Solvability in the large for a class of vector fields on the torus

We study a class of complex vector fields defined on the two-torus of the form L = partial derivative/partial derivative t + (a(x, t) + ib(x, t))partial derivative/partial derivative x, a, b is an element of C-infinity(T-2; R), b not equivalent to 0. We view L as an operator acting on smooth functions and present conditions for L to have either a closed range or a finite-codimensional range. Our results involve, besides condition (P) of Nirenberg and Treves, the behavior of a + ib near each one-dimensional Sussmann orbit homotopic to the unit circle. One of the main goals of our work is to provide some clarification about the role played by the coefficient a in the validity of the above properties of the range. (C) 2006 Elsevier Masson SAS. All rights reserved.