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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.01067 |
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Table of Contents:
- Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb R^N$ associated with a normalized root $R$ and a multiplicity function $k(ν)\ge 0, ν\in R$. In this paper, we first prove that the Besov and Triebel-Lizorkin spaces associated with the Dunkl Laplacian $L$ are identical to the Besov and Triebel-Lizorkin spaces defined in the space of homogeneous type $(\mathbb R^N, \|\cdot\|, dw)$, where $dw({\rm x})=\prod_{ν\in R}\langle ν,{\rm x}\rangle^{k(ν)}d{\rm x}$. Next, consider the Dunkl transform denoted by $\mathcal{F}$. We introduce the multiplier operator $T_m$, defined as $T_mf = \mathcal{F}^{-1}(m\mathcal{F}f)$, where $m$ is a bounded function defined on $\mathbb{R}^N$. Our second aim is to prove multiplier theorems, including the Hörmander multiplier theorem, for $T_m$ on the Besov and Tribel-Lizorkin spaces in the space of homogeneous type $(\mathbb R^N, \|\cdot\|, dw)$. Importantly, our findings present novel results, even in the specific case of the Hardy spaces.