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

TL;DR

This paper extends Nakai's generalized Morrey spaces to one-sided Muckenhoupt weighted cases, studying operator boundedness, inequalities, and applications to fractional differential equations.

Contribution

It introduces a broader class of Morrey spaces incorporating one-sided Muckenhoupt weights and analyzes their operator boundedness and applications to differential equations.

Findings

Established boundedness of one-sided sublinear operators on these spaces.

Proved one-sided Fefferman-Stein inequalities in the new setting.

Demonstrated existence and uniqueness of solutions to fractional differential equations.

Abstract

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.