On multilinear oscillatory singular integrals with rough kernels

AbstractIn this paper, for the multilinear oscillatory singular integral operators TA defined by TAf(x)=p.v.∫RneiP(x,y)Ω(x−y)|x−y|n+mRm+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 Λ̇β (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