Functional Relations, Sharp Mapping Properties, and Regularization of the X-Ray Transform on Disks of Constant Curvature

Abstract

This article surveys recent results aiming at obtaining refined mapping estimates for the X-ray transform on a Riemannian manifold with boundary, which leverage the condition that the boundary be strictly geodesically convex. These questions are motivated by classical inverse problems questions (e.g. range characterization, stability estimates, mapping properties on Hilbert scales), and more recently by uncertainty quantification and operator learning questions.2 A family of Banach (or Hilbert) spaces {(Eα, • α)}α 0 <α<β 0 is a Banach scale in the sense of Krein and Petunin (1966) if (1) E β is densely embedded in Eα when β > α with x α ≤ α,β x β for all x ∈ E β , and (2) interpolation inequalities hold in the sense that there is a function C(α, β, γ) finite at all points of the domain α0 ≤ α < β < γ ≤ β0 such that x β ≤ C(α, β, γ) x

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

“…• In [11] it is shown that in the class of simple geodesic disks with constant curvature, the operator I * 0 µ −1 I 0 is an isomorphism of C ∞ (M ). This is done by constructing a Sobolev scale H k (M ), defined using a degenerate elliptic operator, which have intersection C ∞ (M ) and which satisfy…”

“…and for all k ∈ N 0 } (11). Note that A s (N ) contains functions of the form x s | log x| a for any s with Re s > s, a ∈ R (or Re s ≥ s if a ≤ 0).…”

On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems

“…The proof (see Remark 6.5) relies on recent developments in [35] and [37] and follows from the infinite dimensional Theorem 4.1, which constitutes our main analytical contribution.…”

“…Analytical results. Our results are formulated in a non-standard scale Hs (D, C m ) of Sobolev-type spaces studied systematically in [35]. This so called Zernike scale is defined in terms of the (normalised) Zernike-polynomials Ẑnk :…”

“…5.1]) and generally the operator I 0 is understood to be 1/2-regularising. Taking into account the behaviour near ∂D, it was proved in [35] that I 0 is an isomorphism from a Sobolev-type space H−1/2 (D), defined in (4.6) below, onto L 2 λ (X ). Extending recent developments in [37], we will prove here that this isomorphism property extends to I Φ for a large class of Φ = 0 (see Theorem 4.1).…”