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arXiv · 1910.06245

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

Béchir Amri·Amel Hammi
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Abstract

Let $L_k=-\Delta_k+V$ be the Dunk- Schr\"{o}dinger operators, where $\Delta_k=\sum_{j=1}^dT_j^2$ 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 $1<p\leq 2$. 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.

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Béchir Amri, Amel Hammi. 2019-10-14. Semigroup and Riesz transform for the Dunkl- Schr\"odinger operators. https://arxiv.org/abs/1910.06245

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