Harmonic analysis in Dunkl settings

Abstract

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