Lp maximal regularity for vector-valued Schrödinger operators

Abstract

In this paper we consider the vector-valued Schrödinger operator −Δ+V, where the potential term V is a matrix-valued function whose entries belong to Lloc1(Rd) and, for every x∈Rd, 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 L1(Rd,Rm). Assuming further that the minimal eigenvalue of V belongs to some reverse Hölder class of order q∈(1,∞)∪{∞}, we obtain maximal inequality in Lp(Rd,Rm), for p in between 1 and some q, and generation results.