I am reading a text and having a hard time understanding by what magic the inequality
$$- \int \limits _M \langle \nabla u, \nabla f \rangle u f \le \frac 1 4 \int \limits _M |\nabla u|^2 f^2 + \int \limits _M |\nabla f|^2 u^2$$
has been produced, where $f$ is Lipschitz with compact support and $u$ is smooth in $L^2 (M)$ with $\nabla u \in \vec L^2 (M)$ (as a vector field, that is) and $\Delta u = \lambda u$, and the integration is performed on a Riemannian manifold $M$ with respect to the volume form. For "rigourists", this appears on page 301 of Grigor'yan "Heat kernel and analysis on manifolds", right above formula (11.16).
If you can help me in the case of $f$ smooth and $M = \mathbb R$, I believe that I shall be able to take it over from there. What puzzles me is the inequlity sign (what is missing from a true equality?) and the lack of symmetry in the right-hand side (the $\frac 1 4$ factor), given that the left-hand side is symmetric in $u$ and $f$ (of course, they have different regularity properties). That $u$ is an eigenfunction of $\Delta$ seems to play no role in the equality, since $\lambda$ does not appear in it.
It turns out to be embarassingly easy: the inequality
$$\int \limits _M \left\| \frac 1 {\sqrt 2} f \nabla u + \sqrt 2 u \nabla f\right\| ^2 \ge 0$$
is trivially true, whence the conclusion is immediate.