Oscillatory Integrals and Their Bounds in Lebesgue Spaces

DMS 0300416PI：G Mockenhaupt Inst：GA技术这项调和分析的研究项目涉及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，即有限域上对应于经典高斯和的指数和算子。这类算子的界限是理解n维傅里叶级数之和、傅里叶乘子、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)的压缩现象以及与之相关的极大函数不等式。离散情况在这里用作开发组合工具的模型，以更好地理解欧几里得环境。本课题主要研究用纯音调的叠加来逼近给定信号的问题。根据所采用的逼近方法，可以分析各种振荡积分算子。这些算符也作为数学物理中基本方程的解出现，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精确界密切相关。

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.