Potential kernels for radial Dunkl Laplacians

Abstract

We derive two-sided bounds for the Newton and Poisson kernels of the $W$-invariant Dunkl Laplacian in geometric complex case when the multiplicity $k(\alpha )=1$, i.e. for flat complex symmetric spaces. For the invariant Dunkl-Poisson kernel $P^W(x,y)$, the estimates are $$P^W(x,y)\asymp \frac{P^{{\mathbf{R}}^d}(x,y)}{{\prod }_{\alpha >0}|x-{\sigma }_{\alpha }y{|}^{2k(\alpha )}},$$ where the $\alpha$'s are the positive roots of a root system acting in ${\mathbf{R}}^d$, the ${\sigma }_{\alpha }$'s are the corresponding symmetries and $P^{{\mathbf{R}}^d}$ is the classical Poisson kernel in ${\mathbf{R}}^d$. Analogous bounds are proven for the Newton kernel when $d\ge 3$. The same estimates are derived in the rank one direct product case ${\mathbb{Z}}_2^N$ and conjectured for general $W$-invariant Dunkl processes. As an application, we get a two-sided bound for the Poisson and Newton kernels of the classical Dyson Brownian motion and of the Brownian motions in any Weyl chamber.

31 pages