Regularity for generators of invariant subspaces of the Dirichlet shift

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 subset of partial derivative D we write D-E = {f is an element of D : lim(r -> 1) f(re(it)) = 0 q.e.}, where q. e. stands for "except possibly for e(it) in a set of logarithmic capacity 0 ''. We show that if E is a Carleson set, then there is a function f is an element of D-E that is also in the disc algebra and that generates DE in the sense that D-E = clos {pf : p is a polynomial}. We also show that if phi is an element of D is an extrernal function (i.e. < p phi, phi > = p(0) for every polynomial p), then the limits of vertical bar phi(z)vertical bar exist for every e(it) is an element of partial derivative D as z approaches e(it) from within any polynornially tangential approach region. (C) 2018 Elsevier Inc. All rights reserved.