Collaborative Research: Calderon-Zygmund Operators in Highly Irregular Environments, and Applications

合作研究：高度不规则环境中的 Calderon-Zygmund 算子及其应用

• 批准号: 1600065
• 负责人: Alexander Volberg
• 金额: $ 39万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600065&HistoricalAwards=false
• 关键词: Collaborative; Research; Calderon; Zygmund; Operators

Calderon-Zygmund operators are mathematical objects that play an important role in the understanding of many physical phenomena, ranging from heat transfer to turbulence in dynamical systems. The classical theory of these operators was designed to work on smooth functions. However, nature often provides us with very irregular media with which to engage. This creates the need for a very low-regularity form of the theory of singular integrals, which the principal investigators on this project have constructed. A consequence of the low-regularity theory is that through the action of Calderon-Zygmund operators on a set in a Euclidean space of a very high dimension, one can sometimes conclude that the set itself is of a much lower dimension than the ambient space, an important piece of information from the perspective of data science. To refine this approach to data analysis is one of the main goals of this project. This project considers several problems in nonhomogeneous harmonic analysis, geometric measure theory, and spectral theory. The common theme uniting the problems is the behavior of singular operators with very good (Calderon-Zygmund) kernels in very bad environments (e.g., on sets with no a priori structure, in spaces with matrix weights). Specifically, the project will pursue the following avenues of research: (1) the David-Semmes problem to characterize the rectifiability of sets and measures in high-dimensional Euclidean space in terms of the boundedness of the corresponding Riesz transforms; (2) the geometry of reflection-less measures; (3) the geometric characterization of higher-dimensional analogues of positive analytic capacity; (4) two-weight estimates for very simple singular operators in the non-Hilbert setting; and (5) sharp estimates for classical operators with matrix weights. Singular integral operators with respect to bad measures and very irregular sets appear naturally in many problems of analysis. One of the reasons for their increasing interest in recent years has been the study of analytic capacity. While the theory for the two-dimensional case (i.e., the Cauchy transform on the complex plane) and the theory of analytic capacity that emerged as its by-product are now very well understood, the analogous theory in higher dimensions has not been fully developed. The main roadblock here is the lack of geometric tools in higher dimensions. Additionally, in higher dimensions, nonhomogeneous situations arise more often than in the plane and more often one might expect. For example, boundary value problems in (otherwise smooth) domains with cusps lead to nonhomogeneous problems, because, unlike what happens in the two-dimensional setting, surface measure on the boundary of such a domain is non-doubling. This becomes an even more vexing problem if one wants to consider harmonic measure estimates for domains on whose boundaries "surface measure" is practically arbitrary. This is an important issue that the project seeks to confront.

Calderon-Zygmund操作员是数学对象，在理解许多物理现象中起着重要作用，从热传递到动态系统中的湍流。 这些操作员的经典理论旨在致力于平滑功能。但是，大自然通常会为我们提供非常不规则的媒体参与。这创造了对奇异积分理论的非常低的定型形式，这是该项目的主要研究人员所构建的。低规度理论的结果是，通过Calderon-Zygmund操作员在欧几里得空间中具有非常高维度的欧几里得空间中的集合中的作用，有时可以得出结论，从数据科学的角度来看，该集合本身的维度比环境空间要低得多。要完善这种数据分析方法是该项目的主要目标之一。该项目考虑了非均匀谐波分析，几何测量理论和光谱理论的几个问题。结合问题的共同主题是在非常糟糕的环境中具有非常好的（Calderon-Zygmund）内核的奇异操作员的行为（例如，在没有先验结构的集合中，在具有矩阵重量的空间中）。具体而言，该项目将追求以下研究途径：（1）David-Semmes问题以相应的Riesz变换的界限来表征高维欧几里得空间中集合和度量的重新讨论性； （2）无反射措施的几何形状； （3）阳性分析能力的高维类似物的几何表征； （4）在非希尔伯特环境中非常简单的单数运算符的两个重量估计； （5）具有基质重量的经典操作员的尖锐估计。在许多分析问题中，就不良措施和非常不规则的集合而言，奇异的整体操作员自然而然地出现。近年来，其兴趣日益增加的原因之一是研究能力。尽管现在已经充分了解了二维情况的理论（即，在复杂平面上的Cauchy转换）和作为其副产品的分析能力理论的理解，但较高维度中的类似理论尚未得到充分发展。这里的主要障碍是缺乏更高维度的几何工具。此外，在较高的维度中，非均匀情况的发生频率比在平面中更频繁，人们可能会期望的。例如，具有尖尖的（否则平滑）域中的边界值问题导致了非均匀问题，因为与二维设置中发生的情况不同，此类域的边界上的表面度量是不加倍的。如果一个人想考虑对其边界“表面度量”实际上是任意的域的谐波测量估计，这将成为一个更加烦人的问题。这是该项目寻求面对面的重要问题。