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

TL;DR

This paper explores the sharp regularity of a weighted Sobolev space on the torus and demonstrates that for most vector fields, KAM theory applies with perturbations needing classical derivatives up to a certain order, despite allowing unbounded weak derivatives beyond that.

Contribution

It establishes a novel connection between Sobolev space regularity and the applicability of KAM theory, advancing the understanding of regularity requirements in the field.

Findings

KAM theory holds for almost all vector fields with specific Sobolev regularity.

Perturbations require classical derivatives up to order [n/2], but can have unbounded weak derivatives beyond.

The results contribute to the long-standing KAM regularity conjecture.

Abstract

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/2} \right] $, yet it can admit unbounded weak derivatives from order $ \left[ {n/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.