A Remark on Hörmander Multipliers on Dunkl Hardy Spaces

Let $\mathcal{R}$ be a normalized root system in ${\mathbb{R}}^N$ with a nonnegative multiplicity function $k$, and let $\mathcal{F}$ be the associated Dunkl transform. We prove a Hörmander multiplier theorem on the Hardy spaces $H_{\mathrm{D}\mathrm{u}\mathrm{n}\mathrm{k}\mathrm{l}}^p$, $0<p\le 1$, defined by conical Littlewood–Paley square functions. Let $W_2^s$ denote the classical Sobolev space in ${\mathbb{R}}^N$. More precisely, if a multiplier $m$ satisfies $$\underset{t>0}{\sup }\mathrm{\parallel }\psi (\cdot )m(t\cdot ){\mathrm{\parallel }}_{W_2^s}<\mathrm{\infty }$$ for a nonzero radial cutoff $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^N\setminus \{0\})$ and $$s>\mathbf{N}(\frac{1}{p}-\frac{1}{2}),$$ where $\mathbf{N}$ is the homogeneous dimension, then the Dunkl multiplier $${\mathcal{T}}_{m}f={\mathcal{F}}^{-1}(m\mathcal{F}f)$$ extends to a bounded operator on $H_{\mathrm{D}\mathrm{u}\mathrm{n}\mathrm{k}\mathrm{l}}^p$.