On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

TL;DR

This paper investigates harmonic analysis operators related to the Dunkl harmonic oscillator and Laguerre-symmetrized frameworks, establishing their boundedness on weighted $L^p$ spaces and extending results to new multi-dimensional settings.

Contribution

It introduces boundedness results for various harmonic analysis operators in Dunkl and Laguerre-symmetrized contexts, including cases with negative multiplicity functions.

Findings

Operators are bounded on weighted $L^p$ spaces for $1 < p < $

Results include weak $L^1$ bounds for these operators

Extends boundedness results to Laguerre-symmetrized frameworks

Abstract

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calder\'on-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type.