On Dirichlet Spaces With a Class of Superharmonic Weights. (1st August 2018) - Bao, Guanlong; Göğüş, Nihat Gokhan; Pouliasis, Stamatis
Abstract: In this paper, we investigate Dirichlet spaces ${D_{μ }}$ with superharmonic weights induced by positive Borel measures $μ $ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for ${D_{μ }}$ spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces $H_{μ }^{2}$ via the balayage of the measure $μ $ . We show that ${D_{μ }}$ is equal to $H_{μ }^{2}$ if and only if $μ $ is a Carleson measure for ${D_{μ }}$ . As an application, we obtain the reproducing kernel of ${D_{μ }}$ when $μ $ is an infinite sum of point-mass measures. We consider the boundary behavior and innerouter factorization of functions in ${D_{μ }}$ . We also characterize the boundedness and compactness of composition operators on ${D_{μ }}$ .