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

Let $\mathrm{\Delta }$ be the Dunkl Laplacian on ${\mathbb{R}}^N$ associated with a normalized root system $R$ and a multiplicity function $k(\alpha )\ge 0$. We say that a function $f$ belongs to the Hardy space $H_{\mathrm{\Delta }}^1$ if the nontangential maximal function $\mathcal M_H f(\mathbf x)=\sup_{| \mathbf x-\mathbf y|<t} |\exp(t^2\Delta )f(\mathbf x)|$belongsto$L^1(w(\mathbf x), d\mathbf x)$,where$w(\mathbf x)=\prod_{\alpha\in R} |\langle \alpha,\mathbf x\rangle|^{k(\alpha)}$\mathrm{.}Weprovethat$H^1_\Delta$coincideswiththespace$H^1_{\rm atom}(\mathbb R^N, | \mathbf x-\mathbf y|, w(\mathbf x)d\mathbf x)$ 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 $e^{t\mathrm{\Delta }}$.