spherical form and apply integration-by-parts with respect to the radial coordinate. For the second integral, we bound $\phi_2(x)$ by a constant times $\sup_{|\beta|=k+1}|\partial^\beta\phi(0)||\xi|^{-k-1}$ near the origin, using Taylor’s theorem.

For $s>-d/2$, we define[^7] the space $\mathcal{U}_s({\mathbb{R}}^d)$ to be the space of all functions $\phi\in
C^\infty({\mathbb{R}}^d)$ such that $$x\mapsto
(1+|x|^{d+2s})(\partial^\alpha f)(x) \label{not:yU?}$$ is bounded for all multi-indices $\alpha$. These spaces interpolate between $C^\infty({\mathbb{R}}^d)$ and ${\mathcal{S}}({\mathbb{R}}^d)$, as the derivatives of their elements decay polynomially at a rate indexed by $s$. In particular, ${\mathcal{S}}({\mathbb{R}}^d) \subset
\mathcal{U}_s({\mathbb{R}}^d) \subset \mathcal{U}_{s'}({\mathbb{R}}^d)$ whenever $s>s'$. We equip $\mathcal{U}_s({\mathbb{R}}^d)$ with the topology induced by the family of seminorms $f\mapsto \sup_{x\in {\mathbb{R}}^d} |(1+|x|^{d+2s})(\partial^\alpha
f)(x)|$. By Proposition \[prop:fraclapdecay\], $(-\Delta)^s$ is a continuous map from ${\mathcal{S}}_k({\mathbb{R}}^d)$ to $\mathcal{U}_{s+(k+1)/2}({\mathbb{R}}^d)$ \[not:thatsyU\]. Furthermore, $(-{\Delta})^s\phi=0$ for $\phi
\in {\mathcal{S}}_k({\mathbb{R}}^d)$ implies that ${\widehat}{\phi}$ vanishes except possibly at the origin. This implies that $\phi$ is a polynomial, which in turn implies that $\phi=0$. Therefore, $(-\Delta)^s$ is injective. For all $f$ in the topological dual of the image $(-\Delta)^s {\mathcal{S}}_k({\mathbb{R}}^d)\subset
\mathcal{U}_{s+(k+1)/2}$, we define $(-\Delta)^sf\in {\mathcal{S}}_k'({\mathbb{R}}^d)$ by $$((-\Delta)^sf,\phi) = (f,(-\Delta)^s\phi),$$ It is straightforward to verify that this definition agrees with when $f\in {\mathcal{S}}({\mathbb{R}}^d)$. Observe that $(-{\Delta})^{s_1}(-{\Delta})^{s_2} = (-{\Delta})^{s_1+s_2}$ for all $s_1,s_2\in {\mathbb{R}}$. We will consider two important examples of elements 