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FIOs with cusp singularities and open umbrellas

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Abstract

We study the composition calculus of Fourier integral operators F, with Morin singularities. This composition calculus falls outside of the classical transverse or clean intersection calculus of FIOs and it is motivated by the inverse problems studied in Felea and Greenleaf (Math Res Lett 17(5):867–886, 2010) and Felea and Nolan (J Fourier Anal Appl 21(4):799–821, 2015). In this case, the normal operator \(F^*F\) is an operator with the wave front set in \(\Delta \cup \Lambda \) where \(\Lambda \) is a singular canonical relation called an open umbrella. We show that the operator \(F^*F\) has the same order on \(\Delta \setminus \Lambda \) and on \( \Lambda \setminus \Delta \). This result is a generalization of the results obtained in Felea and Greenleaf (2010) and Felea and Nolan (2015) where the canonical relations have the fold/cusp and cusp/cusp singularities.

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Acknowledgements

The author thanks Prof. Allan Greenleaf for suggesting this problem and for discussions related to this problem. This material is based upon work supported by the National Science Foundation under Grant No. 1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the fall semester 2019.

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  1. School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

    Raluca Felea

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  1. Raluca Felea
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Correspondence to Raluca Felea.

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Felea, R. FIOs with cusp singularities and open umbrellas. J. Pseudo-Differ. Oper. Appl. 12, 38 (2021). https://doi.org/10.1007/s11868-021-00410-1

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