Morrey-type estimates for commutator of fractional integral associated with Schrödinger operators on the Heisenberg group

Let L=−ΔHn+V be a Schrödinger operator on the Heisenberg group Hn, where the nonnegative potential V belongs to the reverse Hölder class RHq1 for some q1≥ Q/ 2 , and Q is the homogeneous dimension of Hn. Let b belong to a new Campanato space Λνθ(ρ), and let IβL be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b,IβL] with b∈Λνθ(ρ) on central generalized Morrey spaces LMp,φα,V(Hn), generalized Morrey spaces Mp,φα,V(Hn), and vanishing generalized Morrey spaces VMp,φα,V(Hn) associated with Schrödinger operator, respectively. When b belongs to Λνθ(ρ) with θ> 0 , 0 < ν< 1 and (φ1, φ2) satisfies some conditions, we show that the commutator operator [b,IβL] is bounded from LMp,φ1α,V(Hn) to LMq,φ2α,V(Hn), from Mp,φ1α,V(Hn) to Mq,φ2α,V(Hn), and from VMp,φ1α,V(Hn) to VMq,φ2α,V(Hn), 1 / p− 1 / q= (β+ ν) / Q. © 2018, The Author(s).