Sobolev Lifting over Invariants
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 clas...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211312 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Sobolev Lifting over Invariants. Adam Parusiński and Armin Rainer. SIGMA 17 (2021), 037, 31 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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.
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| ISSN: | 1815-0659 |