On dirichlet spaces with a class of superharmonic weights

Guanlong Bao, Nihat Gökhan Göǧüş, Stamatis Pouliasis

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In this paper, we investigate Dirichlet spaces Dμ with superharmonic weights induced by positive Borel measures μ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for Dμ spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces H2 via the balayage of the measure μ. We show that Dμ is equal to H2 if and only if μ is a Carleson measure for Dμ. As an application, we obtain the reproducing kernel of Dμ when μ is an infinite sum of point-mass measures. We consider the boundary behavior and inner-outer factorization of functions in Dμ. We also characterize the boundedness and compactness of composition operators on Dμ.

Original languageEnglish
Pages (from-to)721-741
Number of pages21
JournalCanadian Journal of Mathematics
Issue number4
StatePublished - Aug 2018


  • Dirichlet space
  • Hardy space
  • Superharmonic weight


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