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

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.