I am trying to wrap my head around some inequalities which appear in Cedric Villani's book "Hypercoercivity". The inequalities involve mollifiers, Hessian operator and gradient operator. These look simple, but I couldn't understand them yet. So, any help would be appreciated. I've attached the relevant page.
First of all, Condition (7.3) is "There exists a constant $c \ge 0$ such that $\vert \nabla^2 V \vert \le c(1 + \vert \nabla V \vert )$." Second, $V(x)$ is a real-valued function defined on $\mathbb{R}^n$.
(1) With the first inequality, if I replace $\nabla V_\epsilon (x)$ and $\nabla V(x)$ with $f_\epsilon (x)$ and $f(x)$ where $f(x)$ is a real-valued function, then I can understand. However, $\nabla V(x)$ is a multi-valued function. How can I show this inequality holds?
(2) With the second inequality, I really have no idea of how he derived the L-Lipschitz condition from (7.3) only. (Why is there $\vert \nabla V(x)\vert^2$ instead of $\vert \nabla^2 V(x)\vert$?)
(3) I tried the multidimensional mean-value theorem, but it didn't work, especially deriving the term $e^{L\epsilon}$ on the right-hand side. It might come from the second inequality, but I don't know how.
So, I'd appreciate it if you'd give me some help. Thank you.