On invariance of John domains under quasisymmetric mappings

TL;DR

This paper investigates how quasisymmetric mappings affect John domains, establishing conditions under which length and diameter John domains are preserved and characterizing distance John domains.

Contribution

It provides new criteria for when quasisymmetric maps preserve John domain properties and characterizes distance John domains via the weak minimizing property.

Findings

Quasisymmetric maps preserve John domain types under certain conditions.

Necessary and sufficient conditions for a diameter John domain to be length John.

Characterization of distance John domains using the weak minimizing property.

Abstract

In this paper, we prove that if a homeomorphism is quasisymmetric relative to the boundary of the domain then it maps a length John domain to a diameter John domain. Moreover, we prove a necessary and sufficient condition for a diameter John domain to be length John and thereby prove that if $f:G\rightarrow G'$ is $(M, C)-$CQH map, where $G$ is a John domain, and the map extends to the boundary such that the extension is $\eta-$QS relative to $\delta G$ then $G'$ is a John domain. In addition, we characterize distance John domains using the weak minimizing property.