A generalization of the boundedness of certain integral operators in variable Lebesgue spaces

Abstract: Let $A_{1},...A_{m}$ be a $n\times n$ invertible matrices. Let $0 \leq \alpha<n$ and $0<\alpha_{i}<n$ such that $\alpha_1 + ... + \alpha_m = n- \alpha$. We define% \begin{equation*} T_{\alpha}f(x)=\int \frac{1}{\left\vert x-A_{1}y\right\vert ^{\alpha {1}}...\left\vert x-A{m}y\right\vert ^{\alpha {m}}}f(y)dy. \end{equation*}% In \cite{U-V} we obtained the boundedness of this operator from $L^{p(.)}(% \mathbb{R}^{n})$ into $L^{q(.)}(\mathbb{R}^{n})$ for $\frac{1}{q(.)}=\frac{1% }{p(.)}-\frac{\alpha }{n},$ in the case that $A{i}$ is a power of certain fixed matrix $A~\ $and for exponent functions $p$ satisfying log-Holder conditions and $p(Ay)=p(y),$ $y\in \mathbb{R}^{n}$ $.$ We will show now that the hypothesis on $p$, in certain cases, is necessary for the boundedness of $T_{\alpha}$ and we also prove the result for more general matrices $A_{i}.$ \footnote{Partially supported by CONICET and SECYTUNC} \footnote{Math. subject classification: 42B25, 42B35.} \footnote{Key words: Variable Exponents, Fractional Integrals.}