On Dirichlet spaces with a class of superharmonic weights

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Official URL: http://dx.doi.org/10.4153/CJM-2017-005-1

In this paper, we investigate Dirichlet spaces D-mu with superharmonic weights induced by positive Borel measures mu on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for D-mu spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces H-mu(2) via the balayage of the measure mu. We show that D-mu is equal to H-mu(2) if and only if mu is a Carleson measure for D-mu. As an application, we obtain the reproducing kernel of D-mu when mu is an infinite sum of point-mass measures. We consider the boundary behavior and inner-outer factorization of functions in D-mu. We also characterize the boundedness and compactness of composition operators on D-mu.