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Kernel theorems in spaces of generalized functions

100%

Delcroix A.

Banach Center Publications

2010

tom 88

nr 1

77-89

In analogy to the classical isomorphism between 𝓛(𝓓(ℝⁿ), $𝓓 '(ℝ^{m}))$ and $𝓓 '(ℝ^{m+n})$ (resp. $𝓛(𝓢(ℝⁿ),𝓢'(ℝ^{m}))$ and $𝓢'(ℝ^{m+n})$), we show that a large class of moderate linear mappings acting between the space $𝒢_{C}(ℝⁿ) $ of compactly supported generalized functions and 𝒢(ℝⁿ) of generalized functions (resp. the space $𝒢_{𝓢}(ℝⁿ)$ of Colombeau rapidly decreasing generalized functions and the space $𝒢_{τ}(ℝⁿ)$ of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of $𝒢(ℝ^{m+n})$ (resp. $𝒢_{τ}(ℝ^{m+n})$). 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.

Sequence spaces with exponent weights. Realizations of Colombeau type algebras

55%

Delcroix A., Hasler M., Pilipović S., Valmorin V.

Instytut Matematyczny Polskiej Akademii Nauk

We give a description of various algebras of generalized functions based on the introduction of pseudo-ultranorms on spaces of sequences in given locally convex function algebras. We study sheaf properties of these algebras, needed for microlocal analysis, and also consider regularity theory, functoriality and different concepts of association and weak equality in a unified setting. Using this approach, we also give new results on embeddings of ultradistribution and hyperfunction spaces into corresponding algebras of generalized functions.

Ograniczanie wyników

1 Banach Center Publications

2 Delcroix A.

1 Hasler M.

1 Pilipović S.

1 Valmorin V.

1 2010

1 2007

Wyniki wyszukiwania

Kernel theorems in spaces of generalized functions

Sequence spaces with exponent weights. Realizations of Colombeau type algebras