Collaborative research: Weighted Estimates with Matrix Weights and Non-Homogeneous Harmonic Analysis

卡尔德龙-齐格蒙算子在很大程度上帮助我们理解了从热传递到湍流等许多物理现象。最近，这些算子在大数据分析中得到了应用。经典理论是由阿尔贝托·卡尔德隆和安东尼·齐格蒙德在20世纪50年代初建立的，本质上是为了研究光滑的物体。然而，大自然常常用非常不规则的媒介来迷惑我们。因此，有必要建立一个非常低正则性的卡尔德龙-齐格蒙德理论，事实上，这三个pi已经构建了这个理论。这种低正则性理论的一个可能的应用是，通过Calderon-Zygmund算子作用于一个非常高维空间中的集合，我们可以得出这个集合本身结构很好并且可以被分析的结论。这是一个典型的大数据问题。我们最近的另一个观察是，对于这样一个算子，经过充分研究的问题可以被对偶化，从而为超立方体（另一种广泛使用的大数据模型）的分析提供新的信息。本项目将考虑以下问题，在这里以最简单的形式提出：1)齐次设置下平方函数算子的尖锐端点弱加权估计；2)在非齐次情况下出错了什么；3) Calderon—Zygmund算子的矩阵A_2问题，平方函数算子，它们的稀疏算子类比及其尖锐估计；4)如何从平方函数的端点估计到超立方体和高斯空间中的几何不等式；5)协维数大于1的David-Semmes问题；6)协维数大于1的集上的调和测度估计；7)由下估计奇异Riesz变换的正几何量。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Calderon-Zygmund operators are objects that are largely responsible for our understanding of a number of physical phenomena, from heat transfer to turbulence. Recently, these operators have found application in big data analysis. The classical theory was built by Alberto Calderon and Antoni Zygmund in the early 1950s, and was intrinsically designed to work on smooth objects. However, nature often puzzles us with very irregular medium. Thus, the need arose for a very low regularity Calderon-Zygmund theory, which the three PIs have, in fact, constructed. One possible application of such low regularity theory is that by the action of Calderon-Zygmund operators on a set in a space of a very high dimension, we can conclude that the set itself is nicely structured and can be analyzed. This is a typical big data problem. Our other recent observation is that well-studied problems for such an operator can be dualized to provide new information for analysis on the hypercube - another widely used model of big data.This project will consider the following problems, presented here in their simplest by formulation: 1) Sharp end-point weak weighted estimates for the square function operator in the homogeneous setting; 2) What goes wrong in the non-homogeneous case; 3) Matrix A_2 problems for Calderon--Zygmund operators, the square function operator, their sparse operators analogies and their sharp estimates; 4) How to get from end-point estimates of the square function to geometric inequalities on the hypercube and in Gaussian space; 5) The David-Semmes problem for co-dimension higher than one; 6) Harmonic measure estimates on sets of co-dimension bigger than one; and 7) Estimates from below of singular Riesz transforms by positive geometric quantities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.