Variable Fofana's Spaces and their Pre-dual

In this paper, we introduce the variable Fofana’s spaces $(L^{p(\cdot )},L^q{)}^{\alpha }({\mathbb{R}}^n)$ where $1<p(\cdot )<\mathrm{\infty }$ and $1\le q,\alpha \le \mathrm{\infty }$, 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.