Harmonic analysis in Dunkl settings
The Anh Bui
TL;DR
The paper advances harmonic analysis in the Dunkl framework by proving that Besov and Triebel–Lizorkin spaces associated to the Dunkl Laplacian $L$ coincide with their counterparts on the space of homogeneous type $(\mathbb{R}^N,\|\cdot\|,dw)$, enabling familiar heat-kernel and atomic techniques to transfer to the Dunkl setting. It constructs dyadic systems, derives sharp kernel estimates, and develops a robust distributional framework via Calderón reproducing formulae. A central result is the full coincidence of the $(-1,1)$-range Besov/TL spaces defined via the Dunkl operator with the standard homogeneous-type spaces, including Hardy-space identifications like $\\dot F^{0}_{p,2}(dw)\equiv H^p_{CW}(dw)\equiv H^p_{L}(dw)$. Finally, the authors establish Hörmander-type multiplier theorems for the Dunkl transform on these spaces, with refined bounds that extend to Hardy and Riesz-transform settings, broadening the applicability of spectral multipliers in the Dunkl context.
Abstract
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.