Suppose that f : D → D ′ is a quasiconformal mapping, where D and D ′ are domains in 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 → 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 [15].
Suppose that E denotes a real Banach space with the dimension at least 2. The main aim of this paper is to show that a domain D in E is a ψ-uniform domain if and only if D\P is a ψ 1 -uniform domain, and D is a uniform domain if and only if D\P also is a uniform domain, whenever P is a closed countable subset of D satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance (w.r.t. D) between each pair of distinct points in P has a lower bound greater than or equal to 1 2 .
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