Solvability in the large for a class of complex vector fields on the cylinder

This work deals with global solvability of a class of complex vector fields of the form L=∂/∂t+(a(x,t)+ib(x,t))∂/∂x, where a and b are real-valued C∞ functions, defined on the cylinder Ω=R×S1. Relatively compact (Sussmann) orbits are allowed. The connection with Malgrangeʼs notion of L-convexity for supports is investigated.