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 Δ ₚu: = div(|∇u|ᵖ⁻²∇u) . Namely, if ρ is a nonnegative weight such that −Δ ₚρ ⩾0 , then the Hardy inequality c∫ M{|u|ᵖ}{ρ ᵖ}|∇ρ |ᵖ\:dvg⩽∫ M|∇u|ᵖ\:dvg,\: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.