\(L^{p(\cdot)}-L^{q(\cdot)}\) boundedness of some integral operators obtained by extrapolation techniques (Q2200978)

This paper is concerned with the boundedness of a family of ``off-diagonal'' operators on the variable Lebesgue spaces that generalize the classical Riesz potential. More precisely, the authors study operators of the form \[ T_{\alpha}f(x) = \int_{\mathbb{R}^{n}} |x - A_{1} y|^{-\alpha_1} \cdot \cdot \cdot |x - A_{m} y|^{-\alpha_m} f(y) dy, \] where \(0 \leq \alpha < n\), \(\alpha_1 + \cdot \cdot \cdot + \alpha_m = n -\alpha\) (\(m >1\)), \(A_1, . . . , A_m\) are different powers of a matrix \(A\) such that \(A^{M}= I\), \(A_i - A_j\) is invertible for \(i \neq j\), and prove (using extrapolation techniques) that this family of operators is bounded from \(L^{p(\cdot)}\) into \(L^{q(\cdot)}\) for \(\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)} - \frac{\alpha}{n}\). They also obtain weak estimates for this kind of operators.