Baran, Mirosław Complex equilibrium measure and Bernstein type theorems for compact sets in \(\mathbf R^ n\). (English) Zbl 0813.32011 Proc. Am. Math. Soc. 123, No. 2, 485-494 (1995). 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. Reviewer: A.Yu.Rashkovsky (Khar’kov) Cited in 13 Documents MSC: 32U05 Plurisubharmonic functions and generalizations 31C10 Pluriharmonic and plurisubharmonic functions Keywords:Monge-Ampère operator; plurisubharmonic functions; Bernstein inequality Citations:Zbl 0595.32022 PDFBibTeX XMLCite \textit{M. Baran}, Proc. Am. Math. Soc. 123, No. 2, 485--494 (1995; Zbl 0813.32011) Full Text: DOI