  We address some global solvability issues for classes of smooth nonsingular
vector fields $L$ in the plane related to cohomological equations $Lu=f$ in
geometry and dynamical systems. The first main result is that $L$ is not
surjective in $C^\infty(\R^2)$ iff the geometrical condition -- the existence
of separatrix strips -- holds. Next, for nonsurjective vector fields, we
demonstrate that if the RHS $f$ has at most infra-exponential growth in the
separatrix strips we can find a global weak solution $L^1_{loc}$ near the
boundaries of the separatrix strips. Finally we investigate the global
solvability for perturbations with zero order p.d.o. We provide examples
showing that our estimates are sharp.
