Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 037, 31 pages      arXiv:2003.01967      https://doi.org/10.3842/SIGMA.2021.037

Sobolev Lifting over Invariants

Adam Parusiński ^a and Armin Rainer ^b
a) Université Côte d'Azur, CNRS, LJAD, UMR 7351, 06108 Nice, France
^b) Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria

Received November 04, 2020, in final form March 29, 2021; Published online April 10, 2021

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 $\overline{f} \colon {\mathbb R}^m \to V$ 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 $f\colon {\mathbb R} \to \sigma(V) \subseteq {\mathbb C}^n$. 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.

Key words: Sobolev lifting over invariants; complex representations of finite groups; $Q$-valued Sobolev functions.

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