Approximation and Extension of Functions of Vanishing Mean Oscillation

Abstract

We consider various definitions of functions of vanishing mean oscillation on a domain $$\Omega \subset {{{\mathbb {R}}}^n}$$ . If the domain is uniform, we show that there is a single extension operator which extends functions in these spaces to functions in the corresponding spaces on $${{{\mathbb {R}}}^n}$$ , and also extends $$\mathrm{BMO}(\Omega )$$ to $$\mathrm{BMO}({{{\mathbb {R}}}^n})$$ , generalizing the result of Jones. Moreover, this extension maps Lipschitz functions to Lipschitz functions. Conversely, if there is a linear extension map taking Lipschitz functions with compact support in $$\Omega $$ to functions in $$\mathrm{BMO}({{{\mathbb {R}}}^n})$$ , which is bounded in the $$\mathrm{BMO}$$ norm, then the domain must be uniform. In connection with these results we investigate the approximation of functions of vanishing mean oscillation by Lipschitz functions on unbounded domains.