Hölder continuity of the Green function and Markov brothers’ inequality. (English) Zbl 1322.32027
Let \(E\subset\mathbb C^N\) be compact. For \(\gamma\in(0,1]\) and \(B>0\) we say that \(E\) has the Hölder continuity property (\(E\in\text{HCP}(\gamma,B)\)) if \(V_E(z)\leq B\text{dist}^\gamma(z,E)\) for all \(z\in\mathbb C^N\), where \(V_E\) stands for the global extremal function of \(E\). For \(m\geq1\) and \(M>0\) we say that \(E\) has the V. Markov inequality (\(E\in\text{VMI}(m, M)\)) if
\[ \|D^\alpha P\|_E\leq M^{|\alpha|}\frac{(\deg P)^{m|\alpha|}}{(|\alpha|!)^{m-1}}\|P\|_E \]
for every \(\alpha\in\mathbb N_0^N\) and for every complex polynomial \(P\in\mathcal P(\mathbb C^N)\). The main results of the paper are the following theorems.
\[ \|D^\alpha P\|_E\leq M^{|\alpha|}\frac{(\deg P)^{m|\alpha|}}{(|\alpha|!)^{m-1}}\|P\|_E \]
for every \(\alpha\in\mathbb N_0^N\) and for every complex polynomial \(P\in\mathcal P(\mathbb C^N)\). The main results of the paper are the following theorems.
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- \(E\in\text{HCP}(\gamma, B) \Longrightarrow E\in\text{VMI}(1/\gamma, \sqrt{N}(B\gamma e)^{1/\gamma})\);
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- \(E\in\text{VMI}(m, M) \Longrightarrow E\in\text{HCP}(1/m, (MN)^\gamma m)\);
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- \(E\in\text{VMI}(m, M) \Longrightarrow C(E)\geq(N^{3/2}(B\gamma e^2)^{1/\gamma})^{-1}\).
Reviewer: Marek Jarnicki (Kraków)
MSC:
| 32U35 | Plurisubharmonic extremal functions, pluricomplex Green functions |
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
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