Upper and lower bounds for Dunkl heat kernel

TL;DR

This paper establishes precise upper and lower bounds for the Dunkl heat kernel on Euclidean space with reflection group symmetries, involving the distance function, measure, and a rational function expression.

Contribution

It derives explicit bounds for the Dunkl heat kernel, including an exact formula for the auxiliary function, advancing understanding of heat kernel behavior in Dunkl analysis.

Findings

Established bounds for the Dunkl heat kernel involving the measure and distance

Provided an explicit formula for the auxiliary function b3(b1,b2,b3)

Demonstrated the bounds hold uniformly with constants depending on parameters

Abstract

On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(\alpha) > 0$, and the associated measure $$ dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x, $$ let $h_t(\mathbf x,\mathbf y)$ denote the heat kernel of the semigroup generated by the Dunkl Laplace operator $\Delta_k$. Let $d(\mathbf x,\mathbf y)=\min_{\sigma\in G} \| \mathbf x-\sigma(\mathbf y)\|$, 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 $$ C_{l}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{l}\frac{d(\mathbf{x},\mathbf{y})^2}{t}} \Lambda(\mathbf x,\mathbf y,t) \leq h_t(\mathbf{x},\mathbf{y}) \leq C_{u}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{u}\frac{d(\mathbf{x},\mathbf{y})^2}{t}} \Lambda(\mathbf x,\mathbf y,t), $$ where $\Lambda(\mathbf x,\mathbf y,t)$ can be expressed by means of some rational functions of $\| \mathbf x-\sigma(\mathbf y)\|/\sqrt{t}$. An exact formula for $\Lambda(\mathbf x,\mathbf y,t)$ is provided.