A weighted Sobolev space on the torus has sharp regularity that lets KAM theory apply to perturbations with classical differentiability only up to floor(n/2) while higher weak derivatives remain unbounded.
Hu, Quasi-periodic solutions for Schr\"odinger equation with finite smooth quasi-periodic forcing
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Sharp regularity of a weighted Sobolev space over $ \mathbb{T}^n $ and its relation to finitely differentiable KAM theory
A weighted Sobolev space on the torus has sharp regularity that lets KAM theory apply to perturbations with classical differentiability only up to floor(n/2) while higher weak derivatives remain unbounded.