Kernel theorems in spaces of tempered generalized functions

Abstract

AbstractIn analogy to the classical isomorphism between $\mathcal{Lgrave;}(\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.