On quasisymmetry of quasiconformal mappings and its applications

TL;DR

This paper proves that weak quasisymmetry implies quasisymmetry for restrictions of quasiconformal maps on broad domains and applies this to characterize when certain domains are John domains, extending Heinonen's 1989 results.

Contribution

It demonstrates that weak quasisymmetry implies quasisymmetry on broad domains and provides nine equivalent conditions for domains to be John, generalizing Heinonen's 1989 findings.

Findings

Weak quasisymmetry implies quasisymmetry on broad domains.

Established nine equivalent conditions for domains to be John.

Extended Heinonen's 1989 results to broader classes of domains.

Abstract

Suppose that $f: D\to D'$ is a quasiconformal mapping, where $D$ and $D'$ are domains in ${\mathbb R}^n$, and that $D$ is a broad domain. Then for every arcwise connected subset $A$ in $D$, the weak quasisymmetry of the restriction $f|_A: A\to f(A)$ implies its quasisymmetry, and as a consequence, we see that the answer to one of the open problems raised by Heinonen from 1989 is affirmative under the additional condition that $A$ is arcwise connected. As an application, we establish nine equivalent conditions for a bounded domain, which is quasiconformally equivalent to a bounded and simply connected uniform domain, to be John. This result is a generalization of the main result of Heinonen from [Quasiconformal mappings onto John domains, \textit{Rev. Math. Iber.,} {\bf 5} (1989), 97--123].