Let $\hat{f}$, $\hat{g}$ be the the Fourier transform of $f$ and $g$ respectively where $f$ and $g$ are the members of the Schwartz space $\scr{S}{(\mathbb{R}^{N})}$. Then in the process of establishing
$$
\hat{f*g}=\hat{f} \hat{g}
$$
we need to use Fubini's theorem. Indeed,
$$
\hat{f*g}(a)=\int_{\mathbb{R}^{N}}e^{-2\pi iax}(\int_{\mathbb{R}^{N}}f(x-y)g(y)dy)dx
\\
=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}e^{-2\pi iax}f(x-y)g(y)dydx
$$
Then to change the order of integration we need the Fubini's theorem but to apply Fubini's theorem we need that
$$
\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}|e^{-2\pi iax}f(x-y)g(y)|dy\times{dx}<\infty
$$
That is, $F\in{L^{1}}(dy\times{dx})$ where $F(x,y)=e^{-2\pi iax}f(x-y)g(y)$. But I am not sure how to establish that $F\in{L^{1}}(dy\times{dx})$?
Any help/suggestions would be great!
Since $g$ belong to the Schwarz space, for a fixed integer $p$ there is a constant $C_p$ for which the inequality $|g(y)|\leqslant C_p(1+|y|)^{-p}$ holds for any $y\in\mathbb R^N$. We thus are reduced to show that for some $p$ large enough, the integral $$\iint_{\mathbb R^N\times\mathbb R^N}\frac 1{(1+|x-y|)^p(1+|y|)^p}\mathrm dx\mathrm dy$$ is convergent.
Using the Fubini-Tonnelli theorem (for non-negative functions), we are reduced to find a $p$ such that $\int_{\mathbb R^N}\frac 1{(1+|x|)^p}\mathrm dx$ is convergent (for example $p\gt N$).
An "other" way would be to use the Fubini-Tonnelli theorem with the functions $|f|$ and $|g|$ then to prove that each element of the Schwartz space is integrable.