Lp Boundedness for the Multilinear Singular Integral Operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p} (\mathbb{R}^{n} )$$\end{document} boundedness is considered for the maximal multilinear singular integral operator which is defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{*}_{A} f(x) = {\mathop {\sup }\limits_{ \in > 0} }{\left| {{\int_{|x - y| > \in } {\frac{{\Omega (x - y)}} {{|x - y|^{{n + 1}} }}} }(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy} \right|},$$\end{document} where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox{BMO}(\mathbb{R}^{n} ).$$\end{document} A regularity condition on Ω which implies the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p} (\mathbb{R}^{n} )$$\end{document} boundedness of T*A for any 1 < p < ∞ is given.