On quasimöbius maps in real Banach spaces

Abstract

Suppose that $E$ and $E^{\prime }$ denote real Banach spaces with dimension at least 2, that $DE$ and $D^{\prime }E^{\prime }$ are domains, that $f:D\to D^{\prime }$ is an $(M,C)$-CQH homeomorphism, and that $D$ is uniform. The main aim of this paper is to prove that $D^{\prime }$ is a uniform domain if and only if $f$ extends to a homeomorphism $\bar{f}:\bar{D}\to {\bar{D}}^{\prime }$ and $\bar{f}$ is $\eta$-QM relative to $\partial D$. This result shows that the answer to one of the open problems raised by Väisälä from 1991 is affirmative.

17 pages

References in corpus (2)

• Local Convexity Properties of Quasihyperbolic Balls in Punctured Space
• Quasihyperbolic Geometry in Euclidean and Banach Spaces