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A Sharp Entropy Condition For The Density Of Angular Derivatives

Bergman, Alex LU orcid (2024)
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 Carathéodory angular derivatives of f 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, such that its set of Carathéodory angular derivatives 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.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Working paper/Preprint
publication status
published
subject
publisher
arXiv.org
language
English
LU publication?
yes
id
4820e645-5914-4a7a-8e63-f318d47aea84
alternative location
https://arxiv.org/abs/2409.14389
date added to LUP
2024-09-25 11:24:06
date last changed
2025-04-04 15:07:17
@misc{4820e645-5914-4a7a-8e63-f318d47aea84,
  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 Carathéodory angular derivatives of f 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, such that its set of Carathéodory angular derivatives 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.}},
  author       = {{Bergman, Alex}},
  language     = {{eng}},
  month        = {{09}},
  note         = {{Preprint}},
  publisher    = {{arXiv.org}},
  title        = {{A Sharp Entropy Condition For The Density Of Angular Derivatives}},
  url          = {{https://arxiv.org/abs/2409.14389}},
  year         = {{2024}},
}