论文标题
Sobolev举起不变的人
Sobolev Lifting over Invariants
论文作者
论文摘要
我们证明，有限组的复杂表示$ v $ $ g $。令$σ=（σ_1，\ dots，σ_n）$为同质基本不变的最小系统，让$ d $成为其最大程度。我们证明，任何连续的映射$ \ overline {f} \ colon {\ mathbb r}^m \ to v $ to v $，使得$ f =σ\ circ \ circ \ circ \ circy {f} $属于$ c^{d-1,1} $ sobolev class $ w^{1，p} $的$ c^{d-1,1} $。在情况下，$ m = 1 $总是存在连续的选择$ \ edline {f} $，对于给定的$ f \ colon {\ mathbb r} \ toσ（v）\ subseteq {\ subseteq {\ mathbb c}^n $。我们给出$ w^{1，p} $ - $ \ overline {f} $的统一界限，以$ c^{d-1,1} $ - $ f $的规范。结果是最佳的：总的来说，提升$ \叠加{f} $不能具有更高的sobolev规则性，如果$ f $在较大的Hölder类中，它甚至可能没有界限。
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.
