Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus

Let L = partial derivative/partial derivative t + Sigma(N)(j=1) (a(j)+ib(j)) (t) partial derivative/partial derivative x(j) be a vector field defined on the torus TN+1 similar or equal to RN+1 /2 Pi Z(N+1,) where a(j),b(j) are real-valued functions and belonging to the Gevrey class G(s)(T-1), s > 1, for j= 1,..., N. We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets. (C) 2017 Elsevier Inc. All rights reserved.