Commutators of higher order riesz transform associated with schrödinger operators

Let l = - δ + v be a schrödinger operator on r n (n ≥ 3),where v ≢ 0 is a nonnegative potential belonging to certain reverse hölder class b s for s ≥ n / 2. in this paper,we prove the boundedness of commutators r b h f = b r h f - r h (b f) generated by the higher order riesz transform r h = δ 2 (- δ + v) - 1,where b ∈ b m o θ (ρ),which is larger than the space b m o (r n). moreover,we prove that r b h is bounded from the hardy space h l 1 (r n) into weak l w e a k 1 (r n). © 2013 yu liu et al.