The Geometry of Measures and Regularity of Associated Operators

Clemson University, Clemson SC

Investigators

Abstract

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.