Sharp Hardy and Rellich type inequalities on Cartan--Hadamard manifolds and their improvements

TL;DR

This paper establishes new Hardy and Rellich inequalities on Cartan--Hadamard manifolds, showing they are stronger than classical versions and depend on the manifold's curvature, with specific improvements in hyperbolic space.

Contribution

The paper introduces novel Hardy and Rellich inequalities on Cartan--Hadamard manifolds, incorporating curvature effects and demonstrating their superiority over classical inequalities.

Findings

New Hardy and Rellich inequalities are stronger than classical ones.

Curvature influences the strength of these inequalities.

Improved inequalities are specifically established for hyperbolic space.

Abstract

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) for radial derivations (i.e., the derivation along the geodesic curve) on Cartan--Hadamard manifolds. By Gauss lemma, our new Hardy inequality are stronger than the classical one. We also established the improvements of these inequalities in terms of sectional curvature of underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequality in hyperbolic space $\mathbb H^n$. Especially, we show that our new Rellich inequality is indeed stronger the classical one in hyperbolic space $\mathbb H^n$.