Support theorem for the light-ray transform of vector fields on Minkowski spaces (Q1673818)

For a Lorentzian manifold of dimension \(1+n\), \(n\geq 2\), with a Lorentzian metric \(g\) of signature \((-,+,\dots,+)\) and given a certain weight in general, the weighted light-ray transform of a vector field \(f\) can be defined, where \(\gamma\) (see the definition [Equation (1.1) of this paper]) is the family of future pointing light-like geodesics (non-geodesics) under certain prescribed condition. The investigation in this paper promotes the choice of a certain parametrization for the family of light-like geodesics and requires the weight function to be positively homogeneous of degree zero in its second variable. The investigation in this paper focuses on the study of the local and analytic microlocal invertibility of a certain operator on vector fields on the Minkowski spacetime, when the weight is simply 1. Section 3 discusses the Minkowski space for the Fourier analysis of the light-ray transform. The analytic wave front set of a vector-valued distribution is introduced. A support theorem is proved for vector fields on an open set of light-like lines by invoking the analytic continuation across a time-like lines hypersurface in the Minkowski space. Section 6 cites some examples relevant for the present investigation.