  We introduce the Hardy spaces for Fourier integral operators on Riemannian
manifolds with bounded geometry. We then use these spaces to obtain improved
local smoothing estimates for Fourier integral operators satisfying the
cinematic curvature condition, and for wave equations on compact manifolds. The
estimates are essentially sharp, for all $2<p<\infty$ and on each compact
manifold. We also apply our local smoothing estimates to nonlinear wave
equations with initial data outside of $L^{2}$-based Sobolev spaces.
