Sobolev Lifting over Invariants

Abstract

We prove lifting theorems for complex representations 𝑉 of finite groups 𝐺. Let σ = (σ₁,…, σₙ) be a minimal system of homogeneous basic invariants and let 𝑑 be their maximal degree. We prove that any continuous map 𝑓 ̅ : ℝᵐ → 𝑉 such that 𝑓 = σ ∘ 𝑓 ̅ is of class 𝐶ᵈ⁻¹'¹ is locally of Sobolev class 𝑊¹'ᵖ for all 1 ≤ 𝑝 < 𝑑/(𝑑−1). In the case 𝑚 = 1, there always exists a continuous choice 𝑓 ̅ for given f: ℝ →σ(𝑉) ⊆ ℂⁿ. We give uniform bounds for the 𝑊¹'ᵖ-norm of 𝑓 ̅ in terms of the 𝐶ᵈ⁻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.