On a multilinear singular integral operator

In this paper, we prove that the maximal operator $$T_{A_1 ,A_2 }^ bullet fleft( x right) = sup _{varepsilon > 0} left| {int_{left| {x - y } right| > varepsilon } {prodlimits_{j = 1}^2 {P_{m_j } left( {A_j ;x,y} right)frac{{Omega left( {x - y} right)}}{{left| {x - y} right|^{M + n} }}fleft( y right)dy} } } right|,n geqslant 2$$ satisfies $$left| {T_{A_1 ,A_2 }^ bullet f} right|_p leqslant Csumlimits_{left| c right|mm_1 } {left| {D^c A_1 } right|_{BMO} } sumlimits_{left| beta right| = m_2 } {left| {D^beta A_2 } right|_{BMO} } left| f right|p$$ for all 1 $P_{m_1 } left( {A_j ;x,y} right) = A_j left( x right) - sumlimits_{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.