Fractional integral associated with Schrö dinger operator on vanishing generalized Morrey spaces

Abstract

Let L = -?+V be a Schrödinger operator, where the non-negative potential V belongs to the reverse Hölder class RH n/2 , let b belong to a new BMO ? (?) space, and let I L ß be the fractional integral operator associated with L. In this paper, we study the boundedness of the operator IL ß and its commutators [b,I L ß ] with b ? BMO ? (?) on generalized Morrey spaces associated with Schrödinger operator M ?,V p,? and vanishing generalized Morrey spaces associated with Schrödinger operator VM?,V p,? . We find the sufficient conditions on the pair (? 1 ,? 2 ) which ensures the boundedness of the operator I L ß from M?,V p,? 1 to M ?,V q,?2 and from VM ?,V p,?1 to VM ?,V q,?2 , 1/p-1/q =ß /n. When b belongs to BMO ? (?) and (? 1 ,? 2 ) satisfies some conditions, we also show that the commutator operator [b,I L ß ] is bounded from M ?,V p,?1 to M ?,V q,?2 and from VM ?,V p,?1 to VM ?,V q,?2 , 1/p-1/q =ß /n. © 2018, Element, Zagreb.