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

Let L j =∂_t +(a j +ib j )(t j )∂x_j, j=1,…,n, be a system of vector fields defined on the torus T^{n}_{t}×T_{x}, where the coefficients a j and b_j are real-valued functions belonging to the Gevrey class G^{s}(T^{1}), s>1. The global s−hypoellipticity of this system is characterized in terms of Diophantine approximations and the Nirenberg–Treves condition (P).