Harmonic analysis operators in the rational Dunkl settings

TL;DR

This paper investigates harmonic analysis operators in Dunkl settings related to reflection groups, establishing their boundedness in various function spaces despite the unique challenges posed by the dual-metric kernel estimations.

Contribution

It introduces new boundedness results for harmonic analysis operators in Dunkl settings, overcoming limitations of classical Calderón-Zygmund theory due to dual-metric kernel estimations.

Findings

Boundedness of maximal operators in Dunkl spaces

Extension of Littlewood-Paley theory to Dunkl settings

Operators are bounded in Hardy, BMO, and BLO spaces

Abstract

In this paper we study harmonic analysis operators in Dunkl settings associated with finite reflection groups on Euclidean spaces. We consider maximal operators, Littlewood-Paley functions, $\rho$-variation and oscillation operators involving time derivatives of the heat semigroup generated by Dunkl operators. We establish the boundedness properties of these operators in $L^p(\mathbb R^d,\omega _\mathfrak K)$, $1\leq p < 1$, Hardy spaces, BMO and BLO-type spaces in the Dunkl settings. The study of harmonic analysis operators associated to reflection groups need different strategies from the ones used in the Euclidean case since the integral kernels of the operators admit estimations involving two different metrics, namely, the Euclidean and the orbit metrics. For instance, the classical Calder\'on-Zygmund theory for singular integrals does not work in this setting.