Hardy inequalities on Riemannian manifolds and applications

We prove a simple sufficient criteria to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second-order differential operator $\Delta_{p}u := \Div(\abs{\nabla u}^{p-2}\nabla u)$. Namely, if $\rho$ is a nonnegative weight such that $-\Delta_{p}\rho\geq0$, then the Hardy inequality $$c\int_{M}\frac{\abs{u}^{p}}{\rho^{p}}\abs{\nabla \rho}^{p} dv_{g} \leq \int_{M}\abs{\nabla u}^{p} dv_{g}, \quad u\in\Cinfinito_{0}(M)$$ holds. We show concrete examples specializing the function $\rho$.