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