Potentials in pluripotential theory
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 for this, and extends the representation formula to that class. He also gives an example of a measure such that the potential is unbounded but of finite energy.
scientific article; zbMATH DE number 1491232