The question is about the Exercise 9.18 from Folland Real Analysis:
[Folland, Chapter 9, Ex 18] If $n=l+m$, let us write $x \in \mathbb{R}^n$ as $(y,z)$ with $y \in \mathbb{R}^l$ and $z\in \mathbb{R}^m$. Let $\mathcal{F}$ denote the Fourier transform on $\mathbb{R}^n$ and $\mathcal{F}_1$, $\mathcal{F}_2$ the partial Fourier transforms in the first and second sets of variable -i.e., $\mathcal{F}_1f(\eta,z)=\int f(y,z)e^{-2 \pi i \eta \cdot y}dy$ and likewise for $\mathcal{F}_2$. Then $\mathcal{F}_1$ and $\mathcal{F}_2$ are isomorphisms on $\mathcal{S}(\mathbb{R}^n)$ and $\mathcal{S}'(\mathbb{R}^n)$, and $\mathcal{F}=\mathcal{F}_1\mathcal{F}_2=\mathcal{F}_2\mathcal{F}_1$.
$\mathcal{S}(\mathbb{R}^n)$ it is the Schwartz space and $\mathcal{S}'(\mathbb{R}^n)$ the tempered distribuition space.
The last part, it is an application of Fubini's Theorem, because $\mathcal{F}f \in \mathcal{S}(\mathbb{R}^n) \subset L^1(\mathbb{R}^n)=L^1(\mathbb{R}^l\times \mathbb{R}^m)$. I've been struggled with the isomorphism part. First at all, I think I have to prove that the operator $\mathcal{F}_1f \in \mathcal{S}(\mathbb{R}^n)$, I've tried with the equivalence, if $f \in C^\infty$ then $f \in \mathcal{S}(\mathbb{R}^n)$ if and only if, $\partial^\alpha(x^\beta f)$ is bounded $\forall \alpha, \beta$ multi-indices. But it is too hard. Could you give me any ideas? Thaks a lot.