I was reading the lecture note
https://perso.math.u-pem.fr/cannone.marco/harmonic_analysis_tools.pdf (Lecture note by Marco Canone)
In his note page 8, (or exactly before section 1.2), he says that
'if two (scalar) functions $f$ and $g$ are in $H^1$, their product only belongs to $H^{1/2}$ and their derivative $\partial(fg)$ is even less regular as it belongs to $H^{−1/2}$.'
I cannot know why it is true.... even we don't know if $fg\in L^2$. Is it actually true?
I guess I need to estimate $$\int (1+|\xi|^2)^{1/2}\hat{f}(\xi)*\hat{g}(\xi)d\xi$$ but I am not sure how to estimate even.
He is usually considering a function in this note as a tempered distribution but that would not affect my question (I guess).
Thank in advance for any help.