Fractional Sobolev extension and imbedding

Abstract:

Let $\Omega$ be a domain of $\mathbb {R}^n$ with $n\ge 2$ and denote by $W^{s, p}(\Omega )$ the fractional Sobolev space for $s\in (0, 1)$ and $p\in (0, \infty )$. We prove that the following are equivalent:

(i) there exists a constant $C_1>0$ such that for all $x\in \Omega$ and $r\in (0, 1]$, \begin{eqnarray*} |B(x, r)\cap \Omega |\ge C_1 r^n; \end{eqnarray*}

(ii) $\Omega$ is a $W^{s, p}$-extension domain for all $s\in (0, 1)$ and all $p\in (0, \infty )$;

(iii) $\Omega$ is a $W^{s, p}$-extension domain for some $s\in (0, 1)$ and some $p\in (0, \infty )$;

(iv) $\Omega$ is a $W^{s, p}$-imbedding domain for all $s\in (0, 1)$ and all $p\in (0, \infty )$;

(v) $\Omega$ is a $W^{s, p}$-imbedding domain for some $s\in (0, 1)$ and some $p\in (0, \infty )$.