On a product-type operator between Hardy and \u03b1-Bloch spaces of the upper half-plane

Stevo Stević,Ajay K Sharma

doi:10.1186/s13660-018-1867-8

Abstract

Recently we have introduced a product-type operator and studied it on some spaces of analytic functions on the unit disc. Here we start investigating the operator on the space of analytic functions on the upper half-plane. We characterize the boundedness and compactness of the operator between Hardy and α-Bloch spaces on the domain.

Highlights

Let D be the unit disc in the complex plane C, + = {z ∈ C : z > 0} the upper half-plane in C, and + = + ∪ {∞}
To treat product-type operators consisting of exactly one composition, multiplication and differentiation operator in a unified manner, we have recently introduced a generalized operator and studied it on the weighted Bergman spaces
We start investigating the operator on such spaces by characterizing the boundedness and compactness of the operator between the Hardy and α-Bloch spaces on the domain, that is, of Tψ1,ψ2,φ : Hp( +) → Bα( +)

Summary

Introduction

For 0 < p < ∞, the Hardy space of +, denoted by Hp( +), consists of all f ∈ H( +) such that. For 1 ≤ p < ∞, Hp( +) is a Banach space. +, denoted by Bα( +), consists of all f Bα( +) := f (i) + sup ( z)α f (z) < ∞. With the norm (1), the α-Bloch space is a Banach space. For Bloch-type spaces on various domains and operators on them, see, for example, [1,2,3,4,5,6,7,8,9,10,11,12,13] and the references therein.

The product of these two operators

We also have