Local vanishing mean oscillation

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