  We use the Hardy spaces for Fourier integral operators to obtain bounds for
spherical maximal functions in $L^{p}(\mathbb{R}^{n})$, $n\geq2$, where the
radii of the spheres are restricted to a compact subset of $(0,\infty)$. These
bounds extend to general hypersurfaces with non-vanishing Gaussian curvature,
to the complex spherical means, and to geodesic spheres on compact manifolds.
We also obtain improved maximal function bounds and pointwise convergence
statements for wave equations, both on $\mathbb{R}^{n}$ and on compact
manifolds. The maximal function bounds are essentially sharp for all
$p\in[1,2]\cup [\frac{2(n+1)}{n-1},\infty]$, for each such hypersurface, every
complex spherical mean, and on every manifold.
