Fractional integral related to Schrödinger operator on vanishing generalized mixed Morrey spaces.

With b belonging to a new B M O θ (ρ) space, L = − △ + V is a Schrödinger operator on R n with nonnegative potential V belonging to the reverse Hölder class R H n / 2 . The fractional integral operator associated with L is denoted by I β L . We investigate the boundedness of I β L and [ b , I β L ] , which are its commutators with b θ (ρ) on vanishing generalized mixed Morrey spaces V M p → , φ α , V related to Schrödinger operation and generalized mixed Morrey spaces M p → , φ α , V . The boundedness of the operator I β L is ensured by finding sufficient conditions on the pair (φ 1 , φ 2) , which goes from M p → , φ 1 α , V to M q → , φ 2 α , V , and from V M p → , φ 1 α , V to V M q → , φ 2 α , V , ∑ i = 1 n 1 p i − ∑ i = 1 n 1 q i = β . When b belongs to B M O θ (ρ) and (φ 1 , φ 2) satisfies some conditions, we also show that the commutator operator [ b , I β L ] is bounded from M p → , φ 1 α , V to M q → , φ 2 α , V and from V M p → , φ 1 α , V to V M q → , φ 2 α , V . [ABSTRACT FROM AUTHOR]