Hardy inequalities on Riemannian manifolds and applications

D'Ambrosio, Lorenzo; Dipierro, Serena

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 3, pp. 449-475.

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

$$c\underset{M}{\int }\frac{{\left|u\right|}^p}{{\rho }^p}{\left|\nabla \rho \right|}^{p}d{v}_g⩽\underset{M}{\int }{\left|\nabla u\right|}^{p}d{v}_g,u\in {\mathrm{C}}_0^{\infty }\left(M\right),$$

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.

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