Abstract and Applied Analysis
Volume 2009 (2009), Article ID 161528, 8 pages
doi:10.1155/2009/161528

Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane

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.