A note on polynomial convexity: Poletsky disks, Jensen measures and positive currents
\textit{J. Duval} and \textit{N. Sibony} [Duke Math. J. 79, No. 2, 487--513 (1995; Zbl 0838.32006)] proved that a point \(x\) belongs to the polynomial hull \(\widehat{K}\) of a compact set \(K\) in \(\mathbb C^n\) if and only if there exists a positive \((1,1)\)-current \(T\) with \(x\in\text{supp\,}T\) such that \(dd^cT=\widetilde{\sigma}-\delta_x\), where \(\widetilde{\sigma}\) is a representative Jensen measure for (evaluation at) \(x\). Earlier, \textit{E. A. Poletsky} [Indiana Univ. Math. J. 42, No. 1, 85--144 (1993; Zbl 0811.32010)] showed that if \(K\) is a compact subset of a bounded pseudoconvex Runge domain \(\Omega\) in \(\mathbb C^n\), then \(x\in\widehat{K}\) if and only if the following holds: for any open neighborhood \(U\) of \(K\) and for any \(\varepsilon>0\), there exists a holomorphic map \(f: \overline\Delta\to\Omega\) such that \(f(0)=x\) and \[ \sigma\Big(\big\{\zeta\in\partial\Delta\;:\;f(\zeta)\notin U\big\}\Big)<\varepsilon. \] Here \(\sigma\) denotes the normalized Lebesgue measure on the boundary of the unit disk \(\Delta\) in \(\mathbb C\). In this paper, the author shows that the currents \(T\) of Duval and Sibony can be (relatively easily) obtained as weak limits of push-forwards of the Green current \[ G(\omega):=-\iint_{\Delta}\log|\zeta|\cdot\omega,\qquad\omega\in\mathcal E^{1.1}\big(\overline\Delta\big), \] by the Poletsky maps.
positive currents, Poletsky disks, Jensen measures
scientific article; zbMATH DE number 5887812