Global Gevrey hypoellipticity on the torus for a class of systems of complex vector fields

TL;DR

This paper characterizes the global Gevrey hypoellipticity of a system of complex vector fields on the torus, linking it to Diophantine approximations and the Nirenberg-Treves condition (P).

Contribution

It provides a complete characterization of global Gevrey hypoellipticity for a class of vector field systems on the torus, extending previous results.

Findings

Gevrey hypoellipticity characterized in terms of Diophantine conditions.

Connection established between hypoellipticity and Nirenberg-Treves condition (P).

Results applicable to systems with Gevrey class coefficients.

Abstract

Let $L_j = \partial_{t_j} + (a_j+ib_j)(t_j) \partial_x, \, j = 1, \dots, n,$ be a system of vector fields defined on the torus $\mathbb{T}_t^{n}\times\mathbb{T}_x^1$, where the coefficients $a_j$ and $b_j$ are real-valued functions belonging to the Gevrey class $G^s(\mathbb{T}^1)$, with $s>1$. In this paper we were able to characterize the global $s-$hypoellipticity of this system in terms of Diophantine approximations and the Nirenberg-Treves condition (P).