  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.
