Oscillation inequalities for Carleson--Dunkl operator

TL;DR

The paper proves a uniform oscillation inequality for partial sums of the Dunkl transform on weighted $L^p$ spaces and derives corresponding oscillation bounds for the Fourier transform on radial functions in $\mathbb{R}^n$ with a power weight. The approach combines vector-valued projection-estimates in the spirit of Mirek–Szarek–Wright with transplantation methods (Stempak–Trebels) to transfer multipliers across Dunkl orders, ultimately yielding a Rubio de Francia-type transference principle for radial multipliers. The main contributions include quantitative oscillation bounds for Dunkl partial sums under precise weight conditions, and a radial transference principle that extends RdF-type results to the Dunkl/Hankel framework for $p\neq 2$. These results bridge weighted harmonic analysis in the Dunkl setting with classical radial Fourier analysis and offer new vector-valued and transference tools for radial multipliers.

Abstract

In this paper, we establish estimates for the oscillation seminorm for the so-called Carleson--Dunkl operator on weighted $L^p(\mathbb{R},w(x)|x|^{2α+1}{\rm d}x)$ spaces with power weights $w(x)=|x|^β$. As a result, we obtain oscillation estimates for the standard Carleson operator on $L_{\rm rad}^p(\mathbb{R}^n,|x|^β{\rm d}x)$. 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.

Theorems & Definitions (29)

• Theorem 1.4
• Corollary 1.6
• Proposition 1.7
• Theorem 1.8: Rubio de Francia transference theorem for radial multipliers
• Theorem 2.4: Kanjin--Prestini
• Theorem 2.7: Prestini
• Theorem 2.12: El Kamel--Yacoub
• Theorem 2.15
• proof
• Proposition 2.18: Weighted Kanjin--Prestini estimate