In the theory of algebraic $D$-modules, the Fourier transformation is defined as an automorphism $\mathcal{F}$ of $n$-dimensional Weyl algebra $D_n := \mathbb{K}[x_1,\dots,x_n]\langle\partial_{x_1},\dots,\partial_{x_n}\rangle$ that maps $x_i$ and $\partial_{x_i}$ to $-\partial_{x_i}$ and $x_i$, respectively.
This automorphism maps a differential operator
\begin{equation}
P = \sum_{(\alpha, \beta) \in \mathbb{N}^{2n}}c_{\alpha\beta}x^{\alpha}\partial^{\beta} \in D_n
\end{equation}
to its Fourier transform
\begin{equation}
\mathcal{F}[P] = \sum_{(\alpha, \beta) \in \mathbb{N}^{2n}}(-1)^{|\alpha|}c_{\alpha\beta}\partial^{\alpha}x^{\beta} \in D_n,
\end{equation}
where $x = (x_1, \dots, x_n)$, $\partial = (\partial_{x_1},\dots,\partial_{x_n})$, $\alpha$ and $\beta$ are multiindeces in $\mathbb{N}^n$, and $|\alpha| = \alpha_1 + \cdots + \alpha_n$.
On the other hand, the Fourier transform of a function $f(x) \ (x \in \mathbb{R}^n)$ is defined as
\begin{equation}
\mathcal{F}[f](\xi) := \int_{\mathbb{R}^n} e^{-i\xi^\top x} f(x) dx
\end{equation}
My question is about the relationship between these two Fourier transforms. At first, I hoped for the relationship $\mathcal{F}[P \circ f] = \mathcal{F}[P] \circ \mathcal{F}[f](\xi)$, where $\mathcal{F}[P]$ is treated as an element in $\mathbb{K}[\xi_1,\dots,\xi_n]\langle\partial_{\xi_1},\dots,\partial_{\xi_n}\rangle$ which is identical to $D_n$. However, it can be readily seen that this does not hold; Indeed, for the multivariate normal distribution \begin{equation} p(x) = \frac{1}{\sqrt{(2\pi)^n|\Sigma|}}e^{\frac{1}{2}(x-\mu)^\top\Sigma^{-1}(x-\mu)} \end{equation} and a vector of differential operators \begin{equation} P = \partial_x - (x -\mu)^\top\Sigma^{-1} \end{equation} that satisfy $P \circ p(x) = 0$, the characteristic function of the multivariate normal distribution \begin{equation} \mathcal{F}[p](\xi) = e^{i\mu^{\top} \xi - \frac{1}{2}\xi^{\top} \Sigma^{-1} \xi} \end{equation} does not satisfy $\mathcal{F}[P] \circ \mathcal{F}[p](\xi) = 0$ for the Fourier transform \begin{equation} \mathcal{F}[P] = \xi^\top + (\partial_\xi + \mu)^\top \Sigma^{-1}. \end{equation}
Can I somehow get the relationship $\mathcal{F}[P \circ f] = \mathcal{F}[P] \circ \mathcal{F}[f]$ with some modifications if needed? Or am I misunderstanding something? Thanks.