Tensor tomography on asymptotically hyperbolic surfaces

Abstract

We initiate a study of the inversion of the geodesic X-ray transform $I_m$ over symmetric $m$-tensor fields on asymptotically hyperbolic surfaces. This operator has a non-trivial kernel whenever $m\ge 1$. To propose a gauge representative to be reconstructed from X-ray data, we first prove a "tt-potential-conformal" decomposition theorem for $m$-tensor fields (where "tt" stands for transverse traceless), previously used in integral geometry on compact Riemannian manifolds with boundary in Sharafutdinov, 2007; Dairbekov and Sharafutdinov, 2011. The proof is based on elliptic decompositions of the Guillemin-Kazhdan operators $\eta_\pm$ (Guillemin and Kazhdan, 1980) and leverages in the current setting the 0-calculus of Mazzeo-Melrose (Mazzeo and Melrose, 1987; Mazzeo, 1991). Iterating this decomposition gives rise to an "iterated-tt" representative modulo $\ker I_m$ for a tensor field, which is distinct from the often-used solenoidal representative. In the case of the Poincar\'e disk, we show that the X-ray transform of a tensor in iterated-tt form splits into components that are orthogonal relative to a specific $L^2$ structure in data space. For even tensor fields, we provide a full picture of the data space decomposition, in particular a range characterization of $I_{2n}$ for every $n$ in terms of moment conditions and spectral decay. Finally, we give explicit approaches for the reconstruction of even tensors in iterated-tt form from their X-ray transform or its normal operator, using specific knowledge of geodesically invariant distributions with one-sided Fourier content, whose properties are analyzed in detail.