the Fourier transform in the second variable of the Schwartz function $a$, as in the following lemma.

\[lemmaKernel\] Fix a potential $\phi \in C^2$, and let $(Q, P)$ be the associated generalised Weyl pair. For $a \in \mathcal{S}(\mathbb{R}^{2d})$, the integral kernel of the operator $a(Q,P)$ is given by $$k(y,x) = \frac{1}{(2\pi)^{d/2}c^d} \mathcal{F}_2 a\left(\frac{x+y}{2c}, \frac{x-y}{c}\right) \exp\left(\frac{1}{2}(\phi(y)-\phi(x))\right).$$

From Theorem \[thmKernel\], we have for $f \in \mathcal{C}_\phi$, $y \in \mathbb{R}^d$, $$\begin{aligned}
(a(Q,P)f)(y) &= \frac{1}{(2\pi)^{d}c^d} \int_{\mathbb{R}^{2d}}a\left(\frac{x+y}{2c}, \xi\right) \exp\left(-i\xi \left(\frac{x-y}{c}\right)\right) \exp\left(\frac{1}{2}(\phi(y)+\phi(x))\right)f(x) d\xi d\mu(x) \\
&= \frac{1}{(2\pi)^{d}c^d} \int_{\mathbb{R}^{2d}}a\left(\frac{x+y}{2c}, \xi\right) \exp\left(-i\xi \left(\frac{x-y}{c}\right)\right) \exp\left(\frac{1}{2}(\phi(y)-\phi(x))\right)f(x) d\xi dx \\
&= \frac{1}{(2\pi)^{d/2}c^d} \int_{\mathbb{R}^{d}}\mathcal{F}_2a\left(\frac{x+y}{2c}, \frac{x-y}{c}\right) \exp\left(\frac{1}{2}(\phi(y)-\phi(x))\right)f(x) dx, \\\end{aligned}$$ noting that the order of integration can be changed, and the integration in $\xi$ carried out, as everything converges absolutely.

We make use of Lemma \[lemmaKernel\] to extend the generalised Weyl calculus to $HS_0(M)$, as in the following definition:

\[defnWeylHS0M\] For any function $M:\mathbb{R}^d \to \mathbb{R}^d$, and any potential $\phi \in C^2$, define for $a \in HS_0(M)$ the operator $a(Q,P):\mathcal{C}_\phi \to L^1_{loc}(\lambda)$ via the action $$(a(Q,P)f)(y) = \frac{1}{(2\pi)^{d/2}c^d} \int_{\mathbb{R}^d} \mathcal{F}_2 a\left(\frac{x+y}{2c}, \frac{x-y}{c}\right) \exp\left(\frac{1}{2}(\phi(y)-\phi(x))\right)f(x)dx.$$

To ensure this definition makes sense, we check that for $f \in \mathcal{C}_\phi$, $a(Q,P)f \in L^1_{loc}(\mathbb{R}^d)$.

Using the fact that $a \in HS_0(M)$ for some $M:\mathbb{R}^d \to \mathbb{R}^d$, there exists a 