A Sharp Entropy Condition for the Density of Angular Derivatives
Year: 2025
Type: article
Abstract: Let f be a holomorphic self-map of the unit disc. We show that if log ( 1 - | f ( z ) | ) is integrable on a sub-arc of the unit circle, I , then the set of points where the function f has finite Carathéodory angular derivative on I is a countable union of Beurling–Carleson sets of finite entropy. Conversely, given a countable union of Beurling–Carleson sets, E , we construct a holomorphic self-map of the unit disc, f , such that the set of points where the f function has finite Carathéodory angular derivative is equal to E and log ( 1 - | f ( z ) | ) is integrable on the unit circle. Our main technical tools are the Aleksandrov disintegration Theorem and a characterization of countable unions of Beurling–Carleson sets due to Makarov and Nikolski.
Source: Comptes Rendus Mathématique
DOI: 10.5802/crmath.695
Author: Alex Bergman
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4408199257