首先我们将对多线性算子中的多线性奇异积分算子及其相关算子，如极大算子，Littlewood-Pale算子，振荡奇异积分等，以及双线性和三线性Hilbet变换的各种有界性问题进行深入研究。这些算子本身与方程有密切关系，而且此研究涉及到调和分析中的三个公开问题，即Tao提出的三线性Hilbert变换有界性问题和Christ提出的关于带非退化位相函数的多线性振荡积分的有界性问题，以及振荡积分的端点弱型估计问题。通过多线性算子这方面的研究，探索解决三个公开问题的适当途径。其次，我们将对与多线性算子相关的Fourier积分算子中某些低维流形上的算子的L^p范数不等式的极值函数进行研究，主要是想彻底解决申请人和Christ提出的关于Radon型变换的关键点刻画的猜想并研究沿曲线的线性卷积算子的极值函数，并将这些结果推进到多线性情形。最后我们将研究Convex Body 情形下的多线性算子的有界性问题。

First, we will study deeply on the multilinear singular integral operators and related operators, such as maximal singular integral operators，Littlewood-Paley operators and oscillatory integrals. Also,we will make an intensive study of the boundedness problems of bilinear or trilinear Hilbert transform. They are pretty much closely connected with PDE. Moreover, these research involves three open problems in Harmonic analysis, namely, the boundedness problem of trilinear Hilbert transform posed by T.Tao, the boundedness problem of multilinear oscillatory integral with nondegenerate phase functions raised by M. Christ, and the weak type estimate for multilinear oscillatory integrals. We want to explore the way and ideas to deal the three open problems by study multilinear operators we mentioned above. Secondly, we will study some basic operators belong to the type of operators of Fourier integral operators in lowest dimension, which treat questions concerning extremals for certain L^p norm inequalities, whose form is determined by the influence of curvature and singularities. We will consider less fine questions such as the existence of extremizers, precompactness of extremizing sequences, and qualitative andquantiative properties of extremizers. We will focus on solving the conjecture posed by Christ and Xue about the characterization of any critical points of Radon like transform and to study the extremizers of the other Fourier integral operators in cluding the linear and multilinear convolution type operators. Third, efforts will be made to deal with the boundedness of linear and multilinear operators associated with convex bodies.