A breakdown of injectivity for weighted ray transforms in multidimensions

We consider weighted ray-transforms $P_W$ (weighted Radon transforms along straight lines) in $R^d, \,d \geq 2$, 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 $R^d$. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on $R^d \times S^{d-1}$. 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.