Composition of some singular Fourier integral operators and estimates for restricted $X$-ray transforms

Greenleaf, Allan; Uhlmann, Gunther

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 $C\subset (T^{*}X\setminus 0)\times (T^{*}Y\setminus 0)$. 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 $A\in I^m\left(C\right)$, $B\in I^{m^{\prime }}(C^t)$, alors $BA\in I^{m+m^{\prime },0}(\Delta ,\Lambda )$ 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 $C\subset (T^{*}X\setminus 0)\times (T^{*}Y\setminus 0)$. 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 $A\in I^m\left(C\right)$, $B\in I^{m^{\prime }}(C^t)$, then $BA\in I^{m+m^{\prime },0}(\Delta ,\Lambda )$ a class of pseudodifferential operators with singular symbols. From this follows $L^2$ boundedness of $A$ with a loss of 1/4 derivative.

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