Hardy inequalities on Riemannian manifolds and applications

We prove a simple sufficient criteria to obtain some Hardy inequalities on Rie- mannian manifolds related to quasilinear second-order differential operator ∆p u := div | u|p−2 u . Namely, if ρ is a nonnegative weight such that −∆p ρ ≥ 0, then the Hardy inequality c M |u|p | ρ|p dvg ≤ ρp | u|p dvg , ∞ u ∈ C0 (M ). 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.