On quasim\"obius maps in real Banach spaces

Abstract

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\varsubsetneq E$ and $D'\varsubsetneq E'$ are domains, that $f: D\to D'$ 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 $\bar{f}: \bar{D}\to \bar{D}'$ and $\bar{f}$ is $\eta$-QM relative to $\partial D$. This result shows that the answer to one of the open problems raised by V\"ais\"al\"a from 1991 is affirmative.