An endpoint estimate for maximal multilinear singular integral operators

Abstract

A weak type endpoint estimate for the maximal multilinear singular integral operator

$$T_A^* f(x) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int_{\left| {x - y} \right| > \varepsilon } {\frac{{\Omega (x - y)}}{{\left| {x - y} \right|^{n + 1} }}(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy} } \right|$$

is established, where Θ is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and A has derivatives of order one in BMO(ℝⁿ). A regularity condition on Θ which implies an LlogL type estimate of T ^*_A is given.