An inequality involving fractional laplacian

Question

I have to prove that for $s\in(0,1)$ and $\phi\in\mathcal{S}(\mathbb{R}^n)$, ($\phi$ is a Schwartz's function): $$|(-\Delta)^s \phi(x)|\leq c_{n,s}|x|^{-n-2s}, \quad\forall x\in\mathbb{R}^{n}\setminus B_1(0),$$ for some $c_{n,s}>0$. Can you give a me a proof or some reference where i can find the proof. Any help would be appreciated.Where, for all $x\in\mathbb{R}^n$: $$ (-\Delta)^s \phi(x)=-\frac{C(n,s)}{2} \int_{\mathbb{R^n}}\frac{\phi(x+y)+\phi(x-y)-2\phi(x)}{|y|^{n+2s}}\,dy,$$ or, on the whole $\mathbb{R}^n$: $$(-\Delta)^su=\mathcal{F}^{-1}(|\cdot|^{2s}\mathcal{F}u),$$ where $\mathcal{F}$ is the Fourier transform.

Answer 1

Have a look at Getting acquainted with the fractional Laplacian by Nicola Abatangelo and Enrico Valdinoci (you can access it for free on ArXiv). The exact result you have asked for is remarked on in Equation 2.10 and proven in Appendix B.