The John equation for tensor tomography in three-dimensions *The second author was supported by RFBR, Grant 15-01-05929-a. (9th September 2016) - Nadirashvili, Nikolai S; Sharafutdinov, Vladimir A; Vlăduţ, Serge G
Abstract: John proved that a function φ on the manifold of lines in R 3 belongs to the range of the x-ray transform if and only if φ 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 φ 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 φ satisfies some differential equation of order 2 ( m + 1 ) and obeys some smoothness and decay conditions.