Cezhong TONG Junfeng LI

Abstract In this paper,the authors characterize Carleson measures for the weighted Bergman spaces with Bkoll weights on the unit ball.They apply the Carleson embedding theorem to study the properties of Toeplitz-type operators and composition operators acting on such spaces.

Keywords Bkoll weight,Bergman space,Carleson measure,Toeplitz operator,Composition operator

1 Introduction

Recall that for r>0 and z∈Bn,the set

is a Bergman metric ball centered at z with radius r.

The Carleson tent over a non-zero z∈Bnis defined to be the set:

We define the Dp,a,bcharacteristic of two weights u,σ by

Throughout the paper,we will use the notation defined in Section 2 to make this more compactly as:

Let p′be the conjugate number of p.We denote by u∈Bp,bif[u]Bp,b:=[u,σ]Dp,0,b<∞,whereis the dual weight of u.If p>1,according to the Hlder inequality one can obtain that[u]Bp,b≥1.To be more precise,

Constantin proved Carleson-type embedding theorems for weighted Bergman spaces with Bkollweights on the unit disk,and characterized the boundedness,compactness and Schatten class of Toeplitz type operators,integral operators and composition operators in[3].The goal of this paper is to generalize these results to the setting of the unit ball.The key tool is the“test function”(1−z)−sin the weighted Bergman spaces with Bkollweights.

The paper is organized as follows.In Section 2,we briefly give the preliminaries and background information.We recall a covering lemma and prove the key lemma on the norm estimate of the test function(1−z)s.In Section 3,we completely characterize the Carleson embedding theorem from(u)to Lq(dµ).In Section 4,we use the Carleson measure to study the Toeplitz type operators.In Section 5,the boundedness and compactness of composition operators are characterized.

Throughout the paper,for real positive quantities Q1and Q2,we write Q1Q2(or Q2Q1)if there is a positive constant C(independent of the“key”variables)such that Q1C·Q2.And we write Q1Q2if Q1Q2and Q1Q2.

2 Preliminaries

Let Φzbe the involution of Bn.Using Φz,we define the so-called Bergman metric,β on Bn,by:

Let Bβ(z,r)be the ball in the Bergman metric of radius r centered at z.It is well known that for w∈Bβ(z,r)there holds:

and the characteristic functions

We need the following covering lemma in the proofs of our main results.

Lemma 2.1(see[13,Theorem 2.23])There exists a positive N such that for any 0

We will also use the following class of weights which is denoted by Cp,b.A positive locally integrable weight u belongs to Cp,b,or say u satisfies Cp,bcondition if

for some 0−1.To see this,we note that for a given r,there is a a′∈Bnsuch that Bβ(a,r)⊂Ta′with comparable volumes.It follows that

where the constant C>0 may depend on r.See the details in[6].Interested readers can also refer in[11]for further discussions on the Dp,a,bweights.

Lemma 2.2Suppose that u∈Cp,bfor some p>1,and let t,s∈(0,1),and z,w∈Bnwith β(z,w)0.Then we have

where the constant is independent of z and w.

ProofNotice that if Bβ(z,t)⊂Bβ(w,s),then u∈Cp,band β(z,w)

For general case,we have Bβ(z,t),Bβ(w,s)⊂Bβ(w,t+s+r),and hence we have

The proof is completed.

Lemma 2.3(see[6,Lemma 3.1])If p0>1,0

where the constant involved is independent of z∈Bn.

Lemma 2.4Let p>0,p0>1,b>−1 and the weight u∈Bp0,b.We have

where the constant involved is independent of w∈Bn.

ProofIf z∈Tw,then

On the other hand,we firstly consider the case whenDenote by

Then we can obtain the following estimate under this decomposition of Bn.

Denote by

One can easily find that

3 Embedding Theorems

In this section,we will study the boundedness and the compactness of the embedding I:u)→Lq(dµ).We firstly consider the case 0

Theorem 3.1Suppose that q≥p>0,p0>1,u∈Bp0,bis a weight and µ is a positive Borel measure on Bn.Then the following conditions are equivalent.

for all holomorphic f in Bn;

ProofFirstly we prove(a)⇒(b).By choosing an s>n+1+b we get

where we use the condition(a)in the second inequality and Lemma 2.4 in the third inequality.

To prove(b)⇒(c),we let r be sufficiently small and fixed.It will be done to prove

Then we use the condition of Bp0,bto get ub(Bβ(a,r))≃ub(Ta′)as follows,which is analogues to the proof of Lemma 2.2.That is

Hence we have

To prove(a)⇒(e),we denote by

By following Lemma 2.4,we have

The proof of(c)⇒(d)is obvious.

