Let us define the fractional Sobolev space $$ W_{0}^{s,p}(\Omega):=\{u\in W^{s,p}(\mathbb{R}^N):u=0\text{ on }\mathbb{R}^N\setminus\Omega\}, $$ where $$ ||u||_{W^{s,p}(\mathbb{R}^N)}:=||u||_{L^p(\mathbb{R}^N)}+||\frac{u(x)-u(y)}{|x-y|^\frac{N+ps}{p}}||_{L^p(\mathbb{R}^N\times\mathbb{R}^N)}<\infty. $$
Now we define the space $$ V_{0}^{s,p}(\Omega):=\overline{\{C_{c}^\infty(\Omega):||u||_{W^{s,p}(\mathbb{R}^N)}\}}. $$
Are the two spaces $W_0^{s,p}(\Omega)$ and $V_{0}^{s,p}(\Omega)$ same for $0<s<1$ and $p>2$?
Can somebody kindly help. Thanks.