Sobolev lifting over invariants

We prove lifting theorems for complex representations V of finite groups G. Let s = (s ₁, …, s _n) be a minimal system of homogeneous basic invariants and let d be their maximal degree. We prove that any continuous map f: R ^m ? V such that f = s ? f is of class C ^d-1,1 is locally of Sobolev class W ^1,p for all [formula presented]In the case m = 1 there always exists a continuous choice f for given [formula presented]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.