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

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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∞(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 L¹loc 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.