Weighted inequalities for Dunkl–Riesz transforms and Dunkl gradient

Abstract

Over the past 30 years a meaningful harmonic analysis has been construc\-ted in the spaces with Dunkl weights of power type on $\mathbb{R}^d$. The classical Fourier analysis on Euclidean space corresponds to the weightless case. The Dunkl--Riesz potential and the Dunkl--Riesz transforms defined by Thangavelu and Xu play an important role in the Dunkl harmonic analysis.
In particular, they allow one to prove the Sobolev inequalities for the Dunkl gradient. Particular results were obtained here by Amri and Sifi, Abdelkefi and Rachdi, Veliku. Based on the weighted inequalities for the Dunkl--Riesz potential and the Dunkl--Riesz transforms, we prove the general $(L^q,L^p)$ Sobolev inequalities for the Dunkl gradient with radial power weights. The weighted inequalities for the Dunkl--Riesz potential were established earlier.The $L^p$-inequalities for the Dunkl--Riesz transforms with radial power weights are established in this paper. A weightless version of these inequalities was proved by Amri and Sifi.