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

Abstract

Let L2 = (−∆)2 + V2 be the Schrödinger type operator, where V 6= 0 is a nonnegative potential and belongs to the reverse Hölder class RHq1 for q1 > n/2, n ≥ 5. The higher Riesz transform associated with L2 is denoted by R = ∇2L − 2 2 and its dual is denoted by R∗ = L 1 2 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 Λβ(ρ) and 1/q = 1/p−mβ/n.