Spectral projectors, resolvent, and Fourier restriction on the hyperbolic space

We develop a unified approach to proving $L^p-L^q$ boundedness of spectral projectors, the resolvent of the Laplace-Beltrami operator and its derivative on ${\mathbb{H}}^d\mathrm{.}$ In the case of spectral projectors, and when $p$ and $q$ are in duality, the dependence of the implicit constant on $p$ is shown to be sharp. We also give partial results on the question of $L^p-L^q$ boundedness of the Fourier extension operator. As an application, we prove smoothing estimates for the free Schr"{o}dinger equation on ${\mathbb{H}}^d$ and a limiting absorption principle for the electromagnetic Schr"{o}dinger equation with small potentials.