Research Article | Open Access

# Weighted Differentiation Composition Operator from Logarithmic Bloch Spaces to Zygmund-Type Spaces

**Academic Editor:**Giovanni P. Galdi

#### Abstract

Let denote the space of all holomorphic functions on the unit disk of , and let *n* be a positive integer, a holomorphic self-map of , and a weight. In this paper, we investigate the boundedness and compactness of a weighted differentiation composition operator , from the logarithmic Bloch spaces to the Zygmund-type spaces.

#### 1. Introduction

Let denote the open unit disk of the complex plane and the space of all analytic functions in .

The logarithmic Bloch space is defined as follows: The space is a Banach space under the norm . Let denote the subspace of consisting of those such that It is obvious that there are unbounded functions. For example, consider the function . There are also bounded functions that do not belong to . In fact, the interpolating Blaschke products do not belong to . It is easily proved that, for , . first appeared in the study of boundedness of the Hankel operators on the Bergman space. Attele in [1] proved that, for , the Hankel operator is bounded if and only if , thus giving one reason, and not the only reason, why log-Bloch-type spaces are of interest. Ye in [2] proved that is a closed subspace of . Galanopoulos in [3] characterized the boundedness and compactness of the composition operator and the boundedness and compactness of the weighted composition operator . Ye in [4] characterized the boundedness and compactness of the weighted composition operator between the logarithmic Bloch space and the -Bloch spaces on the unit disk and the boundedness and compactness of the weighted composition operator between the little logarithmic Bloch space and the little -Bloch spaces on the unit disk. Li in [5] characterized the boundedness and compactness of the weighted composition operator from Bergman spaces into the logarithmic Bloch space on the unit disk. Ye in [6] characterized the boundedness and compactness of the weighted composition operator from the general function space into the logarithmic Bloch space on the unit disk. Colonna and Li in [7] studied the boundedness and compactness of the weighted composition operators from Hardy space into the logarithmic Bloch space and the little logarithmic Bloch space. Petrov in [8] obtains sharp reverse estimates for the logarithmic Bloch spaces on the unit disk. Castillo et al. in [9] characterized the boundedness and compactness of the composition operator from the logarithmic Bloch spaces into weighted Bloch spaces. García Ortiz and Ramos-Fernández in [10] characterized the boundedness and compactness of the composition operators from logarithmic Bloch spaces into Bloch-type spaces.

Let be a weight; that is, is a positive continuous function on . The Zygmund-type space consists of all such that With the norm , it becomes a Banach space. The little Zygmund-type space is a subspace of consisting of those such that When , the Zygmund-type space becomes the Zygmund space [11], while the little Zygmund-type space becomes the little Zygmund space .

Let be the differentiation operator; that is, . If , then the operator is defined by .

The weighted differentiation composition operator, denoted by , is defined as follows [12, 13]: where and is a nonconstant holomorphic self-map of .

If , then becomes the weighted composition operator , defined by which, for , is reduced to the composition operator for some recent articles on weighted composition operators on some -type spaces, for example, [14–16] and references therein. If , , then , which was studied in [17–21]. When , then , which was studied in [17, 19]. If , , then , that is, the product of differentiation operator and multiplication operator defined by . Zhu in [13] completely characterized the boundedness and compactness of linear operators which are obtained by taking products of differentiation, composition, and multiplication operators from Bergman type spaces to Bers spaces. Stević in [12] studied the boundedness and compactness of the weighted differentiation composition operator from mixed-norm spaces to weighted-type spaces or the little weighted-type space (see also [22–24]). Zhu in [25] studied the boundedness and compactness of the generalized weighted composition operator on weighted Bergman spaces. Yang in [21] studied the boundedness and compactness of the operator and from to and spaces. Liu and Yu in [18] studied the boundedness and compactness of the operator between and Zygmund spaces. Ye and Zhou in [26] studied the boundedness and compactness of the weighted composition operators from Hardy to Zygmund type spaces. Stević in [27] studied the boundedness and compactness of the generalized composition operator from mixed-norm space to the Bloch-type space, the little Bloch-type space, the Zygmund space, and the little Zygmund space. For other recently introduced products of operators on spaces of holomorphic functions see [13, 16]. Motivated by the results [12, 18, 23, 24, 27], we consider the boundedness and compactness of the operators from the logarithmic Bloch spaces to the Zygmund-type spaces and the little Zygmund-type spaces. For the proof, we need different test functions and some complex calculations kills.

Throughout this paper, we will use the letter to denote a positive constant that can change its value at each occurrence.

#### 2. Auxiliary Results

Here we prove and quote some auxiliary results which will be used in the proofs of the main results in this paper.

Lemma 1. *Let be a positive integer. Suppose ; there exists a constant such that
*

*Proof. *We use induction on . Using the definition of the logarithmic Bloch spaces we have
the case holds for . Assume the case holds; since
let ; then we have , so
By the Cauchy integral formula we obtain
Note that
we have
for every . Hence the case holds. The desired result follows. The proof of this lemma is complete.

Lemma 2 (see [4, 28]). *Let
**
then .*

The following criterion for the compactness is a useful tool and it follows from standard arguments (e.g., [29, Proposition 3.11] or [30, Lemma 2.10]).

Lemma 3. *Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. 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 , we have as .*

Lemma 4. *A closed set in is compact if and only if is bounded and satisfies
*

The proof is similar to that of Lemma 1 in [31]; hence we omit it.

#### 3. Boundedness and Compactness of from to Spaces

In this section, we study the boundedness and compactness of .

Theorem 5. *Let , and let n be a nonnegative integer, a holomorphic self-map of , and a weight. Then the following statements are equivalent:*(1)* is bounded;*(2)* is bounded;*(3)

*Proof. *. Suppose that (17), (18), and (19) hold. Then, for every and , by Lemma 1, we have
On the other hand, we have
Applying conditions (20) and (21), we deduce that the operator is bounded.

. This implication is clear.

. Assume that is bounded; that is, there exists a constant , such that
for all . For , we have that
Taking ; we have that
By (23), (24), and the boundedness of the function , we get
In the same way, taking , we have that
By (23), (25), (26), and the boundedness of the function , we have that
For a fixed , set
We get that
By Lemma 2 we have
Hence, and .

On the other hand for each fix , by (30), we obtain that
it follows that for each fix . From (29), we have and
Hence
By (33), we obtain that
And from (23), we have
Thus combining (35) with (34) we get the condition (17).

For a fixed , set
It is easy to see that
Using Lemma 2, we easily get that and with a direct calculation. From (37), we have ,
Hence