Weighted Korenblum-Roberts theory

The classical Korenblum-Roberts Theorem characterizes the cyclic singular inner functions in the Bergman spaces of the unit disk D as those for which the corresponding singular measure vanishes on Beurling-Carleson sets of Lebesgue measure zero. We solve the weighted variant of the problem in which the Bergman space is replaced by a P t ( μ ) space, the closure of analytic polynomials in a Lebesgue space L t ( μ ) corresponding to a measure of the form d A α + w d m , with d A α being the standard weighted area measure on D , dm the Lebesgue measure on the unit circle T , and w a general weight on T . We characterize when P t ( μ ) of this form is a space of analytic functions on D by computing the Thomson decomposition of the measure μ . The structure of the decomposition is expressed in terms of what we call the family of associated Beurling-Carleson sets . We characterize the cyclic singular inner functions in the analytic P t ( μ ) spaces as those for which the corresponding singular measure vanishes on the family of associated Beurling-Carleson sets. Unlike the classical setting, Beurling-Carleson sets of both zero and positive Lebesgue measure appear in our description. As an application of our results, we complete the characterization of the symbols b : D → D which generate a de Branges-Rovnyak space with a dense subset of functions smooth on T . The characterization is given explicitly in terms of the modulus of b on T and the singular measure corresponding to the singular inner factor of b . Our proofs involve Khrushchev's techniques of simultaneous polynomial approximations and linear programming ideas of Korenblum, combined with recently established constrained L 1 -optimization tools.