Calderon-Zygmund Operators in Non-Classical Situations: Weighted Norm Inequalities with Matrix Weights, Operators on Non-Homogeneous Spaces and Analytic Capacity

项目负责人：Fedor Nazarov， Serguei Treil， Alexandre volberg摘要：项目第一部分涉及矩阵权重理论；第二部分是关于非齐次空间上Calderon -Zygmund算子的理论。第一部分的主要动机和应用源于平稳随机过程理论；也就是说，从多变量过程的研究，这可以追溯到Kolmogoroff和Wiener的工作。矩阵权理论与有限维几何和非交换调和分析有许多联系。项目的第二部分与几何测量理论、解析能力、偏微分方程和图像处理等问题密切相关。pi成功地将卡尔德龙-齐格蒙算子理论从一个不必要的同质性假设中解放出来，而同质性假设在历史上是该理论的基石之一。到目前为止，他们的研究在多元随机过程的正则性理论方面取得了许多新的成果。它也对几何测量理论产生了影响。例如，他们的方法产生了另一种（而且更精简的）方法来解决著名的维图什金猜想，即不可校正的一维集合具有零分析能力。现代谐波分析的一个核心部分是处理一种或另一种类型的“奇异算子”。这样的运算符在科学领域无处不在：它们出现在数学物理、概率、工程、图像处理等领域。“奇异运算符”可以有许多不同的含义，但无论它在给定上下文中的确切含义是什么，这个术语几乎总是反映了这样一个事实，即只有微妙的工具才能应用于对这种运算符的研究。他们抵制“蛮力”方法的探测。特别是，为了获得对奇异行为的控制，需要采用对某些内在抵消敏感的技术。所谓的卡尔德隆-齐格蒙德理论出现于20世纪50年代初，作为处理本项目中所研究的奇异算子类型的第一种方法。pi从两个方面扩展了这一理论。首先，它们允许运营商享有的自由度增加，从而暴露出一种全新的取消方式。其次，他们做出了一个相当大胆的举动，他们抛弃了这个理论的一个基本假设，而这个假设早前被认为是不可或缺的。根据pi的发现，这种假设（技术上称为底层空间的同质性）现在似乎完全是多余的。pi的方法已经产生了几个重要的结果：解决了几个长期存在的问题，简化了其他重要但困难的结果的证明，以及使复杂的论证变得更简短、更清晰的一般特征。

Proposal: DMS-9970395Principal Investigators: Fedor Nazarov, Serguei Treil, Alexandre VolbergAbstract: The first part of the project deals with the theory of matrix weights; the second is concerned with the theory of Calderon -Zygmund operators on nonhomogeneous spaces. The primary motivations for and applications of the first part stem from the theory of stationary random processes; namely, from the study of multivariate processes, which dates back to work of Kolmogoroff and Wiener. The theory of matrix weights has many connections with both finite dimensional geometry and noncommutative harmonic analysis. The second part of the project is closely related to problems of geometric measure theory, analytic capacity, partial differential equations, and image processing. The PIs have managed to free the theory of Calderon-Zygmund operators of an unnecessary homogeneity assumption, historically one of the cornerstones of that theory. So far their research has led to many new results in the theory of regularity of multivariate random processes. It has had an impact on geometric measure theory as well. For example, their methods yield an alternative (and rather more streamlined) approach to Guy David's solution of the famous Vitushkin's conjecture that unrectifiable one-dimesional sets have zero analytic capacity.A central part of modern harmonic analysis deals with "singular operators" of one type or another. Such operators are pervasive in the scientific landscape: they turn up in mathematical physics, probability, engineering, image processing, etc. The expression "singular operator" can take many different meanings, but whatever its precise meaning within a given context, the term almost always reflects the fact that only subtle tools can be applied to the investigation of such operators. They are resistant to probing by "brute force" methods. In particular, one needs to employ techniques that are sensitive to certain intrinsic cancellations in order to gain control over the singular behavior. So-called Calderon-Zygmund theory appeared in the early 1950s as a first means of coping with singular operators of the type under scrutiny in this project. The PIs broaden this theory in two directions. First, they allow the number of degrees of freedom enjoyed by the operators to increase, thus exposing a whole new kind of cancellation. Second, they make a rather bold move by disgarding one of the basic and, it was earlier thought, indispensible assumptions of the theory. In light of discoveries by the PIs, this assumption (known technically as the homogeneity of the underlying space) seems now to be completely superflous. The PIs' approach has already had several significant consequences: solutions to several long-standing problems, streamlined proofs for other important but difficult results, and the general feature of making complicated arguments much shorter and more lucid.