The support theorem for the single radius spherical mean transform

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Let f(x) belong to 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. In this article, the support theorem is proven that strengthens this injectivity result. 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 an R-convex domain.