Semigroup and Riesz transform for the Dunkl- Schr\"odinger operators

Let $L_k=-{\mathrm{\Delta }}_k+V$ be the Dunk- Schr"{o}dinger operators, where ${\mathrm{\Delta }}_k={\sum }_{j=1}^d{T}_j^2$ 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 $R_j= T_j L_k^{-1/2}$asan$L^2$-boundedoperatorandweprovethatisofweaktype$(1,1)$andthenisboundedon$L^p($\mathbb{R}$^d,d\mu_k(x))$for$1<p\leq 2$. 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.