Integral operators with rough kernels in variable Lebesgue spaces

TL;DR

This paper investigates the boundedness of integral operators with rough kernels in variable Lebesgue spaces, extending classical results by weakening the conditions on the exponent functions.

Contribution

It establishes boundedness results for a class of integral operators with rough kernels in variable Lebesgue spaces under weaker conditions than traditional log-H"older assumptions.

Findings

Boundedness of operators from L^{p(·)} to L^{q(·)} spaces.

Operators with kernels satisfying size and Dini conditions are bounded.

Results extend classical boundedness to broader variable exponent settings.

Abstract

In this paper we study integral operators with kernels \begin{equation*} K(x,y)= k_1( x- A_1y)...k_m( x-A_my), \end{equation*} $k_i(x)=\frac{\Omega_i(x)}{|x|^{n/q_i}}$ where $\Omega_i: \mathbb{R}^n\to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, $A_{i}$ are certain invertible matrices, and $\frac n{q_1}+\dots\frac n{q_m}=n-\alpha,$ $0\leq \alpha <n.$ We obtain the boundedness of this operator from $L^{p(\cdot)}$ into $% L^{q(\cdot)}$ for $\frac{1}{q(\cdot)}=\frac{1}{p(\cdot)}-\frac{\alpha }{n},$ for certain exponent functions $p$ satisfying weaker conditions than the classical log-H\"older conditions.