Aspects of thinness in harmonic analysis

调和分析员寻求发展数学理论，以帮助找到解决数学物理、电气工程和其他数学分支可能出现的问题的方法。我在谐和分析方面的研究是受测不准原理的谐和分析版本的启发。这一原理在哲学上与海森堡测不准原理有关。粗略地说，它说，关于一个函数获得的信息必须通过相应的傅立叶变换失去控制来“买单”，反之亦然。这在实际应用中具有重要意义。例如，它意味着一个人不能有一个在时间域和频率域都有界的信号。在我的研究中，我对这一原则进行了严格和定量的解释。我研究在某种意义上“薄”的函数、运算符或度量，并试图理解这种“薄”的后果。我的项目有三个主要部分：1.李群经常在物理中使用，因为它们描述真实世界的几何。轨道测量是这种环境中的基本组成部分，支撑力很小。我的项目的第一部分是将它们的变换的大小和相关的几何结构联系起来。这对于李群的分析是很重要的。2.本项目的第二部分涉及对稀疏方向极大算子的研究。极大算子是一种重要的工具，用于从物体的初等分量重建物体。我感兴趣的是，在最大运算符表现良好之前，方向集必须有多小。3.Sidon集可以根据其上支持傅里叶变换的函数的解析性质来定义。尽管它们在某种直观意义上是稀薄的或稀疏的，但它们具有复杂的结构。通过用更简单的稀疏集刻画Sidon集，我们将对相关的函数空间和基础群有更好的理解。

Harmonic analysts seek to develop mathematical theories that help to find solutions to problems that might have arisen from mathematical physics, electrical engineering and other branches of mathematics. My research in harmonic analysis is motivated by the harmonic analysis version of the uncertainty principle. This principle is philosophically related to the Heisenberg uncertainty principle. Roughly speaking, it says information gained about a function has to be "paid for" by a corresponding loss of control on its Fourier transform, or vice versa. This is significant in practical applications. For instance, it implies that one cannot have a signal that is bounded in both the time and frequency domain. In my research, I develop rigorous and quantitative interpretations of this principle. I study functions, operators or measures that are "thin" in some sense and seek to understand the consequences of this thinness. My project has three major parts to it: 1. Lie groups are often used in physics as they describe real-world geometry. Orbital measures are elementary components in this setting and have small support. The first part of my project is to relate the size of their transforms and the associated geometric structures. This is important for analysis on Lie groups. 2. The second part of the project involves the study of thin directional maximal operators. Maximal operators are an important tool, used to reconstruct objects from their elementary components. I am interested in understanding how small the set of directions must be before the maximal operator will be well behaved. 3. Sidon sets can be defined in terms of the analytic properties of the functions whose Fourier transform is supported on the set. Although they are thin or sparse in some intuitive sense, they have a complicated structure. By characterizing Sidon sets in terms of thin sets that are simpler we will have greater understanding of the associated function spaces and underlying groups.