# Factorizations and Hardy-Rellich inequalities on stratified groups

## TL;DR

This paper develops new Hardy and Rellich inequalities on stratified and homogeneous groups using a factorization approach, extending known results and introducing novel inequalities in Euclidean and non-Euclidean settings.

## Contribution

It introduces new Hardy, Hardy-Rellich, and weighted inequalities on stratified and homogeneous groups, including Euclidean cases, using a factorization method inspired by recent work.

## Key findings

- New Hardy and Hardy-Rellich inequalities on stratified groups
- Analogues of Rellich inequalities on stratified and Heisenberg groups
- A novel two-parameter estimate on Euclidean space

## Abstract

In this paper, we obtain Hardy, Hardy-Rellich and refined Hardy inequalities on general stratified groups and weighted Hardy inequalities on general homogeneous groups using the factorization method of differential operators, inspired by the recent work of Gesztesy and Littlejohn \cite{GL17}. We note that some of the obtained inequalities are new also in the usual Euclidean setting. We also obtain analogues of Gesztesy and Littlejohn's 2-parameter version of the Rellich inequality on stratified groups and on the Heisenberg group, and a new two-parameter estimate on $\mathbb{R}^{n}$ which can be regarded as a counterpart to the Gesztesy and Littlejohn's estimate.