we consider are perturbations of the Laplacian by an unbounded drift term, including the classical finite-dimensional Ornstein-Uhlenbeck operator.

In the paper [@vNPWeylGaussian], van Neerven and Portal introduce a Gaussian Weyl calculus to study the classical Ornstein-Uhlenbeck operator $L=-\Delta + x \cdot \nabla$ on $L^p\left(\mathbb{R}^d, (2\pi)^{-\frac{d}{2}}\exp\left(-\frac{x^2}{2}\right)dx\right)$. This approach retrieves important known results about the Ornstein-Uhlenbeck semigroup, such as boundedness $L^p \to L^q$ and optimal domains of holomorphy for the semigroup, using analytic arguments as simple as Schur estimates. With classical approaches, such properties are difficult to prove.

The classical Ornstein-Uhlenbeck operator can be written in terms of a pair of operators P (momentum) and Q (position), satisfying the Heisenberg commutation relations. The Weyl calculus examined by van Neerven and Portal in [@vNPWeylGaussian] is a certain choice of joint functional calculus for this pair $(Q,P)$, that is, a way to assign to a suitable function $a:\mathbb{R}^{2d} \to \mathbb{C}$ a bounded operator $a(Q,P)$. It was their philosophy that studying $L$ via studying the joint functional calculus was more natural, as it separates the strong algebraic properties of the pair $(Q,P)$ from the analytic issues found in proving properties of $L$ directly. Essentially, studying $L$ directly is forgetting that it has its roots in 