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The goal of this paper is to extend Nakai's generalized Morrey spaces to awider function class, the one-sided Muckenhoupt weighted case. Morrey matchingMuckenhoupt enables us to study the weak and strong type boundedness ofone-sided sublinear operators satisfying certain size conditions on theone-sided weighted Morrey spaces. We also establish one-sided Fefferman-Steininequalities on one-sided weighted Morrey spaces in this paper. Meanwhile, theboundedness and compactness of Riemann-Liouville integral operators on locallyone-sided weighted Morrey space are considered. As applications, we establishthe existence and uniqueness of solutions to a Cauchy type problem associatedwith fractional differential equations.

Morrey meets Muckenhoupt: A note on Nakai's generalized Morrey spaces and applications