Semigroup and Riesz transform for the Dunkl- Schrödinger operators

Abstract

Let $L_k=-{\Delta }_k+V$ be the Dunk- Schrödinger operators, where ${\Delta }_k={\sum }_{j=1}^d{T}_j^2$ is the Dunkl Laplace operator associated to the dunkl operators $T_j$ on ${\mathbb{R}}^d$ and $V$ is a nonnegative potential function. In the first part of this paper we introduce the Riesz transform $R_j=T_j{L}_k^{-1/2}$ as an $L^2$- bounded operator and we prove that is of weak type $(1,1)$ and then is bounded on $L^p({\mathbb{R}}^d,d{\mu }_k(x))$ for $1<p\le 2$. The second pat is devoted to the $L^p$ smoothing of the semigroup generated by $L_k$, when $V$ belongs to the standard Koto class.