Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

Antoine Delcroix (1, 2)

1 CRREF - Centre de recherches et de ressources en éducation et formation
2 AOC - Analyse Optimisation Controle

We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{ \infty }-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.

Mots clés: Colombeau generalized functions; rapidly decreasing generalized functions; Fourier transform; microlocal regularity

Domaines: Analyse fonctionnelle [math.FA]

Antoine Delcroix. Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity. Journal of Mathematical Analysis and Applications, 2007, 327, pp.564-584. ⟨10.1016/j.jmaa.2006.04.045⟩. ⟨hal-00020280⟩