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

TL;DR

This paper investigates the boundedness of certain Riesz transforms associated with Schrödinger operators on various L^p spaces, establishing conditions under which these operators are bounded, including for non-negative potentials with specific growth behaviors.

Contribution

It provides new L^p boundedness results for Riesz transforms related to Schrödinger operators, including for the endpoint cases and potentials with power or exponential growth.

Findings

Boundedness on L^p for 1<p≤2 when a≤1/p.

L^∞ boundedness under an integral condition on V.

Counterexample showing possible failure of L^∞ boundedness.

Abstract

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.