Hadamard operators on $\mathscr{D}'(\mathbb{R}^d)$

Abstract

We study continuous linear operators on $\mathscr{D}'(\mathbb{R}^d)$ which admit all monomials as eigenvectors, that is, operators of Hadamard type. Such operators on $C^\infty(\mathbb{R}^d)$ 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 $\mathscr{D}'(\mathbb{R}^d)$ 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 $S \in \mathscr{D}'(\mathbb{R}^d)$. Here $(S \star T)φ= S_y(T_x φ(xy))$ for all $φ\in \mathscr{D}(\mathbb{R}^d)$. To describe the behaviour at infinity we introduce a class $\mathscr{O}_H'(\mathbb{R}^d)$ of distributions defined by the same conditions like in the description of class $\mathscr{O}_C'(\mathbb{R}^d)$ of Laurent Schwartz, but derivatives replaced with Euler derivatives.