等权极点的多复Green函数和Lempert函数
Pluricomplex Green and Lempert functions for equally weighted poles
摘要：
对于$\mathbb C^n$中的一个域$Ω$，具有极点$a_1, ...,a_N \in Ω$的复格林函数被定义为$G(z):=\sup \{u(z): u\in PSH_-(Ω), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$。当单位球中只有一个极点或两个极点时，它等于由$L_S (z) :=\inf \{\sum^N_{j=1}ν_j\log|ζ_j|: \exists ϕ\in \mathcal {O}(\mathbb D,Ω), ϕ(0)=z, ϕ(ζ_j)=a_j, j=1,...,N \}$从解析圆盘定义到$Ω$的Lempert函数。众所周知，我们总是有$L_S (z) \ge G_S(z)$。在更一般的情况下，我们允许加权极点，有一个反例，平等由于卡莱赫德和Wiegerinck，与$Ω$等于bidisk。在这里，我们展示了一个反例，只使用四个不同的平等加权极点的双圆盘。为了做到这一点，我们首先定义了一个更一般的概念，Lempert函数的“多重性”，类似于广义格林函数的Lelong和Rashkovskii，然后我们展示了如何在一些例子中，这可以实现为一个限制的定期Lempert函数时，极点趋于对方。最后，从一个例子，其中$L_S (z) > G_S(z)$在多极的情况下，我们推断，不同的（但足够接近）等权重的极点将提供一个例子，同样的不等式。指出了当极点相互趋向时格林函数和伦珀特函数的极限的悬而未决的问题。
英文摘要：
For $Ω$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in Ω$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(Ω), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $Ω$ by $L_S (z) :=\inf \{\sum^N_{j=1}ν_j\log|ζ_j|: \exists ϕ\in \mathcal {O}(\mathbb D,Ω), ϕ(0)=z, ϕ(ζ_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $Ω$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.
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