Momentum ray transforms, II: range characterization in the Schwartz space

Abstract

The momentum ray transform Ik integrates a rank m symmetric tensor field f over lines of with the weight tk: We give the range characterization for the operator on the Schwartz space of rank m smooth fast decaying tensor fields. In dimensions , the range is characterized by certain differential equations of order which generalize the classical John equations. In the two-dimensional case, the range is characterized by certain integral conditions which generalize the classical Gelfand–Helgason–Ludwig conditions.