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

Abstract

Let L j = ∂ tj + (a j + ib j )(t j )∂ x , j = 1, . . . , n, be a system of vector fields defined on the torus T n t × T 1 x , where the coefficients a j and b j are real-valued functions belonging to the Gevrey class G s (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).