# 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 $\Delta {p}u := \Div(\abs{ abla u}^{p-2} abla u)$. Namely, if $ho$ is a nonnegative weight such that $-\Delta {p} ho\geq0$, then the Hardy inequality $$c\int {M}\frac{\abs{u}^{p}}{ ho^{p}}\abs{ abla ho}^{p} dv {g} \leq \int {M}\abs{ abla u}^{p} dv {g}, \quad u\in\Cinfinito {0}(M)$$ holds. We show concrete examples specializing the function $ho$.

Author: Lorenzo D'Ambrosio; Serena Dipierro

Source: https://archive.org/