Locally uniform domains and extension of bmo functions.

We prove that for a domain $\Omega \subset \mathbb{R}^n$, being $(\epsilon,\delta)$ in the sense of Jones is equivalent to being an extension domain for bmo$(\Omega)$, the nonhonomogeneous version of the space of function of bounded mean oscillation on $\Omega$. In the process we demonstrate that these conditions are equivalent to local versions of two other conditions characterizing uniform domains, one involving the presence of length cigars between nearby points and the other a local version of the quasi-hyperbolic uniform condition. Our results show that the definition of bmo$(\Omega)$ is closely connected to the geometry of the domain.