real-valued functions on ${\mathbb{R}}^d$ whose derivatives of all orders exist and decay faster than any \[not:schw\] polynomial at infinity. A multi-index $\beta = (\beta_1,\ldots,\beta_d)$ is an ordered $d$-tuple of nonnegative integers, and the order of $\beta$ is defined to be $|\beta|\colonequals \sum_{j=1}^d \beta_j$. We equip ${\mathcal{S}}({\mathbb{R}}^d)$ with the topology generated by the family of seminorms $$\left\{\raisebox{2pt}{$\displaystyle\|f\|_{n,\beta} \colonequals \sup_{x\in
      {\mathbb{R}}^d} |x|^n |\partial^\beta f(x)| \,:\,n \geq 0, \: \beta\text{ is a
      multi-index}$} \right\}.$$ The space ${\mathcal{S}}'({\mathbb{R}}^d)$ of tempered distributions is defined to be the space of continuous linear functionals on ${\mathcal{S}}({\mathbb{R}}^d)$ \[not:tdist\].

We take the convention that the Fourier transform $\mathcal F$ acting on a Schwartz function $\phi$ on ${\mathbb{R}}^d$ is the function $$\mathcal{F}[\phi](\xi) = \frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^d} \phi(x)e^{-i\xi \cdot x} \,dx$$ which we will often abbreviate as ${\widehat}{\phi}(\xi)$. The complex Schwartz space (the space of functions whose real and imaginary parts are in ${\mathcal{S}}({\mathbb{R}}^d)$) is closed under the operation of taking the Fourier transform [@tao-epsilon Section 1.13], so the inverse Fourier transform $\mathcal{F}^{-1}$ is well-defined on the complex Schwartz space and satisfies the formula $$\mathcal{F}^{-1}[\phi](x) =
\frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^d}\phi(\xi)e^{ix\cdot\xi}\,d\xi.$$

We define the Fourier transform ${\widehat}{f}$ of a tempered distribution $f$ by setting $({\widehat}{f},\phi)\colonequals (f,{\widehat}\phi)$, so that $\mathcal F$ and $\mathcal F^{-1}$ may be interpreted as 