Ornstein-Uhlenbeck operator, and what we shall refer to as the classical Ornstein-Uhlenbeck operator, is related to the choice $\phi(x) = \frac{x^2}{2}$. In its relationship to Fock spaces, a scaling is introduced to make the classical Ornstein-Uhlenbeck operator appear more symmetric in some sense, which corresponds to taking $c=\sqrt{2}$. However, as we shall see later (take for example Definition \[defnMultiplierExt\]), in many ways it seems more natural to take $c=1$. We will not include the subscript $c$ as part of the notation, but it will always be lurking in the background ready to be set to $1$ or $\sqrt{2}$. Note that the un-tilde’d $U_2$ falls under Definition \[defnUp\], and does not depend on $c$.

To ensure technical correctness, we need some information about the domains of important operators. The next theorem provides a suitable $p$-independent core for our functional calculus

\[thmCore\] The space $\mathcal{C}_\phi = U_2^{-1} C_c^\infty(\mathbb{R}^d)$ is dense in $L^p(\mu)$ for all $p \in [1, \infty)$, and elements of $\mathcal{C}_\phi$ are in $C^2_c(\mathbb{R}^d)$.

Regularity follows from the chain rule and the regularity of $\phi$. To show density in $L^p(\mu)$ we will show density of $U_p\mathcal{C}_\phi$ in $L^p(\lambda)$, employing the isometry of Definition \[defnUp\]. Note that $U_p\mathcal{C}_\phi$ contains every function of 