Copyright © 2009 Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 half-plane, to the Zygmund-type space, where φ is an analytic self-map of the upper half-plane.