\exp(-\phi(x))dx$ for $\phi \in C^2(\mathbb{R}^d)$ approximately quadratic (see Remark \[remAffine\]). We introduce a generalised Weyl pair $(Q,P)$ associated with such a measure via a specific unitary equivalence on $L^2(\mu)$ to the standard Weyl pair on $\mathbb{R}^d$ equipped with Lebesgue measure. Typically, the pair $(Q,P)$ will not generate uniformly bounded groups on $L^p(\mu)$ for $p \neq 2$. Thus, our generalised Weyl calculus will be developed as an extension theory: we work on $L^p(\mu)\cap L^2(\mu)$ and find conditions under which we have a bounded extension to $L^p(\mu)$.

For a function $M:\mathbb{R}^d \to \mathbb{R}^d$, we introduce the normed vector space $HS_0(M) \subset L^\infty\left(\mathbb{R}^{2d}\right)$ of Holomorphic Strip symbols. The main theorem of this paper (Theorem \[thmBoundedness0\]) proves that for a correctly chosen function $M$ based on $p$ and $\phi$, $a \in HS_0(M)$ implies that $a(Q,P)$ extends from $L^p(\mu)\cap L^2(\mu)$ to a bounded operator on $L^p(\mu)$, with norm bounded by $||a||_{HS_0(M)}$. We prove that symbols in $HS_0(M)$ have a certain holomorphic extendability property, which is reminiscent of the optimal functional calculus result for the classical Ornstein-Uhlenbeck operator in the Gaussian case. We also deduce a simple condition on a set $A \subset HS_0(M)$ which implies that $\{a(Q,P); a \in A\}$ is R-bounded on $L^p(\mu)$.

This 