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