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

For a normalized root system $R$ in ${\mathbb{R}}^N$ and a multiplicity function $k\ge 0$ 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 $theassociatedmeasurein$\mathbb R^N$\mathrm{.}Let$\mathcal F$standsfortheDunkltransform\mathrm{.}Givenaboundedfunction$m$on$\mathbb R^N$,weprovethatifthereis$s>\mathbf N$suchthat$m$satisfiestheclassicalH\ddot{o}rmanderconditionwiththesmoothness$s$, then the multiplier operator ${\mathcal{T}}_{m}f={\mathcal{F}}^{-1}(m\mathcal{F}f)$ is of weak type $(1,1)$, strong type $(p,p)$ for $1<p<\mathrm{\infty }$, 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.