- Research Article
- Open Access

# Integral-Type Operators from Spaces to Zygmund-Type Spaces on the Unit Ball

- Congli Yang
^{1, 2}Email author

**2010**:789285

https://doi.org/10.1155/2010/789285

© Congli Yang. 2010

**Received:**7 May 2010**Accepted:**23 December 2010**Published:**9 January 2011

## Abstract

Let denote the space of all holomorphic functions on the unit ball . This paper investigates the following integral-type operator with symbol , , , , where is the radial derivative of . We characterize the boundedness and compactness of the integral-type operators from general function spaces to Zygmund-type spaces , where is normal function on .

## Keywords

- Banach Space
- Function Space
- Compact Subset
- Hardy Space
- Besov Space

## 1. Introduction

Let be the open unit ball of , let be its boundary, and let be the family of all holomorphic functions on . Let and be points in and .

The space is the classical Bergman space (see [3]), is the classical Besov space , and, in particular, is just the Hardy space . The spaces are spaces, introduced by Aulaskari et al. [4, 5]. Further, , the analytic functions of bounded mean oscillation. Note that is the space of constant functions if . More information on the spaces can be found in [6, 7].

Under the norm introduced by is a Banach space and is a closed subspace of . If , we write and for and , respectively.

and will be called the Zygmund-type space.

for all and , where is the ball algebra on .

where and . Stević [9] considered the boundedness of on -Bloch spaces. Lv and Tang got the boundedness and compactness of from to -Bloch spaces for all (see [10]). Recently, Li and Stević discussed the boundedness of from Bloch-type spaces to Zygmund-type spaces in [11]. For more information about Zygmund spaces, see [12, 13].

In this paper, we characterize the boundedness and compactness of the operator from general analytic spaces to Zygmund-type spaces.

In what follows, we always suppose that , , , . Throughout this paper, constants are denoted by ; they are positive and may have different values at different places.

## 2. Some Auxiliary Results

In this section, we quote several auxiliary results which will be used in the proofs of our main results. The following lemma is according to Zhang [14].

Lemma 2.1.

Lemma 2.2 (see [9]).

Lemma 2.3 (see [15]).

For every , it holds .

Lemma 2.4 (see [10]).

Lemma 2.5.

Assume that , , , , and is a normal function on , 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 .

The proof of Lemma 2.5 follows by standard arguments (see, e.g., Lemma 3 in [16]). Hence, we omit the details.

The following lemma is similar to the proof of Lemma 1 in [17]. Hence, we omit it.

Lemma 2.6.

## 3. Main Results and Proofs

Now, we are ready to state and prove the main results in this section.

Theorem 3.1.

Let , , , and let be normal, and , then is bounded if and only if

(i)for ,

(ii)for ,

Hence, (3.1) and (3.2) imply that is bounded.

Then, by [14] and .

Combing (3.9) and (3.10), we get (3.1).

- (ii)

Applying (3.3) and (3.4) in (3.13), for the case , the boundedness of the operator follows.

then by [14].

By (3.14) and (3.15), in the same way as proving (3.1), we get that (3.3) holds.

From (3.15) and (3.17), we see that (3.4) holds. The proof of this theorem is completed.

Theorem 3.2.

Let , , , and let be normal, and , then the following statements are equivalent:

(A) is compact;

(B) is compact;

- (i)
for ,

- (ii)
for ,

Proof.

(A) *⇒* (C). First, for the case
.

which means that (3.18) holds.

Then, for and uniformly on compact subsets of as . we obtain that (3.20) holds for the case .

which implies that (3.21) holds.

Similarly, we obtain that (3.37) holds for the case by (3.20) and (3.21). Exploiting Lemma 2.6, the compactness of the operator follows. The proof of this theorem is completed.

Finally, we consider the case .

Theorem 3.3.

Let , , , and let be normal, , , then the following statements are equivalent:

(A) is bounded;

(B) and

The proof of Theorem 3.3 follows by the proof of Theorem 3.1. So, we omit the details here.

Theorem 3.4.

Let , , , and let be normal, and , then the following statements are equivalent:

(A) is compact;

(B) and

Proof.

*⇒*(B). We assume that is compact. For , we obtain that . Exploiting the test function in (3.22), similarly to the proof of Theorem 3.2, we obtain that (3.39) holds. As a consequence, it follows that

*⇒*(A). Assume that is a sequence in such that , and uniformly on compact of as . By Lemma 2.1 and [18, Lemma 4.2],

Hence, by Lemma 2.5, the compactness of the operator follows. The proof of this theorem is completed.

Theorem 3.5.

Let , , , and let be normal, and , then the following statements are equivalent:

(A) is compact;

(B) and

Proof.

This along with Theorem 3.2 implies that is bounded. Taking the supremum over the unit ball in , letting in (3.46), using the condition ( ), and finally by applying Lemma 2.6, we get the compactness of the operator . This completes the proof of the theorem.

## Declarations

### Acknowledgments

The author wishes to thank Professor Rauno Aulaskari for his helpful suggestions. This research was supported in part by the Academy of Finland 121281.

## Authors’ Affiliations

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