Sublinear operators with rough kernel generated by fractional integrals and commutators on generalized vanishing local Morrey spaces

TL;DR

This paper investigates the boundedness of certain sublinear operators with rough kernels, generated by fractional integrals and commutators, on generalized Morrey spaces, including weak and vanishing variants, under broad size conditions.

Contribution

It establishes new norm inequalities for these operators on generalized local Morrey spaces and their vanishing counterparts, broadening the scope of harmonic analysis tools.

Findings

Operators satisfy norm inequalities on generalized Morrey spaces.

Marcinkiewicz operator meets the theorem conditions.

Results include weak and vanishing local Morrey space cases.

Abstract

In this paper, we consider the norm inequalities for sublinear operators with rough kernel generated by fractional integrals and commutators on generalized local Morrey spaces and on generalized vanishing local Morrey spaces including their weak versions under generic size conditions which are satisfied by most of the operators in harmonic analysis, respectively. As an example to the conditions of these theorems are satisfied, we can consider the Marcinkiewicz operator.