On dimension-free and potential-free estimates for Riesz transforms associated with Schr\"odinger operators

TL;DR

This paper presents a concise proof of dimension-free $L^p$ bounds for Riesz transforms associated with Schrödinger operators, with constants independent of the potential, and also establishes weak type (1,1) estimates.

Contribution

It provides new, simplified proofs of dimension-free $L^p$ estimates and weak type (1,1) bounds for Riesz transforms linked to Schrödinger operators, independent of the potential.

Findings

Dimension-free $L^p$ estimates for $1<p extless=2$

Weak type $(1,1)$ estimates for Riesz transforms

Constants do not depend on the potential $V$

Abstract

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 $\frac{\partial}{\partial x_j}L^{-1/2}$.