For each $f\in C^\infty(\mathbb R^n)$, there exists $u\in C^\infty(\mathbb R^n)$ such that $\Delta u=f$. This, I guess, has been known well before the more general Malgrange-Ehrenpreis theorem that says the same for a general differential operator with constant coefficients.

My question is twofold:

1°) Who proved this first?

2°) Does there exist a continuous operator $S:C^\infty(\mathbb R^n)\to C^\infty(\mathbb R^n)$ such that $\Delta Sf\equiv f$ ?

I thinkhe claimed the general $\Delta u=f$ case. My question was itself areference-request, remember... $\endgroup$ – Jean Duchon Jun 11 '18 at 15:55