A breakdown of injectivity for weighted ray transforms in multidimensions

We consider weighted ray-transforms P_W (weighted Radon transforms along oriented straight lines) in R^d, d≥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 ×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_W ≥n in the space of infinitely smooth compactly supported functions on ℝ^d for arbitrary n∈N∪{∞}, where W are infinitely smooth for d=2 and infinitely smooth almost everywhere for d≥3.