On multilinear oscillatory singular integrals with rough kernels

In this paper, for the multilinear oscillatory singular integral operators TA defined by TAf (x) = p.v. Rn eiP (x,y) Ω(x − y) |x −y|n+m Rm+1(A;x, y)f (y)dy, n 2, where P(x, y) is a nontrivial and real-valued polynomial defined on Rn×Rn, Ω(x) is homogeneous of degree zero on Rn, A(x) has derivatives of order m in Λ˙β (0 < β <1), Rm+1(A;x, y) denotes the (m + 1)th remainder of the Taylor series of A at x expended about y, the author proves that if Ω ∈ Lq(Sn−1) for some q > 1/(1 −β), then for any p ∈ (1,∞), TA is bounded on Lp(Rn). Meanwhile, the weighted Lp-boundedness of TA is also given.  2004 Elsevier Inc. All rights reserved.