Hörmander's multiplier theorem on $H^p$-spaces in the rational Dunkl setting
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arXiv
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| Format: | Preprint |
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2026
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| author | Dziubański, Jacek Hejna-Łyżwa, Agnieszka |
| author_facet | Dziubański, Jacek Hejna-Łyżwa, Agnieszka |
| contents | On $\mathbb{R}^N$ equipped with a normalized root system $\mathcal R$ and a multiplicity function $k\geq 0$, let $dw(\mathbf x)=Π_{α\in \mathcal R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x$, $\mathbf{N}=N+\sum_{α\in \mathcal R}k(α)$ denote the associated measure and the homogeneous dimension of the system $(\mathcal R,k)$ respectively. Let $\mathcal F$ stand for the Dunkl transform. For $0<p\leq 1$, let $m$ be a bounded function on $\mathbb{R}^N$, which satisfies the classical Hörmander's condition with smoothness $s>\mathbf{N}/p$. We show that the multiplier operator $\mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff)$, initially defined on $H^p_{\mathrm{Dunkl}}\cap L^2(dw)$, has a unique extension to a bounded operator in $H^p_{\mathrm{Dunkl}}$, where the space $H^p_{\mathrm{Dunkl}}$ is defined by means of a Littlewood-Paley square function. To prove the theorem, we use special atomic and molecule characterizations of $H^p_{\mathrm{Dunkl}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_20555 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hörmander's multiplier theorem on $H^p$-spaces in the rational Dunkl setting Dziubański, Jacek Hejna-Łyżwa, Agnieszka Functional Analysis primary: 42B30, secondary: 42B35, 33C52, 42B10, 35K08 On $\mathbb{R}^N$ equipped with a normalized root system $\mathcal R$ and a multiplicity function $k\geq 0$, let $dw(\mathbf x)=Π_{α\in \mathcal R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x$, $\mathbf{N}=N+\sum_{α\in \mathcal R}k(α)$ denote the associated measure and the homogeneous dimension of the system $(\mathcal R,k)$ respectively. Let $\mathcal F$ stand for the Dunkl transform. For $0<p\leq 1$, let $m$ be a bounded function on $\mathbb{R}^N$, which satisfies the classical Hörmander's condition with smoothness $s>\mathbf{N}/p$. We show that the multiplier operator $\mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff)$, initially defined on $H^p_{\mathrm{Dunkl}}\cap L^2(dw)$, has a unique extension to a bounded operator in $H^p_{\mathrm{Dunkl}}$, where the space $H^p_{\mathrm{Dunkl}}$ is defined by means of a Littlewood-Paley square function. To prove the theorem, we use special atomic and molecule characterizations of $H^p_{\mathrm{Dunkl}}$. |
| title | Hörmander's multiplier theorem on $H^p$-spaces in the rational Dunkl setting |
| topic | Functional Analysis primary: 42B30, secondary: 42B35, 33C52, 42B10, 35K08 |
| url | https://arxiv.org/abs/2603.20555 |