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

Abstract

We study the $L^{p},$ $1\leqslant p\leqslant \infty,$ boundedness for Riesz transforms of the form $V^{a}(-\frac{1}{2}Δ+V)^{-a},$ where $a>0$ and $V$ is a non-negative potential. We prove that $V^{a}(-\frac{1}{2}Δ+V)^{-a}$ 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.

Theorems & Definitions (52)

• Theorem 1: \ref{['thm:genLp']}
• Theorem 2: \ref{['thm:intL1']}
• Theorem 3
• Theorem 4
• Definition 2.1
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Proposition 2.3