Oscillatory Integrals and Their Bounds in Lebesgue Spaces

勒贝格空间中的振荡积分及其界限

• 批准号: 0300416
• 负责人: Gerd Mockenhaupt
• 金额: $ 9.82万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-15 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300416&HistoricalAwards=false
• 关键词: Oscillatory; Integrals; Their; Bounds; Lebesgue

Abstract DMS 0300416PI: G MockenhauptInst: GA TechThis research project in Harmonic Analysis is concerned with boundedness properties of oscillatory integral operators in Lebesgue spaces as well as analogous problems in a discrete setting, i.e. exponential sum operators corresponding to classical Gaussian sums over finite fields. Bounds on these types of operators are fundamental in understanding summation of n-dimensional Fourier series, Fourier multipliers, regularity properties of solutions of partial differential equation and problems in integral geometry. Related to these problems are questions concerning compression phenomena of families of linear subspaces (Kakeya sets) and associated maximal function inequalities. The discrete case serves here as a model to develop combinatorial tools to get a better understanding in the Euclidean setting. The proposed project focuses on questions arising from the problem of approximating a given signal by a superposition of pure tones.Depending on the approximation method employed one is lead to analyse various oscillatory integral operators. These operators arise also as solutions of fundamental equations in mathematical physics such as e.g. the wave equation and the KdV equation. Regularity and existence problems of this equations are intimately related to sharp bounds on the corresponding oscillatory integral operators.

摘要 DMS 0300416 PI：G MockenhauptInst：GA Tech调和分析的这个研究项目涉及Lebesgue空间中振荡积分算子的有界性以及离散设置中的类似问题，即有限域上对应于经典高斯和的指数和算子。这些类型的运营商的边界是基本的了解n维傅立叶级数，傅立叶乘数，正则性的偏微分方程的解决方案和积分几何问题的总和。与这些问题有关的问题是关于压缩现象的家庭的线性子空间（Kakeya集）和相关的极大函数不等式。离散的情况下，在这里作为一个模型来开发组合工具，以获得更好的理解在欧几里德设置。 该项目的重点是从一个问题所产生的问题，近似一个给定的信号的叠加的纯tones.Dependently采用的近似方法之一是导致分析各种振荡积分算子。这些算子也作为数学物理中的基本方程的解出现，例如波动方程和KdV方程。这类方程的正则性和存在性问题与相应的振动积分算子的上界密切相关。