Note on Pseudo-Harmonic Functions, II
Note on Pseudo-Harmonic Functions, II
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DOI：
10.1112/jlms/s1-22.2.101
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发表时间：
1947-04
影响因子：
1.2
通讯作者：
A. G. Walker
中科院分区：
文献类型：
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作者：
A. G. Walker
在最近的一个notef中，我给出了一系列关于伪调和函数的初等中值定理，即在任意常曲率K的黎曼3-空间C ~ 3中，对于常数A，函数满足V ~ 2 v = X ~ v。其中一个定理提出了下面的匡威，这将在本说明中证明。设v是常曲率K空间C3的区域D中位置的函数;记Ms v为v在中心为0、半径为s的测地球8上的平均值，记v 0为v在0处的值。则v是伪调和函数，满足V ~ 2 v = X ~ v，如果
In a recent notef I gave a number of elementary mean-value theorems for pseudo-harmonic functions, ie functions satisfying V2v= Xv for constant A, in a Riemannian 3-space C3 of any constant curvature K. One of these theorems suggests the following converse, which will be proved in the present note.THEOREM. Let v be a function of position in a domain D of a space C3 of constant curvature K; write Ms v for the mean-value of v over a geodesic sphere 8, of centre 0, and radius s, and v0for the value of v at O. Then v is pseudo-harmonic and satisfies V2v= Xv in D if