On the boundedness of Dunkl multipliers

We use Littlewood–Paley–Stein theory to prove two versions of Dunkl multiplier theorem when the multiplier $m$ satisfies a modified Hörmander condition. When $m$ is radial we give a simple proof of a known result. For general $m$ we prove that the Dunkl multiplier operator takes radial functions in $L^p $ boundedly into $L^p$ for all $p \geq 2$.