Let $n \in \mathbb{N}$ and let $m \in \mathbb{R}$. Define $\langle \xi \rangle := (1 + |\xi|^2)^{1/2}$ for $\xi \in \mathbb{R}^n$, and let $\mathcal{B}(\mathbb{R}^n)$ denote the Borel $\sigma$-algebra on $\mathbb{R}^n$. For each $t \in \mathbb{R}$, let $H^t(\mathbb{R}^n)$ be the Fourier-side [Sobolev space](/page/Sobolev%20Space) consisting of [tempered distributions](/page/Tempered%20Distributions) $u \in \mathcal{S}'(\mathbb{R}^n)$ whose [Fourier transform](/page/Fourier%20Transform) has a measurable representative $\hat{u}: \mathbb{R}^n \to \mathbb{C}$ satisfying $\langle \xi \rangle^t\hat{u}(\xi) \in L^2(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n), \mathcal{L}^n)$, equipped with the norm

\begin{align*} \|u\|_{H^t(\mathbb{R}^n)} := \left(\int_{\mathbb{R}^n} \langle \xi \rangle^{2t}|\hat{u}(\xi)|^2\, d\mathcal{L}^n(\xi)\right)^{1/2}. \end{align*}

Let $a: \mathbb{R}^n \to \mathbb{C}$ be a $\mathcal{B}(\mathbb{R}^n)$-measurable function such that

\begin{align*} M_a := \operatorname{ess\,sup}_{\xi \in \mathbb{R}^n} \langle \xi \rangle^{-m}|a(\xi)| < \infty. \end{align*}

For every $s \in \mathbb{R}$, define $A$ on $H^s(\mathbb{R}^n)$ by requiring that, for every $u \in H^s(\mathbb{R}^n)$ and every measurable Fourier representative $\hat{u}$,

\begin{align*} \widehat{Au}(\xi) = a(\xi)\hat{u}(\xi) \end{align*}

for $\mathcal{L}^n$-a.e. $\xi \in \mathbb{R}^n$. Then this formula defines a well-defined bounded [linear map](/page/Linear%20Map)

\begin{align*} A: H^s(\mathbb{R}^n) \to H^{s-m}(\mathbb{R}^n) \end{align*}

and

\begin{align*} \|A\|_{\mathcal{L}(H^s(\mathbb{R}^n), H^{s-m}(\mathbb{R}^n))} \le M_a. \end{align*}