Sharp regularity of a weighted Sobolev space over ${\mathbb{T}}^n$ and its relation to finitely differentiable KAM theory

In this paper, we investigate the sharp regularity properties of a special weighted Sobolev space defined on the $n$-dimensional torus, which is of independent interest. As a key application, we show that for almost all $n$-dimensional vector fields, the Kolmogorov-Arnold-Moser (KAM) theory holds via this regularity, and in this case, the perturbation must have classical derivatives up to order $\left[n\mathrm{/}2\right]$, yet it can admit unbounded weak derivatives from order $\left[n\mathrm{/}2\right]+1$ to $n$. This result may appear surprising within the classical framework of KAM theory. We also provide further discussion of historical KAM theorems and relevant counterexamples. These findings constitute a new step in the long-standing KAM regularity conjecture.