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

Abstract

Here we introduce the $n$th weighted space on the upper half-plane in the complex plane $ℂ$. For the case $n=2$, we call it the Zygmund-type space, and denote it by $𝒵\left({\Pi }_{+}\right)$. The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator $C_{\phi }f\left(z\right)=f\left(\phi \left(z\right)\right)$ from the Hardy space $H^p\left({\Pi }_{+}\right)$ on the upper half-plane, to the Zygmund-type space, where $\phi$ is an analytic self-map of the upper half-plane.