Hadamard operators on ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)$

We study continuous linear operators on ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)$ which admit all monomials as eigenvectors, that is, operators of Hadamard type. Such operators on $C^{\mathrm{\infty }}({\mathbb{R}}^d)$ and on the space $A({\mathbb{R}}^d)$ of real analytic functions on ${\mathbb{R}}^d$ have been investigated by Domanski, Langenbruch and the author. The situation in the present case, however, is quite different and also the characterization. An operator $L$ on ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)$ is of Hadamard type if there is a distribution T, the support of which has positive distance to all coordinate hyperplanes and which has a certain behaviour at infinity, such that $L(S)=S\star T$ for all $S\in {\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)$. Here $(S \star T)\varphi = S_y(T_x \varphi(xy))$forall$\varphi \in \mathscr{D}($\mathbb{R}$^d)$. To describe the behaviour at infinity we introduce a class ${\mathcal{O}}_H^{\mathrm{\prime }}({\mathbb{R}}^d)$ of distributions defined by the same conditions like in the description of class ${\mathcal{O}}_C^{\mathrm{\prime }}({\mathbb{R}}^d)$ of Laurent Schwartz, but derivatives replaced with Euler derivatives.