Spherical maximal functions and Hardy spaces for Fourier integral operators

We use the Hardy spaces for Fourier integral operators to obtain bounds for spherical maximal functions in L^p(ℝ^n), n≥2, where the radii of the spheres are restricted to a compact subset of (0,∞). These bounds extend to general hypersurfaces with non-vanishing Gaussian curvature, and to geodesic spheres on compact manifolds. We also obtain improved maximal function bounds, and pointwise convergence statements, for wave propagators.