How to recognise extension domains

Abstract

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We prove that there is a bounded extension operator $\dot{W}^{1,p}(\Omega)\to \dot{W}^{1,p}(\mathbb{R}^n)$ if and only if $\Omega$ satisfies the measure density condition and a Bourgain-Brezis-Mironescu type inequality (or limiting formula). As a key ingredient, we establish a fractional Poincar\'e-type inequality under the assumption of Ahlfors regularity alone, improving a result of Ponce (2004). We also prove that, under a mild Hausdorff measure condition on the boundary $\partial \Omega$, fractional extension (from $\dot{W}^{1,p}(\Omega)$ to $\dot{W}^{s,p}(\mathbb{R}^n)$) at a single exponent $s > 1/p$ self-improves to full first-order Sobolev extension (from $\dot{W}^{1,p}(\Omega)$ to $\dot{W}^{1,p}(\mathbb{R}^n)$). These results clarify the role of nonlocal estimates in the geometry of Sobolev extension domains.