On Dunkl Schr\"odinger semigroups with Green bounded potentials

TL;DR

This paper investigates the heat kernel bounds for Dunkl Schr"odinger operators with Green bounded potentials, establishing conditions under which the heat kernel of the associated semigroup is comparable to the Dunkl heat kernel.

Contribution

It provides a characterization of when the heat kernel of the Dunkl Schr"odinger semigroup satisfies lower bounds in terms of the potential's integrability condition.

Findings

Heat kernel lower bounds are established for Dunkl Schr"odinger semigroups.

A necessary and sufficient condition involving the potential V for these bounds is derived.

The results connect the potential's integrability with heat kernel estimates in Dunkl analysis.

Abstract

On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(\alpha) > 0$, and the associated measure $$ dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x, $$ we consider a Dunkl Schr\"odinger operator $L=-\Delta_k+V$, where $\Delta_k$ is the Dunkl Laplace operator and $V\in L^1_{\rm loc} (dw)$ is a non-negative potential. Let $h_t(\mathbf x,\mathbf y)$ and $k^{\{V\}}_t(\mathbf x,\mathbf y)$ denote the Dunkl heat kernel and the integral kernel of the semigroup generated by $-L$ respectively. We prove that $k^{\{V\}}_t(\mathbf x,\mathbf y)$ satisfies the following heat kernel lower bounds: there are constants $C, c>0$ such that $$ h_{ct}(\mathbf x,\mathbf y)\leq C k^{\{V\}}_t(\mathbf x,\mathbf y)$$ if and only if $$ \sup_{\mathbf x\in\mathbb R^N} \int_0^\infty \int_{\mathbb R^N} V(\mathbf y)w(B(\mathbf x,\sqrt{t}))^{-1}e^{-\|\mathbf x-\mathbf y\|^2/t}\, dw(\mathbf y)\, dt<\infty, $$ where $B(\mathbf x,\sqrt{t})$ stands for the Euclidean ball centered at $\mathbf x \in \mathbb{R}^N$ and radius $\sqrt{t}$.