Local vanishing mean oscillation

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

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