Oscillation inequalities for Carleson--Dunkl operator

TL;DR

This paper develops oscillation inequalities for the Carleson--Dunkl operator in weighted $L^p$ spaces, leading to new estimates for the classical Carleson operator and a transference principle for radial multipliers.

Contribution

It introduces oscillation estimates for the Carleson--Dunkl operator and establishes a transference principle for radial multipliers, extending classical harmonic analysis results.

Findings

Oscillation estimates for the Carleson--Dunkl operator in weighted $L^p$ spaces.

Oscillation estimates for the classical Carleson operator on radial $L^p$ spaces.

A transference principle for radial multipliers analogous to Rubio de Francia's.

Abstract

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