Topics in Fourier Analysis

提案号：DMS-0200186PI: Andreas seegerabstract提议的研究将涉及傅里叶分析中的几个相互关联的主题。重点研究了具有退化正则关系的振荡积分算子和傅立叶积分算子的精确正则性。映射属性由标准关系的几何形状控制。在这里，我们特别感兴趣的是在研究与高维曲线相关的平均算子时出现的情况。其他项目包括奇异极大函数和粗糙奇异积分在可积函数类上的行为，波算子在幂零群上的映射性质，以及李群弱可调性失效的问题。本课题主要研究傅里叶分析中的一些基本问题，重点研究了各种振荡积分算子的估计。这种估计对于理解Radon变换和相关傅立叶积分算子的定性和定量行为尤其重要。这些运算符出现在数学的各个领域，如偏微分方程、复分析和积分几何，以及非数学应用，如断层扫描和医学成像。作为一个例子，我们提到x射线断层扫描涉及反转拉东变换算子；也就是说，人们试图从已知的函数的平均值来确定一个图像。所涉及的算子反演这个受限的x射线变换倾向于傅里叶积分算子在这个建议中考虑。

Proposal Number: DMS-0200186PI: Andreas SeegerABSTRACTThe proposed research will be concerned with severalinterrelated topics in Fourier analysis. In particularit is proposed to work on precise regularity propertiesof oscillatory integral and Fourier integral operatorswith degenerate canonical relations. The mappingproperties are governed by the geometry of the canonicalrelations. Here we are, in particular, interested inthe cases that come up when studying averaging operatorsassociated to curves in higher dimensions. Other projects include the behavior of singular maximalfunctions and rough singular integrals on classes ofintegrable functions, the mapping properties of waveoperators on nilpotent groups, and questions concerningthe failure of weak amenability of Lie groups.The research project is focused on some fundamentalquestion in Fourier analysis, with an emphasis on estimates for various oscillatory integral operators.Such estimates are in particular crucial forunderstanding the qualitative and quantitative behaviorof Radon transforms and related Fourier integraloperators. These operators arise in various areas ofmathematics, such as partial differential equations,complex analysis and integral geometry, as well as innon-mathematical applications such as tomography andmedical imaging. As an example we mention that X-raytomography involves inverting a Radon transformoperator; i.e. one seeks to determine an image fromknowledge about the averages of a function for aprescribed family of lines. The operators involved ininverting this restricted X-ray transform tend to beFourier integral operators as considered in thisproposal.