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

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 $e^{t\Delta}$.