Compact weighted composition operators on H^p-spaces

Let u and ψ be two analytic functions on the unit disc D such that ψ(D) ⊂ D. A weighted composition operator uC _ψ induced by u and ψ is defined by uC _ψ f := u · f o ψ for every f in H ^p , the Hardy space of D. We investigate compactness of uC _ψ on H ^p in terms of function-theoretic properties of u and ψ.