Boundedness of high order commutators of Marcinkiewicz integrals associated with Schrodinger operators

Suppose L=−Δ+VL=−Δ+V is a Schrödinger operator on RnRn, where n ≥ 3 and the nonnegative potential V belongs to reverse Holder class RHn. Let b belong to a new Campanato space Λ θ β (ρ), and let μLj be the Marcinkiewicz integrals associated with L. In this paper, we establish the boundedness of the m-order commutators [b^m,μLj] from L^p(R^n)L^p(R^n) to L^q(R^n),L^q(R^n) where 1/q = 1/p−mβ/n and 1 p n/(mβ)