Testing Theorems in Analytic Function Theory, Harmonic Analysis and Operator Theory

解析函数论、调和分析和算子理论中的检验定理

• 批准号: 2349868
• 负责人: Brett Wick
• 金额: $ 31万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349868&HistoricalAwards=false
• 关键词: Testing; Theorems; Analytic; Function; Theory

This proposal involves basic fundamental mathematical research at the intersection of analytic function theory, harmonic analysis, and operator theory. Motivation to study these questions can be found in partial differential equations, which are fundamental to the study of science and engineering. The solution to a partial differential equation is frequently given by an integral operator, a Calderon-Zygmund operator, whose related properties can be used to deduce related properties of these partial differential equations. In general, studying these Calderon-Zygmund operators is challenging and one seeks to study their action on certain spaces of functions, by checking the behavior only on a simpler class of test functions. In analogy, this can be seen as attempting to understand a complicated musical score by simply understanding a simpler finite collection of pure frequencies. The proposed research is based on recent contributions made by the PI, leveraging the skills and knowledge developed through prior National Science Foundation awards. Through this proposal the PI will address open and important questions at the interface of analytic function theory, harmonic analysis, and operator theory. Resolution of questions in these areas will provide for additional lines of inquiry. Funds from this award will support a diverse group of graduate students whom the PI advises; helping to increase the national pipeline of well-trained STEM students for careers in academia, government, or industry.The research program of this proposal couples important open questions with the PI's past work. The general theme will be to use methods around ``testing theorems,'' called ``T1 theorems'' in harmonic analysis or the ``reproducing kernel thesis'' in analytic function theory and operator theory, to study questions that arise in analytic function theory, harmonic analysis, and operator theory. In particular, applications of the proof strategy of testing theorems will: (1) be used to characterize when Calderon-Zygmund operators are bounded between weighted spaces both for continuous and dyadic variants of these operators; (2) serve as motivation for a class of questions related to operators on the Fock space of analytic functions that are intimately connected to Calderon-Zygmund operators; and, (3) be leveraged to provide a method to study Carleson measures in reproducing kernel Hilbert spaces of analytic functions. Results obtained will open the door to other lines of investigation.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.

这个建议涉及基本的基础数学研究的交集解析函数理论，调和分析，算子理论。研究这些问题的动机可以在偏微分方程中找到，这是科学和工程研究的基础。偏微分方程的解通常由积分算子给出，即Calderon-Zygmund算子，其相关性质可以用来推导这些偏微分方程的相关性质。一般来说，研究这些Calderon-Zygmund算子是具有挑战性的，人们试图通过只在一类简单的测试函数上检查其行为来研究它们在某些函数空间上的作用。类似地，这可以被看作是试图通过简单地理解一个更简单的纯频率的有限集合来理解一个复杂的乐谱。拟议的研究是基于PI最近的贡献，利用通过以前的国家科学基金会奖开发的技能和知识。通过这项建议，PI将解决解析函数理论，调和分析和算子理论接口的开放和重要问题。 解决这些领域的问题将提供更多的调查线索。 该奖项的资金将支持PI建议的多元化研究生群体;帮助增加受过良好训练的STEM学生在学术界，政府或工业界的职业生涯的国家管道。该提案的研究计划将重要的开放问题与PI过去的工作结合起来。 总的主题将是使用方法围绕"测试定理“，所谓的”T1定理“在调和分析或"再生核论文”在解析函数理论和算子理论，研究问题，出现在解析函数理论，调和分析和算子理论。 特别地，检验定理的证明策略的应用将：（1）用于刻画Calderon-Zygmund算子在加权空间之间有界的性质，无论是对于这些算子的连续变式还是并矢变式，（2）作为一类与Calderon-Zygmund算子密切相关的解析函数的Fock空间上的算子相关问题的动机;（3）提供了一种研究解析函数再生核Hilbert空间中Carleson测度的方法。 该奖项反映了NSF的法定使命，并被认为是值得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的支持。