bounded $H^\infty$ functional calculus of some angle less than $\pi/2$.

After developing the generalised Weyl calculus associated with the generalised Weyl pair $(Q,P)$ as defined above, we will aim to study the Ornstein-Uhlenbeck operator via the generalised Weyl calculus. In [@vNPWeylGaussian], the formal expression for the classical Ornstein-Uhlenbeck operator $L= \frac{1}{2}(Q^2+P^2-d)$ was very important. This is no longer true in our case. We have as a replacement the following theorem, representing $L$ associated with $\phi$ in the generalised Weyl calculus of the pair $(Q,P)$ associated with $\phi$ (at least formally).

\[thmH\] Take $Q,P$ as above, and let $h:\mathbb{R}^{2d} \to \mathbb{R}$ be the function taking value $h(x, \xi) = \frac{\xi^2}{c^2} + C_\phi (x)$, where $C_\phi(x) = \frac{1}{4}|\nabla\phi(cx)|^2 - \frac{1}{2}\Delta \phi(cx)$. For $f \in \mathcal{C}_\phi$, define $h(Q,P)f \in L^1_{loc}(\mathbb{R}^d)$ by $\left((x,\xi) \mapsto C_\phi(x)\right)(Q,P)$ interpreted as a multiplication operator by $C_\phi(x/c)$ and $\left((x,\xi) \mapsto \frac{\xi^2}{c^2}\right)(Q,P)$ interpreted as $\frac{1}{c^2}P^2$. Then $Lf = h(Q,P)f$.

Fix $f \in \mathcal{C}_\phi$. Let $H = U_2 L U_2^{-1}$. Note that $U_2 f =:g \in C_c^\infty(\mathbb{R}^d)$, so we want to consider $H$ acting on $g \in C_c^\infty(\mathbb{R}^d)$. Calculating $(Hg)(x)$ for all $x\in \mathbb{R}^d$ gives: $$(U_2^{-1} g)(x) = g(x/c)\exp\left(\frac{\phi(x)}{2}\right).$$ So $$\begin{aligned}
(LU_2^{-1}g)(x) &= \sum_{k=1}^d \left(\left(-\left(\partial_k\right)^2 + \partial_k \phi(\cdot) \partial_k \right)g(\cdot/c)\exp\left(\frac{\phi(\cdot)}{2}\right)\right)(x)\\
&= 