The support theorem for the single radius spherical mean transform

2011, Razmadze Mathematical Institute. All rights reserved. Let f Lp(Rn) and R > 0. The transform is considered that integrates the function f over (almost) all spheres of radius R in Rn. This operator is known to be non-injective (as one can see by taking Fourier transform). However, the counterexamples that can be easily constructed using Bessel functions of the 1st kind, only belong to Lp if p > 2n/(n 1). It has been shown previously by S. Thangavelu that for p not exceeding the critical number 2n/(n 1), the transform is indeed injective. A support theorem that strengthens this injectivity result can be deduced from the results of [12], [13]. Namely, if K is a convex bounded domain in Rn, the index p is not above 2n/(n 1), and (almost) all the integrals of f over spheres of radius R not intersecting K are equal to zero, then f is supported in the closure of the domain K. In fact, convexity in this case is too strong a condition, and the result holds for any what we call R-convex domain. We provide a simplified and self-contained proof of this statement.

The support theorem for the single radius spherical mean transform Academic Article