×

A local version of Levenberg’s theorem on determining measures and Leja’s polynomial condition. (English) Zbl 0815.31004

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.

MSC:

31C10 Pluriharmonic and plurisubharmonic functions
32U05 Plurisubharmonic functions and generalizations
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32A17 Special families of functions of several complex variables
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Bedford, E., Taylor, B. A.: A new capacity for plurisubharmonic functions. Acta Math.149, 1–40 (1982). · Zbl 0547.32012
[2] Klimek, M.: Extremal plurisubharmonic functions and L-regular sets in \(\mathbb{C}\) n . Proc. R. Ir. Acad.82A, 217–230 (1982). · Zbl 0494.32005
[3] Klimek, M.: Pluripotential Theory. Oxford: Univ. Press. 1991.
[4] Levenberg, N.: Monge-Ampère measures associated to extremal plurisubharmonic functions in \(\mathbb{C}\) n . Trans Amer. Math. Soc.289, 331–343 (1985). · Zbl 0541.31009
[5] Nguyen Thanh Van, Pleśniak, W.: Invariane ofL-regularity and Leja’s polynomial condition under holomorphic mappings. Proc. R. Ir. Akad.84A, 110–115 (1984). · Zbl 0574.32034
[6] Siciak, J.: Extremal plurisubharmonic functions in \(\mathbb{C}\) N . Ann. Polon. Math.39, 175–211 (1981). · Zbl 0477.32018
[7] Siciak, J.: Extremal plurisubharmonic functions and capacities in \(\mathbb{C}\) n . Sophia Kokyuroku Math.14 (1982) Sophia University, Tokyo (eds.). · Zbl 0579.32025
[8] Siciak, J.: Families of polynomials and determining measures. Ann. Fac. Sci. Toulouse9(2), 193–211 (1988). · Zbl 0634.31005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.