  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 $\mathbb{H}^d.$ 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.
