Combinatorial Questions in Harmonic Analysis

建议：dms-9801410首席研究员：Nets Katz摘要：本项目的目的是研究平面上的定向极大算子。这项研究的主要事实是Besicovitch集的存在，这排除了极大算子在所有方向上的有界性。当方向集受到限制时，理解的就更少。Katz将研究维数大于零的Ahlfors正则方向集上的极大算子在所有L^p空间中的无界性。他还将尝试解决沃尔夫关于在每个方向上都包含一个d维集的集合的猜想。这两个问题都涉及到在Besicovitch类集合的已知结构中寻找弹性。数学的很大一部分(以及物理和信号处理)涉及时间和频率之间的相互作用。在信号处理中，时间是一维连续统，信号是时间的函数。这在很大程度上取决于对给定时间信号的瞬时频率能说些什么。时间和频率在量子力学中也是一个比喻。位置代替时间，动量代替频率。海森堡测不准原理限制了我们可以在多短时间内指定一个频率。它说，一个音符需要一段时间才能被听到--它的波长。在量子力学的比喻中，时间--现在称为位置--不需要是一维的。因此，频率也不会受到如此限制。指定频率所需的相关范围变成了朝向不同方向的矩形。要回答关于这种类型的宇宙的问题，有必要了解不同方向上的矩形之间的相互作用。这个项目的目的是提高对二维空间背景的理解。

Proposal: DMS-9801410 Principal Investigator: Nets Katz Abstract: The purpose of this project is to study directional maximal operators in the plane. The major fact underlying this study is the existence of a Besicovitch set, which rules out the boundedness of maximal operators over all directions. When the set of directions is restricted, less is understood. Katz will work on unboundedness in all L^p spaces for maximal operators over Ahlfors regular sets of directions with dimension greater than zero. He will also try to work out Wolff's conjecture about sets that contain a d-dimensional set in each direction. Both of these problems involve finding elasticity in the known constructions of Besicovitch-like sets. A large part of mathematics (as well as physics and signal processing) involves the interaction between time and frequency. In signal processing, time is a one-dimensional continuum and a signal is a function of time. Much depends upon what can be said about the instantaneous frequency of the signal at a given time. Time and frequency also occur as a metaphor in quantum mechanics. Position takes the place of time and momentum the place of frequency. The Heisenberg uncertainty principle limits how instantaneously we may specify a frequency. It says that a note requires a length of time to be heard -- its wavelength. In the quantum mechanical metaphor, there is no need for time -- now called position -- to be one-dimensional. Thus, neither is frequency so constrained. The relevant ranges needed to specify frequencies become rectangles oriented in various directions. To answer questions about this type of universe, it is necessary to understand the interaction between rectangles in different directions. The purpose of this project is to improve that understanding in the setting of two-dimensional space.