Remark on atomic decompositions for the Hardy space $H^1$ in the rational Dunkl setting
Tom 251 / 2020
Streszczenie
Let $\varDelta$ 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 $H^1_{\varDelta}$ if the nontangential maximal function defined by $\mathcal M_H f(\mathbf x)= \sup_{\| \mathbf x-\mathbf y\| \lt t} |\!\exp(t^2\varDelta )f(\mathbf x)|$ 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 $H^1_\varDelta$ admits atomic decompositions into atoms in the sense of Coifman–Weiss on the space of homogeneous type $\mathbb{R}^N$ equipped with the Euclidean distance $\|\mathbf{x}-\mathbf{y}\|$ and the measure $w(\mathbf{x})d\mathbf{x}$. To this end we improve estimates for the heat kernel of $e^{t\varDelta}$.
