Locally uniform domains and extension of bmo functions

TL;DR

This paper establishes the equivalence between geometric domain conditions and the extension property for bmo functions, revealing a deep connection between domain geometry and function space behavior.

Contribution

It proves that Jones' geometric conditions are equivalent to being an extension domain for bmo functions, linking domain geometry with function space extension properties.

Findings

Equivalence of Jones' geometric conditions and bmo extension domains

Connection between domain geometry and bmo function behavior

Characterization of uniform domains via local geometric conditions

Abstract

We prove that for a domain $\Omega \subset \mathbb{R}^n$, being $(\epsilon,\delta)$ in the sense of Jones is equivalent to being an extension domain for bmo$(\Omega)$, the nonhonomogeneous version of the space of function of bounded mean oscillation on $\Omega$. In the process we demonstrate that these conditions are equivalent to local versions of two other conditions characterizing uniform domains, one involving the presence of length cigars between nearby points and the other a local version of the quasi-hyperbolic uniform condition. Our results show that the definition of bmo$(\Omega)$ is closely connected to the geometry of the domain.