An Endpoint Estimate for the Commutators of Singular Integral Operators with Rough Kernels

Let Ω be homogeneous of degree zero and have mean value zero on the unit sphere Sd− 1, TΩ be the homogeneous singular integral operator with kernel Ω(x)|x|d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {\Omega (x)}{|x|^{d}}$\end{document} and TΩ,b be the commutator of TΩ with symbol b. In this paper, we prove that if Ω∈L(logL)2(Sd−1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Omega }\in L(\log L)^{2}(S^{d-1})$\end{document}, then for b∈BMO(ℝd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b\in \text {BMO}(\mathbb {R}^{d})$\end{document}, TΩ,b satisfies an endpoint estimate of LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L\log L$\end{document} type.