Fourier-Bros-Iagolnitzer transformation and first microlocalization
This first chapter is devoted to the definition of Fourier-Bros-Iagolnitzer (FBI) transformation and to its application to the study of microlocal regularity of distributions. The first section studies FBI transformations with quadratic phases, as those i
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1522
Jean-Marc Delort
R B. I. Transformation Second Microlocalization and Semilinear Caustics
Springer-Verlag Berlin Heidelberg GmbH
Author Delort Departement de Mathematiques Institut Galilee Universite Paris ~ Nord Avenue J.~B. Clement F-93430 Villetaneuse, France
Jean ~ Marc
Mathematics Subject Classification (1991): 35L 70, 35S35 , 58G 17
ISBN 978-3-540-55764-7 ISBN 978-3-662-21539-5 (eBook) DOI 10.1007/978-3-662-21539-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg New York in 1992 Typesetting: Camera ready using Springer lEX macropackage 46/3140-543210 - Printed on acid-free paper
Foreword
This text grew up from lecturcs givcn a t the University of Rennes I during the academic year 1988- 1989. The main topics covered a rc second microlocalization along a la grangian manifold , defined by Sjostrand in [Sj], and its application to the study of conormal singularities for solutions of semilinear hyperbolic partial differential equations, developed by Lebeau [L4]. To give a quite self-contained treatment of these questions, we induded some developments abou t F BI transformations and subanalytic geometry. The text is made oi fou r chapters. In the first one, we define the Fourier-Bros-Ingolnitzer transionnation and study its main proper ties. The second chapter deals with second microlocalization a long a lagrangian subman ifold, and with upper bounds for the wave front set of traces one may obt ai n using it. The third chap ter is devoted to formulas giving geometric upper bounds for the analytic wave front set and for the ser,ond mic:rosllpport of b oundary values of ramified functions. Lastly, the fourth chapter applies the preceding methods to the deriva tion of theorems about the location of microlocal singularities of solutions of scmilinear wave equations with conormw data, in general geometrical situation . Every chapter b egins with a short abstract of its contents, where are collected the bibliographical references. Let me now thank all those who made this writ ing possible. First of all, Gilles Lebeau, from whom I learnt microlocal analysis, especially through lectures he gave with Yves Laurent at Ecole Normale Superieure in 1982- 1983. Some of the notes of these lectures have been used for the writing of parts of Chapter 1. Moreover, h e communicated to me t·he manuscripts of some of his works quoted in the bibliogmphy b efore they
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