no. 2
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms
Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 443-466.

Nous établissons un calcul de composition pour les opérateurs intégraux de Fourier associés à une classe de relations canoniques lisses C⊂(T * X∖0)×(T * Y∖0). Ces relations canoniques, qui se présentent en géométrie intégrale sont telles que π : C→T * Y est un pli de Whitney et ρ : C→T * X est une application blow-down. Si A∈I m (C), B∈I m ′ (C t ), alors BA∈I m+m ′ ,0 (Δ,Λ) qui est une classe d’opérateurs pseudodifférentiels avec des symboles singuliers. Il s’ensuit que A est borné sur L 2 avec une perte de dérivée d’un 1/4.

We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations C⊂(T * X∖0)×(T * Y∖0). These canonical relations, which arise naturally in integral geometry, are such that π : C→T * Y is a Whitney fold and ρ : C→T * X is a blow-down mapping. If A∈I m (C), B∈I m ′ (C t ), then BA∈I m+m ′ ,0 (Δ,Λ) a class of pseudodifferential operators with singular symbols. From this follows L 2 boundedness of A with a loss of 1/4 derivative.

@article{AIF_1990__40_2_443_0,
     author = {Greenleaf, Allan and Uhlmann, Gunther},
     title = {Composition of some singular {Fourier} integral operators and estimates for restricted $X$-ray transforms},
     journal = {Annales de l'Institut Fourier},
     pages = {443--466},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {2},
     year = {1990},
     doi = {10.5802/aif.1220},
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     mrnumber = {91k:58126},
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     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1220/}
}
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Greenleaf, Allan; Uhlmann, Gunther. Composition of some singular Fourier integral operators and estimates for restricted $X$-ray transforms. Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 443-466. doi : 10.5802/aif.1220. https://aif.centre-mersenne.org/articles/10.5802/aif.1220/

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