Regularity, Local and Microlocal Analysis in Theories of Generalized Functions

Jean-André Marti (1)

1 GTSI - Groupe de Technologie des Surfaces et Interfaces [URp2_2]

We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a "frequential microlocal analysis " and into a "microlocal asymptotic analysis". The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations.

Mots clés: Spaces and algebras of generalized functions; sheaf theory; localization; microlocalization; propagation of singularities; nonlinear PDE

Domaines: Analyse fonctionnelle [math.FA]

Jean-André Marti. Regularity, Local and Microlocal Analysis in Theories of Generalized Functions. 2007. ⟨hal-00190006⟩