A breakdown of injectivity for weighted ray transforms in multidimensions

We consider weighted ray-transforms $P\mathrm{\_}W$ (weighted Radon transforms along straight lines) in ${\mathbb{R}}^d,d\ge 2,$ with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on ${\mathbb{R}}^d$. In addition, the constructed weight $W$ is rotation-invariant continuous and is infinitely smooth almost everywhere on ${\mathbb{R}}^d\times {\mathbb{S}}^{d-1}$. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of $W$ is slightly relaxed. We also give examples of continous strictly positive $W$ such that $\dim \ker P\mathrm{\_}W\ge n$ in the space of infinitely smooth compactly supported functions on ${\mathbb{R}}^d$ for arbitrary $n\in \mathbb{N}\cup \{\mathrm{\infty }\}$, where $W$ are infinitely smooth for $d=2$ and infinitely smooth almost everywhere for $d\ge 3$.