Abstract Weighted Morrey Spaces and Applications

We introduce the abstract weighted Morrey spaces Mp,uκ(X,M,μ), where μ is a general measure; investigate the properties of their predual spaces; and prove the boundedness of the Hardy–Littlewood maximal operator. Furthermore, we obtain an extrapolation theorem on Mp,uκ(X,M,μ), and consequently establish the norm inequalities for bounded oscillation (BO) operators on Mp,uκ(X,M,μ). As an application, we verify that BO operators include the maximal operators, Littlewood–Paley square operators, and Carleson operators on classical weighted Morrey spaces, as well as Calderón–Zygmund operators on weighted Morrey spaces of homogeneous type.