Cohomological equation for geodesic flows on flat surfaces

We prove the existence of solutions of the cohomological equation for the geodesic flow on the unit tangent bundle of a compact flat surface with finitely many cone points. We also prove the ergodicity of the holonomy foliation for surfaces with non-rational holonomy, and the cohomology-free property of the horizontal foliated Laplacian under a simultaneous Diophantine condition.