Characterizations of a Class of Dirichlet-Type Spaces and Related Operators

Abstract

In this paper, some characterizations are given in terms of the boundary value and Poisson extension for the Dirichlet-type space . The multipliers of and Hankel-type operators from to are also investigated.

1. Introduction

Let be the unit disk of complex plane . For , the Hardy space, denoted by , is the space consists of all such that

Here, is the space of analytic functions on .

Let denote the boundary of and denote the normalized Lebesgue measure on . Let be a positive Borel measure on . An is said to belong to the space , called the Dirichlet-type space, ifwhere

The space was introduced by Richter in [1] for studying analytic two isometrics. It was shown in [1] that . The norm on is defined as follows:

The space is a Hilbert space with when . If , then coincides with the Dirichlet space . By (Proposition 2.2 in [1]), we have

Here,

Let . We say that if

The norm of the space is given by

The space has been investigated by many authors. In [2], Richter and Sundberg studied the cyclic vectors of . Shimorin studied the reproducing kernels and extremal functions of in [3], see [4–6], for the study of Carleson measure for . The study of composition operators and Toeplitz operators on can be found in [7, 8], respectively, see [9–11], for more study of the space .

In this paper, we provided some characterizations for the space by the boundary value and Poisson extension. Moreover, we study the multipliers of and the Hankel-type operator from to .

In this paper, we always assume that is a positive Borel measure on and is a positive constant that may differ from one occurrence to the other. The notation means that there exists a such that . The notation indicates that and also .

2. Characterizations of the Space

Let . The Poisson extension of , denoted by , is

It is well known that is a harmonic function on .

Let denote the space of all functions on with continuous partial derivatives. For , the gradient of is defined by

First, we state some lemmas.

Lemma 1 (see [6, 8]). Let . Then,if and only if

Remark 1. Let and such that (a.e.) for . Then,For , let denote the boundary value of .

Corollary 1. Let . Then, if and only if .

Proof. Since , then . The desired result follows from Lemma 1.

Lemma 2. Let . Then, the following statements are equivalent:(a).(b).(c), where

Proof. This implication follows by Lemma 1.

Proof.   For , setFrom [11], we see that is subharmonic withBy Green’s formula, we obtainAccording to (17) and (18) and Hardy-Littlewood’s identity (see page 238 in [12]), we haveThe proof is complete.

Theorem 1. Let . Then, the following statements are equivalent:(a).(b).(c) and(d) and there exists a harmonic function such that on and

Proof. This implication follows by Lemma 2 and If , then . SinceWe get from Lemma 2 and Corollary 1.
Inequality (20) impliesLet . Then, . Thus, By Lemma 2,Assume that is a harmonic function such that . Note that is the least harmonic function equal to or greater than (see [12]); hence, . By Lemmas 1 and 2 and Corollary 1, . The proof is complete.

3. Multipliers of

Let . The Carleson box is

Assume that is a positive Borel measure on . If then we say that is a Carleson measure.

If there exists a constant (see [4, 5])then we call that is a -Carleson measure.

Let and . is called the pointwise multipliers of if . We denote the space of all pointwise multipliers of by .

Lemma 3. Let be a positive Borel measure on . Then, is a -Carleson measure if and only iffor all .

Proof. First, we assume that is a -Carleson measure. Suppose that . Without loss of generality, let be a real-valued function. Suppose that is the harmonic conjugate of . Set . Then, by the Cauchy–Riemann equation. From Lemma 2.3 in [7] and Lemma 1, we obtainConversely, for , by Corollary 1, and . Then,which implies that is a -Carleson measure.

Theorem 2. if and only if and is a -Carleson measure.

Proof. Assume that and is a -Carleson measure. Let . By Remark 1, we obtainBy Lemma 1 and Corollary 1, we obtainIn addition, since is a -Carleson measure, by Lemma 3, we haveCombining (32)–(34), we obtain that .
Conversely, assume that . Then, by Theorem 2.7 in [6], we see that . For , by the Closed Graph Theorem, Lemma 1, and Corollary 1, we obtainNext, we show that is a -Carleson measure. From the fact that , we obtainThen, by (35) and (36),which implies that is a -Carleson measure.
By Theorem 2, we obtain the following result.

Corollary 2. Let . Then, if and only if .

4. Hankel-Type Operators on

Let denote the set of all polynomials on . From [1, 2], we see that is dense in . Let

From Theorem 1.10 in [13], we see that is a bounded projection. Here, is the Bergman space which consists of all such that . For , we define a Hankel-type operator on by

Lemma 4 (see Theorem 2.3 in [10]). Let . Then, if and only ifwhere .

Lemma 5 (see Theorem 3.4 in [10]). Let be the operator defined byThen, is bounded.

Theorem 3. Let such that is a -Carleson measure. Then, is a -Carleson measure.

Proof. Suppose that is a -Carleson measure. Then, by Lemma 5,for all . So, it is enough to show thatfor every .
By Hölder’s inequality, we haveConsequently, by Lemma 4, we obtainThe desired result follows.

Theorem 4. Let . Then, the operator is bounded if and only if is a -Carlson measure.

Proof. Suppose that is a -Carlson measure. Let . Then, . By Lemma 4, we get that andSo, is bounded.
Conversely, assume that is bounded. We need to prove thatBy Hölder’s inequality we haveSinceby Lemma 4 and the fact that is bounded, we obtainThe proof is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author was supported by NNSF of China (nos. 11701222 and 11801347), China Postdoctoral Science Foundation (no. 2018M633090), and Key Projects of Fundamental Research in Universities of Guangdong Province (no. 2018KZDXM034).