Wave and Klein–Gordon equations on hyperbolic spaces

Abstract

We consider the Klein–Gordon equation associated with the Laplace–Beltrami operator $\Delta$ on real hyperbolic spaces of dimension $n\ge 2$; as $\Delta$ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well-posedness results for the corresponding semilinear equation with low regularity data.