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 Sharafudtinov, 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. We then provide a full picture of the data space decomposition, in particular a range characterization of $I_{m}$ for every $m$ in terms of moment conditions and spectral decay. Finally, we give explicit approaches for the reconstruction of 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.