  We adapt to $C^r$ Anosov flows on compact manifolds a construction for $C^r$
discrete-time hyperbolic dynamics ($r>1$), obtaining anisotropic Banach or
Hilbert spaces on which the resolvent of the generator of weighted transfer
operators for the flow is quasi-compact. We apply this to study the ergodic
integrals of the horocycle flows $h_\rho$ of $C^r$ codimension one mixing
Anosov flows. In dimension three, for any suitably bunched $C^3$ contact Anosov
flow with orientable strong-stable distribution, we establish power-law
convergence of the ergodic average. We thereby implement the program of
Giulietti-Liverani in the "real-life setting" of geodesic flows in variable
negative curvature, where nontrivial resonances exist.
