The Spherical Mean Value Operators on Euclidean and Hyperbolic Spaces
The Spherical Mean Value Operators on Euclidean and Hyperbolic Spaces
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欧几里得空间和双曲空间上的球均值算子
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Kyung
中科院分区：
文献类型：
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作者：
Kyung
在欧氏空间上，证明了球面中值算子从光滑函数类到其自身和从分布类到其自身的满射性，并分别得到了该算子在紧支撑分布类和紧支撑函数类上的值域特征.在三维双曲空间上，我们得到了紧支撑分布类上球均值算子的值域特征。由此我们证明了该算子是从光滑函数空间到其自身的满射。推广了Fritz John关于球面平均的一些结果.本文给出了双曲空间中迭代球面平均的一个公式。我们还证明了三维双曲空间上的光滑函数f是唯一确定的，如果f的值在某些分裂环上是已知的，其中环的厚度之和是r。最后得到了三维双曲空间中非齐次球平均值方程的显式解。给出了3维和5维欧氏空间中单半径球面平均的补充支撑定理，以及3维双曲空间中单半径球面平均的一个支撑定理.
On Euclidean space, we prove the surjectivity of the spherical mean value operator from the class of smooth functions to itself, and from the class of distributions to itself, and obtain range characterizations of this operator on the class of compactly supported distributions and functions, respectively. On the three dimensional hyperbolic space, we obtain a range characterization of the spherical mean value operator on the class of compactly supported distributions. From this we show that this operator is surjective from the space of smooth functions onto itself. We extend some results on the spherical mean obtained by Fritz John. We derive a formula for the iterated spherical mean in hyperbolic spaces. We also show that a smooth function f on the three dimensional hyperbolic space with known averages over all spheres of a fixed radius r > 0 is uniquely determined, if the values of f are known on certain split annuli where the sum of the thicknesses of the annuli is r. Finally we obtain an explicit solution to the inhomogeneous spherical mean value equation in three dimensional hyperbolic space. We provide supplementary support theorems for the single radius spherical mean in Euclidean space of dimension 3 and 5, and a support theorem for the single radius spherical mean in Hyperbolic space of dimension 3.