Fractional İntegral Related to Schrödinger Operator on Vanishing Generalized Mixed Morrey Spaces

Abstract

With b belonging to a new BMOθ(ρ) space, L=−△+V is a Schrödinger operator on Rn with nonnegative potential V belonging to the reverse Hölder class RHn/2. The fractional integral operator associated with L is denoted by IβL. We investigate the boundedness of IβL and [b,IβL], which are its commutators with bθ(ρ) on vanishing generalized mixed Morrey spaces VMp→,φα,V related to Schrödinger operation and generalized mixed Morrey spaces Mp→,φα,V. The boundedness of the operator IβL is ensured by finding sufficient conditions on the pair (φ1,φ2), which goes from Mp→,φα,V to Mq→,φα,V, and from VMp→,φα,V to VMq→,φα,V, ∑i=1n1pi−∑i=1n1qi=β. When b belongs to BMOθ(ρ) and (φ1,φ2) satisfies some conditions, we also show that the commutator operator [b,IβL] is bounded from Mp→,φα,V to Mq→,φα,V and from VMp→,φα,V to VMq→,φα,V.