Quasiconformal maps with bilipschitz or identity boundary values in Banach spaces

TL;DR

This paper studies quasiconformal maps between Banach space domains with bilipschitz or identity boundary conditions, establishing their quasisymmetry, Hölder continuity, and bilipschitz properties under certain conditions.

Contribution

It demonstrates that $ ext{FQC}$ maps with bilipschitz boundary values are quasisymmetric, and shows bilipschitz equivalence for $M$-QH maps with such boundary conditions, extending understanding of boundary behavior.

Findings

FQC maps with bilipschitz boundary values are $ ext{eta}$-quasisymmetric.

Bounded domains' maps satisfy a two-sided Hölder condition.

$M$-QH maps with bilipschitz boundary values are bilipschitz.

Abstract

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2 and that $D\varsubsetneq E$ and $D'\varsubsetneq E'$ are uniform domains with homogeneously dense boundaries. We consider the class of all $\varphi$-FQC (freely $\varphi$-quasiconformal) maps of $D$ onto $D'$ with bilipschitz boundary values. We show that the maps of this class are $\eta$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided H\"older condition. Moreover, replacing the class $\varphi$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if $f$ is a $\varphi$-FQC map which maps $D$ onto itself with identity boundary values, then there is a constant $C\,,$ depending only on the function $\varphi\,,$ such that for all $x\in D$, the quasihyperbolic distance satisfies $k_D(x,f(x))\leq C$.