Integral operators with rough kernels in variable Lebesgue spaces

In this paper we study integral operators with kernels \begin{equation*} K(x,y)= k_1( x- A_1y)...k_m( x-A_my), \end{equation*} $k_i(x)=\frac{\Omega_i(x)}{|x|^{n/q_i}}$ where $\Omega_i: \mathbb{R}^n\to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, $A_{i}$ are certain invertible matrices, and $\frac n{q_1}+\dots\frac n{q_m}=n-\alpha,$ $0\leq \alpha <n.$ We obtain the boundedness of this operator from $L^{p(\cdot)}$ into $$ for $\frac{1}{q(\cdot)}=\frac{1}{p(\cdot)}-\frac{\alpha }{n},$ for certain exponent functions $p$ satisfying weaker conditions than the classical log-H\"older conditions.