Hardy inequalities on Riemannian manifolds and applications

Abstract

We prove a simple sufficient criterion to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second order differential operator ${\mathrm{\Delta }}_{p}u:=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$. Namely, if $\rho$ is a nonnegative weight such that $-{\mathrm{\Delta }}_p\rho ⩾0$, then the Hardy inequality

holds. We show concrete examples specializing the function $\rho$.

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.