多线性粗糙奇异积分算子与多线性乘子

多线性奇异积分算子理论以及相关的乘子理论是调和分析中的重要课题。受线性粗糙奇异积分算子理论的启发，人们开始关注多线性粗糙奇异积分算子的有界性。本项目将围绕多线性粗糙奇异积分算子和多线性乘子来发展相关的方法以更深入理解这些问题，具体问题包括：多线性粗糙奇异积分算子在各类空间上的有界性，多线性粗糙奇异积分算子的加权估计；不属于经典多线性Hörmander乘子的多线性乘子的有界性，多线性乘子的界、有界指标与乘子光滑性的数量关系。本项目申请人已开展了与该项目相关的部分前期工作，包括与合作者利用双线性乘子和小波分解证明双线性粗糙奇异积分算子的有界性。通过本项目的研究，我们可丰富多线性算子与乘子理论，并有可能推动偏微分方程中相关问题的解决。

Multilinear singular integral operators and multilinear multipliers are important topics in harmonic analysis. Inspired by the theory of linear rough singular integrals, many analysts turn their attention to multilinear rough singular integrals. This project aims to develop methods related to the study of multilinear rough singular integrals and multilinear multipliers, including the boundedness of multilinear rough singular integrals on various spaces, and the weighted estimates of multilinear roughs singular integrals; the boundedness of multilinear multipliers which are not contained in classical multilinear Hörmander category, and the quantitative relation between the bounds, indices of boundedness and smoothness of multilinear multipliers. The applicant has some work related to this project containing the boundedness of bilinear rough singular integrals proved by the applicant and his collaborators via wavelets and bilinear multipliers. We will obtain more results of multilinear singular integrals and multilinear multipliers with the hope of improving the study of related problems in PDE.