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)Δpu:=div(|∇u|p−2∇u). Namely, if ρ   is a nonnegative weight such that −Δpρ⩾0−Δpρ⩾0, then the Hardy inequalityc∫M|u|pρp|∇ρ|pdvg⩽∫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.