# Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities

## TL;DR

This paper explores Hardy inequalities related to Dunkl operators, extending classical and $L^p$ inequalities, and applies these results to derive many-particle Hardy inequalities for general root systems, including improvements for specific cases.

## Contribution

It introduces new Hardy inequalities for Dunkl operators and applies them to establish improved many-particle Hardy inequalities for general root systems.

## Key findings

- Derived various Hardy inequalities for Dunkl operators.
- Established one-dimensional many-particle Hardy inequalities for general root systems.
- Improved known results for root system $A_{N-1}$.

## Abstract

In this paper we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, $L^p$ inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli-Kohn-Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalised root systems are proved, which in the particular case of root systems $A_{N-1}$ improve some well-known results.