\sum_{k=1}^d \left(-\partial_k\left( \frac{1}{c}\partial_k g(\cdot/c)\exp\left(\frac{\phi(\cdot)}{2}\right) + \frac{1}{2}g(\cdot/c)\exp\left(\frac{\phi(\cdot)}{2}\right)\partial_k\phi(\cdot) \right)\right)(x) \\
&+ \partial_k \phi(x) \left( \frac{1}{c}\partial_k g(x/c)\exp\left(\frac{\phi(x)}{2}\right) + \frac{1}{2}g(x/c)\exp\left(\frac{\phi(x)}{2}\right)\partial_k\phi(x) \right)\\
&= \sum_{k=1}^d  -\frac{1}{c^2}\partial^2_k g(x/c)\exp\left(\frac{\phi(x)}{2}\right) -  \frac{1}{2c}\partial_k g(x/c)\exp\left(\frac{\phi(x)}{2}\right)\partial_k\phi(x) \\
&- \frac{1}{2c}\partial_k g(x/c)\exp\left(\frac{\phi(x)}{2}\right)\partial_k\phi(x) - \frac{1}{4}g(x/c)\exp\left(\frac{\phi(x)}{2}\right)(\partial_k\phi(x))^2 - \frac{1}{2}g(x/c)\exp\left(\frac{\phi(x)}{2}\right)\partial^2_k\phi(x) \\
&+  \frac{1}{c}\partial_k g(x/c)\exp\left(\frac{\phi(x)}{2}\right)\partial_k \phi(x) + \frac{1}{2}g(x/c)\exp\left(\frac{\phi(x)}{2}\right)(\partial_k\phi(x))^2 \\
&= \sum_{k=1}^d  -\frac{1}{c^2}\partial^2_k g(x/c)\exp\left(\frac{\phi(x)}{2}\right) + \frac{1}{4}g(x/c)\exp\left(\frac{\phi(x)}{2}\right)(\partial_k\phi(x))^2 - \frac{1}{2}g(x/c)\exp\left(\frac{\phi(x)}{2}\right)\partial^2_k\phi(x) \\
&= \left(-\frac{1}{c^2}\Delta g(x/c) +\left(\frac{1}{4}|\nabla\phi(x)|^2 - \frac{1}{2}\Delta\phi(x)\right)g(x/c)\right)\exp\left(\frac{\phi(x)}{2}\right).\end{aligned}$$ So $$\begin{aligned}
(Hg)(x) &= -\frac{1}{c^2}\Delta g(x) +\left(\frac{1}{4}|\nabla\phi(cx)|^2 - \frac{1}{2}\Delta\phi(cx)\right)g(x)\\
&= \left(\left(-\frac{1}{c^2}\Delta + C_\phi(\cdot)\right)g(\cdot)\right)(x).\end{aligned}$$ This can be expressed directly in the standard Weyl calculus as $h(X,D)$, for $h(x, \xi) := \frac{\xi^2}{c^2} + C_\phi (x)$ with the same interpretation as in the statement of this theorem. But $h(Q,P)f = U_2^{-1} h(X,D) U_2f = Lf$, and so we are done.

$L^p$ bounds on Weyl Pseudodifferential Operators
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In this section we investigate properties of the generalised Weyl calculus associated with a potential $\phi$, insofar as they relate to the functional calculus for the corresponding Ornstein-Uhlenbeck operator. In a later section we will return to the study of the symbol calculus.

Our symbols will correspond to certain functions $a\in L^\infty(\mathbb{R}^{2d})$, which shall be denoted $a(x, \xi)$ for $x, \xi \in \mathbb{R}^d$. To define our symbol class, we need a few intermediate definitions.

\[defnD\] Define the Banach space of $L^1$ 