Fractal Fourier Extension Estimates

分形傅里叶扩展估计

• 批准号: 2107729
• 负责人: Xiumin Du
• 金额: $ 9.99万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2107729&HistoricalAwards=false
• 关键词: Fractal; Fourier; Extension; Estimates

The primary goal of this project is to explore the strength of some newly developed tools in harmonic analysis and seek new applications. Harmonic analysis plays an important role not only in pure mathematics but also in applied math, engineering, physics, and other sciences. The key idea of harmonic analysis is to represent complicated functions as sums of simple functions. Recently, a few new methods in harmonic analysis were developed and as a result, some long-standing open problems in mathematics were solved. It is desirable to get a deeper understanding of these tools and apply them in other settings.By combining the polynomial partitioning method of Guth and decoupling theory of Bourgain-Demeter, the principal investigator (together with Guth and Li) proved a sharp Schrodinger maximal estimate, which is a special case of weighted Fourier extension estimates. As an application, this solved the almost everywhere convergence problem of Schrodinger solutions in dimension two, which was raised by Carleson about 40 years ago. The main novelty in this work is the derivation of linear and bilinear refined Strichartz estimates using decoupling and induction on scales. In other recent work together with Zhang, the principal investigator obtained fractal L^2 estimates, which resolved Carleson's problem in higher dimensions and provided new results on Falconer's distance set problem, spherical average Fourier decay rates of fractal measures, bounding the size of divergence set of Schrodinger solutions, etc. The goal of this project is to make progress towards fully understanding fractal L^p estimates by exploiting ideas from the work mentioned above as well as developing new tools in a more general setting. There will be applications to other problems in analysis.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.

该项目的主要目标是探索一些新开发的工具在谐波分析中的实力，并寻求新的应用。谐波分析不仅在纯数学中，而且在应用数学，工程，物理和其他科学中都起着重要作用。谐波分析的关键思想是将复杂的功能表示为简单函数的总和。最近，开发了一些新的谐波分析方法，因此，解决了一些长期的数学开放问题。希望更深入地了解这些工具并将其应用于其他设置。通过结合Guth的多项式分区方法和Bourgain-Demeter的解耦理论，主要研究人员（与Guth and Li）证明了尖锐的Schrodinger Maximal估计值，这是加重傅立叶量估计的特殊情况。作为一种应用，这解决了施罗丁格解决方案的几乎所有地方的融合问题。这项工作的主要新颖性是使用尺度上的去耦和诱导的线性和双线性精制strichartz估算的推导。在最近与张一起的其他工作中，首席研究者获得了分形L^2估计，从而解决了Carleson在更高维度上的问题，并为Falconer的距离设置问题，球形平均傅立叶衰减速率提供了新的结果，从而依靠该项目的差异，从而使Schrodinger解决方案的大小等于^schrodinger解决方案的范围等。以及在更一般的环境中开发新工具。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准来评估值得支持的。

项目成果

An improved result for Falconer’s distance set problem in even dimensions

• DOI: 10.1007/s00208-021-02170-1
• 发表时间: 2020-06
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Xiumin Du;A. Iosevich;Yumeng Ou;Hong Wang;Ruixiang Zhang

Counterexamples to Lp collapsing estimate

Lp 崩溃估计的反例

• DOI: 10.1215/00192082-8886967
• 发表时间: 2021
• 期刊: Illinois journal of mathematics
• 影响因子: 0.6
• 作者: Du, Xiumin;Machedon, Matei
• 通讯作者: Machedon, Matei