How to Recognise Extension domains

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We characterize $(1,p)$-extension domains in terms of inequalities of Bourgain--Brezis--Mironescu type. More precisely, we show that $\Omega$ is a $(1,p)$-extension domain if and only if it is Ahlfors regular and satisfies, for all $f \in \dot{W}^{1,p}(\Omega)$, $$(1-s)[f]_{W^{s,p}(\Omega)}^p \leq C [f]_{W^{1,p}(\Omega)}^p,$$ for all $s$ sufficiently close to $1$, where $C > 0$ is a constant independent of $s$ and $f$. 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). As a further application, we 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.