Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of power $\alpha \in (0,1)$ by
$$ (-\Delta_g)^\alpha u = \frac{1}{\Gamma(-\alpha)} \int_0^{\infty} t^{-\alpha-1}(e^{t\Delta_g}u-u)\,dt, \quad u\in C^{\infty}_0(\mathbb R^3).$$
I wonder if anyone knows a precise reference for the claim that the above operator can be continuously extended to an operator from $H^s(\mathbb R^3)$ to $H^{s-2\alpha}(\mathbb R^3)$ for all $s\in \mathbb R$.
(The key issue for me to use classical results on PsiDOs is that the above operator is not properly supported and so I don't know if I can use the classical machinery of PsiDO theory)
