  We refine the theory of the cohomological equation for translation flows on
higher genus surfaces with the goal of proving optimal results on the Sobolev
regularity of solutions and of distributional obstructions. For typical
translation surfaces our results are sharp and we find the expected relation
between the regularity of the distributional obstructions and the Lyapunov
exponents of the Kontsevich-Zorich renormalization cocycle. As a consequence we
exactly determine the dimension of the space of obstructions in each Sobolev
regularity class in terms of the Kontsevich-Zorich exponents. For a fixed
arbitrary translation surface and a typical direction, our results are probably
not optimal but are the best which can be achieved with the available harmonic
analysis techniques we have introduced in an earlier paper.
