Locally uniform domains and extension of bmo functions

Abstract

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, the nonhonomogeneous version of the space of functions of bounded mean oscillation on \(\Omega\). Such domains, which can be identified as local versions of uniform domains (defined by requiring the presence of length cigars between nearby points), allow a definition of bmo\((\Omega)\) in terms of "small" and "large" cubes contained in \(\Omega\), where the scale is closely tied to the geometry of the domain.