Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

TL;DR

This paper establishes sharp Hardy and Rellich inequalities with remainders on homogeneous groups, extending classical results to a broad class of nilpotent Lie groups and deriving related identities and higher order inequalities.

Contribution

It provides the first sharp remainder terms and higher order inequalities for Hardy and Rellich inequalities on homogeneous groups, generalizing classical Euclidean results.

Findings

Sharp remainder terms for Hardy and Rellich inequalities on homogeneous groups.

Analogues of classical inequalities and uncertainty principles on these groups.

Higher order Hardy-Rellich inequalities with sharp constants.

Abstract

We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy-Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.