Hardy spaces for the Dunkl harmonic oscillator

TL;DR

This paper introduces a Hardy space associated with the Dunkl harmonic oscillator, characterizes it via Riesz transforms and atomic decompositions, and explores its local atomic structure.

Contribution

It defines a new Hardy space for the Dunkl harmonic oscillator and provides multiple characterizations including atomic and Riesz transform methods.

Findings

Hardy space $\\mathcal{H}^1$ characterized by semigroup maximal functions

Atomic decompositions with local atoms established

Riesz transforms provide alternative characterizations

Abstract

Let $\Delta$ and $L=\Delta -\|\mathbf x\|^2$ be the Dunkl Laplacian and the Dunkl harmonic oscillator respectively. We define the Hardy space $\mathcal H^1$ associated with the Dunkl harmonic oscillator by means of the nontangential maximal function with respect to the semigroup $e^{tL}$. We prove that the space $\mathcal H^1$ admits characterizations by relevant Riesz transforms and atomic decompositions. The atoms which occur in the atomic decompositions are of local type.