The geometry of measures with density bounds in a Holder anisotropic setting

Abstract

Geometric measure theory provides tools for the study of the geometry of sets that are rough, in the sense that they are not smooth enough to support the methods of differential geometry. One approach that has been successfully exploited in this area is based on the idea that the regularity of a set can be characterized by the behavior of quantities that capture small scale geometric properties. An important example of one such quantity is the notion of density of a set or Radon measure. Originally introduced by Besicovitch and later developed by several authors, this concept turns out to be closely tied to the local geometric properties of sets and measures, and has been a source of research problems in geometric measure theory until the present day. In this dissertation we study a problem in this direction, which aims to characterize regularity in terms of a quantitative density condition in a certain anisotropic setting.