On dimension-free and potential-free estimates for Riesz transforms associated with Schrödinger operators

TL;DR

This paper establishes dimension-free $L^p$ bounds for the vector of Schrödinger Riesz transforms $\\mathcal{R}_j = \partial_{x_j} L^{-1/2}$ with $L=-\\Delta+V$, $V\ge 0$, for $1<p\le 2$ in $\mathbb{R}^d$ ($d\ge 3$). The main technique is a factorization $\\mathcal{R}_j = R_j (-\\Delta)^{1/2} L^{-1/2}$ together with Stein’s dimension-free estimates for classical Riesz transforms and a short perturbation (Duhamel) argument to bound $(-\\Delta)^{1/2} L^{-1/2}$ on $L^p$, with a constant independent of $V$. The result yields the sharp bound $\\| (\\sum_{j=1}^d |\\mathcal{R}_j f|^2)^{1/2} \\|_{L^p} \le 2^{(2-p)/p} C_p \\|f\\|_{L^p}$ and a weak-type $(1,1)$ bound, without extra assumptions on $V$, while counterexamples show that the range $p>2$ or $p=1$ generally fails, underscoring the optimality of the result. The methodology also highlights potential extensions to related operators (e.g., Dunkl-Schrödinger-Riesz) under similar frameworks.

Abstract

Let $L=-Δ+ V(x)$ be a Schrödinger 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 $\frac{\partial}{\partial x_j}L^{-1/2}$.

Theorems & Definitions (10)

• Theorem 1.2
• Lemma 2.4
• proof
• Lemma 2.7
• proof
• Corollary 2.14
• proof : Proof of Theorem \ref{['teo']}
• Remark 2.16
• Remark 2.17
• Remark 2.19