Abstract: In Dunkl theory the Dunkl kernel replaces the classical exponential function in its predominant role in harmonic analysis. For regular spectral parameters, we present upper bounds for the Dunkl kernel and its derivatives which are uniform in the spatial variable. These estimates generalize sharp uniform upper bounds for spherical functions of Cartan motion groups and classical Bessel functions. The proof is based on an asymptotic study of the differential system satisfied by the Dunkl kernel with respect to parabolic subgroups of the Weyl group. As a consequence we obtain results regarding the Lebesgue density of the representing measure of Dunkl’s intertwining operator.
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