Hardy inequalities on Riemannian manifolds and applications

We prove a simple sufficient criterion to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second order differential operator Δpu:=div(|∫u|p-2∫u). Namely, if ρ is a nonnegative weight such that -Δpρ≥0, then the Hardy inequalityc∫M|u|pρp|∫ρ|pd vg≤∫M|∫u|pdvg,u∈C0∞(M), holds. We show concrete examples specializing the function ρ. Our approach allows to obtain a characterization of p-hyperbolic manifolds as well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo-Nirenberg inequalities, uncertain principle and first order Caffarelli-Kohn-Nirenberg interpolation inequality.