On $L^p$ estimates for positivity-preserving Riesz transforms related to Schr\"odinger operators

We study the $L^{p},$ $1\leqslant p\leqslant \infty,$ boundedness for Riesz transforms of the form $V^{a}(-\frac{1}{2}\Delta+V)^{-a},$ where $a>0$ and $V$ is a non-negative potential. We prove that $V^{a}(-\frac{1}{2}\Delta+V)^{-a}$ is bounded on $L^p(\mathbb{R}^d)$ with $1< p\leqslant 2$ whenever $a\leqslant 1/p.$ We demonstrate that the $L^{\infty}(\mathbb{R}^d)$ boundedness holds if $V$ satisfies an $a$-dependent integral condition that is resistant to small perturbations. Similar results with stronger assumptions on $V$ are also obtained on $L^{1}(\mathbb{R}^d).$ In particular our $L^{\infty}$ and $L^1$ results apply to non-negative potentials $V$ which globally have a power growth or an exponential growth. We also discuss a counterexample showing that the $L^{\infty}(\mathbb{R}^d)$ boundedness may fail.