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

Zhicheng Tong, Yong Li

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.

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

Abstract

In this paper, we investigate the sharp regularity properties of a special weighted Sobolev space defined on the -dimensional torus, which is of independent interest. As a key application, we show that for almost all -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 , yet it can admit unbounded weak derivatives from order to . 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.

Paper Structure

This paper contains 14 sections, 47 equations.

Table of Contents

  1. Introduction
  2. Sharp regularity of the weighted Sobolev space $\widetilde{\mathcal{L}_{\tau + 1,b}^2}({\mathbb{T}^n})$
  3. Application to the finitely differentiable KAM theorem
  4. Further discussions
  5. Relation to historical KAM theorems and counterexamples
  6. The possible singularity of unbounded weak derivatives
  7. Proofs of all results
  8. Proof of Theorem \ref{['JZT1']}
  9. Proof of Theorem \ref{['JZT2']}
  10. Proof of Theorem \ref{['JZT3']}
  11. Proof of Theorem \ref{['JZT4']} (Main Theorem I)
  12. Proof of Theorem \ref{['TPDLINGLI4']}
  13. Proof of Theorem \ref{['THPOSCHEL']} (Main Theorem II)
  14. Proof of Theorem \ref{['sssss']}