A Remark on Hörmander Multipliers on Dunkl Hardy Spaces

Abstract

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 $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, 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 \[ \sup_{t>0} \|ψ(\cdot)m(t\cdot)\|_{W_2^s}<\infty \] for a nonzero radial cutoff $ψ\in C_c^\infty(\mathbb R^N\setminus\{0\})$ and \[ s>\mathbf N\left(\frac1p-\frac12\right), \] where $\mathbf N$ is the homogeneous dimension, then the Dunkl multiplier \[ \mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff) \] extends to a bounded operator on $H^p_{\mathrm{Dunkl}}$.