I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 48, he wrote "For the second term in (6.9), we use the identity $$\nabla\cdot(Df\nabla u) = f\nabla\cdot(D\nabla u) + \langle D\nabla u,\nabla f\rangle = f\nabla\cdot(D\nabla u) + f\langle D\nabla u,\nabla \log f\rangle.$$ So $$-\int f\left\langle \frac{C\nabla\cdot(Df\nabla u)}{f}, C'u\right\rangle = -\int f\left\langle C\nabla\cdot(Df\nabla u), C'u\right\rangle - \int f\left\langle C\left\langle D\nabla u,\nabla \log f\right\rangle\_{\mathbb{R}^N},C'u\right\rangle\_{\mathbb{R}^m}". $$ Here are some background: $C = (C\_1,\ldots,C\_m)$ and $C' = (C'\_1,\ldots,C'\_m)$ are $m$-turples of derivation operators on $\mathbb{R}^N$, $u = \log(f) + E$ with $E \in C^2(\mathbb{R}^N)$. I think the author made a serious mistake as components of $C$ are derivation operators on $\mathbb{R}^N$, thus (I think) we should have that
\begin{align\*} 
C\nabla\cdot(Df\nabla u) = C\left(f\nabla\cdot(D\nabla u) + f\langle D\nabla u,\nabla \log f\rangle\right) &= f\,C\nabla\cdot(Df\nabla u) + f\,C\left\langle D\nabla u,\nabla \log f\right\rangle \\
&\quad+ \mathrm{two~additional~terms}
\end{align\*}
So why should we have $$-\int f\left\langle \frac{C\nabla\cdot(Df\nabla u)}{f}, C'u\right\rangle = -\int f\left\langle C\nabla\cdot(Df\nabla u), C'u\right\rangle - \int f\left\langle C\left\langle D\nabla u,\nabla \log f\right\rangle\_{\mathbb{R}^N},C'u\right\rangle\_{\mathbb{R}^m} $$
without two additional terms on the right hand side?



A possible error in Villani's monograph "Hypocoercivity"
On [the PDF version on his website](https://cedricvillani.org/sites/dev/files/old_images/2012/07/043.Hypoco.pdf), he writes the left-hand side $$-\int f\left\langle C\left(\frac{\nabla\cdot(Df\nabla u)}{f}\right), C'u\right\rangle$$
which probably means it was a typo, the $C$ should be down the fraction. And there, you don't have this problem.


