Consider $\Omega \subset \mathbb{R}^N$ a bounded and smooth domain. Moreover, consider that $u(x) = 0$ for all $x \in \mathbb{R}^N \backslash \Omega$. It is well known that Laplace's fractional operator is defined by
$$(-\Delta)^s u(x) = c_{n,s}\lim_{\varepsilon \to 0^+}\int_{\mathbb{R}^N \backslash B_\varepsilon (x)} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy.$$
I have been searching with great dedication for a domain $X = D((-\Delta)^s) \subset L^2(\Omega)$ such that $(-\Delta)^s : X \subset L^2(\Omega) \to L^2(\Omega)$ is a closed and densely defined linear operator. Due to the singularity in the integral of the definition of $(-\Delta)^s$, I have not been successful.