A note on solvability of vector fields in Denjoy-Carleman classes

Abstract

We deal with solvability in Denjoy-Carleman classes of complex vector fields defined on Ω = R × S 1 , given by L = ∂/∂t + (a(x, t)+ib(x, t))∂/∂x, b ≡ 0, near the characteristic set Σ = {0}×S 1 . We assume that L vanishes of first order along Σ and that a certain invariant associate to L is an irracional number.