Given $\Omega$ a bounded regular open set in $\Bbb R^d$ we consider
$C_c^\infty(\Omega)$ the space of smooth functions compactly supported in $\Omega$. For $1<p<\infty $ Let's denote by $W^{1,p}(\Omega)$ be the standard Sobolev space. It is common to denote the closure of $C_c^\infty(\Omega)$ in $W^{1,p}(\Omega)$ by $$W_0^{1,p}(\Omega):=\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{1,p}(\Omega)}$$
Note that, by the trivial zero extension the space $C_c^\infty(\Omega)$ can be seen as the space of smooth functions in $\Bbb R^d$ which are compactly supported in $\Omega$ and hence it can be seen as subspaces of $W^{1,p}(\Bbb R^d)$.
I would like to compare the spaces $$W_0^{1,p}(\Omega):=\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{1,p}(\Omega)}$$ and $$\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{1,p}(\Bbb R^d)}$$
More generally if $0<s<1$ is the fractional oder how can we compare the following fractional Sobolev spaces?
$$W_0^{s,p}(\Omega):=\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{s,p}(\Omega)}~~~ and ~~~~\overline{ C_c^{\infty}(\Omega)}^{~~~~W^{s,p}(\Bbb R^d)}$$
Definition: $W^{s,p}(\Omega)$ is the space of class of functions $u$ in $L^p(\Omega)$ such that
$$[u]^p_{{W^{s,p}(\Omega)}}:=\iint\limits_{\Omega\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} dxdy<\infty. $$ Which turns out to be a Banach space endowed with the natural norm $$\|u\|^p_{W^{s,p}(\Omega)}=\|u\|^p_{L^{p}(\Omega)}+[u]^p_{{W^{s,p}(\Omega)}}$$
NB As the second question might less obvious good answer to this may deserve some bounty accordingly. But I will be okay if one just answers the first question.
