Singular integrals on curves, the Beurling-Ahlfors transform, and commutators

曲线上的奇异积分、Beurling-Ahlfors 变换和换向器

• 批准号: 2247234
• 负责人: Henri Martikainen
• 金额: $ 20.92万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247234&HistoricalAwards=false
• 关键词: Singular; integrals; curves; Beurling; Ahlfors

This project addresses properties of singular integrals, an important tool in modern analysis and its applications. Such integrals appear systematically in the context of harmonic analysis, a field of mathematics which studies properties of functions via their decomposition into components according to a suitable notion of vibrational frequency. The study of singular integrals is characterized by delicate cancellations which occur between positive and negative contributions. Some key questions to be considered in this project involve commutators. A commutator is the difference A*B-B*A of two products taken in the opposite order. The commutation property A*B = B*A is rare in general, and the size of the commutator gives a measure for the degree of non-commutativity. Classical examples are related to the uncertainty principle in quantum mechanics. The project will advance state of the art commutator theory involving singular integrals. This includes problems on curves with a geometric flavor and higher parameter variants of recent key estimates with applications to partial differential equations. Part of the project deals with an important singular integral, the Beurling–Ahlfors transform, and particularly with questions related to weighted estimates. Weighted estimates are generally desirable due to their wide use throughout harmonic analysis and its applications. Methodologically, discrete methods involving probability have recently become central in related problems, and such methods will be further developed in combination with new geometric constructions. In addition, the Principal Investigator will supervise graduate students on topics related to the proposed research, and will organize a research workshop for early-career researchers and a separate online conference for undergraduate research. This project revolves around the study of singular integrals, harmonic analysis, and commutators. Among the topics to be considered are weighted estimates for higher powers of the Beurling–Ahlfors transform, an important exemplar of the class of singular integrals. The relevant theory has recently been advanced by the discovery of counterexamples showing the failure of the famous A2 conjecture in this context, but the precise bounds are not completely understood. The investigator will also advance the theory of commutators by studying singular integrals on curves and bi-commutator analogues of recent commutator characterizations. Such commutator problems are also related to weak variants of sparse domination in the multi-parameter setting. Sparse domination methods have been central in the past decade for related problems but up to now have remained a one-parameter tool. On the methodological side, the project will further develop the method of approximate weak factorizations, which has been responsible for many recent breakthroughs, and will combine that method with other novel tools. The project will utilize dyadic-probabilistic analysis, fine-tuned counterexamples, and geometric factorizations to develop applicable harmonic analysis methods suitable for these and other questions.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.

奇异积分是现代分析中的一个重要工具，本课题研究奇异积分的性质及其应用。这种积分系统地出现在谐波分析的背景下，谐波分析是一个数学领域，通过根据适当的振动频率概念将函数分解成分量来研究函数的性质。奇异积分研究的特点是发生在正贡献和负贡献之间的微妙的消去。在这个项目中要考虑的一些关键问题涉及到换向器。换向器是两个乘积按相反顺序取的差A*B-B*A。换易性质A*B = B*A在一般情况下是罕见的，换易子的大小给出了非换易程度的度量。经典的例子与量子力学中的测不准原理有关。该项目将推进涉及奇异积分的最先进的换向子理论。这包括在具有几何特征的曲线上的问题，以及应用于偏微分方程的最新键估计的更高参数变体。项目的一部分涉及一个重要的奇异积分，Beurling-Ahlfors变换，特别是与加权估计相关的问题。由于加权估计在谐波分析及其应用中广泛使用，因此通常需要加权估计。在方法学上，涉及概率的离散方法最近成为相关问题的中心，这种方法将与新的几何结构相结合进一步发展。此外，首席研究员将指导研究生进行与拟议研究相关的主题，并将为早期职业研究人员组织一个研究研讨会，并为本科生研究组织一个单独的在线会议。这个项目围绕着奇异积分、谐波分析和换向子的研究展开。其中要考虑的主题是加权估计的Beurling-Ahlfors变换，一类奇异积分的一个重要的例子。最近，通过发现反例来证明著名的A2猜想在这种情况下的失败，相关理论得到了发展，但精确的界限尚未完全理解。研究人员还将通过研究曲线上的奇异积分和最近换向子表征的双换向子类似物来推进换向子理论。这类换向子问题还涉及到多参数设置中稀疏支配的弱变体。在过去的十年中，稀疏支配方法已经成为相关问题的核心，但到目前为止仍然是一个单参数工具。在方法方面，该项目将进一步发展近似弱分解方法，该方法最近取得了许多突破，并将该方法与其他新工具结合起来。该项目将利用二元概率分析、微调反例和几何分解来开发适用于这些问题和其他问题的谐波分析方法。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。