dominated functions $\mathcal{D}_{\infty,1} = \{ g \in L^\infty(\mathbb{R}^d; L^1(\mathbb{R}^d)); \exists G \in L^1(\mathbb{R}^d), |g(x,k)|<G(k) \text{ for a.e. } x,k \in \mathbb{R}^d\}$, equipped with the norm $$||g||_{\mathcal{D}_{\infty,1}} = \inf\left\{ ||G||_{L^1(\mathbb{R}^d)}; G \in L^1(\mathbb{R}^d), |g(x,k)|<G(k) \text{ for a.e. } x,k \in \mathbb{R}^d\right\}.$$

We won’t prove that $\mathcal{D}_{\infty,1}$ is a Banach space, although it is easy. Subadditivity and homogeneity of $||\cdot||_{\mathcal{D}_{\infty,1}}$ is obvious. That $||\cdot||_{\mathcal{D}_{\infty,1}}$ is positive definite can be seen by noting it is bounded below by the $L^\infty(\mathbb{R}^d; L^1(\mathbb{R}^d))$ norm. Checking that $\mathcal{D}_{\infty,1}$ is complete is a standard exercise in telescoping sums, and using the fact that $L^\infty(\mathbb{R}^d; L^1(\mathbb{R}^d))$ is complete.

For $a\in L^\infty(\mathbb{R}^{2d})$, define $I_a:\mathbb{R}^d \to \mathcal{S}'(\mathbb{R}^d)$, via the action at $x \in \mathbb{R}^d$ and for $\varphi \in \mathcal{S}(\mathbb{R}^d)$ as $$\left\langle I_a(x), \varphi \right\rangle = \int_{\mathbb{R}^d} a(x, \xi)\varphi(\xi)d\xi.$$

We will make use of the Fourier transform $\mathcal{F}$ acting on tempered distributions $\sigma \in \mathcal{S}'(\mathbb{R}^d)$, which we normalise such that $\langle \mathcal{F}\sigma, \varphi \rangle = \langle \sigma, \mathcal{F}^*\varphi \rangle$ for all $\varphi \in \mathcal{S}(\mathbb{R}^d)$, where $$\mathcal{F}^*\varphi(k) = (2\pi)^{-\frac{d}{2}}\int_{\mathbb{R}^d} \varphi(\xi)\exp(i\xi k)d\xi.$$

Now we can define our symbol class.

\[defnHS0M\] Fix $M:\mathbb{R}^d \to \mathbb{R}^d$. The space $HS_0(M)$ (standing for Holomorphic Strip) is a subspace of $L^\infty(\mathbb{R}^{2d})$, with $$HS_0(M) = \left\{ 
\begin{array}{c}
 a \in 