Support theorem for the Radon–Kipriyanov $K_\gamma$-transform

The support theorem for the Radon transform was obtained by L. Helgason. The Radon transform of functions of groups of variables related by spherical symmetry is a special case of a more general transformation, namely, Radon–Kipriyanov transform $K_\gamma$. This transformation corresponds to a weight multi-index $\gamma=(\gamma_1,\ldots,\gamma_m)$ and coincides with Radon transform if all components of the multi-index $\gamma$ are natural numbers. In general, the $K_\gamma$-transformation can be interpreted as a transformation of functions of a fractional argument. In this paper, we prove a general support theorem. In a special case where $\gamma=0$, this theorem coincides with the Helgason theorem.