On the extension of VMO functions

Abstract

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

26 pages