Hilbert Transforms Associated with Dunkl-Hermite Polynomials

TL;DR

This paper studies Hilbert transforms and related integral operators associated with Dunkl-Hermite functions in one dimension, establishing their properties and mapping behavior within a weighted $L^p$ framework.

Contribution

It introduces and analyzes Hilbert transforms and conjugate Poisson integrals in the Dunkl-Hermite setting, extending classical harmonic analysis tools to this context.

Findings

Hilbert transforms are Calderón-Zygmund operators

Mapping properties follow from Calderón-Zygmund theory

Established properties of heat-diffusion and Poisson integrals in Dunkl setting

Abstract

We consider expansions of functions in $L^{p}(\mathbb{R},|x|^{2k}dx)$, $1\leq p<+\infty$ with respect to Dunkl-Hermite functions in the rank-one setting. We actually define the heat-diffusion and Poisson integrals in the one-dimensional Dunkl setting and study their properties. Next, we define and deal with Hilbert transforms and conjugate Poisson integrals in the same setting. The formers occur to be Calder\'on-Zygmund operators and hence their mapping properties follow from general results.