It remains to prove(d)⇒(a).If f is holomorphic in Bn,then by Lemma 2.3 we have

where the last inequality is deduced by Lemma 2.1.The proof is completed.

Let us turn to the case 0

Then Khinchine’s inequality is the following.

Khinchine’s InequalityFor 0

if and only if the function

ProofSince u∈Bp0,bfor some p0>1,by Lemma 2.3 we have

The sufficiency will be clarified by the following computation:

It remains to prove the necessity.For{cj}∈ℓp,we define

where{aj}⊂Bnand r0>0 satisfy the conditions in Lemma 2.1.It is followed by[6,Theorem 4.1]that

According to the embedding condition,we now get

Applying Fubini’s theorem and Khinchine’s inequality,we deduce that

where the last inequality follows the condition(a)and(3.1).

It is easy to see that

where the constant involved depends on r only.Then we can obtain

Since the sequence{cj}is chosen arbitrarily fromℓp,the sequence

That is

That completes the proof.

The measure µ characterized in Theorems 3.1–3.2 is called a((u),q)-Carleson measure.Using very similar methods to those above,one can also characterize the compactness of the embedding map fromu)to Lq(dµ),where the measure µ is also called the((u),q)-vanishing Carleson measure.We just include the statements without the proofs.

Theorem 3.3Suppose that q≥p>0,p0>1,r>0,u∈Bp0,bis a weight and µ is a positive Borel measure on Bn.Then the following conditions are equivalent.

whenever{fk}is bounded in(u)that converges to 0 uniformly on compact subsets of Bn;

(b)

(c)

(d)

where{ak}is the sequence described in Lemma 2.1.

Theorem 3.4Suppose that p>q>0,r>0,u∈Bp0,bis a weight and µ is a positive Borel measure on Bn.Then the embedding I:(u)→Lq(dµ)is compact if and only if I is bounded.

4 Toeplitz-Type Operators

In this section,we will characterize the boundedness,compactness and Schatten class of Toeplitz type operators onu)for the Bkollweight u.

According to Lemma 2.3,the reproducing kernel of(u)will be denoted by K(z,w).Given a positive Borel measure µ on Bn,the Toeplitz operator Tµassociated with µ on(u)is the linear transformation defined by

By the straightforward computation above,one can get

According to that observation and applying Theorems 3.1 and 3.3,one can get the following characterization of Toeplitz operators.

Theorem 4.1If p0>1 and u∈Bp0,b.Let µ be a positive Borel measure on Bn.Then the following are equivalent:

Furthermore,the following are equivalent:

where{ak}is the sequence described in Lemma 2.1.

ProofWe note that Tµ=I*I from(4.1),where I*is the adjoint operator of the embedding I:(u)→L2(dµ).The proof is completed.

It is well known that the Berezin trasform plays a role in the theory of Toeplitz operator.The Berezin transform of the Toeplitz operator Tµis given by

Proposition 4.1If p0>1 and u∈Bp0,b.Let µ be a positive Borel measure on Bn.If Tµis bounded onu),then the Berezin transformis bounded on Bn.

ProofSince u∈Bp0,b,by Lemma 2.3 one gets that:

where z∈Bnand r>0.Let{aj}and r>0 be chosen as in Lemma 2.1.We have

The proof is completed.

Recall that if{ek}is an orthonormal basis ofthen the Bergman kernel inis given by

and

If p≥1,the Toeplitz operator Tµ∈Spif and only if

Lemma 4.1Suppose that p0>1 and u∈Bp0,b.Then there is an r∈(0,1)such that

ProofBy Lemma 2.3,we have

Then we will get one of the inequalities in(4.3).

To prove the reverse inequality,by choosing s≥(n+1+b)the function

The proof is completed.

Now we turn to characterize the measure µ so that Tµbelongs to the Schatten class Sp,p≥1.We aim to extend the results in the setting of standard Bergman spaces(see[7,9])to the Bergman spaces with Bkollweights.

Theorem 4.2If p0>1 and u∈Bp0,b.Let µ be a positive Borel measure on Bn.Then Toeplitz operator Tµbelongs to the Schatten class Spfor some p≥1,if and only if

where the sequence{aj}and r>0 satisfy the conditions in Lemma 2.1.

ProofWe firstly prove the sufficiency.Since u∈Bp0,b,one can employ Lemma 2.3 to get that

for every positive integer k.Since(4.4)implies that Tµis compact on(u),we have

Plugging this into(4.2),we have

which proves the sufficiency.

