Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

Details Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

2006-03-08

arxiv.org/abs/math/0603183v1

J. Math. Anal. Appl. 327 (2007) 564-584

Mathematics

Functional Analysis

Submitted to the Journal of Mathematical Analysis and Applications

Scientific paper

10.1016/j.jmaa.2006.04.045

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.

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