Tempered fundamental solutions

Question

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a tempered fundamental solution, namely a $E\in\mathscr{S}'$ with $P(\partial) E=\delta_0$.

Answer 1

Indeed, this has been demonstrated by L. Hörmander in 1958 ('On the division of distributions by polynomials', Arkiv för Matematik Band 3, nr 53; Theorem 5). In fact he showed the division problem tempered distributions; and proved that if $P$ is a polynomial that does not vanish identically, the multiplication mapping $\mathscr{S}\rightarrow\mathscr{S}$ defined by $f\mapsto Pf$ admits a continuous inverse. From this result the rest easily follows.