Weighted Korenblum-Roberts Theory

The classical Korenblum-Roberts Theorem characterizes the cyclic singular inner functions in the Bergman spaces of the unit disk $\mathbb{D}$ 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 $\mathcal{P}^t(\mu)$ space, the closure of analytic polynomials in a Lebesgue space $\mathcal{L}^t(\mu)$ corresponding to a measure of the form $dA_\alpha + w\, dm$, with $dA_\alpha$ being the standard weighted area measure on $\mathbb{D}$, $dm$ the Lebesgue measure on the unit circle $\mathbb{T}$, and $w$ a general weight on $\mathbb{T}$. We characterize when $\mathcal{P}^t(\mu)$ of this form is a space of analytic functions on $\mathbb{D}$ by computing the Thomson decomposition of the measure $\mu$. 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 $\mathcal{P}^t(\mu)$ 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 $b:\mathbb{D} \to \mathbb{D}$ which generate a de Branges-Rovnyak space with a dense subset of functions smooth on $\mathbb{T}$. The characterization is given explicitly in terms of the modulus of $b$ on $\mathbb{T}$ 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 $\mathcal{L}^1$-optimization tools.