Local duality in Loewner equations
Local duality in Loewner equations
复制标题
Loewner 方程中的局部对偶性
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
P. Gumenyuk
中科院分区：
文献类型：
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作者：
Manuel D. Contreras;S. Díaz;P. Gumenyuk
在不同作者在Loewner理论中引入的框架和结构的多样性中，可以区分两种密切相关但仍然不同的推理方式，通俗地说，可以描述为“增加”和“减少”。在本文中，我们简要回顾了主要类型的（确定性）Loewner演化在文献中讨论，并详细描述了最近在[Bracci et al. to appear in J Reine Angew Math; arXiv：0807.1594v1]，[Bracci et al. in Math Ann 344：947-962，2009; arXiv：0807.1715v1]，[Contreras et al. in Revista Matem 'atica Iberoamericana 26：975-1012，2010; arXiv：0807.1715v1]，[Contreras et al. in Revista Matem'atica Iberoamericana 26：975-1012，2010; arXiv：0807.1715v1]中提出的Loewner理论中的一般统一方法内的“增加”和“减少”情况之间的局部对偶性。0902.3116v1]。特别地，我们将R. O. Bauer [in J Math Anal Appl 302：484-501，2005; arXiv：math/0306130 v1]的几个结果扩展到这个一般设置，这些结果处理弦Loewner演化。虽然对偶性是通过简单地改变参数来给出的，但并不是所有的“减”情形的结果都可以通过仅仅平移“增”情形的相应结果来得到。特别是，作为一个副产品，建立本地的对偶进化家庭和他们的“减少”对应，我们得到了一个新的广义Loewner链的特征。
Among diversity of frameworks and constructions introduced in Loewner Theory by different authors, one can distinguish two closely related but still different ways of reasoning, which colloquially may be described as "increasing" and "decreasing". In this paper we review in short the main types of (deterministic) Loewner evolution discussed in the literature and describe in detail the local duality between "increasing" and "decreasing" cases within the general unifying approach in Loewner Theory proposed recently in [Bracci et al. to appear in J Reine Angew Math; arXiv:0807.1594v1], [Bracci et al. in Math Ann 344:947-962, 2009; arXiv:0807.1715v1], [Contreras et al. in Revista Matem'atica Iberoamericana 26:975-1012, 2010; arXiv:0902.3116v1]. In particular, we extend several results of R.O.Bauer [in J Math Anal Appl 302:484-501, 2005; arXiv:math/0306130v1], which deal with the chordal Loewner evolution, to this general setting. Although the duality is given by a simple change of the parameter, not all the results for the "decreasing" case can be obtained by mere translating the corresponding results for the "increasing" case. In particular, as a byproduct of establishing local duality between evolution families and their "decreasing" counterparts we obtain a new characterization of generalized Loewner chains.