Marcinkiewicz integrals associated with Schrödinger operator and their commutators on vanishing generalized Morrey spaces

Let L= − Δ + V be a Schrödinger operator, where Δ is the Laplacian on Rn and the non-negative potential V belongs to the reverse Hölder class RHq for q≥ n/ 2. In this paper, we study the boundedness of the Marcinkiewicz integral operators μjL and their commutators [b,μjL] with b∈ BMOθ(ρ) on generalized Morrey spaces Mp,φα,V(Rn) associated with Schrödinger operator and vanishing generalized Morrey spaces VMp,φα,V(Rn) associated with Schrödinger operator. We find the sufficient conditions on the pair (φ1, φ2) which ensure the boundedness of the operators μjL from one vanishing generalized Morrey space VMp,V to another VMp,V, 1 < p< ∞ and from the space VM1,V to the weak space VWM1,V. When b belongs to BMOθ(ρ) and (φ1, φ2) satisfies some conditions, we also show that [b,μjL] is bounded from Mp,V to Mp,V and from VMp,V to VMp,V, 1 < p< ∞. © 2017, The Author(s).