operators from ${\mathcal{S}}'({\mathbb{R}}^d)$ to ${\mathcal{S}}'({\mathbb{R}}^d)$. Regarding $\phi \in {\mathcal{S}}({\mathbb{R}}^d)$ as a tempered distribution via $\phi(\psi)\colonequals \int_{{\mathbb{R}}^d}\phi(x)\,\psi(x)\,dx$, we have the continuous, dense inclusion ${\mathcal{S}}({\mathbb{R}}^d) \subset {\mathcal{S}}'({\mathbb{R}}^d)$. For the fundamentals of the theory of distributions, we refer the reader to [@lax  Appendix B] or [@tao-epsilon]. For a more detailed introduction to distribution theory we refer to [@friedlander] and [@hormanderV1].

For $r\in {\mathbb{R}}$, define ${\mathcal{S}}_r({\mathbb{R}}^d) \subset {\mathcal{S}}({\mathbb{R}}^d)$ to be the set of Schwartz functions $\phi$ such that $(\partial^\alpha {\widehat}{\phi})(0) = 0$ (or, equivalently, $\int_{{\mathbb{R}}^d} x^\alpha \phi(x)\,dx=0$) for all multi-indices $\alpha$ such that $|\alpha|\leq r$ \[not:schwM\]. We equip ${\mathcal{S}}_r({\mathbb{R}}^d)$ with the subspace topology inherited from ${\mathcal{S}}({\mathbb{R}}^d)$ and denote by ${\mathcal{S}}_r'({\mathbb{R}}^d)$ the topological dual \[not:tdistM\] of ${\mathcal{S}}_r({\mathbb{R}}^d)$. Observe that ${\mathcal{S}}_r'({\mathbb{R}}^d)$ is canonically isomorphic to ${\mathcal{S}}'({\mathbb{R}}^d)/\mathcal{T}_r({\mathbb{R}}^d)$, where $\mathcal{T}_r({\mathbb{R}}^d)$ denotes the space of polynomials of degree at most $r$ \[not:poly\] on ${\mathbb{R}}^d$. Observe also that ${\mathcal{S}}_{r}({\mathbb{R}}^d)={\mathcal{S}}({\mathbb{R}}^d)$ whenever $r$ is negative, and that ${\mathcal{S}}_0({\mathbb{R}}^d)=\{\phi \in {\mathcal{S}}({\mathbb{R}}^d) \,:\, \int_{{\mathbb{R}}^d}\phi(x)\,dx
=0\}$.

Given $r\in {\mathbb{R}}$, we also consider the space $${\widetilde}{{\mathcal{S}}}_r({\mathbb{R}}^d) = \{\phi \in {\mathcal{S}}({\mathbb{R}}^d)\,:\, (\partial^\alpha \phi)(0) = 0
\text{ for all }|\alpha| \leq r\},$$ which is equal to the image of ${\mathcal{S}}_r({\mathbb{R}}^d)$ under the inverse Fourier transform operator\[not:schwMFI\]. We define the Fourier transform of an element of ${\mathcal{S}}_r'({\mathbb{R}}^d)$ as an 