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

这一建议涉及解析函数论、调和分析和算子理论交集的基础数学研究。研究这些问题的动机可以在偏微分方程式中找到，这是学习科学和工程的基础。偏微分方程解通常由一个积分算子Calderon-Zygmund算子给出，它的相关性质可以用来推导这些偏微分方程解的相关性质。一般来说，研究这些Calderon-Zygmund算子是具有挑战性的，人们试图通过只检查更简单的测试函数类上的行为来研究它们在某些函数空间上的作用。以此类推，这可以被视为试图通过简单地理解更简单的纯频率有限集合来理解复杂的乐谱。这项拟议的研究基于国际和平研究所最近做出的贡献，利用了之前获得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之间的开放和重要问题。解决这些领域的问题将提供更多的调查线索。这一奖项的资金将支持国际和平协会建议的不同的研究生群体；帮助增加全国训练有素的STEM学生在学术界、政府或行业就业的渠道。该计划的研究计划将重要的公开问题与国际和平协会过去的工作结合起来。总的主题将是使用围绕“检验定理”的方法，在调和分析中称为“T1定理”，或在解析函数论和算子理论中的“再生核命题”，来研究在解析函数论、调和分析和算子理论中出现的问题。特别地，检验定理的证明策略的应用将：(1)被用来刻画Calderon-Zygmund算子何时在这些算子的连续和并进变体的加权空间之间有界；(2)作为与Calderon-Zygmund算子密切相连的解析函数的Fock空间上的算子相关的一类问题的动机；以及(3)被用来提供一种方法来研究再生解析函数的核Hilbert空间中的Carleson度量。所获得的结果将为其他方面的调查打开大门。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.