Topics in Harmonic Analysis: Interplay between Time-Frequency Analysis, Additive Combinatorics and Partial Differential Equations

该研究项目的重点是数学中三个主要领域之间的相互作用：谐波分析，添加剂组合学和偏微分方程（PDE），其中谐波分析组件起主导作用。 该研究计划具有以下特点：（1）它被结构化为几类问题（有些将在下一段中详细介绍），其中一些主要研究者已经通过开发新的分析方法做出了相关贡献/进步;（2）它涵盖了广泛的技术，难度取决于主题的性质，也允许部分进展;（3）它有两个主要组成部分，即，一个纯理论的一个处理的问题，集中在理解各种具有高振荡内核的所谓的最大算子的行为和更应用的一面，流体动力学领域的一部分，研究两种流体界面的奇异性形成的问题。在后一个方向上的进步将有助于更好地理解我们周围的物理现实。该项目的其他方面是通过参加会议、讲座和研究生课程、与谐波分析和偏微分方程领域的其他研究人员/专家互动、培训研究生和博士后研究员来传播主要研究员的工作。这个项目研究了几个问题，定位自己的界面之间的谐波分析和添加剂组合，或谐波分析和流体动力学。报告的结构沿着几个主要主题，其中我们提到以下几个主题（第（2）、（3）和（4）项代表与合作者的联合项目）：（1）傅立叶级数在L^1附近的逐点收敛;（2）最大薛定谔算子、Kakeya极值和和积估计;（3）在没有涡量的情况下，为增加对“极限”水波情况的理解，二维两流体界面的奇点的形成;（4）旋转二维水波问题和具有非平凡涡量的三维Klein-Gordon方程的拟线性方程组，重点讨论了小初值条件下的存在时间。在第一个主题上，作者已经做出了贡献，开发了新的方法，或者改进了已知的结果（例如，逐点收敛沿着缺额连续的部分傅立叶和）或恢复最著名的结果源于著名的工作伦纳特Carleson。第一个和第二个主题的方法结合了联合收割机的时频分析和加法组合学技术。第三个主题调查，对于二维，两种流体的问题，如何改变的界面的几何形状的边界接近一个“飞溅”的情况下，从水波的情况;第四章探讨解的存在时间（相对于小的初始数据）的二维水波情况和三维Klein-Gordon系统的涡的存在的影响，流体力学文献中一个新的、迄今为止尚未探索的主题。

This research project focuses on the interplay between three major areas in mathematics: harmonic analysis, additive combinatorics, and partial differential equations (PDE), with the harmonic analysis component playing the dominant role. The research plan has the following characteristics: (1) it is structured into several classes of problems (some to be detailed in the next paragraph), on some of which the principal investigator has already made relevant contribution/advancements by developing new analysis methods; (2) it covers a wide range of techniques, with levels of difficulty varying depending on the nature of the subject that also allow partial progress; (3) it has two main components, namely, a purely theoretical one dealing with problems that focus on understanding the behavior of various so-called maximal operators with highly oscillatory kernels and a more applied side, part of the area of fluid dynamics, studying the problem of singularity formation for two-fluid interfaces. An advancement in this latter direction will contribute to a better understanding of the physical reality around us. Further aspects of the project are the dissemination of the principal investigator's work through conference participation, lectures, and graduate courses, interaction with other researchers/experts in the areas of harmonic analysis and PDE, the training of graduate students and postdoctoral fellows. This project investigates several problems that position themselves at the interface either between harmonic analysis and additive combinatorics, or between harmonic analysis and fluid dynamics. It is structured along several major themes, among which we mention the following (items (2), (3), and (4) represent joint projects with collaborators): (1) pointwise convergence of Fourier series near L^1; (2) the maximal Schrodinger operator, Kakeya extremizers, and sum-product estimates; (3) formation of singularities for the two-dimensional, two-fluid interface in the absence of vorticity in order to increase understanding of the "limiting" water-wave case; (4) the rotational two-dimensional water-wave problem and quasilinear systems of Klein--Gordon equations in three dimensions with nontrivial vorticity, with a focus on the time of existence for small initial data. On the first topic the author has already made contributions, developing new methods that either improved on known results (e.g., the pointwise convergence along lacunary subsequences of partial Fourier sums) or recovered the best known results stemming from the celebrated work of Lennart Carleson. The approaches for both the first and second topics combine techniques from time-frequency analysis and additive combinatorics. The third theme investigates, for the two-dimensional, two-fluid problem, how the geometry of the interface changes as the boundary approaches a "splash" scenario from the water-wave case; the fourth one explores how the time of existence for solutions (relative to small initial data) of the two-dimensional water-wave case and three dimensional Klein-Gordon systems are affected by the presence of vorticity, a new and hitherto mostly unexplored theme in the fluid dynamics literature.