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Abstract and Applied Analysis
Volume 2010, Article ID 214762, 14 pages
http://dx.doi.org/10.1155/2010/214762
Research Article

On an Integral-Type Operator Acting between Bloch-Type Spaces on the Unit Ball

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
2Faculty of Engineering, Ibaraki University, Hitachi 316-8511, Japan

Received 30 January 2010; Accepted 18 March 2010

Academic Editor: Narcisa C. Apreutesei

Copyright © 2010 Stevo Stević and Sei-Ichiro Ueki. 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

Let denote the open unit ball of . For a holomorphic self-map of and a holomorphic function in with , we define the following integral-type operator: , . Here denotes the radial derivative of a holomorphic function in . We study the boundedness and compactness of the operator between Bloch-type spaces and , where is a normal weight function and is a weight function. Also we consider the operator between the little Bloch-type spaces and .

1. Introduction

Let denote the open unit ball of the -dimensional complex vector space and the space of all holomorphic functions on . For with the Taylor expansion let be the radial derivative of where is a multi-index, and . It is well known that where is the usual gradient on

Let be a holomorphic self-map of and with . Then and define an operator on as follows:

The following important formula involving and was proved, for example, in [1]

Motivated by papers [2, 3], operators were introduced by the first author of the present paper and Zhu in [1, 46], where its boundedness and compactness from the -Bloch space, the Zygmund space, the mixed-norm space, and the generalized weighted Bergman space into the Bloch-type space on the unit ball are studied. In our previous work [7], we studied the boundedness and compactness of acting between weighted-type spaces. For related operators on see, for example, [821] and the references therein.

Let be a strictly positive continuous function on (weight). If for every , we call it radial weight. A weight is called normal ([9, 22]) if it is radial and there are and , such that is decreasing on , is increasing on ,

A radial weight is called typical if it is nonincreasing with respect to and as If is normal, then by the monotonicity of , for we have that that is, is decreasing on . On the other hand, from the first equality in (1.5), we have that for any there is a such that which implies Hence every normal weight is also typical.

For a weight the associated weight ([23]) is defined by

Here denotes the weighted-type space consisting of all with (see, e.g., [23, 24]). Associated weights assist us in studying of weighted-type spaces of holomorphic functions. It is known that associated weights are also continuous, and for each we can find an such that . Let be the little weighted-type space, that is, the space of all such that as . If is typical, then the unit ball is the closure of for the compact open topology. Hence we have and so for each we can choose an such that . A weight is called essential if it satisfies that for some positive constant . By the arguments in [25], we see that a normal weight function is also essential. For some examples of essential weights, see, for example, [25]. Related results can also be found in [22, 26].

The Bloch-type space is the space of all holomorphic functions on such that where is a weight (see, e.g., [20]). The little Bloch-type space consists of all such that Both spaces and are Banach spaces with the norm and is a closed subspace of . When the space is a classical Bloch space.

The purpose of this paper is to characterize the boundedness and compactness of the operators and .

Throughout this paper, we assume that is a holomorphic self-map of and with . Furthermore, some constants are denoted by they are positive and may differ from one occurrence to the other. The notation means that there exists a positive constant such that . Moreover, if both and hold, then one says that .

2. Auxiliary Results

Here we formulate and prove some auxiliary results which are used in the proofs of the main ones.

The following lemma was proved in [20, Theorem  2.1].

Lemma 2.1. Let be a normal weight function and . Then if and only if and it holds that Moreover, if and only if .

As an application of Lemma 2.1, we have the following result.

Lemma 2.2. Let be a normal weight function and Then if and only if it holds that where

Proof. Take an . For a fixed by Lemma 2.1, we can choose a such that for any . Since for , and , we have Since we see that the second term in (2.3) converges to as
If and then by (2.2) we have By (1.6) we have that for .
Hence we have for all This proves that whenever
Conversely, the normality of implies that for any we have so that for any On the other hand, by the assumption we have that for every there is an such that for
By letting in the following inequality, which easily follows from Lemma 2.1: then using (2.7) and (2.8), we get , as claimed.

Corollary 2.3. Let be a normal weight function. Then the set of all holomorphic polynomials is dense in .

Proof. For the homogeneous expansion of an we set for each Since uniformly on compact subsets of as , we see that uniformly on for any . Moreover, we have Combining this with Lemma 2.2, we get the desired result.

The following lemma can be found in [1, Lemma  3]. Its proof is similar to the proof of the corresponding one-dimensional result in [27], for the case of the little Bloch space . Hence we omit the proof.

Lemma 2.4. A closed subset in is compact if and only if it is bounded and

The following lemma is very useful for estimating the norm of the Bloch-type space.

Lemma 2.5. Assume that is a positive integer and is normal. Then for every

Proof. For the details of the proof, we can refer [9] or [28].

3. The Boundedness of Operator

In this section we consider the boundedness of the operator or .

