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Abstract

In analogy to the classical isomorphism between $\mathcal{L}(\mathcal{D}(\mathbb{R}^{n}) ,\mathcal{D}^{\prime}(\mathbb{R}^{m})) $ and $\mathcal{D}^{\prime}(\mathbb{R}^{m+n})$ (resp. $\mathcal{L}(\mathcal{S}(\mathbb{R}^{n}),\mathcal{S} ^{\prime}(\mathbb{R}^{m}))$ and $\mathcal{S}^{\prime}(\mathbb{R}^{m+n})$), we show that a large class of moderate linear mappings acting between the space $\mathcal{G}_{C}(\mathbb{R}^{n}) $ of compactly supported generalized functions and $\mathcal{G}(\mathbb{R}^{n}) $ of generalized functions (resp. 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 specific regular subspaces of $\mathcal{G}(\mathbb{R}^{m+n})$ (resp. $\mathcal{G} _{\tau}(\mathbb{R}^{m+n})$). The main novelty is to use accelerated $\delta$-nets, which are unit elements for the convolution product in these regular subspaces, to construct the kernels. Finally, we establish a strong relationship between these results and the classical ones.