A generalization of the boundedness of certain integral operators in variable Lebesgue spaces
Lucas Alejandro Vallejos, Marta Susana Urciuolo
Abstract
Let $A_{1},...A_{m}$ be a $n\times n$ invertible matrices. Let $0 \leq α<n$ and $0<α_{i}<n$ such that $α_1 + ... + α_m = n- α$. 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 $L^{p(.)}(% \mathbb{R}^{n})$ 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 $T_α$ 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.}