Hadamard operators on $\mathscr {D}’(\mathbb {R}^d)$

Dietmar Vogt¹

¹ FB Math.-Nat. Bergische Universität Wuppertal Gauss-Str. 20 D-42119 Wuppertal, Germany

Studia Mathematica, Tome 237 (2017) no. 2, pp. 137-152

Résumé

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.

Détail

DOI : 10.4064/sm8481-1-2017

Keywords: study continuous linear operators mathscr mathbb which have monomials eigenvectors operators hadamard type operators infty mathbb space mathscr mathbb real analytic functions mathbb have investigated doma ski langenbruch author situation present however quite different characterization operator mathscr mathbb hadamard type there distribution support which has positive distance coordinate hyperplanes which has certain behaviour infinity star mathscr mathbb here star varphi x varphi varphi mathscr mathbb describe behaviour infinity introduce class mathscr mathbb distributions defined conditions description class mathscr mathbb laurent schwartz derivatives replaced euler derivatives