  We prove bounds for twisted ergodic averages for horocycle flows of
hyperbolic surfaces, both in the compact and in the non-compact finite area
case. From these bounds we derive effective equidistribution results for
horocycle maps. As an application of our main theorems in the compact case we
further improve on a result of A. Venkatesh, recently already improved by J.
Tanis and P. Vishe, on a sparse equidistribution problem for classical
horocycle flows proposed by N. Shah and G. Margulis, and in the general
non-compact, finite area case we prove bounds on Fourier coefficients of cups
forms which are off the best known bounds of A. Good only by a logarithmic
term. Our approach is based on Sobolev estimates for solutions of the
cohomological equation and on scaling of invariant distributions for twisted
horocycle flows.
