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 L(p) multiplier theorems for Dunkl transform on the Euclidean space R(d) can be transferred from the corresponding results for h-harmonic expansions on the unit sphere S(d) of R(d+1) The result is then applied to establish a Hormander 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 L(p) spaces. (C) 2010 Elsevier Inc. All rights reserved.