Kernel Theorems in Spaces of Tempered Generalized Functions

Antoine Delcroix (1)

1 AOC - Analyse Optimisation Controle

In analogy to the classical isomorphism between $\mathcal{L}\left( \mathcal{S}\left( \mathbb{R}^{n}\right) ,\mathcal{S}^{\prime}\left( \mathbb{R}^{m}\right) \right) $ and $\mathcal{S}^{\prime}\left( \mathbb{R}^{n+m}\right) $, we show that a large class of moderate linear mappings acting between the space $\mathcal{G}_{\mathcal{S}}\left( \mathbb{R}^{n}\right) $ of Colombeau rapidly decreasing generalized functions and the space $\mathcal{G}_{\tau}\left( \mathbb{R}^{n}\right) $ of temperate ones admits generalized integral representations, with kernels belonging to $\mathcal{G}_{\tau}\left( \mathbb{R}^{n+m}\right) $. Furthermore, this result contains the classical one in the sense of the generalized distribution equality.

Mots clés: kernel Theorem; Colombeau temperate generalized functions; integral operator; temperate distributions

Domaines: Analyse fonctionnelle [math.FA]

Antoine Delcroix. Kernel Theorems in Spaces of Tempered Generalized Functions. Math. Proc. Camb. Philos. Soc., 2007, 142 (3), pp.557-572. ⟨10.1017/S0305004107000011⟩. ⟨hal-00019916⟩