A local version of Levenberg's theorem on determining measures and Leja's polynomial condition
Let \(E\) be a bounded subset of \(\mathbb{C}^ N\) and \(\mu\) be a non-negative weakly increasing set function on \(E\) such that \(\mu (\emptyset) =0\). The pair \((E,\mu)\) is said to satisfy Leja's polynomial condition \((L^*)\) at a point \(a\in \mathbb{C}^ N\) if for every family \({\mathcal F}\) of polynomials in \(\mathbb{C}^ N\) with \(\mu (\{ z\in E\): \(\sup_{f\in {\mathcal F}} | f(z)| =\infty\})=0\) and for any \(b>1\), the family \({\mathcal F}_ b:= \{b^{-\deg f} \cdot f\): \(f\in{\mathcal F}\}\) is uniformly bounded in a neighbourhood of \(a\). Let \(V_ E(z)= \sup\{ u(z)\): \(u\in {\mathcal L}\), \(u\leq 0\) on \(E\}\), where \({\mathcal L}\) is the Lelong class of all plurisubharmonic functions in \(\mathbb{C}^ N\) of logarithmic growth, \(V^*_ E= \limsup_{w\to z} V_ E(w)\). The following theorem is proved. Let \(\mu\) vanish on pluripolar subsets of \(E\), and for each \(F\subset E\), \(\mu (E\setminus F)=0\) if and only if \(\mu(F)= \mu(E)\). Then \((E,\mu)\) satisfies \((L^*)\) at \(a\in \mathbb{C}^ N\) if and only if \(V^*_ F(a)= V^*_ E (a)=0\) for any \(F\subset E\) with \(\mu(F)= \mu(E)\). This provides a short proof of the invariance of the condition \((L^*)\) under nondegenerate holomorphic mappings.
scientific article; zbMATH DE number 733132
scientific article