# Generalized fractional integral operators and their commutators with functions in generalized Campanato spaces on Orlicz spaces

## TL;DR

This paper studies the boundedness of commutators of generalized fractional integral operators with functions in generalized Campanato spaces on Orlicz spaces, establishing necessary and sufficient conditions for their boundedness.

## Contribution

It introduces a new framework for analyzing commutators on Orlicz spaces using generalized Young functions and fractional maximal operators.

## Key findings

- Established boundedness criteria for commutators on Orlicz spaces.
- Proved boundedness of generalized fractional maximal operators on Orlicz spaces.
- Provided a characterization linking Campanato functions and operator boundedness.

## Abstract

We investigate the commutators $[b,I_{\rho}]$ of generalized fractional integral operators $I_{\rho}$ with functions $b$ in generalized Campanato spaces and give a necessary and sufficient condition for the boundedness of the commutators on Orlicz spaces. To do this we define Orlicz spaces with generalized Young functions and prove the boundedness of generalized fractional maximal operators on the Orlicz spaces.