# A simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fields

## TL;DR

This paper provides an elementary proof of the Hardy inequality on Carnot groups using integration by parts and analyzes its potential extension to hypoelliptic vector fields, highlighting open problems in the field.

## Contribution

It introduces a simple, elementary proof of the Hardy inequality on Carnot groups and explores the possibility of generalizing this approach to hypoelliptic vector fields.

## Key findings

- Elementary proof of Hardy inequality on Carnot groups
- Analysis of commutator structure for generalization
- Identification of open problems related to hypoelliptic vector fields

## Abstract

We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.