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 script Z sign(Π +). The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator C φ f(z)=f(φ(z)) from the Hardy space H p (Π +) on the upper half-plane, to the Zygmund-type space, where φ is an analytic self-map of the upper half-plane.