M. Agranovsky and P. Kuchment

The Support Theorem for the Single Radius Spherical Mean Transform

abstract:
Let $f\in L^p(\mathbb{R}^n)$ and $R>0$. The transform is considered that integrates the function $f$ over (almost) all spheres of radius $R$ in $\mathbb{R}^n$. 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.
A support theorem that strengthens this injectivity result can be deduced from the results of  [V. Volchkov, Integral geometry and convolution equations. Kluwer Academic Publishers, Dordrecht, 2003], [Valery V. Volchkov and Vitaly V. Volchkov}, Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2009]. 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 $R$-convex domain.
We provide a simplified and self-contained proof of this statement.

Mathematics Subject Classification: 35L05, 92C55, 65R32, 44A12

Key words and phrases: Spherical mean, Radon transform, support