Boundedness of Commutators of Integral Operators of Fractional Type and $M_{\alpha , L^{r}\log L}$ Maximal Operator in Variable Lebesgue Spaces

Abstract

In this paper we study the commutators of integral operator T in variable Lebesgue spaces $L^{p(\cdot )}({\mathbb {R}}^n)$, with $p(\cdot )\in K_0({{\mathbb {R}}^n}) \cap N_{\infty }({{\mathbb {R}}^n})$, where T is the operator with kernel

$$\begin{aligned} K(x,y)=k_1(x-A_1y)\dots k_m(x-A_my), \end{aligned}$$

$A_1,\dots ,A_m$ are invertible matrices and each $k_i$ satisfies certain fractional size condition $S_{n-\alpha _i,\Psi _i}$, and certain fractional Hörmander condition $H_{n-\alpha _i,\Psi _i}$, with $\alpha _1+\dots +\alpha _m =n-\alpha$, $0\le \alpha < n$ and $\Psi _i$ are Young functions. We obtain the maximal operator $M_{\alpha ,L^r\log L^{\lambda }}$, with $1\le r< \frac{\alpha }{n}$ and $\lambda \ge 0$, and the commutators of T are bounded from $L^{p(\cdot )}({\mathbb {R}}^n)$ into $L^{q(\cdot )}({\mathbb {R}}^n)$, for $\frac{1}{q(\cdot )}=\frac{1}{p(\cdot )}-\frac{\alpha }{n}$ and certain $p(\cdot )$. Also, in the case $\alpha =0$ we obtain that the commutator of T satisfies a $L \log L$-type endpoint estimate.