Boundedness of High Order Commutators of Riesz Transforms Associated with Schr(o)dinger Type Operators

Let L2 =(-Δ)2 + V2 be the Schr(o)dinger type operator,where V ≠ 0 is a nonnegative potential and belongs to the reverse H(o)lder class RHq1 for q1 ＞ n/2,n ≥5.The higher Riesz transform associated with L2 is denoted by R =▽72L2-1/2 and its dual is denoted by R* =L2-1/2▽2.In this paper,we consider the m-order commutators[bm,R]and[bm,R*],and establish the (Lp,Lq)-boundedness of these commutators when b belongs to the new Campanato space Δθβ(ρ) and 1/q =1/p-mβ/n.