I know that it holds this embeding $$H^{s}(\Omega) \hookrightarrow H^{s'}(\Omega)$$ for $s>s'$ and for any $\Omega \subset \mathbb{R}^N$. In this case, anyway, the fractional laplacian is defined on $\Omega$.
I wonder if it is possible, roughly, to obtain the same result, but with the fractional laplacian defined on $\mathbb{R}^N$. Or, said in a different way, if $H^{s}(\mathbb{R}^N) \hookrightarrow H^{s'}(\mathbb{R}^N)$ holds if one restricts the integration over $\Omega$.
More precisely, I wonder if it holds this integral-inequality estimate
$$\int_{\Omega} |u|^2 + \int_{\Omega} |(-\Delta)^{s'/2} u|^2 \leq C \left( \int_{\Omega} |u|^2 + \int_{\Omega} |(-\Delta)^{s/2} u|^2 \right) $$ where $$(-\Delta)^{s/2} u (x)= \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+s}} dy$$ for $x \in \Omega$, and $u \in H^{s}(\mathbb{R}^N)\subset H^{s'}(\mathbb{R}^N)$. I know it holds for $\Omega = \mathbb{R}^N$ (by use of Fourier transform), thus I'm interested in the case $\Omega \subsetneq \mathbb{R}^N$.
ATTEMPT
Adapting the proof of Proposition 2.1 in "Hitchhiker's guide to the fractional Sobolev spaces" it is easy to prove that
$$\int_{\Omega} |u|^2 + \int_{\Omega} \int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s'}} \leq C \left( \int_{\Omega} |u|^2 + \int_{\Omega} \int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} \right) $$
but clearly is not the same thing.
I've thought to some cutoff function, but the nonlocality does not help.
I've tried also to use the relations with the regional and spectral laplacian, but it didn't work.