Well-posedness and scattering for wave equations on hyperbolic spaces with singular data

We consider the wave and Klein-Gordon equations on the real hyperbolic space ${\mathbb{H}}^n$ ($n\ge 2$) 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.