Potentials in pluripotential theory
type: Item
instance of: scholarly article
title: Potentials in pluripotential theory
zbMATH Open document ID: 0961.31005
author: Magnus Carlehed
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2001-05-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73495
review text: The author discusses potentials of positive measures in domains in \({\mathbb{C}}^n\), where the kernel is the logarithm or the pluricomplex Green function. Let \(\mu\) be a finite, positive measure with support in \(\Omega\), where \(\Omega\) is a bounded domain in \({\mathbb{C}}^n\). Let \(g(z,w)\) be the pluricomplex Green function for \(\Omega\) with pole in \(w\in\Omega\). The pluricomplex potential of \(\mu\) is defined as \[ p_\mu(z)=\int_\Omega g(z,w) d\mu(w) , \] and the logarithmic potential of \(\mu\) is defined as \[ lp_\mu(z)=\int_\Omega \log|z-w|d\mu(w) . \] The author proves a characterization of logarithmic potentials, which can be viewed as a counterpart to the Riesz decomposition theorem, and gives some examples of pluricomplex potentials in the unit ball. He extends the comparison principle and partial integration formulae to certain classes of unbounded plurisubharmonic functions. The author defines so-called integrated measures. Typically we will have a family of positive measures \(\nu(v_1,\dots,v_n)\), depending on the parameters \(v_1,\dots,v_n\), such that the total mass is bounded, uniformly in all the parameters. He then defines measures \[ \int_{\Omega^k}\nu(v_1,\dots,v_n) d\mu(v_1)\cdots d\mu(v_k) , \] where \(1\leq k\leq n\), and \(\mu\) is a given positive measure. After that he proves representation formulae for the Monge-Ampère measures of bounded potentials. For the pluricomplex potential, the formula is \[ (dd^cp_\mu(z))^n=\int_{\Omega^n}dd^c_zg(z,v_1)\wedge\dots\wedge dd^c_zg(z,v_n) d\mu(v_1)\cdots d\mu(v_n) . \] The author performs the procedure \(\mu\mapsto p_mu\mapsto (dd^cp_\mu)^n\). It turns out that if \(g(\cdot,v)\in C^2\) and \(p_\mu\) is bounded, then the procedure is ``smoothing'', i.e. the resulting measure is absolutely continuous, even if \(\mu\) is not. Finally, he investigates measures whose pluricomplex potentials in the unit ball have finite energy in the sense of Cegrell. The author gives a sufficient condition f…
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 32U15
Mathematics Subject Classification ID: 31-02
zbMATH DE Number: 1491232
zbMATH Keywords: pluripotential theory
zbMATH Keywords: plurisubharmonic function
zbMATH Keywords: pluricomplex Green function
zbMATH Keywords: Monge-Ampère measure
zbMATH Keywords: pluricomplex potential
zbMATH Keywords: logarithmic potential
MaRDI profile type: Publication
OpenAlex ID: W2321523804
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DOI: 10.5802/AFST.940
description: scientific article; zbMATH DE number 1491232