Mathematical Sciences: Oscillatory Integrals in Harmonic Analysis and Their Applications

小行星9401277 该奖项支持对谐波分析理论中出现的问题进行数学研究。 这项工作围绕着思想有关的振荡积分及其应用。 振荡积分是调和分析的重要组成部分。 有两种。 第一种方法是将一个函数作为具有非线性指数的广义傅里叶变换。 第二个是由一个积分算子与核携带振荡因子。 该项目涉及两种类型的研究。 第二类振荡积分的一个重要的类别是振荡奇异积分。 作为振荡积分和奇异积分的结合，这些算子与Fourier级数的收敛、Heisenberg群和其它幂零李群上的奇异调和分析以及奇异Radon变换等问题密切相关。 该项目涉及系统的研究振荡积分检查其有界性的性质时，适用于勒贝格空间，弱（1，1）-有界性，有界性的哈代空间，最佳衰减的二次范数和应用卷积与措施支持的子流形。 调和分析结合了这些数学元素，最好地体现了综合的思想。 一种是试图将复杂的问题分解为基本的组成部分。 然后分析这些组件的基本特性。 最后，通过组件的重组来重构解决方案。 傅立叶级数和傅立叶变换是在这种情况下使用的工具的例子;一个离散，另一个表示连续分解。最近的振荡积分理论增加了新的层面，一些更经典的方法，调和分析。 ***

9401277 Pan This award supports mathematical research on problems arising in the theory of harmonic analysis. The work revolves around ideas relating to oscillatory integrals and their applications. Oscillatory integrals are an essential part of harmonic analysis. There are two kinds. The first involves a single function acting as a generalized Fourier transform with nonlinear exponent. The second is given by an integral operator with a kernel carrying oscillatory factors. This project involves studies of both kinds. An important class of oscillatory integrals of the second kind is the class oscillatory singular integrals. As a combination of oscillatory integral and singular integral, these operators are intimately connected to problems concerning the convergence of Fourier series, singular harmonic analysis on the Heisenberg group and other nilpotent Lie groups and singular Radon transforms. The project involves a systematic study of oscillatory integrals to examine their boundedness properties when applied to Lebesgue spaces, weak (1,1)-boundedness, boundedness on Hardy spaces, optimal decay in the quadratic norm and applications to convolution with measures supported on submanifolds. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. More recently the theory of oscillatory integrals added new dimensions to some of the more classical approaches to harmonic analysis. ***