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; 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.

该项目解决了单数积分的属性，这是现代分析及其应用中的重要工具。这些积分在谐波分析的背景下系统地出现，这是数学领域，该领域根据振动频率的合适概念，通过其分解为组件研究功能的性质。奇异积分的研究的特征是在正面和负面贡献之间发生微妙的取消。该项目中要考虑的一些关键问题涉及换向器。换向器是以相反顺序采集的两种产品的差异A*B-B*a。换向性能A*B = B*A通常很少，并且换向器的大小为非交换性程度提供了衡量。经典示例与量子力学中的不确定性原理有关。该项目将推进艺术换向器理论的状态涉及奇异积分。这包括曲线上的几何风味和较高的参数变体的最新关键估计值，并应用于部分微分方程。该项目的一部分涉及一个重要的奇异积分，beurling -ahlfors变换，尤其是与加权估计有关的问题。加权估计通常是由于它们在整个谐波分析及其应用中广泛使用而是可取的。从方法上讲，涉及概率的离散方法最近已成为相关问题的核心，并且将与新的几何结构结合进一步开发此类方法。此外，首席研究员将监督研究生与拟议研究有关的主题，并将组织一个针对早期研究人员的研究研讨会，并为本科研究提供单独的在线会议。该项目围绕着奇异积分，谐波分析和换向器的研究。在要考虑的主题中，是对Beurling -ahlfors Transform较高力量的加权估计，这是单数积分类别的重要典范。在这种情况下，发现了著名A2概念的失败，但相关理论最近是通过发现的，但尚未完全理解精确的界限。研究者还将通过研究最近的换向器字符的曲线和双交换器类似物的奇异积分来推动换向因子的理论。问题也与多参数环境中稀疏支配的弱变体有关。在过去的十年中，对于相关问题而言，稀疏的统治方法一直是核心，但现在仍然是一个参数工具。在方法论方面，该项目将进一步开发近似弱分解的方法，这是造成许多最近突破的原因，并将该方法与其他新型工具相结合。该项目将利用二元型核能分析，微调的反例和几何因素来开发适用于这些问题和其他问题的适用谐波分析方法。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛影响的审查标准来通过评估而被认为是珍贵的支持。