In his book Singular Integrals and Differentiability Properties of Functions, Stein constructs an extension operator $\mathcal{E}:W^{m,p}(\Omega)\rightarrow W^{m,p}(\mathbb{R}^{N})$, $m\in\mathbb{N}$, $1\le p<\infty$, in domains of the form $$ \Omega:=\{(x,y)\in\mathbb{R}^{N-1}\times\mathbb{R}:\,y>f(x)\}, $$ where $f:\mathbb{R}^{N-1}\rightarrow\mathbb{R}$ is a Lipschitz continuous function. Let $\phi:[1,\infty)\rightarrow\mathbb{R}$ be a continuous function be such that \begin{equation} \int_{1}^{\infty}\phi(t)\,dt=1,\quad\int_{1}^{\infty}t^{n}\phi(t)\,dt=0,\quad \lim_{t\rightarrow\infty}t^{n}\phi(t)=0\label{sIII s5}% \end{equation} for all $n\in\mathbb{N}$. Given $u\in W^{m,p}(\Omega)$, the extension $\mathcal{E}(u)$ to $\mathbb{R}^{N}\setminus\overline{\Omega}$ is given by by \begin{equation} \mathcal{E}(u)(x,y):=% %TCIMACRO{\dint _{1}^{\infty}}% %BeginExpansion {\displaystyle\int_{1}^{\infty}} %EndExpansion \phi(t)u(x,y+3t\sqrt{1+L^{2}}d_{\operatorname*{reg}}(x,y))\,dt,\label{sIII s6}% \end{equation} where $L$ is the Lipschitz constant of $f$ and $d_{\operatorname*{reg}}$ is the regularized distance to $\partial\Omega$.
Since the fractional Sobolev space $W^{s,p}(\Omega)$, $0<s<1$, is obtained by interpolating $L^{p}(\Omega)$ and $W^{1,p}(\Omega)$ and since $\mathcal{E}% :L^{p}(\Omega)\rightarrow L^{p}(\mathbb{R}^{N})$ and $\mathcal{E}% :W^{1,p}(\Omega)\rightarrow W^{1,p}(\mathbb{R}^{N})$ are continuous, one can use interpolation theory to show that $\mathcal{E}:W^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^{N})$ is a continuous extension operator.
I was trying to give a direct proof of this fact by using the explicit seminorm $$ v\mapsto\int_{\mathbb{R}^{N}\setminus\Omega}\int_{\mathbb{R}^{N}% \setminus\Omega}\frac{|v(X)-v(Y)|^{p}}{|X-Y|^{N+sp}}dXdY, $$ where $X=(x,y)$. I tried adding and subtracting stuff but I would need a change of variables to get back to $\Omega$
