On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

TL;DR

This paper studies various harmonic analysis operators related to the Dunkl Laplacian in multiple dimensions, establishing their boundedness in weighted L^p spaces and weak L^1 bounds using Calderón-Zygmund theory.

Contribution

It extends the boundedness results of harmonic analysis operators to the Dunkl setting with reflection group ^n, including negative multiplicities.

Findings

Maximal operators are bounded on weighted L^p spaces.

g-functions and Lusin area integrals are bounded in the same setting.

Riesz transforms and multipliers are also bounded, with weak L^1 estimates.

Abstract

We consider several harmonic analysis operators in the multi-dimensional context of the Dunkl Laplacian with the underlying group of reflections isomorphic to $\mathbb{Z}_2^n$ (also negative values of the multiplicity function are admitted). Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes transform type. Using the general Calder\'on-Zygmund theory we prove that these objects are bounded in weighted $L^p$ spaces, $1<p<\infty$, and from $L^1$ into weak $L^{1}$.