Kernel theorems in spaces of generalized functions

Antoine Delcroix

Banach Center Publications, Tome 89 (2010), p. 77-89 / Harvested from The Polish Digital Mathematics Library

Résumé

In analogy to the classical isomorphism between ((ℝⁿ), $ ^{\prime }(ℝ^{m}))$ and $ ^{\prime }(ℝ^{m+n})$ (resp. $((ℝⁿ),^{\prime }(ℝ^{m}))$ and $^{\prime }(ℝ^{m+n})$), we show that a large class of moderate linear mappings acting between the space $_{C}(ℝⁿ) $ of compactly supported generalized functions and (ℝⁿ) of generalized functions (resp. the space $_{}(ℝⁿ)$ of Colombeau rapidly decreasing generalized functions and the space $_{τ}(ℝⁿ)$ of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of $(ℝ^{m+n})$ (resp. $_{τ}(ℝ^{m+n})$). The main novelty is to use accelerated δ-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.

Détail

Publié le : 2010-01-01

Zbl 1203.46029

EUDML-ID : urn:eudml:doc:282250