A note on the multilinear singular integral operators

Abstract

L p (R n ) boundedness is considered for the multilinear singular integral operator defined by $$T_A f(x) = \smallint _{\mathbb{R}^n } \frac{{\Omega (x - y)}}{{|x - y|^{n + 1} }}(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy$$ 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 BMO(R n ). We give a smoothness condition which is fairly weaker than that Ω∈Lip α(Sn−1) (0<α⪯1) and implies theL p (R n ) (1<p<∞) boundedness for the operator TA. Some endpoint estimates are also established.