Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane

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-infinity (R-2) if 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-loc(1) 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.