Hölder Continuity of the Green Function and Markov Brothers’ Inequality

Let VE be the pluricomplex Green function associated with a compact subset E of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{C}^{N}$\end{document}. The well-known Hölder continuity property of E means that there exist constants B>0,γ∈(0,1] such that VE(z)≤B dist(z,E)γ. The main result of this paper says that this condition is equivalent to a Vladimir Markov-type inequality; i.e., ∥DαP∥E≤M|α|(degP)m|α|(|α|!)1−m∥P∥E, where m,M>0 are independent of the polynomial P of N variables. We give some applications of this equivalence, e.g., for convex bodies in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N}$\end{document}, for uniformly polynomially cuspidal sets and for some disconnect compact sets.