Abstract

In this paper, we characterize the closure of the Morrey space in the Bloch space. Furthermore, the boundedness and compactness of composition operators from the Bloch space to the closure of the Morrey space in the Bloch space are investigated.

1. Introduction

Let be the open unit disk in the complex plane and be the space of analytic functions on . For a fixed point , let denote the Mbius transformation on . Recall that the Bloch space, denoted by , is the space of all for which It is a Banach space with the above norm . The little Bloch space consists of all such that It is easy to see that the little Bloch space is the subspace of . It is well known that is the closure of polynomials in .

For , the Hardy space consists of all functions with Denote by the space of bounded analytic functions on .

For an arc , let be the normalized length of and be the corresponding Carleson box; i.e., Clearly, if , then . Let . A nonnegative measure on is said to be an -Carleson measure (see [1]) if If , a bounded -Carleson measure is the classical Carleson measure.

For , the Morrey space is the set of all such that Here Clearly, , the space of analytic functions whose boundary functions have bounded mean oscillation. From [2] or [3], the norm of functions can be defined as follows. We remark that and It is well known that the function After a calculation, we see that , but . See [2–6] for the study of Morrey space and related operators.

For every self-map on , the composition operator is defined on by It is a simple consequence of the Schwartz-Pick lemma that any analytic self-mapping of induces a bounded composition operator on the Bloch space. Madigan and Matheson in [7] proved that is compact if and only if . Here and henceforth See, for example, [7–14] for more characterizations of the boundedness and compactness of composition operators on the Bloch space.

In 1974, Anderson, Clunie, and Pommerenke posed the problem of how to describe the closure of in the Bloch norm (see [15]). This is still an open problem. Anderson in [16] mentioned that Jones gave a characterization of , the closure of in the Bloch norm (an unpublished result). A complete proof was provided by Ghatage and Zheng in [17]. Zhao in [18] studied the closures of some Möbius invariant spaces in the Bloch space. Monreal Galán and Nicolau in [19] characterized the closure in the Bloch norm of the space for . Later, Galanopoulos et al. in [20] studied the closure in the Bloch norm of on the unit ball in . Moreover, they have extended this result to the whole range . Bao and Göğüş [21] studied the closure of Dirichlet type spaces in the Bloch space. See [22–26] for some related results.

It is well known that (when ) Hence, From [19], we see that a Bloch function is in if and only if, for every , Here and From [16, 17], we see that a Bloch function is in if and only if, for every , It is natural to ask what is , the closure of the Morrey type space () in the Bloch norm?

The purpose of this paper is to characterize . Moreover, we study the boundedness and compactness of composition operators and .

Throughout this paper, we say that if there exists a constant such that . The symbol means that .

2. Main Results and Proofs

In this section we give our main results and proofs. For this purpose, we need the following well-known estimate which can be found in [27] or [18].

Lemma 1. Let and . If , then

The following lemma is Lemma 3.1.1 in [1].

Lemma 2. Let and a nonnegative measure on . Then is a -Carleson measure if and only if

Lemma 3. Let . Then if and only if Moreover,

Proof. Denote Then, from (7) we have that if and only if is a -Carleson measure. Hence, by Lemma 2, we get that if and only if Let . We get the desired result.

Now we present and prove our main results in this paper.

Theorem 4. Let and . Then if and only if, for any ,

Proof. Take and . Then there exists a such that . Since we see that . Then by Lemma 3 we get as desired.
Conversely, suppose that (23) holds. Fix and let satisfy (23). Without loss of generality, we may assume that For any , by Proposition 4.27 in [28], where . Following [18], we decompose as , where and After a calculation, we get and Let . Then ; we obtain Then by Lemma 3.10 of [28] we get Hence . Applying Fubini’s theorem and Lemma 1, we deduce that that is, . Thus, for any , there exists a function such that ; i.e., . The proof is complete.

Next, we consider the boundedness and compactness of composition operators from to .

Theorem 5. Let and let be an analytic self-map of . Then is bounded if and only if, for any ,

Proof. Assume that is bounded. From [29], we see that there exists two functions such that By the boundedness of , we get Hence, Theorem 4 implies that, for any , and When , we get which implies that either or Hence, Conversely, suppose that (34) holds. Let . Then Therefore, for any , we obtainFrom Theorem 4, we have , i.e., is bounded. The proof is complete.

Theorem 6. Let and let be an analytic self-map of . Then is automatically bounded.

Proof. Since and , we see that . Let . For any , there is a constant () such that whenever . Let . Then, by the assumption and Schwarz-Pick Lemma, we have which implies that Thus, Let . Then Hence, . Since , by Theorem 4 we get By Theorem 4 again, we see that . Hence is bounded. The proof is complete.

Theorem 7. Let and let be an analytic self-map of . Then the following statements are equivalent. (i) is compact;(ii) is compact;(iii) 

Proof. . It is clear.
Since , we see that is compact. Using [9, Theorem 1], (48) follows.
By the assumption, we see that there exists , such that Let such that Then, . Therefore, which implies that Let . Then . Hence Since , by Theorem 4 we have By (52), (53), and Theorem 5, is bounded.
Since (48) holds, from [7, Theorem 2], we see that is compact. Therefore is compact. The proof is complete.

Theorem 8. Let and let be an analytic self-map of . Then is compact if and only if

Proof. Suppose that (54) holds. By Theorem 7, is compact. Since , we get that is compact, as desired.
Conversely, assume that is compact. It is clear that since . Since is closure of all polynomials in and the space contains all polynomials, hence, is compact. By Theorem 7 we see that (54) holds. The proof is complete.

From Theorems 7 and 8 and [13, Theorem 3], we immediately get the following corollary.

Corollary 9. Let and let be an analytic self-map of . Then the following statements are equivalent. (i) is compact;(ii) is compact;(iii) is compact;(iv) is compact;(v) is compact;(vi);(vii)

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.