Injectivity and stability for the sub-Riemannian X-ray transform on H-type groups

Abstract

We prove injectivity and a stability estimate for the X-ray transform along sub-Riemannian geodesics of $H$-type groups, a setting in which almost every geodesic carries conjugate points. The geodesics are helices labeled by a charge $\lambda$ valued in the dual of the center. We prove a Fourier slice theorem, which identifies the X-ray transform over geodesics with charge $\lambda$ as the restriction of the group Fourier transform to their visible frequencies, followed by an explicit multiplier. We characterize the sets of charges whose geodesics determine an integrable function, as exactly those whose visible frequencies are dense. In particular, when the center of the group has dimension at least two, charges of a single magnitude suffice provided their directions fill a great circle. Laguerre asymptotics on the spectrum of the normal operator at fixed charge yields a stability estimate at every nonzero visible frequency with the loss of exactly half a derivative in the sub-Laplacian, except on the first Heisenberg group. Finally, we prove that data limited to a band or a cap of charges determines all sufficiently high central frequencies of an unknown function.

arXiv comment45 pages, 1 figure. v3: new title, shorter abstract and introduction. Minor corrections throughout