Carleson Measures and Toeplitz Operators on Doubling Fock Spaces?

Xiaofen LV

Abstract Given φ a subharmonic function on the complex plane C,with ?φdA being a doubling measure,the author studies Fock Carleson measures and some characterizations onμsuch that the induced positive Toeplitz operator Tμis bounded or compact between the doubling Fock space Fpφand F∞φwith 0

Keywords Doubling Fock spaces,Carleson measure,Toeplitz operators

1 Introduction

Suppose ν is a positive Borel measure on C,denoted by ν≥ 0.We call ν is doubling,if there exists some constant C>0 such that

for z∈C and r>0,where D(z,r)={w∈C:|w?z|

Suppose H(C)is the collection of all holomorphic functions on C.For 0

if 0

As far as we know,these doubling Fock spaces were first introduced by Christ[5].In 2003,Marco,Massaneda and Ortega-Cerd`a[15]studied the interpolating and sampling sequences for the doubling Fock spaces.After that,Marzo and Ortega-Cerd`a[16]gave quite sharp pointwise estimates of the Bergman kernel associated to these spaces.Let K(·,·)be the reproducing kernel for.The orthogonal projection P fromtocan be represented as

Givenμ≥0,Toeplitz operator Tμonis defined to be

if it can be well(densely)defined.

The behaviors of positive Toeplitz operators on Fock spaces have been studied by many authors.In 2010,Isralowitz and Zhu[13]discussed the characterizations onμ≥0 such that Tμis bounded,compact and in Schatten classes on the classical Fock space.Wang,Cao and Xia[23]studied the same problems on the Fock-Sobolev space F2,m.In[9],Hu and Lv characterized the boundedness and compacteness of Tμfrom one Fock spaceto anotherfor 10,Schuster and Varolin[22]obtained the necessary and sufficient conditions such that Tμis bounded or compact onfor 1

Carleson measures have been extensively applied to various problems in Hardy(and Bergman)space theory.On the classical Fock space,Carleson measures were first introduced in[13].The reference[4]is the first one where the so-called Carleson measures for Fock-Sobolev spaces were studied.See also[9–10,17,22].

In this paper,with 0

However,these two points are not available for doubling Fock spaces.For example,for φ(z)=|z|4,Constantin and Pel′aez[8]concluded

when p ≠q.

We always use C to denote positive constants whose value may change from line to line but does not depend on the functions being considered.Two quantities A and B are called equivalent if there exists some C such that C?1A ≤ B ≤ CA,written as“A ? B”.

2 Carleson Measures

In this section,we are going to introduce the Fock Carleson measure,which will be used in the following sections.First,we list some notations and preliminary results.These results can be found in[12]and[15].

Recall that,φ is a subharmonic,real-valued function on C,which satisfies dν= ?φdA a doubling measure,and ρ(·)is the positive radius such that ν(D(z,ρ(z)))=1 for z ∈ C.Given r>0,write Dr(z)=D(z,rρ(z)).There exists some constant C>0 such that for z ∈ C and w∈Dr(z),

Moreover,for fixed r>0 we have m1,m2>0 such that

which follows from the triangle inequality.Given r>0,we say a sequencein C is an r-lattice if{Dr(ak)}kcovers C and the disks ofare pairwise disjoint,see[15]for details.For m>0,there exists some positive integer N such that

For our later use,we need the concepts of averaging function and Berezin transform.Givenμ≥0,the average ofμis defined as

For 00,there is some C>0 such that for f∈H(C)and z∈C,we obtain

Thus,in a way similar to Lemma 2.2 in[9],we get

For t>0,we set the t-Berezin transform ofμto be

where kt,z(w)=K(w,z)/kK(·,z)kt,φis the normalized Bergman kernel for.When φ(z)=,the t-Berezin transform is closely connected with the heat flow as mentioned in[1].

We also need some other spaces.Let 0

The space Lpis defined as

and lpconsists of all sequencewith

To prove the main results,we need some lemmas.Lemma 2.1 lists some well-known results about the Bergman kernel for.Most of them can be seen in[12,16,19].We only need to show the statements of(3)and(4)for p=∞.Notice that 1/p=0 if p=∞.

Lemma 2.1 The Bergman kernel K(·,·)satisfies:

(1)There exist positive constants C and ? such that

for w,z∈C.

(2)There exists some r0>0 such that

whenever z∈C and w∈Dr0(z).

(3)For 0

(4)For 0

Proof Given z∈C,by[19],there holds

This,together with(2.4),we have

for z∈ C.Hence,by(2.6)and(5)in[15,p.869],if|z|is large enough,we have some β ∈(0,1)such that

for w∈C.This tells us the statement(4)is true.The proof is completed.

Next,we are going to introduce(vanishing)(p,q)-Fock Carleson measures.When n=1,all the spaces studied in[4,9–10,13,17,22]are special cases of ours here.When p=q>1,this is just the Fock Carleson measure discussed in[19].

Definition 2.1 Let 0

And also,we callμa vanishing(p,q)-Fock Carleson measure if

The following three theorems characterize(vanishing)(p,q)-Fock Carleson measures for all possible 0

Theorem 2.1 Let 0

(1)μis a(p,q)-Fock Carleson measure.

