• cohomological equation
• horocycle flow
• invariant distributions

In this thesis we give an account of the theory of the cohomological equation for horocycle flow as it has been developed in a seminal paper by Flaminio and Forni.
Let U be the generator of the horocycle flow and let h be a regular function, we ask about the existence and the regularity of a solution to the cohomological equation U f = g.
Since we are dealing with a homogeneous space, the solution is based on Fourier theory over SL(2, R). As in ordinary Fourier theory, a function g can be decomposed into the sum of countably many functions that are eigenvectors of a second-order elliptic operator. Thus, we are reduced to studying the cohomological equation only in the case where g is an eigenvector of the aforementioned operator.
To do so, we introduce a special class of generalized measures called invariant distributions: these are functionals that are invariant under the action of U. Then it is evident that the necessary condition for a function g to be of the form U f is given by being in the kernel of all invariant distributions.
Finally, we show that this condition is indeed sufficient: that is, invariant distributions are the only obstructions to the solution of the cohomological equation and we get sharp estimates of the regularity of this solution.