  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].

Complex Variables