Invertibility of the Pestov–Uhlmann reconstruction operator
Let $(M, g)$ be a smooth compact oriented simple Riemannian surface: its boundary is strictly convex and any two points are joined by a unique geodesic, depending smoothly on its endpoints. On the unit tangent bundle $SM$ let $X$ be the geodesic vector field, $V$ the vertical vector field generating rotation in each fibre $S_xM$, and $X_\perp = [X, V]$. For $f \in C^\infty(M)$ let $u^f$ solve $X u^f = -f$ on $SM$ with $u^f = 0$ on the incoming boundary, and put $$Wf(x) = \frac{1}{2\pi} \int_{S_xM} X_\perp u^f(x, v)\, dS_x(v).$$ Pestov and Uhlmann proved that $W$ extends to a smoothing operator on $L^2(M)$ and that $f + W^2 f$ is given by an explicit formula in the geodesic X-ray transform of $f$. If $g$ has constant curvature then $W = 0$, and Krishnan showed that $W$ has $L^2$ norm less than one when the gradient of the Gaussian curvature of $g$ is small enough, and Guillarmou and Monard proved the same bound, with explicit constants, for metrics close to one of constant negative curvature.
Problem. Is $\operatorname{Id} + W^2$ invertible on $L^2(M; \mathbb{C})$ for every such surface? Equivalently, is $\ker(\operatorname{Id} + W^2) = \{0\}$ always?
Source: Paternain, Salo and Uhlmann (2023), §15.1, Open problem 3, on the Fredholm inversion formula of Pestov and Uhlmann (2004); stated by AIM — Open Applied Problems as invertibility of Id + W² on L²(M; ℂ).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, geodesic X-ray transforms, inversion formulas, simple surfaces
References
@book{paternain2023,
author = {Paternain, Gabriel P. and Salo, Mikko and Uhlmann, Gunther},
title = {Geometric Inverse Problems: With Emphasis on Two Dimensions},
series = {Cambridge Studies in Advanced Mathematics},
volume = {204},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2023},
doi = {10.1017/9781009039901},
note = {\S 15.1, Open problem 3},
}
@article{pestov2004,
author = {Pestov, Leonid and Uhlmann, Gunther},
title = {On characterization of the range and inversion formulas for the geodesic {X}-ray transform},
journal = {Int. Math. Res. Not.},
volume = {2004},
number = {80},
year = {2004},
pages = {4331--4347},
doi = {10.1155/S1073792804142116},
}
@article{krishnan2010,
author = {Krishnan, Venkateswaran P.},
title = {On the inversion formulas of {P}estov and {U}hlmann for the geodesic ray transform},
journal = {J. Inverse Ill-Posed Probl.},
volume = {18},
number = {4},
year = {2010},
pages = {401--408},
doi = {10.1515/jiip.2010.017},
}
@article{monard2014,
author = {Monard, Fran{\c{c}}ois},
title = {Numerical implementation of geodesic {X}-ray transforms and their inversion},
journal = {SIAM J. Imaging Sci.},
volume = {7},
number = {2},
year = {2014},
pages = {1335--1357},
doi = {10.1137/130938657},
eprint = {1309.6042},
archivePrefix = {arXiv},
}
@article{guillarmou2017,
author = {Guillarmou, Colin and Monard, Fran{\c{c}}ois},
title = {Reconstruction formulas for {X}-ray transforms in negative curvature},
journal = {Ann. Inst. Fourier (Grenoble)},
volume = {67},
number = {4},
year = {2017},
pages = {1353--1392},
doi = {10.5802/aif.3112},
eprint = {1511.05516},
archivePrefix = {arXiv},
}
@misc{aim-pestov-uhlmann-fredholm-invertibility,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Invertibility of the Pestov–Uhlmann reconstruction operator}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/061-pestov-uhlmann-fredholm-invertibility.md},
note = {Entry 061 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00545, "Invertibility of the Pestov–Uhlmann reconstruction operator", clopen.solutions.