Abstract: 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<∞and on each compact manifold. We also apply our local smoothing estimates to nonlinear wave equations with initial data outside of L2-based Sobolev spaces.