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

• Autorzy: Leokadia Bialas-Ciez
• Języki publikacji: EN

Abstrakty

EN

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E ⊂ ℂ^{N}$. 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 $Φ_{E}$. Moreover, we show that one of these extremal-like functions is equal to $Φ_{E}$ if E is a nonpluripolar set with $lim_{n→∞} Mₙ(E)^{1/n} = 1$ where
$Mₙ(E) := sup{|| |grad P| ||_{E}/||P||_{E}}$,
the supremum is taken over all polynomials P of N variables of total degree at most n and $||·||_{E}$ 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 $ℂ^{N}$.

Kategorie tematyczne

• 32U35: Pluricomplex Green functions
• 41A44: Best constants

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2012
• Tom: 106
• Numer: 1
• Strony: 41-51

Daty

wydano

2012

Twórcy

autor

Leokadia Bialas-Ciez

• Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland

Identyfikatory

DOI

10.4064/ap106-0-4