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On Dirichlet Spaces with a Class of Superharmonic Weights

Published:2017-05-19
Printed: Aug 2018
• Guanlong Bao,
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
• Nihat Gökhan Göğüş,
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul, 34956 Turkey
• Stamatis Pouliasis,
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul, 34956 Turkey
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Abstract

In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with superharmonic weights induced by positive Borel measures $\mu$ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for $\mathcal{D}_\mu$ 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 $\mathcal{D}_\mu$ is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite sum of point mass measures. We consider the boundary behavior and inner-outer factorization of functions in $\mathcal{D}_\mu$. We also characterize the boundedness and compactness of composition operators on $\mathcal{D}_\mu$.
 Keywords: Dirichlet space, Hardy space, superharmonic weight
 MSC Classifications: 30H10 - Hardy spaces 31C25 - Dirichlet spaces 46E15 - Banach spaces of continuous, differentiable or analytic functions

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