New developments in geometric Fourier analysis

几何傅里叶分析的新进展

• 批准号: 2620030
• 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: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2620030
• 关键词: New; developments; geometric; Fourier; analysis

This project aims to harness the power of a variety of newly available tools in harmonic analysis to study classical objects such as geometric maximal functions. One possible direction is to study variants of Bougain's circular maximal function. This operator acts on functions on the Euclidean plane by taking maximal averages over concentric circles. It is intimatelyrelated to the behaviour of space/time averages of solutions to the linear wave equation. Recently, the local smoothing conjecture for the wave equation was established by Guth--Wang--Zhang. This conjecture implies (and is substantially stronger than) Bourgain's circular maximal function theorem, as well as many other classical results in harmonic analysissuch as the Bochner--Riesz and restriction conjectures in 2 dimensions. The proof of the local smoothing conjecture involves a powerful Littlewood--Paley square function inequality for functions frequency supported near the lightcone. This inequality, and the methods used to prove it, are likely to have a broad range of further applications and it is of great interest to explore other situations where they may apply.

该项目旨在利用调和分析中各种新工具的力量来研究经典对象，如几何极大函数。一个可能的方向是研究Bougain循环极大函数的变体。该算子通过在同心圆上取最大平均值来作用于欧几里得平面上的函数。它与线性波动方程解的空间/时间平均的行为密切相关。最近，Guth-Wang-Zhang建立了波动方程的局部光滑猜想。这一猜想暗示了（并且比之强得多）布尔甘的循环极大函数定理，以及调和分析中的许多其他经典结果，如Bochner-Riesz和二维限制定理。局部光滑化猜想的证明涉及到一个强有力的Littlewood-Paley平方函数不等式，它适用于频率在光锥附近的函数。这个不等式，以及用来证明它的方法，很可能有广泛的进一步的应用，它是非常感兴趣的探索其他情况下，他们可能适用。