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

Abstract

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 show that a naïve use of the pushforward Theorem leads to nonsharp index sets. We then refine these results using Mellin functions, showing that certain coefficients vanish; this recovers the sharpness of known results. A number of consequences for the mapping properties of the X-ray transform and its normal operator(s) follow.

The $C^\infty$-isomorphism property for a class of singularly-weighted X-ray transforms

“…Our object of study is a family of singularly weighted X-ray transforms: given γ > −1, define the operator I 0 d γ (f ) := I 0 (d γ f ) defined on C(D). By [14,Prop. 2.7], such an operator can be extended as a bounded operator in the Hilbert setting…”

“…Normal operators associated with X-ray transforms in "simple" geometries (e.g., I ♯ 0 I 0 ) have been long known to be elliptic pseudodifferential operators in the interior [26], and although this provides local stability estimates in the interior, obtaining full invertibility all the way to the boundary requires deeper study. In recent works [18,19,20,17,14], it has been observed that considering singularly weighted L 2 topologies on M and ∂ + SM could lead to normal operators (all of the form I ♯ 0 w 2 I 0 w 1 with w 1,2 two weight functions) with similar ellipticity property in the interior, while having desirable global functional properties (e.g., invertibility all the way to the boundary), and providing new ways to reconstruct f from I 0 f . Depending on the choice of weights w 1,2 , one becomes able to find scales of Sobolev spaces on M where continuity and stability estimates can be formulated, even sometimes in an isometric manner.…”

“…Recently, forward mapping properties have been obtained for I 0 and I ♯ 0 on strictly geodesically convex Riemannian manifolds [14], allowing one to predict the index set 1 of I 0 f from that of f , and the index set of I ♯ 0 g from that of g. This allows for a systematic understanding of the forward mapping properties of operators of the form I ♯ 0 w 2 I 0 w 1 on Fréchet subspaces of C ∞ (M int ) with polyhomogeneous conormal expansions off of ∂M , where the weights w 1 , w 2 have specific singular boundary behavior. Depending on the choice of weights, there are several (desirable or less desirable) possible outcomes: a normal operator may (i) map a Fréchet space F into itself, (ii) cycle through a finite sequence of Fréchet spaces or (iii) yield a non-repeating chain of Fréchet spaces F 1 → F 2 → F 3 .…”

“…µ I 0 falls into the first one, with F = C ∞ (M ). To make this last example more general, it is observed in [14] that if (d, t) denote boundary defining functions for M and ∂ + SM (with an additional requirement on t restricting its index set, see [14,Def. 4.4]), for any γ > −1, the operator…”

“…For purposes of inversion, one naturally asks whether the converse inclusion holds, and Conjecture 2.9 in [14] states: Conjecture 1.1. Given (M, g) a simple Riemannian manifold with boundary and γ > −1, there exist (d, t) boundary defining functions for M and ∂ + SM making…”