On bilipschitz extensions in real Banach spaces

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\not=E$ and $D'\not=E'$ are bounded domains with connected boundaries, that $f: D\to D'$ is an $M$-QH homeomorphism, and that $D'$ 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.

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