Let $n\in\mathbb N$. Let $f\in C_c^\infty(\mathbb R^n)$ and let $\Omega\subseteq\mathbb R^n$ be open such that it contains the support of $f$. I have encountered the following inequality
$$-\int_\Omega\left(f^*(x)\partial_nf(x)+\partial_nf^*(x)f(x)\right)\text{d}x\le2\|f\|_{L_2(\Omega)}\|\nabla f\|_{L_2(\Omega)^n},$$
where $\nabla f$ denotes the gradient of $f$ and $\partial_n f$ denotes the first derivative of $f$ with respect to the $n$-th coordinate. I believe that this should be simple enough to derive, but I am having difficulty doing so myself. I am not sure how to go from the expression on the left - which is close to the $L_1$- scenario to the $L_2$ - scenario on the right. One thought I had was to look at $f(x)f^*(x)=|f(x)|^2$ and to argue that both $f(x)\le|f(x)|^2$ and $f^*(x)\le|f(x)|^2$, but I realise that this is nonsensical since both $f(x)$ and $f^*(x)$ might be complex, whilst the right hand since is always purely real.
How does one derive this bound? What am I missing?
The integrand is $$ f^*(x)\partial_nf(x)+\partial_nf^*(x)f(x) = \partial_n |f(x)|^2 $$ and thus it is a real number. Therefore we can estimate $$ -(f^*(x)\partial_nf(x)+\partial_nf^*(x)f(x)) \le |f^*(x)\partial_nf(x)+\partial_nf^*(x)f(x)| \le 2 |\partial_n f(x)||f(x)| $$ since $|f| = |f^\ast|$ and $|\partial_n f| = |\partial_n f^\ast|$. By Cauchy-Schwarz, $$ -\int_\Omega\left(f^*(x)\partial_nf(x)+\partial_nf^*(x)f(x)\right)\text{d}x\le 2 \int_\Omega|\partial_n f(x)||f(x)| \text{d}x \le 2\|f\|_{L_2(\Omega)}\|\partial_n f\|_{L_2(\Omega)} $$ which implies the desired estimate.