Characterizations of $H^1$ and Fefferman-Stein decompositions of ${\rm BMO}$ functions by systems of singular integrals in the Dunkl setting

Abstract

We extend the classical theorem of Uchiyama about constructive Fefferman-Stein decompositions of ${\rm BMO}$ 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 $k \geq 0$, let \[ dw(\mathbf{x}) = \prod_{\alpha \in R} |\langle \alpha, \mathbf{x} \rangle|^{k(\alpha)} \, d\mathbf{x} \] 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 \[ \text{rank} \left( \begin{array}{ccccc} 1 & \theta_1(\xi) & \theta_2(\xi) & \ldots & \theta_d(\xi) \\ 1 & \theta_1(-\xi) & \theta_2(-\xi) & \ldots & \theta_d(-\xi) \end{array} \right) = 2 \quad \text{for all } \xi \in \mathbb{R}^N \text{ with } \|\xi\| = 1, \] then any compactly supported ${\rm BMO}(\mathbb{R}^N, \|\mathbf{x} - \mathbf{y}\|, dw)$ function $f$ can be decomposed into \[ f = g_0 + \sum_{j=1}^d \mathbf{S}^{\{j\}} g_j, \quad \left\| \sum_{j=0}^d g_j \right\|_{L^\infty} \leq C \|f\|_{\rm BMO}, \] where $\mathbf{S}^{\{j\}} g = \mathcal{F}^{-1}(\theta_j \mathcal{F}g)$. As a corollary, we obtain characterizations of the Hardy space $H^1_{\rm Dunkl}$ by the system of singular integral operators $({\rm Id}, \mathbf{S}^{\{1\}}, \mathbf{S}^{\{2\}}, \dots, \mathbf{S}^{\{d\}})$.