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}\_τ(\mathbb{R}^{n}) $ of temperate ones admits generalized integral representations, with kernels belonging to $\mathcal{G}\_τ(\mathbb{R}^{n+m}) $. Furthermore, this result contains the classical one in the sense of the generalized distribution equality.