I have a question on fractional Sobolev Hilbert space $H^s$ with fractional $s$: I have a bounded Lipschitz domain $\Omega\subseteq\mathbb{R}^d$ that is the union of two Lipschitz domains $\Omega_1,\Omega_2$.

My question:

Can I define the Norm on $H^s(\Omega)$ equivalently as the sum of the norms $H^2(\Omega_1)$ and $H^2(\Omega_2)$?

This is trivial for integer order spaces, but I just don't get the point for fractionals.

And if I have two smooth diffeomorphisms $\phi_1,\phi_2$ bounded in suitably many derivatives that map two other bounded Lipschitz domains $A_1,A_2$ to $\Omega_1,\Omega_2$ by $\phi_i(A_i)=\Omega_i$, can I maybe even define the norm on $\|\cdot\|_{H^2(\Omega_i)}$ by the norm $\|\cdot\|_{H^2(A_i)}$?


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