Quasiconformal maps with bilipschitz or identity boundary values in Banach spaces

Suppose that $E$ and $E$ denote real Banach spaces with dimension at least 2 and that $Dvarsubsetneq E$ and $Dvarsubsetneq E$ 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 Holder 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 $xin D$, the quasihyperbolic distance satisfies $k_D(x,f(x))leq C$.