An alternative proof of Tataru's dispersive estimates

TL;DR

This paper provides a new, more self-contained proof of Tataru's dispersive estimates for wave equations on hyperbolic space, utilizing Bessel potentials and special functions.

Contribution

It offers an alternative proof based on Bessel potentials, Gamma functions, and Bessel functions, making the argument more self-contained than previous proofs.

Findings

Proof is more self-contained than Tataru's original

Utilizes Bessel potentials and special functions

Simplifies understanding of dispersive estimates

Abstract

The aim of this article is to give an alternative proof of Tataru's dispersive estimates for wave equations posed on the hyperbolic space. Based on the formula for the wave kernel on hyperbolic spaces, in \cite{MR2743652}, we give the proof from the perspective of Bessel potentials, by exploiting various facts about Gamma functions, modified Bessel functions, and Bessel potentials. This leads to our proof being more self-contained than that in Tataru \cite{MR1804518}.