Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

Abstract

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.

Submitted to the Journal of Mathematical Analysis and Applications

References in corpus (2)

• Generalized Integral Operators and Schwartz Kernel Theorem
• Remarks on the embedding of spaces of distributions into spaces of Colombeau generalized functions

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• A new approach to temperate generalized functions
• Topological properties of regular generalized function algebras
• Regularity, Local and Microlocal Analysis in Theories of Generalized Functions
• Geometrical embeddings for distributions into algebras of generalized functions