Kernel Theorems in Spaces of Tempered Generalized Functions

Abstract

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

15 pages

References in corpus (1)

• Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity