The paper determines the exact range of s and p for which singular projections preserve W^{s,p} regularity onto manifolds, proving full validity for sp < ℓ when s ≥ 1 and optimality of p < ℓ when s < 1 via counterexamples.
Van Schaftingen, The extension of traces for Sobolev mappings between manifolds , 2024, arXiv: https://arxiv.org/abs/2403.18738 2403.18738 [math.AP]
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A surprising threshold for the validity of the method of singular projection
The paper determines the exact range of s and p for which singular projections preserve W^{s,p} regularity onto manifolds, proving full validity for sp < ℓ when s ≥ 1 and optimality of p < ℓ when s < 1 via counterexamples.