫Euclid Math TwoBA@n（g（w）-g（z））f（z）（1-|z|2）α1-〈w，z〉n+1+αdv（z）.

Choosing β ≥ 0 such that α-β>0， by Lemmas 2.3 and 2.5， we get

|I2|=cα-1cα∫Euclid Math TwoBA@n∫Euclid Math TwoBA@n（g（w）-g（z））（1-|z|2）α（1-|w|2）α-11-〈w，z〉n+1+αdv（z）dμ（w）=

cα-1cα∫Euclid Math TwoBA@nf（z）（1-|z|2）α∫Euclid Math TwoBA@n（g（w）-g（z））（1-|w|2）α-11-〈w，z〉n+1+αdμ（w）dv（z）≤

cα-1cα∫Euclid Math TwoBA@n|f（z）|（1-|z|2）β∫Euclid Math TwoBA@n（1-|z|2）α-β（1-|w|2）β|g（w）-g（z）||w-z|·

（1-|w|2）α-β-1w-z1-〈w，z〉n+1+αdμ（w）dv（z）.

Since μ satisfies the condition Sα，β， hence |I2|≤C‖f‖A1（Bn）‖g‖Bα（Bn）.

Next， we consider I1. By Fubinis Theorem， we have

I1=cα-1∫Euclid Math TwoBA@nPα（fg-）（w）（1-|w|2）α-1dμ（w）=

cα-1∫Euclid Math TwoBA@ncα∫Euclid Math TwoBA@nf（z）g（z）（1-|z|2）α1-〈w，z〉n+1+αdv（z）（1-|w|2）α-1dμ（w）=

cα-1cα∫Euclid Math TwoBA@nf（z）g（z）∫Euclid Math TwoBA@n（1-|w|2）α-11-〈z，w〉n+1+αdμ（w）（1-|z|2）αdv（z）.

Let

Qα（μ）（z）=cα-1∫Euclid Math TwoBA@n（1-|w|2）α-11-〈z，w〉n+1+αdμ（w）. Then we have

I1=cα∫Euclid Math TwoBA@nf（z）g（z）Qα（μ）（z）（1-|z|2）αdv（z）.

By some elementary calculation， we obtain the following relation between Qα（μ） and Pα-1（μ）：

Qα（μ）（z）=Pα-1（μ）（z）+1n+αRPα-1（μ）（z）.

Since g（z） and Pα-1（μ） belong to Bα（Euclid Math TwoBA@n）， by Lemma 2.4， there exist constant C1 and C2 satisfying the following inequalities， respectively， |g（z）|≤C1‖g（z）‖Bα（Bn）， |Pα-1（μ）|≤C2Pα-1（μ）Bα（Euclid Math TwoBA@n）. Then

（1-|z|2）αg（z）Qα（μ）（z）=

|（1-|z|2）αg（z）Pα-1（μ）（z）+

g（z）n+α·（1-|z|2）αRPα-1（μ）（z）|<

（1-|z|2）α|g（z）｜·｜Pα-1（μ）|+

1n+α|g（z）‖·|Pα-1（μ）‖Bα（Euclid Math TwoBA@n）≤

C1C2‖g（z）‖Bα（Euclid Math TwoBA@n）‖Pα-1（μ）‖Bα（Euclid Math TwoBA@n）+

C1n+α‖g（z）‖Bα（Euclid Math TwoBA@n）‖Pα-1（μ）‖Bα（Euclid Math TwoBA@n）≤

C‖g（z）‖Bα（Euclid Math TwoBA@n）.

Thus we conclude that

|I1|≤C‖f‖A1（Euclid Math TwoBA@n）‖g（z）‖Bα（Euclid Math TwoBA@n）.

Therefore， Tαμ  is bounded on Bα（Euclid Math TwoBA@n）.

Conversely， if Tαμ is bounded on Bα（Euclid Math TwoBA@n）， then Tαμ（1）=Pα-1（μ）∈Bα（Euclid Math TwoBA@n）. This completes the proof.

4 Compact Toeplitz operators

In this section we present our main characterization of compact Toeplitz operator on Bα（Euclid Math TwoBA@n） with 0<α<1.

Theorem 4.1 Let 0<α<1. If the positive Borel measure μ satisfies lim|z|→1Sα，β（μ）（z）=0 then Tαμ is compact on Bα（Euclid Math TwoBA@n） if and only if Pα-1（μ）∈Bα（Euclid Math TwoBA@n）.

