The sharp estimate for the Weyl-invariant Dunkl heat kernel

Let $\Sigma$ be a root system in $\mathbb{R}^d$, let $W$ be its Weyl group, and let $k(\alpha)$ be the multiplicity attached to a positive root $\alpha$. For $t>0$ and $X,Y\in\overline{\mathfrak{a}^+}$, let $p_t^W(X,Y)$ be the Weyl-invariant Dunkl heat kernel, and let $E_k^W(X,Y)$ be the Weyl-invariant Dunkl kernel. Write $f\asymp g$ for two-sided bounds by constants independent of the variables.

The Weyl-invariant Dunkl-kernel estimate. The heat kernel satisfies

$$p_t^W(X,Y)\asymp t^{-\frac d2}\frac{e^{-\frac{|X-Y|^2}{4t}}}{\prod_{\alpha>0}(t+\alpha(X)\alpha(Y))^{k(\alpha)}}.$$

Consequently, the source also states the corresponding Dunkl-kernel estimate

$$E_k^W(X,Y)\asymp \frac{e^{\lambda(X)}}{\prod_{\alpha>0}(1+\alpha(X)\alpha(\lambda))^{k(\alpha)}}.$$

These estimates are motivated by known general heat-kernel bounds and by the sharp $A_n$ radial result. Their validity in the full Dunkl setting is left as a conjecture.