Topics in Fourier Analysis

提案编号：DMS-0200186 PI：Andreas Seeger摘要所提出的研究将涉及傅立叶分析中的几个相互关联的主题。特别地，本文提出研究具有退化正则关系的振荡积分算子和Fourier积分算子的精确正则性。映射属性由正则关系的几何形状决定。在这里，我们特别感兴趣的情况下，出现在研究平均算子与曲线在更高的维度。 其他研究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性质，幂零群上波算子的映射性质，李群弱顺从性失效问题等。重点是各种振荡积分算子的估计。这些估计对于理解定性和定量的行为的Radon变换和相关的Fourier积分算子。这些算子出现在各种数学领域，如偏微分方程、复分析和积分几何，以及非数学应用，如断层摄影和医学成像。作为一个例子，我们提到，X射线断层扫描涉及到反转氡transformoperator;即一个试图确定一个图像从知识的平均值的函数为aprescribed家庭的线。在反演这个限制的X射线变换涉及的运营商往往是傅立叶积分运营商在这个建议中考虑。

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.