Let $L=-\Delta + V(x)$ be a Schr\"odinger operator on $\mathbb R^d$, where $V(x)\geq 0$, $V\in L^2_{ \rm loc} (\mathbb R^d)$. We give a short proof of dimension free $L^p(\mathbb R^d)$ estimates, $1<p\leq 2$, for the vector of the Riesz transforms $$\big(\frac{ \partial}{ \partial x_1}L^{ -1/2}, \frac{ \partial}{ \partial x_2}L^{ -1/2},\dots,\frac{ \partial}{ \partial x_d}L^{ -1/2}\Big).$$ 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 $\frac{ \partial}{ \partial x_j}L^{ -1/2}$.
Categories: math.FA