Commutator of fractional integral with Lipschitz functions associated with Schrödinger operator on local generalized Morrey spaces

Let L= − Δ + V be a Schrödinger operator on Rn, where n≥ 3 and the nonnegative potential V belongs to the reverse Hölder class RHq1 for some q1> n/ 2. Let b belong to a new Campanato space and IβL be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b,IβL] with b on local generalized Morrey spaces LMp,φα,V,{x0}, generalized Morrey spaces Mp,φα,V and vanishing generalized Morrey spaces VMp,φα,V associated with Schrödinger operator, respectively. When b belongs to with θ> 0 , 0 < ν< 1 and (φ1, φ2) satisfies some conditions, we show that the commutator operator [b,IβL] are bounded from LMp,V,{x0} to LMq,V,{x0}, from Mp,V to Mq,V and from VMp,V to VMq,V, 1 / p− 1 / q= (β+ ν) / n. © 2018, The Author(s).