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

Abstract: We consider the mapping properties of singularly-weighted normal operators associated to X-ray transforms on manifolds with boundary. While normal operators associated to geodesic X-ray transforms in "simple" settings are known to be elliptic pseudodifferential operators in the interior, their behavior near the boundary is more subtle; in particular the normal operators need to be precomposed with weights in order to even map $C^\infty$ of a manifold with boundary back to itself. This motivates asking which choice of weights guarantee the normal operator to be an isomorphism of $C^\infty$; such questions arise in considering theoretical guarantees for the consistency and uncertainty quantification of statistical recovery algorithms, where one needs to know on what spaces the operator can be considered invertible. In this talk, we will show that a particular family of weights on the Euclidean disk and on simple disks of constant curvature do give rise to normal operators which are isomorphisms on $C^\infty$. The proof involves deriving the Singular Value Decomposition of a weighted X-ray transform and studying certain function spaces based on the singular vectors of the X-ray transform, which coincides with the eigenfunctions of a particular degenerately elliptic Kimura-type differential operator. Joint work with Rohit Kumar Mishra and Francois Monard.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine