QIAN Rui-shen,YANG Liu

(1.School of Mathematics and Statistics,Lingnan Normal University,Zhanjiang 524048,China)

(2.Department of Mathematics,Shaanxi Xueqian Normal University,Xi’an 710100,China)

Abstract:In this paper,by using Hardy space’s properties and elementary calculations,we study boundary characterization and boundary multipliers of analytic Moerry space,and Toeplitz operators acting on Hardy space to analytic Moerry space are also investigated.For the above questions,the necessary and sufficient conditions are obtained.

Keywords:boundary multiplies;Toeplitz operators;Hardy spaces;Morrey spaces

1 Introduction

Denote by T the boundary of the open unit disk D in the complex plane C.Let H(D)be the space of analytic functions in D.For 0

Let H∞be the space of bounded analytic function on D consisting of functions f∈H(D)with

We refer to[1,2]for Hpand H∞spaces.

For λ ∈ (0,1],denote by L2,λ(T)the Morrey space of all Lebesgue measurable functions f on T that satisfy

where|I|denotes the length of the arc I and

Clearly,L2,1(T)coincides with BMO(T),the space of functions with bounded mean oscillation on T(cf.[3,4]).Similar to a norm on BMO(T)given in[4,p.68],a norm on L2,λ(T)can be defined by

From Xiao’s monograph[5,p.52],

It is well known that if f ∈ H2,then its non-tangential limit f(ζ)exists almost everywhere for ζ∈ T.For λ ∈ (0,1],the analytic Morrey space L2,λ(D)is the set of f ∈ H2with f(ζ) ∈ L2,λ(T).It is clear that L2,1(D)is BMOA,the analytic space of functions with bounded mean oscillation(cf.[3,4]).For λ ∈ (0,1],L2,λ(D)is located between BMOA and H2.It is worth mentioning that there exists a isomorphism relation between analytic Morrey spaces and M?bius invariant Qpspaces via fractional order derivatives of functions(see[6]).Recall that for 0

where dA is the area Lebesgue measure on D and σa(z)=is the M?bius transformation of the unit disk D interchanging a and 0.See[5,7]for a general exposition on Qpspaces.Recently,the interest in L2,λ(D)spaces grew rapidly(cf.[8–13]).

An important problem of studying function spaces is to characterize the multipliers of such spaces.For a Banach function space X,denote by M(X)the class of all multipliers on X.Namely,

Bao and Pau[14]characterized boundary multipliers of Qpspaces.Stegenga[15]described multipliers of BMO(T)which is equal to L2,1(T).It is natural to consider multipliers of L2,λ(T)with λ ∈ (0,1)in this paper.

Given a function ? ∈ L2(T).Let T?be the Toeplitz operator on H2with symbol ? defined by

For the study of Toeplitz operators on Hardy spaces and Bergman spaces,see,for example,[16,17].We refer to[9]for the results of Toeplitz operators on L2,λ(D)spaces.

The aim of this paper is to consider boundary multiplies and Toeplitz operators associated with analytic Morrey spaces.In Section 2,using a characterization of L2,λ(T)in terms of functions with absolute values,we characterize the multipliers of L2,λ(T).In Section 3,we characterize the boundedness and compactness of Toeplitz operators from Hardy spaces to analytic Morrey spaces.

Throughout this paper,we write a.b if there exists a constant C such that a≤Cb.Also,the symbol a≈b means that a.b.a.

2 Boundary Multiplies of Analytic Morrey Spaces

By the study of certain integral operators on analytic Morrey spaces,Li,Liu and Lou[8]proved that M(L2,λ(D))=H∞.In this section,applying a characterization of L2,λ(T)in terms of absolute values of functions,we characterize M(L2,λ(T)),boundary multiplies of analytic Morrey spaces.

Given f∈L2(T),letbe the Poisson extension of f.Namely,

where

Let 0< λ <1.From[5,p.52],f ∈ L2,λ(T)if and only if

where ? is the Laplace operator.Also,f ∈ L2,λ(D)if and only if

We need the following useful inequality(see[18,Lemma 2.5]).

Lemma A Suppose that s>?1,r,t≥0,and r+t?s>2.If t

for all z,ζ∈D.

