I see a statement that: for every tempered function $f$ and a "polynomial" of derivative operator $P(D)$ with constant coefficient, we can prove that the constant coefficient PDE
$P(D)u=f$,
has a tempered solution $u$ by using the following result:
For every tempered distribution $F$ and non zero polynomial $P$, we can find a tempered distribution $S$ such that:
$F=PS$,
And applying fourier transform on both sides of the equation.
I've seen it in corollary 1 and corollary 3 of Atiyah's paper: https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160230202
Or a more straight forward reference, in the page 1 of this paper by Hormander: https://math.uchicago.edu/~shmuel/lg-readings/Hormander,%20division.pdf
Maybe it's a trivial question but I'm confused because fourier transform would only carry distribution to distribution, I wonder how the do we obtain the function $P(D)u, f$ by applying fourier transform to distribution $PS, F$?
Thanks you for your answer!