H\"ormander's multiplier theorem for the Dunkl transform

Abstract

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Denote by $dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x $ the associated measure in $\mathbb R^N$. Let $\mathcal F$ stands for the Dunkl transform. Given a bounded function $m$ on $\mathbb R^N$, we prove that if there is $s>\mathbf N$ such that $m$ satisfies the classical H\"ormander condition with the smoothness $s$, then the multiplier operator $\mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff)$ is of weak type $(1,1)$, strong type $(p,p)$ for $1<p<\infty$, and bounded on a relevant Hardy space $H^1$. To this end we study the Dunkl translations and the Dunkl convolution operators and prove that if $F$ is sufficiently regular, for example its certain Schwartz class seminorm is finite, then the Dunkl convolution operator with the function $F$ is bounded on $L^p(dw)$ for $1\leq p\leq \infty$. We also consider boundedness of maximal operators associated with the Dunkl convolutions with Schwartz class functions.