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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.

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    $\begingroup$ I do not think there is a simple answer, but Chapter 6: Domains, Uniqueness and the Cauchy Problem in [Martínez, Sanz, The Theory of Fractional Powers of Operators, Elsevier, 2001] might be a good starting point. $\endgroup$ – Mateusz Kwaśnicki Jan 14 at 12:03
  • $\begingroup$ Thank you! I will check it. $\endgroup$ – Migalobe Jan 14 at 12:17
  • $\begingroup$ It depends a lot on the situation. For example, you can look at this paper sciencedirect.com/science/article/pii/S0022123618301046 where the operator consider is a delta-like perturbation of the Laplacian. $\endgroup$ – Capublanca Jan 14 at 13:21
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The problem can be, in general, very hard. A famous example is the Kato's conjecture (en.wikipedia.org/wiki/Kato%27s_conjecture) 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.

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