Boundedness of self-map composition operators for two types of weights on the upper half-plane

In this paper we find conditions for boundedness of self-map composition operators on weighted spaces of holomorphic functions on the upper half-plane for two kinds of weights which are of moderate growth.


Introduction
Different properties of composition operators between weighted spaces of holomorphic functions on the unit disc or upper half-plane have been the subject of many papers in recent decades (Ardalani, 2014;Ardalani & Lusky, 2011, 2012a, 2012bBonet, 2003;Bonet, Domanski, & Lindstrom, 1998, 1999Bonet, Fritz, & Jorda, 2005;Cowen, 1995;Madigan, 1993, Shapiro, 1987Zhu, 1990). In Theorem 2.3 of Bonet et al. (1998), authors have characterized boundedness of self-map composition operators on weighted spaces of holomorphic functions on the unit disc in terms of associated weight which satisfies well-known growth condition that is used by Lusky (1995). Indeed they have found a condition under which all self-map composition operators on weighted spaces of holomorphic functions on the unit disc are bounded. In this paper we intend to find conditions for boundedness of all self-map composition operators on weighted spaces of holomorphic functions on the upper half-plane for standard weights in the sense of Ardalani (2014), Ardalani andLusky (2011, 2012a) and for a new type of weights on the upper half-plane which we call it type(II) weights. For

PUBLIC INTEREST STATEMENT
The present paper is devoted to the problem of continuity of composition operators on the weighted spaces of holomorphic functions on the upper half-plane equipped with sup-norms. These spaces of holomorphic functions (on the unit disc) with controlled growth are natural classes studied by Shields and Williams in seventies and later by large variety of authors. Composition operators are very natural operators and their study is by now a true industry which is interesting and worth studying. standard weights we use the results of Ardalani and Lusky (2012b) in order to prove Theorem 2.1. For weights of type(II) we make an isomorphism between weighted spaces of holomorphic functions on the unit disc and weighted spaces of holomorphic functions on the upper half-plane. Then we use this isomorphism and Theorem 2.3 of Bonet et al. (1998) to obtain a sufficient condition for boundedness of self-map composition operators on weighted spaces of holomorphic functions on the upper half-plane. This isomorphism is constructed under a certain growth condition which we call it ( * ) � throughout this paper. Although, we use the concept of associated weight to prove Theorem 3.2, the associated weight does not appear in the assertion of Theorem 3.2 and that is important because it is difficult to calculate an associated weight. We continue with the preliminaries which are required in the rest of this paper.

Definition 1.2 A continuous function
Note that Standard weights are increasing thorough the imaginary axis while type(II) weights are increasing on the imaginary axis whenever the imaginary part is in (0, 1]. Also type(II) weights has symmetric property which makes them interesting (see Remark 1.13). Definition 1.6 Let be a type(II) weight on G. We say satisfies condition ( * ) � if there are constants Im 2 ) whenever Im 2 ≤ Im 1 and | 1 |, | 2 | ≤ 1.
Lemma 1.15 Let be a radial weight on . Then the following are equivalent: Remark 1.16 Following example shows that there are standard weights satisfying ( * ), but not all composition operators are bounded. Therefore, the situation on the upper half-plane is essentially different from the unit disc.

Boundedness of composition operators for standard weights
Although Remark 1.15 and Example 1.17 show that we cannot expect to obtain a result similar to Lemma 1.15 for standard weights but we are able to characterize all the analytic maps such that the self-map composition operators on the upper half-plane are bounded.

Composition maps C :H (G) ⟶ H (G) are bounded if and only if
Proof By Corollary 1.5 of Ardalani and Lusky (2012b)
Following example shows that Theorem 2.1 is not true if does not satisfy condition ( * * ).

Main results
We begin this section with Lemma 3.1 which makes an isomorphism between weighted spaces of holomorphic functions on the upper half-plane (for type(II) weights) and weighted spaces of holomorphic functions on the unit disc. This isomorphism is our main tool to prove Theorem 3.2.
Proof First assertion of the lemma is obvious. By Remark 1.13 there is a constant C > 0 such that Consider a fixed z ∈ . Firstly, assume Re z ≤ 0. Since | z | ≥ −Re z, we have