Potential kernels for radial Dunkl Laplacians. (20th August 2022)

• Record Type: Journal Article
• Authors: Graczyk, P.; Luks, T.; Sawyer, P.
• Is Part Of: Canadian journal of mathematics. Volume 74:Number 4(2022)
• Page Start: 1005
• Page End: 1033
• Publication Date: 2022-08-20
• Subjects: 17B22 -- 60J45; Potential kernel -- Newton kernel -- Dunkl setting -- complex symmetric space -- root system; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X21000195
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: We derive two-sided bounds for the Newton and Poisson kernels of the W -invariant Dunkl Laplacian in the 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 $$ \begin{align*} P^{W}(x, y)\asymp \frac{P^{\mathbf{R}^{d}}(x, y)}{\prod_{\alpha> 0 \ }|x-\sigma_{\alpha} y|^{2k(\alpha)}}, \end{align*} $$ 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.