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

In this paper, we transform a formula for the $A_2$ Dunkl kernel by Béchir 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.