Regularity for generators of invariant subspaces of the Dirichlet shift

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Richter S., Yilmaz F.

Journal of Functional Analysis, cilt.277, sa.7, ss.2117-2132, 2019 (SCI-Expanded, Scopus)

• Yayın Türü: Makale / Tam Makale
• Cilt numarası: 277 Sayı: 7
• Basım Tarihi: 2019
• Doi Numarası: 10.1016/j.jfa.2018.10.006
• Dergi Adı: Journal of Functional Analysis
• Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
• Sayfa Sayıları: ss.2117-2132
• Anahtar Kelimeler: Carleson set, Dirichlet space, Extremal function, Logarithmic capacity
• Ankara Yıldırım Beyazıt Üniversitesi Adresli: Hayır

Özet

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.