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 $k(\alpha )\ge 0$. We say that a function $f$ belongs to the Hardy space $H_{\Delta }^1$ if the nontangential maximal function ${\mathcal{M}}_{H}f(\mathbf{x})={\sup }_{\Vert \mathbf{x}-\mathbf{y}\Vert <t}|\exp (t^2\Delta )f(\mathbf{x})|$ belongs to $L^1(w(\mathbf{x})d\mathbf{x})$, where $w(\mathbf{x})={\prod }_{\alpha \in R}|⟨\alpha ,\mathbf{x}⟩{|}^{k(\alpha )}$. We prove that $H_{\Delta }^1$ coincides with the space $H_{\mathrm{atom}}^1({\mathbb{R}}^N,\Vert \mathbf{x}-\mathbf{y}\Vert ,w(\mathbf{x})d\mathbf{x})$ 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 $e^{t\Delta }$.

arXiv comment2 figures