Weak Solutions of the Cohomological Equation on R2 for Regular Vector Fields

In a recent article (De Leo, R., Ann. Glob. Anal. Geom., 39, 3, 231-248 2011), we studied the global solvability of the so-called cohomological equation L(xi)f = g in C-infinity(R-2), where xi is a regular vector field on the plane and L-xi the corresponding Lie derivative operator. In a joint article with T. Gramchev and A. Kirilov (2011), we studied the existence of global L-loc(1) weak solutions of the cohomological equation for planar vector fields depending only on one coordinate. Here we generalize the results of both articles by providing explicit conditions for the existence of global weak solutions to the cohomological equation when xi is intrinsically Hamiltonian or of finite type.