polynomial decay at infinity:

\[prop:fraclapdecay\] Let $k\in
  \{-1,0,1,2,\ldots\}$, $\phi \in {\mathcal{S}}_k({\mathbb{R}}^d)$, and $s>-\frac{1}{2}(d+k+1)$. If $\alpha$ is a multi-index, then there exists a constant $C$ such that $\phi \in {\mathcal{S}}_k({\mathbb{R}}^d)$ implies $$\label{eq:polydecay} \sup_{x\in
      {\mathbb{R}}^d}(1+|x|^{d+2s+k+1})|\partial^\alpha(-{\Delta})^s\phi(x)| \leq C
    \sup_{|\beta|\leq \max(k+1,|\alpha|)}\|\partial^\beta \phi\|_{L^\infty({\mathbb{R}}^d)}.$$ Furthermore, $(-{\Delta})^s\phi$ is real-valued and smooth.

The proof of smoothness is routine: we write the inverse Fourier transform using its definition and differentiate under the integral sign. To show that $(-{\Delta})^s\phi$ is real-valued, we note that a function $\psi\in L^1({\mathbb{R}}^d)$ is the Fourier transform of a real-valued function in $L^1({\mathbb{R}}^d)$ if and only if $\psi(-\xi)$ and $\psi(\xi)$ are complex conjugates for all $\xi \in {\mathbb{R}}^d$. The function $\xi\mapsto
  |\xi|^{-2s}{\widehat}{\phi}(\xi)$ has this property whenever ${\widehat}{\phi}$ does, so we may conclude that $(-{\Delta})^s\phi$ is real-valued.

To prove , let $\{f_1,f_2\}$ be a partition of unity subordinate to the open cover $\left\{{\mathbb{R}}^d\setminus
  B\left(0,\frac{1}{|x|}\right),B\left(0,\frac{2}{|x|}\right)\right\}$ of ${\mathbb{R}}^d$, and define $\phi_i(\xi)= f_i(\xi) {\widehat}{\phi}(\xi)/|\xi|^{k+1}$ for $i \in
  \{1,2\}$. We calculate $$\begin{aligned}
    \partial^{\alpha}(-{\Delta})^s \phi(x) &= C \int_{{\mathbb{R}}^d} e^{i x\cdot \xi}
    \xi^\alpha|\xi|^{2s+k+1}
    {\widehat}{\phi}(\xi)/|\xi|^{k+1}\,d\xi \\
    &= C \int_{{\mathbb{R}}^d\setminus B(0,1/|x|)} e^{i x\cdot \xi}
    \xi^\alpha|\xi|^{2s+k+1}
    \phi_1(\xi)\,d\xi + \\
    &\hspace{2cm}C\int_{B(0,2/|x|)} e^{i x\cdot \xi}
    \xi^\alpha|\xi|^{2s+k+1} \phi_2(\xi)\,d\xi,
  \end{aligned}$$ where $C$ is some constant. To obtain the desired bound for the first integral, we write the integral in 