On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators

Abstract

We study the $L^p,$ $1⩽p⩽\infty ,$ 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 $1<p⩽2$ whenever $a⩽1/p.$ 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.

33 pages, accepted for publication in Annales de l'Institut Fourier