An Example of Non-uniqueness for Radon Transforms with Continuous Positive Rotation Invariant Weights

We consider weighted Radon transforms 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 and with continuous weight. Moreover, in this example the weight W is rotation invariant. In particular, by this result we continue studies of Quinto (J Math Anal Appl 91(2): 510-522, 1983), Markoe and Quinto (SIAM J Math Anal 16(5), 1114-1119, 1985), Boman (J Anal Math 61(1), 395-401, 1993) and Goncharov and Novikov (An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. arXiv:1709.04194v2, 2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in R-d, d >= 3.