I will leave a screen capture of the page in question. The author defines a sheaf mapping $D_{X} \to End_{\mathbb{C}}(\Omega_{X})^{op}$ which is a sheaf of rings morphism (here $^{op}$ means opposite ring, i.e., the same ring with opposite product: $a \times^{op} b = b \times a$). Here, the multiplication is taken to be composition, and the mapping is defined by, $a \mapsto (\omega \mapsto a\omega)$ and for $v \in \Theta_{X}$, $v \mapsto -L_{v}$. So, for example, if $P \in D_{X}$ is locally written as $P|_{U} = \sum \limits_{\alpha = (\alpha_{1},...,\alpha_{n})} a_{\alpha} \partial^{\alpha}$, then we have $P|_{U} \mapsto (\omega \mapsto \sum \limits_{\alpha = (\alpha_{1},...,\alpha_{n})}(-1)^{|\alpha|} L_{\partial_{x_{n}}}^{\alpha_{n}} ... L_{\partial_{x_1}}^{\alpha_{1}} (a_{\alpha} \omega))$, if I'm not mistaken.
The author points out that, by defining $wP$ as the action of the image of $P$ to $\omega$, we have a sort of generalization of formal integration by parts. This can be achieved by showing that for any top degree diffeential form $\omega$:
$$ (\omega P)a - \omega P(a) = d \eta$$ for some differential form $\eta$. However, I can't seem to show this. I know this can be checked locally and it is enough to check it for some $P = c \partial_{x_1}^{\alpha_{1}}...\partial_{x_n}^{\alpha_{n}} = c \partial^{\alpha}$ i think. Since $\omega$ is a top degree form, locally $\omega = f dx_{1} \wedge ... \wedge dx_{n}$, so the only thing I got out of this was that $\omega P = \partial^{\alpha}(cf) dx_{1} \wedge ... \wedge dx_{n}$, but this doesn't seem to help.
If anyone has any ideas, I appreciate any help in advance :)
