Hadamard operators on ${\mathcal{D}}^{\prime }(\Omega )$

Abstract

For open sets $\Omega \subset {\mathbb{R}}^d$ we study Hadamard operators on ${\mathcal{D}}^{\prime }(\Omega )$, 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.