CAREER: Weighted Fourier extension estimates and interactions with PDEs and geometric measure theory

职业：加权傅里叶扩展估计以及与偏微分方程和几何测度理论的相互作用

• 批准号: 2237349
• 负责人: Xiumin Du
• 金额: $ 49.84万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2237349&HistoricalAwards=false
• 关键词: CAREER; Weighted; Fourier; extension; estimates

Harmonic analysis is an important branch of mathematics. A key idea behind harmonic analysis is to express a general function or operator as a sum of simpler parts. Harmonic analysis has countless practical applications in signal processing, tomography, quantum mechanics, etc. It is also a powerful tool to study many theoretical aspects of mathematics. Fourier restriction theory is a subfield of harmonic analysis, which asks if one can meaningfully restrict the Fourier transform of a function onto a hypersurface, for example, a sphere. One then studies how small pieces of a function with different frequencies interfere with each other. Fourier restriction theory is a central topic in harmonic analysis and plays a fundamental role in certain problems in number theory, differential equations, and geometric measure theory. Thanks to the development of new ideas and techniques in restriction theory, several new state-of-the-art results in harmonic analysis and related fields have been established recently. However, a large portion of the new results are still not sharp, or are unknown in the general dimensions. This project will further develop these new ideas and techniques in restriction theory, and push forward the current best results for related questions, especially in the high-dimension case. The project’s multifaceted activities will include a new component for the existing Northwestern Emerging Scholars Program, work with the Math Alliance, involvement in the Chicago Symposium series, as well as the initiation of a summer program: Harmonic Analysis Reading and Research in Summer (HARRIS).More specifically, in the project weighted Fourier extension estimates (WFEE), and their variants and applications in partial differential equations and geometric measure theory, will be studied. One important case of WFEE is the boundedness of the Schrödinger maximal function. Such estimates are motivated by the recent proof of the almost everywhere convergence problem of Schrödinger solutions, a question which was raised by Carleson four decades ago. The full range of boundedness of the Schrödinger maximal function has been established when the spatial dimension is 1 or 2, but it remains open in higher dimensions. Another special case of WFEE that will be investigated is connected to a difficult problem in geometric measure theory: Falconer's distance set problem. The project will not only apply but also improve various tools and techniques from harmonic analysis, geometric measure theory, incidence geometry, combinatorics, and other related areas.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.

调和分析是数学的一个重要分支。调和分析背后的一个关键思想是将通用函数或算子表示为更简单部分的总和。调和分析在信号处理、断层扫描、量子力学等领域有着无数的实际应用。傅里叶限制理论是研究数学许多理论方面的一个强大工具，它是调和分析的一个子领域，它询问人们是否可以将函数的傅里叶变换有意义地限制在超曲面（例如球体）上。不同频率的函数傅里叶限制理论是调和分析的一个中心主题，并且在数论、微分方程和几何测度论的某些问题中发挥着基础作用，这得益于限制理论中新思想和技术的发展，一些新的理论。谐波分析和相关领域的最先进成果最近已经建立，但是，很大一部分新成果仍然不够清晰，或者在总体维度上是未知的。限制理论中的技术，并推动相关的当前最佳结果该项目的多方面活动将包括现有西北新兴学者计划的新组成部分、与数学联盟的合作、参与芝加哥研讨会系列以及启动夏季计划：夏季调和分析阅读和研究（HARRIS）。更具体地说，该项目将研究加权傅立叶扩展估计（WFEE）及其在偏微分方程和几何测度理论中的变体和应用。 WFEE 是薛定谔极大函数的有界性，这种估计是由最近证明薛定谔解的几乎处处收敛问题所激发的，这是卡尔森在四十年前提出的问题。当空间维度为 1 或 2 时已经成立，但在更高维度中它仍然是开放的，将要研究的 WFEE 的另一个特殊情况与几何测度论中的一个难题有关：该项目不仅适用于调和分析、几何测量理论、重合几何、组合学和其他相关领域的各种工具和技术，而且还改进了 Falconer 距离集问题。该奖项反映了 NSF 的法定使命，并被认为值得通过以下方式获得支持：使用基金会的智力价值和更广泛的影响审查标准进行评估。