Sobolev Lifting over Invariants

Abstract

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 $\overline{f}:{\mathbb{R}}^m\to V$ such that $f=\sigma \circ \overline{f}$ is of class $C^{d-1,1}$ is locally of Sobolev class $W^{1,p}$ for all $1\le p<d/(d-1)$. In the case $m=1$ there always exists a continuous choice $\overline{f}$ for given $f:\mathbb{R}\to \sigma (V)\subseteq {\mathbb{C}}^n$. We give uniform bounds for the $W^{1,p}$-norm of $\overline{f}$ in terms of the $C^{d-1,1}$-norm of $f$. The result is optimal: in general a lifting $\overline{f}$ cannot have a higher Sobolev regularity and it even might not have bounded variation if $f$ is in a larger Hölder class.