On bilipschitz extensions in real Banach spaces

Abstract

Suppose that $E$ and $E^{\prime }$ denote real Banach spaces with dimension at least 2, that $D\ne E$ and $D^{\prime }\ne E^{\prime }$ are bounded domains with connected boundaries, that $f:D\to D^{\prime }$ is an $M$-QH homeomorphism, and that $D^{\prime }$ is uniform. The main aim of this paper is to prove that $f$ extends to a homeomorphism $\bar \bar{D}\to \bar{D}'$ and $\bar{f}|\partial D$ is bilipschitz if and only if $f$ is bilipschitz in $\bar{D}$. The answer to some open problem of Väisälä is affirmative under an natural additional condition.

arXiv admin note: substantial text overlap with arXiv:1105.4684