Sobolev Lifting over Invariants

Abstract

We prove lifting theorems for complex representations V of finite groups G. Let σ = (σ 1 , . . . , σ n ) be a minimal system of homogeneous basic invariants and let d be their maximal degree. We prove that any continuous map f :In the case m = 1 there always exists a continuous choice f for given f : R → σ(V ) ⊆ C n . We give uniform bounds for the W 1,p -norm of f in terms of the C d−1,1 -norm of f . The result is optimal: in general a lifting f cannot have a higher Sobolev regularity and it even might not have bounded variation if f is in a larger Hölder class.