Boundedness of High Order Commutators of Riesz Transforms Associated with Schrodinger Type Operators

Let L-2 = (-Delta)(2) + V-2 be the Schrodinger type operator, where V not equal 0 is a nonnegative potential and belongs to the reverse Holder class RHq1 for q(1) > n/2, n >= 5. The higher Riesz transform associated with L-2 is denoted by R = del L-2(2)-1/2 and its dual is denoted by R* = L-2(-1/2)del(2). In this paper, we consider the m-order commutators [b(m), R] and [b(m), R*], and establish the (L-p , L-q)-boundedness of these commutators when b belongs to the new Campanato space Lambda(theta)(beta)(rho) and 1/q = 1/p - m beta/n.