Abstract

The boundedness, compactness, and essential norm of weighted composition operators from Besov Zygmund-type spaces into Zygmund-type spaces are investigated in this paper.

1. Introduction

Let denote the open unit disk in the complex plane and the space of all analytic functions in . For an analytic self-map of and , the weighted composition operator is defined as follows:

When , is just the composition operator, denoted by . In the past several decades, composition operators and weighted composition operators have received much attention and appear in various settings in the literature (see, for example, [2–5, 8, 10, 13, 15, 16, 19]).

Let . The Bloch type space consists of those functions for which

is a Banach space under the above norm. It is known that when , is the classical Bloch space.

For , an is said to be in the Zygmund-type space , if

It is easy to check that is a Banach space under the norm . When , is the Zygmund space. When , is just the Bloch type space . In particular, when , is just the Bloch space . Hence, the Zygmund space is the space of all such that with norm

Let be the normalized area measure on . For , the Besov space, denoted by , is the space of all such that

This space is a Banach space with the following norm In particular, is the classical Dirichlet space. Besov spaces are Möbius invariant in the sense that for all and , the set of all Möbius maps of (see [1, 19]).

In [4], Colonna and Tjani introduced a new class type space , called the Besov Zygmund-type space, which consists of all such that . Since the Besov space is contained in the Bloch space, it follows that the Besov Zygmund-type space is a subset of the Zygmund space, and hence, it is contained in the disk algebra.

Colonna and Li studied the boundedness and compactness of the operator and in [2, 3], respectively. Here, is the space of bounded analytic functions. (See [2, 3, 6–14, 16, 17] for more results of composition operators, weighted composition operators, and related operators on the Zygmund space and Zygmund-type spaces.) Colonna and Tjani characterized the boundedness and compactness of in [4].

In this work, we give some characterizations for the boundedness, compactness, and the essential norm of the operator .

Throughout the paper, we denote by a positive constant which may differ from one occurrence to the next. In addition, we say that if there exists a constant such that . The symbol means that .

2. Main Results and Proofs

In this section, we formulate and prove our main results in this paper. For this purpose, we need the following lemmas.

Lemma 1. Suppose . Then, there exists a positive constant such that for every .

Proof. For , it is well known that Then, the inequalities in (6) follow from the definition of the Besov Zygmund-type space. Since the Zygmund space is continuously embedded into , as shown in Lemma 2.1 of [18], we get that . The proof is complete.

Lemma 2 (see [4]). Let . Every sequence in bounded in norm has a subsequence which converges uniformly in to a function in .

Lemma 3 (see [4]). Let be a Banach space that is continuously contained in the disk algebra, and let be any Banach space of analytic functions on . Suppose that (i)the point evaluation functionals on are continuous(ii)for every sequence in the unit ball of that exists and a subsequence such that uniformly on (iii)the operator is continuous if has the supremum norm and is given the topology of uniform convergence on compact setsThen, is a compact operator if and only if given a bounded sequence in such that uniformly on , then the sequence as .

The following result is a direct consequence of Lemmas 2 and 3.

Lemma 4. Let and . If is bounded, then is compact if and only if as for any sequence in bounded in norm which converge to uniformly in .

The following estimates are fundamental in operator theory and function spaces on the unit disk (see ([19], Lemma 3.10)).

Lemma 5 (see [19]). Suppose that is real, and (i)If then as a function of is bounded on (ii)If then (iii)If then Now, we are in a position to give the following characterization of bound composition operators from to .

Theorem 6. Let , , , and be an analytic self-map of . Then, is bounded if and only if ,

Proof. First, suppose that , (11) and (12) hold. For arbitrary and , by Lemma 1, we have Therefore, is bounded.
Conversely, suppose that is bounded. Applying the operator to with , and using the boundedness of , we get that , , and . Hence, For such that , set Then, Thus, for , we have and by Lemma 5, So, Therefore, by the boundedness of , we get Since , for any such that , we have which implies that By (14), we get From (23) and (24), we see that (11) holds.
For , define So, By Lemma 5, we see that and . By the boundedness of , we get After a calculation, Hence, for any , On the one hand, from (28), we obtain On the other hand, by (15), we get From (29) and (30), we see that (12) holds. The proof is complete.