Harmonic analysis in Dunkl settings

TL;DR

This paper establishes the equivalence of certain function spaces associated with the Dunkl Laplacian to those in a homogeneous type space and proves multiplier theorems, including Hörmander’s, for Dunkl transforms, with new results even for Hardy spaces.

Contribution

It introduces the equivalence of Dunkl-related function spaces with homogeneous type spaces and proves new multiplier theorems for Dunkl transforms, extending known results.

Findings

Dunkl Besov and Triebel-Lizorkin spaces are equivalent to those in homogeneous type spaces.

Established Hörmander multiplier theorem for Dunkl transform on these spaces.

Provided new results for Hardy spaces in the Dunkl setting.

Abstract

Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb R^N$ associated with a normalized root $R$ and a multiplicity function $k(\nu)\ge 0, \nu\in R$. 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 $(\mathbb R^N, \|\cdot\|, dw)$, where $dw({\rm x})=\prod_{\nu\in R}\langle \nu,{\rm x}\rangle^{k(\nu)}d{\rm x}$. 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 H\"ormander 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.