Hadamard operators on $\mathscr{D}'(\Omega)$

TL;DR

This paper characterizes Hadamard operators on the space of distributions over open sets in Euclidean space, showing they are convolution operators with distributions, extending known results from the entire space to open subsets.

Contribution

It extends the characterization of Hadamard operators from the whole space to arbitrary open subsets, using novel methods for the local case.

Findings

Hadamard operators are convolution operators with distributions.

Characterization applies to open subsets, not just the whole space.

New methods are developed for the local setting.

Abstract

For open sets $\Omega\subset \mathbb{R}^d$ we study Hadamard operators on $\mathscr{D}'(\Omega)$, that is, continuous linear operators which admit all monomials as eigenvectors. We characterize them as operators of the form $L(S)=S\star T$ where $T$ is a distribution and $\star$ the multiplicative convolution. This extends previous results for the case of $\Omega=\mathbb{R}^d$ but requires essentially different methods.