$L^p$ Maximal regularity for vector-valued Schr\"{o}dinger operators

Abstract

In this paper we consider the vector-valued Schr\"{o}dinger operator $-\Delta + V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in $L^1(\mathbb{R}^d,\mathbb{R}^m).$ Assuming further that the minimal eigenvalue of $V$ belongs to some reverse H\"older class of order $q\in(1,\infty)\cup\{\infty\}$, we obtain maximal inequality in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $p$ in between $1$ and some $q$.