非各向同性平衡区域、Lempert函数估计和谱Nevanlinna-Pick问题
Nonisotropically balanced domains, Lempert function estimates, and the spectral Nevanlinna-Pick problem
摘要：
引入λ-非各向同性平衡域的概念，并证明C^n，n \geq 2中的对称化多圆盘是这种域的一个例子.给定一个λ-非各向同性平衡区域Ω，我们从上和从下得到了Lempert函数在（0，z）\in Ω \times Ω处的有效估计.我们使用这些估计，以获得一定的条件，实现两点Nevanlinna-Pick插值的对称polydisc。将我们的Lempert函数估计的推导思想应用到所谓的谱单位球Ω_n上，得到了：a）在（0，W）\in Ω_n \times Ω_n上的Lempert函数公式; b）在谱单位球上实现两点Nevanlinna-Pick插值的充要条件.
英文摘要：
We introduce the notion of a λ-nonisotropically balanced domain and show that the symmetrized polydisc in C^n, n \geq 2, is an example of such a domain. Given a λ-nonisotropically balanced domain Ω, we derive effective estimates from above and from below for the Lempert function at (0,z)\inΩ\timesΩ. We use these estimates to derive certain conditions for realising a two-point Nevanlinna-Pick interpolation in the symmetrized polydisc. Applying the ideas used in the derivation of our Lempert function estimates to the so-called spectral unit ball Ω_n, we deduce: a) a formula for the Lempert function at (0,W)\inΩ_n\timesΩ_n; and b) a necessary and sufficient condition for realising a two-point Nevanlinna- Pick interpolation in the spectral unit ball.
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