On Dirichlet Spaces With a Class of Superharmonic Weights

Abstract

AbstractIn this paper, we investigate Dirichlet spaces with superharmonic weights induced by positive Borel measures μ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces via the balayage of the measure μ. We show that is equal to if and only if μ is a Carleson measure for . As an application, we obtain the reproducing kernel of when μ is an infinite sum of point-mass measures. We consider the boundary behavior and innerouter factorization of functions in . We also characterize the boundedness and compactness of composition operators on .