On the extension of VMO functions

We consider functions of vanishing mean oscillation on a bounded domain $\mathrm{\Omega }$ and prove a $\mathrm{V}\mathrm{M}\mathrm{O}$ analogue of the extension theorem of P. Jones for $\mathrm{B}\mathrm{M}\mathrm{O}(\mathrm{\Omega })$. We show that if $\mathrm{\Omega }$ satisfies the same condition imposed by Jones (i.e.\ is a uniform domain), there is a linear extension map from $\mathrm{V}\mathrm{M}\mathrm{O}(\mathrm{\Omega })$ to $\mathrm{V}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$ which is bounded in the $\mathrm{B}\mathrm{M}\mathrm{O}$ norm. Moreover, if such an extension map exists from $\mathrm{V}\mathrm{M}\mathrm{O}(\mathrm{\Omega })$ to $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$, then the domain is uniform.