Improved Hardy inequalities on Riemannian Manifolds

TL;DR

This paper generalizes Hardy inequalities on Riemannian manifolds, providing conditions under which these inequalities hold, covering various geometries like Euclidean space and hyperbolic space.

Contribution

It introduces a broad framework for Hardy inequalities on Riemannian manifolds, extending previous results with new conditions involving weights and potentials.

Findings

Established sufficient conditions for Hardy inequalities on Riemannian manifolds.

Unified various cases like Euclidean space and hyperbolic space under a common framework.

Extended classical Hardy inequalities to weighted and geometric contexts.

Abstract

We study the following version of Hardy-type inequality on a domain $\Omega$ in a Riemannian manifold $(M,g)$: $$ \int{\Omega}|\nabla u|_g^p\rho^\alpha dV_g \geq \left(\frac{|p-1+\beta|}{p}\right)^p\int{\Omega}\frac{|u|^p|\nabla \rho|_g^p}{|\rho|^p}\rho^\alpha dV_g +\int{\Omega} V|u|^p\rho^\alpha dV_g, \quad \forall\ u\in C_c^\infty (\Omega). $$ We provide sufficient conditions on $p, \alpha, \beta,\rho$ and $V$ for which the above inequality holds. This generalizes earlier well-known works on Hardy inequalities on Riemannian manifolds. The functional setup covers a wide variety of particular cases, which are discussed briefly: for example, $\mathbb{R}^N$ with $p<N$, $\mathbb{R}^N\setminus \{0\}$ with $p\geq N$, $\mathbb{H}^N$, etc.