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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. 

Abstract Following the work of Mazzeo–Swoboda–Weiß–Witt [Duke Math. J. 165 (2016), 2227–2271] and Mochizuki [J. Topol. 9 (2016), 1021–1073], there is a map$$\overline{\Xi }$$between the algebraic compactification of the Dolbeault moduli space of$${\rm SL}(2,\mathbb{C})$$Higgs bundles on a smooth projective curve coming from the$$\mathbb{C}^\ast$$action and the analytic compactification of Hitchin’s moduli space of solutions to the$$\mathsf{SU}(2)$$self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ‘limiting configurations’. This map extends the classical Kobayashi–Hitchin correspondence. The main result that this article will show is that$$\overline{\Xi }$$fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. 

Abstract We continue our study, initiated in [34], of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $$2\pi $$. Deformations are obstructed precisely when the number $$2$$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in [37, 38].