Inequalities for Dunkl–Riesz transforms and Dunkl gradient with radial piecewise power weights

A beautiful and meaningful harmonic analysis has been constructed on the Euclidean space $\mathbb{R}^d$ with Dunkl weight. The classical Fourier analysis on $\mathbb{R}^d$ corresponds to the weightless case. The Dunkl–Riesz potential and the Dunkl–Riesz transforms play an important role in the Dunkl harmonic analysis. In particular, they allow one to prove the Sobolev type inequalities for the Dunkl gradient. Earlier we proved $(L^q,L^p)$-inequalities for the Dunkl–Riesz potential with two radial piecewise power weights. For the Dunkl–Riesz transforms, we proved $L^p$-inequality with one radial power weight and, as a consequence, we obtained $(L^q,L^p)$-inequalities for the Dunkl gradient with two radial power weights. In this paper, these results for the Dunkl–Riesz transforms and the Dunkl gradient for radial power weights are generalized to the case of radial piecewise power weights.