Hardy inequalities for general elliptic operators with improvements

TL;DR

This paper develops optimal generalized Hardy inequalities for elliptic operators, including conditions for improvements with non-negative potentials, extending classical inequalities and their boundary-related variants.

Contribution

It introduces new optimal generalized Hardy inequalities for elliptic operators and provides necessary and sufficient conditions for their improvements with potentials.

Findings

Derived optimal generalized Hardy inequalities

Established conditions for adding non-negative potentials

Unified classical and boundary Hardy inequalities

Abstract

We obtain optimal generalized versions of Hardy inequalities, which as special cases contain Hardy's inequality and Hardy's inequality involving the distance function to the boundary of $ \Omega$. In addition we obtain neccesary and sufficient conditions to add improvements in the form of non negative potentials.