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

TL;DR

This paper establishes sharp $L^p-L^q$ bounds for spectral projectors, resolvent operators, and Fourier restriction on hyperbolic space, with applications to Schrödinger equations and electromagnetic problems.

Contribution

It provides a unified framework for $L^p-L^q$ estimates on hyperbolic space, including sharp bounds for spectral projectors and partial results on Fourier extension.

Findings

Sharp $L^p-L^q$ bounds for spectral projectors when $p$ and $q$ are dual

Partial results on Fourier extension operator boundedness

Smoothing estimates for Schrödinger equations on 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 Schr\"{o}dinger equation on $\mathbb{H}^d$ and a limiting absorption principle for the electromagnetic Schr\"{o}dinger equation with small potentials.