I have a general function $u(x,y,z,t)$. Then,
(1) what would be the space-time Fourier transform of $$G \star \frac{\partial^n u}{ \partial t^n }$$
and
(2) would the relation $$G \star \frac{\partial^n u}{ \partial t^n } = \frac{\partial^n (G \star u)}{ \partial t^n }$$ hold true?
Here, note that (a) the symbol $\star$ represents convolution in space and (b) $G$ is a function of $(x,y,z)$ only (i.e. it does not depend on time).