The ray transform of symmetric tensor fields with incomplete projection data, I: The kernel of the ray transform

We consider the ray transform $I_\Gamma$ that integrates symmetric rank $m$ tensor fields on ${\mathbb{R}}^n$ supported in a bounded convex domain $D\subset{\mathbb{R}}^n$ over lines. The integrals are known for the family $\Gamma$ of lines $l$ such that endpoints of the segment $l\cap D$ belong to a given part $\gamma=\partial D\cap{\mathbb{R}}^n_+$ of the boundary, for some half-space ${\mathbb{R}}^n_+\subset{\mathbb{R}}^n$. We prove that the kernel of the operator $I_\Gamma$ coincides with the space of $\gamma$-potential tensor fields.