The division problem for tempered distributions of one variable (Q765927)

The authors deal with the following division problem: suppose \(F\in C^{\infty}(\mathbb{R})\) satisfies \(F\mathcal{S}(\mathbb{R})\subset\mathcal{S}(\mathbb{R})\) (such an \(F\) is called a multiplier). For each tempered distribution \(T\in\mathcal{S}'(\mathbb{R})\) find another \(S\in\mathcal{S}'(\mathbb{R})\) so that \(T=FS\). In the first section, the authors show that a positive solution to the division problem for \(F\) is equivalent to the multiplication operator \(M_F:\mathcal{S}(\mathbb{R})\to\mathcal{S}(\mathbb{R}),\,f\mapsto Ff\) having a closed range. In Theorem 2.1 they characterize, in terms of the properties of \(F\), when \(M_F\) has closed range. And in Theorem 2.3 they show that this is the case if and only if \(M_F\) has a linear and continuous left inverse. This last statement does no longer hold in the several variable case even for polynomials as follows from results due to \textit{M. Langenbruch} [Proc. R. Soc. Edinb., Sect. A 114, No. 3--4, 169--179 (1990; Zbl 0711.35032)].