Hardy's inequality in the scope of Dirichlet forms

TL;DR

This paper explores Hardy's inequality within the framework of regular Dirichlet forms, providing new conditions for its validity and extending results to strongly local forms, especially in Euclidean spaces.

Contribution

It offers an alternative necessary and sufficient condition for Hardy's inequality in Dirichlet forms, extending known Euclidean results to more general settings.

Findings

New necessary and sufficient condition for Hardy's inequality in Dirichlet forms

Extension of results to strongly local Dirichlet forms

Application to Euclidean spaces with improved understanding

Abstract

We revisit Hardy's inequality in the scope of regular Dirichlet forms following an analytical method. We shall give an alternative necessary and sufficient condition for the occurrence of Hardy's inequality. A special emphasis will be given for the case where the Dirichlet form under consideration is strongly local, extending therefore some known results in the Euclidean case.