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

TL;DR

This paper extends the boundedness results of certain fractional integral operators in variable Lebesgue spaces to more general matrices, establishing necessary conditions on the exponent functions for boundedness.

Contribution

It generalizes previous boundedness results of integral operators to broader classes of matrices and clarifies the necessity of conditions on variable exponents.

Findings

Boundedness of operators for more general matrices $A_i$.

Necessity of certain conditions on variable exponents.

Extension of previous results to broader matrix classes.

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.}