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

Effective equidistribution of twisted horocycle flows and horocycle maps