Improved Hardy inequalities on Riemannian Manifolds

Abstract

We study the following version of Hardy-type inequality on a domain $\Omega$ in a Riemannian manifold $(M,g)$: $$\int \Omega |\nabla u{|}_g^p{\rho }^{\alpha }d{V}_g\ge {\left(\frac{|p-1+\beta |}{p}\right)}^p\int \Omega \frac{|u{|}^p|\nabla \rho {|}_g^p}{|\rho {|}^p}{\rho }^{\alpha }d{V}_g+\int \Omega V|u{|}^p{\rho }^{\alpha }d{V}_g,\forall u\in C_c^{\infty }(\Omega ).$$ 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 $p\ge N$, ${\mathbb{H}}^N$, etc.

Accepted for publication in Complex Variables and Elliptic Equations

References in corpus (1)

• On the best possible remaining term in the Hardy Inequality