Geometric Hardy inequalities via integration on flows

TL;DR

This paper presents a geometric method using integral curves to derive Hardy inequalities on differentiable manifolds, leading to new inequalities and explicit optimal potentials.

Contribution

It introduces a novel geometric approach for Hardy inequalities on manifolds, providing explicit optimal potentials and unifying known results with new findings.

Findings

Derived explicit optimal Hardy potentials.

Reproduced known Hardy inequalities using the geometric method.

Established new Hardy inequalities on manifolds.

Abstract

We introduce a geometric approach of integral curves for functional inequalities involving directional derivatives in the general context of differentiable manifolds that are equipped with a volume form. We focus on Hardy-type inequalities and the explicit optimal Hardy potentials that are induced by this method. We then apply the method to retrieve some known inequalities and establish some new ones.