Remark on atomic decompositions for Hardy space $H^1$ in the rational Dunkl setting

TL;DR

This paper establishes the equivalence of Hardy spaces defined via maximal functions and atomic decompositions in the Dunkl setting, extending classical harmonic analysis results to a more general framework involving root systems and multiplicity functions.

Contribution

It proves the equivalence of Hardy space definitions in the Dunkl setting and improves heat kernel estimates, advancing the understanding of harmonic analysis in this context.

Findings

Hardy space $H^1_ riangle$ coincides with atomic Hardy space.

Improved estimates for the heat kernel of $e^{t riangle}$.

Extension of classical Hardy space theory to Dunkl operators.

Abstract

Let $\Delta$ be the Dunkl Laplacian on $\mathbb R^N$ associated with a normalized root system $R$ and a multiplicity function $k(\alpha)\geq 0$. We say that a function $f$ belongs to the Hardy space $H^1_{\Delta}$ if the nontangential maximal function $\mathcal M_H f(\mathbf x)=\sup_{\| \mathbf x-\mathbf y\|<t} |\exp(t^2\Delta )f(\mathbf x)|$ belongs to $L^1(w(\mathbf x)\, d\mathbf x)$, where $w(\mathbf x)=\prod_{\alpha\in R} |\langle \alpha,\mathbf x\rangle|^{k(\alpha)}$. We prove that $H^1_\Delta$ coincides with the space $H^1_{\rm atom}(\mathbb R^N, \| \mathbf x-\mathbf y\|, w(\mathbf x)d\mathbf x)$ understood as the atomic Hardy space on the space of homogeneous type in the sense of Coifman--Weiss. To this end we improve estimates for the heat kernel of $e^{t\Delta}$.