Remark on atomic decompositions for the Hardy space 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}.
