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

卡尔德龙-齐格蒙算子是数学对象，在理解许多物理现象中起着重要作用，从热传递到动力系统中的湍流。这些算子的经典理论被设计用于光滑函数。然而，大自然经常为我们提供非常不规则的媒介。这就产生了对奇异积分理论的一种非常低规则形式的需求，这个项目的主要研究人员已经构建了这种形式。低正则理论的一个结果是，通过Calderon-Zygmund算子在高维欧几里得空间中的集合上的作用，人们有时可以得出结论，该集合本身的维数比周围空间低得多，从数据科学的角度来看，这是一个重要的信息。改进这种数据分析方法是本项目的主要目标之一。本计画考虑了非齐次谐波分析、几何测量理论和频谱理论中的几个问题。统一这些问题的共同主题是非常好的（Calderon-Zygmund）核的奇异算子在非常恶劣的环境下（例如，在没有先验结构的集合上，在具有矩阵权重的空间上）的行为。具体而言，该项目将追求以下研究途径：(1)David-Semmes问题，以相应Riesz变换的有界性来表征高维欧几里德空间中集合和测度的可纠偏性；(2)无反射措施的几何形状；(3)具有正解析能力的高维类似物的几何表征；(4)非hilbert条件下非常简单奇异算子的双权估计；(5)具有矩阵权值的经典算子的尖锐估计。关于不良测度和非常不规则集的奇异积分算子自然出现在许多分析问题中。近年来他们越来越感兴趣的原因之一是对分析能力的研究。虽然二维情况下的理论（即复平面上的柯西变换）和作为其副产品出现的分析能力理论现在已经很好地理解了，但高维的类似理论还没有完全发展起来。这里的主要障碍是缺乏高维的几何工具。此外，在更高的维度中，非齐次情况比在平面中出现得更频繁，也更频繁。例如，在具有顶点的（光滑的）域中的边值问题会导致非齐次问题，因为，与二维环境中发生的情况不同，这种域边界上的表面度量是非加倍的。如果要考虑边界上的“表面测度”实际上是任意的域的调和测度估计，这将成为一个更加令人烦恼的问题。这是该项目试图解决的一个重要问题。

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.