Regularity for generators of invariant subspaces of the Dirichlet shift

Richter S., Yilmaz F.

Journal of Functional Analysis, vol.277, no.7, pp.2117-2132, 2019 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 277 Issue: 7
  • Publication Date: 2019
  • Doi Number: 10.1016/j.jfa.2018.10.006
  • Journal Name: Journal of Functional Analysis
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2117-2132
  • Keywords: Carleson set, Dirichlet space, Extremal function, Logarithmic capacity
  • Ankara Yıldırım Beyazıt University Affiliated: No


Let D denote the classical Dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable. For a set E⊆∂D we write DE={f∈D:limr→1−⁡f(reit)=0q.e.}, where q.e. stands for “except possibly for eit in a set of logarithmic capacity 0”. We show that if E is a Carleson set, then there is a function f∈DE that is also in the disc algebra and that generates DE in the sense that DE=clos{pf:p is a polynomial}. We also show that if φ∈D is an extremal function (i.e. 〈pφ,φ〉=p(0) for every polynomial p), then the limits of |φ(z)| exist for every eit∈∂D as z approaches eit from within any polynomially tangential approach region.