Blaschke Products and Zero Sets in Weighted Dirichlet Spaces


In this paper, we deal with superharmonically weighted Dirichlet spaces \(\mathcal {D}_{\omega }\). First, we prove that the classical Dirichlet space is the largest, among all these spaces, which contains no infinite Blaschke product. Next, we give new sufficient conditions on a Blaschke sequence to be a zero set for \(\mathcal {D}_{\omega }\). Our conditions improve Shapiro-Shields condition for \(\mathcal {D}_{\alpha }\), when α ∈ (0,1).

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The authors are grateful to the referee for his valuable remarks and suggestions.

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Correspondence to O. El-Fallah.

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Research partially supported by “Hassan II Academy of Science and Technology”.

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Idrissi, H.BE., El-Fallah, O. Blaschke Products and Zero Sets in Weighted Dirichlet Spaces. Potential Anal 53, 1299–1316 (2020).

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  • Blaschke product
  • Dirichlet space
  • Capacity

Mathematics Subject Classification (2010)

  • 31C25
  • 30J10
  • 31C15