I`ll present a recent joint work with A. Lawrie, J. Lührmann and S. Shahshahani on the global-in-time local(-in-space) smoothing estimates for the linear Schrödinger-type equation on hyperbolic space. This result is a crucial ingredient in our current investigation of asymptotic stability of solitons for nonlinear dispersive equations on hyperbolic space, which I`ll also briefly describe. At the end of the talk, I`ll point out outstanding issues for pursuing this program for more general curved background, which seem to require a suitable generalization of harmonic analytic techniques.