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

Abstract

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 Schrödinger equation on ${\mathbb{H}}^d$ and a limiting absorption principle for the electromagnetic Schrödinger equation with small potentials.

arXiv commentFinal version, to appear