Beurling-Carleson sets, inner functions and a semi-linear equation

TL;DR

This paper explores Beurling-Carleson sets, their properties, and their connections to inner functions and semi-linear equations, revealing new characterizations and optimal examples in complex analysis.

Contribution

It provides a general definition of Beurling-Carleson sets, characterizes measures not charging these sets, and links measure properties to solutions of a semi-linear PDE.

Findings

Measures not charging Beurling-Carleson sets are characterized by Roberts decomposition.

Properties like Nevanlinna class membership imply measures are on countable unions of Beurling-Carleson sets.

Measures on unions of α-Beurling-Carleson sets correspond to solutions of a specific semi-linear PDE.

Abstract

Beurling-Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups of Widom-type and the corona problem in quotient Banach algebras. After surveying these developments, we give a general definition of Beurling-Carleson sets and discuss some of their basic properties. We show that the Roberts decomposition characterizes measures that do not charge Beurling-Carleson sets. For a positive singular measure $\mu$ on the unit circle, let $S_\mu$ denote the singular inner function with singular measure $\mu$. In the second part of the paper, we use a corona-type decomposition to relate a number of properties of singular measures on the unit circle such as membership of $S'_\mu$ in the Nevanlinna class $\mathcal N$, area conditions on level sets of $S_\mu$ and wepability. It was known that each of these properties holds for measures concentrated on Beurling-Carleson sets. We show that each of these properties implies that $\mu$ lives on a countable union of Beurling-Carleson sets. We also describe partial relations involving the membership of $S'_\mu$ in the Hardy space $H^p$, membership of $S_\mu$ in the Besov space $B^p$ and $(1-p)$-Beurling-Carleson sets and give a number of examples which show that our results are optimal. Finally, we show that measures that live on countable unions of $\alpha$-Beurling-Carleson sets are almost in bijection with nearly-maximal solutions of $\Delta u = u^p \cdot \chi_{u > 0}$ when $p > 3$ and $\alpha = \frac{p-3}{p-1}$.