Qingze LIN

Abstract The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Pelez,who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces.However,their characterizations for the boundedness are not complete.In this paper,the author completely characterizes the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces Dpα to Dqβ(−1<α,β and 0

Keywords Volterra type operator,Boundedness,Compactness,Weighted Dirichlet space,Order boundedness

1 Introduction

Let D be the unit disk of a complex plane and let H(D)be the space consisting of all the analytic functions on D.For 0

For any fixed function g∈H(D),the Volterra type operator Tgand its companion operator Sgare defined,respectively,by

for any f∈H(D).

Let|I|be the normalized Lebesgue length of I,which is an interval of∂D.The Carleson square S(I)is defined by

For any s>0 and any positive Borel measureµin D,we say thatµis an s-Carleson measure if there is a positive constant C such that

For a space X of analytic functions on D,it is often useful to know the integrability properties of the functions f∈X.That is to determine for which positive Borel measure µ on D there is a continuous inclusion X⊂Lp(dµ),or equivalently,by the closed graph theorem,there exists a positive constant C such that for any f∈X,

Theorem 1.1Suppose that 0

For the case of p≥q,the corresponding characterizations were partly investigated in[9,26,31],where several questions were still open.

In Section 2,we completely characterize the boundedness of Volterra type operators Tgand Sgfrom the weighted Dirichlet spacesto(−1<α,β and 0

2 Boundedness of Volterra Type Operators

The Volterra type operator Tgwas introduced by Pommerenke[27]to study the exponentials of BMOA functions and in the meantime,he proved that Tgacting on the Hardy-Hilbert space H2is bounded if and only if g∈BMOA.After his work,Aleman,Siskakis and Cima[1–2]studied the boundedness and compactness of Tgon the Hardy space Hp,where they showed that Tgis bounded(compact)on Hp,0

Recently,Lin et al.[20–22]characterized the boundedness and the strict singularities of the Volterra type operators acting on the(derivative)Hardy spaces and weighted Banach spaces with general weights.Li and Stevi[18–19]introduced the generalized composition operators(also called generalized Volterra type operators)acting on Zygmund spaces and Bloch type spaces and so forth,which had attracted intensive attentions.For instance,Mengestie[24]obtained a complete description of the boundedness and compactness of the product of the Volterra type operators and composition operators on the weighted Fock spaces,and recently,he studied the topological structure of the space of Volterra-type integral operators on the Fock spaces endowed with the operator norm(see[25]).Furthermore,by applying the Carleson embedding theorem and the Littlewood-Paley formula,Constantin and Pelez[5]obtained the boundedness and compactness of Tgon the weighted Fock spaces and investigated the invariant subspaces of the classical Volterra operator Tzon such spaces.

The multiplication operator Mgis defined by

The following relation holds:

Then we characterize the boundedness of these operators.

ProofThis follows directly from Theorem 1.1 and the closed graph theorem.

Then it holds that

It is easy to prove that fa∈and there exists a positive constant C such that for all a∈D,‖fa‖≤C.Denoting∆(a,r)as the pseudo-hyperbolic disk with center a and radius r,we have

As an immediate corollary,we obtain the known results originally proven by Zhao[33].

ProofThis follows immediately from the fact that DSg=MgD,where D is the differential operator.

ProofSince(Mgf)(z)=f(0)g(0)+(Tgf)(z)+(Sgf)(z),the sufficiency follows immediately from Theorems 2.1–2.2.It remains to prove the necessity.In this case,it is obvious that if we can prove thatas|z|→1−,then all the other statements follow immediately from Theorems 2.1–2.2 again.

Given a∈D,define the function Faby

Then Fa(a)=0,and the remainder of the proof is essentially similar to the converse part of the proof of Theorem 2.2.

3 Compactness of Volterra Type Operators

For any s>0 and µ a positive Borel measure in D,we say µ is a vanishing s-Carleson measure if

Theorem 3.1Suppose that 0

Proof(1)is known(see[15]for example).

For(2),we notice that this condition is,in deed,a vanishing-logarithmic Carleson measure and the proof of it is basically similar to that of[26,Theorem 3.1(ii)].

Now for(3),since when p>α+2,it holds that⊂H∞,where H∞is the space of all the bounded analytic functions on D,the compactness follows easily by the standard arguments.

Then we characterize the compactness of these operators.

Theorem 3.2Let−1<α,β,g∈H(D)and 0

ProofThis follows directly from Theorem 3.1.

As an immediate corollary,we obtain the known results originally proven byukoviand Zhao[6].

Given a∈D,define the function Faby

Then Fa(a)=0,and the remainder of the proof is similar to that of Theorem 3.3.

4 Order Boundedness of Volterra Type Operators

Let X be a Banach space of holomorphic functions defined on D,q>0,(Ω,A,µ)be a measure space and

An operator T:X→Lp(Ω,A,µ)is said to be order bounded if there exists a nonnegative function g∈Lp(Ω,A,µ)such that for all f∈X with‖f‖X≤1,it holds that

Order boundedness plays an important role in studying the properties of many concrete operators acting between Banach spaces like Hardy spaces,weighted Bergman spaces and so forth(see[13–14,29–30]).Recently,order boundedness of weighted composition operators between weighted Dirichlet spaces were studied in[10,28].In this section,we investigate the order boundedness of Volterra type operators between weighted Dirichlet spaces.Recall that in this case,if we define the measure Aβby dAβ(z)=(1−|z|2)βdA(z),then an operator T:→is order bounded if and only if there exists a nonnegative function g∈Lq(Aβ)such that for all f∈with‖f‖≤1,it holds that

Before proving the results,we first give some auxiliary lemmas.

Now we are ready to prove our results.

Theorem 4.1Let−1<α,β,g∈H(D)and 0

Hence,by Lemma 4.1,the inequality holds

Then by Lemma 4.1,for all f∈with‖f‖Dpα≤1,

The proofs of(2)and(3)are almost similar to that of(1),thus we omit the details.

By Theorems 2.1,3.2 and 4.1,we obtain the following corollary.

Corollary 4.1Let−1<α,β,g∈H(D)and α+2

Theorem 4.2Let−1<α,β,g∈H(D)and 0

ProofThe proof is similar to that of Theorem 4.1 except that in this case,we resort to Lemma 4.2 instead of Lemma 4.1.

Theorem 4.3Let−1<α,β,g∈H(D)and 0

By taking

then h∈Lq(Aβ)since

Accordingly,Mg:is order bounded.

For any z∈D,we consider the function

In conclusion,for all z∈D,

which implies that

The proofs of(2)and(3)are similar to that of(1)by some minor modifications.For example,in(2),we take the test function

Thus the proof is complete.

By Theorems 4.1–4.3,we obtain the following corollary.

Corollary 4.2Let−1<α,β,g∈H(D)and 0

AcknowledgementsThe author is grateful to the referee for his(or her)valuable comments and suggestions.Also,he would like to thank the brilliant mathematician,Loo-Keng Hua,for his excellent books which had inspired him into mathematics.At last,he wants to express his gratitude to the great star,Bruce Lee,for inspiring him with the fighting spirit.