Singularly-weighted x-ray tensor tomography

Abstract If d is a boundary defining function for the Euclidean unit disk and I denotes the geodesic x-ray transform, for γ ∈ ( − 1 , 1 ) , we study the singularly-weighted x-ray transforms I m d γ acting on symmetric m -tensors. For any m , we provide a sharp range decomposition and characterization in terms of a distinguished Hilbert basis of the data space, that comes from earlier studies of the Singular Value Decomposition for the case m = 0 , see Mishra R K et al (2022 Inverse Problems 39 024001). Since for m ⩾ 1 , the transform considered has an infinite-dimensional kernel, we fully characterize this kernel, and propose a representative for an m -tensor to be reconstructed modulo kernel, along with efficient procedures to do so. This representative is based on a new generalization of the potential/conformal/transverse-tracefree decomposition of tensor fields in the context of singularly weighted L 2 -topologies.