The Geometry of Measures and Regularity of Associated Operators

该项目涉及以下基本物理问题背后的一些数学研究：物体的几何形状在多大程度上可以由与物体相关的力场（例如，它的引力场）的信息来确定？势理论中的这类逆问题有着丰富的历史，但正确回答这一问题所需的数学工具，特别是当将力场与物体质量分布联系起来的算符对远程相互作用很敏感时，目前还不发达。在这个项目中，首席研究员将开发工具来进一步理解这个问题，特别是集中在如果一个人只知道这个领域有有限的数量级时可以说什么。更具体地说，该项目主要关注度量的几何形状与相关微分或奇异积分算子的规律性之间的关系。自从引入柯西变换和里兹变换作为研究解析函数和调和函数行为的工具以来，这个问题一直吸引着数学家。通过对无反射测度的研究，提出了一种解决这类问题的综合方法。这种方法最近产生了几个新的结果，并可能解决一些悬而未决的问题，特别是那些涉及具有有界Riesz变换的度量的支持平滑性的问题。为了取得进展，需要开发定量几何和高阶偏微分方程方面的新工具。此外，首席研究员寻求建立在拟线性微分方程理论的最新创新，以考虑各种非线性微分算子的类似问题，其中没有积分表示可用。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。

This project concerns a study of the some of the mathematics behind the following basic physical question: To what extent can the geometry of a body be determined from information about a force field associated to the body (for instance, its gravitational field)? Such inverse problems in potential theory have a rich history, but the mathematical tools needed to properly answer this question, especially in the case when the operator relating the force field to the mass distribution of the body is sensitive to long-range interactions, are currently underdeveloped. In this project, the principal investigator will develop tools to further understand this problem, concentrating especially on what can be said if one knows only that the field has bounded magnitude.More specifically, the project primarily concerns the relationship between the geometry of a measure and the regularity of an associated differential or singular integral operator. This is is a question that has attracted mathematicians ever since the Cauchy and Riesz transforms were introduced as tools to study the behavior of analytic and harmonic functions, respectively. An integrated approach to such problems is proposed that goes through the study of reflectionless measures. This approach has recently yielded several new results and could potentially address a number of open problems, especially those concerning the smoothness of the support of a measure that has a bounded Riesz transform. Here new tools in quantitative geometry and higher order partial differential equations need to be developed in order to make progress. Furthermore, the principal investigator seeks to build upon recent innovations in the theory of quasilinear differential equations to consider analogous problems for a wide range of nonlinear differential operators, where no integral representation is available.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.