Boundedness for Riesz transform associated with Schr?dinger operators and its commutator on weighted Morrey spaces related to certain nonnegative potentials

摘要

Let L = ? Δ + V be a Schr?dinger operator, where Δ is the Laplacian on R n and the nonnegative potential V belongs to the reverse H?lder class B q for q ≥ n / 2 . The Riesz transform associated with the operator L is denoted by T = ? ( ? Δ + V ) ? 1 2 and the dual Riesz transform is denoted by T ? = ( ? Δ + V ) ? 1 2 ? . In this paper, we establish the boundedness for the operator T ? and its commutator on the weighted Morrey spaces L α , V , ω p , λ ( R n ) related to certain nonnegative potentials belonging to the reverse H?lder class B q for n / 2 ≤ q < n , where p 0 ′ < p < ∞ and 1 p 0 = 1 q ? 1 n .