Global integral criteria for composition operators

Let Dα denote the Dirichlet-type space of functions analytic on the unit disk U and Qα the conformal invariant version of this space. Any analytic self-map φ of U induces a composition operator Cφ acting on Dα, respectively, Qα by Cφf = f ◦ φ, where f ∈ Dα, respectively, f ∈ Qα. The aim of this paper is to characterize boundedness and compactness of such operators in terms of global area integrals of φ.  2002 Elsevier Science (USA). All rights reserved