preprint Lukas Langen author 73664 For an arbitrary reduced root system, we give upper bounds for the Dunkl kernel with regular spectral parameter and its derivatives, which are uniform in the spatial variable. These estimates generalize well-known sharp upper bounds for classical one-variable Bessel functions and for spherical functions of Cartan motion groups. As a consequence, we prove that the representing measure of Dunkl's intertwining operator is absolutely continuous with respect to the Lebesgue measure for multiplicities $k> 1/2$ and generic spectral parameter. This settles a conjecture posed in [RdJ02] at least for $k>1/2$. 2026 eng 2607.02176 31 L. Langen, ArXiv:2607.02176 (2026). Langen, Lukas. “Uniform Bounds on the Dunkl Kernel.” <i>ArXiv:2607.02176</i>, 2026. L. Langen, “Uniform bounds on the Dunkl kernel,” <i>arXiv:2607.02176</i>. 2026. Langen, L. (2026). Uniform bounds on the Dunkl kernel. In <i>arXiv:2607.02176</i>. @article{Langen_2026, title={Uniform bounds on the Dunkl kernel}, journal={arXiv:2607.02176}, author={Langen, Lukas}, year={2026} } Langen L. Uniform bounds on the Dunkl kernel. <i>arXiv:260702176</i>. Published online 2026. Langen, Lukas. “Uniform Bounds on the Dunkl Kernel.” <i>ArXiv:2607.02176</i>, 2026. 665502026-07-21T13:08:08Z2026-07-21T13:11:05Z