Characterizations of $H^1$ and Fefferman-Stein decompositions of $\mathrm{B}\mathrm{M}\mathrm{O}$ functions by systems of singular integrals in the Dunkl setting

We extend the classical theorem of Uchiyama about constructive Fefferman-Stein decompositions of $\mathrm{B}\mathrm{M}\mathrm{O}$ functions by systems of singular integrals to the rational Dunkl setting. On ${\mathbb{R}}^N$ equipped with a root system $R$ and a multiplicity function $k\ge 0$, let $$dw(\mathbf{x})=\prod _{\alpha \in R}\mathrm{\mid }⟨\alpha ,\mathbf{x}⟩{\mathrm{\mid }}^{k(\alpha )}d\mathbf{x}$$ denote the associated measure, and let $\mathcal{F}$ stand for the Dunkl transform. Consider a system $({\theta }_0,{\theta }_1,{\theta }_2,\dots ,{\theta }_d)$ of functions on ${\mathbb{R}}^N$ that are smooth away from the origin and homogeneous of degree zero, with ${\theta }_0(\xi )\equiv 1$. We prove that if $$\text{rank}\left(\begin{array}{ccccc}1& {\theta }_1(\xi )& {\theta }_2(\xi )& \dots & {\theta }_d(\xi )\\ 1& {\theta }_1(-\xi )& {\theta }_2(-\xi )& \dots & {\theta }_d(-\xi )\end{array}\right)=2\text{for all }\xi \in {\mathbb{R}}^N\text{ with }\mathrm{\parallel }\xi \mathrm{\parallel }=1,$$ then any compactly supported $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^N,\mathrm{\parallel }\mathbf{x}-\mathbf{y}\mathrm{\parallel },dw)$ function $f$ can be decomposed into $$f=g_0+\sum _{j=1}^d{\mathbf{S}}^{\{j\}}{g}_j,{\parallel \sum _{j=0}^d{g}_j\parallel }_{L^{\mathrm{\infty }}}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}},$$ where ${\mathbf{S}}^{\{j\}}g={\mathcal{F}}^{-1}({\theta }_j\mathcal{F}g)$. As a corollary, we obtain characterizations of the Hardy space $H_{\mathrm{D}\mathrm{u}\mathrm{n}\mathrm{k}\mathrm{l}}^1$ by the system of singular integral operators $(\mathrm{I}\mathrm{d},{\mathbf{S}}^{\{1\}},{\mathbf{S}}^{\{2\}},\dots ,{\mathbf{S}}^{\{d\}})$.