On Bilipschitz Extensions in Real Banach Spaces

Abstract

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