LINEAR AND NON-LINEAR THEORY OF GENERALIZED FUNCTIONS AND ITS APPLICATIONS BANACH CENTER PUBLICATIONS, VOLUME 88 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2010 SHEAF THEORY AND REGULARITY. APPLICATION TO LOCAL AND MICROLOCAL ANALYSIS JEAN-ANDR ´ E MARTI Equipe Analyse Alg´ ebrique Non Lin´ eaire Laboratoire CEREGMIA Universit´ e des Antilles et de la Guyane 97275 Schoelcher Cedex, Martinique (France) E-mail: jean-andre.marti@univ-ag.fr Abstract. A review of some methods in sheaf theory is presented to make precise a general concept of regularity in algebras or spaces of generalized functions. This leads to the local analysis of the sections of sheaves or presheaves under consideration and then to microlocal analysis and microlocal asymptotic analysis. 1. Introduction. We are going to present in this article a review of certain methods in sheaf theory and to discuss the notion of regularity in algebras or spaces of generalized functions. We dissociate the microlocal analysis of the sections of sheaves or presheaves under consideration into so-called frequential microlocal analysis and a microlocal asymp- totic analysis. The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear operators (including pseudodifferential operators) in the theories described further on. However the study has been extended to certain non-linear cases in the classical theories involving Sobolev’s techniques. The microlocal asymptotic analysis is a new spectral study of singularities which gives some results involving nonlinear operations. In terms of sheaf theory, the notion of regularity in algebras or spaces of generalized functions can be formulated in a general way. If A is a presheaf of algebras or vector spaces on a topological space X, B is a subpresheaf of A and Ω is an arbitrary open set in X, then B(Ω) can be considered as the space or algebra of some regular elements of 2000 Mathematics Subject Classification : Primary 35A27; Secondary 35A18, 46F30, 35D10. Key words and phrases : sheaf theory, algebras of generalized functions, localization, microlocal- ization and wave front sets, nonlinear partial differential equations. The paper is in final form and no version of it will be published elsewhere. DOI: 10.4064/bc88-0-17 [211] c  Instytut Matematyczny PAN, 2010