TITLE: Domain of $(-\Delta)^\alpha$, $\alpha \in (0,1)$
QUESTION [5 upvotes]: Is there any simple way to characterize explicitly the domain of fractional powers for a given operator? For example, the domain of Dirichlet Laplacian on a bounded nice domain $\Omega \subset \mathbb{R}^n$:
$$(-\Delta)^\alpha, \quad \alpha \in (0,1).$$
I know that there is a characterization using interpolation theory, so one can find $H^{2\alpha}(\Omega)$ or $H^{2\alpha}_0(\Omega)$. But I'm asking if there is a systematic and simple way to do that, so we can use it for more complicated operators, for example matrix valued operators.
Any reference which gives the proof of the above characterization or a related topic will be very helpful.

REPLY [2 votes]: The problem can be, in general, very hard. A famous example is the Kato's conjecture (Wikipedia link) concerning the domain of the square root of certain elliptic operators, which was solved only in 2001, almost a half-century after Kato posed the question.