We know that
$$\mathscr{F}\left\{f*g\right\}=\mathscr{F}\left\{f\right\}\mathscr{F}\{g\}$$
so I was wondering whether the inverse is true:
$$\mathscr{F}\left\{fg\right\}=\mathscr{F}\left\{f\right\}*\mathscr{F}\{g\}$$
I can't seem to find this formula anywhere, but here's how it is derived. Let's suppose that we are in the domain of distributions (thus the inversion theorem is always valid). Antitransforming the former relation, you get
$$f*g=\mathscr{F}^{-1}\{\mathscr{F}\left\{f\right\}\mathscr{F}\{g\}\}$$
Then you postulate $f$ and $g$ to be the antitransforms of some distributions $\phi$ and $\psi$ (which is always the case), thus
$$\mathscr{F}^{-1}\{\phi\}*\mathscr{F}^{-1}\{\psi\}=\mathscr{F}^{-1}\{\phi\psi\}$$
Now, suppose that the product in the second term is well defined in the domain of distributions (which is almost always the case if any of the two is $C^{\infty}$). By replacing the antitransform with the transform, you get
$$\mathscr{F}\{\phi\psi\}=\mathscr{F}\{\phi\}*\mathscr{F}\{\psi\}$$
So, if I take into account every hypothesis, is the formula valid?