Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież

Banach Center Publications (2011)

  • Volume: 92, Issue: 1, page 27-36
  • ISSN: 0137-6934

Abstract

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Let E be a compact set in the complex plane, g E be the Green function of the unbounded component of ℂ ∞ ∖ E with pole at infinity and M ₙ ( E ) = s u p ( | | P ' | | E ) / ( | | P | | E ) where the supremum is taken over all polynomials P | E ≢ 0 of degree at most n, and | | f | | E = s u p | f ( z ) | : z ∈ E . The paper deals with recent results concerning a connection between the smoothness of g E (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence M ₙ ( E ) n = 1 , 2 , . . . . Some additional conditions are given for special classes of sets.

How to cite

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Leokadia Białas-Cież. "Smoothness of Green's functions and Markov-type inequalities." Banach Center Publications 92.1 (2011): 27-36. <http://eudml.org/doc/281729>.

@article{LeokadiaBiałas2011,
abstract = {Let E be a compact set in the complex plane, $g_\{E\}$ be the Green function of the unbounded component of $ℂ_\{∞\}∖E$ with pole at infinity and $Mₙ(E) = sup (||P^\{\prime \}||_\{E\})/(||P||_\{E\})$ where the supremum is taken over all polynomials $P|_\{E\} ≢ 0$ of degree at most n, and $||f||_\{E\} = sup\{|f(z)| : z ∈ E\}$. The paper deals with recent results concerning a connection between the smoothness of $g_\{E\}$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence $\{Mₙ(E)\}_\{n = 1,2,...\}$. Some additional conditions are given for special classes of sets.},
author = {Leokadia Białas-Cież},
journal = {Banach Center Publications},
keywords = {Green function; polynomial; Markov inequality; polar set; Cantor set},
language = {eng},
number = {1},
pages = {27-36},
title = {Smoothness of Green's functions and Markov-type inequalities},
url = {http://eudml.org/doc/281729},
volume = {92},
year = {2011},
}

TY - JOUR
AU - Leokadia Białas-Cież
TI - Smoothness of Green's functions and Markov-type inequalities
JO - Banach Center Publications
PY - 2011
VL - 92
IS - 1
SP - 27
EP - 36
AB - Let E be a compact set in the complex plane, $g_{E}$ be the Green function of the unbounded component of $ℂ_{∞}∖E$ with pole at infinity and $Mₙ(E) = sup (||P^{\prime }||_{E})/(||P||_{E})$ where the supremum is taken over all polynomials $P|_{E} ≢ 0$ of degree at most n, and $||f||_{E} = sup{|f(z)| : z ∈ E}$. The paper deals with recent results concerning a connection between the smoothness of $g_{E}$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence ${Mₙ(E)}_{n = 1,2,...}$. Some additional conditions are given for special classes of sets.
LA - eng
KW - Green function; polynomial; Markov inequality; polar set; Cantor set
UR - http://eudml.org/doc/281729
ER -

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