Non-Standard Sparse Estimates and Weighted Inequalities

对物理和工程等领域中出现的模型进行数学分析通常需要从定量（大小，增长率）和定性（有界性）的角度理解函数。谐波分析提供了回答这些问题的工具，通过以方便的方式分解函数，分析组件，然后重新组合有关组件的信息以提供整体理解。该项目研究并进一步开发了一种新的，令人惊讶的强大的谐波分析方法，称为稀疏支配原则。从广义上讲，稀疏支配提供了一种研究一般函数的方法，通过将它们与正的、局部的、易于理解的版本进行比较，从而获得与更复杂的技术相同的强度和清晰度的结果。这种新方法的主要优点之一是它的多功能性：它不仅可以用于经典环境，而且在对所涉及的对象进行潜在的重新定义之后，可以用于各种更一般的场景。本计画旨在探讨稀疏支配技术的完整强度与多功能性，包含三个研究方向，旨在深入分析稀疏界限的概念。 一个方向涉及向量值函数空间的矩阵加权估计，这是一个重新引起人们兴趣的领域，部分原因是它与椭圆偏微分方程的联系。最近的一些稀疏支配结果重新解释了函数平均值的传统定义，表明解决所谓的A2猜想的进展可能是触手可及的。首席研究员计划继续研究矩阵A2猜想，寻求支持结果的进一步证据和潜在的反例。第二个方向，离散算子理论，涉及解析数论和调和分析。直到最近，没有加权界已建立在离散设置，但使用稀疏支配产生了新的结果和问题，如加权估计的问题的离散振荡希尔伯特变换与一般多项式相位和问题的加权界离散分数奇异积分。主要研究人员打算研究这些功能的目标是提供一个更广泛的加权理论的算术运算符。第三个方向涉及稀疏集合的不同解释，以前只考虑立方体。主要研究者的兴趣在于开发用长方形代替立方体的方法，以回答有关Bochner-Riesz算子的问题，以及涉及与Kakeya极大函数相关的平均值的有界性结果。该奖项反映了NSF的法定使命，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，被认为值得支持。

Mathematical analysis of models arising in fields such as physics and engineering often requires an understanding of functions both from a quantitative (size, rate of growth) and a qualitative (boundedness) point of view. Harmonic analysis provides tools for answering these questions through decomposing a function in a convenient way, analyzing the components, and then reassembling the information about the components to provide an overall understanding. This project investigates and further develops a new, surprisingly powerful approach in harmonic analysis known as sparse domination principles. In broad terms, sparse domination gives a way of studying general functions by comparing them to positive, localized, easy-to-understand versions, thereby obtaining results of the same strength and sharpness as with much more involved techniques. One of the main advantages of this new approach is its versatility: it can be employed not only in the classical setting, but also, after a potential redefinition of the objects involved, in a variety of more general scenarios. This project intends to explore the full strength and versatility of sparse domination techniques.The project comprises three research directions aimed at a deeper analysis of the concept of sparse bounds. One direction concerns matrix weighted estimates on vector-valued function spaces, an area that has enjoyed renewed interest, partially due to its connections to elliptic partial differential equations. Some recent sparse domination results, which reinterpret the traditional definition of function averages, suggest that progress on resolving the so-called A2 conjecture may be within reach. The principal investigator plans to continue research into the matrix A2 conjecture, seeking both further evidence in support of the result and potential counterexamples. A second direction, the theory of discrete operators, relates analytic number theory and harmonic analysis. Until recently, no weighted bounds had been established in the discrete setting, but the use of sparse domination has yielded new results and problems, such as the question of weighted estimates for the discrete oscillatory Hilbert transform with a general polynomial phase and the problem of weighted bounds for discrete fractional singular integrals. The principal investigator intends to study such functions with the goal of providing a broader weighted theory for arithmetic operators. A third direction concerns a different interpretation of sparse collections, previously considered only with respect to cubes. The principal investigator is interested in developing methods with cubes replaced by rectangles to answer questions regarding Bochner-Riesz operators and boundedness results involving averages associated to the Kakeya maximal function.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.