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

TL;DR

This paper characterizes Hadamard-type operators on the space of distributions, showing they are convolution operators with specific distributions that have controlled support and asymptotic behavior, differing from classical cases.

Contribution

It provides a new characterization of Hadamard operators on distributions using convolution with distributions having particular support and growth conditions, extending previous results on smooth and analytic functions.

Findings

Operators are convolution with distributions supported away from coordinate hyperplanes.

Introduces a class of distributions with specified support and asymptotic properties.

Provides a complete description of Hadamard operators on $\

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 $(S \star T)\varphi = S_y(T_x \varphi(xy))$ for all $\varphi \in \mathscr{D}(\mathbb{R}^d)$. 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.