Fourier Integral Operators and Maximal Functions in Harmonic Analysis

这个数学项目的重点是进一步发展谐波分析的现代技术。谐波分析已被证明是其他数学分支的宝贵工具，在物理学，自然科学，信号和数字图像处理以及工程中有应用。该项目旨在研究振荡积分和极大函数。这里考虑的振荡积分类包括各种积分变换，例如傅立叶变换，这是科学中的基本对象，因为它允许将信号分解为其基频。关于与一个对象族相关联的极大函数的定量陈述常常导致关于该对象族的极限行为的定性结论。特别是，极大函数在理解函数的可微性和支配物理和自然定律的偏微分方程的解的性质方面发挥着重要作用。该项目将通过指导本科生促进劳动力发展，并通过组织会议促进基础设施发展。研究者和合作者将从事调和分析中的几个相互关联的项目，这些项目涉及傅里叶积分算子（FIO）和极大函数的研究。主要有三个研究方向。第一个对应的应用程序的解耦不等式和局部平滑估计FIO对建立尖锐的LP-Sobolev边界的平均运营商的曲线和尖锐的Lp界相关的最大函数，与家庭的这种平均值。相关的问题，如变系数类似物和平均算子退化流形也将研究。本研究也将探讨一阶Sobolev空间中函数的极大函数的正则性。这与本项目的第二个主题有关，建立Hardy-Littlewood极大函数及其分数阶对应函数的端点Sobolev正则性。这里的技术是非傅立叶分析的。第三个也是最后一个研究方向是致力于研究半波传播算子和非退化傅里叶积分算子的Festhiman-Stein型双加权不等式，该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估，更广泛的影响审查标准。

This mathematical project focuses on further development of modern techniques in harmonic analysis. Harmonic analysis has proven to be an invaluable tool for other branches of mathematics, with applications in physics, natural sciences, signal and digital image processing, and engineering. The project aims to study oscillatory integrals and maximal functions. The class of oscillatory integrals considered here encompasses various integral transforms such as the Fourier transform, which is a basic object in science as it allows the decomposition of a signal into its fundamental frequencies. Quantitative statements about the maximal functions associated to a family of objects often lead to qualitative conclusions about the limiting behavior of that family. In particular, maximal functions play an important role in understanding differentiability properties of functions and properties of the solutions of the partial differential equations that govern the laws of physics and nature. This project will contribute to workforce development through mentoring of undergraduate students and to infrastructure development via conference organization. The investigator and collaborators will work on several interrelated projects in harmonic analysis that involve the study of Fourier integral operators (FIOs) and maximal functions. There are three main research directions. The first corresponds to the applications of decoupling inequalities and local smoothing estimates for FIOs towards establishing sharp Lp-Sobolev bounds of averaging operators over curves and sharp Lp bounds for related maximal functions, associated to families of such averages. Related questions such as variable coefficient analogues and averaging operators over degenerate manifolds will also be studied. The research will also investigate regularity properties of those maximal functions for functions in first order Sobolev spaces. This connects with the second theme of this project, establishing endpoint Sobolev regularity properties of the Hardy-Littlewood maximal function and its fractional counterpart. The techniques here are non-Fourier analytic. The third and final direction of research is devoted to the study of two-weighted inequalities of Fefferman-Stein type for the half-wave propagator and non-degenerate Fourier integral operators, in which the weights are related via a novel maximal function associated to the wave front set of the FIO.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.