Reducibility of first order linear operators on tori via Moser's theorem

TL;DR

This paper proves the reducibility of certain linear first order operators on tori by extending Moser's theorem, using Nash-Moser techniques to handle small perturbations and improve regularity results.

Contribution

It introduces a generalized Moser's theorem for straightening vector fields on tori, achieving higher regularity in conjugation through Nash-Moser methods.

Findings

Linear first order operators on tori can be conjugated to constant flows under small perturbations.

The conjugation is achieved via a $C^ ablafty$ torus diffeomorphism.

The method improves regularity results compared to classical approaches.

Abstract

In this paper we prove reducibility of classes of linear first order operators on tori by applying a generalization of Moser's theorem on straightening of vector fields on a torus. We consider vector fields which are a $C^\infty$ perturbations of a constant vector field, and prove that they are conjugated --by a $C^\infty$ torus diffeomorphism-- to a constant diophantine flow, provided that the perturbation is small in some given $H^{s_1}$ norm and that the initial frequency is in some Cantor-like set. Actually in the classical results of this type the regularity of the change of coordinates which straightens the perturbed vector field coincides with the class of regularity in which the perturbation is required to be small. This improvement is achieved thanks to ideas and techniques coming from the Nash-Moser theory.