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).
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 ifMoreover,
Proof. Suppose is bounded. LetThenLemma 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 thatIf is bounded on , then for we obtainTaking 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).