Some estimates for commutators of Riesz transforms associated with Schrِdinger operators

We consider the Schrödinger operator L = − Δ + V on R n , where the nonnegative potential V belongs to the reverse Hölder class B q 1 for some q 1 ≥ n 2 . Let q 2 = 1 when q 1 ≥ n and 1 q 2 = 1 − 1 q 1 + 1 n when n 2 < q 1 < n . Set δ ′ = min { 1 , 2 − n q 1 } . Let H L p ( R n ) be the Hardy space related to the Schrödinger operator L for n n + δ ′ < p ≤ 1 . The commutator [ b , R ] is generated by a function b ∈ Λ ν θ , where Λ ν θ is a function space which is larger than the classical Companato space, and the Riesz transform R ≐ ∇ ( − Δ + V ) − 1 2 . We show that the commutator [ b , R ] is bounded from L p ( R n ) into L q ( R n ) for 1 < p < q 2 ′ , where 1 q = 1 p − ν n and bounded from H L p ( R n ) into L q ( R n ) for n n + ν < p ≤ 1 , where 1 q = 1 p − ν n . Moreover, we prove that the commutator [ b , R ] maps H L n n + ν ( R n ) continuously into weak L 1 ( R n ) . At last, we give a characterization for the boundedness of the commutator [ b , R ] in an extreme case.