of $((-\Delta)^s
{\mathcal{S}}_k({\mathbb{R}}^d))'$:

\(i) Elements of homogeneous Sobolev spaces. Let $s\in {\mathbb{R}}$ and $H=s-d/2$. It is straightforward to verify that $f\in \dot{H}^s({\mathbb{R}}^d)$ determines an element of $((-\Delta)^s {\mathcal{S}}_H({\mathbb{R}}^d))'$ via $(f,\phi)\colonequals ({\widehat}{f},
{\widehat}{\phi})$. Furthermore, the definition of $(-\Delta)^sf$ arising from this correspondence satisfies ${\widehat}{(-{\Delta})^sf}(\xi)=|\xi|^{2s}{\widehat}{f}(\xi)$. It follows that $(-{\Delta})^s$ is an isometric isomorphism from ${\dot{H}}^{s_0}({\mathbb{R}}^d)$ to ${\dot{H}}^{s_0-2s}({\mathbb{R}}^d)$.

\(ii) Measurable functions $f:{\mathbb{R}}^d \to \mathbb{C}$ satisfying $$\label{eq:finite_integral}
\int_{{\mathbb{R}}^d}|f(x)|(1+|x|^{d+2s+k+1})^{-1}\,dx < \infty.$$ Interpreting $f$ as a linear functional on $(-\Delta)^s {\mathcal{S}}_k({\mathbb{R}}^d)$ by integration against a test function, the continuity of $f$ with respect the $\mathcal{U}_{s+(k+1)/2}({\mathbb{R}}^d)$ topology follows from Proposition \[prop:fraclapdecay\].

The following proposition gives an alternative representation of the fractional Laplacian in the case $0<s<1$.

\[prop:diff\_quo\] For all $f\in {\mathcal{S}}({\mathbb{R}}^d)$, $x\in
  {\mathbb{R}}^d$, and $s\in(0,1)$, we have $$\begin{aligned}
    (-{\Delta})^s f(x) = -\frac{1}{2}C(d,s)\int_{{\mathbb{R}}^d} \frac{f(x+y)-2f(x)
      +f(x-y) }{|y|^{d+2s}}\,dy,
  \end{aligned}$$ where $1/C(d,s) = \int_{{\mathbb{R}}^d} (1-\cos x_1) |x|^{-d-2s} \,dx$.

Combine Lemma 3.2 and Proposition 3.3 in [@dihitchhiker].

When $s\in \{0,1,2,\ldots\}$, the fractional Laplacian $(-{\Delta})^s$ coincides with the poly-Laplacian, a fundamental example of a higher order elliptic operator, obtained by iterating the Laplacian operator. For $s\in (0,1)$, $(-{\Delta})^s$ is a classical example of a non-local pseudo-differential operator. These two classes generate all the operators of the form $(-\Delta)^s$ for $s \geq 0$ in 