Relationship between Schwartz space and fractional Sobolev space

Question

Let $u$ be an element of the Schwartz space $\mathcal{S}(\mathbb{R}^N)$. Given the fractional Laplacian operator $(-\Delta)^s:\mathcal{S}(\mathbb{R}^N) \to L^2(\Omega)$, with $s \in (0,1)$ , defined by $$(-\Delta)^s u(x) := -\dfrac{1}{2}C(N,s)\int_{\mathbb{R}^N} \dfrac{u(x + y) + u(x - y) - 2u(x)}{|y|^{N + 2s}}dy,$$ where $C := C(N,s)$ is a real constant, is it possible to conclude that $(-\Delta)^s u \in \mathcal{S}(\mathbb{R}^N)$?

Answer 1

It’s not true that $(-\Delta )^s u \in \mathcal S (\mathbb R^n)$ given that $u \in \mathcal S (\mathbb R^n)$. It is true, however, that $u\in C^\infty (\mathbb R^n) \cap L^1(\mathbb R^n)$, see Corollary 2.10 in Fractional Thoughts by Nicola Garofalo.