A LIPSCHITZ ESTIMATE FOR MULTILINEAR OSCILLATORY SINGULAR INTEGRALS WITH ROUGH KERNELS

In this paper, for the multilinear oscillatory singular integral operators TA1,A2,..,Ar defined by (x) where P(x,y) is a nontrivial and real-valued polynomial defined on Rn × Rn, Ω(x) is homogeneous of degree zero on Rn, As(x) has derivatives of order ms in ∧βs (0 ＜βs ＜1),Rms 1 (As; x, y) denotes the (ms 1)-st remainder of the Taylor series of As at x expended about y(s=1,2,…,r),M=∑rs=1ms, the author proves that if 0<β=∑rs=1βs<1, and Ω∈ Lq(Sn-1) for some q ＞ 1/(1 - β), then for any p ∈ (1, ∞), and some appropriate 0 ＜ β ＜ 1, TA1,A2,…,Ar is bounded on Lp(Rn).