On the best exponent in Markov inequality
type: Item
instance of: scholarly article
title: On the best exponent in Markov inequality
zbMATH Open document ID: 1273.41008
author: Mirosław Baran
author: Leokadia Białas-Cież
author: Beata Milówka
published in: Potential Analysis
publication date: 2013-02-15T00:00:00Z
review text: Recall that a compact set \(E\subset\mathbb{C}^{N}\) is said to be a Markov set if there are two constants \(M>0\) and \(m>0\) such that the following inequality \[ \||\nabla P|\|\leq M(\deg P)^m\| P\|_E,\tag{(*)} \] for each polynomial \(P\) holds, where \(\nabla P=(\frac{\partial P}{\partial z_1},\dotsc, \frac{\partial P}{\partial z_N})\), \(|\nabla P|=\left(\sum_{j=1}^N\left| \frac{\partial P}{\partial z_j}\right|^2\right)^{1/2}\) and \(\|\cdot\|_E\) is the sup-norm on \(E\). If \(E\) is a Markov set, then the Markov exponent of \(E\) is defined by \[ m(E):=\inf\{m>0:\exists M>0\text{ such that for all } P \text{ inequality (*) holds}\}. \] (If \(E\) is not a Markov set, then \(m(E):=\infty\).) (see [\textit{M. Baran} and \textit{W. Pleśniak}, Proc. Am. Math. Soc. 123, No. 9, 2785--2791 (1995; Zbl 0841.41011)]). The present paper contains several new results on Markov exponents. The main results of them are (i) for each \(N\geq2\) and an arbitrary \(p\geq1\) a compact set \(E\subset\mathbb{C}^N\) is constructed such \(m(E)=p\) and the Markov inequality (*) does not hold with exponent \(m=p\); (ii) a compact set \(E\subset\mathbb{C}^2\) is constructed with the property that the Markov inequality holds pointwise for each point of \(E\), but \(E\) is not a Markov set; (iii) in the case \(N=1\), it is shown that holomorphic functions transform Markov sets into Markov sets and an estimate of the transformation of the Markov exponent is given.
reviewed by: Radu Paltanea
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A44
zbMATH DE Number: 6136710
zbMATH Keywords: Green function
zbMATH Keywords: Markov inequality
zbMATH Keywords: Markov exponent
zbMATH Keywords: best constants
zbMATH Keywords: holomorphic mappings
MaRDI profile type: Publication
full work available at URL: s11118-012-9290-0
OpenAlex ID: W2031914300
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DOI: 10.1007/S11118-012-9290-0
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description: scientific article; zbMATH DE number 6136710