Geodesic
Regular nonlinear generalized functions and applications
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 31 (2006), p. 163 .

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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.
DOI : 10.2298/BMAT0631163D
Classification : 35A18 35A27 42B10 46E10 46F30
Keywords: Colombeau generalized functions, Schwartz kernel theorem, rapidly decreasing generalized functions, Fourier transform, microlocal analysis 
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     title = {Regular nonlinear generalized functions and applications},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
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A. Delcroix. Regular nonlinear generalized functions and applications. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 31 (2006), p. 163 . doi : 10.2298/BMAT0631163D. http://geodesic.mathdoc.fr/articles/10.2298/BMAT0631163D/

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