#### 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., [1–4] 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 [12–16]. On related operators on the unit polydisk see, for example, [17–21] and references therein. The boundedness and compactness of operator (1.6) between various spaces of holomorphic functions has been extensively studied recently, see, [17, 22–36]. 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).