Local vanishing mean oscillation

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 $\mathrm{vmo}(\mathrm{\Omega })$. We also study $\mathrm{cmo}(\mathrm{\Omega })$, the closure in $\mathrm{bmo}(\mathrm{\Omega })$ of the continuous functions with compact support in $\Omega$. Using these approximation results, we prove that there is a bounded extension from $\mathrm{vmo}(\mathrm{\Omega })$ and $\mathrm{cmo}(\mathrm{\Omega })$ to the corresponding spaces on ${\mathbb{R}}^n$, if and only if $\Omega$ is a locally uniform domain.