On Penrose integral formula and series expansion of k-regular functions on the quaternionic space Hn☆
On Penrose integral formula and series expansion of k-regular functions on the quaternionic space Hn☆
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DOI：
10.1016/j.geomphys.2012.11.002
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发表时间：
2013-02
影响因子：
1.5
通讯作者：
Qianqian Kang;Wei Wang
中科院分区：
文献类型：
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作者：
Qianqian Kang;Wei Wang
k-Cauchy-Fueter算子可以看作是C4n上全纯k-Cauchy-Fueter算子对四元数空间n的限制。一个广义的Penrose积分公式给出了全纯k-Cauchy-Fueter方程的解，反过来，这些方程的任何全纯解都可以由这个积分公式给出。通过对四元数空间Hn≤C4n的限制，得到了所有的k正则函数。该积分公式还给出了k正则函数通过齐次k正则多项式的级数展开式。特别地，结果适用于左正则函数，它恰好是1正则函数。用积分公式或这样的级数来表示函数的k正则性几乎是初等的，但要证明任何k正则函数可以用积分公式或这样的级数来表示的逆部分，需要用到一些套理论的工具。
The k-Cauchy–Fueter operator can be viewed as the restriction to the quaternionic space Hnof the holomorphic k-Cauchy–Fueter operator on C4n. A generalized Penrose integral formula gives the solutions to the holomorphic k-Cauchy–Fueter equations, and conversely, any holomorphic solution to these equations is given by this integral formula. By restriction to the quaternionic space Hn⊆C4n, we find all k-regular functions. The integral formula also gives the series expansion of a k-regular function by homogeneous k-regular polynomials. In particular, the result holds for left regular functions, which are exactly 1-regular. It is almost elementary to show the k-regularity of the function given by the integral formula or such series, but the proof of the inverse part that any k-regular function can be provided by the integral formula or such series involves some tools of sheaf theory.