A transference theorem for the Dunkl transform and its applications

For a family of weight functions invariant under a finite reflection group, we show how weighted Lp multiplier theorems for Dunkl transform on the Euclidean space Rd can be transferred from the corresponding results for h-harmonic expansions on the unit sphere Sd of Rd+1. The result is then applied to establish a Hörmander type multiplier theorem for the Dunkl transform and to show the convergence of the Bochner–Riesz means of the Dunkl transform of order above the critical index in weighted Lp spaces.