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Global Wave-Front Properties for Fourier Integral Operators and Hyperbolic Problems

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Abstract

We illustrate the composition properties for an extended family of \({\text {SG}}\) Fourier integral operators. We prove continuity results on modulation spaces, and study mapping properties of global wave-front sets for such operators. These extend classical results to more general situations. For example, there are no requirements on homogeneity for the phase functions. Finally, we apply our results to the study of the propagation of singularities, in the context of modulation spaces, for the solutions to the Cauchy problems for the corresponding linear hyperbolic operators.

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Notes

  1. Let us observe that (5.25) needs to be fulfilled only for a single value of m.

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Acknowledgments

The first author gratefully acknowledges the partial support from the PRIN Project “Aspetti variazionali e perturbativi nei problemi differenziali nonlineari” (coordinator at Universitàdegli Studi di Torino: Prof. S. Terracini) during the development of the present paper. We also wish to thank the anonymous referees for the useful suggestions, aimed at improving the content and readability of the paper.

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Authors and Affiliations

  1. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Turin, Italy

    Sandro Coriasco

  2. Department of Computer science, Physics and Mathematics, Linnæus University, Växjö, Sweden

    Karoline Johansson & Joachim Toft

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  1. Sandro Coriasco
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  2. Karoline Johansson
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  3. Joachim Toft
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Correspondence to Joachim Toft.

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Communicated by Hans G. Feichtinger.

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Coriasco, S., Johansson, K. & Toft, J. Global Wave-Front Properties for Fourier Integral Operators and Hyperbolic Problems. J Fourier Anal Appl 22, 285–333 (2016). https://doi.org/10.1007/s00041-015-9422-1

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