Approximation and Extension of Functions of Vanishing Mean Oscillation

We consider various definitions of functions of vanishing mean oscillation on a domain Ω ⊂ 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 R n , and also extends BMO ( Ω ) to BMO ( 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 Ω to functions in BMO ( R n ) , which is bounded in the 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.