Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary

Mazzeo, Rafe; Monard, François

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 809-847

We study the mapping properties of the X-ray transform and its adjoint on spaces of conormal functions on Riemannian manifolds with strictly convex boundary. After desingularizing the double fibration, and expressing the X-ray transform and its adjoint using b-fibrations operations, we employ tools related to Melrose’s pushforward theorem to describe the mapping properties of these operators on various classes of polyhomogeneous functions, with special focus to computing how leading order coefficients are transformed. The appendix explains that a naive use of the pushforward theorem leads to a suboptimal result with non-sharp index sets. Our improved results are obtained by closely inspecting Mellin functions which arise in the process, showing that certain coefficients vanish. This recovers some sharp results known by other methods. A number of consequences for the mapping properties of the X-ray transform and its normal operator(s) follow.

Nous étudions les propriétés fonctionnelles de la transformée en rayons X et de son adjointe dans les espaces conormaux, sur les variétés riemanniennes a bord strictement convexe. Après une désingularisation préalable de la double fibration sous-jacente en une double b-fibration, nous exprimons les deux opérateurs étudiés comme des compositions d’intégration le long des fibres et de précompositions par des b-fibrations. Nous utilisons ensuite des techniques reliées aux théorèmes d’intégration le long des fibres et de précomposition de Melrose pour en déduire l’action de ces opérateurs sur des espaces de fonctions polyhomogènes conormales, décrivant notamment comment les développements ainsi que les termes principaux sont transformés. Nous expliquons dans l’appendice qu’une application naïve du théorème d’intégration le long des fibres donne une surestimation des ensembles d’indices qui ne permet pas d’obtenir la précision de certains résultats existant dans la littérature. Notre approche retrouve cette précision et la généralise, en inspectant de plus près les fonctionnelles de Mellin qui entrent en jeu, et en montrant l’annulation de certains coefficients. Nous discutons de quelques applications des résultats principaux, concernant la transformation en rayons X et ses opérateurs normaux associés.

Classification : 44A12, 53C65, 35B40, 35S99
Keywords: Geodesic X-ray transform, mapping properties, pushforward theorem, polyhomogeneous conormal spaces, b-fibration
Mots-clés : Transformée en rayons X, propriétés fonctionnelles, espaces polyhomogènes, théorème d’intégration le long des fibres, b-fibration

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