Proof Let {gn} be a sequence in Bα（Euclid Math TwoBA@n） such that ‖gn‖Bα（Euclid Math TwoBA@n）≤1 and gn（z）→0 uniformly on compact subsets ofEuclid Math TwoBA@n. Let  f be in the unit ball  of A1（Euclid Math TwoBA@n）， by a similar discussion as Theorem 3.1， we have

〈f，Taμ（gn）〉α-1=

cα-1∫Euclid Math TwoBA@nPα（fg-n）（w）（1-|w|2）α-1dμ（w）+

cα-1∫Bn（I-Pα）（fg-n）（w）（1-|w|2）α-1dμ（w）=

I1，n+I2，n，

where

I1，n=cα∫Euclid Math TwoBA@nf（z）gn（z）Qα（μ）（z）·

（1-|z|2）αdv（z），

I2，n=cα-1cα∫Euclid Math TwoBA@n∫Euclid Math TwoBA@n

（gn（w）-gn（z））f（z）（1-|z|2）α（1-|w|2）α-11-〈w，z〉n+1+αdv（z）dμ（w）.

Firstly， we consider I2，n. Let Bδ={z：|z|≤δ}， where 0<δ<1. We will divide the integral into two parts， say，

limn→∞|I2，n|=limn→∞cα-1cα∫Euclid Math TwoBA@n∫Euclid Math TwoBA@n（gn（w）-gn（z））f（z）（1-|z|2）α（1-|w|2）α-11-〈w，z〉n+1+αdv（z）dμ（w）=

limn→∞cα-1cα∫Euclid Math TwoBA@nf（z）（1-|z|2）α∫Euclid Math TwoBA@n（gn（w）-gn（z））（1-|w|2）α-11-〈w，z〉n+1+αdμ（w）dv（z）≤

limn→∞C∫Euclid Math TwoBA@n＼Euclid Math TwoBA@δ|f（z）|（1-|z|2）α∫Euclid Math TwoBA@n|gn（w）-gn（z）|（1-|w|2）α-11-〈w，z〉n+1+αdμ（w）dv（z）+

limn→∞C∫Euclid Math TwoBA@δ|f（z）|（1-|z|2）α∫Euclid Math TwoBA@n|gn（w）-gn（z）|（1-|w|2）α-11-〈w，z〉n+1+αdμ（w）dv（z）J1，n+J2，n.

For J1，n， since

lim|z|→1Sα，β（μ）（z）=lim|z|→1（1-|z|2）β∫Bn（1-|w|2）α-β-11-〈w，z〉n+α+1/2dμ（w）=0，

where β≥0 and α-β>0， for a fixed ε>0， let δ get sufficiently close to 1 such that Sα，β（μ）（z）<ε， combining with Lemmas 2.3 and 2.5， we have

J1，n≤limn→∞C∫Euclid Math TwoBA@n＼Euclid Math TwoBA@δ|f（z）|（1-|z|2）β∫Euclid Math TwoBA@n（1-|z|2）α-β（1-|w|2）β·

|gn（w）-gn（z）||w-z|（1-|w|2）α-β-1|w-z|1-〈w，z〉n+1+αdμ（w）dv（z）≤

limn→∞C∫Euclid Math TwoBA@n＼Euclid Math TwoBA@δ|f（z）‖|gn‖Bα（Euclid Math TwoBA@n）（1-|z|2）β∫Euclid Math TwoBA@n（1-|w|2）α-β-1|w-z|1-〈w，z〉n+1+αdμ （w）dv（z）≤

C∫Euclid Math TwoBA@n＼Euclid Math TwoBA@δ|f（z）|（1-|z|2）β∫Euclid Math TwoBA@n（1-|w|2）α-β-11-〈w，z〉n+α+1/2dμ（w）dv（z）≤

Cε∫Euclid Math TwoBA@n＼Euclid Math TwoBA@δ|f（z）|dv（z）≤Cε‖f‖A1（Euclid Math TwoBA@n）≤Cε.

For  J2，n， letEuclid Math TwoBA@r={z：|z|

J2，n≤limn→∞C∫Euclid Math TwoBA@δ|f（z）|（1-|z|2）α∫Euclid Math TwoBA@n＼Euclid Math TwoBA@r|gn（w）-gn（z）|（1-|w|2）α-11-〈w，z〉n+1+αdμ（w）dv（z）+

limn→∞C∫Euclid Math TwoBA@δ|f（z）|（1-|z|2）α∫Euclid Math TwoBA@n＼Euclid Math TwoBA@r|gn（w）-gn（z）|（1-|w|2）α-11-〈w，z〉n+1+αdμ（w）dv（z）K1，n+K2，n.

