On quasimöbius maps and uniform domains in real Banach spaces

Details On quasimöbius maps and uniform domains in real Banach spaces On quasimöbius maps and uniform domains in real Banach spaces

2011-05-24

arxiv.org/abs/1105.4684v2

Mathematics

Complex Variables

27 pages

Scientific paper

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 in 1991 is affirmative. As an application of the obtained result, we show that the answer to another open problem of V\"ais\"al\"a on the bilipschitz extension of QH homeomorphisms in Banach spaces from 1999 is also affirmative under a natural additional condition.

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