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Local properties of Colombeau generalized functions. (English) Zbl 1034.46037
The authors introduce and compare new notions describing various aspects of local behavior of elements in Colombeau algebras of generalized functions. The new regularity notions are based substantially on the intrinsically Colombeau-theoretic concepts of sharp topology and generalized points. Thus they are capable of describing particular non-classical features of Colombeau generalized functions, such as differentiability of functions at generalized points and regularity at a point.
Several explicit examples are provided which either discern subtle differences in regularity notions or show existence of sharply differentiable functions on generalized points that are not induced by Colombeau functions. Moreover, a deeper analysis of sheaf theoretic properties of Colombeau algebras and Colombeau microfunctions on open subsets is provided, where suppleness and non-flabbiness is proven.

46F10 Operations with distributions and generalized functions
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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