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 Lp(R,w(x)|x|2α+1dx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p(\mathbb {R},w(x)|x|^{2\alpha +1}\textrm{d}x)$$\end{document} spaces with power weights w(x)=|x|β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(x)=|x|^\beta $$\end{document}. As a result, we obtain oscillation estimates for the standard Carleson operator on Lradp(Rn,|x|βdx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{rad}^p(\mathbb {R}^n,|x|^\beta \textrm{d}x)$$\end{document}. As a byproduct, we obtain a transference principle for radial multipliers on Lradp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{rad}^p$$\end{document} spaces, in the spirit of the Rubio de Francia transference principle.