The support theorem for the single radius spherical mean transform

Details The support theorem for the single radius spherical mean transform The support theorem for the single radius spherical mean transform

2009-05-08

arxiv.org/abs/0905.1310v2

Memoirs on Differential Equations and Mathematical Physics 52 (2011), 1-16

Physics

Mathematical Physics

Scientific paper

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.

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