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

TL;DR

This paper demonstrates that certain singularly weighted normal operators associated with the X-ray transform on disks and symmetric surfaces are smooth-function isomorphisms, enabling invertibility in various functional spaces.

Contribution

It introduces a family of normal operators for the X-ray transform that are proven to be $C^ abla$-isomorphisms, extending known results to singular weights and curved geometries.

Findings

Normal operators are $C^ abla$-isomorphisms on the disk.

Range characterizations for the X-ray transform are provided.

Isomorphism property extends to constant-curvature surfaces.

Abstract

We study a one-parameter family of self-adjoint normal operators for the X-ray transform on the closed Euclidean disk ${\mathbb D}$, obtained by considering specific singularly weighted $L^2$ topologies. We first recover the well-known Singular Value Decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of $C^\infty({\mathbb D})$. As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the X-ray transform are provably invertible, in Fr\'echet and Hilbert spaces encoding specific boundary behavior.