$\mathcal{E}'$ as an algebra by multiplicative convolution

Abstract

We study the algebra $\mathcal{E}'(\mathbb{R}^d)$ of distributions with compact support equipped with the multiplication $(T\star S)(f)=T_x(S_y(f(xy))$ where $xy=(x_1 y_1,\dots,x_d y_d)$. This allows us a very elegant access to the theory of Hadamard type operators on $C^\infty(\Omega)$, $\Omega$ open in $\mathbb{R}^d$, that is, of operators which admit all monomials as eigenvectors. We obtain a representation of the algebra of such operators as an algebra of holomorphic functions with classical Hadamard multiplication. Finally we study global solvability for such operators, in particular of Euler differential operators, on open subsets of $\mathbb{R}_+^d$.