## The Hyperbolic Cauchy ProblemThe approach to the Cauchy problem taken here by the authors is based on theuse of Fourier integral operators with a complex-valued phase function, which is a time function chosen suitably according to the geometry of the multiple characteristics. The correctness of the Cauchy problem in the Gevrey classes for operators with hyperbolic principal part is shown in the first part. In the second part, the correctness of the Cauchy problem for effectively hyperbolic operators is proved with a precise estimate of the loss of derivatives. This method can be applied to other (non) hyperbolic problems. The text is based on a course of lectures given for graduate students but will be of interest to researchers interested in hyperbolic partial differential equations. In the latter part the reader is expected to be familiar with some theory of pseudo-differential operators. |

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### Contents

Interpolation theorems | 12 |

Fourier integral operators with a complexvalued phase function | 25 |

Wave front sets in Gevrey classes | 43 |

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assertion Assume that p(x bounded Cauchy problem Cauchy-Schwarz inequality change of coordinates clear coefficients compact set constant c independent Corollaries 4.4.4 define DEFINITION denote doubly characteristic set effectively hyperbolic equation finite propagation speed follows from Lemma follows from Proposition Fourier integral operators G 1R G Hn G IRn G lRn G S(fi,g Gevrey classes Hahn-Banach theorem Hamilton map homogeneous of degree homogeneous symplectic coordinates Hormander hyperbolic operators hyperbolic with respect implies inequality Kajitani Let p(x Lipschitz continuous Malgrange preparation theorem maps continuously Math modulo Moreover open set operators in Gevrey parametrix partition of unity pm(x positive constant pp(X principal symbol proof of Lemma propagation of singularities Proposition 2.1.2 Proposition 6.6 proves the lemma pseudo differential operator pseudo-differential quadratic form real-valued resp right-hand side small conic neighbourhood Sobolev space solution subsection sufficiently small supp[x symbols satisfying 6.5 Theorem 1.1 wave front sets write