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)的观点来理解函数。调和分析提供了回答这些问题的工具，方法是以一种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。这个项目研究并进一步发展了一种新的，令人惊讶的强大的调和分析方法，称为稀疏支配原理。广义地说，稀疏支配提供了一种学习一般函数的方法，通过将它们与正的、本地化的、易于理解的版本进行比较，从而获得与使用更复杂的技术相同的强度和锐度的结果。这种新方法的主要优点之一是它的多功能性：它不仅可以用于古典场景，而且在可能重新定义所涉及的对象之后，还可以用于各种更一般的场景。该项目旨在探索稀疏控制技术的全面性和通用性。该项目包括三个研究方向，旨在更深入地分析稀疏界的概念。一个方向涉及向量值函数空间上的矩阵加权估计，这一领域重新引起了人们的兴趣，部分原因是它与椭圆型偏微分方程的联系。最近的一些稀疏支配结果重新解释了函数平均的传统定义，表明在解决所谓的A2猜想方面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，寻找支持这一结果的进一步证据和潜在的反例。第二个方向是离散算符理论，将解析数理论和调和分析联系起来。直到最近，还没有在离散环境下建立加权边界，但稀疏支配的使用已经产生了新的结果和问题，例如具有一般多项式相位的离散振荡Hilbert变换的加权估计问题和离散分数次奇异积分的加权边界问题。主要研究人员打算研究这类函数，目的是为算术运算符提供更广泛的加权理论。第三个方向涉及对稀疏集合的不同解释，以前只考虑多维数据集。首席研究员感兴趣的是开发用矩形代替立方体的方法，以回答有关Bochner-Riesz算子和涉及Kakeya最大函数相关平均的有界性结果的问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.