Hankel Operators on the Weighted 𝐿^𝑃-Bergman Spaces with Exponential Type Weights

Abstract

We characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weighted -Bergman spaces with exponential type weights.

1. Introduction

Let be the unit disc in the complex plane and the area measure on , and denote by the space of all analytic functions in . Let with . For , the weighted Bergman space is the space of functions such that Note that is the closed subspace of consisting of analytic functions. Since the space is a reproducing kernel Hilbert space, for each , there are functions with , where is the usual inner product in . The orthogonal projection from to is given by where .

Given so that there exists a dense subset of with for , the big Hankel operator with symbol is densely defined by where is the orthogonal projection of onto .

We write . Then the -equation can be written by For , we look for a solution of minimal -norm. Notice that the solution of minimal norm is the one that is orthogonal to the kernel of on ; that is, . Then, if solves (4), we get The linear operator given by is called the canonical solution operator to on .

For any , obviously and That is, the canonical solution operator coincides with the big Hankel operator acting on with symbol . Motivated by this fact, we now consider Hankel operators with conjugate analytic symbols on . For , we do not necessarily have . Let . Then is dense in . For symbol such that we consider the densely defined big Hankel operator on given by

A positive function on is said to belong to the class if it satisfies the following three properties.(a)There exists a constant such that (b)There exists a constant such that (c)For each , there are constants and such that

In this paper, we characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weighted -Bergman spaces with exponential type weights as follows.

Theorem 1. Let and . Let with , and the function is in the class . Then extends to a bounded linear operator on if and only if

Theorem 2. Let and . Let with , and the function is in the class . Then extends to a compact linear operator on if and only if

In [1], Luecking firstly proved the same results in the context of the ordinary -Bergman spaces. For -Bergman spaces with exponential type weights, the same results were proved in [2–4]. Moreover, Schatten-class Hankel operators are also indicated in their papers.

The expression means that there is a constant independent of the relevant variables such that , and means that and .

2. Preliminaries

From now on we assume that , , and the function is in the class . The following notations will be frequently used: where and are the constants in the conditions (a) and (b) in Section 1 and

Lemma 3 (see [5]). For each , there exists a constant (depending on ) such that for , one has

By using the upper estimates for in Lemma 3, Arroussi and Pau [5] proved that the orthogonal projection projects boundedly onto for .

Lemma 4 (see [6]). Let and . Then,

By using the third Green formula, we get the following two approximation results.

Lemma 5 (see [7]). For small , there exists such that where is a harmonic function in with .

Lemma 6 (see [7]). For small , one has the estimate where is defined in Lemma 5.

The following is a certain submean value property of . We follow the proof of ([8], Lemma 19).

Lemma 7. Let . For any small , there exists such that for any and (a);(b), provided .

Proof. (a) By Lemma 5, there exists some constant such that for . Since is harmonic, there is an analytic function on such that and on . Thus we have . Hence by the submean value property together with Lemma 5, we get
(b) We begin as follows: Since is harmonic, there is an analytic function such that Note that On the other hand, For , we have Hence we have Thus

Despite that the next result was proved in [4], we give the proof of different method by using Lemma 7.

Proposition 8. There is an independent of such that

Proof. Let . By (b) of Lemma 7, we have Hence it follows that
Note that by Lemma 3 and (a) of Lemma 7, we have Thus we have if we choose small .

3. Hankel Operators on

For the proof of boundedness of Hankel operator on -Bergman spaces with exponential type weights in [2–4], they used Hörmander's -estimates for . However, for -Bergman spaces, we need the following -estimates for .

Theorem 9 (see [6]). Let with . Suppose that the function satisfies conditions (a) and (b) in Section 1. Let . Then there is a solution to the equation such that provided the right hand side integral is finite.

Let be an analytic function in satisfying the condition (8). Let By the reproducing formula in we get

Lemma 10. Let . Then

Proof. Consider First, Now, We take a large constant so that . Then
Thus we get the result.

Theorem 11. Let . Let . Then extends to a bounded linear operator on if and only if

Proof. Assume first that By Theorem 9, there is a solution of the equation such that Since is the minimal -norm solution of the -equation, we have . In [5], Arroussi and Pau proved that the orthogonal projection projects boundedly onto for . Thus we have By (45) and (46), we have which shows that can be extended to a bounded linear operator on .
Conversely, assume that is bounded on . Then we have Using Proposition 8 and Lemma 3, there exists such that Hence we have Since is an analytic function in , by the Cauchy estimates applied to , we can now conclude Thus we get the result.

Lemma 12. Let . Then

Proof. Let be a compact subset of . We choose a large constant so that . Then we have for where .

Theorem 13. Let . Let . Then extends to a compact linear operator on if and only if

Proof. Suppose now that is compact on . Then by Riesz-Tamarkin compactness theorem, we have uniformly in . Now, by Lemma 12, as . Thus we have
We choose so that
Then This implies that
For , the inclusion holds, and This implies that
Assume now that It is enough to show that for any sequence that is bounded in norm and converges uniformly to zero on compact subsets, we have as . As in relation (46), we have
Now Since converges uniformly to zero on compact subsets, Now Hence is compact.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This is supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (NRF-2011-0013740).