For K1，n， by a similar discussion as J1，n， we obtain K1，n≤Cε. For K2，n， since gn（z）→0 uniformly on any compact subsets ofEuclid Math TwoBA@n， we can choose n large enough such that |gn（w）-gn（z）|（1-|w|2）α-1≤ε uniformly for z belongs to compact subsets ofEuclid Math TwoBA@n， therefore

K2，n=limn→∞C∫Bδf（z）|（1-|z|2）α·

∫Euclid Math TwoBA@r|gn（w）-gn（z）|（1-|w|2）α-11-〈w，z〉n+1+α·

dμ（w）dv（z）≤

Cε∫Euclid Math TwoBA@δ|f（z）|（1-|z|2）α·

∫Euclid Math TwoBA@r11-〈w，z〉n+1+αdμ（w）dv（z）≤

Cε‖f‖A1（Bn）≤Cε.

Consequently， we have limn→∞|I2，n|≤Cε， which yields that limn→∞|I2，n|=0.

For  I1，n， since ‖gn（z）‖Bα（Bn）≤1， gn（z）→0 uniformly on any compact subsets ofEuclid Math TwoBA@n， we can choose n large enough so that |gn（z）|≤ε uniformly for z belongs to compact subsets ofEuclid Math TwoBA@n. Combined this with what we have estimated in the proof of Theorem 3.1， we obtain

limn→∞|I1，n|=

∫Euclid Math TwoBA@nf（z）gn（z）Qα（μ）（z）（1-|z|2）αdv（z）≤

Cε‖f‖A1（Euclid Math TwoBA@n）‖Pα-1（μ）‖Bα（Euclid Math TwoBA@n）≤Cε.

Thus limn→∞|I1，n|=0. Therefore， Tαμ is compact on Bα（Euclid Math TwoBA@n）.

Conversely， let Tαμ be compact on Bα（Euclid Math TwoBA@n）. Then Tαμ is bounded on Bα（Euclid Math TwoBA@n）. By Theorem 31， we have Pα-1（μ）∈Bα（Euclid Math TwoBA@n）. This completes the proof.

5 Characterization fractional derivatives on Bloch-type spaces

In this section， we will give a characterization of functions on Bα（Euclid Math TwoBA@n） in terms of fractional derivatives and its module with α>1.

For 0<α<1， the Lipschitz space Λα（Euclid Math TwoBA@n） consists of all holomorphic functions  f onEuclid Math TwoBA@n such that

‖f‖*Λα（Euclid Math TwoBA@n）=

sup|f（z）-f（w）|z-w|α：z，w∈Euclid Math TwoBA@n，z≠w<∞.

The space Λα（Euclid Math TwoBA@n） is called the holomorphic Lipschitz space of order α. It is well known that each space Λα（Euclid Math TwoBA@n） is contained in the ball algebra and contains the polynomials. For each α∈（0，1）， the holomorphic Lipschitz space Λα（Euclid Math TwoBA@n） is a Banach space with the norm ‖f‖Λα（Euclid Math TwoBA@n）=|f（0）|+‖f‖*Λα（Euclid Math TwoBA@n）. Please refer to Ref. ［1， Theorem 78］ for the detailed proof.

Lemma 5.1［1］ Suppose that 0<α<1，β>1 and f is  holomorphic inEuclid Math TwoBA@n. Then the following conditions are equivalent：

（i） f∈Λα（Euclid Math TwoBA@n）;

（ii） f is in the ball algebra and its boundary values satisfy

sup|f（ζ）-f（ξ）||ζ-ξ|α：ζ，ξ∈Euclid Math TwoBA@n，ζ≠ξ<∞;

（iii） （1-|z|2）1-α|Rf（z）| is bounded inEuclid Math TwoBA@n;

（iv） There exists a function g∈L∞（Euclid Math TwoBA@n） such that

f（z）=∫Euclid Math TwoBA@ng（w）dvβ（w）1-〈z，w〉n+1+β-α， z∈Euclid Math TwoBA@n;

（v） （1-|z|2）1-α|SymbolQC@f（z）| is bounded inEuclid Math TwoBA@n.

