Regularity for generators of invariant subspaces of the Dirichlet shift

Abstract

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 writeDE={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.