  We consider the wave and Klein-Gordon equations on the real hyperbolic space
$\mathbb{H}^{n}$ ($n \geq2$) in a framework based on weak-$L^{p}$ spaces.
First, we establish dispersive estimates on Lorentz spaces in the context of
$\mathbb{H}^{n}$. Then, employing those estimates, we prove global
well-posedness of solutions and an exponential asymptotic stability property.
Moreover, we develop a scattering theory and construct wave operators in such
singular framework.
