L^\infty(\mathbb{R}^{2d}); \exists ! g_a \in \mathcal{D}_{\infty,1}, \forall x \in \mathbb{R}^d, \forall \varphi \in \mathcal{S}(\mathbb{R}^d),\\
   \left\langle \mathcal{F}(I_a(x)), \varphi \right\rangle = (2\pi)^\frac{d}{2}\int_{\mathbb{R}^d}\exp\left(-|M(x)k|\right)g_a(x,k)\varphi(k)dk
\end{array} \right\}$$ and norm defined by $$||a||_{HS_0(M)} = ||g_a||_{\mathcal{D}_{\infty,1}}.$$ For $a \in HS_0(M)$, we define $\mathcal{F}_2a$ to be the measurable function $\mathbb{R}^{2d} \to \mathbb{C}$ with action $$(x, k) \mapsto (2\pi)^\frac{d}{2} \exp\left(-|M(x)k|\right)g_a(x,k).$$

In some sense, $\mathcal{F}_2a(x,k)$ is the “Fourier transform in the second ($\xi$) variable" with $k$ the variable dual to $\xi$, which is our reason for using this notation. It is easy to verify that if $a(x, \cdot) \in L^1(\mathbb{R}^d)$ for each $x\in \mathbb{R}^d$, then $\mathcal{F}_2a$ is indeed the Fourier transform in the second variable for each fixed $x$. However, not all symbols we will consider will be integrable in the second variable, which makes our definition via the space of tempered distributions useful.

We will also often refer to the $g_a \in \mathcal{D}_{\infty,1}, G_a \in L^1(\mathbb{R}^d)$ associated with $a \in HS_0(M)$, by which we mean the unique such $g_a$ as seen in Definition \[defnHS0M\], and $G_a$ dominating $g_a$ as in Definition \[defnD\]. The main reason for using such a complicated definition is that the integral kernel of $a(Q,P)$ for $a \in \mathbb{S}(\mathbb{R}^{2d})$ from Theorem \[thmKernel\] is closely related to 