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Uniform bounds on the Dunkl kernel

  • Lukas Langen
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{langen2026uniform,
  title = {{Uniform bounds on the Dunkl kernel}},
  author = {Lukas Langen},
  howpublished = {arXiv},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2607.02176},
  eprint = {2607.02176},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.02176},
}
RIS
TY  - UNPB
TI  - Uniform bounds on the Dunkl kernel
AU  - Lukas Langen
PY  - 2026
DA  - 2026/07/16
PB  - arXiv
DO  - 10.48550/arxiv.2607.02176
UR  - https://arxiv.org/abs/2607.02176
AB  - 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.
ER  -