Fractal Fourier Extension Estimates

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优势，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占有重要的地位。调和分析的关键思想是将复杂函数表示为简单函数的和。近年来，谐波分析中出现了一些新的方法，从而解决了一些长期存在的数学难题。更深入地了解这些工具并在其他环境中应用它们是可取的。将Guth的多项式分划方法与bourgin - demeter的解耦理论相结合，证明了一个尖锐的薛定谔极大估计，这是加权傅里叶扩展估计的一种特殊情况。作为一个应用，它解决了大约40年前Carleson提出的二维薛定谔解的几乎处处收敛问题。这项工作的主要新颖之处是利用尺度上的解耦和归纳推导线性和双线性精细Strichartz估计。在最近与Zhang的其他工作中，首席研究员获得了分形L^2估计，解决了更高维度的Carleson问题，并在Falconer距离集问题、分形测度的球面平均傅里叶衰减率、薛定谔解散度集的边界大小等方面提供了新的结果。该项目的目标是通过利用上述工作中的思想以及在更一般的环境中开发新工具，在充分理解分形L^p估计方面取得进展。分析中的其他问题也会有应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.