The John equation for tensor tomography in three-dimensions
Résumé
John proved that a function $j$ on the manifold of lines in $R^3$ belongs to the
range of the x-ray transform if and only if $j$ satisfies some second order
differential equation and obeys some smoothness and decay conditions. We
generalize the John equation to the case of the x-ray transform on arbitrary
rank 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 fields
if and only if $j$ satisfies some differential equation of order 2(m + 1) and
obeys some smoothness and decay conditions.