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

Abstract

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