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Siciak's extremal function via Bernstein and Markov constants for compact sets in $\mathbb{C}^N$

Tom 106 / 2012

Annales Polonici Mathematici 106 (2012), 41-51 MSC: Primary 32U35; Secondary 41A44. DOI: 10.4064/ap106-0-4

Streszczenie

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E\subset \mathbb{C}^N$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $\varPhi_E$. Moreover, we show that one of these extremal-like functions is equal to $\varPhi_E$ if $E$ is a nonpluripolar set with $\lim_{n\rightarrow\infty} M_n(E)^{1/n} =1$ where $$M_n (E) := \sup {\left\| |{\rm grad}\,P|\right\|_E}/{\| P\|_E}, \label{Markov}%(0.1)$$ the supremum is taken over all polynomials $P$ of $N$ variables of total degree at most $n$ and $\|\cdot \|_E$ is the uniform norm on $E$. The above condition is fulfilled e.g. for all regular (in the sense of the continuity of the pluricomplex Green function) compact sets in $\mathbb{C}^N$.