The Wedge-of-the-edge Theorem: Edge-of-the-wedge Type Phenomenon Within the Common Real Boundary
The Wedge-of-the-edge Theorem: Edge-of-the-wedge Type Phenomenon Within the Common Real Boundary
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边楔定理：共同实边界内的楔边型现象
DOI：
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发表时间：
2017
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通讯作者：
J. Pascoe
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文献类型：
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作者：
J. Pascoe
多复变函数中的楔边定理给出了定义在多上半平面和多下半平面上的函数的解析延拓，即所有坐标分别位于上半平面和下半平面的$mathbb{C}^{n}$中的点集通过真实的空间中的一个集$mathbb{R}^{n}$。真实的空间中集合的几何可以迫使函数在边界本身内解析地连续，这在我们的边缘楔定理中是有条件的。例如，如果一个函数扩展到$mathbb{R}^{n}$中两个立方体的并集，这两个立方体都是正方向的，并且有一些小的重叠，那么函数必须解析地继续到该重叠的一个固定大小的邻域，而不依赖于重叠的大小。
Abstract The edge-of-the-wedge theorem in several complex variables gives the analytic continuation of functions defined on the poly upper half plane and the poly lower half plane, the set of points in $mathbb{C}^{n}$ with all coordinates in the upper and lower half planes respectively, through a set in real space, $mathbb{R}^{n}$ . The geometry of the set in the real space can force the function to analytically continue within the boundary itself, which is qualified in our wedge-of-the-edge theorem. For example, if a function extends to the union of two cubes in $mathbb{R}^{n}$ that are positively oriented with some small overlap, the functions must analytically continue to a neighborhood of that overlap of a fixed size not depending of the size of the overlap.