Fourier multipliers, square functions and incidence theory.

傅里叶乘数、平方函数和关联理论。

• 批准号: 2444701
• 金额: --
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2444701
• 关键词: Fourier; multipliers; square; functions; incidence

Proposed research proposal*:The goal of this project is to apply a number of newly developed techniques in harmonic analysis to the studyof Fourier multipliers. In a recent breakthrough, Guth--Wang--Zhang resolved the longstanding L^4 cone squarefunction conjecture and local smoothing conjecture for the wave equation in 2 spatial dimensions. One difficultyin proving estimates for the cone square function is that the setup is not directly amenable to the Lorentzrescaling, which features prominently in powerful induction-on-scale techniques which pervade modernharmonic analysis. To circumvent this issue, a new kind of estimate was introduced which subsumes thesquare function as a special case. This general framework is more robust and, in particular, is stable underLorentz transformation. There are many other examples of problems in which the scaling structure breaks downin a similar fashion, and it would be interesting to attempt to apply these methods, for instance, in the study ofBochner--Reisz-type multipliers associated to curves. In another direction, it is known that the local smoothingconjecture is closely related to the radial multiplier conjecture, which aims to characterise the L^p boundednessof radial Fourier multipliers in terms of the finiteness of the L^p norm of the associated kernel. It is natural toinvestigate whether advances in the understanding of local smoothing yield any new insights into radialmultipliers. These questions are also related to the incidence geometry of circular annuli, as expounded inworks of Heo--Nazarov--Seeger and Cladek. Recent advances in incidence geometry, such as the developmentof the polynomial partitioning method, may have some relevance to the investigation.

拟议的研究建议*：该项目的目的是将许多新开发的技术应用于谐波分析中，并将其用于傅立叶乘数研究。在最近的突破中，Guth-Wang-Zhang解决了长期以来的l^4锥形方形的猜想和局部平滑猜想，以在2个空间维度中为波方程式进行了猜想。锥形方形功能的一个难度证明估计值是，该设置不直接适合LorentzRessing，该设置以强大的启动式在尺度技术中突出具有遍布现代性分析的功能。为了避免此问题，引入了一种新型的估计，该估计值将其作为特殊情况。该一般框架更强大，尤其是在氯伦茨转化下稳定的。还有许多其他问题的例子，其中缩放结构在类似的方式中分解了，并且尝试应用这些方法，例如，在Bochner的研究 - reisz-type乘数​​与曲线相关的研究中。在另一个方向上，众所周知，局部平滑概念与径向乘数猜想密切相关，该猜想的目的是根据相关核的L^p norm的有限性来表征径向傅立叶乘数的l^p界面。自然要对理解局部平滑的进步是否产生任何新见解，这是自然而然的。这些问题也与圆形环体的发生率几何相关，如heo-nazarov-seeger and cladek所述。发病率几何形状的最新进展，例如多项式分配方法的开发，可能与研究有一定的意义。