Variable Fofana's Spaces and their Pre-dual

TL;DR

This paper introduces variable Fofana's spaces, explores their properties, establishes their pre-dual, and applies these results to characterize fractional integral operators and commutators, extending classical results.

Contribution

It defines variable Fofana's spaces, proves their pre-dual, and applies these findings to fractional integral operators, providing new insights even for classical spaces.

Findings

Established the pre-dual of variable Fofana's spaces.

Characterized fractional integral operators on these spaces.

Extended classical results to variable exponent settings.

Abstract

In this paper, we introduce the variable Fofana's spaces $(L^{p(\cdot)},L^q)^\alpha (\mathbb{R}^n)$ where $1< p(\cdot)<\infty$ and $1\leq q,\alpha\leq\infty$, then show some properties and establish the pre-dual of those spaces which are contributed to prove the necessary conditions of fractional integral commutators' boundedness. As applications, the characterization of fractional integral operators and commutators on variable Fofana's spaces are discussed, which are new result even for the classical Fofana's spaces.