Overview
This research investigates the Dunkl kernel, providing uniform upper bounds for the kernel itself and its derivatives. These bounds are established for a general reduced root system and are observed to be independent of the spatial variable. The findings generalize existing sharp upper bounds applicable to classical one-variable Bessel functions and spherical functions found in Cartan motion groups.
A key consequence of this analysis is the demonstration that the representing measure of Dunkl's intertwining operator exhibits absolute continuity with respect to the Lebesgue measure. This holds under specific conditions: for multiplicities $k > 1/2$ and for a generic spectral parameter. This outcome directly addresses and settles a conjecture originally presented in [RdJ02], at least within the specified range of multiplicities $k > 1/2$.
Research Context
The study centers on the Dunkl kernel, a fundamental object in harmonic analysis associated with root systems. Previous work has established sharp upper bounds for specific related functions, including classical one-variable Bessel functions and spherical functions pertinent to Cartan motion groups. The current research aims to extend such uniform bounding properties to the more general context of the Dunkl kernel across arbitrary reduced root systems.
Another area of focus is Dunkl's intertwining operator, for which properties of its representing measure are of interest. A conjecture regarding the absolute continuity of this measure, posed in [RdJ02], provided a specific open problem that this research aimed to address.
Approach
The methodology involved developing upper bounds for the Dunkl kernel and its derivatives. These bounds were designed to be uniform with respect to the spatial variable, considering an arbitrary reduced root system and a regular spectral parameter. The derived estimates were then compared to known sharp upper bounds for classical one-variable Bessel functions and spherical functions of Cartan motion groups, indicating a successful generalization.
The established uniform bounds were subsequently utilized to investigate the properties of the representing measure of Dunkl's intertwining operator. Specifically, the analysis focused on determining conditions under which this measure would exhibit absolute continuity relative to the Lebesgue measure. The conditions identified were multiplicities $k > 1/2$ and a generic spectral parameter.
Findings
- Uniform upper bounds for the Dunkl kernel were established for an arbitrary reduced root system with a regular spectral parameter. These bounds are uniform in the spatial variable.
- Uniform upper bounds were also derived for the derivatives of the Dunkl kernel under the same conditions.
- These estimates generalize existing sharp upper bounds for classical one-variable Bessel functions.
- The derived estimates also generalize existing sharp upper bounds for spherical functions of Cartan motion groups.
- It was demonstrated that the representing measure of Dunkl's intertwining operator is absolutely continuous with respect to the Lebesgue measure.
- This absolute continuity holds for multiplicities $k > 1/2$ and a generic spectral parameter.
- The findings settle a conjecture from [RdJ02] for multiplicities $k > 1/2$.