On functions of bounded mean oscillation with bounded negative part

Let b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b$$\end{document} be a locally integrable function and M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{M}$$\end{document} be the bilinear maximal function M(f,g)(x)=supQ is not an element of x1|Q|integral Q|f(y)g(2x-y)|dy.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{M}(f,g)(x)=\sup_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f(y)g(2x-y)|dy.$$\end{document} In this paper, characterization of the BMO function in terms of commutator Mb(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{M}<^>{(1)}_{b}$$\end{document} is established. Also, we obtain the necessary and sufficient conditions for the boundedness of the commutator [b,M]1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[b, \mathfrak{M}]_{1}$$\end{document}. Moreover, some new characterizations of Lipschitz and non-negative Lipschitz functions are obtained.