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

Abstract

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