Denjoy Domains and BMOA

TL;DR

This paper proves that for Carleson-homogeneous Denjoy domains, the logarithm of the derivative of their universal covering functions belongs to BMOA, linking geometric domain properties with function space regularity.

Contribution

It establishes a new connection between Carleson-homogeneous Denjoy domains and BMOA regularity of universal coverings, extending previous ideas to a broader class of planar domains.

Findings

If $U$ is a Carleson-homogeneous Denjoy domain, then $ ext{log} {f'} otin BMOA$ for its universal covering.

Develops a general theorem providing sufficient conditions for $ ext{log} {f'} o BMOA$ in planar domains.

Links geometric properties of domains with analytic function space regularity.

Abstract

A Denjoy domain is a plane domain whose complement is a closed subset $E$ of the extended real line $\bar{R}$ containing $\infty$ : such a domain is called Carleson-homogeneous if there exists $C>0$ such that for all $z\in E$ and $r>0$, one has $\vert E\cap [z-r,z+r]\vert\geq Cr$, where $\vert\cdot\vert$ is the Lebesgue measure on the line. We prove that if $U=\bar{ \mathbb C}\backslash K$ is a Carleson-homogeneous Denjoy domain then, if $f$ stands for one of its universal coverings, $\log {f'}\in BMOA.$ In order to prove this result, we develop ideas from [On Carleson measures induced by Beltrami coefficients being compatible with Fuchsian groups, Ann. Fenn. Math. 46(2021),67-77] leading to a general theorem about planar domains giving sufficient conditions ensuring that $\log {f'}\in BMOA$ for any universal covering $f.$