Lemma 5.2［1］ Suppose that α>0，β>1 and f is holomorphic inEuclid Math TwoBA@n. Then the following conditions are equivalent：

（i） f∈Euclid Math TwoBA@α（Bn）;

（ii） The function （1-|z|2）α|Rf（z）| is bounded inEuclid Math TwoBA@n;

（iii） There exists a function g∈L∞（Euclid Math TwoBA@n） such that

f（z）=∫Euclid Math TwoBA@ng（w）dvβ（w）1-〈z，w〉n+α+β， z∈Euclid Math TwoBA@n.

In view of Lemma 5.1 and Lemma 5.2， we clearly see that Λ1-α（Euclid Math TwoBA@n）=Bα（Euclid Math TwoBA@n）. for any 0<α<1. Therefore， in order to obtain a characterization of the functions on Bα（Euclid Math TwoBA@n） in terms of fractional derivatives with 0<α<1， we only need to get the corresponding result for Λα（Euclid Math TwoBA@n）， and Zhu in Ref. ［1， Theorem 7.17］ has gotten this， which is  shown in the following Lemma.

Lemma 5.3 Suppose that t>α>0. If γ is a real parameter such that neither n+γ nor n+γ+t is a negative integer， then a holomorphic function f inEuclid Math TwoBA@n belongs toEuclid Math TwoBA@α（Bn） if and only if the function （1-|z|2）t+α-1Rγ，tf（z） is bounded inEuclid Math TwoBA@n.

By using the relation Λ1-α（Euclid Math TwoBA@n）=Bα（Euclid Math TwoBA@n）， we give the characterization of functions on Bα（Euclid Math TwoBA@n） in terms of fractional derivatives with 0<α<1.

Theorem 5.4 Suppose that 0<α<1， t+α>1. If γ is a real parameter such that neither n+γ nor n+γ+t is a negative integer， then a holomorphic function f inEuclid Math TwoBA@n  belongs to Bα（Euclid Math TwoBA@n） if and only if the function （1-|z|2）t+α-1Rγ，tf（z） is bounded inEuclid Math TwoBA@n.

Lemma 5.5［1］ Suppose neither n+γ nor n+γ+t is a negative integer. If β=γ+N for  some positive integer N， then there exists a one-variable polynomial h of degree N such that

Rγ，t11-〈z，w〉n+1+β=h〈z，w〉1-〈z，w〉n+1+β+t.

There also exists a polynomial P（z，w） such that

Rγ，t11-〈z，w〉n+1+β+t=P（z，w）1-〈z，w〉n+1+β.

Lemma 5.6［1］ Suppose neither n+γ nor n+γ+t is a negative integer. Then the operator Rγ，t is the unique continuous linear operator on H（Euclid Math TwoBA@n） satisfying

Rγ，t11-〈z，w〉n+1+γ+t=11-〈z，w〉n+1+γ

for all w∈Euclid Math TwoBA@n. Similarly， the operator Rγ，t is the unique continuous linear operator on H（Euclid Math TwoBA@n） satisfying

Rγ，t11-〈z，w〉n+1+γ+t=11-〈z，w〉n+1+γ

for all w∈Euclid Math TwoBA@n.

Next we give the characterization of functions on Bα（Bn） in terms of fractional derivatives with α>1.

Theorem 5.7 Suppose that α>1 and t>0. If γ is a real parameter such that neither n+γ nor n+γ+t is a negative integer. Then a holomorphic function f  onEuclid Math TwoBA@n belongs to Bα（Euclid Math TwoBA@n） if and only if supz∈Euclid Math TwoBA@n（1-|z|2）α-1+tRγ，tf（z） is bounded onEuclid Math TwoBA@n.

Proof If f∈Bα（Euclid Math TwoBA@n）， then by Lemma 5.2 there exists a function f∈Bα（Euclid Math TwoBA@n） such that

f（z）=∫Euclid Math TwoBA@ng（w）dvβ（w）1-〈z，w〉n+α+β，z∈Euclid Math TwoBA@n，

here β=γ-α+N+1 and N is a large enough positive integer such that  β>-1. It follows from Lemma 5.5 that

Rγ，tf（z）=cα∫Euclid Math TwoBA@nh〈z，w〉g（w）1-〈z，w〉n+α+β+tdvβ（w），

z∈Euclid Math TwoBA@n，

where h is a one-variable polynomial of degree N-α+1. An application of Lemma 2.2 then shows the function 1-|z|2α-1+tRγ，tf（z） is bounded onEuclid Math TwoBA@n.

