Fractional integral associated to Schrodinger operator on the Heisenberg groups in central generalized Morrey spaces

Let L = −∆Hn + V be a Schrodinger operator on the Heisenberg groups Hn, where the non-negative potential V belongs to the reverse Holder class RHQ/2 and Q is the homogeneous dimension of Hn. Let b belong to a new BMOθ(Hn, ρ) space, and let IL be the fractional integral operator associated with L. In this paper, we study the boundedness of the operator IL β and its commutators [b, IL ] with b ∈ BMOθ(Hn, ρ) on central generalized Morrey spaces LMα,V (H β ) and generalized Morrey β p,ϕ n spaces Mα,V (H ) associated with Schrodinger operator. We find the sufficient conditions on the pair (ϕ , ϕ ) which ensures p,ϕ n 1 2 the boundedness of the operator IL from LMα,V (H ) to LMα,V (H ) and from Mα,V (H ) to Mα,V (H ), 1/p − 1/q = β/Q. β p,ϕ1 n q,ϕ2 n p,ϕ1 n q,ϕ2 n When b belongs to BMOθ(Hn, ρ) and (ϕ1, ϕ2) satisfies some conditions, we also show that the commutator operator [b, IL ] is bounded from LMα,V (Hn ) to LMα,V (Hn ) and from Mα,V to Mα,V , 1/p − 1/q = β/Q.