Local vanishing mean oscillation

TL;DR

This paper investigates various notions of vanishing mean oscillation on domains, establishing approximation and extension results, and characterizing when bounded extensions exist based on domain geometry.

Contribution

It provides new conditions for the density of Lipschitz functions in VMO and characterizes when bounded extensions are possible on domains.

Findings

Sufficient conditions for density of Lipschitz functions in VMO.

Characterization of domains allowing bounded extensions of VMO and CMO.

Extension results hold if and only if the domain is locally uniform.

Abstract

We consider various notions of vanishing mean oscillation on a (possibly unbounded) domain $\Omega \subset \mathbb{R}^n$, and prove an analogue of Sarason's theorem, giving sufficient conditions for the density of bounded Lipschitz functions in the nonhomogeneous space $\rm{vmo}(\Omega)$. We also study $\rm{cmo}(\Omega)$, the closure in $\rm{bmo}(\Omega)$ of the continuous functions with compact support in $\Omega$. Using these approximation results, we prove that there is a bounded extension from $\rm{vmo}(\Omega)$ and $\rm{cmo}(\Omega)$ to the corresponding spaces on $\mathbb{R}^n$, if and only if $\Omega$ is a locally uniform domain.