Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer: $$ [u]_{1,\Omega}^2:= \int_{\Omega} \int_{\Omega} \frac{|u(x) - u(y)|^2}{|x-y|^{n+2}} \,dxdy. $$ Is this equivalent to the Sobolev semi-norm: $$ |u|_{1,\Omega}^2:= \int_{\Omega} |Du|^2 dx\,? $$
My guess is no because, when we divide $\overline{\Omega} = \overline{\Omega}_1 \cup \overline{\Omega}_2$ into two non-overlapping subdomain, for the Sobolev semi-norm it is just adding broken norms on the subdomains: $$ |u|_{1,\Omega}^2 = |u|_{1,\Omega_1}^2 + |u|_{1,\Omega_2}^2. $$ However for the Slobodeckij norm, it has some cross term like: $$ \int_{\Omega_1} \int_{\Omega_2} \frac{|u(x) - u(y)|^2}{|x-y|^{n+2}} \,dxdy, $$ and apparently, $$ [u]_{1,\Omega}^2 \neq [u]_{1,\Omega_1}^2 + [u]_{1,\Omega_2}^2. $$
I am guessing the equivalence relies on the regularities of the $\Omega$, but I could not find any reference on this. Any comments are welcome.