\in {\mathcal{S}}'({\mathbb{R}}^d)$ by $\psi h(f) =
h(\psi f)$. Furthermore, if $h\in H^{s}({\mathbb{R}}^d)$, then $\psi h\in H^s({\mathbb{R}}^d)$ as well [@ban-crainic Lemma 4.3.16]. For $h\in C_c^\infty(D)'$, we say that $h \in H_{\text{loc}}^{-s}(D)$ if $\psi h \in H^{-s}({\mathbb{R}}^d)$ for every $\psi \in C_c^\infty(D)$. We equip $H_{\text{loc}}^{-s}(D)$ with a topology generated by the seminorms $\|\psi\cdot\|_{H^{-s}({\mathbb{R}}^d)}$, which implies that $h_n \to h$ in $H_{\text{loc}}^{-s}(D)$ if and only if $\psi
h_n \to \psi h$ in $H^{-s}({\mathbb{R}}^d)$ for all $\psi \in C_c^\infty(D)$.

The Fractional Laplacian {#sec: fractional laplacian}
------------------------

The fractional Laplacian generalizes the notion of a power $(-{\Delta})^s$ of the Laplacian from nonnegative integer values of $s$, for which it is defined as a local operator by iterating the Laplacian, to all real values of $s$. A standard reference for the fractional Laplacian is [@landkof1972foundations]. Here we use ideas from Section 2 of [@silvestre2007regularity]. Let $k\in \{-1,0,1,2,\ldots\}$, and let $\phi \in {\mathcal{S}}_k({\mathbb{R}}^d)$. If $s>-\frac{1}{2}(d+k+1)$, then we set $$\label{eq:fraclapdef_smooth}
(-{\Delta})^s\phi \colonequals \mathcal{F}^{-1}\left[ \xi\mapsto
  |\xi|^{2s}{\widehat}{\phi}(\xi)\right],$$ which is well-defined because $\xi\mapsto |\xi|^{2s}{\widehat}{\phi}(\xi)$ is in $L^1({\mathbb{R}}^d)$. Note that agrees with the local definition of $-\Delta$ when $s=1$. Because of the singularity at the origin in its Fourier transform, $(-{\Delta})^s\phi$ is not necessarily Schwartz. However, it is real-valued, smooth, and has 