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We introduce a general context involving a presheaf A and a subpresheaf B ofA. We show that all previously considered cases of local analysis ofgeneralized functions (defined from duality or algebraic techniques) can beinterpretated as the B-local analysis of sections of A. But the microlocalanalysis of the sections of sheaves or presheaves under consideration isdissociated into a "frequential microlocal analysis " and into a "microlocalasymptotic analysis". The frequential microlocal analysis based on the Fouriertransform leads to the study of propagation of singularities under only linear(including pseudodifferential) operators in the theories described here, buthas been extended to some non linear cases in classical theories involvingSobolev techniques. The microlocal asymptotic analysis can inherit from thealgebraic structure of B some good properties with respect to nonlinearoperations.

Regularity, Local and Microlocal Analysis in Theories of Generalized Functions