Fourier multipliers, square functions and incidence theory.

建议研究方案*：本项目的目标是将一些最新发展的谐波分析技术应用到傅里叶乘法器的研究中。在最近的突破中，Guth-Wang-Zhang解决了二维波动方程的长期存在的L^4锥平方函数猜想和局部光滑化猜想。证明圆锥平方函数估计的一个困难是设置不能直接服从洛伦兹标度，洛伦兹标度法是现代调和分析中普遍采用的强大的尺度归纳技术的显著特征。为了避免这个问题，引入了一种新的估计，它将平方函数作为特例包含在内。这种一般框架更健壮，特别是在洛伦兹变换下是稳定的。还有许多其他问题的例子，其中标度结构以类似的方式分解，尝试应用这些方法将是有趣的，例如，在研究与曲线相关的Bochner-Reisz型乘子。另一方面，我们知道局部光滑猜想与径向乘子猜想密切相关，后者的目的是利用相伴核的L^p范数的有限性刻画径向傅立叶乘子的L^p有界性。研究在理解局部平滑方面的进步是否会对辐射乘数产生任何新的见解是很自然的。这些问题也与圆形环的入射几何有关，正如Heo--Nazarov-Seeger和Cladek在著作中所阐述的那样。入射几何方面的最新进展，如多项式分割法的发展，可能与这一研究有一定的相关性。

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.