The sub-Riemannian X-ray transform on $H$-type groups: Fourier slice theorems, injectivity sets and frequency support

Abstract

We study the X-ray transform along sub-Riemannian geodesics of $H$-type groups, a setting in which every geodesic apart from the horizontal lines carries conjugate points. The geodesics are helices labelled by a charge $\lambda$ in the dual of the center. Using the group Fourier transform we prove a Fourier slice theorem, which identifies the transform restricted to the geodesics of charge $\lambda$ as a Fourier restriction to their visible frequencies $\Gamma_\lambda^*$, a family of parallel hyperplanes of central frequencies dual to the central advance of the helices, followed by an explicit operator-valued multiplier. We characterize the sets of charges $Z$ whose geodesics determine an integrable function, one direction for each $\lambda\in Z$ sufficing, as exactly those for which $\bigcup_{\lambda\in Z}\Gamma_\lambda^*$ is dense. In particular the full transform is injective, and when the center has dimension at least two, charges of a single magnitude suffice provided their directions fill a circle. Averaging the normal operator over the horizontal directions gives a joint spectral multiplier of the sub-Laplacian and the central derivatives, whose eigenvalues we compute. This yields, outside the first Heisenberg group, a sharp stability estimate at fixed charge and central frequency with the loss of exactly half a derivative in the sub-Laplacian, as for the Euclidean X-ray transform. Finally we describe which central frequencies of $f$ are determined by a band or a cap of charges, namely all sufficiently high ones, the bounded remainder being recovered only by analytic continuation, in contrast with Euclidean limited-angle tomography.