Sobolev Lifting over Invariants

We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $\sigma =({\sigma }_1,\dots ,{\sigma }_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map $\bar{f}\text{\hspace{0.17em}⁣}:{\mathbb{R}}^m\to V$ such that $f = \sigma \circ \overline{f}$isofclass$C^{d-1,1}$islocallyofSobolevclass$W^{1,p}$forall$1 \le p<d/(d-1)$\mathrm{.}Inthecase$m=1$ there always exists a continuous choice $\bar{f}$ for given $f\colon {\mathbb R} \to \sigma(V) \subseteq {\mathbb C}^n$\mathrm{.}Wegiveuniformboundsforthe$W^{1,p}$-normof$\overline{f}$intermsofthe$C^{d-1,1}$-normof$f$. The result is optimal: in general a lifting $\bar{f}$ cannot have a higher Sobolev regularity and it even might not have bounded variation if $f$ is in a larger H"older class.