Compactness of the commutators of homogeneous singular integral operators

Let TΩ be the singular integral operator with kernel \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\Omega (x)}} {{\left| x \right|^n }}$$\end{document}, where Ω is homogeneous of degree zero, integrable and has mean value zero on the unit sphere Sn−1. In this paper, by Fourier transform estimates, Littlewood-Paley theory and approximation, the authors prove that if Ω ∈ L(lnL)2(Sn−1), then the commutator generated by TΩ and CMO(ℝn) function, and the corresponding discrete maximal operator, are compact on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p \left( {\mathbb{R}^n ,\left| x \right|^{\gamma _p } } \right)$$\end{document} for p ∈ (1, ∞) and γp ∈ (−1, p − 1).