Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$ Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\Omega); (-\Delta)^su \in L^2(\Omega)\},$$
for some $s \in (0,1)$. I want to show that the operator $(-\Delta)^s : D((-\Delta)^s) \subset L^2(\Omega) \to L^2(\Omega)$ is closed. Here, $\Omega \subset \mathbb{R}^N$ is a bounded and smooth domain with $u = 0$ in $\mathbb{R}^N \backslash \Omega$.
Attempt:
Let $(u_n) \subset D((-\Delta)^s) $ with $u_n\to u$ in $L^2(\Omega)$ and $(-\Delta)^su_n \to v$ in $L^2(\Omega)$ . By Fatou's Lemma, I was able to show that $u \in H^s(\Omega)$, where
$$H^s(\Omega) = \left\{u \in L^2(\Omega); \int \int_{\Omega \times \Omega}\frac{|u(x) - u(y)|^2}{|x - y|^{(N + 2s)}}\right\}.$$
Now, I need to show that $\int_{\Omega}|(-\Delta)^su(x)|^2 dx < \infty$. I have tried in different ways, but without success. For example, the most I can do is show, by Fatou's Lemma, that $\int_{\Omega}|(-\Delta)^su(x)|^2 dx \geqslant \|v\|^2_{L^2(\Omega)}$.