Fractal Fourier Extension Estimates

分形傅里叶扩展估计

基本信息

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

项目摘要

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方面的优势并寻求新的应用。调和分析不仅在纯数学中发挥着重要作用，而且在应用数学、工程、物理和其他科学中也发挥着重要作用。调和分析的关键思想是将复杂函数表示为简单函数的和。最近，调和分析中的一些新方法被开发出来，从而解决了数学中一些长期悬而未决的问题。希望更深入地了解这些工具并将其应用到其他环境中。通过结合 Guth 的多项式划分方法和 Bourgain-Demeter 的解耦理论，主要研究者（与 Guth 和 Li 一起）证明了尖锐的薛定谔极大估计，这是加权傅立叶扩展估计的特例。作为一种应用，这解决了卡尔森大约 40 年前提出的二维薛定谔解几乎处处收敛的问题。这项工作的主要新颖之处在于使用尺度上的解耦和归纳来推导线性和双线性精化 Strichartz 估计。在最近与张合作的其他工作中，首席研究员获得了分形 L^2 估计，这解决了更高维度的卡尔森问题，并为 Falconer 距离集问题、分形测度的球形平均傅立叶衰减率、限制散度集的大小提供了新结果该项目的目标是通过利用上述工作中的想法以及开发更通用的新工具，在充分理解分形 L^p 估计方面取得进展环境。分析中还将适用于其他问题。该奖项反映了 NSF 的法定使命，并且通过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，被认为值得支持。