On bilipschitz extensions in real Banach spaces

TL;DR

This paper investigates conditions under which bilipschitz homeomorphisms between certain domains in real Banach spaces extend to boundary homeomorphisms that are also bilipschitz, resolving an open problem in the field.

Contribution

It proves that a bilipschitz extension exists for $M$-QH homeomorphisms between specific domains, affirming an open problem with an added natural condition.

Findings

Extension of bilipschitz homeomorphisms to boundary is equivalent to their bilipschitz property in the domain.

Affirmative answer to V"ais"al"a's open problem under an additional natural condition.

Extension preserves bilipschitz constants under specified conditions.

Abstract

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 V\"ais\"al\"a is affirmative under an natural additional condition.