Now we characterize L2,λ(T)via absolute values of functions as follows.See[9]for the analytic version of the following result.

Theorem 2.1 Let 0<λ<1 and f∈L2(T).Then f∈L2,λ(T)if and only if

Proof Let f∈L2(T).It is well known(cf.[19,p.564])that

for all z∈D.Combining this with the Fubini theorem,we obtain that for any a∈D,

By Lemma A and the same argument in[19,p.563],we get that

Thus,

for all a∈D.

Let

From(2.1)and(2.3),we get that f ∈ L2,λ(T).

On the other hand,let f ∈ L2,λ(T).Without loss of generality,we may assume that f is real valued.Denote bythe harmonic conjugate function of.Set.The Cauchy-Riemann equations give.Thus g ∈ L2,λ(D).By the growth estimates of functions in L2,λ(D)(cf.[8,Lemma 2]),one gets that

for all z∈D.Consequently,

Combining this with(2.3),f ∈ L2,λ(T),we get that

The proof is completed.

Denote by L∞(T)the space of essentially bounded measurable functions on T.Using Theorem 2.1,we characterize multipliers of L2,λ(T)as follows.

Theorem 2.2 Let 0< λ <1.Then M(L2,λ(T))=L∞(T).

Proof Let f ∈ L∞(T)and g ∈ L2,λ(T).From Theorem 2.1,one gets that

Applying Theorem again,we know that fg ∈ L2,λ(T).Thus L∞(T)? M(L2,λ(T)).

On the other hand,let f ∈ M(L2,λ(T)).By the closed graph theorem,there exists a positive constant C such that|||fg|||L2,λ(T)≤ C|||g|||L2,λ(T)for any g ∈ L2,λ(T).Set h=f/C.Clearly,h ∈ L2,λ(T).We deduce that|||hn|||L2,λ(T)≤ |||h|||L2,λ(T)for all positive integer n.As mentioned in Section 1,L2,λ(T) ? L2(T).Form the closed graph theorem again,there exists a positive constant C1satisfying

for all f ∈ L2,λ(T).Consequently,

Since n is arbitrary,we know that h∈L∞(T).Hence f∈L∞(T).The proof is completed.

3 Toeplitz Operators from Hardy Spaces to Analytic Morrey Spaces

In this section,we characterize the boundedness and compactness of Toeplitz operators from the Hardy space Hpto the analytic Morrey space L2,1?2p(D)for 2

Following[9],we use a norm of L2,λ(D),λ ∈ (0,1),defined by

The following well-known lemma can be found in[17].

Lemma B Suppose s>0 and t>?1.Then there exists a positive constant C such that

for all z∈D.

Applying some well-known results of Toeplitz operators and composition operators on Hardy spaces,we characterize the boundedness of T?from Hardy spaces to analytic Morrey spaces as follows.

Theorem 3.3 Let 2

Proof Suppose that T?is bounded from Hpto L2,1?2(D).For b∈D,let

Note that p>2.By the well known estimates in[20,p.9],one gets that

Thus functions fbbelong to Hpuniformly for all b∈D.Consequently,

Note that

On the other hand,let ? ∈ L∞(T).It is well known that T?is bounded on Hp(cf.[21–23]).Namely,kT?gkHp.k?kL∞(T)kgkHpfor all g ∈ Hp.Let f ∈ Hp,we deduce that

By the well-known characterization of composition operators on Hp(cf.[17,Theorem 11.12]),we get that

Thus,

Note that p>2.By Lemma B,we get that.The proof is completed.

We characterize the compactness of Toeplitz operators from Hptoas follows.

Theorem 3.4 Let 2

Proof It suffices to prove the necessity.Let? D be a sequence such that|an|→1 as n→∞.Set

As explained in the proof of Theorem 3.3,.Clearly,fn→0 uniformly on compact subsets of D as n→∞.Since T?is compact,we get that

By the proof of Theorem 3.3,one gets thatfor all n.Consequently,→0,n→∞.Since anis arbitrary and?? is harmonic,by the maximum principle,≡0 on D.Hence ?=0 on T.We finish the proof.