单半径球面平均变换的支撑定理
The support theorem for the single radius spherical mean transform
摘要：
设f（x）属于L^p（R^n）且R>0.该变换被认为是在R^n中半径为R的（几乎）所有球面上对函数f进行积分。该算子已知是非内射的（通过傅立叶变换可以看出）。然而，可以容易地利用第一类贝塞尔函数构造的反例仅当p>2n/（n-1）时属于L^p。S.之前已经证明了这一点。Thangavelu的p不超过临界数2n/（n-1），变换确实是单射。在这篇文章中，支持定理证明，加强这一内射性的结果。也就是说，如果K是R^n中的凸有界区域，指数p不大于2n/（n-1），并且$f$在不与K相交的半径为$R$的球面上的（几乎）所有积分都等于零，则f在区域K的闭包中被支撑。事实上，凸性在这种情况下是一个太强的条件，并且结果对任何我们称之为R-凸域的区域都成立。
英文摘要：
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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