Let $f:\mathbb{R}\rightarrow \mathbb{C}$ and $g:\mathbb{R}\rightarrow \mathbb{C}$. I would like to known if any of the following formula can fail when each of the quantities are well defined: $$ \mathcal{F}(f\ast g) = \mathcal{F}(f)\cdot\mathcal{F}(g) $$ $$ \mathcal{F}(f\cdot g) = \mathcal{F}(f)\ast\mathcal{F}(g) $$ $$ \mathcal{F}^{-1}(f\ast g) = \mathcal{F}^{-1}(f)\cdot\mathcal{F}^{-1}(g) $$ $$ \mathcal{F}^{-1}(f\cdot g) = \mathcal{F}^{-1}(f)\ast\mathcal{F}^{-1}(g) .$$
In case they do not always hold can someone provide a counter example (where each quantity is well defined)?