Convolution operator and maximal function for the Dunkl transform

Abstract

For a family of weight functions h_K invariant under a finite reflection group on R^d, analysis related to the Dunkl transform is carried out for the weighted L^p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence.