Unique continuation and a schwarz reflection principle for analytic sets
Unique continuation and a schwarz reflection principle for analytic sets
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解析集的独特延拓和施瓦茨反射原理
DOI：
10.1080/03605309308820999
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发表时间：
1993
影响因子：
1.9
通讯作者：
L. Rothschild
中科院分区：
文献类型：
--
作者：
M. S. Baouendi;L. Rothschild
本文讨论了全纯函数在边界点上的唯一延拓问题。我们的第一个结果（定理1）表明，如果h是复上半平面中0邻域内的全纯函数，连续到边界，在0处无穷级为零，并且将实数映射到半空间中，则h必须同向为零。然后，定理1被用来锐化和简化复分析中唯一延拓的先前结果。在与Alinhac [2]的联合工作中，作者证明了：任何在复上半平面中0邻域内的向量值全纯函数，连续到边界，在0处无穷级为零，并将实数映射到Cn中C2类的全真实的流形M上，必须同向为零。(See[3]也是Bell-Lempert [3]的另一个证明。在最近的工作中，Huang和Krantz 161能够通过用a> 0的C1 pa代替CZ来削弱上述结果中的假设。我们的定理2涉及一个全纯向量值函数，它把实数映射到Cn的一个子集上，其形式为（I9z15@1Xz1），其中05 @< 1。作为结果（见推论（2.4）），当全真实的流形M仅为C1类时，我们得到了上述唯一延拓结果的一个非常简单的证明。(In特别地，当M是C1曲线时，这给出了亚历山大[I]结果的完全不同的证明。）使用经典的定理，
This paper deals with unique continuation of holomorphic functions at boundary points. Our first result (Theorem 1) shows that if h is a holomorphic function in a neighborhood of 0 in the complex upper half plane, continuous up to the boundary, vanishing of infinite order at 0, and mapping the reals into a half space, then h must vanish identically. Theorem 1 is then used to sharpen and simplify previous results of unique continuation in complex analysis. In a joint work with Alinhac [2], the authors showed that any vector valued holomorphic function in a neighborhood of 0 in the complex upper half plane, continuous up to the boundary, vanishing of infinite order at 0, and mapping the reals into M, a totally real manifold in Cn of class C2, must vanish identically.(See also Bell-Lempert [3] for a different proof.) In recent work, Huang and Krantz 161 were able to weaken the assumption in the result above by replacing CZ by C1pa with a> 0. Our Theorem 2 deals with a holomorphic vector valued function mapping the reals into a subset of Cn of the form (I9zl 5@ 1Xz1) with 0 5@< 1. As a consequence (see Corollary (2.4)), we obtain a very simple proof of the unique continuation result mentioned above when the totally real manifold M is only of class C1.(In particular this gives a completely different proof of a result of Alexander [I] when M is a C1 curve.) Using a classical theorem of