Integral operators with rough kernels in variable Lebesgue spaces

Abstract

We study integral operators with kernels

$$K(x,y)= k_{1}( x- A_1y) \cdots k_{m}( x-A_my),$$

$k_{i}(x)=\frac{\Omega_{i}(x)}{|x|^{n/q_i}}$ where $\Omega_{i} \colon \mathbb{R}^{n} \to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A_i are certain invertible matrices, and $\frac n{q_1}+\cdots+ \frac n{q_m} = n - \alpha, 0 \leq \alpha <n$. 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-Hölder conditions.