In this paper, we investigate the relationship between semisolidity and locally weak quasisymmetry of homeomorphisms in quasiconvex and complete metric spaces. Our main objectives are to (1) generalize the main result in [14] together with other related results, and (2) give a complete answer to the open problem given in [14]. As an application, we prove that the composition of two locally weakly quasisymmetric mappings is a locally weakly quasisymmetric mapping and that it is quasiconformal.2000 Mathematics Subject Classification. Primary: 30C65, 30F45; Secondary: 30C20.

Suppose that E and E ′ denote real Banach spaces with dimension at least 2, that D E and D ′ E ′ are domains, that f : D → D ′ is an (M, C)-CQH homeomorphism, and that D is uniform. The main aim of this paper is to prove that D ′ is a uniform domain if and only if f extends to a homeomorphism f : D → D ′ and f is η-QM relative to ∂D. This result shows that the answer to one of the open problems raised by Väisälä from 1991 is affirmative.2000 Mathematics Subject Classification. Primary: 30C65, 30F45; Secondary: 30C20.

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].

The main aim of this paper is to give a complete solution to one of the open problems, raised by Heinonen from 1989, concerning the subinvariance of John domains under quasiconformal mappings in R n . As application, the quasisymmetry of quasiconformal mappings is discussed.The reader is referred to [11] for the definition of QED domains. By Theorem A and [34, Theorem 5.6], the following is obvious.