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

The momentum ray transform I ^k integrates a rank m symmetric tensor field f  over lines of with the weight t ^k: 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.