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 H^d. 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 Schrödinger equation on H^d and a limiting absorption principle for the electromagnetic Schrödinger equation with small potentials.