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

We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations $C\subset \left({T}^{*}X\setminus 0\right)×\left({T}^{*}Y\setminus 0\right)$. These canonical relations, which arise naturally in integral geometry, are such that $\pi$ : $C\to {T}^{*}Y$ is a Whitney fold and $\rho$ : $C\to {T}^{*}X$ is a blow-down mapping. If $A\in {I}^{m}\left(C\right)$, $B\in {I}^{{m}^{\prime }}\left({C}^{t}\right)$, then $BA\in {I}^{m+{m}^{\prime },0}\left(\Delta ,\Lambda \right)$ a class of pseudodifferential operators with singular symbols. From this follows ${L}^{2}$ boundedness of $A$ with a loss of 1/4 derivative.

Nous établissons un calcul de composition pour les opérateurs intégraux de Fourier associés à une classe de relations canoniques lisses $C\subset \left({T}^{*}X\setminus 0\right)×\left({T}^{*}Y\setminus 0\right)$. Ces relations canoniques, qui se présentent en géométrie intégrale sont telles que $\pi$ : $C\to {T}^{*}Y$ est un pli de Whitney et $\rho$ : $C\to {T}^{*}X$ est une application blow-down. Si $A\in {I}^{m}\left(C\right)$, $B\in {I}^{{m}^{\prime }}\left({C}^{t}\right)$, alors $BA\in {I}^{m+{m}^{\prime },0}\left(\Delta ,\Lambda \right)$ 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.

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title = {Composition of some singular {Fourier} integral operators and estimates for restricted $X$-ray transforms},
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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, Volume 40 (1990) no. 2, pp. 443-466. doi : 10.5802/aif.1220. http://archive.numdam.org/articles/10.5802/aif.1220/

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