On dimension-free and potential-free estimates for Riesz transforms associated with Schrödinger operators

Abstract

Let $L=-\hat{I}”+V(x)$ be a Schrödinger operator on ${\mathbb{R}}^d$, where $V(x)\ge 0$, $V\in L_{\mathrm{loc}}^2({\mathbb{R}}^d)$. We give a short proof of dimension free $L^p({\mathbb{R}}^d)$ estimates, $1<p\le 2$, for the vector of the Riesz transforms $$(\frac{\partial }{\partial x_1}{L}^{-1/2},\frac{\partial }{\partial x_2}{L}^{-1/2},\dots ,\frac{\partial }{\partial x_d}{L}^{-1/2}).$$ The constant in the estimates does not depend on the potential $V$. We simultaneously provide a short proof of the weak type $(1,1)$ estimates for $\frac{\partial }{\partial x_j}{L}^{-1/2}$.