Journal of Mathematics
Volume 2021 (2021), Article ID 4398397, 8 pages
https://doi.org/10.1155/2021/4398397

Research Article

Weighted Composition Operators from Derivative Hardy Spaces into -th Weighted-Type Spaces

Nanhui Hu1,2

¹Department of Mathematics, Shantou University, Shantou 515063, Guangdong, China
²Department of Mathematics, Jiaying University, Meizhou 514015, Guangdong, China

Correspondence should be addressed to Nanhui Hu; hunhjyu@163.com

Received 15 April 2021; Accepted 3 June 2021; Published 17 June 2021

Academic Editor: Sei Ichiro Ueki

Copyright © 2021 Nanhui Hu. 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.

The boundedness, compactness, and the essential norm of weighted composition operators from derivative Hardy spaces into -th weighted-type spaces are investigated in this paper.

1. Introduction

Let denote the space of analytic functions on the open unit disk . Let denote the set of all analytic self-maps of . Let . The composition operator with the symbol is defined by

Let and . The weighted composition operator is defined on by

It is important to give function theoretic descriptions when and induce a bounded or compact weighted composition operator on various function spaces. See references [1, 2] for more information of this research field.

For , the Hardy space, denoted by , consists of all functions such that (see [3])

As usual, denotes the space of bounded analytic functions in . If , we say that belongs to the derivative Hardy space, denoted by . When , the space is a Banach space under the norm defined by

When , is a Hilbert space. In [4], Roan started the study of composition operators on the space . In [5], MacCluer investigated composition operators on the space in terms of Carleson measure. The boundedness and compactness of weighted composition operators on were studied in [6]. See references [4–7] and references therein for more study of composition operators and weighted composition operators on the space .

If is a radial, positive, and continuous function on , then is called a radial weight. Let be a radial weight and , the set of all positive integers. Let denote the -th weighted space, which consists of all such that

It is a Banach space with the norm . When and , becomes the Bloch-type space and the Zygmund-type space , respectively. Furthermore, when , is the Bloch space and is the Zygmund space. For some results on the space , see references [2, 8–13].

Let with . Recall that the partial Bell polynomials are defined as follows:

where the sum taken over all sequences of nonnegative integers such that the following two conditions hold:

See reference [14] for more information about Bell polynomials.

In [15], Stević studied the boundedness and compactness of the composition operator from to on the unit disk. In [12], Stević studied the boundedness and compactness of weighted composition operators from and the Bloch space to . See references [8–10] for more characterizations for weighted composition operators from and the Bloch space to . In [16], Zhu and Du studied the boundedness, compactness, and essential norm of weighted composition operators from weighted Bergman spaces with doubling weight to . Recall that the essential norm of a bounded linear operator is its distance to the set of compact operators mapping into , that is,

Here, are Banach spaces and denotes the operator norm. In [17], Colonna and Tjani studied the boundedness and compactness of weighted composition operators from derivative Hardy spaces to Bloch-type space .

In this paper, we use Bell polynomials to study the boundedness and compactness of weighted composition operators from derivative Hardy spaces to -th weighted-type spaces . Moreover, we give an estimate for the essential norm of weighted composition operators from to .

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. Boundedness

In this section, the boundedness of weighted composition operators from to is characterized.

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

for every .

Proof. The first inequality follows from the fact that are contained in the disk algebra for . In addition, it is well known that for every , there exists a positive constant such that

which implies the second inequality.
For any , , and , set

After a calculation, for each ,

Lemma 2. Let and . For any , there exist constants such that

Moreover, converges to 0 uniformly in .

Proof. The proof is similar to the proof of Theorem 1 in [16]. Hence, we omit the details of the proof.
For the simplicity of this paper, we define

From the definition, we see that, for example,

This fact will be used in the proof of the following theorem.
Now, we are in a position to state and prove the first result in this paper.

Theorem 1. Let , , , , and be a weight. Then, the operator is bounded if and only if and

Proof (sufficiency). Let . After a calculation, we have (see, e.g., [12])

Since and , we see that . Hence, by (14) and Lemma 1, we get

From (18), for each ,

Therefore, from (19) and (20) and the fact that , we get that is bounded.
Necessity. Assume that is bounded. It is clear that , and for each and ,

by the boundedness of .
Next, we show that for ,

Applying the operator for , by (17) and (14), we obtain

Therefore, by the triangle inequality and the boundedness of , we obtain

Now, we assume that for , To get the desired result, we only need to show that

Applying the operator for , we obtain

Hence, from the boundedness of and triangle inequality again, we get the desired result.
For any and , by Lemma 2,

where are independent of the choice of . From the last inequality and (22), for any , we get

Therefore,

The proof is complete.
Let . We get the following result, which was first obtained in [17].

Corollary 1. Let , , , and be a weight. Assume that is bounded if and only if and

Let . We get the following result.

Corollary 2. Let , , , and be a weight. Then, the operator is bounded if and only if ,

3. Essential Norm

In this section, we obtain some estimates for the essential norm of the weighted composition operator , and we need the following lemma.

Lemma 3. (see [17]). 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

(1)The point evaluation functionals on are continuous(2)For every sequence in the unit ball of , there exists and a subsequence such that uniformly on (3)The operator is continuous if has the supremum norm and is given the topology of uniform convergence on compact sets

Then, is a compact operator if and only if, given a bounded sequence in such that uniformly on , then\ as .

Lemma 4. (see [17]). Let . Every sequence in bounded in norm has a subsequence which converges uniformly in to a function in .
The following result is a direct consequence of Lemma 3 and Lemma 4.

Lemma 5. Let , , and be a weight. If is a bounded linear operator from into , then is compact if and only if as for any sequence in bounded in norm which convergences to 0 uniformly in .

Theorem 2. Let , , , , and be a weight such that is bounded. Then,

Proof. First, we prove that

If , there is a number such that . In this case, (32) is vacuously satisfied and the desired result follows.
Assume that . Let be a sequence in such that as . Since is bounded, for any compact operator and , by using Lemma 2 and Lemma 5, we obtain

Hence,

which implies that

as desired.
Now, we show that

Let and define . Then, is compact and . It is clear that uniformly on compact subsets of as . Let be a sequence such that as . Then, for any positive integer , the operator is compact. By the definition of the essential norm, we get

Hence, it is sufficient to show that

For any such that ,

where such that for all . Since for any nonnegative integer , uniformly on compact subsets of as . It is clear that

From Lemma 4,

while

For any , by Lemma 1 and Lemma 2,

Taking the limit as , by Lemma 1, we get

By the same reason, we have

Hence, by (40)–(43), (45), and (46), we obtain

Hence, from the last two inequalities, we obtain (39). The proof is complete.
From Theorem 2 and the well-known result that if and only if is compact, we obtain the following corollary.

Corollary 3. Let , , , , and be a weight. Assume that is bounded, then the operator is compact if and only if

Let . We get the following result, which was first obtained in [17].

Corollary 4. Let , , , and be a weight such that is bounded. Then, the operator is compact if and only if

Let . We get the following result.

Corollary 5. Let , , , and be a weight such that is bounded. Then, the operator is compact if and only if

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

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