The Geometry of Measures and Regularity of Associated Operators

这个项目涉及对以下基本物理问题背后的一些数学问题的研究：从与物体相关的力场(例如，其引力场)的信息可以在多大程度上确定物体的几何形状？位势理论中的这类反问题有着丰富的历史，但正确回答这个问题所需的数学工具目前还不发达，特别是在将力场与物体的质量分布联系起来的算符对远程相互作用敏感的情况下。在这个项目中，首席研究人员将开发工具来进一步了解这个问题，特别是当一个人只知道场有界值时可以说些什么。更具体地说，这个项目主要涉及度量的几何和相关的微分或奇异积分算子的正则性之间的关系。自从柯西变换和Riesz变换分别被引入作为研究解析函数和调和函数行为的工具以来，这个问题就一直吸引着数学家。通过对无反射措施的研究，提出了一种综合解决这类问题的方法。这种方法最近已经产生了几个新的结果，并可能解决一些公开的问题，特别是那些关于具有有界Riesz变换的度量的支撑点的光滑性的问题。在这里，为了取得进展，需要开发定量几何和高阶偏微分方程式的新工具。此外，主要研究人员试图建立在拟线性微分方程组理论的最新创新的基础上，以考虑范围广泛的非线性微分算子的类似问题，其中没有可用的积分表示。

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.