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

TL;DR

This paper explores the algebraic structure of distributions on Euclidean space with a multiplicative convolution, linking it to Hadamard operators and their representation as holomorphic functions, and investigates their global solvability.

Contribution

It introduces a novel algebraic framework for $ ext{E}'( ext{R}^d)$ with multiplicative convolution, connecting it to Hadamard operators and their functional representations.

Findings

Representation of Hadamard operators as holomorphic functions

Characterization of the algebra of such operators

Results on global solvability on open subsets of $ ext{R}_+^d$

Abstract

We study the algebra $\mathscr{E}'(\mathbb{R}^d)$ equipped with the multiplication $(T\star S)(f)=T_x(S_y(f(xy))$ where $xy=(x_1y_1,\dots,x_dy_d)$. 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 on open subsets of $\mathbb{R}_+^d$.