Composition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-Plane

Stevo Stević

doi:10.1155/2009/161528

Abstract

Here we introduce thenth weighted space on the upper half-planeΠ+={z∈ℂ:Im z>0}in the complex planeℂ. For the casen=2, we call it the Zygmund-type space, and denote it by&#x1D4B5;(Π+). The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operatorCφf(z)=f(φ(z))from the Hardy spaceHp(Π+)on the upper half-plane, to the Zygmund-type space, whereφis an analytic self-map of the upper half-plane.

Highlights

Let Π be the upper half-plane, that is, the set {z ∈ C : Im z > 0} and H Π the space of all analytic functions on Π
To clarify the notation we have just introduced, we have to say that the main reason for this name is found in the fact that for the case of the unit disk D {z : |z| < 1} in the complex palne C, Zygmund see, e.g., 1, Theorem 5.3 proved that a holomorphic function on D continuous on the closed unit disk D satisfies the following condition: f ei θ h f ei θ−h − 2f eiθ sup h>0, θ∈ 0,2π h
Motivated by Theorem A, here we investigate the boundedness of the operator Cφ : Hp Π → Z Π

Summary

Stevo Stevic

Recommended by Simeon Reich Here we introduce the nth weighted space on the upper half-plane Π {z ∈ C : Im z > 0} in the complex plane C. For the case n 2, we call it the Zygmund-type space, and denote it by Z Π. The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator Cφf z f φ z from the Hardy space Hp Π on the upper halfplane, to the Zygmund-type space, where φ is an analytic self-map of the upper half-plane.

Introduction

Im z Im φ z

Cφ gw