New developments in geometric Fourier analysis

该项目旨在利用谐波分析中各种新工具的力量来研究几何极大函数等经典对象。一个可能的方向是研究布干圆极大函数的变体。这个算子通过在同心圆上取极大平均值作用于欧几里德平面上的函数。它与线性波动方程解的时空平均行为密切相关。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（并且实质上强于）布尔甘的圆极大函数定理，以及谐波分析中的许多其他经典结果，如Bochner- Riesz和二维中的限制猜想。局部平滑猜想的证明涉及到一个强大的Littlewood—Paley平方函数不等式，该不等式适用于光锥附近支持频率的函数。这个不等式，以及用来证明它的方法，可能有广泛的进一步应用，探索它们可能适用的其他情况是非常有趣的。

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.