Theorem 2.2 Let 0

Theorem 2.3 Let 0

Remark 2.1 In the setting of classical Fock spaces,μis a(p,q)-Fock Carleson measure for some p≤q if and only ifμis a(vanishing)(p,q)-Fock Carleson measure for all possible p≤q(see[9]).This is still available forwith,which can be seen in[10].Now,Theorems 2.1–2.2 tell us that this phenomenon can only occur when ρ(·) ? 1.Theorems 2.1–2.3 extend the results in[4,9–10,13,17,19,22].From these theorems above,μis a(p,q)-Fock Carleson measure if and only if it is a(tp,tq)-Fock Carleson measure,t>0.So(p,q)-Fock Carleson measure can be simply called-Fock Carleson measure,and written asfor simplicity.

3 Toeplitz Operators

In this section,for 0

Lemma 3.1 Letμ be a t-Fock Carleson measure for some t>0.Toeplitz operator Tμis well-defined onfor all 00 anddμ for measurable set E ? C.

Proof In a way similar to the proof of[12,Lemma 3.1],we can conclude that Tμis welldefined onfor 0

In a way similar to the proof of(2.5),we obtain

Sinceμ is a t-Fock Carleson measure,Theorems 2.1–2.3 tell us that there exists some t1∈ R such that

Thus

Notice that,for p,s>0 and real number k,there is C>0 such that

(see[12,Lemma 2.1]).This,together with(3.2)shows

as j→∞.Hence,(3.1)is true.The proof is ended.

In this position,we will characterize the boundedness and compactness of positive Toeplitz operators fromtoor fromtowith 0

Theorem 3.1 Let 0

(1)Tμ:→is bounded if and only ifμis aCarleson measure.Furthermore,

(2)Tμ:→is compact if and only ifμis a vanishingCarleson measure.

Proof(1)First,we assume that Tμ:→is bounded.For z∈C,Lemma 2.1 yields

This,together with Theorem 2.1,shows thatμis aCarleson measure,and

On the other hand,suppose thatμis aCarleson measure.Thenis bounded on C for δ>0,which follows from Theorem 2.1.Given f∈,by(2.4),we get

Given z ∈ C,since K(·,z)f(·)∈ H(C),applying(2.5)to the weight 2φ,we have

(3.6),Lemma 2.1 and Theorem 2.1 imply

Therefore,Tμis bounded fromto,and

This,combined with(3.5),gives(3.3).

(2)Suppose thatμis a vanishingCarleson measure.By Theorem 2.2,we know

SettingμRas in Lemma 3.1,TμRis compact fromto.Moreover,μ?μR≥0,Tμ?μRis bounded from Fp(φ)to,and for r>0,

Thus,(3.3)and Theorem 2.1 tell us

as R→∞.So we can conclude that Tμ:→is also compact.Conversely,we assume that Tμ:→is compact.Thenis bounded for δ>0.Notice that{kp,z:z ∈ C}is bounded in.So{Tμkp,z:z∈C}is relatively compact in.For any sequencewith,there exists a subsequence ofconverging to some h in.Without loss of generality,we may assume

We only need to show h≡0.For any w∈C,(3.7)implies

By Lemma 2.1 and(3.2),we have

So,for any ε>0,there is some R>0 such that

Since kp,z→0 uniformly on compact subsets of C as z→∞,we get

while j is large enough.By H¨older’s inequality,Lemma 2.1 and(3.2),we obtain

where C is independent of j and ε.Therefore

On the other hand,(3.8)implies

for w∈C.Hence,h≡0,which means

This,combined with(3.4),yields that as j→∞,

Thus,

We conclude thatμis a vanishingCarleson measure,which follows from Theorem 2.2.The proof is complete.

Theorem 3.2 Let 0

Furthermore,

Proof By[12,Lemma 2.4],we get the equivalence of(3),(4)and(5).To prove(1)?(5),we suppose that Tμis bounded fromto.Given any bounded sequence{λk}kand r0-lattice{ak}k,where r0as in Lemma 2.1,set

In a way similar to[10,Lemma 2.4],we haveandSince Tμis bounded fromto,we have Tμf∈.Khinchine’s inequality and Fubini’s theorem show

where ψkis the k-th Rademacher function on[0,1].Since Tμ:→is bounded,the inequality above is no more than

Since the balls{Dr0(ak)}kcover C,(2.3)gives

By(2.4),we know

These yield

Notice that Lemma 2.1 shows

Setting βk=|λk|p,we know{βk}k∈ l∞.Hence

Therefore

and

To prove(3)?(2),we need to give

for some δ>0.We deal with the case 10,(3.7),Lemma 2.1 and Ho¨lder’s inequality show

We now deal with the case p≤1.For some r-lattice{ak}kand f∈,(2.1)and(2.4)show

By the triangle inequality,we have m1>0 such that Dr(ak)?Dm1r(w)if w∈Dmr(ak).Hence,(2.1)and(2.3)yield

Fubini’s theorem and(3.2)give

Therefore,(3.11)comes true.TakingμRas in Lemma 3.1,then

and

The estimate(3.9)follows from(3.10)–(3.11).The proof is completed.

By Theorems 2.3 and 3.2,we obtain the following corollary.

Corollary 3.1 Let 1

(1)Tμ:→is bounded.

(2)Tμ:→is compact.

AcknowledgementThe author would like to thank the referees for making some very good suggestions.