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arXiv 2003.019672020-03math.CAmath.DGmath.FAmath.RT

Sobolev Lifting over Invariants

Adam Parusiński, Armin Rainer

发表信息：: SIGMA 17 (2021), 037, 31 pages
DOI：: 10.3842/SIGMA.2021.037

英文摘要：

We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $σ=(σ_1,\dots,σ_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map $\overline{f} \colon {\mathbb R}^m \to V$ such that $f = σ\circ \overline{f}$ 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 $f\colon {\mathbb R} \to σ(V) \subseteq {\mathbb C}^n$. 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.

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