Hardy inequalities, Rellich inequalities and local Dirichlet forms

TL;DR

This paper explores Hardy and Rellich inequalities within the framework of local Dirichlet forms, providing conditions to derive Rellich inequalities from Hardy inequalities, and applies these results to various weighted second-order operators.

Contribution

It introduces general conditions linking Hardy and Rellich inequalities and applies these to a broad class of weighted second-order operators, including Grushin type.

Findings

Rellich inequality constants are derived from Hardy constants.

Conditions for Rellich inequalities are verified for many weighted operators.

Applications include operators on punctured Euclidean space and Grushin-type operators.

Abstract

First the Hardy and Rellich inequalities are defined for the submarkovian operator associated with a local Dirichlet form. Secondly, two general conditions are derived which are sufficient to deduce the Rellich inequality from the Hardy inequality. In addition the Rellich constant is calculated from the Hardy constant. Thirdly, we establish that the criteria for the Rellich inequality are verified for a large class of weighted second-order operators on a domain $\Omega\subseteq \Ri^d$. The weighting near the boundary $\partial \Omega$ can be different from the weighting at infinity. Finally these results are applied to weighted second-order operators on $\Ri^d\backslash\{0\}$ and to a general class of operators of Grushin type.