Sparseness and Bellman Functions in Harmonic Analysis

谐波分析可以被广泛地描述为通过将复杂的算子或函数分解成更小、更简单的部分来解开它们——尽管是无限多的部分。部件的类型取决于手头对象的性质。这一建议的重点是适合于二元方法的问题：简单的部分通常是一个小的、只有两个值的正方形信号的一些变化。自从Stefanie Petermichl在2000年发现分析中被研究最多的算子之一希尔伯特变换可以用一种有意义的方式表示为这些简单算子的平均值以来，已经取得了巨大的进步。这个项目的核心将是通过将Petermichl的工作与谐波分析中几个长期存在的开放问题联系起来来推进Petermichl的工作。一个特别令人兴奋的途径将是在现代调和分析中两种“竞争”的方法，Bellman函数方法和稀疏算子控制方法之间建立桥梁。这本身就有可能成为一种新的“方法”，因为它涉及了谐波分析中两种主要方法的新思想。该项目还包括组织谐波分析活动，以及与研究生合作和培训。出发点是困扰学界相当长时间的并矢平方函数的一个开放问题。这里，加权不等式，特别是与Muckenhoupt权重，将解决。这些类型的不等式最近主导了现代调和分析，并矢平方函数是该领域最重要的算子之一-特别是因为它可以被视为Littlewood-Paley理论的起点。然后要解决的问题将是希尔伯特空间设置（参数p=2）中弱范数中的Muckenhoupt特征的锐功率。除了希尔伯特空间设置（p不等于2）之外的所有其他情况都是已知的，这使得这个问题仍然开放变得更加奇怪-通常，希尔伯特空间设置是一个“简单”的情况，可以从中提取所有其他值。然而，并矢平方函数的非线性性质在一种非常独特的情况下，把这一切都颠倒过来。计划中的方法，将Bellman函数的新思想与稀疏算子的新思想结合起来，最近取得了重大进展。预计这种方法还将得到更广泛的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Harmonic analysis can be broadly described as unlocking difficult operators or functions by breaking them down into smaller, simpler parts – albeit infinitely many. The type of parts depends on the nature of the object at hand. This proposal is focused on questions amenable to a dyadic approach: the simpler parts are usually some variation of a small, square signal which takes only two values. Huge advancements have been made since Stefanie Petermichl’s discovery in 2000, that one of the most studied operators in analysis, the Hilbert transform, can be expressed in a meaningful way as an average of such simple operators. Central to this project will be the advancement of Petermichl’s work by relating it to several long-standing open questions in harmonic analysis. An especially exciting path will be building a bridge between two “competing” methods in modern harmonic analysis, the Bellman function method and the sparse operator domination approach. This has the likelihood of becoming a new “method” in its own, as it involves new ideas from two leading methods in harmonic analysis. The project further includes organization of activities in harmonic analysis, as well as working with and training graduate students. The starting point is an open problem on the dyadic square function which has baffled the field for quite some time. Here, weighted inequalities, specifically with Muckenhoupt weights, will be settled. These types of inequalities have recently dominated modern harmonic analysis, and the dyadic square function is one of the most important operators in the field - especially because it can be viewed as the starting point of Littlewood-Paley theory. The question to then tackle will be the sharp power of the Muckenhoupt characteristic in the weak norm in the Hilbert space setting (parameter p=2). All other situations except the Hilbert space setting (p not equal to 2) are known, which makes it even more strange that this question is still open – usually, the Hilbert space setting is the “simple” case one extracts all other values from. However, the nonlinear nature of the dyadic square function turns this all on its head, in a quite unique situation. The planned approach, to merge new ideas for Bellman functions with new ideas for sparse operators, has seen significant recent progress. Much broader applications of this approach are anticipated as well.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.