Next， we will assume that the function 1-|z|2α-1+tRγ，tf（z） is bounded onEuclid Math TwoBA@n. It follows from the remark of Ref. ［1， Lemma 2.18.］ that  Rγ，tf and Rγ+N，tf are comparable for any holomorphic function f， hence the function （cβ/cβ+α-1+t）（1-|z|2）α-1-tRγ+N，tf（z） is also bounded in Bn， where N is the same as the previous paragraph. By Lemma 2.1， we have

Rγ+N，tf（z）=

cβ∫Euclid Math TwoBA@n1-|w|2β+α-1+tRγ+N，tf（w）1-〈z，w〉n+1+β+α-1+tdv（w）=

∫Euclid Math TwoBA@n1-|w|2α-1+tRγ+N，tf（w）1-〈z，w〉n+1+γ+N+tdvβ（w），

where β=γ-α+N+1 is also as in the previous paragraph. Apply the operator Rγ+N，t inside the integral sign and use Lemma 5.6， we have

f（z）=∫Euclid Math TwoBA@n（1-|w|2）α-1+tRγ+N，tf（w）1-〈z，w〉n+1+r+Ndvβ（w）=

∫Euclid Math TwoBA@n（1-|w|2）α-1+tRγ+N，tf（w）1-〈z，w〉n+β+αdvβ（w）.

Since the function （1-|w|2）α-1+tRγ+N，tf（w） belongs to L∞（Euclid Math TwoBA@n） by Lemma 5.1， we see that f is in Bα（Euclid Math TwoBA@n） in view of Lemma 5.2.

Finally， we give the characterization of Bα（Euclid Math TwoBA@n） in terms of its module with α>1.

Theorem 5.8 Suppose that α>1 and f is holomorphic in Bn. Then f∈Bα（Euclid Math TwoBA@n） if and only if the function （1-|z|2）α-1|f（z）| is bounded inEuclid Math TwoBA@n.

Proof If f∈Bα（Euclid Math TwoBA@n）， then by Lemma 5.2， there exists a function g∈L∞（Euclid Math TwoBA@n） such that

f（z）=∫Euclid Math TwoBA@ng（w）dvβ（w）1-〈z，w〉n+α+β， z∈Euclid Math TwoBA@n，

where β>-1. Thus， by Lemma 2.2， for every z∈Euclid Math TwoBA@n， there exists a constant C>0 such that

|f（z）|=∫Euclid Math TwoBA@ng（w）1-〈z，w〉n+α+βdvβ（w）=

Cβ∫Euclid Math TwoBA@n（1-|z|2）βg（w）1-〈z，w〉n+α+βdv（w）≤

Cβ‖g‖∞∫Euclid Math TwoBA@n（1-|w|2）β1-〈z，w〉n+α+βdv（w）≤

C（1-|z|2）-（α-1）.

Thus （1-|z|2）α-1|f（z）| is bounded inEuclid Math TwoBA@n.

Conversely， if （1-|z|2）α-1|f（z）|≤M for some constant M>0， then by Lemma 2.1 we have

f（z）=cα-1∫Euclid Math TwoBA@n（1-|w|2）α-1f（w）1-〈z，w〉n+αdv（w），

z∈Euclid Math TwoBA@n.

Thus

Rf（z）=∑nk=1zkfzk（z）=cα-1∑nk=1zkzk∫Euclid Math TwoBA@n（1-|w|2）α-1f（w）1-〈z，w〉n+αdv（w）=

cα-1（n+α）∑nk=1zk∫Euclid Math TwoBA@n（1-|w|2）α-1f（w）wk1-〈z，w〉n+α+1dv（w）=

cα-1（n+α）∑nk=1zk∫Euclid Math TwoBA@n（1-|w|2）α-1f（w）〈z，w〉1-〈z，w〉n+α+1dv（w）.

By Lemma 2.2， there exists a constant C>0 such that

|Rf（z）|≤cα-1（n+α）·

∫Euclid Math TwoBA@n（1-|w|2）α-1|f（w）|1-〈z，w〉n+α+1dv（w）≤

CM（1-|z|2）-α

for all z∈Euclid Math TwoBA@n. This shows that f∈Bα（Euclid Math TwoBA@n）. The proof is end.

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引用本文格式：

中 文：  贾策，  王晓峰， 张艺渊. Bloch型空间上的Toeplitz 算子及分数阶导数刻画［J］. 四川大学学报：  自然科学版， 2023， 60：  031001.

英 文：  Jia C，  Wang X F， Zhang Y Y. On the characterization of Toeplitz operators and fractional derivatives  on Bloch-type space ［J］. J Sichuan Univ：  Nat Sci Ed， 2023， 60：  031001.