Hadamard operators on $\mathscr {D}’(\mathbb {R}^d)$
Volume 237 / 2017
Abstract
We study continuous linear operators on $\mathscr {D}’(\mathbb {R}^d)$ which have all monomials as eigenvectors, that is, operators of Hadamard type. Such operators on $C^\infty (\mathbb {R}^d)$ and on the space $\mathscr {A}(\mathbb {R}^d)$ of real analytic functions on $\mathbb {R}^d$ have been investigated by Domański, Langenbruch and the author. The situation in the present case, however, is quite different, as also is the characterization. An operator $L$ on $\mathscr {D}’(\mathbb {R}^d)$ is of Hadamard type if there is a distribution $T$, the support of which has positive distance to all coordinate hyperplanes and which has a certain behaviour at infinity, such that $L(S) = S\star T$ for all $S\in \mathscr {D}’(\mathbb {R}^d)$. Here $(S\star T)\varphi = S_y(T_x\varphi (xy))$ for all $\varphi \in \mathscr {D}(\mathbb {R}^d)$. To describe the behaviour at infinity we introduce a class $\mathscr {O}_H’(\mathbb {R}^d)$ of distributions defined by the same conditions as in the description of the class $\mathscr {O}_C’(\mathbb {R}^d)$ of Laurent Schwartz, but with derivatives replaced by Euler derivatives.
