On the Range Characterization of the two dimensional attenuated Doppler Transform

We characterize the range of the attenuated and non-attenuated $X$-ray transform of compactly supported vector fields in the plane. The characterization is in terms of a Hilbert transform associated with the $A$-analytic functions à la Bukhgeim. As an application we determine necessary and sufficient conditions for the attenuated Doppler and $X$-ray data to be mistaken for each other.

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