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Allan Greenleaf; Gunther Uhlmann
Composition of some singular Fourier integral operators and estimates for restricted $X$-ray transforms
Annales de l'institut Fourier, 40 no. 2 (1990), p. 443-466, doi: 10.5802/aif.1220
Article PDF | Reviews MR 91k:58126 | Zbl 0695.58026

Résumé - 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.

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