This is part of the getting the Diamagnetic Kinetic Energy in "Large Fermionic Systems" by Fournais, Lewin & Solovej (2015) p.17
I will summarise the question I have.
Define $$ \cal{F}[g](p)= (2\pi\hbar)^{-d} \int_{R^d} g(x)e^{-\frac{i}{\hbar}p\cdot x} dx $$ be a $\hbar$-scaled Fourier Transform.
Furthermore let $f_{x,0}^\hbar$ be a real-valued function. And $u\in L^2(R^d)$ be some complex function (wave function).
My question is, how did the author get from line 3 to line 4 in the proof: $$ \int_{R^d} p\ |\cal{F}[f_{x,0}^\hbar u](p)|^2\ dp= \hbar \ \Im \bigg(\int_{R^d}f_{x,0}^\hbar (y) \bar{u}(y)\ \triangledown(f_{x,0}^\hbar u)(y)\ dy \bigg), $$ where $\Im$ is the imarginary part.
I ran out of ways. I thought of integration by parts and/or use the Fourier transformation of a derivative, but to no avails. Help is appreciated! I am especially confused where the $\Im$ comes from.
P.s. I'm equally confuse how he got from line 4 to line 5 :(
Solved it.
We utilise the Fourier transformation of gradient $\cal{F}[\triangledown g](p) = \frac{i}{\hbar}p \cal{F}[g](p)$ as well as the $\cal{F} [gh](p) = (g\ast h)(y) $