# Rellich, Gagliardo-Nirenberg, Trudinger and Caffarelli-Kohn-Nirenberg inequalities for Dunkl operators and applications

## TL;DR

This paper extends classical inequalities like Rellich, Gagliardo-Nirenberg, Trudinger, and Caffarelli-Kohn-Nirenberg to Dunkl operators, providing new weighted inequalities and applications to nonlinear wave equations.

## Contribution

It introduces weighted higher order inequalities for Dunkl operators and extends classical inequalities, with applications to nonlinear damped wave equations.

## Key findings

- Established new weighted inequalities for Dunkl operators.
- Extended classical inequalities to the Dunkl setting.
- Applied Gagliardo-Nirenberg inequality to nonlinear wave equations.

## Abstract

In this paper we obtain weighted higher order Rellich, weighted Gagliardo-Nirenberg, Trudinger, Caffarelli-Kohn-Nirenberg inequalities and the uncertainty principle for Dunkl operators. Moreover, we introduce an extension of the classical Caffarelli-Kohn-Nirenberg inequalities. Furthermore, we give an application of Gagliardo-Nirenberg inequality to the Cauchy problem for the nonlinear damped wave equations for the Dunkl Laplacian.