Kakeya Maximal Operators and Oscillatory Integrals

摘要陶氏将研究几何分析猜想和结果，并将它们与振荡积分陈述联系起来，如限制猜想。前者的一个典型例子是Kakeya猜想，即在R^n中每个方向都包含一条直线段的集合一定有n维，后者的一个典型例子是对各种偏微分方程组(波动方程、薛定谔方程等)的广义Strichartz估计。自1970年代以来，人们就知道这两类声明之间存在密切关系，但没有系统的办法，已知的为数不多的具体联系也不太令人满意。我们希望在这个方向上收集、简化、统一和扩展以前的成果。陶渊明还将使用一些新的和有前途的技术直接攻击其中一些猜想(特别是Kakeya猜想)；例如，陶渊明将利用Kakeya猜想的仿射不变性。这项工作的目的之一是加深我们对振荡积分的理解，振荡积分是物理(光学、量子力学、声学和任何其他涉及波的物理领域)中许多地方都存在的一种数学表达式，在其他数学领域也具有理论上的重要性。了解这些积分，特别是知道它们可以变得多大，最终可能会导致物理应用的新设计(例如，最大化“最佳位置”面积的网球拍，或在广泛频率范围内具有大量焦点的弯曲反射器)，或者至少在理论上对此类设计施加限制。在模拟某些物理系统(如地球的地震行为)时也有数值应用；如果知道某个振荡积分在某种技术意义上永远不会变得很“大”，那么这将为物理系统的计算机模拟的准确性提供理论保证。陶渊明将借助几何学来研究这些振荡积分；这两个数学领域之间的联系是已知的(有点类似于几何光学和光波动理论之间的关系)，但还没有完全理解。如果这种联系发展得足够彻底，我们也许能够将振荡积分中的困难问题归结为更简单的几何问题，或者至少利用几何技巧在振荡积分问题上取得部分进展。

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.