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Generalized oscillatory integrals and Fourier integral operators

Part of: Partial differential equations Distributions, generalized functions, distribution spaces Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  28 May 2009

Claudia Garetto ,
Günther Hörmann  and
Michael Oberguggenberger
Claudia Garetto
Affiliation:
Institut für Grundlagen der Bauingenieurwissenschaften, Leopold-Franzens-Universität Innsbruck, Technikerstrasse 13, 6020 Innsbruck, Austria; Email: (claudia@mat1.uibk.ac.at, michael.oberguggenberger@uibk.ac.at)
Günther Hörmann
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria; Email: (guenther.hoermann@univie.ac.at)
Michael Oberguggenberger
Affiliation:
Institut für Grundlagen der Bauingenieurwissenschaften, Leopold-Franzens-Universität Innsbruck, Technikerstrasse 13, 6020 Innsbruck, Austria; Email: (claudia@mat1.uibk.ac.at, michael.oberguggenberger@uibk.ac.at)
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Abstract

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In this paper, a theory is developed of generalized oscillatory integrals (OIs) whose phase functions and amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need for a general framework for partial differential operators with non-smooth coefficients and distribution dataffi The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wavefront sets.

Keywords

algebras of generalized functionsFourier integral operatorsmicrolocal analysis

MSC classification

Secondary: 35S30: Fourier integral operators 46F30: Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) 35B65: Smoothness and regularity of solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 52 , Issue 2 , June 2009 , pp. 351 - 386
Copyright
Copyright © Edinburgh Mathematical Society 2009