How to recognise extension domains

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ be a bounded domain and $1<p<\infty$. We prove that there is a bounded extension operator ${\dot{W}}^{1,p}(\hat{I}\text{c◯})\to {\dot{W}}^{1,p}({\mathbb{R}}^n)$ if and only if $\hat{I}\text{c◯}$ 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 $\partial \hat{I}\text{c◯}$, fractional extension (from ${\dot{W}}^{1,p}(\hat{I}\text{c◯})$ to ${\dot{W}}^{s,p}({\mathbb{R}}^n)$) at a single exponent $s>1/p$ self-improves to full first-order Sobolev extension (from ${\dot{W}}^{1,p}(\hat{I}\text{c◯})$ to ${\dot{W}}^{1,p}({\mathbb{R}}^n)$). These results clarify the role of nonlocal estimates in the geometry of Sobolev extension domains.

18 pages