Journal of Function Spaces
Volume 2018 (2018), Article ID 4983827, 5 pages
https://doi.org/10.1155/2018/4983827

Research Article

Analytic Morrey Spaces and Bloch-Type Spaces

Ofori Samuel,¹ Jianfei Wang,¹ and Yile Zhao²

¹Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
²Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Correspondence should be addressed to Jianfei Wang; wjfustc@zjnu.cn

Received 25 September 2017; Accepted 24 January 2018; Published 11 March 2018

Academic Editor: Alberto Fiorenza

Copyright © 2018 Ofori Samuel et al. 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.

This paper is devoted to characterizing the boundedness of the Riemann-Stieltjes operators from analytic Morrey spaces to Bloch-type spaces. Moreover, the boundedness of the superposition operator and weighted composition operator on analytic Morrey spaces is discussed, respectively.

1. Introduction

The open unit disk, the unit circle, and the area measure in the complex are denoted by , , and , respectively. The space of all analytic functions in will be written , and denotes the Hardy space on the unit disk .

For , the space of analytic Campanato space, denoted by , is the set of all satisfying

where is taken over all subarcs of and

Table 1 will help us to understand the structure of (see, e.g., [1–3] and [4, p. 209–217] for the real counterparts).

Table 1: The structure of analytic Campanato space.

Index
Analytic Campanato space

Analytic Hardy space

Analytic Morrey space

Analytic John-Nirenberg space

Analytic Lipschitz space

Space of constants

Of course, this value defines a seminorm on . A complete norm on can be equipped by

For , the -Bloch space, denoted by , is the set of all for which

The expression defines a seminorm while the natural norm is given by

The Riemann-Stieltjes operator with analytic function symbol is defined by

The corresponding integral operator is defined by

Obviously, the multiplication operator is given as follows:

Continuing from [5–14], in this paper, we consider three operators associated with the analytic Morrey spaces. More precisely, in Section 2, we give two lemmas which are used to prove our main results in the paper. In Section 3, we obtain the boundedness of the Riemann-Stieltjes operators and from analytic Morrey spaces to Bloch-type spaces. In Section 4, the boundedness of the defined superposition operator on analytic Morrey spaces is proved. In Section 5, we consider the boundedness of the weighted composition operator from analytic Morrey spaces to Bloch-type spaces.

Notation. Let and denote that there exists an absolute constant such that and , respectively. means that and hold.

2. Some Lemmas

We now recall some auxiliary results which will be used throughout this paper. Lemmas 1 and 2 are proved, the first by Xiao and Yuan in [15] and the second by Wang in [16].

Lemma 1. Let . If , then

hold for all .

Lemma 2. For any fixed . Let . Then functions

belong to . Moreover, is uniformly bounded in ; that is,

3. Boundedness of the Operators

In this section, we give the boundedness of and from , respectively.

Theorem 3. Let . Suppose that . Then, is bounded if and only if

Moreover,

Proof. Suppose is bounded. Taking , then

Note that

It implies that

Hence,

Conversely, suppose

By Lemma 1, we have that

Hence,

implies is bounded and

Theorem 4. Let and . Then is bounded if and only if

Moreover,

Proof. Suppose is bounded. Let

Then

Lemma 2 shows that

Taking supremum in the last inequality over the set and applying to the maximum modulus principle we have

Hence

That is,

Conversely, suppose

By Lemma 1, we have that

Then

which implies is bounded and

Remark 5. When in Theorem 4, the conclusion is equivalent to .

As an application of Theorems 3 and 4, we can obtain the following corollary.

Corollary 6. Let and . Then is bounded if and only if .

4. Superposition on

Let and represent two subspaces of . If is a complex-valued function such that whenever , then we call that acts by superposition from into . If and contain the linear functions, then must be an entire function. The superposition with symbol is then defined by . A basic question is when map into continuously. This question has been studied for many distinct pairs —see, for example, [17–20]. In this section, we are interested in the analytic Morrey space and have the following result which extends the case of in [3].

Theorem 7. Let . Then is bounded on if and only if for some .

Proof. Lemma 1 shows that

If is bounded on , then for we obtain

Taking the following -function

in the last inequality, we have

Since

so there is a positive independent of such that

In particular, setting yields

Setting in the last estimate and noticing , we obtain that the entire function is bounded on . By using the maximum principle, we get that must be a linear function.
Conversely, if for some , then , and hence . Hence is bounded on .

5. Weighted Composition Operator from to

Suppose that and is an analytic self-map of , . These maps induce a linear weighted composition operator which is defined by

where is the operator of pointwise multiplication by and is the composition operator . Our next result will be as follows.

Theorem 8. Let , , and be an analytic self-map of . Then

if and only if

Proof. The growth of functions in shows that if then

It is easy to see

If

then and . It yields that which proves that the sufficiency holds. On the other hand, the necessary follows by taking the test function in

Remark 9. When , the result reduces to Xiao and Yuan in [15]. When , denote the multiplication operator.

if and only if

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors were partially supported by the National Natural Science Foundation of China (nos. 11471111, 11571105, and 11671362) and the Natural Science Foundation of Zhejiang Province (nos. LY16A010004 and LY13A010021).

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