Convolution operator and maximal function for Dunkl transform

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For a family of weight functions, hκ, invariant under a finite reflection group on ᵈ, analysis related to the Dunkl transform is carried out for the weighted Lᵖ 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.