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

Let L = − Δ H n + V be a Schrödinger operator on the Heisenberg group H n , where the nonnegative potential V belongs to the reverse Hölder class R H q 1 for some q 1 ≥ Q / 2 , and Q is the homogeneous dimension of H n . Let b belong to a new Campanato space Λ ν θ ( ρ ) , and let I β L be the fractional integral operator associated with L . In this paper, we study the boundedness of the commutators [ b , I β L ] with b ∈ Λ ν θ ( ρ ) on central generalized Morrey spaces L M p , φ α , V ( H n ) , generalized Morrey spaces M p , φ α , V ( H n ) , and vanishing generalized Morrey spaces V M p , φ α , V ( H n ) associated with Schrödinger operator, respectively. When b belongs to Λ ν θ ( ρ ) with θ > 0 , 0 < ν < 1 and ( φ 1 , φ 2 ) satisfies some conditions, we show that the commutator operator [ b , I β L ] is bounded from L M p , φ 1 α , V ( H n ) to L M q , φ 2 α , V ( H n ) , from M p , φ 1 α , V ( H n ) to M q , φ 2 α , V ( H n ) , and from V M p , φ 1 α , V ( H n ) to V M q , φ 2 α , V ( H n ) , 1 / p − 1 / q = ( β + ν ) / Q .