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 $0\le \alpha <n$ and $0<{\alpha }_i<n$ such that ${\alpha }_1+...+{\alpha }_m=n-\alpha$. We define% \begin{equation*} T_αf(x)=\int \frac{1}{\left\vert x-A_{1}y\right\vert ^{α _{1}}...\left\vert x-A_{m}y\right\vert ^{α_{m}}}f(y)dy. \end{equation*}% In \cite{U-V} we obtained the boundedness of this operator from $L^{p(.)}($ into $L^{q(.)}({\mathbb{R}}^n)$ for $\frac{1}{q(.)}=\frac{1% }{p(.)}-\frac{α}{n},$ 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.}

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