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

Let $L=-\mathrm{\Delta }+V(x)$ be a Schrödinger operator on ${\mathbb{R}}^d$, where $V(x)\ge 0$, $V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2({\mathbb{R}}^d)$. 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 $$(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}{L}^{-1\mathrm{/}2},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}{L}^{-1\mathrm{/}2},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_d}{L}^{-1\mathrm{/}2})\mathrm{.}$$ 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 $\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}{L}^{-1\mathrm{/}2}$.