Kernel Theorems in Spaces of Tempered Generalized Functions

Details Kernel Theorems in Spaces of Tempered Generalized Functions Kernel Theorems in Spaces of Tempered Generalized Functions

2006-03-02

arxiv.org/abs/math/0603035v1

Math. Proc. Camb. Philos. Soc. 142, 3 (2007) 557-572

Mathematics

Functional Analysis

15 pages

Scientific paper

10.1017/S0305004107000011

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.

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