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可用工具的强大功能来研究几何最大函数等经典对象。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/时间平均值的行为密切相关。最近，Guth--Wang--Zhang 建立了波动方程的局部平滑猜想。这个猜想暗示了（并且明显强于）布尔干的圆极大函数定理，以及调和分析中的许多其他经典结果，例如 Bochner--Riesz 和二维限制猜想。局部平滑猜想的证明涉及光锥附近支持的函数频率的强大的 Littlewood-Paley 平方函数不等式。这种不等式以及用于证明它的方法可能会具有广泛的进一步应用，并且探索它们可能适用的其他情况非常有趣。