There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu=f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann surface M of finite area and f is a given function on SM. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the ergodic averages of horocycle flows.
"Invariant distributions and time averages for horocycle flows." Duke Math. J. 119 (3) 465 - 526, 15 September 2003. https://doi.org/10.1215/S0012-7094-03-11932-8