Maximal inequalities and Riesz transforms for vector-valued magnetic Schr\"odinger operators

TL;DR

This paper establishes maximal inequalities and boundedness of Riesz transforms for vector-valued magnetic Schrödinger operators with potentials in $L^1_{loc}$, extending analysis in vector-valued $L^p$ spaces.

Contribution

It proves new maximal inequalities and Riesz transform bounds for vector-valued magnetic Schrödinger operators with less regular potentials.

Findings

Maximal inequalities hold in $L^p$ for $p o 1$

Riesz transforms are bounded on $L^p$ for $p o 2$

Results extend to matrix-valued potentials in $L^1_{loc}$

Abstract

We consider vector-valued magnetic Schr\"odinger operators $-\bm \Delta_{\bm a}+V$ with magnetic potential $\bm a \in L^2_{\mathrm{loc}}(\mathbb{R}^d;\mathbb{R}^d)$ and electric potential $V$ given by a matrix-valued function whose entries belong to $L^1_{\mathrm{loc}}(\mathbb{R}^d)$. We prove maximal inequalities in $L^p(\mathbb{R}^d;\mathbb{C}^m)$, $p\in[1,\infty)$ and the boundedness of the Riesz transforms $(\nabla - i\bm a)(-\bm \Delta_{\bm a}+V)^{-\frac{1}{2}}$ and $V^{\alpha}(-\bm \Delta_{\bm a}+V)^{-\alpha}$ on $L^p(\mathbb{R}^d;\mathbb{C}^m)$ for every $p \in (1,2]$ and every $\alpha\in[0,1/p]$.