$\exp(\pm \epsilon(x))$, which is bounded above and below by positive constants. Thus boundedness of operators in our generalised Weyl calculus should only depend on $c, p, N$ and $l$. However, the operators which we should care about (such as the relevant Ornstein-Uhlenbeck operator $L$), will depend on all of $\phi$, not just its unbounded terms.

It should be noted that the way this functional calculus is defined is quite different to the standard methods. Rather than a convergence lemma or a density argument, we have found an integral operator expression for “nice" symbols, and then extended to a large class of symbols for which the integral representation can be made sense of. Thus when we say that the generalised Weyl calculus extends to $HS_0(M)$, we mean both that for $a \in HS_0(M)$, $a(Q,P):\mathcal{C}_\phi \to L^1_{loc}(\lambda)$ as defined in Definition \[defnWeylHS0M\] extends uniquely to a bounded operator on $L^p(\mu)$ (by density of $\mathcal{C}_\phi$ as in Theorem \[thmCore\]), and also that if $a \in \mathcal{S}(\mathbb{R}^{2d}) \cap HS_0(M)$, then the expressions for $a(Q,P)$ from Definitions \[defnWeylCalc\] and \[defnWeylHS0M\] agree (by Lemma \[lemmaKernel\]).

\[remInitial\] Suppose that $a:\mathbb{R}^{2d} \to \mathbb{C}$ is such that for each $x \in \mathbb{R}^d$, $a(x, \xi + i \eta)$ can be 