Spanne‐Type and Adams‐Type Estimates for Generalized Fractional Integral Operators on Bourgain–Morrey Spaces Over Spaces of Homogeneous Type, With Applications to Schrödinger Operators

ABSTRACT Let be an Ahlfors regular quasi‐metric measure space of dimension (a space of homogeneous type) equipped with a system of dyadic cubes in the sense of Christ and Hytönen–Kairema. We study the boundedness of the generalized fractional integral operator on the Bourgain–Morrey spaces of Hatano, Nogayama, Sawano, and Hakim and, more generally, on the Besov–Bourgain–Morrey scale of Zhao, Sawano, Tao, Yang and Yuan, which contains the Bourgain–Morrey spaces () and the Fofana spaces (). The kernel is controlled by power‐law envelopes with indices and satisfies a Zygmund‐type integral condition at infinity, which we show to be necessary for estimates with power‐type target exponents. Two principal results are proved. First, a Spanne‐type estimate: maps into , where and , for every and every ; the restriction cannot be removed. Second, an Adams‐type estimate, proved by the Hedberg method: maps into , where , whenever and , and the relation is the best that the Hedberg method can give. As applications we derive an Olsen‐type inequality on these spaces, a mapping property of on Bourgain–Morrey spaces for potentials in the scale‐invariant Morrey class of Fefferman and Phong, and an explicit form bound for Schrödinger operators with ; this space is shown to be contained in the Kato class.