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

本课题研究现代分析及其应用中的重要工具--奇异积分的性质。这样的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，根据适当的振动频率概念，通过将函数分解成分量来研究函数的性质。奇异积分的研究特点是在正贡献和负贡献之间发生微妙的抵消。在这个项目中要考虑的一些关键问题涉及换向器。换向器是按相反顺序取的两个乘积的差A*B-B*A。换位性质A*B=B*A一般很少见，换位器的大小给出了非对易程度的一个度量。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交换子理论。这包括具有几何味道的曲线上的问题，以及最近关键估计的更高参数变量，以及应用于偏微分方程的问题。该项目的一部分涉及一个重要的奇异积分，即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，因为它们在整个调和分析及其应用中被广泛使用。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，这种方法将结合新的几何构造得到进一步发展。此外，首席调查员将就与拟议研究有关的主题指导研究生，并将为职业早期研究人员组织一次研究讲习班，并为本科生研究举办一次单独的在线会议。这个项目围绕着奇异积分、调和分析和换向器的研究展开。其中要考虑的主题是Beurling-Ahlfors变换的高次方的加权估计，Beurling-Ahlfors变换是一类奇异积分的重要范例。最近发现的反例表明著名的A2猜想在这种情况下是失败的，从而提出了相关的理论，但确切的界限还没有完全理解。研究人员还将通过研究曲线上的奇异积分和最近交换子刻画的双交换子类似来推进交换子理论。这样的交换子问题也与多参数设置中稀疏支配的弱变体有关。在过去的十年里，稀疏控制方法一直是相关问题的核心，但到目前为止，它仍然是一种单参数工具。在方法论方面，该项目将进一步发展近似弱因式分解方法，这是最近许多突破的原因，并将该方法与其他新工具相结合。该项目将利用并元概率分析、微调反例和几何分解来开发适用于这些和其他问题的调和分析方法。该奖项反映了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 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.