数学学报 1959, 9(3) 306-314 DOI:   cnki:ISSN:0583-1431.0.1959-03-007   ISSN: 0583-1431 CN: 11-2038/O1

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华罗庚
陆启铿
典型域的调和函数论(Ⅲ)——斜对称方阵双曲空间的调和函数
华罗庚;陆启铿
中国科学院数学研究所 ,中国科学院数学研究所
摘要: <正> 3.1.斜对称方阵双曲空间的调和函数 命 Z 代表 n×n 斜对称方阵(?)(?)个复变数 z_(12),z_(13),…z_(1n),z_(23),…,(?)…,z_(n-1,n)空间的域我们引进运算子
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MSC2000 调和函数:7901,双曲空间:4261,典型域:3025,对称方阵:2571,斜对称:1383,华罗庚:941,微分方程:488,类函数:466
THEORY OF HARMONIC FUNCTIONS OF THE CLASSICAL DOMAINS(Ⅲ)——HARMONIC FUNCTIONS IN THE HYPERBOLIC SPACE.OF SKEW SYMMETRIC MATRICES
L.K.HUA AND K.H.LooK(Institute of Mathematics,Academia Sinica)
Abstract: Before to establish the boundary value problem of the partial differential equation ofsecond order(?)which is of elliptic type in the domain (?)(n):(?)αβ=1,…n,and is degenerated on the bourdary (?)(n)of R_m(n),we first considerthe geometrical structure of the botundary.(?)Let(?)(n)be the points Z of(n)(the closure of (?))such that(?)of rank n—2r(r=0.1,…,(n/2))Then(?)where(?)is the characteristic manifold of (?)(n)(c.f.Hua andLook~[1]The points of(?)(n)are of the type K=U′F(n)U,U running over all unitarymatrices of order n and F(2v)=(?)For a matrix U(uαβ)of order n,we define a matrix U~{2}of order 1/2n(n—1)such the element of U~{2} at the(αβ)row and the(λμ)colume is given by(?)(c.f.Hua~[3],p,93 and Look~[1],p.625).when U runs over all unitary matrices,the matrices U~{2} form a group U~{2} whichhas a subgroup,(?)generating by the elements(?)where V~(2r)are unitary matrices,of order 2r satisfying the relation V′F(2r)V=F(2r)and(?)(n)can be mapped by anU(n—2r)are arbitrary unitary matrices of order n—2r.(?)(n)can be mapped by an one-to-one real analytic transformation onto the topological product (?)(n—2r)×m~(2r)where m~(2r) denotes the quotient space(?)When n is even, for h real-valued func.tion cp(K) continuous on (?)m(n),we have,asin the two previous parts of this paper,(?)where V(?)and K are the total volume and the volume element of(?)respectively.When n is odd,the closure of Rm(n)can be imbedded into the closure of R_m(n+1)such that (?)(n)appears as a submanifold of(?)(n+1).Then we can apply the resultsfor even n and obtain the corresponding formulae for odd n.After the above consideration,it is clear how to suggest and then solve the boundaryvalue problem of the equation(1)in (?)_m(n).Denote (?)the class of real-valued functions(?)u(Z), each ou(Z).each of Which in (?)_m(n)is continuousand on(?)is harmonic with respect to the coordinates of R_m(n—2r),i.e.,u(Z)satisfies the dif-equation(1)corresponding to R_m(n—2r)(r≡O,1,…,[n/2]—1).Then,ferential equation(1)corresponding to (?)(n—2).Then,for any real-valued functionψ(K)continuous on(?)tll(n),there is a unique function u(Z)of(?)which takes the given boundary values ψ(K)on(?)(n).Moreover,this functioncan be represented explicitly by the“Poisson integral” of ψ(K):where a=(n-1)/2 for even n and a=n/2 for odd n,and V(?))is the total volumeof (?)(n).
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收稿日期 1959-03-06 修回日期 1900-01-01 网络版发布日期  
DOI: cnki:ISSN:0583-1431.0.1959-03-007
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