Global solvability for a class of pseudodifferential operators on the torus

We give a complete characterization for the global solvability of a pseudodifferential operator P=D_t + c(t,D_x) on the (N+1)-dimensional torus T^{N+1} = S^1_t x T^N_x. Our characterization is given in terms of diophantine conditions and a notion of super-logarithmic oscilation of the symbol of the imaginary part of c(t,D_x).