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Abstract and Applied Analysis
Volume 2010, Article ID 198608, 7 pages
http://dx.doi.org/10.1155/2010/198608
Research Article

On an Integral-Type Operator from Zygmund-Type Spaces to Mixed-Norm Spaces on the Unit Ball

Mathematical Institute of the Serbian Academy of Sciences and Arts, 36/III Knez Mihailova, 11000 Belgrade, Serbia

Received 23 June 2010; Accepted 27 October 2010

Academic Editor: H. B. Thompson

Copyright © 2010 Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The boundedness and compactness of an integral-type operator recently introduced by the author from Zygmund-type spaces to the mixed-norm space on the unit ball are characterized here.

1. Introduction

Let be the open unit ball in , its boundary, the normalized volume measure on , and the class of all holomorphic functions on . Strictly positive, bounded, continuous functions on are called weights.

For an with the Taylor expansion , let be the radial derivative of , where is a multi-index, and .

A positive, continuous function on the interval is called normal [1] if there are and and , such that If we say that a function is normal, we also assume that it is radial, that is, ,.

Let be a weight. By , we denote the class of all such that and call it the Zygmund-type class. The quantity is a seminorm. A norm on can be introduced by . Zygmund-type class with this norm will be called the Zygmund-type space.

The little Zygmund-type space on , denoted by , is the closed subspace of consisting of functions satisfying the following condition

For , , and normal, the mixed-norm space consists of all functions such that where and is the normalized surface measure on . For , , and , the space is equivalent with the weighted Bergman space .

In [2], the present author has introduced products of integral and composition operators on as follows (see also [35]). Assume ,, and is a holomorphic self-map of , then we define an operator on by The operator is an extension of the operator introduced in [6]. Here, we continue to study operator by characterizing the boundedness and compactness of the operator between Zygmund-type spaces and the mixed-norm space. For some results on related integral-type operators mostly in , see, for example, [3, 627] and the references therein.

In this paper, constants are denoted by ; they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that . If both and hold, then one says that .

2. Auxiliary Results

In this section, we quote several lemmas which are used in the proofs of the main results.

The first lemma was proved in [2].

Lemma 2.1. Assume that is a holomorphic self-map of , and . Then, for every it holds

The next Schwartz-type characterization of compactness [28] is proved in a standard way (see, e.g., the proof of the corresponding lemma in [11]), hence we omit its proof.

Lemma 2.2. Assume , is a holomorphic self-map of ,, is normal, and is a weight. Then, the operator is compact if and only if for every bounded sequence converging to 0 uniformly on compacts of we have .

The next lemma is folklore and can be found, for example, in [6] (one-dimensional case for standard power weights is due to Flett [29, Theorems 6 and 7]).

Lemma 2.3. Assume that , , is normal, and . Then, the following asymptotic relationship holds for every ,

Lemma 2.4. Assume that is normal and . Then, Moreover, if then for any .

Proof. By Lemma 3.1 in [21] applied to we have that Hence, for , we have that so that where .
If , then by the mean value property of the function (see [30]), Jensen’s inequality, and Parseval’s formula, we obtain
From (2.9) and (2.6), we obtain From (2.8) and (2.10), (2.3) follows, from which by (2.4) the second statement follows.

Lemma 2.5. Assume is normal and (2.4) holds. Then, for every bounded sequence converging to 0 uniformly on compacts of , we have that

Proof. From (2.4), we have that for every , there is a such that for .
Hence, from (2.12) it follows that for each and
From (2.12) and (2.13), we obtain Letting in this inequality, using the assumption that converges to 0 on the compact , and using the fact that is an arbitrary positive number, the lemma follows.

3. The Boundedness and Compactness of

The boundedness and compactness of the operator are characterized in this section.

Theorem 3.1. Assume that , is a holomorphic self-map of ,, and are normal, and   satisfies condition (2.4). Let Then, the following statements are equivalent:
(a) is bounded;
(b) is bounded;
(c) is compact;
(d) is compact;
(e) .
Moreover, if is bounded, then the following asymptotic relations hold:

Proof. The implications ,,, and are obvious.
Since is bounded and , by Lemma 2.1 we have that . Moreover,
Assume that is a bounded sequence converging to 0 uniformly on compacts of . Then, by Lemmas 2.1, 2.3, and 2.5, we have which along with Lemma 2.2 implies the compactness of .
From (2.4) and by Lemmas 2.3 and 2.4, we have This, together with (3.3) and the inequality implies the asymptotic relations in (3.2), as desired.

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