Maximal inequalities and Riesz transforms for vector-valued magnetic Schrödinger operators

Abstract

We consider vector-valued magnetic Schrödinger operators $-\bm Δ_{\bm a}+V$ 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 $(\nabla - i\bm a)(-\bm Δ_{\bm a}+V)^{-\frac{1}{2}}$ and $V^α(-\bm Δ_{\bm a}+V)^{-α}$ on $L^p(\mathbb{R}^d;\mathbb{C}^m)$ for every $p \in (1,2]$ and every $α\in[0,1/p]$.