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

Calderon-Zygmund算子是对我们理解从热传递到湍流的许多物理现象负有主要责任的对象。最近，这些运算符在大数据分析中得到了应用。经典理论是由Alberto Calderon和Antoni Zygmund在20世纪50年代初建立的，本质上是设计用于光滑物体的。然而，大自然经常用非常不规则的介质来迷惑我们。因此，需要一个非常低的规律性卡尔德龙-齐格蒙德理论，这三个PI，事实上，构建。这种低正则性理论的一个可能的应用是，通过Calderon-Zygmund算子在高维空间中的集合上的作用，我们可以得出结论，集合本身是结构良好的，可以分析。这是一个典型的大数据问题。我们最近的另一个观察是，这种算子的充分研究的问题可以被对偶化，为超立方体的分析提供新的信息-另一个广泛使用的大数据模型。这个项目将考虑以下问题，在这里以最简单的形式提出：1）齐次设置中平方函数算子的尖锐端点弱加权估计; 2）在非齐次情形下的问题; 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精确估计; 4）如何从平方函数的端点估计得到超立方体和高斯空间上的几何不等式; 5）余维大于1的David-Semmes问题; 6）余维大于1的集合上的调和测度估计;以及7）奇异Riesz变换的正几何量的自下而上的估计。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.