The Quasisymmetric Mappings under the Quasihyperbolic Metric in Real Banach Spaces

Abstract

Suppose that $E,E^{\prime }$ are $Q$ -regular Banach spaces, that $G\mathrm{\subset }E$ and $G^{\prime }\mathrm{\subset }{E}^{\prime }$ are domains, and that $f:G⟶G^{\prime }$ is a homeomorphism. The aim of this study is to prove that the quasisymmetric mapping under the quasihyperbolic metric is equivalent to the freely quasiconformal mapping under the condition $E,E^{\prime }$ being $Q$ -regular Banach spaces. This result is a partial solution to an open problem raised by Väisälä.