EMIS ELibM Electronic Journals Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiques
Vol. CXXXIII, No. 31, pp. 163–174 (2006)

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Regular nonlinear generalized functions and applications

A. Delcroix

Universite des Antilles et de la Guyane, Laboratoire AOC, Campus de Fouillole, 97159 Pointe-a-Pitre, Guadeloupe, E-mail: Antoine.Delcroix@univ-ag.fr

Abstract: We present new types of regularity for Colombeau nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the simplified model. This generalizes the notion of ${\mathcal G}^{\infty}$-regularity introduced by M. Oberguggenberger. As a first application, we show that these new spaces are useful in a problem of representation of linear maps by integral operators, giving an analogon to Schwartz kernel theorem in the framework of nonlinear generalized functions. Secondly, we remark that these new regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microlocal analysis of singularities of generalized functions, with respect to these regularities.

Keywords: Colombeau generalized functions, Schwartz kernel theorem, rapidly decreasing generalized functions, Fourier transform, microlocal analysis

Classification (MSC2000): 35A18, 35A27, 42B10, 46E10, 46F30

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Electronic fulltext finalized on: 10 Jun 2006. This page was last modified: 20 Jun 2011.

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