$L^p$ Maximal regularity for vector-valued Schrödinger operators

Abstract

In this paper we consider the vector-valued Schrödinger operator $-\Delta +V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L_{\mathrm{loc}}^1({\mathbb{R}}^d)$ 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 Hölder 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 $q$.