Lp 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 $L_{\mathrm{loc}}^1\left(R^d\right)$ and, for every $x\in R^d$, $V\left(x\right)$ 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 $L^1(R^d,R^m)$. 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 $L^p(R^d,R^m)$, for p in between 1 and some q, and generation results.