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arXiv · 1105.4684

On quasim\"obius maps in real Banach spaces

Manzi Huang·Yaxiang Li·Matti Vuorinen·Xiantao Wang
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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.

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Manzi Huang, Yaxiang Li, Matti Vuorinen, Xiantao Wang. 2011-05-24. On quasim\"obius maps in real Banach spaces. https://arxiv.org/abs/1105.4684

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