The John equation for tensor tomography in three-dimensions

John proved that a function $j$ on the manifold of lines in $R^3$ belongs to therange of the x-ray transform if and only if $j$ satisfies some second orderdifferential equation and obeys some smoothness and decay conditions. Wegeneralize the John equation to the case of the x-ray transform on arbitraryrank symmetric tensor fields: a function j on the manifold of lines in$R^3$belongs to the range of the x-ray transform on rank m symmetric tensor fieldsif and only if $j$ satisfies some differential equation of order 2(m + 1) andobeys some smoothness and decay conditions.