Well-Posedness and Scattering for Wave Equations on Hyperbolic Spaces With Singular Data

We consider wave equations with critical shift on the real hyperbolic space H n ( n ≥ 2 ) in the framework of weak- L p spaces. First, we establish dispersive estimates on Lorentz spaces in the context of 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.