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

建议：DMS-9970395主要研究人员：Fedor Nazarov，Serguei Treil，Alexandre Volberg摘要：该项目的第一部分涉及矩阵权理论;第二部分涉及非齐次空间上的Calderon -Zygmund算子理论。第一部分的主要动机和应用源于平稳随机过程的理论;也就是说，从多变量过程的研究，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。该项目的第二部分是密切相关的几何测量理论，分析能力，偏微分方程和图像处理的问题。PI成功地将Calderon-Zygmund算子理论从不必要的同质性假设中解放出来，这是该理论的历史基石之一。迄今为止，他们的研究在多元随机过程的正则性理论方面已取得了许多新的成果。它对几何测量理论也产生了影响。例如，他们的方法产生了一种替代（而不是更精简）的方法，以盖伊大卫的解决方案的著名Vitushkin的猜想，不可求长的一维集有零的分析能力。现代调和分析的中心部分处理“奇异算子”的一种类型或另一种。这种算子在科学领域中是普遍存在的：它们出现在数学物理、概率论、工程学、图像处理等领域。“奇异算子”的表达可以有许多不同的含义，但无论它在给定的上下文中的确切含义如何，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算子的研究。它们抵抗“蛮力”方法的探测。特别地，需要采用对某些固有抵消敏感的技术，以便获得对奇异行为的控制。所谓的Calderon-Zygmund理论出现在20世纪50年代初，作为处理该项目中仔细审查的奇异算子类型的第一种方法。PI在两个方向上扩展了这一理论。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，他们采取了相当大胆的举措，抛弃了该理论的一个基本假设，而且人们早期认为这是不可或缺的假设。根据PI的发现，这个假设（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，简化了对其他重要但困难结果的证明，以及使复杂论证更简短，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.