On the Fractional Laplace-Beltrami operator

Question

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?

Answer 1

This is false even for the euclidean fractional Laplacian. For a cheap proof, the Fourier transform of $(-\Delta)^{a/2}u$ is $|\xi|^a\widehat u(\xi)$ which is not smooth at 0. For a more solid proof, it is not difficult to prove that $(-\Delta)^{a/2} u$ for $u$ a compactly supported function decays like $\sim|x|^{-n-a}$.