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), p. 443-466 / Harvested from Numdam

Résumé

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 $ : $C\rightarrow T^*Y$ est un pli de Whitney et $\rho $ : $C\rightarrow T^*X$ est une application blow-down. Si $A\in I^m(C)$, $B\in I^{m^{\prime }}(C^ t)$, 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.

Abstract

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 $ : $C\rightarrow T^*Y$ is a Whitney fold and $\rho $ : $C\rightarrow T^*X$ is a blow-down mapping. If $A\in I^ m(C)$, $B\in I^{m^{\prime }}(C^ t)$, 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.

Détail

Publié le : 1990-01-01

MR 91k:58126 | Zbl 0695.58026

DOI : https://doi.org/10.5802/aif.1220

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