The support theorem for the single radius spherical mean transform

Let f(x) belong to L^p(R^n) and R>0. The transform is considered that integrates the function f over (almost) all spheres of radius R in R^n. 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 L^p 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. In this article, the support theorem is proven that strengthens this injectivity result. Namely, if K is a convex bounded domain in R^n, 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 an R-convex domain.