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.

本计画探讨奇异积分的性质，奇异积分是现代分析及其应用的重要工具。这种积分系统地出现在调和分析的背景下，调和分析是一个数学领域，它通过根据振动频率的适当概念将函数分解为分量来研究函数的性质。奇异积分的研究的特点是微妙的取消之间发生的积极和消极的贡献。在这个项目中要考虑的一些关键问题涉及制冷剂。换位子是以相反顺序取的两个乘积的差A*B-B*A。一般情况下，对易性A*B = B*A是罕见的，而对易子的大小给出了非对易性程度的度量。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新换位子理论。这包括具有几何风格的曲线问题以及最近关键估计的更高参数变体及其在偏微分方程中的应用。该项目的一部分涉及一个重要的奇异积分，Beurling-Ahlfors变换，特别是与加权估计有关的问题。加权估计通常是可取的，由于其广泛使用在整个谐波分析及其应用。在方法论上，涉及概率的离散方法最近已成为相关问题的核心，并且这种方法将结合新的几何构造得到进一步发展。此外，首席研究员将监督研究生与拟议的研究相关的主题，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，调和分析，和分解器的研究。其中要考虑的主题是加权估计的Beurling-Ahlfors变换，一类奇异积分的重要范例的更高的权力。相关的理论最近被反例的发现所推进，反例显示了著名的A2猜想在这方面的失败，但精确的界限还没有完全理解。研究者也将通过研究曲线上的奇异积分和最近的换位子刻画的双换位子类似物来推进换位子理论。这样的交换子问题也与多参数设置中的稀疏支配的弱变体有关。稀疏支配方法在过去的十年中一直是相关问题的核心，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最近取得许多突破的近似弱因式分解方法，并将把该方法与其他新工具联合收割机结合起来。该项目将利用二元概率分析，微调反例，几何因式分解，以开发适用的谐波分析方法适合这些和其他question.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。