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Abstract and Applied Analysis
Volume 2011, Article ID 152635, 9 pages
http://dx.doi.org/10.1155/2011/152635
Research Article

Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to Space on the Unit Ball

Department of Mathematics, Tianjin University, Tianjin 300072, China

Received 16 January 2011; Accepted 16 March 2011

Academic Editor: Ljubisa Kocinac

Copyright © 2011 Yu-Xia Liang and Ze-Hua Zhou. 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

This paper characterizes the boundedness and compactness of the product of extended Cesàro operator and composition operator from Lipschitz space to space on the unit ball of .

1. Introduction

Let be the unit ball in the -dimensional complex space , the closure of will be written as . By we denote the Lebesgue measure on normalized so that and by the normalized rotation invariant measure on the boundary of . Let be the class of all holomorphic functions on and the collection of all the holomorphic self-mappings of . Denote by the unit ball algebra of all continuous functions on that are holomorphic on .

For , let be the radial derivative of .

We recall that the -Bloch space for consists of all such that The expression defines a seminorm while the natural norm is given by . This norm makes into a Banach space. When , is the well known Bloch space.

For , denotes the holomorphic Lipschitz space of order which is the set of all such that, for some , for every . It is clear that each space contains the polynomials and is contained in the ball algebra . It is well known that is endowed with a complete norm that is given by See [1, 2] for more information of the Lipschitz spaces on .

For , let be Green's function on with logarithmic singularity at , where is the Möbius transformation of with , , and .

Let , , a function is said to belong to if (see, e.g., [35])

If is a Banach space of holomorphic functions on a domain and if is a (holomorphic) self-map of , the composition operator of symbol is defined by . The study of composition operators consists in the comparison of the properties of the operator with that of the function itself, which is called the symbol of . One can characterize boundedness and compactness of and many other properties. We refer to the books in [6, 7] and to some recent papers in [4, 5, 8] to learn much more on this subject.

Let , the following integral-type operator was first introduced in [9] This operator is called generalized Cesàro operator. It has been well studied in many papers, see, for example, [3, 924] as well as the related references therein.

It is natural to discuss the product of extended Cesàro operator and composition operator. For and , the product can be expressed as It is interesting to characterize the boundedness and compactness of the product operator on all kinds of function spaces. Even on the disk of , some properties are not easily managed; see some recent papers in [18, 2528].

Building on those foundations, the present paper continues this line of research and discusses the operator in high dimension. The remainder is assembled as follows: in Section 2, we state a couple of lemmas. In Section 3, we characterize the boundedness and compactness of the product of extended Cesàro operator and composition operator from Lipschitz spaces to spaces on the unit ball of .

Throughout the remainder of this paper, will denote a positive constant, the exact value of which will vary from one appearance to the next. The notation means that there is a positive constant such that .

2. Some Lemmas

To begin the discussion, let us state a couple of lemmas, which are used in the proofs of the main results.

Lemma 2.1. Suppose that . Then,

Proof. The proof of this Lemma follows by standard arguments (see, e.g., [9, 29, 30]).

Lemma 2.2 (see [2, 31]). If , then ; furthermore, as varies through .

The following criterion for compactness follows from standard arguments similar to the corresponding lemma in [6]. Hence, we omit the details.

Lemma 2.3. Assume that and . Suppose that or   is one of the following spaces , . 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 .

Lemma 2.4 (see [4, 5]). If , then

The next lemma was obtained in [32].

Lemma 2.5. If , , then the elementary inequality holds

It is obvious that Lemma 2.5 holds for the sum of finite number , that is, where and is a positive constant.

Lemma 2.6 (see [4, 5]). For , , , , there exists such that for every .

Lemma 2.7 (see [4]). There is a constant so that, for all and , one has

Lemma 2.8 (see [4, 5]). Suppose that , , , and . If , then , and .

Lemma 2.9. Let be a bounded sequence in which converges to zero uniformly on compact subsets of the unit ball , where . Then, .

Proof. It follows from Lemma 2.8 that and for any . So, when , the proof of this lemma is similar to that of Lemma 3.6 of [33], hence the proof is omitted.

3. The Boundedness and Compactness of the Operator

Theorem 3.1. Assume that , , ,   , , , and . Then, is bounded if and only if .

Proof. Assume that . Since , by Lemmas 2.2 and 2.4, for any , we have Since , by using Lemma 2.1 and relations (2.3) and (3.1), we have Thus is bounded.
Conversely, suppose that is bounded. Taking the function , then From which, the boundedness of implies that . This completes the proof of this theorem.

Next, we characterize the compactness of .

Theorem 3.2. Assume that , , , , , , and . Then, is compact if and only if is bounded, and

Proof. Assume that is bounded and (3.4) holds. It follows from Theorem 3.1 that .
Now, let be a bounded sequence of functions in such that uniformly on the compact subsets of as . Suppose that . It follows from (3.4) that, for any , there exists such that, for every , Set , then where
Let , then is a compact subset of . Since uniformly on compact subsets of as and , we get
On the other hand, by (3.5) and Lemmas 2.2 and 2.4, it follows that Since is arbitrary, from the above inequalities, we get Hence, by (3.10) and Lemma 2.3, we conclude that is compact.
For the converse direction, we suppose that is compact. It is obvious that is bounded.
Now, we prove (3.4). Setting the test functions for fixed , where and . It is easy to check that , and uniformly on the compact subsets of as . Write , since is compact, by Lemma 2.3, it follows that, as , Note that ; by the relation (3.11) and Lemma 2.5, we have This means that, for every , there is such that, for every , Thus, when , by the above inequality, we obtain From which, the desired result (3.4) holds. This completes the proof of this theorem.

Remark 3.3. When , the product of extended Cesàro operator is the generalized extended Cesàro operator ; thus, by Theorems 3.1 and 3.2, we have the following two corollaries.

Corollary 3.4. Assume that , , , , , and . Then, is bounded if and only if .

Corollary 3.5. Assume that , , , , , and . Then, is compact if and only if is bounded, and

Acknowledgments

The authors would like to thank the editor and referees for carefully reading the paper and providing corrections and suggestions for improvements. Z.-H. Zhou was supported in part by the National Natural Science Foundation of China (Grant nos. 10971153 and 10671141).

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