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.

该建议涉及分析功能理论，谐波分析和操作者理论的交集中的基本基本数学研究。研究这些问题的动机可以在部分微分方程中找到，这是科学和工程研究至关重要的。偏微分方程的解决方案通常是由Calderon-Zygmund运算符的积分运算符给出的，其相关属性可用于推断这些部分微分方程的相关属性。通常，研究这些Calderon-Zygmund操作员具有挑战性，人们试图通过仅在更简单的测试功能上检查行为来研究其对某些功能空间的行为。从类似的角度来看，这可以看作是试图通过简单地了解更简单的纯频率收集来理解复杂的乐谱。拟议的研究是基于PI的最新贡献，利用了通过先前的国家科学基金会奖开发的技能和知识。通过该建议，PI将在分析功能理论，谐波分析和操作者理论的界面上解决开放和重要的问题。 在这些领域中解决问题将提供其他查询。 该奖项的资金将支持PI建议的一群不同的研究生；有助于增加训练有素的STEM学生的全国性渠道，从事学术界，政府或行业的职业。 一般的主题将是在谐波分析中使用``测试定理''的方法，称为``T1 t1定理''或在分析功能理论和操作者理论中``复制内核论文''中的``复制内核论文''，以研究在分析功能理论，和谐分析和操作者理论中出现的问题。 特别是，测试定理的证明策略的应用将：（1）用于表征何时Calderon-Zygmund运算符在这些操作员的连续和二元变体的加权空间之间进行界定； （2）作为与Calderon-Zygmund运营商密切相关的分析功能的FOCK空间相关的一系列问题的动机； （3）被利用以提供一种研究Carleson测量方法来重现分析功能的内核空间的方法。 获得的结果将打开其他调查范围的大门。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。