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Abstract and Applied Analysis
Volume 2009 (2009), Article ID 246521, 9 pages
http://dx.doi.org/10.1155/2009/246521
Research Article

Extended Cesàro Operators from Logarithmic-Type Spaces to Bloch-Type Spaces

Department of Mathematics, JiaYing University, Meizhou, Guangdong 514015, China

Received 27 March 2009; Accepted 12 April 2009

Academic Editor: Stevo Stević

Copyright © 2009 Dinggui Gu. 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 the extended Cesàro operator from logarithmic-type spaces to Bloch-type spaces on the unit ball are completely characterized in this paper.

1. Introduction

Let be the unit ball of the unit sphere of , the space of all holomorphic functions in , and the space of all bounded holomorphic functions on . For , let denote the radial derivative of .

A positive continuous function on the interval is called normal if there is and and , such that If we say that is normal we will also assume that

Let be normal. The Bloch-type space is the space of all functions such that is Banach space with the norm The little Bloch-type space consists of all such that It is easy to see that is a closed subspace of . When , , we obtain so-called -Bloch spaces and little -Bloch spaces, respectively, which for are reduced to classical Bloch spaces (see, e.g., [14] and the references therein). When , we obtain the logarithmic Bloch space and the little logarithmic Bloch space , respectively, (see [5]). It was shown that is a multiplier of if and only if and in [6].

An is said to belong to the logarithmic-type space , if It is easy to see that becomes a Banach space under the norm , and that the inclusions hold. For some information of the space see [7, 8].

Let . The extended Cesàro operator on is defined by This operator is a natural extension of a one-dimensional operator defined in [9]. Some other results on the one-dimensional operator can be found, for example, in [10, 11] (see also the references therein). For some extensions of operator (1.6) on the unit disk see [1216]. On related operators on the unit polydisk see, for example, [1721] and references therein. The boundedness and compactness of operator (1.6) between various spaces of holomorphic functions has been extensively studied recently, see, [17, 2236]. For some integral operators on spaces of harmonic functions see, for example, [37] as well as the references therein. A new extension of operator (1.6) in the unit ball case have been recently introduced by Stević in [38] (see also [5, 39]).

In this paper, we study the extended Cesàro operator from to Bloch-type spaces and . Sufficient and necessary conditions for the extended Cesàro operator to be bounded and compact are given.

Throughout the paper, constants are denoted by , they are positive and may not be the same in every occurrence.

2. Main Results and Proofs

In this section, we give our main results and their proofs. Before stating these results, we need some auxiliary results, which are incorporated in the lemmas which follows.

Lemma 2.1. Assume that and are normal. Then is compact if and only if is bounded and for any bounded sequence in which converges to zero uniformly on compact subsets of as , one has as

The proof of Lemma 2.1 follows by standard arguments (see, e.g., Lemmas in [20, 21, 29]). Hence, we omit the details.

Lemma 2.2. Assume that is normal. A closed set in is compact if and only if it is bounded and satisfies

This lemma can be found in [5], and its proof is similar to the proof of Lemma in [40]. Hence, it will be omitted.

The following result was proved in [8].

Lemma 2.3. There exist two functions such that

Now we are in a position to state and prove our main results.

Theorem 2.4. Assume that and is normal. Then is bounded if and only if Moreover, if is bounded then the following asymptotic relation holds

Proof. Assume that (2.3) holds. Then, for any , we have In addition, it is easy to see that . Therefore we have as desired.
Conversely, assume that is bounded. For , set It is easy to see that and .
For any , we have from which (2.3) follows, moreover From (2.6) and (2.9), we see that (2.4) holds. The proof is completed.

Theorem 2.5. Assume that and is normal. Then is compact if and only if

Proof. Suppose that is compact. Let be a sequence in such that . Set It is easy to see that . Moreover uniformly on compact subsets of as . By Lemma 2.1, In addition, which together with (2.12) implies that From the above inequality we see that (2.10) holds.
Conversely, assume that (2.10) holds. From Theorem 2.4 we see that is bounded. In order to prove that is compact, according to Lemma 2.1, it suffices to show that if is a bounded sequence in converging to uniformly on compact subsets of , then
Let be a bounded sequence in such that uniformly on compact subsets of as By (2.10) we have that for any , there is a constant such that whenever . Let . From (2.10) we see that . Equality (2.16) along with the fact that implies Observe that is a compact subset of , so that Therefore Since is an arbitrary positive number it follows that the last limit is equal to zero. Therefore, is compact. The proof is completed.

Remark 2.6. From [24] and Theorems 2.4 and 2.5, we see that is bounded if and only if is bounded; is compact if and only if is compact.

Theorem 2.7. Assume that and is normal. Then the following statements are equivalent:(a) is bounded;(b) is compact;(c)

Proof. (b)(a). This implication is obvious.
(a)(c). Assume that is bounded. Now we prove that (2.20) holds. Note that (2.20) is equivalent with Hence we only need to show that (2.21) holds. This can be done by contradiction. Now assume that the condition (2.21) does not hold. If it was, then it would exist and a sequence such that , and for sufficiently large . We may assume that and also According to Lemma 2.3 we know that there exist two functions such that Let Then clearly . By the boundedness of we have On the other hand, for sufficiently large . Since , from the above inequality we obtain that , which is a contradiction.
(c)(b). From (1.5) we have that Taking the supremum in the above inequality over all such that then letting , by (2.20) we arrive at From this and by employing Lemma 2.2, we see that is compact. The proof is completed.

From Theorems 2.4, 2.5, and 2.7, we have the following corollary.

Corollary 2.8. Let . Then(1) is bounded if and only if (2) is compact if and only if is bounded if and only if is compact if and only if

Acknowledgment

The author is supported partly by the Natural Science Foundation of Guangdong Province (no. 07006700).

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