Composition Operators on the Weighted Bergman Space in the Unit Polydisk

HU Rong,HU Peng-yan

(1.College of Mathematics and Finance-Economics,Sichuan University of Arts and Science,Dazhou 635000,China;2.College of Mathematics and Computational Science,Shenzhen University,Shenzhen 518060,China)

Composition Operators on the Weighted Bergman Space in the Unit Polydisk

HU Rong1,HU Peng-yan2

(1.College of Mathematics and Finance-Economics,Sichuan University of Arts and Science,Dazhou 635000,China;2.College of Mathematics and Computational Science,Shenzhen University,Shenzhen 518060,China)

We consider the boundedness of composition operators on the Bergman space, and shows that when it is induced by automorphism is always bounded.At first we got a change of variables formula,which is very important for the proof of the boundedness of composition operators,and then obtain an upper bound for the special operator norm on Bergman space.

bergman space;polydisk;composition operators;automorphism

§1. Introduction

Let U be the unit disk in the complex plane and dA be the normalized area measure on U. Forα＞−1,z∈U,let

More generally,let Unbe the unit polydisk inℂnand forα＞−1,z=(z1,z2,···,zn)∈Un,

Let a=(a1,a2,···,an)∈Unandϕabe the involution of Un,thenϕa(0)=a and

Defi nition 1[2]Supposeα＞−1 and p＞0.f defined on Unis said to be in Lp(Un,d Vα) if

Defi nition 2[2]Let H(Un)be the class ofallholomorphic functions with domain Un.For α＞−1 and p＞0 the weighted Bergman spaceis defined as

which is called the Bergman norm of f in

Letϕbe a holomorphic map of Un.For f∈H(Un),the composition operators Cϕis defined by Cϕ(f)=f◦ϕ.The boundedness of Cϕon functions space have been studied by many mathematicians,see[3-9].For example,it is proved that Cϕis bounded on the Bergman space in unit disk and the Bloch space in unit polydisk.When n=1,it is well-known that Cϕis bounded on each of spaceand

But this result does not extend to the higher dimensional polydisk.For example,the map ϕ(z1,z2,···,zn)=(z1,z2,···,zn)is known to induce an unbounded composition operator onsee[11].

In this paper we obtain that the composition operator Cϕinduced by the automorphism of Unis bounded on the Bergman spaceand also get an upper bound of this kind of the operator.Before proving the main result,we give a change of variables formula which is very important for us.

§2.Main Results

On the unit ball Bnofℂn,we have

whereϕis a automorphism of Bn,f∈L1(Bn,d Vα)and a=ϕ(0).

In this paper,as the first main result,we obtain a similar formula on the unit polydisk which have great help for the second.

Theorem 1 Let Aut(Un)denote the class of all automorphism with domain Un.Suppose f∈L1(Un,d Vα),there existsθ1,θ2,···,θnand permutationτ:(1,2,···,n)→(1,2,···,n),such that

whereϕ∈Aut(Un)and a=(a1,a2,···,an)=ϕ(0).

The boundedness of composition operator Cϕwhich induced by the automorphism of Unis the following.

Theorem 2 Let 0＜p＜∞andα＞−1.For anyϕ∈Aut(Un),the composition operator Cϕis bounded onand the operator norm satisfies

where a=(a1,a2,···,an)=ϕ(0).

In order to prove the main results,we need the following lemmas.

Lemma 1[13]Let JRϕadenote the real Jacobian determinant of involutionϕaof Un,then

Lemma 2[14]For anyϕ∈Aut(Un),there exists a linear mapping F(z)=(eiθ1zτ(1),···, eiθnzτ(n))and a point a∈Un,such that

whereϕais a involution of Un,θ1,θ2,···,θnare real numbers andτ:(1,2,···,n)→(1,2,···,n)is a permutation.

We are now in a position to prove the main results.

Proof of Theorem 1 We discuss this in two different conditions.

(a)If a=0,by the proof of Lemma 2,there exists a permutationτ:(1,2,···,n)→(1,2,···,n)such that

It is wellknown that

So itcould be proved that d V(z)is invariant under the action ofϕ(z)=(eiθ1zτ(1),···,eiθnzτ(n)). Since

then by a natural change of variables z=ϕ−1(w)and taking notice thatϕ−1andϕhave the same structure,we have d V(ϕ−1(w))=d V(w).The desired result could be produced.

(b)If a/=0,by Lemma 2,there exists a linear mapping F(z)=(eiθ1zτ(1),···,eiθnzτ(n)) such that

Let z=ϕa◦F(w)=ϕ−1a◦F(w),then

Since

where JRϕa[F(w)]is the real Jacobian determinant ofϕaat point F(w),then by Lemma 1,we have

Also we can easily obtain that

So we have

Proof of Theorem 2 For f∈Apα(Un),by Definition 2 and Theorem 1 we have

where

Then

This shows that the composition operators Cϕ(f)=f◦ϕis bounded on Apα(Un)and the desired result is proved.

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tion:32A30,47B38

1002–0462(2014)01–0009–05

Chin.Quart.J.of Math. 2014,29(1):9—13

date:2013-10-17

Supported by the Scientific Research Fund of Sichuan Provincial Education Department(13ZB0101)

Biography:HU Rong(1985-),female,native of Nanchong,Sichuan,a lecturer of Sichuan University of Arts and Science,M.S.D.,engages in several complex.

CLC number:O174.56 Document code:A