Oscillatory Integrals and Their Bounds in Lebesgue Spaces

本谐分析项目研究Lebesgue空间中振荡积分算子的有界性以及离散环境下的类似问题，即有限域上经典高斯和对应的指数和算子。这些算子的界是理解n维傅里叶级数和、傅里叶乘数、偏微分方程解的正则性和积分几何问题的基础。与这些问题相关的是线性子空间族（Kakeya集合）的压缩现象和相关的极大函数不等式问题。离散的情况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得设置。提议的项目集中于通过纯音叠加近似给定信号所产生的问题。根据所采用的近似方法，可以对各种振荡积分算子进行分析。这些算符也出现在数学物理中基本方程的解中，例如波动方程和KdV方程。该方程的正则性和存在性问题与相应的振荡积分算子的锐界密切相关。

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.