Wave and Klein-Gordon equations on hyperbolic spaces

We consider the Klein–Gordon equation associated with the Laplace–Beltrami operator $\mathrm{\Delta }$ on real hyperbolic spaces of dimension $n\text{\hspace{0.17em}⁣}\ge \text{\hspace{0.17em}⁣}2$; as $\mathrm{\Delta }$ 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.