As I was reading the note https://www-dimat.unipv.it/savare/Ravello2010/ravelloC.pdf (page 9), I was not sure why the last equality holds. It seems to me that the last expression can be obtained by integration by parts. However, I don't know if there is any boundary conditions assumed implicitly. Any help will be greatly appreciated!
Yes, this equality was obtained by integration by parts, and yes, it is implicitly assumed that all boundary terms vanish (which means that all functions involved decay sufficiently fast at infinity, since the integral is over the whole $\mathbb{R}^d$). This is very typical for formal computations like this, and would normally only be mentioned orally (if at all) in talks.