I would appreciate it if a reference could be given for the following claim.

Let $g$ be a Riemannian metric on $\mathbb R^n$, $n\geq 2$ that is equal to the Euclidean metric outside some compact set. Let us define for each $\alpha\in (0,1)$ the fractional Laplace--Beltrami operator on $\mathbb R^n$ via
$$ (-\Delta_g)^\alpha u = \int_0^\infty t^{-1-\alpha} (e^{\Delta_g}-1)u \,dt.$$
Prove that the operator $(-\Delta_g)^\alpha$ maps the **Schwartz space** $S(\mathbb R^n)$ continuously into itself.

I suppose that the following estimate on the heat kernel $$ |e^{t\Delta_g}(x,y)| \leq C t^{-\frac{n}{2}} e^{-c\frac{|x-y|^2}{t}} \quad \forall \, t>0 x,y \in \mathbb R^n,$$ is used to prove this but would anyone have a reference for this?