Improved Hardy inequalities on Riemannian Manifolds

Abstract

We study the following version of Hardy-type inequality on a domain $Ω$ in a Riemannian manifold $(M,g)$: $$ \intΩ|\nabla u|_g^pρ^αdV_g \geq \left(\frac{|p-1+β|}{p}\right)^p\intΩ\frac{|u|^p|\nabla ρ|_g^p}{|ρ|^p}ρ^αdV_g +\intΩ V|u|^pρ^αdV_g, \quad \forall\ u\in C_c^\infty (Ω). $$ We provide sufficient conditions on $p, α, β,ρ$ 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.