Harmonic analysis in Dunkl settings

Let $L$ be the Dunkl Laplacian on the Euclidean space ${\mathbb{R}}^N$ associated with a normalized root $R$ and a multiplicity function $k(\nu )\ge 0,\nu \in R$. 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 $({\mathbb{R}}^N,\mathrm{\parallel }\cdot \mathrm{\parallel },dw)$, where $dw(\mathrm{x})={\prod }_{\nu \in R}⟨\nu ,\mathrm{x}{⟩}^{k(\nu )}d\mathrm{x}$. Next, consider the Dunkl transform denoted by $\mathcal{F}$. We introduce the multiplier operator $T_m$, defined as $T_{m}f={\mathcal{F}}^{-1}(m\mathcal{F}f)$, 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 $({\mathbb{R}}^N,\mathrm{\parallel }\cdot \mathrm{\parallel },dw)$. Importantly, our findings present novel results, even in the specific case of the Hardy spaces.