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arXiv · 1509.05759

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

Dietmar Vogt
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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 $\mathbb{R}_+^d$.

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Dietmar Vogt. 2015-09-18. $\mathscr{E}'$ as an algebra by multiplicative convolution. https://doi.org/10.7169/facm%2F1719

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