Characterizations of $H^1$ and Fefferman-Stein decompositions of ${\rm BMO}$ functions by systems of singular integrals in the Dunkl setting
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2025
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| _version_ | 1866910910313922560 |
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| author | Dziubański, Jacek Hejna, Agnieszka |
| author_facet | Dziubański, Jacek Hejna, Agnieszka |
| contents | 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 \[ dw(\mathbf{x}) = \prod_{α\in R} |\langle α, \mathbf{x} \rangle|^{k(α)} \, d\mathbf{x} \] denote the associated measure, and let $\mathcal{F}$ stand for the Dunkl transform. Consider a system $(θ_0, θ_1, θ_2, \dots, θ_d)$ of functions on $\mathbb{R}^N$ that are smooth away from the origin and homogeneous of degree zero, with $θ_0(ξ) \equiv 1$. We prove that if \[ \text{rank} \left( \begin{array}{ccccc} 1 & θ_1(ξ) & θ_2(ξ) & \ldots & θ_d(ξ) \\ 1 & θ_1(-ξ) & θ_2(-ξ) & \ldots & θ_d(-ξ) \end{array} \right) = 2 \quad \text{for all } ξ\in \mathbb{R}^N \text{ with } \|ξ\| = 1, \] 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 $\mathbf{S}^{\{j\}} g = \mathcal{F}^{-1}(θ_j \mathcal{F}g)$. As a corollary, we obtain characterizations of the Hardy space $H^1_{\rm Dunkl}$ by the system of singular integral operators $({\rm Id}, \mathbf{S}^{\{1\}}, \mathbf{S}^{\{2\}}, \dots, \mathbf{S}^{\{d\}})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_04964 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizations of $H^1$ and Fefferman-Stein decompositions of ${\rm BMO}$ functions by systems of singular integrals in the Dunkl setting Dziubański, Jacek Hejna, Agnieszka Functional Analysis 44A20, 42B20, 42B25, 47B38, 35K08, 33C52, 39A70 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 \[ dw(\mathbf{x}) = \prod_{α\in R} |\langle α, \mathbf{x} \rangle|^{k(α)} \, d\mathbf{x} \] denote the associated measure, and let $\mathcal{F}$ stand for the Dunkl transform. Consider a system $(θ_0, θ_1, θ_2, \dots, θ_d)$ of functions on $\mathbb{R}^N$ that are smooth away from the origin and homogeneous of degree zero, with $θ_0(ξ) \equiv 1$. We prove that if \[ \text{rank} \left( \begin{array}{ccccc} 1 & θ_1(ξ) & θ_2(ξ) & \ldots & θ_d(ξ) \\ 1 & θ_1(-ξ) & θ_2(-ξ) & \ldots & θ_d(-ξ) \end{array} \right) = 2 \quad \text{for all } ξ\in \mathbb{R}^N \text{ with } \|ξ\| = 1, \] 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 $\mathbf{S}^{\{j\}} g = \mathcal{F}^{-1}(θ_j \mathcal{F}g)$. As a corollary, we obtain characterizations of the Hardy space $H^1_{\rm Dunkl}$ by the system of singular integral operators $({\rm Id}, \mathbf{S}^{\{1\}}, \mathbf{S}^{\{2\}}, \dots, \mathbf{S}^{\{d\}})$. |
| title | Characterizations of $H^1$ and Fefferman-Stein decompositions of ${\rm BMO}$ functions by systems of singular integrals in the Dunkl setting |
| topic | Functional Analysis 44A20, 42B20, 42B25, 47B38, 35K08, 33C52, 39A70 |
| url | https://arxiv.org/abs/2503.04964 |