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

TL;DR

This paper studies Dunkl- Schr"odinger operators, establishing the boundedness of the associated Riesz transform and analyzing the semigroup's smoothing properties for certain potentials.

Contribution

It introduces the Riesz transform for Dunkl- Schr"odinger operators as an $L^2$-bounded operator and proves its weak type and $L^p$ boundedness, along with semigroup smoothing results.

Findings

Riesz transform is of weak type (1,1)

Riesz transform is bounded on $L^p$ for $1<p extless=2$

Semigroup generated by $L_k$ exhibits $L^p$ smoothing for potentials in Kato class

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.