Quasiconformal maps with bilipschitz or identity boundary values in Banach spaces

Abstract

Suppose that $E$ and $E^{\prime }$ denote real Banach spaces with dimension at least 2 and that $DE$ and $D^{\prime }E^{\prime }$ are uniform domains with homogeneously dense boundaries. We consider the class of all $\phi$-FQC (freely $\phi$-quasiconformal) maps of $D$ onto $D^{\prime }$ 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 Hölder condition. Moreover, replacing the class $\phi$-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 $\phi$-FQC map which maps $D$ onto itself with identity boundary values, then there is a constant $C,$ depending only on the function $\phi ,$ such that for all $x\in D$, the quasihyperbolic distance satisfies $k_D(x,f(x))\le C$.

13 pages

References in corpus (2)

• Local Convexity Properties of Quasihyperbolic Balls in Punctured Space
• Close-to-convexity of quasihyperbolic and $j$-metric balls