Lp(·)–Lq(·) boundedness of some integral operators obtained by extrapolation techniques

Given a matrix A such that {A^{M}=I} and {0\leq\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\,)}\rightarrow L^{q(\,\cdot\,)}} boundedness, {\frac{1}{q(\,\cdot\,)}=\frac{1}{p(\,\cdot\,)}-\frac{\alpha}{n}}, and weak type estimates for integral operators of the formT_{\alpha}f(x)=\int\frac{f(y)}{|x-A_{1}y|^{\alpha_{1}}\cdots|x-A_{m}y|^{\alpha% _{m}}}\,dy,where {A_{1},\dots,A_{m}} are different powers of A such that {A_{i}-A_{j}} is invertible for {i\neq j}, {\alpha_{1}+\cdots+\alpha_{m}=n-\alpha}. We give some generalizations of these results.