Blaschke Products and Zero Sets in Weighted Dirichlet Spaces

  • H. Bahajji-El Idrissi
  • O. El-FallahEmail author


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).


Blaschke product Dirichlet space Capacity 

Mathematics Subject Classification (2010)

31C25 30J10 31C15 


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


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratory of Mathematical Analysis and ApplicationsFaculty of Sciences, Mohammed 5 University in RabatRabatMorocco

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