A LIPSCHITZ ESTIMATE FOR MULTILINEAR OSCILLATORY SINGULAR INTEGRALS WITH ROUGH KERNELS

In this paper, for the multilinear oscillatory singular integral operatorsTA1,A2,···,Ar defined byTA1,A2,···,Ar f(x) = p.v.RneiP(x,y) ?(x ? y)|x ? y|n+Mrs=1Rms+1(As; x, y)f(y)dy, n 2,where P (x, y) is a nontrivial and real-valued polynomial defined on Rn × Rn, ?(x) ishomogeneous of degree zero on Rn, As(x) has derivatives of order ms in Λ˙βs (0 < βs < 1),Rms+1(As; x, y) denotes the (ms + 1)-st remainder of the Taylor series of As at x expendedabout y (s = 1, 2, ··· , r), M =èrs=1 ms, the author proves that if 0 < β =èrs=1 βs < 1,and ? ∈ Lq(Sn?1) for some q > 1/(1 ? β), then for any p ∈ (1, ∞), and some appropriate0 < β < 1, TA1,A2,···,Ar is bounded on Lp(Rn).