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Regularity, local and microlocal analysis in theories of generalized functions. (English) Zbl 1196.35027
Author’s abstract: We introduce a general context involving a presheaf \(\mathcal{A}\) and a subpresheaf \(\mathcal B\) of \(\mathcal{A}\). We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the \(\mathcal B\)-local analysis of sections of \(\mathcal{A}\).
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. Microlocal asymptotic analysis is a new spectral study of singularities. It can inherit from the algebraic structure of \(\mathcal B\) some good properties with respect to nonlinear operations.

MSC:
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A18 Wave front sets in context of PDEs
35L60 First-order nonlinear hyperbolic equations
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
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