On families of optimal Hardy-weights for linear second-order elliptic operators

TL;DR

This paper develops a method to construct optimal Hardy-weights for second-order elliptic operators by reducing the problem to one-dimensional Sturm-Liouville operators, leading to new inequalities including a Rellich inequality.

Contribution

It introduces a novel reduction technique to derive families of optimal Hardy-weights for elliptic operators, extending known results to higher dimensions.

Findings

Characterization of all optimal Hardy-weights for Sturm-Liouville operators

Construction of families of optimal Hardy inequalities in higher dimensions

Proof of a new Rellich inequality

Abstract

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterise all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on a given interval, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.