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Complex equilibrium measure and Bernstein type theorems for compact sets in \(\mathbf R^ n\). (English) Zbl 0813.32011

For a compact set \(E \subset \mathbb{C}^ n\), \(u^*_ E(z)\) denotes the plurisubharmonic extremal function associated with \(E:u^*_ E (z) = \limsup_{w \to z} \sup \{(w) : u \in {\mathcal L}\), \(u \leq 0\) on \(E\}\), where \({\mathcal L}\) is the class of plurisubharmonic functions in \(\mathbb{C}^ n\) with logarithmic growth: \(u(z) \leq \text{const} + \log (1 + | z |)\). Developing some ideas from E. Bedford and B. A. Taylor [Trans. Am. Math. Soc. 294, 705-717 (1986; Zbl 0595.32022)], the author studies the measure \(\lambda_ E = (dd^ cu^*_ E)^ n\) for \(E\) being a compact subset of \(\mathbb{R}^ n \subset \mathbb{C}^ n\). The main result is a lower bound for the density of \(\lambda_ E\). It can be considered as an improvement of the classical Bernstein inequality for polynomials to the multivariate case.

MSC:

32U05 Plurisubharmonic functions and generalizations
31C10 Pluriharmonic and plurisubharmonic functions

Citations:

Zbl 0595.32022
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