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

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.