## Abstract

Given a matrix *A* such that ${A}^{M}=I$ and $0\le \alpha <n$, for an exponent *p* satisfying $p(Ax)=p(x)$ for a.e. $x\in {\mathbb{R}}^{n}$, using extrapolation techniques, we obtain ${L}^{p(\cdot )}\to {L}^{q(\cdot )}$ boundedness, $\frac{1}{q(\cdot )}=\frac{1}{p(\cdot )}-\frac{\alpha}{n}$, and weak type estimates for integral operators of the form

${T}_{\alpha}f(x)=\int \frac{f(y)}{{|x-{A}_{1}y|}^{{\alpha}_{1}}\mathrm{\cdots}{|x-{A}_{m}y|}^{{\alpha}_{m}}}\mathit{d}y,$

where ${A}_{1},\mathrm{\dots},{A}_{m}$ are different powers of *A* such that ${A}_{i}-{A}_{j}$ is invertible for $i\ne j$, ${\alpha}_{1}+\mathrm{\cdots}+{\alpha}_{m}=n-\alpha $.
We give some generalizations of these results.

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