By[6,Theorem 4.1],A is surjective,hence A*TµA∈Sp.That is

On the other hand,we have

which completes the proof.

Proposition 4.2If p0>1 and u∈Bp0,b.Let µ be a positive Borel measure on Bn.If Tµ∈Sp((u)),then the Berezin transform(u)for 0

ProofFirstly,we fix r as stated in Lemma 4.1.Suppose Tµ∈Sp,that is

where{aj}and r>0 satisfy the conditions in Lemma 2.1.By Lemma 4.1,we have

That completes the proof.

5 Composition Operators

Every holomorphic ϕ:Bn→Bninduces a composition operator

namely,Cϕf=f◦ϕ.When n=1,it is well known that Cϕis always bounded on(D),and Cϕis compact onD)if and only if

When n>1,there are lots of unbounded composition operators on classical Bergman spaces with standard weights Ap(Bn,dvb).Interested readers can see more details in[4,13].

For a positive weight function u on Bn,we consider the pullback measure of dub=udvbunder the map ϕ:Bn→Bn,given by

for any Borel subset E of Bn.

By the embedding theorems in Section 3,one can get the characterization of bounded and compact composition operators between different weighted Bergman spaces.

Theorem 5.1Let 01 and u be a Bp0,bweight.If ϕ:Bn→Bnis a holomorphic map,and let µϕ,u,bbe the pullback measure defined above.Then the following are equivalent:

(a)Composition operator Cϕ:is bounded;

(b)the pullback measure µϕ,u,bis a Carleson measure:

holds for all w∈Bn.

ProofThe proof of(b)⇒(a)follows directly from the definition of the pullback measures and the characterization of the Carleson measures in Theorem 3.1.

To prove(a)⇒(c),we denote by

It remains to prove(c)⇒(b).Just note that

which completes the proof.

Remark 5.1We can deduce from the proof above that

A similar argument gives the following characterization of the compactness of Cϕon(u).

Theorem 5.2Let 01 and u be a Bp0,bweight.If ϕ:Bn→Bnis a holomorphic map,and let µϕ,u,bbe the pullback measure.Then Cϕ:is compact if and only if µϕ,u,bis a vanishing Carleson measure if and only if

Now we turn to the case 0

Theorem 5.3Let 01 and u be a Bp0,bweight.If ϕ:Bn→Bnis a holomorphic map,and let µϕ,u,bbe the pullback measure defined above.Then the following are equivalent:

where the sequence{aj}and r>0 satisfy the conditions in Lemma 2.1.

ProofNote that K(z,w)is the reproducing kernel of(u).The adjoint operatormay be computed as

So we have

Then it is clear that Cϕ∈Spif and only if Tµϕ,u,b∈The proof is completed.

6 Final Remarks

In[14],Zhu characterizes the Schatten p class of Toeplitz operator on the standard weighted Bergman spaceswhen 0

(1)We have the local conditions on the weight u.So the weight u behaves“stable”if we only do the analysis on the small local pieces of the unit ball.

(2)We lack the global property of the weight u.The mbius automorphism can map 0 to any other point in the unit ball.So u could be hard to control when we change the variables by mbius automorphisms.The original piece can be transferred to any other new piece of the ball.

It is well known that the Berezin transform is a powerful tool to study the Toeplitz operators on the standard weighted Bergman spaces.Our results Propositions 4.1–4.2 give the necessary conditions of the boundedness and Schatten class of the Toeplitz operators in terms of the Berezin transforms.The sufficient parts seem to be rather different to the case of the standard weighted Bergman spaces,especially in several complex variables.In the proof of the standard weighted Bergman spaces,the reproducing kernel ofIt is relatively easy to estimate the kernel from below.In the setting of Bkollweights,we also need to estimate the reproducing kernel of)from below.Because it is uncertain that the kernel coincides with any explicit function.We have to estimate the kernel without any explicit computation.By generalizing the results in[5],on the unit disk,we have settled the estimate of the kernel from below and completely characterized the Toeplitz operator in terms of the Berezin transform(see[12]).Unfortunately,the method seems to be invalid on the unit ball,because the zeros distribution of the holomorphic functions in several complex variables varies the case in the one complex variable.

AcknowledgementsTong thanks to Prof.Brett D.Wick for discussions and Department of Mathematics of Washington University in St.Louis for its hospitality and support.The authors thank the anonymous referee(s)for the careful review and suggestions.