On a multilinear singular integral operator

In this paper, we prove that the maximal operator $$T_{A_1 ,A_2 }^ \bullet f\left( x \right) = \sup _{\varepsilon > 0} \left| {\int_{\left| {x - y } \right| > \varepsilon } {\prod\limits_{j = 1}^2 {P_{m_j } \left( {A_j ;x,y} \right)\frac{{\Omega \left( {x - y} \right)}}{{\left| {x - y} \right|^{M + n} }}f\left( y \right)dy} } } \right|,n \geqslant 2$$ satisfies $$\left\| {T_{A_1 ,A_2 }^ \bullet f} \right\|_p \leqslant C\sum\limits_{\left| c \right|mm_1 } {\left\| {D^c A_1 } \right\|_{BMO} } \sum\limits_{\left| \beta \right| = m_2 } {\left\| {D^\beta A_2 } \right\|_{BMO} } \left\| f \right\|p$$ for all 1 $P_{m_1 } \left( {A_j ;x,y} \right) = A_j \left( x \right) - \sum\limits_{\left| m \right| \leqslant m_j } {\frac{1}{{a!}}D^a A} \left( y \right)\left( {x - y} \right)^2 \left( {j = 1,2} \right),M = m_1 + m_2 ;\Omega $ is homogeneous of degree 0, has vanishing moment up to order M and satisfies L^q-Dini condition for some q>1.