Convolution operator and maximal function for Dunkl transform

TL;DR

This paper develops analysis tools for the Dunkl transform on weighted $L^p$ spaces, including summability, maximal functions, and convergence results, extending classical harmonic analysis to reflection-invariant settings.

Contribution

It introduces a maximal function and studies summability and convergence of the inverse Dunkl transform using generalized translation and convolution, broadening harmonic analysis methods.

Findings

Established almost everywhere convergence of Dunkl transform inverses.

Analyzed summability of Poisson integrals and Bochner-Riesz means.

Developed maximal function techniques for Dunkl analysis.

Abstract

For a family of weight functions, $h_\kappa$, invariant under a finite reflection group on $\RR^d$, analysis related to the Dunkl transform is carried out for the weighted $L^p$ spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence.