TITLE: Continuous right inverse to the Laplacian operator on $C^\infty$
QUESTION [8 upvotes]: 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 $\Delta Sf\equiv f$ ?

REPLY [5 votes]: Assuming Question 2 asks about a continuous linear right inverse, the answer is no. In section 3, Theorem 3.3 of the article "Some results on continuous linear maps between Fréchet spaces" by Dietmar Vogt it is proved that no hypoelliptic operator on $C^{\infty}(\Omega)$ admits a continuous linear right inverse for any open $\Omega \subset \mathbb{R}^n$ with $n>1$.