Geometric Measure Theory

Abstract

AbstractIn this chapter we discuss basic concepts and results from Geometric Measure Theory, including sets of locally finite perimeter, Ahlfors regularity, the corkscrew condition, uniformly rectifiable (UR) sets, the local John condition, the class of Ahlfors regular domains, the class of UR domain, nontangentially accessible (NTA) domains, Reifenberg flat sets and Reifenberg flat domains, chord-arc curves (CAC) and chord-arc domains (CAD), the John-Nirenberg space of functions of bounded mean oscillations (BMO), the Sarason space of functions of vanishing mean oscillations (VMO), Muckenhoupt weights on Ahlfors regular sets, and Sobolev spaces on Ahlfors regular sets. Here we also introduce the category of δ-flat Ahlfors regular domains (or δ-flat AR domains, or simply δ-AR domains) which plays a basic role in this work, and show that these are ubiquitous in applications, by providing a multitude of concrete examples. Finally, we present a decomposition theorem to the effect that the boundaries of δ-AR domains may be well approximated locally by Lipschitz graphs.