In this paper, we consider a kind of multilinear operators related to oscillatory singular integrals with rough kernels andgive a criterion of certain boundedness for fthis kind of operators.

Let f Е Hp (R) or Hp (T) (0<p<1), again let Srj(..) and σ δRf(x) (δ=1/P-1) bethe Riesz means of f at the critical index on T anci R respectively. In this paper, by usr ofgrand maximal functions and the atom decomposition method, the fundamental estimares on approximation tor SδR f and σδRf on the respective real Hardy spaces Hp(T) ant Hp (R) ar estanloshed: ‖f-Sσδf‖H(P T)≤Cpω(f; 1/R) Hp(T), ‖f-Sσδf‖H(PR)≤Cpω(f; 1/R) Hp(R), where H (P, ∞) are Lorentz-Hardy spaces and ω(f; h)HP=sup ‖f(·+u)-f(·)‖HP.

This paper is devoted to the study on the LP -mapping prnperlies for a class of multilinear oscillatory singular integrals with polynomial phase and rough kernel By means of the method of block decomposition tot the kernel funcion. the authors show that for any non trivial polynomial phase, the Lp(Rn) boundedness of the muhilinear oseillatory singular integral operators and that of the corresponding local multilinear singular integral opterators are equivalent; and for any real-valued polynomial phase, the Lp(Rn) bonundedness of the multi linear oscillatory integral operators can be deduced from that of the cprresponding multilinear singular integral operators.