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Fourier Integral Operators Defined by Classical Symbols with Exit Behaviour
Author(s) -
Coriasco Sandro,
Panarese Paolo
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200207)242:1<61::aid-mana61>3.0.co;2-4
Subject(s) - mathematics , fourier integral operator , gravitational singularity , microlocal analysis , operator theory , class (philosophy) , fourier transform , pure mathematics , cauchy distribution , cauchy's integral formula , order (exchange) , mathematical analysis , algebra over a field , cauchy problem , initial value problem , finance , artificial intelligence , computer science , economics
We present the detailed construction of the classical version of the calculus for Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour ( SG symbols). In particular, we analyse what happens when one restricts the choice of amplitude and phase functions to the subset of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a “classical” counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1), refining the known results about the analogous problem for general SG hyperbolic operators. The theory developed here will be used in forthcoming papers to study the propagation of singularities and the Weyl formula for suitableclasses of operators defined on manifolds with cylindrical ends.