Local regularity in non-linear generalized functions

In this review article we present regularity properties of generalized functions which are useful in the analysis of non-linear problems. It is shown that Schwartz distributions embedded into our new spaces of generalized functions, with additional properties described through the association, belong to various classical spaces with finite or infinite type of regularities.Comment: 17 page

Asymptotic ideals (ideals in the ring of Colombeau generalized constants with continuous parametrization)

We study the asymptotics at zero of continuous functions on (0, 1] by means of their asymptotic ideals, i.e., ideals in the ring of continuous functions on (0, 1] satisfying a polynomial growth condition at 0 modulo rapidly decreasing functions at 0. As our main result, we characterize maximal and prime ideals in terms of maximal and prime filters

Generalized Integral Operators and Schwartz Kernel Theorem

In connection with the classical Schwartz kernel theorem, we show that in the framework of Colombeau generalized functions a large class of linear mappings admit integral kernels. To do this, we need to introduce news spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that in some sense Schwartz' result is contained in our main theorem.Comment: 18 page

Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

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.Comment: Submitted to the Journal of Mathematical Analysis and Application

Symmetric hyperbolic systems in algebras of generalized functions and distributional limits

We study existence, uniqueness, and distributional aspects of generalized solutions to the Cauchy problem for first-order symmetric (or Hermitian) hyperbolic systems of partial differential equations with Colombeau generalized functions as coefficients and data. The proofs of solvability are based on refined energy estimates on lens-shaped regions with spacelike boundaries. We obtain several variants and also partial extensions of previous results and provide aspects accompanying related recent work by C. Garetto and M. Oberguggenberger

Asymptotic algebras and applications

International audienceStarting from a locally convex metrisable topological space and from any asymptotic scale, we construct a generalized extension of this space. To those extensions, we associate Hausdorff topologies. We introduce the notion of a temperate map, with respect to a given asymptotic scale, between two locally convex metrisable semi-normed spaces. We show that such mappings extend in a canonical way to mappings between the respective generalized extensions. We give an application to nonlinear Dirichlet boundary value problems with singular data in the framework of generalized extension