we wish to apply the Weyl calculus to - the original motivation of study for van Neerven and Portal in [@vNPWeylGaussian] - is the natural position and momentum operator pair in the standard Gaussian weighted spaces $L^p\left(\mathbb{R}^d, \exp\left(-\frac{x^2}{2}\right)dx\right)$, in which the momentum operator $P$ does not satisfy the bounded group generation property unless $p=2$. However, in this case we can define the Weyl calculus on $L^2$ via Definition \[defnWeylCalc\] and then determine conditions under which $a(Q,P)$ extends to a bounded operator on $L^p$. The cases we will consider will be based over $\mathbb{R}^d$ equipped with Borel measures of the following form.

A function $\phi \in C^2(\mathbb{R}^d)$ will be referred to as a potential. associated with a potential $\phi$ is the Borel measure $\mu$ on $\mathbb{R}^d$ with $d\mu = \exp(-\phi(x))dx$.

We will generally assume $\phi$ to be a twice differentiable function throughout the paper. Although the initial definitions work for any $C^2$ function $\phi$, to obtain boundedness of operators we will later have to restrict to $\phi$ which is approximately quadratic (see Remark \[remAffine\]). To be able to relate things to the standard pair, we proceed via unitary equivalence:

\[defnU2\] For $c>0$, define the (multiples of) unitary transformations $$\tilde{U}_2 = A 