Analytic Morrey Spaces and Bloch-Type Spaces

Table 1 will help us to understand the structure ofALp,η (see, e.g., [1–3] and [4, p. 209–217] for the real counterparts). Of course, this value defines a seminorm on ALp,η. A complete norm onALp,η can be equipped by 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩ALp,η = 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩p + 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩ALp,η,∗ . (3) For α > 0, the α-Bloch space, denoted by Bα = Bα(D), is the set of all f ∈ A(D) for which Bα (f) = sup z∈D (1 − |z|2)α 󵄨󵄨󵄨󵄨󵄨f󸀠 (z)󵄨󵄨󵄨󵄨󵄨 < ∞. (4)


Introduction
The open unit disk, the unit circle, and the area measure in the complex C are denoted by D, T, and (), respectively.The space of all analytic functions in D will be written A(D), and H  denotes the Hardy space on the unit disk D.

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].
belong to AL , .Moreover,   is uniformly bounded in AL , ; that is, 3. Boundedness of the Operators   ,   : In this section, we give the boundedness of   and   from AL , → B  , respectively. Moreover, Proof.Suppose   : Note that Hence, sup Conversely, suppose sup By Lemma 1, we have that Then Taking supremum in the last inequality over the set 1/2 ≤ || < 1 and applying to the maximum modulus principle we have sup which implies   : AL , → B  is bounded and Remark 5. When  < 1 + (1 − )/ in Theorem 4, the conclusion is equivalent to  ≡ 0.
As an application of Theorems 3 and 4, we can obtain the following corollary.

Superposition on AL 𝑝,𝜂
Let X and Y represent two subspaces of A(D).If  is a complex-valued function C such that  ∘  ∈ Y whenever  ∈ X, then we call that  acts by superposition from X into Y.If X and Y contain the linear functions, then  must be an entire function.The superposition S  : X → Y with symbol  is then defined by S  () = ∘.A basic question is when S  map X into Y continuously.This question has been studied for many distinct pairs (X, Y)-see, for example, [17][18][19][20].In this section, we are interested in the analytic Morrey space and have the following result which extends the case of  = 2 in [3].
Taking the following AL , -function Setting || → 1 in the last estimate and noticing  ∈ (0, 1), we obtain that the entire function  is bounded on C. By using the maximum principle, we get that  must be a linear function.

Weighted Composition Operator from
AL , to B  Suppose that  ∈ A(D) and  is an analytic self-map of D, (D) ⊂ D. 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.

Table 1 :
The structure of analytic Campanato space.Notation.Let A ≲ B and A ≳ B denote that there exists an absolute constant  > 0 such that A ≤ B and A ≥ B, respectively.A ≈ B means that A ≲ B and A ≳ B hold.