Weighted Norm Inequalities for the Opdam--Cherednik Transform

TL;DR

This paper develops weighted norm inequalities for the Opdam--Cherednik transform, extending classical inequalities like Heisenberg--Pauli--Weyl and Hardy--Littlewood with new weight and norm variations.

Contribution

It introduces several new weighted inequalities and variations of classical inequalities specifically for the Opdam--Cherednik transform, expanding the theoretical framework.

Findings

Extended Heisenberg--Pauli--Weyl inequalities with weights

Proved Hardy--Littlewood inequality for the transform

Derived Nash-type and Clarkson-type inequalities

Abstract

In this paper, we study several weighted norm inequalities for the Opdam--Cherednik transform. We establish different versions of the Heisenberg--Pauli--Weyl inequality for this transform. In particular, we give an extension of this inequality using weights with different exponents and present a variation of the inequality that incorporates $L^p$-norms for the Opdam--Cherednik transform. Also, we prove a version of the Hardy--Littlewood inequality for this transform. Finally, we give other variations of the Heisenberg--Pauli--Weyl inequality such as the Nash-type and Clarkson-type inequalities for the Opdam--Cherednik transform.