The cohomological equation for area-preserving flows on compact surfaces

We study the equation $Xu=f$ where $X$ belongs to a class of area-preserving vector fields, having saddle-type singularities, on a compact orientable surface $M$ of genus $g\geq 2$. For a “full measure" set of such vector fields we prove the existence, for any sufficiently smooth complex valued function $f$ in a finite codimensional subspace, of a finitely differentiable solution $u$. The loss of derivatives is finite, but the codimension increases as the differentiability required for the solution increases, so that there are a countable number of necessary and sufficient conditions which must be imposed on $f$, in addition to infinite differentiability, to obtain infinitely differentiable solutions. This is related to the fact that the "Keane conjecture" (proved by several authors such as H.Masur, W.Veech, M.Rees, S.Kerckhoff, M.Boshernitzan), which implies that for "almost all" $X$ the unique ergodicity of the flow generated by $X$ on the complement of its singularity set, does not extend to distributions. Indeed, our approach proves that, for “almost all" $X$, the vector space of invariant distributions not supported at the singularities has infinite (countable) dimension, while according to the Keane conjecture the cone of invariant measures is generated by the invariant area form $\omega$.