Composition operators in some function spaces of hyperbolic type

Abstract

In this paper, we introduce natural metrics in the weighted hyperbolic (α,ω)-Bloch class and F∗(p,q,s;ω) classes. These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, Lipschitz continuous and boundedness of the composition operator Cφ acting from the hyperbolic (α,ω)-Bloch class to the classes F∗(p,q,s;ω) are characterized by conditions depending on an analytic self-map φ : D → D.