Composition Operators and the Closure of Morrey Space in the Bloch Space

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.


Introduction
Clearly, if  = D, then () = D. Let  > 0. A nonnegative measure  on D is said to be an -Carleson measure (see [1]) if Here , the space of analytic functions whose boundary functions have bounded mean oscillation.From [2] or [3], the norm of functions  ∈ L 2, (D) can be defined as follows. L We remark that B ̸ ⊆ L 2, and L 2, ̸ ⊆ B. It is well known that the function After a calculation, we see that . See [2][3][4][5][6] for the study of Morrey space and related operators.
For every self-map  on D, the composition operator   is defined on (D) by It is a simple consequence of the Schwartz-Pick lemma that any analytic self-mapping  of D induces a bounded composition operator   on the Bloch space.Madigan and Matheson in [7] See, for example, [7][8][9][10][11][12][13][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problem.Anderson in [16] mentioned that Jones gave a characterization of C B (), the closure of  in the Bloch norm (an unpublished result).A complete proof was provided by Ghatage and Zheng in [17].Zhao in [18] studied the closures of some Möbius invariant spaces in the Bloch space.Monreal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 1 <  < ∞.Later, Galanopoulos et al. in [20] studied the closure in the Bloch norm of   on the unit ball in C  .Moreover, they have extended this result to the whole range 0 <  < ∞.Bao and Gögüs ¸ [21] studied the closure of Dirichlet type spaces D  (−1 <  ≤ 1) in the Bloch space.See [22][23][24][25][26] for some related results.
It is well known that (when 0 <  < 1) Hence, From [19], we see that a Bloch function  is in C B ( 2 ∩ B) if and only if, for every  > 0, Here and From [16,17], we see that a Bloch function  is in C B () if and only if, for every  > 0, It is natural to ask what is C B (L 2, ∩ B), the closure of the Morrey type space L 2, (0 <  < 1) in the Bloch norm?
The purpose of this paper is to characterize C B (L 2, ∩ B).Moreover, we study the boundedness and compactness of composition operators Throughout this paper, we say that  ≲  if there exists a constant  such that  ≤ .The symbol  ≈  means that  ≲  ≲ .

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].
The following lemma is Lemma 3.1. Proof.Denote Then, from (7) we have that  ∈ L Let  = .We get the desired result.
Now we present and prove our main results in this paper.
The proof is complete.
Next, we consider the boundedness and compactness of composition operators from B to C B (L 2, ∩ B).
Theorem 5. Let 0 <  < 1 and let  be an analytic self-map of D. Then   : B → C B (L 2, ∩ B) is bounded if and only if, for any  > 0, Proof.Assume that   : From [29], we see that there exists two functions By the boundedness of   , we get  1 ∘ ,  2 ∘  ∈ C B (L 2, ∩ B).Hence, Theorem 4 implies that, for any  > 0, and Therefore, for any  > 0, we obtain From Theorem 4, we have  ∘  ∈ C B (L