Composition operators from the Hardy space to the nth weighted-type space on the unit disk and the half-plane

Abstract

The boundedness of the composition operator Cφ f (z) = f (φ (z)) from the Hardy space Hp (X), p > 0, where X is the upper half-plane Π+ = {z ∈ C : Im z > 0} or the unit disk D = {z ∈ C : | z | < 1} in the complex plane C, to the nth weighted-type space, where φ is an analytic self-map of X, is characterized.