A formula and sharp estimates for the Dunkl kernel for the root system $A_2$

TL;DR

This paper refines the formula for the $A_2$ Dunkl kernel, expressing it via the $A_1$ kernel with positive terms, enabling sharp estimates for both the Dunkl and heat kernels.

Contribution

It transforms the $A_2$ Dunkl kernel formula to involve only positive terms and derives sharp estimates, improving understanding of these kernels.

Findings

Derived a new formula for the $A_2$ Dunkl kernel with positive terms.

Established sharp estimates for the $A_2$ Dunkl kernel.

Obtained sharp estimates for the associated heat kernel.

Abstract

In this paper, we transform a formula for the $A_2$ Dunkl kernel by B\'echir Amri. The resulting formula expresses the $A_2$ Dunkl kernel in terms of the $A_1$ Dunkl kernel involving only positive terms. This result allows us to derive sharp estimates for the $A_2$ Dunkl kernel. As an interesting by-product, we obtain sharp esitmates for the corresponding heat kernel.