Integral operators with rough kernels in variable Lebesgue spaces

We study integral operators with kernels K(x,y)=k1(x-A1y)⋯km(x-Amy),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K(x,y)= k_{1}( x- A_1y) \cdots k_{m}( x-A_my),$$\end{document}ki(x)=Ωi(x)|x|n/qi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{i}(x)=\frac{\Omega_{i}(x)}{|x|^{n/q_i}}$$\end{document} where Ωi:Rn→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega_{i} \colon \mathbb{R}^{n} \to \mathbb{R}$$\end{document} are homogeneous functions of degree zero, satisfying a size and a Dini condition, Ai are certain invertible matrices, and nq1+⋯+nqm=n-α,0≤α<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac n{q_1}+\cdots+ \frac n{q_m} = n - \alpha, 0 \leq \alpha <n$$\end{document}. We obtain the boundedness of this operator from Lp(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p(\cdot)}$$\end{document} into Lq(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{q(\cdot)}$$\end{document} for 1q(·)=1p(·)-αn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)} - \frac{\alpha}{n}$$\end{document}, for certain exponent functions p satisfying weaker conditions than the classical log-Hölder conditions.