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

Calderon-Zygmund算符是一种数学对象，在理解动力系统中从热传递到湍流的许多物理现象中发挥着重要作用。这些算子的经典理论是为处理光滑函数而设计的。然而，大自然经常为我们提供非常不规律的媒体来参与。这就需要一种非常低正则形式的奇异积分理论，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算子对高维欧几里德空间中的集合的作用，有时可以得出结论，该集合本身的维度比环境空间低得多，从数据科学的角度来看，环境空间是一条重要的信息。改进这种数据分析方法是该项目的主要目标之一。这个项目考虑了非齐次调和分析、几何测度论和谱理论中的几个问题。统一这些问题的共同主题是具有非常好(Calderon-Zygmund)核的奇异算子在非常糟糕的环境中的行为(例如，在没有先验结构的集合上，在具有矩阵权重的空间中)。具体地说，该项目将追求以下研究途径：(1)David-Semmes问题，以相应的Riesz变换的有界性来刻画高维欧氏空间中集合和测度的可正性；(2)无反射测度的几何；(3)正分析能力的高维类似物的几何特征；(4)非常简单奇异算子在非Hilbert环境下的双权估计；以及(5)带矩阵权的经典算子的尖锐估计。在许多分析问题中，关于不良测度和非常不规则集的奇异积分算子是很自然地出现的。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。虽然二维情形的理论(即复平面上的柯西变换)和作为其副产品出现的解析能力理论现在已经被很好地理解，但更高维的类似理论还没有得到充分的发展。这里的主要障碍是缺乏更高维度的几何工具。此外，在更高的维度中，不均匀的情况比在平面中更频繁地出现，也更多地出现在人们可能预期的情况中。例如，在具有尖点的(否则光滑的)区域中的边值问题会导致非齐次问题，因为与在二维环境中发生的情况不同，这种区域边界上的曲面测量是非加倍的。如果人们想要考虑其边界上的“表面度量”实际上是任意的区域的调和度量估计，这就成为一个更令人烦恼的问题。这是该项目试图面对的一个重要问题。

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.