Integral operators with rough kernels in variable Lebesgue spaces

Abstract

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 $\frac{n}{q_1}+\dots \frac{n}{q_m}=n-\alpha ,$ $0\le \alpha <n.$ 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-Hölder conditions.