How to recognise extension domains

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