Fourier积分算子的局部光滑性及其相关研究

Local smoothness of Fourier integral operators and related research

本文综述满足电影型(cinematic)曲率条件的Fourier积分算子的局部光滑性及其相关研究.电影型曲率条件包含非退化条件及曲率条件.作为范例重点讨论如何通过双线性方法建立变系数版本的平方函数不等式,进而改进了Mockenhaupt-Seeger-Sogge局部光滑性的结果.与此同时,本文还分析了解决局部光滑性猜想的困难、可能的途径,以及它与其他数学猜想之间的联系.

In this survey,we are devoted to reviewing local smoothing estimates and the related research of Fourier integral operators satisfying the cinematic curvature conditions,which contain non-degenerate conditions and curvature conditions.We focus on how to establish a variable coefficient version of the square function inequality through the bilinear method,and then improve the local smoothing estimates established by MockenhauptSeeger-Sogge.At the same time,the difficulty and the possible ways of solving the local smoothing conjecture,and connections with other well-known mathematical conjectures are also discussed.