On quasisymmetry of quasiconformal mappings and its applications

Suppose that $f:D\to D^{\mathrm{\prime }}$ is a quasiconformal mapping, where $D$ and $D^{\mathrm{\prime }}$ 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 $f{\mathrm{\mid }}_A:A\to f(A)$ 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].