Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and $\overline{\mathcal{F}}$ co-fourier-transform).
Prove: (1) $(TD_j - D_jT)\phi =0$ and (2) $T(x_j\phi)-x_jT(\phi)=0$ for $\phi\in \mathcal{S}$.
What's there to say other than "it's true because it's true". ?
Take for granted: For $\phi\in \mathcal{S}$ with $\phi(y) = 0$ we have $$\phi(x)=\sum_{j=1}^n(x_j-y_j)\phi_j(x), $$
for appropriate $\phi_j\in \mathcal{S}$. Show that (3) $\phi(x_0)=0$ implies $T(\phi)(x_0)=0$. Pick a positive function $\phi_0\in \mathcal{S}$, and apply this to $\phi(x_0)\phi_0-\phi_0(x_0)\phi$ to show there exist a function $c(x)$ so that $(T\phi)(x)=c(x)\phi(x)$ for all $\phi\in \mathcal{S}$.
Use relation (1) to show that $c(x)$ is a constant $c$. Take $\phi=e^{-|x|^2}$ to determine $c$ and derive the inversion formula for $\mathcal{S}$.
I understand why (3) holds by application of (2) but I don't see how to make an inversion-formula. I can also show $c= \frac{T(\phi_0)} {\phi_0}$,and why it is a constant. But i don't quite understand how to show (1),(2). Thanks for help and/or suggestions.