Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929451805179904 |
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| author | Mazzeo, Rafe Monard, François |
| author_facet | Mazzeo, Rafe Monard, François |
| contents | 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 naïve 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_14904 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary Mazzeo, Rafe Monard, François Analysis of PDEs Differential Geometry 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 naïve 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. |
| title | Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2112.14904 |