On Dunkl Schrödinger semigroups with Green bounded potentials

Abstract

On ${\mathbb{R}}^N$ equipped with a normalized root system $R$, a multiplicity function $k(\alpha )>0$, and the associated measure $$dw(\mathbf{x})=\prod _{\alpha \in R}|⟨\mathbf{x},\alpha ⟩{|}^{k(\alpha )}d\mathbf{x},$$ we consider a Dunkl Schrödinger operator $L=-{\Delta }_k+V$, where ${\Delta }_k$ is the Dunkl Laplace operator and $V\in L_{\mathrm{loc}}^1(dw)$ is a non-negative potential. Let $h_t(\mathbf{x},\mathbf{y})$ and $k_t^{\{V\}}(\mathbf{x},\mathbf{y})$ denote the Dunkl heat kernel and the integral kernel of the semigroup generated by $-L$ respectively. We prove that $k_t^{\{V\}}(\mathbf{x},\mathbf{y})$ satisfies the following heat kernel lower bounds: there are constants $C,c>0$ such that $$h_{ct}(\mathbf{x},\mathbf{y})\le C{k}_t^{\{V\}}(\mathbf{x},\mathbf{y})$$ if and only if $$\underset{\mathbf{x}\in {\mathbb{R}}^N}{\sup }{\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}V(\mathbf{y})w(B(\mathbf{x},\sqrt{t}){)}^{-1}{e}^{-\Vert \mathbf{x}-\mathbf{y}{\Vert }^2/t}dw(\mathbf{y})dt<\infty ,$$ where $B(\mathbf{x},\sqrt{t})$ stands for the Euclidean ball centered at $\mathbf{x}\in {\mathbb{R}}^N$ and radius $\sqrt{t}$.

22 pages