On quasim\"obius maps in real Banach spaces

TL;DR

This paper proves that under certain conditions, a homeomorphism between domains in real Banach spaces extends to a quasim"obius map on the boundary, affirmatively resolving an open problem from 1991.

Contribution

It establishes a characterization of uniform domains via boundary extension and quasim"obius maps in real Banach spaces, solving an open problem from V"ais"al"a's 1991 work.

Findings

D' is uniform iff f extends to a boundary homeomorphism that is η-quasim"obius

Confirms the boundary extension property characterizes uniform domains in this setting

Addresses and resolves an open problem from V"ais"al"a (1991)

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.