Oscillation inequalities for Carleson--Dunkl operator

In this paper, we establish estimates for the oscillation seminorm for the so-called Carleson–Dunkl operator on weighted $L^p(\mathbb{R},w(x)\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha +1}\mathrm{d}x)$ spaces with power weights $w(x)=\mathrm{\mid }x{\mathrm{\mid }}^{\beta }$. As a result, we obtain oscillation estimates for the standard Carleson operator on $L_{\mathrm{r}\mathrm{a}\mathrm{d}}^p({\mathbb{R}}^n,\mathrm{\mid }x{\mathrm{\mid }}^{\beta }\mathrm{d}x)$. As a byproduct, we obtain a transference principle for radial multipliers on $L_{\mathrm{r}\mathrm{a}\mathrm{d}}^p$ spaces, in the spirit of the Rubio de Francia transference principle.