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A Sharp Entropy Condition for The Density of Angular Derivatives

Bergman, Alex LU orcid (2025) In Comptes Rendus Mathématique p.29-33
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.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Comptes Rendus Mathématique
pages
5 pages
publisher
Academie des sciences
external identifiers
  • scopus:105002414097
ISSN
1631-073X
DOI
10.5802/crmath.695
language
English
LU publication?
yes
id
76e2f9c7-03aa-42e5-9912-7c10f64003fc
date added to LUP
2025-03-13 17:43:05
date last changed
2025-06-03 04:05:33
@article{76e2f9c7-03aa-42e5-9912-7c10f64003fc,
  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.}},
  author       = {{Bergman, Alex}},
  issn         = {{1631-073X}},
  language     = {{eng}},
  pages        = {{29--33}},
  publisher    = {{Academie des sciences}},
  series       = {{Comptes Rendus Mathématique}},
  title        = {{A Sharp Entropy Condition for The Density of Angular Derivatives}},
  url          = {{http://dx.doi.org/10.5802/crmath.695}},
  doi          = {{10.5802/crmath.695}},
  year         = {{2025}},
}