Kakeya Maximal Operators and Oscillatory Integrals

摘要 陶将学习几何分析的原理和结果， 并将它们与振荡积分语句（如限制猜想）联系起来。 前者的一个典型例子是Kakeya猜想，即一个集合，其中包含一个线段，在每个 R ^n中的方向必须有维数n。 的典型例子 后者是各种偏微分方程（ 波动方程、薛定谔方程等）。 人们已经知道 自20世纪70年代以来， 两种类型的声明，但不存在系统的方法， 已知的几个具体连接不是很令人满意。 我们希望收集、简化、统一和扩展以前的结果， 这个方向。陶还将攻击其中的一些错误（特别是 Kakeya猜想）直接，使用一些新的和有前途的 技术;例如，陶将利用仿射不变性， 挂谷猜想 这项工作的目的之一是加深我们对 振荡积分是一种数学表达式 这发生在物理学的许多地方（光学，量子力学， 声学和任何其他处理波的物理学领域），以及 在其他数学领域具有重要的理论意义。 理解这些积分，特别是知道 他们可以得到，最终可能导致新的设计， 应用（例如，使网球拍的面积最大化的网球拍） "甜蜜点"，或弯曲的反射器，有大量的焦点 点的频率范围很广），或至少在理论上 限制这种设计。 也有数值应用时， 模拟某些物理系统（例如， 如果一个人知道某个振荡积分永远不会 在某种技术意义上变得非常“大”，那么这将 为计算机的准确性提供了理论保证 物理系统的模拟。涛会研究这些 借助几何学的振荡积分; 这两个数学领域是已知的（有点类似于 几何光学和波动理论之间的关系 光），但不完全理解。 如果这种联系是 发展得足够彻底，我们也许能够减少困难， 振荡积分中的问题转化为更简单的几何问题， 或者至少使用几何技术来获得部分进展， 振荡积分问题

ABSTRACT Tao will study geometrical analysis conjectures and results, and relate them to oscillatory integral statements such as the restriction conjecture. A typical example of the former is the Kakeya conjecture, that a set which contains a line segment in every direction in R^n must have dimension n. A typical example of the latter is the generalized Strichartz estimate for various PDE (the wave equation, the Schrodinger equation, etc.). It has been known since the 1970s that there is an intimate relationship between the two types of statements, but no systematic approach exists, and the few concrete connections that are known are not very satisfactory. We hope to collect, simplify, unify, and extend previous results in this direction. Tao will also attack some of these conjectures (notably the Kakeya conjecture) directly, using some new and promising techniques; for example, Tao will exploit the affine invariance of the Kakeya conjecture. One of the aims of this work is to deepen our understanding of oscillatory integrals, which are a type of mathematical expression which occur in many places in physics (optics, quantum mechanics, acoustics, and any other field of physics dealing with waves), as well as having theoretical importance in other fields of mathematics. Understanding these integrals, and in particular knowing how large they can get, may ultimately lead to new designs for physical applications (e.g. tennis rackets that maximize the area of the "sweet spot", or curved reflectors that have a large number of focus points for a wide range of frequencies), or at least place theoretical limits on such designs. There are also numerical applications when modeling certain physical systems (e.g. the seismic behavior of the Earth); if one knows that a certain oscillatory integral will never become very "large" in a certain technical sense, then this will provide a theoretical guarantee to the accuracy of the computer sim ulation of the physical system. Tao will study these oscillatory integrals with the aid of geometry; a connection between these two fields of mathematics is known (being somewhat similar to the relationship between geometrical optics and the wave theory of light), but is not understood completely. If this connection is developed thoroughly enough, we may be able to reduce difficult questions in oscillatory integrals to simpler problems in geometry, or at least use geometrical techniques to obtain partial progress on the oscillatory integral problems.