  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 $%
L^{q(\cdot)}$ 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.
