Global solvability and global hypoellipticity for a class of complex vector fields on the 3-torus

Abstract

This work deals with global solvability and global hypoellipticity of complex vector fields of the form \(L=\partial /\partial t+ib_1(t)\partial /\partial x_1+ib_2(t)\partial /\partial x_2\), defined on \(\mathbb {T}^3\simeq \mathbb {R}^{3}/2\pi \mathbb {Z}^{3}\), where both \(b_1\) and \(b_2\) belong to \(\mathcal {C}^{\infty }(\mathbb {T}^1;\mathbb {R}).\) The solvability and hypoellipticity depend on condition (\(\mathcal P\)) and also on Diophantine properties of the coefficients.