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We study the algebra \(\mathscr{E}'(\mathbb{R}^d)\) equipped with the multiplication \((T\star S)(\varphi)=T_x(S_y(f(xy))\) where \(xy=(x_1y_1,\dots,x_dy_d)\). For open sets \(\Omega'\), \(\Omega\subset \mathbb{R}^d\) we determine the distributions \(T\in\mathscr{E}'(\mathbb{R}^d)\) such that \(T\star\mathscr{E}'(\Omega')\subset \mathscr{E}'(\Omega)\), in particular, the \(T\) which operate on \(\mathscr{E}'(\Omega)\). They are distributions whose support is contained in the dilation sets \(V(\Omega',\Omega)\) or \(V(\Omega)\), respectively. By transposition we obtain a characterization of the \(\star\)-convolution operators which send \(C^\infty(\Omega)\) to \(C^\infty(\Omega')\) and, in particular, the \(\star\)-convolution operators on \(C^\infty(\Omega)\). These are called Hadamard-type operators and they are characterized by the property that they admit all monomials as eigenvectors. The algebra \(M(\Omega)\) of such operators is a closed subalgebra of \(L(C^\infty(\Omega))\) and we determine the topology induced from \(L_b(C^\infty(\Omega))\) on \(\mathscr{E}'(V(\Omega))\). We show that the algebra \(M(\Omega)\) is isomorphic to an algebra of holomorphic functions around zero where multiplication is the classical Hadamard multiplication, that is, multiplication of the Taylor coefficients. Hadamard-type operators assigned to distributions with support \(\{(1,\dots,1)\}\) are called Euler operators and we study global solvability for such operators on open subsets of \(\mathbb{R}_+^d\).
Analogous problems for real analytic functions are studied in papers of Domański-Langenbruch and a paper of Domański-Langenbruch-Vogt. While the problems are analogous, the results, the methods and the difficulties to overcome are, in part, quite different.
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