Theorem 3.1. Let be a normal weight function and a weight function. Then the following conditions are equivalent: (a) is bounded;(b) is bounded;(c) and satisfy Moreover, if is bounded, then

Proof. The implication is clear, so we only prove and .
: assume that is bounded and fix . We may assume that . For there exists such that and . We define the function as follows: Since we see that and . Hence, by (1.4), we have and so condition (3.1) is true.
: we assume (3.1) and take an . Since is an essential weight (due to its normality), (1.4) gives for any . By Lemma 2.1, we have , and so we obtain This implies that is bounded. The relation (3.2) follows from (3.4) and (3.6). This completes the proof.

Theorem 3.2. Let be a normal weight function and a weight function. Then the following conditions are equivalent: (a)  :  is bounded; (b) and satisfy

Proof. : as in the proof of Theorem 3.1, for fixed and we see that and satisfy the condition
On the other hand, since the normality of implies that the function belongs to , we obtain that for each , and so as
: the assumption shows that for any polynomial . For each , by Corollary 2.3, we can choose a sequence of polynomials such that as . Furthermore, the assumption shows that : is bounded by Theorem 3.1. Thus we obtain Since and is closed in , we have for any . Hence which means that is bounded. The proof is accomplished.

The following corollary is an immediate consequence of Theorems 3.1 and 3.2.

Corollary 3.3. Let be a normal weight function and a weight function. Then is bounded if and only if and is bounded.

4. The Compactness of Operator

In this section we characterize the compactness of or . To do this, we need the following standard lemma (see, e.g., [13, Lemma  3]).

Lemma 4.1. Let and be weight functions. Suppose that the operator is bounded. Then is compact if and only if for every bounded sequence in which converges to uniformly on compact subsets of , as .

Theorem 4.2. Let and be weight functions. Suppose that is a holomorphic self-map of such that and the operator is bounded. Then is compact. Here denotes the supremum .

Proof. Since , we see that for some and any . From the proof of Theorem 3.1, we see that the boundedness of implies Thus we obtain that Take a bounded sequence in such that uniformly on compact subsets of as . By (1.4), we have Since also converges to uniformly on as (4.2) and (4.3) show that as . From Lemma 4.1, it follows that is compact, and so we get the assertions.

Lemma 4.3. Suppose that is a weight function. Then there exists a sequence in the closed unit ball of such that uniformly on compact subsets of as .

Proof. Let with as . For each there exists such that and . We define as follows: Since and , we have and for each . For any compact subset of , we can choose an such that . Hence we obtain that for any From the above inequality, it follows that converges to uniformly on compact subsets of as . This completes the proof.

Remark 4.4. If we assume that is typical in Lemma 4.3, then we can choose . In this case, hence, we see that belongs to for each .

Theorem 4.5. Let be a normal weight function and a weight function. Suppose that the operator is bounded and . Then the following conditions are equivalent: (a) is compact;(b) is compact; (c) and satisfy

Proof. : this implication is obvious.
: take a sequence in with as and put for each . Then, by Remark 4.4 after Lemma 4.3, there exists a sequence in such that and uniformly on compact subsets of as . By Lemma 4.1, the compactness of implies that as .
On the other hand, (1.4) gives , and so we have From the construction (4.4) of , we obtain for each . Combining this with (4.7), we have Letting , we have for any sequence with This proves that (4.6) is true.
: we will prove the following estimate: Here denotes the essential norm of , namely, Now we take a sequence which increasingly converges to and put Since , Theorem 4.2 implies that is compact for each . For any with , from (1.4) it follows that for some fixed . The essentiality of and Lemma 2.1 give Similarly, we also have for each The normality of implies that for each and some and so by the essentiality, Thus (4.16) and (4.18) give for each . By (4.15) and (4.19), we obtain
When , by using the mean value theorem, we have Since is also normal, by Lemmas 2.1 and 2.5, we have Hence we obtain Since the boundedness of implies , letting in the above inequality, we have By using (4.14), (4.20), and (4.24) and letting we obtain the desired estimate So if condition (4.6) is true, then , which means that is compact. Our proof is accomplished.

Theorem 4.6. Let be a normal weight function and a weight function. Suppose that the operator is bounded. Then is compact if and only if

Proof. Suppose that (4.26) holds. For any , by Lemma 2.1 and (1.4), we have Combining this with (4.26), we obtain Hence it follows from Lemma 2.4 that is compact.
Conversely, we assume that is compact. By Theorem 3.2, we see that Thus this implies (4.26) if .
Now assume . We claim that Further assume that is a sequence in such that If , then from this and (4.29) we have that both limits in (4.30) are equal to zero. If , then there is a subsequence such that as . Hence we have and so (4.30) holds.
Since is also compact, by Theorem 4.5, we see that the second limit in (4.30) is equal to zero, so that (4.26) holds. This completes the proof.

Acknowledgment

This work was supported by JSPS Grant-in-Aid for Young Scientists (Start-up: no. 20840004) and Ibaraki University.

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