Weighted Korenblum-Roberts Theory

TL;DR

This paper extends the classical Korenblum–Roberts cyclicity theory to weighted analytic function spaces $\,\mathcal{P}^t(\mu)$ with measures $\mu=dA_\alpha+w\,dm$, characterizing when the space consists of analytic functions on the disk via a Thomson decomposition. It introduces Beurling-Carleson sets and their $w$-associated variants to describe the analytic structure, proving that a singular inner function $S_\nu$ is cyclic in $\mathcal{P}^t(\mu)$ if and only if $\nu$ vanishes on all sets in $\mathrm{Assoc}\mathcal{BC}(w)$. The authors develop an optimization framework using Hausdorff functionals and constrained $\mathcal{L}^1$-programming, adapting Korenblum’s linear programming to establish sufficiency, while a permanence argument and Khrushchev-type lemmas underpin necessity results. As an application, they obtain a complete characterization for when de Branges–Rovnyak spaces $\mathcal{H}(b)$ admit a dense subset of boundary-smooth functions, in terms of the modulus of $b$ on the circle and the singular inner factor, with broader implications for Nevanlinna-class cyclicity. Overall, the work blends Thomson decomposition, constrained optimization, and classical cyclicity techniques to yield explicit, sharp criteria in a weighted analytic setting with potential extensions to more general gauges and $H(b)$-type spaces.

Abstract

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 $\mathcal{P}^t(μ)$ space, the closure of analytic polynomials in a Lebesgue space $\mathcal{L}^t(μ)$ corresponding to a measure of the form $dA_α+ w\, dm$, with $dA_α$ 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 $\mathcal{P}^t(μ)$ of this form is a space of analytic functions on $\mathbb{D}$ 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 $\mathcal{P}^t(μ)$ 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 $b:\mathbb{D} \to \mathbb{D}$ 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.

Theorems & Definitions (47)

• Theorem A
• Theorem B
• Corollary
• Theorem : Korenblum-Roberts cyclicity theorem
• Theorem C
• Proposition 2.1
• Lemma 2.2
• proof : Proof of \ref{['L:ResBoundLemma']}
• Lemma 3.1
• proof