Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces

Abstract

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.

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“…The main advantage of this approach avoids to make use of volume integrals and conformal modulus, which allows one to study the quasiconformality of mappings in Banach spaces with dimension infinity and metric spaces without volume measures. This research has recently attracted substantial interest in the research community (see e.g., [3,7,8,10,17] and reference therein).…”

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