Regularity, Local and Microlocal Analysis in Theories of Generalized Functions

TL;DR

This paper develops a unified framework for local and microlocal analysis of generalized functions using presheaves, distinguishing between frequential and asymptotic microlocal methods, and exploring their properties and applications.

Contribution

It introduces a general setting for analyzing generalized functions via presheaves, unifies previous local analysis approaches, and separates microlocal analysis into frequential and asymptotic components.

Findings

Frequential microlocal analysis relates to propagation of singularities under linear operators.

Microlocal asymptotic analysis inherits algebraic properties suitable for nonlinear operations.

The framework extends classical microlocal analysis to generalized functions with new structural insights.

Abstract

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.