Smoothness of Green's functions and Markov-type inequalities

Volume 92 / 2011

Leokadia Białas-Cież Banach Center Publications 92 (2011), 27-36 MSC: 31-02, 30-02, 31C15, 30A10 DOI: 10.4064/bc92-0-2

Abstract

Let $E$ be a compact set in the complex plane, $g_E$ be the Green function of the unbounded component of $\mathbb{C}_\infty\setminus E$ with pole at infinity and $M_n(E)=\sup \frac{\|P'\|_E}{\|P\|_E}$ where the {supremum} is taken over all polynomials $P|_E\not\equiv 0$ of degree at most $n$, and $\|f\|_E=\sup \{|f(z)| : z\in E\}$. 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 $\{M_n(E)\}_{n=1,2,\dots}$. Some additional conditions are given for special classes of sets.

Authors

  • Leokadia Białas-CieżInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail

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