美国科学院院士、Wolf奖得主Stein把振荡积分算子总结为调和分析中最重要的三类算子之一。事实上，Fourier变换和Bochner-Riesz平均都是经典的振荡积分算子。Stein和Phong研究了一类带有紧支集光滑函数为核的振荡积分算子，得到了这类算子的L2范的最佳衰减估计,回答了Wolf奖得主 Arnold 关于最佳衰减数由相位函数的牛顿多面体得到的一个猜想。受Stein等人关于振荡积分研究的启发，本项目将研究下面几个问题。(1)研究一维空间上带多项式相位的振荡积分算子的Lp范衰减估计及刻画p的最大范围；(2) 探讨带实解析函数相位的振荡积分算子的Lp范最佳衰减估计；（3）关于振荡积分算子的一致性估计；(4) 广义Fourier变换的限制性定理；(5) 多线性振荡积分；(6) 高维空间上的振荡积分算子。对这些问题的研究将扩展调和分析的研究方法和范围，极有可能带来方法上的创新。

E.M.Stein who won Wolf Prize attributed the oscillatory integral operator to one of the most important three operators in Harmonic Analysis. Actually,the Fourier transform and Bochner-Riesz means are two classical oscillatory integral operators. Stein and Phong investigated a class of oscillatory integral operators with the kernel being smooth function and compact support, and obtained that the sharp L^2 decay estimates of the operator. This work answered the an important conjecture which was put by the distinguished mathematician Arnold.That is, the sharp decay estimate is determinated by Newton polyhedron resulted from the phase function of the oscillatory integral. Motivated by the research results obtained by Stein, we will raise the following questions in the project. (1) Sharp L^p estimates of oscillatory integral operators with polynomial phases and the characterizations of the maximum range of p. (2) Sharp L^p estimates of oscillatory integral operators with real analytic phases and the characterizations of the maximum range of p.(3) On the uniform estimates of the oscillatory integral.(4) The restriction theorem of the generalized Fourier transform.(5) On the multilinear oscillatory integral.(6) The oscillatory integral operators on high-dimension space. The study of those questions can extend the range of research in Harmonic analysis and is possible to give some innovative ideas.