Upper and lower bounds for Dunkl heat kernel

Abstract

On ${\mathbb{R}}^N$ equipped with a normalized root system $R$, a multiplicity function $k(\alpha )>0$, and the associated measure $$dw(\mathbf{x})=\prod _{\alpha \in R}|⟨\mathbf{x},\alpha ⟩{|}^{k(\alpha )}d\mathbf{x},$$ let $h_t(\mathbf{x},\mathbf{y})$ denote the heat kernel of the semigroup generated by the Dunkl Laplace operator ${\Delta }_k$. Let $d(\mathbf{x},\mathbf{y})={\min }_{\sigma \in G}\Vert \mathbf{x}-\sigma (\mathbf{y})\Vert$, where $G$ is the reflection group associated with $R$. We derive the following upper and lower bounds for $h_t(\mathbf{x},\mathbf{y})$: for all $c_l>1/4$ and $0<c_u<1/4$ there are constants $C_l,C_u>0$ such that $$C_{l}w(B(\mathbf{x},\sqrt{t}){)}^{-1}{e}^{-c_l\frac{d(\mathbf{x},\mathbf{y}{)}^2}{t}}\Lambda (\mathbf{x},\mathbf{y},t)\le h_t(\mathbf{x},\mathbf{y})\le C_{u}w(B(\mathbf{x},\sqrt{t}){)}^{-1}{e}^{-c_u\frac{d(\mathbf{x},\mathbf{y}{)}^2}{t}}\Lambda (\mathbf{x},\mathbf{y},t),$$ where $\Lambda (\mathbf{x},\mathbf{y},t)$ can be expressed by means of some rational functions of $\Vert \mathbf{x}-\sigma (\mathbf{y})\Vert /\sqrt{t}$. An exact formula for $\Lambda (\mathbf{x},\mathbf{y},t)$ is provided.

arXiv comment17 pages, we corrected some typos