# Continuous right inverse to the Laplacian operator on $C^\infty$

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 know how to prove the existence of a solution of $\Delta u=f$, but the solution involves partition of unity and the Wlash theorem on harmonic approximation so the solution does not depend linearly on $f$. I do not see how to prove the existence of solution to $Pu=f$ for general operators with constant coefficients. The Malgrange-Ehrenpreis theorem proves the existence of a fundamental solution, but the convolution of $f$ with the fundamental solution does not make sense if $f$ growths fast. Can you provide references to the proof of the existence of a solution to $Pu=f$ in the general case? – Piotr Hajlasz Jun 1 '18 at 0:18
• Sorry I'm late. I somehow checked some original stuff by Malgrange before I asked, and I think he claimed the general $\Delta u=f$ case. My question was itself a reference-request, remember... – Jean Duchon Jun 11 '18 at 15:55