Quasiconformal and HQC mappings between Lyapunov Jordan domains

TL;DR

This paper proves that quasiconformal mappings from the unit disk to Lyapunov domains preserve Lyapunov-type subdomains touching the boundary and establishes that harmonic quasiconformal maps are co-Lipschitz, solving an open problem.

Contribution

It demonstrates the boundary behavior of quasiconformal maps between Lyapunov domains and proves harmonic quasiconformal maps are co-Lipschitz, addressing an open question in the field.

Findings

Quasiconformal maps preserve Lyapunov-type subdomains touching the boundary.

Harmonic quasiconformal maps are co-Lipschitz on the unit disk.

The results settle an open problem regarding boundary regularity.

Abstract

Let $h$ be a quasiconformal (qc) mapping of the unit disk $\mathbb{U}$ onto a Lyapunov domain. We show that $h$ maps subdomains of Lyapunov type of $\mathbb{U}$, which touch the boundary of $\mathbb{U}$, onto domains of similar type. In particular if $h$ is a harmonic qc (hqc) mapping of $\mathbb{U}$ onto a Lyapunov domain, using it, we prove that $h$ is co-Lipschitz (co-Lip) on $\mathbb{U}$. This settles an open intriguing problem.