Fourier Integral Operators and Maximal Functions in Harmonic Analysis

这项数学项目着眼于现代谐波分析技术的进一步发展。谐波分析已被证明是其他数学分支的宝贵工具，在物理、自然科学、信号和数字图像处理以及工程中都有应用。该项目旨在研究振荡积分和极大函数。这里讨论的振荡积分包括各种积分变换，例如傅里叶变换，这是科学中的一个基本目标，因为它允许将信号分解为其基频。关于与一族对象相关的最大函数的定量陈述通常会导致关于该族的限制行为的定性结论。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解的性质方面起着重要的作用。该项目将通过指导本科生促进劳动力发展，并通过会议组织促进基础设施发展。研究人员和合作者将致力于调和分析中几个相互关联的项目，这些项目涉及傅立叶积分算子(FIOS)和极大函数的研究。主要研究方向有三个。第一个对应于FIOS的解耦不等式和局部光滑估计在建立曲线上平均算子的Lp-Sobolev界和相关极大函数的Lp界方面的应用，这些Lp-Sobolev界与这些平均族相关。相关的问题，如变系数相似和退化流形上的平均算子也将被研究。本文还将研究一阶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，即建立Hardy-Littlewood极大函数及其分数次对应函数的端点Soblev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致力于研究半波传播子和非退化傅立叶积分算子的Fefferman-Stein型两个加权不等式，其中权重通过与FIO波前集相关联的新的最大函数相关。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.