On invariance of John domains under quasisymmetric mappings

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.