与极值函数有关的凸性
Convexity properties related to extremal functions
摘要：
1962年，J. Siciak引入了多项式极值函数。Siciak极值函数是研究多项式导数的重要工具之一。W.在纸圈Pleopenniak（最重要的联合与W。Pawalucki）揭示了紧$K$附近的Siciak极值函数的行为与多项式导数的界之间的深刻联系。1990年W. Pleumniak引入了与紧$K$的马尔可夫性等价的条件（P）.在同一篇文章中，还提出了一个问题，即Siciak极值函数的哪些性质是马尔可夫性质所必需的。一个更强的问题是霍尔德连续性的对数的Siciak极值函数。这个问题可以用多项式空间上任意范数$q$的更一般的情形来表示。本文研究了Siciak函数的推广的性质与多项式导数的范数的性质之间的关系。特别是我们得到了与主要问题相关的马尔可夫因子$M_n(q,k)$的一些深层次的性质。主要结果之一是$M_n(q,k)^{1/k}$的Kolmogorov-Landau型性质，它是关于多项式导数族的三角形序列的一个条件，而不是关于特殊多项式的Kolmogorov-Landau显著不等式的直接类比：$q$ .单独地，这个条件（一个较弱的版本就足够了）什么也没有给出。但是如果我们假设$q$有A.证明了关于$q$的马氏性，满足条件$C(q)>0$则$q$具有Vladimir马尔可夫性.在$q(P)=||P||_E$的情况下，这意味着非多极马尔可夫集具有Hölder连续多复格林函数。
英文摘要：
In 1962 J. Siciak introduced polynomial extremal function. The Siciak extremal function is one of the most important tool to investigate derivatives of polynomials. W. Pleśniak in the circle of papers (most important joint with W. Pawłucki) showed deep connections between behavior of Siciak extremal function near compact $K$ and bounds for derivatives of polynomials. In 1990 W. Pleśniak introduced condition (P) which is equivalent to Markov property of compact $K$. In the same paper there was stated a problem which property of Siciak's extremal function are necessary to Markov's property. A stronger question is on Hölder continuity of the logarithm of the Siciak extremal function. This problem can be formulate in more general case of arbitrary norms $q$ on the space of polynomials. In the present paper we investigate the connection between behavior of generalizations of Siciak's function and the behavior of norms of derivatives of polynomials. In particular we get some deep properties of Markov factors $M_n(q,k)$ related to the main problems. One of the main result is the Kolmogorov-Landau type property of $M_n(q,k)^{1/k}$ which is a condition on the triangle sequence of family of derivatives of polynomials not for particular polynomials as for direct analogon of the Kolmogorov-Landau remarkable inequalities: $\log M_n(q,k)^{1/k}\leq \log const. + (1-\frac{\log k}{\log n})\log M_n(q,1)+\frac{\log k}{\log n}M_n(q,n)^{1/n},\ 1\leq k\leq n$.It seems that this condition is satisfied for arbitrary norm $q$. Separately this condition (a weaker version is sufficient) gives nothing. But if we assume that $q$ has A. Markov's property with respect to $q$ and satisfies a condition $C(q)>0$ then $q$ posseses Vladimir Markov property. In the case $q(P)=||P||_E$ this means that non pluripolar Markov sets possese Hölder continuous pluricomplex Green function.
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