On the extension of VMO functions

TL;DR

This paper extends the classical Jones extension theorem from BMO to VMO functions on uniform domains, establishing a linear bounded extension operator and characterizing uniform domains via extension properties.

Contribution

It proves a VMO analogue of Jones' extension theorem and characterizes uniform domains through the existence of bounded extension operators for VMO functions.

Findings

Existence of a bounded linear extension map from VMO(Ω) to VMO(ℝ^n) for uniform domains.

Characterization of uniform domains via the existence of extension maps for VMO functions.

Extension map boundedness in the BMO norm implies the domain is uniform.

Abstract

We consider functions of vanishing mean oscillation on a bounded domain $\Omega$ and prove a $\rm{VMO}$ analogue of the extension theorem of P. Jones for $\rm{BMO}(\Omega)$. 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 $\rm{VMO}(\Omega)$ to $\rm{VMO}(\mathbb{R}^n)$ which is bounded in the $\rm{BMO}$ norm. Moreover, if such an extension map exists from $\rm{VMO}(\Omega)$ to $\rm{BMO}(\mathbb{R}^n)$, then the domain is uniform.