  We introduce a twisted cohomology cocycle over the Teichmueller flow and
prove a "spectral gap" for its Lyapunov spectrum with respect to the
Masur-Veech measures. We then derive Hoelder estimates on spectral measures and
bounds on the speed of weak mixing for almost all translation flows in every
stratum of Abelian differentials on Riemann surfaces, as well as bounds on the
deviation of ergodic averages for product translation flows on the product of a
translation surface with a circle.
