The support theorem for the single radius spherical mean transform

Mark Agranovsky, Peter Kuchment

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)1-16
Number of pages16
JournalMemoirs on Differential Equations and Mathematical Physics
Volume52
StatePublished - 2011

Bibliographical note

Publisher Copyright:
© 2011, Razmadze Mathematical Institute. All rights reserved.

Funding

The work of the first author was performed when he was visiting Texas A&M University and was partially supported by the ISF (Israel Science Foundation) grant 688/08. The second author was partially supported by the NSF grants DMS 0604778 and 0908208 and by the IAMCS. The authors express their gratitude to ISF, NSF, Texas A&M University, and IAMCS for the support. We are also grateful to the reviewer of the first version of the text for pointing to us the reference to [13] and to Y. Lyubarskii for useful comments and references.

FundersFunder number
IAMCS
National Science FoundationDMS 0604778, 0908208
Texas A and M University
Israel Science Foundation688/08

    Keywords

    • Radon transform
    • Spherical mean
    • Support

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