Aspects of thinness in harmonic analysis

谐波分析者寻求发展数学理论，帮助找到数学物理、电子工程和其他数学分支中可能出现的问题的解决方案。我在谐波分析方面的研究是由不确定性原理的谐波分析版本所激发的。这个原理在哲学上与海森堡测不准原理有关。粗略地说，它说关于一个函数的信息必须通过对其傅里叶变换的相应控制的损失来“支付”，反之亦然。这在实际应用中具有重要意义。例如，它意味着一个信号不可能在时域和频域都有界。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。李群在物理学中经常使用，因为它们描述了现实世界的几何形状。在这种情况下，轨道测量是基本的组成部分，支持很小。我项目的第一部分是将它们的变换大小和相关的几何结构联系起来。这对李群的分析是很重要的。2. 项目的第二部分涉及到薄方向极大算子的研究。极大算子是一种重要的工具，用于从初等分量重构对象。我感兴趣的是理解在最大算子表现良好之前方向集合必须有多小。3. 西顿集可以根据在该集合上支持傅里叶变换的函数的解析性质来定义。虽然它们在某种直观上是薄的或稀疏的，但它们具有复杂的结构。通过用更简单的瘦集来描述西顿集，我们将更好地理解相关的函数空间和底层群。

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.