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

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 $\alpha = \frac{p-3}{p-1}$.